chapter 21
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Chapter 21. The Kinetic Theory of Gases. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 21Chapter 21
The Kinetic Theory of GasesThe Kinetic Theory of Gases
The description of the behavior of a gas in The description of the behavior of a gas in terms of the macroscopic state variables terms of the macroscopic state variables PP VV and and TT can be related to simple averages of microscopic can be related to simple averages of microscopic quantities such as the quantities such as the massmass and and speedspeed of the of the molecules in the gas The resulting theory is called molecules in the gas The resulting theory is called the the Kinetic Theory of GasesKinetic Theory of Gases
From the point of view of kinetic theory a gas From the point of view of kinetic theory a gas consist of a large number of molecules making consist of a large number of molecules making elastic collisions with each other and with the walls elastic collisions with each other and with the walls of containerof container
In the absence of external forces (we may In the absence of external forces (we may neglect gravity) there no preferred position of the neglect gravity) there no preferred position of the molecule in the container and no preferred molecule in the container and no preferred directions for itrsquos velocity vector directions for itrsquos velocity vector
The molecules are separated on the The molecules are separated on the average by distances that are large compared with average by distances that are large compared with their diameters and they exert no forces on each their diameters and they exert no forces on each other except when they collideother except when they collide
This final assumption is equivalent to This final assumption is equivalent to assuming a very low gas density which is the assuming a very low gas density which is the same as assuming that the gas is an ideal gas same as assuming that the gas is an ideal gas
Because momentum is conserved the Because momentum is conserved the collisions the molecules make with each other collisions the molecules make with each other have no effect on the total momentum in any have no effect on the total momentum in any directions ndash thus such collisions may be directions ndash thus such collisions may be neglectedneglected
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
bull The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
bull It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
The description of the behavior of a gas in The description of the behavior of a gas in terms of the macroscopic state variables terms of the macroscopic state variables PP VV and and TT can be related to simple averages of microscopic can be related to simple averages of microscopic quantities such as the quantities such as the massmass and and speedspeed of the of the molecules in the gas The resulting theory is called molecules in the gas The resulting theory is called the the Kinetic Theory of GasesKinetic Theory of Gases
From the point of view of kinetic theory a gas From the point of view of kinetic theory a gas consist of a large number of molecules making consist of a large number of molecules making elastic collisions with each other and with the walls elastic collisions with each other and with the walls of containerof container
In the absence of external forces (we may In the absence of external forces (we may neglect gravity) there no preferred position of the neglect gravity) there no preferred position of the molecule in the container and no preferred molecule in the container and no preferred directions for itrsquos velocity vector directions for itrsquos velocity vector
The molecules are separated on the The molecules are separated on the average by distances that are large compared with average by distances that are large compared with their diameters and they exert no forces on each their diameters and they exert no forces on each other except when they collideother except when they collide
This final assumption is equivalent to This final assumption is equivalent to assuming a very low gas density which is the assuming a very low gas density which is the same as assuming that the gas is an ideal gas same as assuming that the gas is an ideal gas
Because momentum is conserved the Because momentum is conserved the collisions the molecules make with each other collisions the molecules make with each other have no effect on the total momentum in any have no effect on the total momentum in any directions ndash thus such collisions may be directions ndash thus such collisions may be neglectedneglected
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
bull The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
bull It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
In the absence of external forces (we may In the absence of external forces (we may neglect gravity) there no preferred position of the neglect gravity) there no preferred position of the molecule in the container and no preferred molecule in the container and no preferred directions for itrsquos velocity vector directions for itrsquos velocity vector
The molecules are separated on the The molecules are separated on the average by distances that are large compared with average by distances that are large compared with their diameters and they exert no forces on each their diameters and they exert no forces on each other except when they collideother except when they collide
This final assumption is equivalent to This final assumption is equivalent to assuming a very low gas density which is the assuming a very low gas density which is the same as assuming that the gas is an ideal gas same as assuming that the gas is an ideal gas
Because momentum is conserved the Because momentum is conserved the collisions the molecules make with each other collisions the molecules make with each other have no effect on the total momentum in any have no effect on the total momentum in any directions ndash thus such collisions may be directions ndash thus such collisions may be neglectedneglected
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
bull The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
bull It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
This final assumption is equivalent to This final assumption is equivalent to assuming a very low gas density which is the assuming a very low gas density which is the same as assuming that the gas is an ideal gas same as assuming that the gas is an ideal gas
Because momentum is conserved the Because momentum is conserved the collisions the molecules make with each other collisions the molecules make with each other have no effect on the total momentum in any have no effect on the total momentum in any directions ndash thus such collisions may be directions ndash thus such collisions may be neglectedneglected
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
bull The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
bull It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molecular Model of an Ideal GasMolecular Model of an Ideal Gas
bull The model shows that the pressure that a gas The model shows that the pressure that a gas exerts on the walls of its container is a exerts on the walls of its container is a consequence of the collisions of the gas molecules consequence of the collisions of the gas molecules with the wallswith the walls
bull It is consistent with the macroscopic description It is consistent with the macroscopic description developed earlierdeveloped earlier
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The gas consist of a very large number of identical The gas consist of a very large number of identical
molecules each with mass molecules each with mass mm but with negligible size (this but with negligible size (this
assumption is approximately true when the distance between assumption is approximately true when the distance between
the molecules is large compared to the size) the molecules is large compared to the size)
The consequence The consequence rarrrarr for negligible size molecules we can for negligible size molecules we can
neglect the intermolecular collisions neglect the intermolecular collisions
bull The molecules donrsquot exert any action-at-distance forces The molecules donrsquot exert any action-at-distance forces
on each other This means there are no potential energy on each other This means there are no potential energy
changes to be considered so each molecules kinetic energy changes to be considered so each molecules kinetic energy
remains unchanged This assumption is fundamental to the remains unchanged This assumption is fundamental to the
nature of an ideal gasnature of an ideal gas
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Assumptions for Ideal Gas TheoryAssumptions for Ideal Gas Theory
bull The molecules are moving in random The molecules are moving in random directions with a distribution of speeds that is directions with a distribution of speeds that is independent of directionindependent of direction
bull Collisions with the container walls are elastic Collisions with the container walls are elastic conserving the moleculersquos energy and conserving the moleculersquos energy and momentummomentum
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull Assume a container Assume a container is a cubeis a cube
Edges are length Edges are length dd
bull Look at the motion Look at the motion of the molecule in of the molecule in terms of its velocity terms of its velocity componentscomponents
bull Look at its Look at its momentum and the momentum and the average forceaverage force
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energybull Since the collision is elastic the Since the collision is elastic the y y - -
component of moleculersquos velocity component of moleculersquos velocity remains unchanged while the remains unchanged while the xx - - component reverses sign Thus the component reverses sign Thus the molecule undergoes the momentum molecule undergoes the momentum change of magnitude change of magnitude 2mv2mvxx
bull After colliding with the right hand After colliding with the right hand
wall the wall the x x - component of - component of moleculersquos velocity will not change moleculersquos velocity will not change until it hits the left-hand wall and its until it hits the left-hand wall and its x x - velocity will again reverses - velocity will again reverses
ΔΔt = 2d vt = 2d vxx
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The average force due to the The average force due to the each molecule on the walleach molecule on the wall
bull To get the total force on the To get the total force on the wall we sum over all wall we sum over all N N molecules Dividing by the wall molecules Dividing by the wall area area AA then gives the force per then gives the force per unit area or pressure unit area or pressure
d
mv
vd
mv
t
pF x
x
xi
2
)2(
2
V
vm
Ad
vm
Ad
mv
A
F
A
FP xx
x
i 22
2
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
N
vx 2
2xv
V
mNP
N
v
V
mN
V
vmP xx
22
SinceSince is just the average of the squares of is just the average of the squares of xx ndash ndash components of velocities components of velocities
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
2xv
2222zyx vvvv
2yv 2
zvSince the molecules are moving in random directions the Since the molecules are moving in random directions the
average quantities and must be average quantities and must be
equal and the average of the molecular speed equal and the average of the molecular speed
and and vv22 = = 3v3vxx22 or or vvxx
22 = = vv2233 Then the expression for Then the expression for
pressurepressure
2
3v
V
mNP
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull The relationship can be writtenThe relationship can be written
bull This tells us that pressure is proportional to the number This tells us that pressure is proportional to the number of molecules per unit volume (of molecules per unit volume (NNVV) and to the average ) and to the average translational kinetic energy of the moleculestranslational kinetic energy of the molecules
2
3v
V
mNP
___22 1
3 2
NP m v
V
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Pressure and Kinetic EnergyPressure and Kinetic Energy
bull This equation also relates the macroscopic This equation also relates the macroscopic quantity of pressure with a microscopic quantity quantity of pressure with a microscopic quantity of the average value of the square of the of the average value of the square of the molecular speedmolecular speed
bull One way to increase the pressure is to increase One way to increase the pressure is to increase the number of molecules per unit volumethe number of molecules per unit volume
bull The pressure can also be increased by The pressure can also be increased by increasing the speed (kinetic energy) of the increasing the speed (kinetic energy) of the moleculesmolecules
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
A A 200-mol200-mol sample of oxygen gas is confined to a sample of oxygen gas is confined to a 500-L500-L vessel at a pressure of vessel at a pressure of 800 atm800 atm Find the Find the average translational kinetic energy of an oxygen average translational kinetic energy of an oxygen molecule under these conditions molecule under these conditions
223 2N mv
PV
2
av A A
5 3 3
av 23A
21av
3 where 2
2 2
3 800 atm 1013 10 Pa atm 500 10 m32 2 2 2 mol 602 10 molecules mol
505 10 J molecule
mv PVK N nN N
N
PVK
N
K
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull We can take the pressure as it relates to the kinetic We can take the pressure as it relates to the kinetic energy and compare it to the pressure from the equation energy and compare it to the pressure from the equation of state for an ideal gasof state for an ideal gas
bull Therefore the temperature is a direct measure of the Therefore the temperature is a direct measure of the average molecular kinetic energyaverage molecular kinetic energy
V
TNkm v
V
NP B
2
2
1
3
2
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molecular Interpretation of TemperatureMolecular Interpretation of Temperature
bull Simplifying the equation relating temperature and Simplifying the equation relating temperature and kinetic energy giveskinetic energy gives
bull This can be applied to each direction This can be applied to each direction
with similar expressions for with similar expressions for vvyy and and vvzz
___2
B
1 3
2 2m v k T
___2
B
1 1
2 2xm v k T
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
A Microscopic Description of TemperatureA Microscopic Description of Temperature
bull Each translational degree of freedom Each translational degree of freedom contributes an equal amount to the contributes an equal amount to the energy of the gasenergy of the gas
bull A generalization of this result is called A generalization of this result is called the the theorem of equipartition of energytheorem of equipartition of energy
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Theorem of Equipartition of EnergyTheorem of Equipartition of Energy
bull Each degree of freedom contributes Each degree of freedom contributes frac12frac12kkBBTT to the energy of a system where to the energy of a system where
possible degrees of freedom in addition to possible degrees of freedom in addition to those associated with translation arise those associated with translation arise from rotation and vibration of moleculesfrom rotation and vibration of molecules
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Total Kinetic EnergyTotal Kinetic Energy
bull The total kinetic energy is just The total kinetic energy is just NN times the kinetic energy times the kinetic energy of each moleculeof each molecule
bull If we have a gas with only translational energy this is the If we have a gas with only translational energy this is the internal energy of the gasinternal energy of the gas
bull This tells us that the internal energy of an ideal gas This tells us that the internal energy of an ideal gas depends only on the temperaturedepends only on the temperature
___2
tot trans B
1 3 3
2 2 2K N m v Nk T nRT
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific HeatMolar Specific Heat
bull Several processes can Several processes can change the temperature of change the temperature of an ideal gasan ideal gas
bull Since Since ΔΔTT is the same for is the same for each process each process ΔΔEEintint is also is also the samethe same
bull The heat is different for the The heat is different for the different pathsdifferent paths
bull The heat associated with a The heat associated with a particular change in particular change in temperature is temperature is notnot unique unique
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific HeatMolar Specific Heat
bull We define specific heats for two processes that We define specific heats for two processes that frequently occurfrequently occurndash Changes with constant volumeChanges with constant volumendash Changes with constant pressureChanges with constant pressure
bull Using the number of moles Using the number of moles nn we can define we can define molar specific heats for these processesmolar specific heats for these processes
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific HeatMolar Specific Heat
bull Molar specific heatsMolar specific heats
QQ = = nCnCVV ΔΔTT for constant-volume processesfor constant-volume processes
QQ = = nCnCPP ΔΔTT for constant-pressure processesfor constant-pressure processesbull QQ (under constant pressure) must account for both (under constant pressure) must account for both
the increase in internal energy and the transfer of the increase in internal energy and the transfer of energy out of the system by workenergy out of the system by work
bull QQ (under constant volume) account just for change (under constant volume) account just for change the internal energy the internal energy
bull QQconstant constant PP gt Q gt Qconstant constant VV for given values of for given values of nn and and ΔΔTT
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Ideal Monatomic GasIdeal Monatomic Gas
bull A monatomic gas contains one atom per A monatomic gas contains one atom per moleculemolecule
bull When energy is added to a monatomic gas in When energy is added to a monatomic gas in a container with a fixed volume all of the a container with a fixed volume all of the energy goes into increasing the translational energy goes into increasing the translational kinetic energy of the gaskinetic energy of the gasndash There is no other way to store energy in There is no other way to store energy in
such a gassuch a gas
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Ideal Monatomic GasIdeal Monatomic Gas
bull Therefore Therefore
bull ΔΔEEintint is a function of is a function of TT only onlybull At constant volumeAt constant volume
QQ = = ΔΔEEintint = = nCnCVV ΔΔTTThis applies to all ideal gases not just monatomic onesThis applies to all ideal gases not just monatomic ones
nRTTNkKE Btranstot 2
3
2
3in t
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Monatomic GasesMonatomic Gases
bull Solving Solving
for for CCVV gives gives CCVV = 32 = 32 RR = 125 Jmol = 125 Jmol K K for all for all
monatomic gasesmonatomic gasesbull This is in good agreement with experimental This is in good agreement with experimental
results for monatomic gasesresults for monatomic gases
TnRTnCE V 2
3in t
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Monatomic GasesMonatomic Gasesbull In a constant-pressure process In a constant-pressure process ΔΔEEintint = = QQ + + WW and and
bull Change in internal energy depends only on Change in internal energy depends only on temperature for an ideal gas and therefore are the temperature for an ideal gas and therefore are the same for the constant volume process and for same for the constant volume process and for constant pressure process constant pressure process
CCPP ndash C ndash CVV = R = R
VPTn CWQE P i n t
Tn RTn CTn C PV
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Monatomic GasesMonatomic Gases
CP ndash CV = R
bull This also applies to any ideal gasThis also applies to any ideal gas
CP = 52 R = 208 Jmol K
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Ratio of Molar Specific HeatsRatio of Molar Specific Heats
bull We can also define We can also define
bull Theoretical values of Theoretical values of CCVV CCPP and and are in are in excellent agreement for monatomic gasesexcellent agreement for monatomic gases
bull But they are in serious disagreement with the But they are in serious disagreement with the values for more complex moleculesvalues for more complex moleculesndash Not surprising since the analysis was for Not surprising since the analysis was for
monatomic gasesmonatomic gases
5 2167
3 2P
V
C R
C R
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Sample Values of Molar Specific HeatsSample Values of Molar Specific Heats
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
bull A A 100-mol sample of air (a diatomic ideal gas) sample of air (a diatomic ideal gas) atat 300 K confined in a cylinder under a heavy confined in a cylinder under a heavy piston occupies a volume of piston occupies a volume of 500 L Determine Determine the final volume of the gas after the final volume of the gas after 440 kJ of of energy is transferred to the air by heatenergy is transferred to the air by heat
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the of energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
nR
Q
RR
n
Q
RCn
Q
nC
QT
VP 5
2
2
3)(
The piston moves to keep pressure constant The piston moves to keep pressure constant
nR TV
P
P VQ nC T n C R T
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
bull A A 100-mol 100-mol sample of air (a diatomic ideal gas) atsample of air (a diatomic ideal gas) at 300 K300 K confined in a cylinder under a heavy piston occupies a confined in a cylinder under a heavy piston occupies a volume ofvolume of 500 L500 L Determine the final volume of the gas Determine the final volume of the gas afterafter 440 kJ440 kJ of energy is transferred to the air by heatof energy is transferred to the air by heat
nRT
QV
P
Q
nR
Q
P
nRT
P
nRV
5
2
5
2
5
2
LK
Kmol
Jmol
LJV 533
)300(3148)1(
)5)(10404(
5
2 3
LLLVVV if 538533005
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific Heats of Other MaterialsMolar Specific Heats of Other Materials
bull The internal energy of more complex gases The internal energy of more complex gases must include contributions from the rotational must include contributions from the rotational and vibrational motions of the moleculesand vibrational motions of the molecules
bull In the cases of solids and liquids heated at In the cases of solids and liquids heated at constant pressure very little work is done since constant pressure very little work is done since the thermal expansion is small and the thermal expansion is small and CCPP and and CCVV are approximately equalare approximately equal
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Adiabatic Processes for an Ideal GasAdiabatic Processes for an Ideal Gas
bull Assume an ideal gas is in an equilibrium state Assume an ideal gas is in an equilibrium state and so and so PVPV = = nRTnRT is validis valid
bull The pressure and volume of an ideal gas at any The pressure and volume of an ideal gas at any time during an adiabatic process are related by time during an adiabatic process are related by PV PV = constant= constant
bull = = CCPP CCVV is assumed to be constant during the is assumed to be constant during the processprocess
bull All three variables in the ideal gas law (All three variables in the ideal gas law (PP VV TT ) ) can change during an adiabatic processcan change during an adiabatic process
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Equipartition of EnergyEquipartition of Energy
bull With complex With complex molecules other molecules other contributions to internal contributions to internal energy must be taken energy must be taken into accountinto account
bull One possible way to One possible way to energy change is the energy change is the translational motion of translational motion of the center of massthe center of mass
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Equipartition of EnergyEquipartition of Energy
bull Rotational motion Rotational motion about the various axes about the various axes also contributesalso contributes
We can neglect the We can neglect the rotation around the rotation around the yy axis since it is negligible axis since it is negligible compared to thecompared to the xx and and zz axesaxes
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Equipartition of EnergyEquipartition of Energy
bull The molecule can The molecule can also vibratealso vibrate
bull There is kinetic There is kinetic energy and potential energy and potential energy associated energy associated with the vibrationswith the vibrations
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Equipartition of EnergyEquipartition of Energy
bull The translational motion adds three degrees of The translational motion adds three degrees of freedomfreedom
bull The rotational motion adds two degrees of The rotational motion adds two degrees of freedomfreedom
bull The vibrational motion adds two more degrees The vibrational motion adds two more degrees of freedomof freedom
bull Therefore Therefore EEintint = 72 = 72 nRTnRT and and CCVV = 72 = 72 RR
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Agreement with ExperimentAgreement with Experiment
bull Molar specific heat is a function of temperatureMolar specific heat is a function of temperaturebull At low temperatures a diatomic gas acts like a At low temperatures a diatomic gas acts like a
monatomic gas monatomic gas CCVV = 32 = 32 RR
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Agreement with ExperimentAgreement with Experiment
bull At about room temperature the value increases At about room temperature the value increases to to CCVV = 52 = 52 RR
This is consistent with adding rotational energy but This is consistent with adding rotational energy but not vibrational energynot vibrational energy
bull At high temperatures the value increases At high temperatures the value increases to to CCVV = 72 R = 72 R
This includes vibrational energy as well as rotational This includes vibrational energy as well as rotational and translationaland translational
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Complex MoleculesComplex Molecules
bull For molecules with more than two atoms the For molecules with more than two atoms the vibrations are more complexvibrations are more complex
bull The number of degrees of freedom is largerThe number of degrees of freedom is largerbull The more degrees of freedom available to a The more degrees of freedom available to a
molecule the more ldquowaysrdquo there are to store molecule the more ldquowaysrdquo there are to store energyenergy
This results in a higher molar specific heatThis results in a higher molar specific heat
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Quantization of EnergyQuantization of Energy
bull To explain the results of the various molar To explain the results of the various molar specific heats we must use some quantum specific heats we must use some quantum mechanicsmechanics
Classical mechanics is not sufficientClassical mechanics is not sufficient
bull In quantum mechanics the energy is In quantum mechanics the energy is proportional to the frequency of the wave proportional to the frequency of the wave representing the particlerepresenting the particle
The energies of atoms and molecules are The energies of atoms and molecules are quantizedquantized
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Quantization of EnergyQuantization of Energy
bull This energy level diagram This energy level diagram shows the rotational and shows the rotational and vibrational states of a vibrational states of a diatomic moleculediatomic molecule
bull The lowest allowed state is The lowest allowed state is the the ground stateground state
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Quantization of EnergyQuantization of Energy
bull The vibrational states are separated by larger The vibrational states are separated by larger energy gaps than are rotational statesenergy gaps than are rotational states
bull At low temperatures the energy gained during At low temperatures the energy gained during collisions is generally not enough to raise it to collisions is generally not enough to raise it to the first excited state of either rotation or the first excited state of either rotation or vibrationvibration
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Quantization of EnergyQuantization of Energy
bull Even though rotation and vibration are Even though rotation and vibration are classically allowed they do not occurclassically allowed they do not occur
bull As the temperature increases the energy of the As the temperature increases the energy of the molecules increasesmolecules increases
bull In some collisions the molecules have enough In some collisions the molecules have enough energy to excite to the first excited stateenergy to excite to the first excited state
bull As the temperature continues to increase more As the temperature continues to increase more molecules are in excited states molecules are in excited states
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Quantization of EnergyQuantization of Energy
bull At about room temperature rotational energy is At about room temperature rotational energy is contributing fullycontributing fully
bull At about At about 1000 K1000 K vibrational energy levels are vibrational energy levels are reachedreached
bull At about At about 10 000 K10 000 K vibration is contributing fully vibration is contributing fully to the internal energyto the internal energy
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific Heat of SolidsMolar Specific Heat of Solids
bull Molar specific heats in solids also demonstrate a Molar specific heats in solids also demonstrate a marked temperature dependencemarked temperature dependence
bull Solids have molar specific heats that generally Solids have molar specific heats that generally decrease in a nonlinear manner with decreasing decrease in a nonlinear manner with decreasing temperaturetemperature
bull It approaches zero as the temperature It approaches zero as the temperature approaches absolute zeroapproaches absolute zero
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
DuLong-Petit LawDuLong-Petit Law
bull At high temperatures the molar specific heats At high temperatures the molar specific heats approach the value of approach the value of 3R
This occurs above This occurs above 300 K300 K
bull The molar specific heat of a solid at high The molar specific heat of a solid at high temperature can be explained by the equipartition temperature can be explained by the equipartition theoremtheorem
Each atom of the solid has six degrees of freedomEach atom of the solid has six degrees of freedom
The internal energy is The internal energy is 33nRTnRT andand CCvv = 3 = 3 RR
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Molar Specific Heat of Solids
bull As As TT approaches approaches 00 the molar specific heat the molar specific heat approaches approaches 00
bull At high temperatures At high temperatures CCVV becomes a becomes a
constant at constant at ~3~3RR
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Specific Heat and Molar Specific Heat of Some Solids and LiquidsSpecific Heat and Molar Specific Heat of Some Solids and Liquids
SubstanceSubstance c kJkgc kJkgKK c Jmolc JmolmiddotmiddotKK
AluminumAluminum 09000900 243243
BismuthBismuth 01230123 257257
CopperCopper 03860386 245245
GoldGold 01260126 256256
Ice (-10Ice (-1000C)C) 205205 369369
LeadLead 01280128 264264
SilverSilver 02330233 249249
TungstenTungsten 01340134 248248
ZinkZink 03870387 252252
MercuryMercury 01400140 283283
WaterWater 418418 752752
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Use the Dulong-Petit law to calculate the specific heat of Dulong-Petit law to calculate the specific heat of coppercopper
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
The molar mass of copper is The molar mass of copper is 635635 gmolgmol Use the Dulong- Use the Dulong-Petit law to calculate the specific heat of copperPetit law to calculate the specific heat of copper
The molar specific heat is the heat capacity per moleThe molar specific heat is the heat capacity per mole
The Dulong-Petit law gives The Dulong-Petit law gives cc in terms of R in terms of R
cc= 3R= 3R
Mcn
mc
n
Cc
KkgkJKgJmolg
KmolJ
M
cc
39203920
563
)3148(3
This differs from the measured value of 0386 kJkgThis differs from the measured value of 0386 kJkgK given K given in Table by less than 2in Table by less than 2
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
The specific heat of a certain metal is measured to The specific heat of a certain metal is measured to be be 102 kJkg102 kJkgKK (a) Calculate the molar mass of (a) Calculate the molar mass of this metal assuming that the metal obeys the this metal assuming that the metal obeys the Dulong-Petit law (b) What is the metalDulong-Petit law (b) What is the metal
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
The specific heat of a certain metal is measured to be The specific heat of a certain metal is measured to be 102 102 kJkgkJkgKK (a) Calculate the molar mass of this metal assuming (a) Calculate the molar mass of this metal assuming that the metal obeys the Dulong-Petit law (b) What is the that the metal obeys the Dulong-Petit law (b) What is the metalmetal
(a)(a)
gmolkgmolKkgJ
KmolJ
c
RM
RMc
Rc
Mcc
452402445010021
)3148(33
3
3
3
(b) The metal must be magnesium which has a molar (b) The metal must be magnesium which has a molar mass of 2431 gmolmass of 2431 gmol
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Boltzmann Distribution LawBoltzmann Distribution Law
bull The motion of molecules is extremely chaoticThe motion of molecules is extremely chaoticbull Any individual molecule is colliding with others at Any individual molecule is colliding with others at
an enormous ratean enormous rateTypically at a rate of a billion times per secondTypically at a rate of a billion times per second
bull We add the We add the number densitynumber density nnV V ((E E ))
This is called a distribution functionThis is called a distribution function
It is defined so that It is defined so that nnV V ((E E ) ) dEdE is the number of is the number of
molecules per unit volume with energy between molecules per unit volume with energy between EE and and EE + + dE dE
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Number Density and Boltzmann Distribution Number Density and Boltzmann Distribution LawLaw
bull From statistical mechanics the number density is From statistical mechanics the number density is
bull This equation is known as the This equation is known as the Boltzmann Boltzmann Distribution LawDistribution Law
bull It states that the probability of finding the It states that the probability of finding the molecule in a particular energy state varies molecule in a particular energy state varies exponentially as the energy exponentially as the energy ((frac12mvfrac12mv22) ) divided by divided by kkBBTT
Tk
E
VBenEn
0)(
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull The observed speed The observed speed distribution of gas distribution of gas molecules in thermal molecules in thermal equilibrium is shown in equilibrium is shown in figure The number of figure The number of molecules having speeds molecules having speeds in the range in the range vv to to v+dvv+dv is is equal to the area of the equal to the area of the shaded rectangle shaded rectangle NNvvdvdv
The functionThe function NNvv
approaches zero as approaches zero as vv approaches infinityapproaches infinity
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Distribution of Molecular SpeedsDistribution of Molecular Speeds
bull As indicated in figure the As indicated in figure the average speedaverage speed is somewhat is somewhat lower than the lower than the rmsrms speed speed The The most probable speedmost probable speed vvmpmp is the speed at which the is the speed at which the
distribution curve reaches a distribution curve reaches a peakpeak
00
00
00
2
4112
618
7313
m
Tk
m
Tkv
m
Tk
m
Tkv
m
Tk
m
Tkvv
BBmp
BBavg
BBavgrms
m pa vgr m s vvv
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Distribution FunctionDistribution Function
bull The fundamental expression that describes the The fundamental expression that describes the distribution of speeds indistribution of speeds in NN gas molecules is gas molecules is
mm is the mass of a gas molecule is the mass of a gas molecule kkBB is is Boltzmannrsquos constant and Boltzmannrsquos constant and TT is the absolute is the absolute temperaturetemperature
2
3 2
22
B
42
Bmv k TV
mN N v e
k T
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Speed DistributionSpeed Distributionbull The peak shifts to the right as The peak shifts to the right as TT increases increases
This shows that the average speed increases with This shows that the average speed increases with increasing temperatureincreasing temperature
bull The asymmetric shape occurs because the lowest The asymmetric shape occurs because the lowest possible speed is possible speed is 00 and the highest is and the highest is infinityinfinity
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Speed DistributionSpeed Distribution
bull The distribution of molecular speeds depends The distribution of molecular speeds depends both on the mass and on temperatureboth on the mass and on temperature
bull The speed distribution for liquids is similar to The speed distribution for liquids is similar to that of gasesthat of gases
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
EvaporationEvaporationbull Some molecules in the liquid are more energetic Some molecules in the liquid are more energetic
than othersthan othersbull Some of the faster moving molecules penetrate Some of the faster moving molecules penetrate
the surface and leave the liquidthe surface and leave the liquidThis occurs even before the boiling point is reachedThis occurs even before the boiling point is reached
bull The molecules that escape are those that have The molecules that escape are those that have enough energy to overcome the attractive forces enough energy to overcome the attractive forces of the molecules in the liquid phaseof the molecules in the liquid phase
bull The molecules left behind have lower kinetic The molecules left behind have lower kinetic energiesenergies
Therefore evaporation is a cooling processTherefore evaporation is a cooling process
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Mean Free PathMean Free Path
bull The molecules move with constant speed The molecules move with constant speed along straight lines between collisionsalong straight lines between collisions
bull The average distance between collisions is The average distance between collisions is called the called the mean free pathmean free path
bull The path of an individual molecule is randomThe path of an individual molecule is randomThe motion is not confined to the plane of the The motion is not confined to the plane of the paperpaper
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Mean Free PathMean Free Path
bull A molecule moving through a A molecule moving through a gas collides with other gas collides with other molecules in a random molecules in a random fashionfashion
bull This behavior is sometimes This behavior is sometimes referred to as a referred to as a random-walk random-walk processprocess
bull The The mean free pathmean free path increases as the number of increases as the number of molecules per unit volume molecules per unit volume decreasesdecreases
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Mean Free PathMean Free Path
bull The The mean free pathmean free path is related to the is related to the diameter of the molecules and the density of diameter of the molecules and the density of the gasthe gas
bull We assume that the molecules are spheres We assume that the molecules are spheres of diameter of diameter dd
bull No two molecules will collide unless their No two molecules will collide unless their paths are less than a distance paths are less than a distance dd apart as the apart as the molecules approach each othermolecules approach each other
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Mean Free PathMean Free Path
bull The mean free path The mean free path ℓℓ equals the average equals the average distance distance vvavgavgΔΔtt traveled in a time interval traveled in a time interval ΔΔtt
divided by the number of collisions that occur in divided by the number of collisions that occur in that time intervalthat time interval
22
1
VV
v t
d nd v t n
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-
Collision FrequencyCollision Frequency
bull The number of collisions per unit time is the The number of collisions per unit time is the collision frequencycollision frequency
bull The inverse of the collision frequency is the The inverse of the collision frequency is the collision collision mean free timemean free time
2ƒ Vd vn
- Chapter 21
- PowerPoint Presentation
- Slide 3
- Slide 4
- Molecular Model of an Ideal Gas
- Assumptions for Ideal Gas Theory
- Slide 7
- Pressure and Kinetic Energy
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Molecular Interpretation of Temperature
- Slide 18
- A Microscopic Description of Temperature
- Theorem of Equipartition of Energy
- Total Kinetic Energy
- Molar Specific Heat
- Slide 23
- Slide 24
- Ideal Monatomic Gas
- Slide 26
- Monatomic Gases
- Slide 28
- Slide 29
- Ratio of Molar Specific Heats
- Sample Values of Molar Specific Heats
- Slide 32
- Slide 33
- Slide 34
- Molar Specific Heats of Other Materials
- Adiabatic Processes for an Ideal Gas
- Equipartition of Energy
- Slide 38
- Slide 39
- Slide 40
- Agreement with Experiment
- Slide 42
- Complex Molecules
- Quantization of Energy
- Slide 45
- Slide 46
- Slide 47
- Slide 48
- Molar Specific Heat of Solids
- DuLong-Petit Law
- Slide 51
- Specific Heat and Molar Specific Heat of Some Solids and Liquids
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Boltzmann Distribution Law
- Number Density and Boltzmann Distribution Law
- Distribution of Molecular Speeds
- Slide 60
- Distribution Function
- Speed Distribution
- Slide 63
- Evaporation
- Mean Free Path
- Slide 66
- Slide 67
- Slide 68
- Collision Frequency
-