1 gases chapter 5. 2 gas properties four properties determine the physical behavior of any gas:...
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GasesGases
Chapter 5
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Gas PropertiesGas Properties
Four properties determine the physical behavior of any gas:Amount of gasGas pressureGas volumeGas temperature
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Gas pressureGas pressure
Gas molecules exert a force on the walls of their container when they collide with it
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Gas pressureGas pressure
Gas pressure can support a column of liquidPliquid = g•h•d
g = acceleration due to the force of gravity (constant)h = height of the liquid columnd = density of the liquid
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Atmospheric Atmospheric pressurepressure
Torricelli barometer In the closed tube, the liquid
falls until the pressure exerted by the column of liquid just balances the pressure exerted by the atmosphere.
Patmosphere = Pliquid = ghd
Patmosphere liquid heightStandard atmospheric
pressure (1 atm) is 760 mm Hg
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Units for pressureUnits for pressure
In this course we usually convert to atm
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Gas pressureGas pressure
Pliquid = g•h•dPressure exerted by a column
of liquid is proportional to the height of the column and the density of the liquid
Container shape and volume do not affect pressure
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ExampleExample
A barometer filled with perchloroethylene (d = 1.62 g/cm3) has a liquid height of 6.38 m. What is this pressure in mm Hg (d = 13.6 g/cm3)?P = ghd = g hpce dpce = g hHg dHg
hpce dpce = hHg dHg
hHg = hpce d pce = (6.38 m)(1.62 g/cm3) = 0.760 mdHg 13.6 g/cm3
hHg = 760 mm Hg
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Gas pressureGas pressure
A manometer compares the pressure of a gas in a container to the atmospheric pressure
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Gas Laws: BoyleGas Laws: Boyle
In 1662, Robert Boyle discovered the first of the simple gas laws
PV = constant
For a fixed amount of gas at constant temperature, gas pressure
and gas volume are inversely proportional
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Gas Laws: CharlesGas Laws: Charles
In 1787, Jacques Charles discovered a relationship between gas volume and gas temperature:
volume (mL)
temperature (°C)
• relationship between volume and temperature is always linear
• all gases reach V = 0 at same temperature, –273.15 °C
• this temperature is ABSOLUTE ZEROABSOLUTE ZERO
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A temperature scale for gases:A temperature scale for gases:the Kelvin scalethe Kelvin scale
A new temperature scale was invented: the Kelvin or absolute temperature scale
K = °C + 273.15Zero Kelvins = absolute zero
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Gas laws: CharlesGas laws: Charles
Using the Kelvin scale, Charles’ results isFor a fixed amount of gas at constant pressure, gas
volume and gas temperature are directly proportional
A similar relationship was found for pressure and temperature:
V
T=constant
P
T=constant
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Standard conditions for gasesStandard conditions for gases
Certain conditions of pressure and temperature have been chosen as standard conditions for gasesStandard temperature is 273.15 K (0 °C)Standard pressure is exactly 1 atm (760 mm Hg)
These conditions are referred to as STP (standard temperature and pressure)
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Gas laws: AvogadroGas laws: Avogadro
In 1811, Avogadro proposed that equal volumes of gases at the same temperature and pressure contain equal numbers of particles.At constant temperature and pressure, gas volume is
directly proportional to the number of moles of gas
Standard molar volume: at STP, one mole of gas occupies 22.4 L
V
n=constant
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Putting it all together:Putting it all together:Ideal Gas EquationIdeal Gas Equation
Combining Boyle’s Law, Charles’ Law, and Avogadro’s Law give one equation that includes all four gas variables:
R is the ideal or universal gas constantR = 0.08206 atm L/mol K
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PVnT
=R or PV =nRT
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Using the Ideal Gas EquationUsing the Ideal Gas Equation
Ideal gas equation may be expressed two ways:One set of conditions: ideal gas law
Two sets of conditions: general gas equation
PV =nRT
P1V1
n1T1
=P2V2
n2T2
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Ideal Gas Equation and molar massIdeal Gas Equation and molar mass
Solving for molar mass (M)
PV =nRTn =
mM
PV =mRTM
M =mRTPV
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Ideal Gas Equation and gas densityIdeal Gas Equation and gas density
PV =nRT =mRTM
P =mRTVM
d =mV
P =dRTM
d =MPRT
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Gas densityGas density
Gas density depends directly on pressure and inversely on temperature
Gas density is directly proportional to molar mass
d =MPRT
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Mixtures of GasesMixtures of Gases
Ideal gas law applies to pure gases and to mixturesIn a gas mixture, each gas occupies the entire
container volume, at its own pressureThe pressure contributed by a gas in a mixture is
the partial pressure of that gasPtotal = PA + PB (Dalton’s Law of Partial
Pressures)
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Mixtures of Mixtures of GasesGases
When a gas is collected over water, it is always “wet” (mixed with water vapor).Ptotal = Pbarometric = Pgas + Pwater vapor
Example: If 35.5 mL of H2 are collected over water at 26 °C and a barometric pressure of 755 mm Hg, what is the pressure of the H2 gas? The water vapor pressure at 26 °C is 25.2 mm Hg.
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Gas mixturesGas mixtures
The mole fraction represents the contribution of each gas to the total number of moles.XA = mole fraction of A
XA =nA
ntotal
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Gas MixturesGas Mixtures
For gas mixtures,
mole fractionequals
pressure fractionequals
volume fraction
Each gas occupiesthe entire container.
The volume fraction describesthe % composition by volume.
nAntotal
=PA
Ptotal
=VA
Vtotal
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Gases in Chemical ReactionsGases in Chemical Reactions
To convert gas volume into moles for stoichiometry, use the ideal gas equation:
If both substances in the problem are gases, at the same T and P, gas volume ratios = mole ratios.
P2 = P1 and T2 = T1
n2
n1
=P2V2RT2
P1V1RT1
=V2
V1
n =PVRT
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A Model for Gas BehaviorA Model for Gas Behavior
Gas laws describe what gases do, but not why.Kinetic Molecular Theory of Gases (KMT) is the
model that explains gas behavior.developed by Maxwell & Boltzmann in the mid-1800sbased on the concept of an ideal or perfect gas
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Ideal gasIdeal gas
Composed of tiny particles in constant, random, straight-line motion Gas molecules are point masses, so gas volume is just the empty
space between the molecules Molecules collide with each other and with the walls of their
container The molecules are completely independent of each other, with no
attractive or repulsive forces between them. Individual molecules may gain or lose energy during collisions, but
the total energy of the gas sample depends only on the absolute temperature.
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Molecular collisions and pressureMolecular collisions and pressure
Force of molecular collisions depends oncollision frequencymolecule kinetic energy, ek
ek depends on molecule mass m and molecule speed u
molecules move at various speeds in all directions
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ek =12mu2
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Molecular speedMolecular speed
Molecules move at various speedsImagine 3 cars going 40 mph, 50 mph, and 60 mph
Mean speed = u = (40 + 50 + 60) ÷ 3 = 50 mphMean square speed (average of speeds squared)
u2 = (402 + 502 + 602) ÷ 3 = 2567 m2/hr2 Root mean square speed
urms = √2567 m2/hr2 = 50.7 mph
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Distribution of molecule speedsDistribution of molecule speeds
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The basic equation The basic equation of KMTof KMT
Combining collision frequency, molecule kinetic energy, and the distribution of molecule speeds gives the basic equation of KMTP = gas pressure and V = gas volumeN = number of moleculesm = molecule massu2 = mean square molecule speed (average of speeds squared)
€
P =13NVmu2
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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas
€
P =13NA
Vmu2
PV=13NAmu
2 =RT
If n = 1,N = NA
andPV = RT
€
P =13NVmu2
€
PV=nRTAvogadro’s number
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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas
€
PV=13NAmu
2 =RT
NAmu2 =3RT=Mu2
NA x m (Avogadro’s number x mass of one molecule)= mass of one mole of molecules (molar mass M)
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Combine the Equations ofCombine the Equations ofKMT and Ideal GasKMT and Ideal Gas
€
3RT=Mu2
u2 =urms=3RTM
We can calculate the root mean square speedfrom temperature and molar mass
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Calculating root mean square speedCalculating root mean square speedTo calculate root mean square speed from
temperature and molar mass:Units must agree!Speed is in m/s, so
R must be 8.3145 J/mol K M must be in kg per mole, because Joule = kg m2 / s2
Speed is inversely related to molar mass: light molecules are faster, heavy molecules are slower
€
urms=3RTM
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Interpreting temperatureInterpreting temperature
Combine the KMT and ideal gas equations again
€
ek =32
RNA
T=constant×T
€
PV=13NAmu
2 =23 ×1
2NAmu2
€
PV=23NA
12( mu2
)=23NAek =RT
Again assume n=1, so N = NA and PV = RT
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Interpreting Interpreting temperaturetemperature
Absolute (Kelvin) temperature is directly proportional to average molecular kinetic energy
At T = 0, ek = 0
€
ek =32
RNA
T
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Diffusion and EffusionDiffusion and EffusionDiffusion (a) is
migration or mixing due to random molecular motion
Effusion (b) is escape of gas molecules through a tiny hole
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Rates of diffusion/effusionRates of diffusion/effusion
The rate of diffusion or effusion is directly proportional to molecular speed:
The rates of diffusion/effusion of two different gases are inversely proportional to the square roots of their molar masses (Graham’s Law)
€
rate of effusion of Arate of effusion of B
=(urms)A
(urms)B
=3RT MA
3RT MB
=MB
MA
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Using Graham’s LawUsing Graham’s Law
Graham’s Law applies to relative rates, speeds, amounts of gas effused in a given time, or distances traveled in a given time.
€
rate of effusion of Arate of effusion of B
=(urms)A
(urms)B
=MB
MA
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Using Graham’s Law with timesUsing Graham’s Law with times
Graham’s law can be confusing when applied to times
rate = amount of gas (n) time (t)
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Use common senseUse common sensewith Graham’s Lawwith Graham’s Law
When you compare two gases, the lighter gasescapes at a greater rate has a greater root mean square speedcan effuse a larger amount in a given timecan travel farther in a given timeneeds less time for a given amount to escape or travel
Make sure your answer reflects this reality!
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Reality CheckReality Check
Ideal gas molecules Real gas molecules constant, random, same
straight-line motion point masses are NOT points – molecules
have volume; Vreal gas > Videal
gas
independent of each other are NOT independent – molecules are attracted to
each other, so Preal gas < Pideal gas
gain / lose energy during same (some energy may becollisions, but total energy absorbed in moleculardepends only on T ek))
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Real gas correctionsReal gas corrections
For a real gas,a corrects for attractions between gas molecules, which
tend to decrease the force and/or frequency of collisions (so Preal < Pideal)
b corrects for the actual volume of each gas molecule, which increases the amount of space the gas occupies (so Vreal > Videal)
The values of a and b depend on the type of gas
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An equation for real gases:An equation for real gases:the van der Waals equationthe van der Waals equation
€
Preal+n2aV 2
⎛
⎝ ⎜ ⎞
⎠ ⎟ Vreal– nb( ) =nRT=PidealVideal
Add correction to Preal to make it equal to Pideal,because intermolecular attractions decrease real pressure
Subtract correction to Vreal to make it equal to Videal,because molecular volume increases real volume
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When do I needWhen do I needthe van der Waals equation?the van der Waals equation?
Deviations from ideality become significant whenmolecules are close together (high pressure)molecules are slow (low temperature)
At low pressure and high temperature, real gases tend to behave ideally
At high pressure and low temperature, real gases do not tend to behave ideally
}non-idealconditions