chapter 16 temperature and the kinetic theory of gases
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Chapter 16
Temperature
and the
Kinetic Theory of Gases
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Overview of Thermodynamics Extends the ideas of temperature and internal
energy Concerned with concepts of energy transfers
between a system and its environment And the resulting variations in temperature or
changes in state Explains the bulk properties of matter and the
correlation between them and the mechanics of atoms and molecules
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Temperature We associate the concept of temperature with
how hot or cold an objects feels Our senses provide us with a qualitative
indication of temperature Our senses are unreliable for this purpose We need a reliable and reproducible way for
establishing the relative hotness or coldness of objects that is related solely to the temperature of the object Thermometers are used for these measurements
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Thermal Contact Two objects are in thermal contact
with each other if energy can be exchanged between them The exchanges can be in the form of heat
or electromagnetic radiation The energy is exchanged due to a
temperature difference
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Thermal Equilibrium Thermal equilibrium is a situation in
which two object would not exchange energy by heat or electromagnetic radiation if they were placed in thermal contact The thermal contact does not have to also
be physical contact
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Zeroth Law of Thermodynamics If objects A and B are separately in
thermal equilibrium with a third object C, then A and B are in thermal equilibrium with each other Let object C be the thermometer Since they are in thermal equilibrium with
each other, there is no energy exchanged among them
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Zeroth Law of Thermodynamics, Example
Object C (thermometer) is placed in contact with A until it they achieve thermal equilibrium
The reading on C is recorded Object C is then placed in contact with object B until they
achieve thermal equilibrium The reading on C is recorded again
If the two readings are the same, A and B are also in thermal equilibrium
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Temperature Temperature can be thought of as the
property that determines whether an object is in thermal equilibrium with other objects
Two objects in thermal equilibrium with each other are at the same temperature If two objects have different temperatures, they
are not in thermal equilibrium with each other
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Thermometers A thermometer is a device that is used
to measure the temperature of a system Thermometers are based on the
principle that some physical property of a system changes as the system’s temperature changes
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Thermometers, cont The properties include
The volume of a liquid The length of a solid The pressure of a gas at a constant volume The volume of a gas at a constant pressure The electric resistance of a conductor The color of an object
A temperature scale can be established on the basis of any of these physical properties
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Thermometer, Liquid in Glass A common type
of thermometer is a liquid-in-glass
The material in the capillary tube expands as it is heated
The liquid is usually mercury or alcohol
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Calibrating a Thermometer A thermometer can be calibrated by
placing it in contact with some environments that remain at constant temperature
Common systems involve water A mixture of ice and water at atmospheric
pressure Called the ice point or freezing point of water
A mixture of water and steam in equilibrium Called the steam point or boiling point of water
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Celsius Scale The ice point of water is defined to be
0oC The steam point of water is defined to
be 100oC The length of the column between these
two points is divided into 100 equal segments, called degrees
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Problems with Liquid-in-Glass Thermometers An alcohol thermometer and a mercury
thermometer may agree only at the calibration points
The discrepancies between thermometers are especially large when the temperatures being measured are far from the calibration points
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Gas Thermometer The gas thermometer offers a way to
define temperature Also directly relates temperature to internal
energy Temperature readings are nearly
independent of the substance used in the thermometer
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Constant Volume Gas Thermometer
The physical change exploited is the variation of pressure of a fixed volume gas as its temperature changes
The volume of the gas is kept constant by raising or lowering the reservoir B to keep the mercury level at A constant
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Constant Volume Gas Thermometer, cont The thermometer is calibrated by using
a ice water bath and a steam water bath The pressures of the mercury under
each situation are recorded The volume is kept constant by adjusting A
The information is plotted
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Constant Volume Gas Thermometer, final To find the
temperature of a substance, the gas flask is placed in thermal contact with the substance
The pressure is found on the graph
The temperature is read from the graph
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Absolute Zero The thermometer
readings are virtually independent of the gas used
If the lines for various gases are extended, the pressure is always zero when the temperature is –273.15o C
This temperature is called absolute zero
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Absolute Temperature Scale Absolute zero is used as the basis of
the absolute temperature scale The size of the degree on the absolute
scale is the same as the size of the degree on the Celsius scale
To convert: TC = T – 273.15 TC is the temperature in Celsius T is the Kelvin (absolute) temperature
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The absolute temperature scale is now based on two new fixed points Adopted in 1954 by the International
Committee on Weights and Measures One point is absolute zero The other point is the triple point of water
This is the combination of temperature and pressure where ice, water, and steam can all coexist
Absolute Temperature Scale, 2
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Absolute Temperature Scale, 3 The triple point of water occurs at 0.01o
C and 4.58 mm of mercury This temperature was set to be 273.16
on the absolute temperature scale This made the old absolute scale agree
closely with the new one The unit of the absolute scale is the kelvin
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Absolute Temperature Scale, 4 The absolute scale is also called the
Kelvin scale Named for William Thomson, Lord Kelvin
The triple point temperature is 273.16 K No degree symbol is used with kelvins
The kelvin is defined as 1/273.16 of the temperature of the triple point of water
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Some Examples of Absolute Temperatures
This figure gives some absolute temperatures at which various physical processes occur
The scale is logarithmic The temperature of
absolute cannot be achieved
Experiments have come close
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Energy at Absolute Zero According to classical physics, the kinetic
energy of the gas molecules would become zero at absolute zero
The molecular motion would cease Therefore, the molecules would settle out on the
bottom of the container Quantum theory modifies this and shows
some residual energy would remain This energy is called the zero-point energy
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Fahrenheit Scale A common scale in everyday use in
the US Named for Daniel Fahrenheit Temperature of the ice point is 32oF Temperature of the steam point is 212oF There are 180 divisions (degrees)
between the two reference points
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Comparison of Scales Celsius and Kelvin have the same size
degrees, but different starting pointsTC = T – 273.15
Celsius and Fahrenheit have difference sized degrees and different starting points
932
5F CT T F
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Comparison of Scales, cont To compare changes in temperature
Ice point temperatures 0oC = 273.15 K = 32oF
steam point temperatures 100oC = 373.15 K = 212oF
5
9C FT T T
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Thermal Expansion Thermal expansion is the increase in the
size of an object with an increase in its temperature
Thermal expansion is a consequence of the change in the average separation between the atoms in an object
If the expansion is small relative to the original dimensions of the object, the change in any dimension is, to a good approximation, proportional to the first power of the change in temperature
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Thermal Expansion, example As the washer is heated, all
the dimensions will increase A cavity in a piece of
material expands in the same way as if the cavity were filled with the material
The expansion is exaggerated in this figure
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Linear Expansion Assume an object has an initial length L That length increases by L as the
temperature changes by T The change in length can be found by
L = Li T is the average coefficient of linear
expansion
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Linear Expansion, cont This equation can also be written in
terms of the initial and final conditions of the object: Lf – Li = Li(Tf – Ti)
The coefficient of linear expansion has units of (oC)-1
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Linear Expansion, final Some materials expand along one
dimension, but contract along another as the temperature increases
Since the linear dimensions change, it follows that the surface area and volume also change with a change in temperature
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Thermal Expansion
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Volume Expansion The change in volume is proportional to the
original volume and to the change in temperature
V = Vi T is the average coefficient of volume
expansion For a solid, 3
This assumes the material is isotropic, the same in all directions
For a liquid or gas, is given in the table
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Area Expansion The change in area is proportional to
the original area and to the change in temperature
A = Ai T is the average coefficient of area
expansion = 2
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Thermal Expansion, Example In many situations,
joints are used to allow room for thermal expansion
The long, vertical joint is filled with a soft material that allows the wall to expand and contract as the temperature of the bricks changes
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Bimetallic Strip Each substance has
its own characteristic average coefficient of expansion
This can be made use of in the device shown, called a bimetallic strip
It can be used in a thermostat
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Water’s Unusual Behavior As the temperature
increases from 0o C to 4o C, water contracts
Its density increases Above 4o C, water
expands with increasing temperature
Its density decreases The maximum density
of water (1 000 kg/m3) occurs at 4oC
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Gas: Equation of State It is useful to know how the volume, pressure
and temperature of the gas of mass m are related
The equation that interrelates these quantities is called the equation of state These are generally quite complicated If the gas is maintained at a low pressure, the
equation of state becomes much easier This type of a low density gas is commonly
referred to as an ideal gas
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Ideal Gas – Details A collection of atoms or molecules that
Move randomly Exert no long-range forces on one another Are so small that they occupy a negligible
fraction of the volume of their container
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The Mole The amount of gas in a given volume is
conveniently expressed in terms of the number of moles
One mole of any substance is that amount of the substance that contains Avogadro’s number of molecules Avogadro’s number, NA = 6.022 x 1023
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Moles, cont The number of moles can be
determined from the mass of the substance: n = m / M M is the molar mass of the substance
Commonly expressed in g/mole m is the mass of the sample n is the number of moles
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Gas Laws When a gas is kept at a constant
temperature, its pressure is inversely proportional to its volume (Boyle’s Law)
When a gas is kept at a constant pressure, the volume is directly proportional to the temperature (Charles’ Laws)
When the volume of the gas is kept constant, the pressure is directly proportional to the temperature (Guy-Lussac’s Law)
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Ideal Gas Law The equation of state for an ideal gas
combines and summarizes the other gas lawsPV = n R T
This is known as the ideal gas law R is a constant, called the Universal Gas
Constant R = 8.314 J/ mol K = 0.08214 L atm/mol K
From this, you can determine that 1 mole of any gas at atmospheric pressure and at 0o C is 22.4 L
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Ideal Gas Law, cont The ideal gas law is often expressed in
terms of the total number of molecules, N, present in the sample
P V = n R T = (N / NA) R T = N kB T kB is Boltzmann’s constant
kB = 1.38 x 10-23 J / K
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Ludwid Boltzmann 1844 – 1906 Contributions to
Kinetic theory of gases Electromagnetism Thermodynamics
Work in kinetic theory led to the branch of physics called statistical mechanics
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Kinetic Theory of Gases Uses a structural model based on the
ideal gas model Combines the structural model and its
predictions Pressure and temperature of an ideal
gas are interpreted in terms of microscopic variables
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Structural Model Assumptions The number of molecules in the gas is
large, and the average separation between them is large compared with their dimensions The molecules occupy a negligible volume
within the container This is consistent with the macroscopic
model where we assumed the molecules were point-like
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Structural Model Assumptions, 2 The molecules obey Newton’s laws of
motion, but as a whole their motion is isotropic Any molecule can move in any direction
with any speed Meaning of isotropic
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Structural Model Assumptions, 3 The molecules interact only by short-range
forces during elastic collisions This is consistent with the ideal gas model, in
which the molecules exert no long-range forces on each other
The molecules make elastic collisions with the walls
The gas under consideration is a pure substance All molecules are identical
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Ideal Gas Notes An ideal gas is often pictured as
consisting of single atoms However, the behavior of molecular
gases approximate that of ideal gases quite well Molecular rotations and vibrations have no
effect, on average, on the motions considered
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Pressure and Kinetic Energy Assume a container
is a cube Edges are length d
Look at the motion of the molecule in terms of its velocity components
Look at its momentum and the average force
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Pressure and Kinetic Energy, 2 Assume perfectly
elastic collisions with the walls of the container
The relationship between the pressure and the molecular kinetic energy comes from momentum and Newton’s Laws
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Pressure and Kinetic Energy, 3 The relationship is
This tells us that pressure is proportional to the number of molecules per unit volume (N/V) and to the average translational kinetic energy of the molecules
___22 1
3 2
NP mv
V
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Pressure and Kinetic Energy, final This equation also relates the macroscopic
quantity of pressure with a microscopic quantity of the average value of the molecular translational kinetic energy
One way to increase the pressure is to increase the number of molecules per unit volume
The pressure can also be increased by increasing the speed (kinetic energy) of the molecules
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A Molecular Interpretation of Temperature We can take the pressure as it relates to the
kinetic energy and compare it to the pressure from the equation of state for an idea gas
Therefore, the temperature is a direct measure of the average translational molecular kinetic energy
___22 1
3 2 B
NP mv Nk T
V
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A Microscopic Description of Temperature, cont Simplifying the equation relating
temperature and kinetic energy gives
This can be applied to each direction,
with similar expressions for vy and vz
___
21 3
2 2 Bmv k T
___
21 1
2 2x Bmv k T
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A Microscopic Description of Temperature, final Each translational degree of freedom
contributes an equal amount to the energy of the gas In general, a degree of freedom refers to
an independent means by which a molecule can possess energy
A generalization of this result is called the theorem of equipartition of energy
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Theorem of Equipartition of Energy The theorem states that the energy of a
system in thermal equilibrium is equally divided among all degrees of freedom
Each degree of freedom contributes ½ kBT per molecule to the energy of the system
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Total Kinetic Energy The total translational kinetic energy is just N
times the kinetic energy of each molecule
This tells us that the total translational kinetic energy of a system of molecules is proportional to the absolute temperature of the system
___21 3 3
2 2 2total BE N mv Nk T nRT
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Monatomic Gas For a monatomic gas, translational kinetic
energy is the only type of energy the particles of the gas can have
Therefore, the total energy is the internal energy:
For polyatomic molecules, additional forms of energy storage are available, but the proportionality between Eint and T remains
int
3
2E nRT
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Root Mean Square Speed The root mean square (rms) speed is
the square root of the average of the squares of the speeds Square, average, take the square root
Solving for vrms we find
M is the molar mass in kg/mole
___
2 3 3Brms
k T RTv v
m M
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Some Example vrms Values
At a given temperature, lighter molecules move faster, on the average, than heavier molecules
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Distribution of Molecular Speeds The observed speed
distribution of gas molecules in thermal equilibrium is shown
NV is called the Maxwell-Boltzmann distribution function
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Distribution Function The fundamental expression that describes
the distribution of speeds in N gas molecules is
mo is the mass of a gas molecule, kB is Boltzmann’s constant and T is the absolute temperature
23
2224
2B
mvk To
VB
mN N v e
k T
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Average and Most Probable Speeds
The average speed is somewhat lower than the rms speed
The most probable speed, vmp is the speed at which the distribution curve reaches a peak
2
1.41B Bmp
o o
k T k Tv
m m
81.60B B
o o
k T k Tv
m m
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Speed Distribution The peak shifts to the right
as T increases This shows that the average
speed increases with increasing temperature
The width of the curve increases with temperature
The asymmetric shape occurs because the lowest possible speed is 0 and the upper classical limit is infinity
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Speed Distribution, final The distribution of molecular speeds
depends both on the mass and on temperature
The speed distribution for liquids is similar to that of gasses
rms mpv v v
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Evaporation Some molecules in the liquid are more energetic
than others Some of the faster moving molecules penetrate
the surface and leave the liquid This occurs even before the boiling point is reached
The molecules that escape are those that have enough energy to overcome the attractive forces of the molecules in the liquid phase
The molecules left behind have lower kinetic energies
Therefore, evaporation is a cooling process
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Atmosphere For such a huge volume of gas as the
atmosphere, the assumption of a uniform temperature throughout the gas is not valid
There are variations in temperature Over the surface of the Earth At different heights in the atmosphere
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Temperature and Height At each location, there is a
decrease in temperature with an increase in height
As the height increases, the pressure decreases
The air parcel does work on its surroundings and its energy decreases
The decrease in energy is manifested as a decrease in temperature
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Lapse Rate The atmospheric lapse rate is the
decrease in temperature with height The lapse rate is similar at various
locations across the surface of the earth The average global lapse rate is about –
6.5o C / km This is for the area of the atmosphere
called the troposphere
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Layers of the Atmosphere Troposphere
The lower part of the atmosphere Where weather occurs and airplanes fly
Tropopause The imaginary boundary between the troposphere
and the next layer Stratosphere
Layer above the tropopause Temperature remains relatively constant with
height