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16.5 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 Building a structural model based on the
ideal gas model The structure and the predictions of a
gas made by this structural model Pressure and temperature of an ideal
gas are interpreted in terms of microscopic variables
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Assumptions of the Structural Model
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 ideal-gas model,
in which we assumed the molecules to be 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 Meaning of “isotropic”: Any molecule can
move in any direction with any speed
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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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16.6 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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Exercises of Chapter 16
6, 17, 28, 33, 43, 48, 52, 55, 59, 60, 69, 78