Chapter 19
The Kinetic Theory of Gases
In this chapter we will introduce the kinetic theory of
gases, which relates the motion of the constituent atoms to
the volume, pressure, and temperature of the gas. The
following topics will be covered:
Ideal gas law
Internal energy of an ideal gas
Distribution of speeds among the atoms in a gas
Specific heat under constant volume
Specific heat under constant volume
Adiabatic expansion of an ideal gas
(19–1)
Example 1:
Example 2:
Example 3:
Example 4:
Work Done by an Ideal Gas at Constant Volume
p
Consider process a  f . During this process
the volume of the ideal gas is kept constant.
Thus the work W done by the gas is
Vi
Vf
W   pdV  0.
Work Done by an Ideal Gas at Constant Pressure
Consider process i  a. During this process
the pressure is kept constant at p and the volume
changes from Vi to V f . The work W done by
Vf
the gas is W 
Vf
 pdV  p  dV  p V
Vi
f
 Vi .
Vi
(19–5)
Example 5:
Example 6:
Example 7:
The molar mass of argon is 39.95 g/mol.
vrms
3  8.31J/mol  K  313K 
3RT


 442 m/s.
3
M
39.95  10 kg/mol
Example 8:
K avg 
Translational Kinetic Energy
mv 2
The kinetic energy of a gas molecule K 
.
2
2
 mv 2 
mvrms
Its average kinetic energy K avg  
.
 
2
 2 avg
Thus K avg 
3kT
2
m 3RT 3RT

.
2 M
2NA
We finally get:
K avg 
3kT
2
At a given temperature T all ideal gas molecules, no matter what their mass,
have the same average translational kinetic energy. When we measure the
temperature of a gas, we are also measuring the average translational
kinetic energy of its molecules.
(19–8)
Example 9:
K avg
3
 (1.38  1023 J/K) (1600 K) = 3.31  1020 J .
2
3nRT
Eint 
2
Internal Energy of an Ideal Gas
Consider a monatomic gas such as He, Ar, or Kr. In this case the internal energy
Eint of the gas is the sum of the translational kinetic energies of the constituent
atoms. The average translational kinetic energy of a single atom is given by the
equation K avg 
3kT
.
2
A gas sample of n moles contains N  nN A atoms.
The internal energy of the gas Eint  NK avg 
nN A 3kT 3nRT

.
2
2
The equation above expresses the following important result:
The internal energy Eint of an ideal gas is a function of gas temperature
only; it does not depend on any other parameter.
(19–11)
Example 10:
P44

pV
i i  p f Vf
TV
i i
 1

 Tf V f
Adiabatic Expansion of an Ideal Gas
 1
Consider the ideal gas in fig. a. The container is well
insulated. When the gas expands, no heat is transferred
to or from the gas. This process is called adiabatic.
Such a process is indicated on the p - V diagram of fig. b
by the red line. The gas starts at an initial pressure pi and
initial volume Vi . The corresponding final parameters
are p f and V f . The process is described by the equation


piVi  p f V f . Here the constant  
Cp
CV
.
Using the ideal gas law we can get the equation
TV
i i
 1
 Tf V f
 1
Vi  1
 T f  Ti  1
Vf
If V f  Vi we have adiabatic expansion and T f  Ti .
If V f  Vi we have adiabatic compression and T f  Ti .
(19–15)
Ti  T f
piVi  p f V f
Free Expansion
In a free expansion, a gas of initial volume Vi and initial
pressure pi is allowed to expand in an empty container
so that the final volume is V f and the final pressure p f .
In a free expansion Q  0 because the gas container is insulated. Furthermore,
since the expansion takes place in vacuum the net work W  0.
The first law of thermodynamics predicts that Eint  0.
Since the gas is assumed to be ideal there is no change in temperature:
Ti  T f . Using the law of ideal gases we get the following equation,
which connects the initial with the final state of the gas:
piVi  p f V f .
(19–16)
Checkpoint 5:
Rank paths 1,2 and 3 in Fig according to the energy transfer to
the gas as heat Greatest First.
Example 18:
Example 19:
Example 20:
Example 21:
Example 22:
Example 23:
Example 24:
Example 25:
Example 26:
Example 27:
Example 28:
Example 29:
Example 30:
Example 31:
Example 32:
Example 33:
Example 34:
Example 35:
Example 36:
Example 37:
Example 38: