1. CHAPTER 14
THE CLASSICAL STATISTICAL TREATMENT OF AN IDEAL GAS

2. This chapter is a continuation of the previous one. We show
how a partition function is used, in this case with an ideal
gas.
We also obtain a formula for the distribution of speeds
in a gas, a result that is often obtained from the kinetic theory
of gases.

3. In the previous chapter we derived the following equations
appropriate for a system fo r which the Maxwell-Boltzmann
Distribution is applicable.

4. Thermodynamic properties from the partition function
In this chapter we will concentrate on an ideal gas. The
Maxwell-Boltzmann distribution is used. We develop expressions
for various thermodynamic variables in terms of ln(Z), a process
that we began in Chapter 13.
Internal energy:
We recall that
Differentiating Z gives
Therefore
or more compactly

5. Gibbs Function
The Gibbs thermodynamic potential has previously been
written in terms of the chemical potential. For a single component
system so
Enthalpy: We have (see diagram on Fact Sheet) H=G+TS
and using the expression for U
Pressure:
From the VFT-VUS diagram dF=-PdV-SdT
Considering F=F(V,T)

6. This gives
(reciprocity relationship)
Hence, evaluating this partial derivative using the expression for
F=F(lnZ), that is, F = -NkT[lnZ-lnN-1]
results in

7. Summary
(M-B distribution!)
potentials

8. Partition function for a gas.( See text for different approach.)
We first note that, instead of using
in which the sum is over the energy levels, we can use
in which the sum is over energy states.
Consider an ideal monatomic gas in a cubical container of volume V.
The energy states are given by
The partition function is then a sum over the n’s, with the integers
going from 1 to infinity. We temporarily set
This permits us to write

9. For this system the partition function is then
Unless T or V is extremely small the energy levels are very closely
spaced and a good approximation is to replace sums by integrals.
The integral is well-known (see Fact Sheet)
Using the expression for a: or
constant

10. Properties of a monatomic ideal gas.
Once we have lnZ we can rapidly compute the other thermodynamic
variables. Consider the pressure:
But so PV=NkT
(equation of state)
Next let us calculate the internal energy. We have
which quickly yields
This again reminds us that the average kinetic energy per particle is
(3/2)kT
Hence, using a model of a gas and statistical concepts, we have
derived the empirical relationships for P and U. These results can
be checked by experiments.

11. Now we consider the entropy
Consider 1 mole of an ideal monatomic gas:
This is called the Sackur-Tetrode Equation. Before a discussion of
statistical mechanics we obtained an expression for s which included
a constant, , whose value was unknown. The equation was

12. Statistical mechanics provides us with an expression for this constant.
This equation agrees well with experiment. The theoretical expression
is very important in a study of the thermodynamics of chemical
reactions.
We should note that the expression for s is not valid as T goes to zero
as, in this expression, s does not approach zero. This is not surprising
as we have used Maxwell-Boltzmann statistics, which are not valid
at low temperatures.

13. Distribution of speeds in an ideal monatomic gas.
It is often necessary to compute the averages of various
functions of the molecular speeds (in astrophysics, for example).
To do so, we must know the probability distribution of molecular
speeds, the so-called Maxwell-Boltzmann distribution of speeds.
This was first derived by Maxwell on the basis of molecular
collisions and later by Boltzmann using statistical methods. For a
continuous distribution we calculated the number of states in an
energy region For bosons with s=0, the density of states is:
We also have and
Hence

14. In addition, for an ideal gas,
Using (1) and (2) in
is the number of molecules per unit energy interval.
is then the number of molecules in an infinitesimal
energy interval.
Notice that Planck’s constant does not appear in the expression.
The work of Planck came after the death of Maxwell.

15. What is measured experimentally is the distribution of speeds.
We can convert the formula to the number of molecules per
unit speed.
and so
The distribution expression is then or

16. (see next slide)
With these expressions one can calculate averages for functions of
the energy or the speed.
We will write down some averages. Students should go through
the calculations in detail.
Mean speed

18. Root-mean-square speed
Most probable speed
is that speed for which
See Table 11.2 for some values.
The Maxwell speed distribution has been checked by careful
experimental measurements and the agreement is excellent.

19. Equipartition of energy
The principle of equipartition of energy states that:
At a temperature T, the average energy of any
quadratic degree of freedom is (1/2)kT
If a system has N molecules, each with f degrees of freedom, and there
are no other (non-quadratic) temperature-dependent forms of energy,
then the total thermal energy is
This is not the total energy of the system. There are “static” terms,
such as the energy stored in chemical bonds, mass energy, etc. A
gas in a gravitational field would have an energy contribution mgΔz,
which is not quadratic.
{Note: The concept of degree of freedom is different (broader) in
thermodynamics than in classical mechanics.}

20. The principle does not take into consideration quantum effects,
such as the quantization of rotational and vibrational energies. It
is applicable to indistinguishable particles in equilibrium. The
essential condition for validity is that the energy levels must be
continuous (classical!).
It does not work for liquids and solids: in these cases the interactions among the particles must be taken into consideration when calculating the partition function.
The quadratic forms of energy include translational kinetic energy (along three axes), rotational kinetic energy and
vibrational terms.
We will then write the energy of a molecule as and there will be f terms.

21. The partition function is then
Because of the properties of exponentials, this is a product of terms,
each term corresponding to one degree of freedom.
Since we are dealing with a continuous energy distribution, the sums
are replaced by integrals.
We make a change of variables
Since there are f terms in the product
The important point to notice is that the do not contain
Taking the logarithm of Z,

22. We have, for the internal energy,
The average internal energy is
Hence
The average value of each independent quadratic energy term is then

23. This is a classical result. In a strictly correct quantum mechanical
description there is a discrete set of energy states and in general
sums cannot be replaced by integrals. At temperatures for which
the spacing between levels is <<kT, the equipartition theorem is
valid. This condition is not valid at low temperatures.
In using the equipartition theorem, it is necessary to decide
upon the appropriate number of degrees of freedom. This is simple
for a monatomic molecule, which has only translational motion and
so f=3. A diatomic molecule ( like a dumbbell) has an additional
two degrees of rotational motion and so f=5. A diatomic molecule
can also vibrate which introduces two additional degrees of freedom
(one for kinetic energy and one for potential energy). However these
vibrational modes do not contribute to a molecule’s thermal energy
except at high temperatures. We say that these vibrational modes
are “frozen out” at room temperature. Clearly a complete discussion
of the thermal properties ( such as the specific heat) of gases
requires the use of quantum mechanics.

24. If we confine ourselves to “normal” temperatures and use
we obtain:
monatomic ideal gas:
diatomic ideal gas:

25. These values for gases are in accord with experimental
results at room temperature, but these classical results
are incorrect at low and high temperatures. The next
slide shows experimental values for the specific
heat of a gas as a function of temperature. The need
for a quantum mechanical discussion is obvious. Chapter
15 of the textbook is devoted to this important topic.

26. Experimental results for the specific heat at constant volume for
H2 as a function of temperature. θrot (85.5K) and θvib (6140K)
are characteristic temperatures at which rotational and
vibrational modes are activated.