Physical Chemistry Third Edition

25.4 The Calculation of Molecular Partition Functions 1077

Summary of the Chapter

The properties of macroscopic systems are determined by the behavior of molecules

making up the system. Through statistical mechanics we can in principle calculate these

properties from molecular properties.

The two principal postulates of statistical mechanics are: (1) Observed macroscopic

properties of a system held at a fixed energy can be equated to an average over system

mechanical states corresponding to that energy. (2) All system states of the same energy

are given equal weight in this average.

For a dilute gas, we can average over molecular states with a molecular probability

distribution. The most probable distribution was found as an approximation to the

average distribution. It is the Boltzmann distribution

pj

1

z

gje−εj/kBT

wherepjis the probability that a randomly selected molecule would be found in

molecule energy levelj, of degeneracygj, and energyj. The quantityzis the molecular

partition function. It can be written as a sum over levels,

z

∑

j

gje−εj/kBT

and can also be written as a sum over states, in which case the sum is the same except

for the absence of the degeneracygj.

For atomic gases, to an excellent approximation

zztrzel

and for diatomic or polyatomic gases, to a good approximation,

zztrzrotzvibzel

All factors except for the electronic partition function can be expressed with general

formulas. The electronic partition function must be summed up term by term, but can

often be approximated by a single term.

ADDITIONAL PROBLEMS

25.41a.Show that the values ofWin Table 25.1 are equal to
4
N 0 !N 1 !N 2 !N 3 !N 4 !, whereN^0 is the number of times 0
appears in the list of quantum numbers,N 1 is the
number of times 1 appears in the list, etc. Remember
that 0! is defined to equal 1.
b.Find the value ofΩfor the example system of four
oscillators if the system energy equals 5hν.
c.Find the value ofΩfor the example system
of four oscillators if the system energy equals 3hν.

25.42For a gas of diatomic molecules:

a.What is the effect on the translational partition

function if the Kelvin temperature is doubled at

constant volume?

b.What is the effect on the translational partition

function if the Kelvin temperature is doubled at

constant pressure?

c.What is the effect on the rotational partition function if

the Kelvin temperature is doubled at constant volume?

At constant pressure?