Physical Chemistry Third Edition

27.6 The Classical Statistical Mechanics of Dense Gases and Liquids 1151

in phase space, which has both coordinates and momentum components on its axes.

This probability density was obtained, and formulas for thermodynamic variables were

obtained in terms of a classical partition function. For a dilute gas, corrections for par-

ticle indistinguishability and the relation between volume in phase space and quantum

states must be made in order to recover the same formulas as were obtained with the

quantum canonical ensemble, but no advantage over quantum statistical mechanics is

obtained. A brief introduction to the classical statistical mechanics of dense gases and

liquids was presented.

ADDITIONAL PROBLEMS

27.22In Chapter 9 of this textbook, a crude equation of state for
a hard-sphere gas was written:

P(Vm−b)PVm(1−b/Vm)RT

where

bNAv

2

3

πd^3

and whereNAvis Avogadro’s constant,Vmis the molar

volumeV/n, and wheredis the hard-sphere diameter.

Convert this equation into the virial form

PVm

RT

 1 +

B 2

Vm

+

B 3

Vm^2

+...

using the identity

1

1

−x 1 +x+x^2 +...

Compare your result for the second virial coefficient with

that of Eq. (27.6-11).

27.23Identify the following statements as either true or false. If
a statement is true only under special circumstances, label
it as false.

a.Molecular partition functions are not useful for the

canonical ensemble approach to statistical

mechanics.

b.The canonical ensemble approach is more easily used

than the approach of Chapters 25 and 26 for dilute

gases.

c.The canonical ensemble approach can be used only

for dilute gases.

d.The statistical mechanical approach used in Chapters

25 and 26 is equivalent to using the microcanonical

ensemble.

e.Classical statistical mechanics applies accurately to

translations of molecules.

f.Classical statistical mechanics applies accurately to

rotations of molecules.

g.Classical statistical mechanics applies accurately to

vibrations of molecules.

h.Nonideal gases are more easily treated in classical

statistical mechanics than in quantum statistical

mechanics.