Physical Chemistry Third Edition

27.6 The Classical Statistical Mechanics of Dense Gases and Liquids 1147

27.6 The Classical Statistical Mechanics of

Dense Gases and Liquids

Although the classical statistical mechanics of dilute gases provides no advantage

over quantum statistical mechanics, its application to liquids and dense gases is more

productive than the application of quantum statistical mechanics to these systems.

Unfortunately, a full discussion of this application is beyond the scope of this book,

and we present only a few basic facts and formulas.

We consider a model monatomic substance that has apairwise intermolecular

potential energy:

V 

N∑− 1

i 1

∑N

ij+ 1

u(rij) (27.6-1)

whereu(rij) is the potential energy of particlesiandj(the pair potential function) and

rijis the distance between the centers of moleculesiandj. This is a good approximation

for dense gases. In liquids there can be contributions to the potential energy that

involve three particles at a time, but it is still a fair approximation for liquids. The

limits on the sums are chosen so that each pair of particles appears only once in the

double sum.

The general equations for the canonical partition function and its application to

thermodynamics functions are valid. From Eq. (27.4-6)

Zcl(2πmkBT)^3 N/^2

∫

exp

⎡

⎣−

N∑− 1

j 1

∑N

ij+ 1

u(rij)

kBT

⎤

⎦dq(2πmkBT)^3 N/^2 ζ (27.6-2)

whereζis the configuration integral. Mayer showed that the configuration integral can

be expressed as a series of powers and logarithms of the density of the gas or liquid.

The leading term is that for a dilute gas, and the coefficients of the other terms are

integrals calledclusterintegrals.^2

Joseph E. MayerwasanAmerican
physicalchemist who madevarious
contributionstostatisticalmechanics.
Hemet his futurewife,MariaGoeppert
(1906–1972) inGöttingen,Germany,
whenhewasapostdoctoralresearcher.
Theymovedto theUnitedStatesin
1930 andworkedattheJohns Hopkins
University,ColumbiaUniversity,
theUniversity of Chicago,andthe
University of CaliforniaatSanDiego.
Mrs. Goeppert Mayer workedwithout
payuntil 1959,whentheymovedto
UCSD.Shereceivedthe 1963 Nobel
Prizeinphysics for herworkontheshell
modelofthenucleus,which shehad
carriedout whileworking without pay.

The potential energy of our model fluid is given by

〈V〉

1

ζ

∫

⎡

⎣

N∑− 1

b 1

∑N

ab+ 1

u(rab)

⎤

⎦exp

⎡

⎣−

N∑− 1

j 1

∑N

ij+ 1

u(rij)

kBT

⎤

⎦d^3 r 1 ...d^3 rN (27.6-3)

After the integration, all of the terms in theaandbsums are identical, so that

〈V〉

N(N−1)

ζ

∫

u(r 12 ) exp

⎡

⎣−

N∑− 1

j 1

∑N

ij+ 1

u(rij)

kBT

⎤

⎦d^3 r 1 ...d^3 rN

N(N−1)

∫

u(r 12 )f(2)(r 1 ,r 2 )d^3 r 1 d^3 r 2 (27.6-4)

(^2) J. E. Mayer and M. G. Mayer,Statistical Mechanics, Wiley, New York, 1940. See T. Hill,Statistical
Thermodynamics, Addison-Wesley, Reading, MA., 1960, p. 261ff for a readable discussion.