Physical Chemistry Third Edition

1148 27 Equilibrium Statistical Mechanics. III. Ensembles

wheref(2)(r 1 ,r 2 ) is the two-bodyreduced coordinate distribution function

f(2)(r 1 ,r 2 )

1

ζ

∫

exp

⎡

⎣−

N∑− 1

j 1

∑N

ij+ 1

u(rij)

kBT

⎤

⎦d^3 r 3 ...d^3 rN (27.6-5)

The one-body reduced coordinate distribution function is defined similarly:

f(1)(r 1 )

1

ζ

∫

exp

⎡

⎣−

N∑− 1

j 1

∑N

ij+ 1

u(rij)

kBT

⎤

⎦d^3 r 2 d^3 r 3 ...d^3 rN (27.6-6)

Mayer showed that the two-body reduced coordinate distribution function can be

expressed as a series of powers and logarithms of the density, with a second type

of cluster integrals as coefficients.^3

The radial distribution function, or pair correlation function,g(r), is defined by

g(r 12 )

f(2)(1, 2)

f(1)(1)f(1)(2)

(27.6-7)

The radial distribution function,g(r), is the probability of finding a second molecule at

a location at a distancerfrom a given molecule divided by the probability of finding

a molecule far from the given molecule. For a system of particles without intermolec-

ular forces, it is equal to unity for all values ofr. For a dense gas or a liquid with

intermolecular forces, it is equal to zero forr0 and goes through one or more max-

ima and minima and approaches unity for large values ofr.^4

The pressure can be expressed in terms of the radial distribution function^5

P

NkBT

V

[

1 −

N

6 VkBT

∫

r

(

du

dr

)

g(r)d^3 r

]

(27.6-8)

Exercise 27.9

Show that Eq. (27.6-8) gives the correct pressure of an ideal gas, for whichu(r)0.

The pressure of a nonideal gas can also be described by thevirial equation of state,

which is a power series in the reciprocal of the molar volume,Vm:

PVm

RT

B 1 +

B 2

Vm

+

B 3

Vm^2

+

B 4

Vm^3

+ ··· (27.6-9)

where the first virial coefficient,B 1 , is equal to unity. The virial coefficients can be

expressed as sums of cluster integrals of a third type.^6 The second virial coefficient,

B 2 , is given by

B 2 −

NAV

2

∞∫

0

[e−u(r)/kBT−1]4πr^2 dr (27.6-10)

whereris the distance between the particles andNAvis Avogadro’s constant.

(^3) J. E. Mayer and M. G. Mayer,op. cit. (note 2).
(^4) The radial distribution function unfortunately has the same name as the probability density for finding
an electron at a specified distance from the nucleus in an atom, defined in Chapter 17.
(^5) P. A. Egelstaff,An Introduction to the Liquid State, Academic Press, New York, 1967, p. 20.
(^6) J. E. Mayer and M. G. Mayer,op. cit. (note 2).