Physical Chemistry Third Edition

27.4 Classical Statistical Mechanics 1135

The integral over the momentum components factors into a product of integrals, one

for each momentum component, such as

∫∞

−∞

exp

(

−p^2 x 1

2 mkBT

)

dpx 1 (2πmkBT)^1 /^2 (27.4-5)

where we have looked up the integral in Appendix C. Every momentum component

gives the same result after integration, so that

Zcl(2πmkBT)^3 N/^2

∫

e−V(q)/kBTdq(2πmkBT)^3 N/^2 ζ (27.4-6)

The integralζis called theconfiguration integral

ζ

∫

e−V(q)/kBTdq (27.4-7)

This is a 3N-fold integral over all of the coordinates.

Exercise 27.3

Carry out the derivation of the classical canonical probability density by considering a system

made up of two subsystems.

Dilute Gases in the Classical Canonical Ensemble

The equations we have written apply to any system of atoms without electronic exci-

tation, whether the system is a solid, liquid, or gas. If the system consists of a single

particle in a rectangular box with its lower left rear corner at the origin and with dimen-

sionsa,b, andcin thex,y, andzdirections, the configuration integral would be

ζ

∫c

0

∫b

0

∫a

0

e−V(x,y,z)/kBTdxdydz

(

system of a single particle

in a rectangular box

)

(27.4-8)

If there are no forces on the particle inside the box, its potential energy is equal to a

constant, which we can set equal to zero:

ζ

∫c

0

∫b

0

∫a

0

e^0 dxdydzabcV

⎛

⎝

single particle

in a box with

zero potential energy

⎞

⎠ (27.4-9)

If the system is a dilute gas ofNatoms we can ignore the intermolecular interactions

and each particle moves as though alone in the box. If we set the constant potential

energy equal to zero the configuration integral is a product ofNintegrals like that of

Eq. (27.4-9):

ζ

∫

e^0 d^3 r 1 d^3 r 2 ...d^3 rNVN

(

dilute monatomic gas

with zero potential energy

)

(27.4-10)

The classical canonical partition function is

Zcl(2πmkBT)^3 N/^2 VNzNcl

(

dilute monatomic gas

with zero potential energy

)

(27.4-11)