Physical Chemistry Third Edition

1136 27 Equilibrium Statistical Mechanics. III. Ensembles

where zcl is the molecular phase integral or theclassical molecular partition

function:

zcl(2πmkBT)^3 /^2 V

(

dilute monatomic gas

with zero potential energy

)

(27.4-12)

This is the entire classical molecular partition function for a monatomic gas without

electronic excitation.

The classical molecular partition function for dilute diatomic and polyatomic gases

without electronic excitation contains three factors. The translational factor is the same

as given by the formula in Eq. (27.4-12), since the translational motion of a molecule

is the same as that of an atom:

ztr,cl(2πmkBT)^3 /^2 V

(

any dilute gas without

electronic excitation

)

(27.4-13)

The derivation of the rotational factor in the classical molecular partition function

of a diatomic molecule is a little more complicated, and is carried out in the following

example:

EXAMPLE27.2

To obtain the rotational factor in the classical molecular partition function of a diatomic or

linear polyatomic gas, we must find the conjugate coordinates and momenta as discussed in

Appendix E. We consider a rigid rotor. In spherical polar coordinates withrre(fixed) and

withV0 (fixed) the Lagrangian is

L

1

2

mr^2 e ̇θ^2 +

1

2

mre^2 sin^2 (θ)φ ̇^2

The conjugate momenta toθandφare obtained through Eq. (E-20) of Appendix E:

pθ

∂L

∂θ ̇

mre^2 θ ̇^2

pφ

∂L

∂φ ̇

mre^2 sin^2 (θ)φ ̇^2

The classical Hamiltonian is

H

p^2 θ

2 mre^2

+

p^2 φ

2 mre^2 sin^2 (θ)

The classical rotational partition function is obtained by integrating over all values of the

conjugate momenta, overθfrom 0 toπ, and overφfrom0to2π. Note that since we are

integrating in phase space, no Jacobian is necessary.

zrot,cl

∫π

0

∫ 2 π

0

∫∞

−∞

∫∞

−∞

exp

(

p^2 θ

2 mre^2 kBT

+

p^2 φ

2 mr^2 esin^2 (θ)kBT

)

dpφdpθdφdθ



∫π

0

∫ 2 π

0

√

2 mr^2 ekBT

√

π

√

2 mr^2 esin^2 (θ)kBT

√

πdφdθ



√

2 mre^2 kBT

√

π

√

2 mre^2 kBT

√

π

∫π

0

∫ 2 π

0

sin^2 (θ)dφdθ

 2 πmr^2 ekBT 4 π 8 π^2 mr^2 ekBT 8 π^2 IekBT