Physical Chemistry Third Edition

27.4 Classical Statistical Mechanics 1137

The result of this example is

zrot,cl 8 π^2 IekBT

(

diatomic or linear polyatomic substance

)

(27.4-14)

whereIeis the moment of inertia of the molecule. The derivation of the rotational

factor in the classical partition function for a nonlinear polyatomic substance is even

more complicated, and we simply present the result:

zrot,cl

√

π(8π^2 kBT)^3 /^2 (IAIBIC)^1 /^2

(

nonlinear polyatomic substance

)

(27.4-15)

whereIA,IB, andICare the principal moments of inertia as defined in Chapter 22.

We represent the vibration of a diatomic molecule by a harmonic oscillator with

massμ(the reduced mass of the molecule) and frequencyν. The classical vibrational

partition function for a harmonic oscillator is

zvib,cl

∫∞

−∞

exp

[

−

p^2 / 2 μ

kBT

]

dp

∫∞

−∞

exp

[

−

kx^2 / 2

kBT

]

dx

(2πμkBT)^1 /^2 (2πkBT/k)^1 /^2

(2πkBT)

√

μ

k



kBT

ν

(dilute diatomic gas) (27.4-16)

whereμis the reduced mass,kis the force constant, andνis the frequency of the

oscillator, from Eq. (14.2-24). We have used infinite limits for the integrations since

the harmonic oscillator is not limited to a finite range of oscillation. For a polyatomic

substance, there is one factor like that of Eq. (27.4-16) for each normal mode:

zvib,cl

3 n∏−5(6)

i 1

kBT

νi

(dilute polyatomic gas) (27.4-17)

where the limit on the product indicates 3n−5 normal modes for a linear molecule

and 3n−6 normal modes for a nonlinear molecule. We will not attempt to discuss

electronic motion classically, but assume that electronic excitation can be ignored.

For diatomic and polyatomic dilute gases without electronic excitation we have the

same relation as for monatomic gases:

ZclzNcl (27.4-18)

where

zclztr,clzrot,clzvib,cl

(

any dilute gas without eletronic excitation

)

(27.4-19)

Comparison of the Classical and Quantum Partition

Functions

The translational factor of the classical molecular partition function is proportional to

the quantum mechanical translational factor:

ztr,qm

(

2 πmkBT

h^2

) 3 / 2

V

ztr,cl

h^3

(27.4-20)