Physical Chemistry Third Edition

1074 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

The Vibrational Partition Function

In the harmonic oscillator approximation, the vibrational energy of a polyatomic mole-

cule is the sum of the energies of normal modes, each of which acts like a harmonic

oscillator, as in Eq. (22.4-19). A nonlinear polyatomic molecule has 3n−6 normal

modes, and a linear polyatomic molecule has 3n−5 normal modes, wherenis the

number of nuclei in the molecule. Choosing the energy of the ground vibrational state

to equal zero, we have

εvib

3 n∑− 5 ( 6 )

i 1

h ̃νivi

3 n∑− 5 ( 6 )

i 1

hcν ̃ivi (25.4-23)

whereνirepresents the classical vibration frequency of normal mode numberi, and

viis the quantum number for this normal mode. The parameterν ̃is the frequency of

vibration divided by the speed of light, and is the quantity usually found in tables. The

upper limit of the sum is equal to 3n−5 for linear polyatomic molecules and equal to

3 n−6 for nonlinear molecules.

Since the vibrational energy is a sum of 3n−5or3n−6 terms and since the

quantum numbers for each vibration are independent of each other, the vibrational

partition function of a polyatomic substance is a product of factors, and each factor is

analogous to that of a diatomic molecule:

zvib

3 n∏− 5 ( 6 )

i 1

1

1 −e−hνi/kBT



3 n−∏ 5 ( 6 )

i 1

1

1 −e−hc ̃νi/kBT

(25.4-24)

where the upper limit of the product indicates 3n−5 for a linear polyatomic molecule

and 3n−6 for a nonlinear polyatomic molecule.

EXAMPLE25.12

The vibrational frequencies of carbon dioxide areν 1  4. 162 × 1013 s−^1 ,ν 2  2. 000 ×

1013 s−^1 (two bends at this frequency), andν 3  7. 043 × 1013 s−^1. Calculate the vibrational

partition function at 298.15 K.

Solution

Let

x 1 

hν 1

kBT



(6. 6261 × 10 −^34 J s)(4. 162 × 1013 s−^1 )

(

1. 3807 × 10 −^23 JK−^1

)

( 298 .15 K)

 6. 699

By similar calculationsx 2  3 .219 andx 3  11 .337. The vibrational partition function is

zvib

1

1 −e−^6.^699

(

1

1 −e−^3.^219

) 2

1

1 −e−^11.^337

(1.0012)(1.0417)^2 (1.000012) 1. 0865

The lowest-frequency vibrations (the two bending modes) make the largest contributions

tozvib, but the size of the vibrational partition function indicates that there is very little

vibrational excitation.