Physical Chemistry , 1st ed.

nonzero minimum energy of vibration (the “zero-point energy”). Because vi-

brational motions of molecules represent another form of energy, we can de-

fine a vibrational partition function qvibfor a molecule such that

qvib 



ifirst

giei/kT (18.9)

vibrational level

(Vibrational energy levels do not exist for monatomic gaseous species, because

at least two atoms must be bonded together in order to have a vibration.)

We will consider a simple diatomic molecule first, then generalize our final

equations for a polyatomic molecule that has 3N6 (or 3N5 for linear

molecules) vibrational motions, where Nis the number of atoms in the mol-

ecule. If we make the assumption that the single vibration of a diatomic mol-

ecule is an ideal harmonic oscillator, then quantum mechanics gives us an

equation for the quantized energy of that harmonic oscillator:

Eh (v ^12 ) (18.10)

where Eis the energy of the harmonic oscillator,his Planck’s constant, is the

classical frequency of the oscillator, and vis the vibrational quantum number.

The classical frequency can be expressed as



2

1

 

k

(18.11)

where kis the force constant of the oscillator (in units of N/m) and is the

reduced mass of the oscillator (in units of kg). Finally, recall that for a diatomic

molecule having two atoms with masses m 1 and m 2 , the definition of reduced

mass is





m

m

1

1

m

m

2

2

 (18.12)

which can also be written as



1



m

1

1

 

m

1

2

 (18.13)

Having re-established the definitions of the terms in equation 18.10, we will

use that equation to rewrite our vibrational partition function. Making the

substitution for iin equation 18.9:

qvib 



ifirst

giexp  (18.14)

vibrational level

Here, we are using the summation index ias the index on the quantum num-

ber v. Because of the sum of two terms in the power of the exponential, we can

rewrite equation 18.14 as

qvib 



ifirst

giexpv

k

ih

T

^

exp  (18.15)

vibrational level

The exp(^12 h /kT) part is the same for all of the infinite terms in the summa-

tion.* Therefore, we can factor it out of the summation (and make the ^12 term

^12 h



kT

h (vi ^12 )



kT

624 CHAPTER 18 More Statistical Thermodynamics

*Of course, for real molecules the vibrational quantum number never reaches infinity.

However, the final expressions still do a remarkable job of predicting thermodynamic prop-

erties for molecules.