Physical Chemistry , 1st ed.

a 2 in the denominator of the exponent) and assume that each vibrational level

is singly degenerate so that giequals 1:

qvibeh^ /2kT 



ifirst

evih^ /kT (18.16)

vibrational level

There is no need to approximate the infinite sum in equation 18.16 as an in-

tegral, for it turns out that the summation is a well-known infinite sum that

converges to a known value: (1 eh^ /kT)^1. We can substitute for the sum-

mation and get as an expression for qvib:

qvibeh^ /2kT(1 eh^ /kT)^1 

1

e



h

e



/2

h

k

T

/kT (18.17)

This expression is applicable at any temperature.

There are several ways we can deal with equation 18.17. The first thing to

point out is that in order for the exponent in the exponentials to be unitless,

the expression h /kmust have units of temperature, kelvins. We define the vi-

brational temperatureof a diatomic molecule,v,as

v 

h

k

 (18.18)

Equation 18.17 can be rewritten as

qvib

1

e





e

v



/2



T

v/T (18.19)

Additionally, in the limit of high temperature (and by high we mean at least

well above the vibrational temperature v), the exponentials in equation 18.19

have small exponents. Under these conditions, exponentials can be approxi-

mated by the Taylor-series expansion

ex 1 x x

4

2

   

Truncating at the second term, we have for equation 18.19:

qvib

1 

1

(



1





v/



2

v

T

/T)



At high temperatures, the v/2Tterm is negligible with respect to 1 and is ig-

nored in the numerator, and in the denominator the 1’s cancel. We get

qvib

v

1

/T



This is usually rewritten as

qvib



T

v



k

h

T

 (at high T) (18.20)

This is an extremely simple expression for a partition function. Table 18.1

lists a few vvalues for some diatomic molecules. At temperatures well

above vfor each gas-phase molecule, the vibrational partition function is

given simply by equation 18.20. At temperatures near vor lower, the

more complete expression in equation 18.19 must be used. (But be careful:

for some of the molecules listed, the stable phase is not the gas phase at

Tv!)

18.4 Molecules: Vibrations 625

Table 18.1 Vibrational temperatures vfor
some diatomic molecules
Molecule v(K)
H 2 6215
HCl 4227
CO 3100
N 2 3374
HBr 3700
Cl 2 810
NO 2690
I 2 310
O 2 2230
HI 3200