Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Phvsics 211

Solution:
The wave function of N2 is symmetric as "14 is a boson. The spin
wave functions of N2 consist of six symmetric and three antisymmetric
functions. We know that the rotating wave function is symmetric when the
spin wave function is symmetric, and the rotating wavefunction is antisym-
metric when the spin wave function is antisymmetric. Hence, the partition
function of ortho-N2 is

and I is the rotational moment of inertia of N2. Similarly,

A2
where B, = -
2kT'

Zpara = 3(21+ l)e-or'(i+l)/T.

I= 1,3,5....

As B,/T << 1 at ordinary temperatures, the sums can be replaced by inte-

grals:

3T

e-erx/Tdx = -

20,

Therefore , the relative abundance is given by

At equilibrium, portho = ppara, the above ratio is 2.

B,/T B l,exp[-B,l(I + 1)/T] << 1. Hence

When the temperature is lowered towards the absolute zero, we have

The relative abundance is

Northo = (:) exp(2Br/T)

Npara