Physical Chemistry , 1st ed.

symmetric nuclear wavefunctions and 2I^2 I2(^12 )^2 ^12 1 antisymmet-

ric nuclear wavefunction. Since all rotational states have the same degener-

acy, 2J 1, there will be roughly three times as many H 2 molecules in odd

rotational states as there are H 2 molecules in even rotational states.

The choice of hydrogen as an example is not random. Because hydrogen’s
rotational temperature is much higher than its boiling point, hydrogen exhibits
unusual thermodynamic properties at low temperatures. This is caused by an
extremely slow conversion between symmetric and antisymmetric nuclear
states, which effectively limits the transitions between adjacent rotational
states. In fact, diatomic hydrogen with the antisymmetric nuclear state is called
para-hydrogen,and diatomic hydrogen with the symmetric nuclear state is
called ortho-hydrogen.Ortho- and para-hydrogen have different thermo-
dynamic properties at low temperatures as a consequence of the population
differences of the rotational levels, and can effectively behave as two different
substances. (For details, consult a statistical thermodynamics textbook.)
For homonuclear diatomic molecules, then, the nuclear and rotational par-
tition functions must be considered together. The only real difference is that
the nuclear partition function introduces an additional degeneracy to the over-
all partition function. Therefore, we have

For boson nuclei:
qrot,nuc
(2I^2 I) 
Jodd

(2J 1)eJ(J+1)r/T (I 1)(2I 1) 

Jeven

(2J 1)eJ(J+1)r/T

For fermion nuclei:
qrot,nuc
(2I^2 I) 
Jeven

(2J 1)eJ(J+1)r/T (I 1)(2I 1) 

Jodd

(2J 1)eJ(J+1)r/T

Notice that the only difference between the two partition functions is the
index on the summations. For boson nuclei, odd Jvalues have a certain nu-
clear degeneracy and even Jvalues have another; for fermion nuclei, the nu-
clear degeneracies are switched.
In the limit of high temperature (Tr), we recognize that the summa-
tion over even Jvalues is approximately the same as the summation over odd
Jvalues, so we can rewrite both equations above as

qrot,nuc
(2I^2 I) 
half of

(2J 1)eJ(J+1)r/T (I 1)(2I 1) 

half of

(2J 1)eJ(J+1)r/T

the J’s the J’s

and we can factor out the summation from both of the terms:

qrot,nuc(2I^2 I) (I 1)(2I 1)





half of

(2J 1)eJ(J+1)r/T

the J’s 

which can be simplified to

qrot,nuc(2I 1)^2 





half of

(2J 1)eJ(J+1)r/T

the J’s 

The summation over half of the Jvalues is equal to one-half of the summation

18.5 Diatomic Molecules: Rotations 633