Statistical Physics, Second Revised and Enlarged Edition

The heat capacity ofhydrogen 79

fivedegrees offreedom a totalof^52 kkkBTtoU,andthereforeCV=^52 NkkkB,inagreement

with(7.10).

At room temperature vibrationisinthe extreme quantumlimit(kkkBTε)since

the characteristic temperature of vibration (around 6 000 K) is so high. However, at

elevated temperatures, one expects the heat capacity to rise towards^72 NkkkBas vibration

becomes excited.Thisisobserved,althoughthe gasdissociatesbefore theclassical

limit is reached. (The detailed comparison with experiment is also complicated by

significant coupling between vibrational and rotational states.)

7. 3 .1 Theonsetofrotation

Although the treatment so fargives a satisfactoryoutline of the properties of hydrogen,
the details of the onset of rotation around 50 –2 00 K are notwell described. Ifwe use
the rotationalpartitionfunction of(7.8) to calculateCV,rot,weobtain curve A of
Fig. 7.2. This is at variance with the experimental results for H 2 ,which are more like
curveC.
Whythediscrepancy? Actuallyan experimenter canhelphere. Natureiskindin
that there are several sorts of ‘hydrogen’. Using isotopic separation, it is possible to
deduceCVfor the gasesH 2 ,HD andD 2. Even allowingfor thedifferent values of
(H 2 85 K,HD 64 K,D 24 3K)arisingfrom thedifferent moments ofinertia, the curves
forCVdiffer markedly. And in fact our theoretical curve A is in good agreement with
experimentfor HD.
There is agreat difference between a heteronuclear and a homonuclear molecule.
It is a matter of the identity of the two nuclei. If the nuclei are different, and therefore
distinguishable, thenwhatwehavedonesofariscorrect. HencetheagreementforHD.
At first sight the modification toZfor identical nuclei might seem to be a simple
one, merely to allow for the fact that a rotation ofπrather than 2 πleads to identity.

A

B

C

1

2

0

T

CVVV,rot

NkkkB

 2  3 

Fig. 7. 2 The rotationalheat capacity ofadiatomic gas. Thethree curves A, B andC are explainedin

the text.