Statistical Physics, Second Revised and Enlarged Edition

Summary 81

Hence the ratioofortho- to para-hydrogen molecules when thegas equilibrates at
high temperature is 3:1. When the hydrogen is then cooled, and experiments are
performedwithno ortho–para conversion, then the resultislike a 3:1 mixture oftwo
dissimilargases, andwefindthat

CV=0.7 5 CV,ortho+0.2 5 CV, para (7.12)

where the orthoandparaheat capacities are calculatedrespectivelyonlyfrom theodd
andeven terms ofthe partitionfunction. This resultgives curve C, atlastinagreement
with the usual experiment.
Twofinalpoints. Ortho–para conversionis a technicalprobleminH 2 inits use
inrefrigeration. Ifitislefttoequilibrate atlow temperatures, then muchheat will
bereleased as thel=1 states gradually relax tol= 0. In addition all the thermal
properties, even theboiling point, change withthe ortho:para ratio. Hydrogenisbest
avoided in cryogenics! The second point is that similar effects occur in thegasD 2 ,
although the details are quite different fromH 2 since the deuteron is a boson(S
wavefunction) rather than afermion. Weleave this topic as an exercise.

7.4 SUMMARY

In this chapter we stop to examine the thermal properties of diatomicgases, and see
that this brings together results from several earlier chapters.

1 .Adiatomicgas molecule canbe consideredtohaveindependent energycontribu-
tionsfromtranslation(asamonatomicmolecule), rotation, vibrationandelectronic
excitation.

2. The contribution to thethermalproperties (notablytoFandCV)from translational
   motion is identical to that of a monatomic gas.
   3 .Theother contributions areidenticalto thosederivedfrom Boltzmann statistics of
   similar localizedparticles.


3. The partition function, summing Boltzmann factors (exp(−εi/kkkBT))over all
   states, againplays a usefulrole.


4. This topicgives agood example of energyscales and the excitation of degrees of
   freedom (equipartition of energy at highT, quantum ‘freezing out’ at lowT).


5. In practice, electronic excitation and vibration playa minor role for almost all
   gases.


6. Two-dimensionalrotationisfully excitedin mostdiatomic gases, to giveCCCV=
   5
   2 NkkkB.


7. The onset of rotation is seen in H 2 , but it holds additionalquantum surprises
   because the two atomsinthe rotating molecule areidenticalandthereforeindis-
   tinguishable. Nuclear spinaffects theweighting(andsometimes the accessibility)
   of the odd and even rotational states, and hence the thermal properties.