Heat capacity ofa diatomic gas 75eachinternaldegree offreedomis particularlysimple. Itgives merely a−NkkkBTlnZ
additiontoF;and this is identical to the free energy of a set oflocalizedparticles
(Chapter 2) withappropriate one-particle states. Therefore, we shall findthat most of
thehardgroundworkfor thischapterisalready done, so that we mayconcentrate on
the results.7.2 Heat capacityof a diatomicgas
Tobe specific,let us use theideas ofthe previous section to calculate theheat capacity
CVof a typical diatomic gas.7 .2.1 Translational contributionAs already indicated, this term holds nothing new above the content of Chapter 6.
The translationalmotion ofanymoleculeisdescribedbythek-states of‘fittingwaves
into boxes’ fame. And this treatment leads to apartition function, ( 6 .6)
ZZZtrans=V( 2 πMkkkBT/h^2 )^3 /^2 (7.6)withMbeing the mass ofthemolecule. Hence asbeforeUUUtrans= 23 NkkkBTandthe
heat capacitycontributionisgivenbyCV,trans=3
2
NkkkB7 .2.2 Electronic contributionWhen we come to the internal contributions,(7.4)and(7.5)tell us that we can
forget about themoleculesbeingspecificallythose ofagas. Rather we can treat the
contributions as identical to those from an assemblyof distinguishable particles.
Allowance for thermal excitation into a higher electronic state of energyεabove
thegroundstateisidenticalto the treatment ofthe two-state system ofsection 3.1.
The result in the heat capacity is to add toCVa ‘Schottky anomaly’ term as given in
(3.4) andFig. 3.3.
Inpractice this contribution toCVisnegligibleinalmost allcases. Thisisbecause
typical values ofεare around 10 eV, so that we are in the extremequantum limit
(kkkBTε)at any realistic temperature. HenceCV,elec≈0. Theonly exceptional
everydaygases are NO andO 2 .In theseεhappens tobe particularlysmallandthe
Schottky anomaly peaks are calculated to appear around 7 5 K for NO and 5 000 K for
O 2 .However, withthese exceptionsitisjustifiabletoneglect theelectronic term, and
we shall dosointhe rest ofthischapter.