Statistical Physics

4.7 Diatomic Molecules 63Fig. 4.9. Contribution to the heat capacity from vibration for a diatomic molecule.C V /N k B is plotted as a function of k BT/ω = k BT/(hω/2π)where 〈n〉 is the thermal average of the quantum number n, andisgivenby〈n〉 =∞∑n=1∞∑n=0[n exp[exp−βE (V)n−βE (V)n]] = 1e βω − 1 . (4.50)The result for U (V) (4.48) can be recovered from these equations, since U (V) =N〈E (V) 〉. This distribution of n is called the Bose distribution. It has alreadyappeared in (2.9), where we considered a distribution of money. There, theunit of energy ω was replaced by a unit of money, 1 yen.When the thermal energy (1/2)k B T is much smaller than the unit ofenergy ω, the system cannot accept energy from a heat bath. In sucha case, we can neglect the corresponding degree of freedom, and say thatthat degree of freedom is “dead”; in this situation, that degree of freedommakes a negligible contribution to any observable. For nitrogen or oxygen,hω/k B ≃ 2000 K ≫ T ≃ 300 K, and so the vibrational degree of freedomis dead at 300 K. Likewise, the units of energy for the motion of electronsin a molecule and for the motions within a nucleus are much higher thanthe thermal energy. This is the reason we can neglect the degrees of freedomassociated with those motions. The energy of the center of gravity isalso quantized, as can be seen from (4.4). However, the unit of energy in thiscase, (1/2m)(h/2L) 2 , is very small for macroscopic values of L, andsothetranslational motion of the center of gravity is not dead even at low temperature.Thus, the law of equipartition is obeyed in this case down to the lowesttemperatures.