This Demonstration considers the variation of heat capacity of an ideal diatomic gas, specifically hydrogen, with temperature. Both classical and quantum points of view are considered. The quantum-mechanical theory shows a monotonic increase in molar heat capacity with temperature. This approaches a constant value at high temperature, namely
J/mol K. This limit corresponds, in fact, to the prediction of classical statistical mechanics, which takes into account translational, rotational and vibrational contributions to heat capacity, equal to
, respectively, independent of temperature. In quantum statistical mechanics, on the other hand, the rotational and vibrational contributions become active only above certain characteristic temperatures, which leads to the observed, somewhat stepwise increase of heat capacity with increasing temperature.