Past exam of the thermal and statistical physics course of the University of Cambridge / 2014 / Paper 1 / 1

For one oscillator, set $x=\hslash \Omega /(k_{B}T)$. Its quantum-oscillator energy and heat capacity areAt low temperature, $U\to \hslash \Omega /2$ and $C_V\sim k_B{x}^2{e}^{-x}\to 0$: excitations freeze out but zero-point energy remains. At high temperature, $U\sim k_{B}T$ and $C_V\to k_B$, recovering equipartition. The zero-point term does not contribute to heat capacity.

Using $C_V=T(\partial S/\partial T{)}_V$ and the stated Maxwell relation, interchange derivatives within a smooth single phase:Therefore $C_V$ depends only on temperature for the given temperature-linear equation of state. It need not be constant in temperature.

For $N$ identical classical noninteracting particles, use $Z_N=Z_1^N/N!$. At fixed $N$,so the canonical partition-function identities giveThus, for one mole,The supplied $T^{3/2}$ dependence includes only three translational degrees of freedom; additional thermally active internal modes would change the heat capacity.