Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 34D (Statistical Physics)

A fixed-axis diatomic molecule has one bond-stretch vibrational coordinate $q$, reduced mass $m_r$, and angular frequency $\omega$. In the classical harmonic approximation its internal Hamiltonian is $p^2/(2m_r)+m_r{\omega }^2{q}^2/2$. The phase-space contribution per molecule isThe Gaussian integrals cancel the reduced mass. Extending the small bond displacement to the whole real line is the usual harmonic approximation. There is no rotational factor because the molecular orientation is fixed.

For $N$ identical dilute molecules of total mass $M$ in volume $V$, translation supplies $V(2\pi MkT/h^2{)}^{3/2}$. Including indistinguishability givesUsing Stirling's approximation and defining $Q=(V/N)(2\pi MkT/h^2{)}^{3/2}kT/(\hslash \omega )$,The $7/2$ is $1+5/2$: the logarithm contains $T^{5/2}$, combining three translational quadratic degrees with two vibrational ones. This is a classical result, valid when vibration is thermally classical rather than quantum frozen.