Past exam of the physics B course of the University of Cambridge / 2021 / Paper 2 / Section C / C9 / d

At low temperature, $T\ll {\theta }_r$, retain the ground level and the first excited level. Their energies are $0$ and ${\hslash }^2/I$, with degeneracies $1$ and $3$. Thereforeso in the notation of the question,$A$ is dimensionless and $B$ has units of kelvins. The first omitted level has three times the first excitation energy, explaining the exponent in the error term.

At high temperature, $T\gg {\theta }_r$, many rotational levels are occupied and the sum can be replaced to leading order by an integral:Set $u=J(J+1)$, so that $du=(2J+1)dJ$. ThenThusThe high-temperature approximation here concerns the rotational spectrum. For a real molecule it should still be used below temperatures where neglected vibrational excitation or dissociation invalidates the rigid-rotor model.