Vibrational heat capacity of a diatomic gas ($C_{V,\mathrm{vib}}=N{k}_B{x}^2{e}^x/(e^x-1{)}^2$)

In the independent harmonic approximation, each diatomic molecule has one bond-stretching quantum harmonic oscillator with energies $(n+1/2)\hslash \omega$. Write $x=\hslash \omega /(k_{B}T)$. Its canonical partition function is $z=e^{-x/2}/(1-e^{-x})$, and differentiating its logarithm givesFor $T\ll \hslash \omega /k_B$, excitation is exponentially suppressed and the heat capacity tends to zero. At high temperature it tends to $N{k}_B$, one $k_B/2$ contribution from each quadratic kinetic and potential term. The zero-point energy does not contribute to the derivative with temperature. For a dilute gas whose rotations are already active, adding translation and rotation gives molar $C_V\simeq 5R/2$ before vibration is activated and $7R/2$ afterwards, provided dissociation and electronic excitations are negligible.