= Solution
A <diatomic molecule> has one internal bond-stretching vibrational coordinate. Near its equilibrium separation, approximate the bond by a spring of stiffness $K$ and use the <reduced mass> $\mu=m_1m_2/(m_1+m_2)$. Its <frequency> is $\omega=\sqrt{K/\mu}$ and its <quantum harmonic oscillator> levels are $\epsilon_r=(r+1/2)\hbar\omega$.
For independent molecules in <thermal equilibrium>, set $x=\hbar\omega/(k_BT)=\theta_v/T$. The single-molecule <canonical partition function> is a <geometric series>:
$$
z_v=\sum_{r=0}^\infty e^{-(r+1/2)x}=\frac{e^{-x/2}}{1-e^{-x}}.
$$
Using $U=-\partial\log(z_v^N)/\partial\beta$, where $\beta=1/(k_BT)$, gives
$$
U_v=N\hbar\omega\left(\frac12+\frac1{e^x-1}\right),\qquad \boxed{C_{V,v}=\left(\frac{\partial U_v}{\partial T}\right)_V=Nk_B\frac{x^2e^x}{(e^x-1)^2}.}
$$
Per mole replace $Nk_B$ by $R$. The <zero-point energy> is present in $U_v$ but is temperature-independent and contributes nothing to the <heat capacity>.
For $T\ll\theta_v$, occupation of the first excited vibrational state is exponentially suppressed: $C_{V,v}\simeq Nk_Bx^2e^{-x}\to0$. The vibrational degree of freedom is frozen out. For $T\gg\theta_v$, $C_{V,v}\to Nk_B$. This recovers classical equipartition: the quadratic kinetic term and quadratic elastic term each contribute $k_B/2$ per molecule.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-1-vibrational-heat-capacity.png]
{title=Activation of a diatomic molecule's harmonic vibrational contribution to heat capacity}
The <vibrational heat capacity of a diatomic gas> must be distinguished from the total <heat capacity>. In a dilute gas with rotations already active and vibration still frozen, translation contributes $3R/2$ and rotation $R$, so molar $C_V\simeq5R/2$. Once vibration is active it adds $R$, giving $C_V\simeq7R/2$. For an ideal gas of a fixed number of intact molecules, $C_P=C_V+R$, so the corresponding values are $7R/2$ and $9R/2$. Very low temperatures also freeze rotations, while sufficiently high temperatures bring anharmonicity, dissociation or electronic excitations; the harmonic plateau is restricted to the range where those effects are negligible.
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