Frequently in the literature, a characteristic temperature for vibration is defined as \(\theta_{\text {vib }}=h v / k\). Express \(e\) and \(c_{v}\) for a diatomic molecule [Eqs. (16.49) and (16.51)] in terms of \(\theta_{\text {vib. }}\).

\[\underbrace{e}_{\begin{array}{c}
\text { Sensible }  \tag{16.49}\\
\text { energy }
\end{array}}=\underbrace{\frac{3}{2} R T}_{\begin{array}{c}
\text { Translational } \\
\text { energy }
\end{array}}+\underbrace{R T}_{\begin{array}{c}
\text { Rotational } \\
\text { energy }
\end{array}}+\underbrace{\frac{h v / k T}{e^{h v / k T}-1} R T}_{\begin{array}{c}
\text { Vibrational } \\
\text { energy }
\end{array}}+\underbrace{e_{\mathrm{el}}}_{\begin{array}{c}
\text { Electronic } \\
\text { energy }
\end{array}}\]

\[\begin{equation*}
c_{v}=\frac{3}{2} R+R+\frac{(h v / k T)^{2} e^{h v / k T}}{\left(e^{h v / k T}-1\right)^{2}} R+\frac{\partial e_{\mathrm{el}}}{\partial T} \tag{16.51}
\end{equation*}\]