[NOTES/SM-04011] Specific Heat of Diatomic Gases --- Quantum Effects

The energy of a harmonic oscillator is quantized and the energy levels are given by
\begin{align*}
\epsilon =& nh\nu +\underbrace{\frac{1}{2} h\nu} \qquad\qquad
\boxed{h\nu\approx
10^{-14}ergs}
\end{align*}
The partition function corresponding to vibrational modes is
\begin{align*}
Z =&\sum_{n=0}^\infty e^{-\beta nb\nu}
=\left(\frac{1}{1-e^{-\beta h\nu}}\right)
\end{align*}
Thus the contribution of vibrational modes to the average energy at temperature is
\begin{align*}
U = -\frac{\partial}{\partial\beta}\ln Z =&
\frac{\partial}{\partial\beta} \ln (1-e^{-\beta h\nu})\\
=& \frac{h\nu}{e^{\beta h \nu}-1}
\end{align*}

It is seen that the contribution to the average energy at high temperatures agree with the classical result. Thus for \(\beta h\nu\ll 1, \text{ or } T\gg \frac{h\nu}{k}\), we have
\begin{align*}
U = \frac{h\nu}{1+\beta h\nu-1} =&\frac{1}{\beta} =kT\,.
\end{align*}

Classical and quantum results agree at low temperatures. For this to happen the difference between the energy levels much smaller than $kT$. The difference between the classical and quantum values is significant at low temperatures if $kT\ll h\nu$ or \(\beta h \nu \gg 1\). In this case we have

\begin{align*}
U =& \frac{h\nu}{\exp(\beta h\nu)-1} = \frac{h\nu}{\exp(\beta
h\nu)},\qquad\qquad \beta h\nu\gg 1\\
=&h\nu\exp(-\beta h\nu)
\end{align*}
Similar comments apply to rotational degrees of freedom.