4.4: Partition Functions and Average Energies at High Temperatures
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It is enlightening to find the integral approximations to the partition functions and average energies for our simple quantum-mechanical models of translational, rotational, and vibrational motions. In doing so, however, it is important to remember that the use of integrals to approximate Boltzmann-equation sums assumes that there are a large number of energy levels, \(\epsilon_i\), for which \(\epsilon_i\ll kT\). If we select a high enough temperature, the energy levels for any motion will always satisfy this condition. The energy levels for translational motion satisfy this condition even at sub-ambient temperatures. This is the reason that Maxwell’s derivation of the probability density function for translational motion is successful.
Rotational motion is an intermediate case. At sub-ambient temperatures, the classical-mechanical derivation can be inadequate; at ordinary temperatures, it is a good approximation. This can be seen by comparing the classical-theory prediction to experimental values for diatomic molecules. For diatomic molecules, the classical model predicts a constant-volume heat capacity of \({5k}/{2}\) from \(3\) degrees of translational and \(2\) degrees of rotational freedom. Since this does not include the contributions from vibrational motions, constant-volume heat capacities for diatomic molecules must be greater than \({5k}/{2}\) if both the translational and rotational contributions are accounted for by the classical model. For diatomic molecules at \(298\) K, the experimental values are indeed somewhat larger than \({5k}/{2}\). (Hydrogen is an exception; its value is \(2.47\ k\).)
Vibrational energies are usually so big that only a minor fraction of the molecules can be in higher vibrational levels at reasonable temperatures. If we try to increase the temperature enough to make the high-temperature approximation describe vibrational motions, most molecules decompose. Likewise, electronic partition functions must be evaluated from the defining equation.
The high-temperature limiting average energies can also be calculated from the Boltzmann equation and the appropriate quantum-mechanical energies. Recall that we find the following quantum-mechanical energies for simple models of translational, rotational, and vibrational motions:
Translation
\[\epsilon^{\left(n\right)}_{\mathrm{trans}}=\frac{n^2h^2}{8m{\ell }^2} \nonumber \]
(\(\mathrm{n\ =\ 1,\ 2,\ 3,\dots .}\) Derived for a particle in a box)
Rotation
\[\epsilon^{\left(m\right)}_{\mathrm{rot}}=\frac{m^2h^2}{8{\pi }^2I} \nonumber \] (\(\mathrm{m\ =\ 1,\ 2,\ 3,\ \dots .}\) Derived for rotation about one axis—each energy level is doubly degenerate)
Vibration
\[\epsilon^{\left(n\right)}_{\mathrm{vibration}}=h\nu \left(n+\frac{1}{2}\right) \nonumber \] (\(\mathrm{n\ =\ 0,\ 1,\ 2,\ 3,\dots .}\) Derived for simple harmonic motion in one dimension)
When we assume that the temperature is so high that many \(\epsilon_i\) are small compared to \(kT\), we find the following high-temperature limiting partition functions for these motions:
\[z_{\mathrm{translation}}=\sum^{\infty }_{n=1}{\mathrm{exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)\approx \int^{\infty }_0{\mathrm{exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)dn={\left(\frac{2\pi mkT{\ell }^2}{h^2}\right)}^{1/2} \nonumber \]
\[z_{\mathrm{rotation}}=\sum^{\infty }_{m=1}{\mathrm{2\ exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)\approx 2\int^{\infty }_0{\mathrm{exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)dn={\left(\frac{8{\pi }^3IkT}{h^2}\right)}^{1/2} \nonumber \] \[z_{\mathrm{vibration}}=\sum^{\infty }_{n=0}{\mathrm{exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)\approx \int^{\infty }_0{\mathrm{exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)dn=\frac{kT}{h\nu }\mathrm{exp}\ \left(\frac{-h\nu }{2kT}\right) \nonumber \]
We can then calculate the average energy for each mode as
\[\left\langle \epsilon \right\rangle =z^{-1}\int^{\infty }_0{\epsilon_n}{\mathrm{exp} \left(\frac{-\epsilon_n}{kT}\right)\ }dn \nonumber \]
and find
\[\begin{align*} \left\langle \epsilon_{\mathrm{translation}}\right\rangle &=z^{-1}_{\mathrm{translation}}\int^{\infty }_0{\left(\frac{n^2h^2}{8m{\ell }^2}\right)\mathrm{\ exp}}\left(\frac{-n^2h^2}{8m{\ell }^2kT}\right)dn \\[4pt] &=\frac{kT}{2} \\[4pt] \left\langle \epsilon_{\mathrm{rotation}}\right\rangle &=z^{-1}_{\mathrm{rotation}}\int^{\infty }_0{2\left(\frac{m^2h^2}{8{\pi }^2I}\right)\mathrm{\ exp}}\left(\frac{-m^2h^2}{8{\pi }^2IkT}\right)dm \\[4pt] &=\frac{kT}{2} \\[4pt] \left\langle \epsilon_{\mathrm{vibration}}\right\rangle &=z^{-1}_{\mathrm{vibration}} \times \int^{\infty }_0{h\nu \left(n+\frac{1}{2}\right) \mathrm{\ exp}}\left(\frac{-h\nu }{kT}\left(n+\frac{1}{2}\right)\right)dn \\[4pt] &=kT+\frac{h\nu }{2} \\[4pt] &\approx kT \end{align*} \nonumber \]
where the last approximation assumes that \({h\nu }/{2}\ll kT\). In the limit as \(T\to 0\), the average energy of the vibrational mode becomes just \({h\nu }/{2}\). This is just the energy of the lowest vibrational state, implying that all of the molecules are in the lowest vibrational energy level at absolute zero.