24.9: The Rotational Partition Function of A Diatomic Ideal Gas

For a diatomic molecule that is free to rotate in three dimensions, we can distinguish two rotational motions; however, their wave equations are intertwined, and the quantum mechanical result is that there is one set of degenerate rotational energy levels. The energy levels are

\[{\epsilon }_{r,J}=\frac{J\left(J+1\right)h^2}{8{\pi }^2I} \nonumber \]

with degeneracies \(g_J=2J+1\), where \(J=0,\ 1,\ 2,\ 3,\dots\).

(Recall that \(I\) is the moment of inertia, defined as \(I=\sum{m_ir^2_i}\), where \(r_i\) is the distance of the \(i^{th}\) nucleus from the molecule’s center of mass. For a diatomic molecule, \(XY\), whose internuclear distance is \(r_{XY}\), the values of \(r_X\) and \(r_Y\) must satisfy the conditions \(r_X+r_Y=r_{XY}\) and \(m_Xr_X=m_Yr_Y\). From these relationships, it follows that the moment of inertia is \(I=\mu r^2_{XY}\), where \(\mu\) is the reduced mass.) For heteronuclear diatomic molecules, the rotational partition function is

\[z_r=\sum^{\infty }_{J=0}{\left(2J+1\right)}{\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ } \nonumber \]

For homonuclear diatomic molecules, there is a complication. This complication occurs in the quantum mechanical description of the rotation of any molecule for which there is more than one indistinguishable orientation in space. When we specify the locations of the atoms in a homonuclear diatomic molecule, like \(H_2\), we must specify the coordinates of each atom. If we rotate this molecule by \({360}^{\mathrm{o}}\) in a plane, the molecule and the coordinates are unaffected. If we rotate it by only \({180}^{\mathrm{o}}\) in a plane, the coordinates of the nuclei change, but the rotated molecule is indistinguishable from the original molecule. Our mathematical model distinguishes the \({180}^{\mathrm{o}}\)-rotated molecule from the original, unrotated molecule, but nature does not.

This means that there are twice as many energy levels in the mathematical model as actually occur in nature. The rotational partition function for a homonuclear diatomic molecule is exactly one-half of the rotational partition function for an “otherwise identical” heteronuclear diatomic molecule. To cope with this complication in general, it proves to be useful to define a quantity that we call the symmetry number for any molecule. The symmetry number is usually given the symbol \(\sigma\); it is just the number of ways that the molecule can be rotated into indistinguishable orientations. For a homonuclear diatomic molecule, \(\sigma =2\); for a heteronuclear diatomic molecule, \(\sigma =1\).

Making use of the symmetry number, the rotational partition function for any diatomic molecule becomes

\[z_r=\left(\frac{1}{\sigma }\right)\sum^{\infty }_{J=0}{\left(2J+1\right)}{\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ } \label{exact} \]

For most molecules at ordinary temperatures, the lowest rotational energy level is much less than \(kT\), and this infinite sum can be approximated to good accuracy as the corresponding integral. That is

\[z_r \approx \left(\frac{1}{\sigma }\right)\int^{\infty }_{J=0}{\left(2J+1\right){\mathrm{exp} \left[\frac{J\left(J+1\right)h^2}{8{\pi }^2IkT}\right]\ }}dJ \nonumber \]

Initial impressions notwithstanding, this integral is easily evaluated. The substitutions \(a={h^2}/{8{\pi }^2IkT}\) and \(u=J\left(J+1\right)\) yield

\[ \begin{align} z_r & \approx \left(\frac{1}{\sigma }\right)\int^{\infty }_{u=0} \mathrm{exp} \left(-au\right) du \\[4pt] & \approx \left(\frac{1}{\sigma }\right)\left(\frac{1}{a}\right)=\frac{8{\pi }^2IkT}{\sigma h^2} \label{approx}\end{align} \]

To see that this is a good approximation for most molecules at ordinary temperatures, we calculate the successive terms in the partition function of the hydrogen molecule at \(25\ \mathrm{C}\). The results are shown in Table 1. We choose hydrogen because the energy difference between successive rotational energy levels becomes greater the smaller the values of \(I\) and \(T\). Since hydrogen has the smallest angular momentum of any molecule, the integral approximation will be less accurate for hydrogen than for any other molecule at the same temperature. For hydrogen, summing the first seven terms in the exact calculation (Equation \ref{exact}) gives \(z_{\mathrm{rotation}}=1.87989\), whereas the approximate calculation (Equation \ref{approx}) gives \(1.70284\). This difference corresponds to a difference of \(245\ \mathrm{J}\) in the rotational contribution to the standard Gibbs free energy of molecular hydrogen.

Table 1: Rotational Partition Function Contributions for Molecular Hydrogen at 298 K

J	\(=\frac{\left(2J+1 \right)}{ \sigma} exp ^{Z_J} \left( - \frac{J \left( J+1 \right) h^2}{8 \pi^2 IkT} \right)\)	\(\approx \sum^{Z_r} Z_J\)
0	0.50000	0.50000
1	0.83378	1.33378
2	0.42935	1.76313
3	0.10323	1.86637
4	0.01267	1.87904
5	0.00082	1.87986
6	0.00003	1.87989