I was reading about the equipartition theorem and I got the following quotations from my books:
A diatomic molecule like oxygen can rotate about two different axes. But rotation about the axis down the length of the molecule doesn't count. - Daniel V. Schröder's Thermal Physics.
A diatomic molecule can rotate like a top only about axes perpendicular to the line connecting the atoms but not about that line itself. - Resnick, Halliday, Walker s' Fundamentals of Physics.
Why is it so? Doesn't the rotation take place that way?
Answer
The energy levels of a diatomic molecule are $E = 2B, 6B, 12B$ and so on, where $B$ is:
$$ B = \frac{\hbar^2}{2I} $$
Most of the mass of the molecule is in the nuclei, so when calculating the moment of inertia $I$ we can ignore the electrons and just use the nuclei. But the size of the nuclei is around $10^{-5}$ times smaller than the bond length. This means the moment of inertia around an axis along the bond is going to be about $10^{10}$ smaller than the moment of inertia around an axis normal to the bond. Therefore the energy level spacings will be around $10^{10}$ times bigger along the bond than normal to it.
In principle we can still excite rotations about the axis along the bond, but you'd need huge energies to do it.
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