Equipartition theorem. Neglecting one rotational degree of freedom

Question

My teacher said that a rotational degree of freedom for a molecule in a gas can be neglected if the associated moment of inertia is very small, because the associated frequency of rotation will be very high. Could somebody explain this to me ?

Answer 1

If for some rotational degree of freedom the component of the momentum of inertia $I$ is very small, the component of angular momentum is of the order of $I\omega$, where $\omega$ is the relevant angular velocity. On the other hand, for lower excited levels (for this degree of freedom) this component of angular momentum is of the order of $\hbar$, therefore, $\omega$ is of the order of $\frac{\hbar}{I}$. The energy for these levels is of the order of $I\omega^2$, therefore, it is of the order of $\frac{\hbar^2}{I}$. For these levels to be significantly filled, this energy should be at least comparable to the temperature (in the energy units) $kT$. As $I$ is very small, this does not happen at reasonable temperatures (or, in other words, it only happens at temperatures that are much higher than the temperatures at which other rotational degrees of freedom become important).