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From the book Chemical Principles The Quest for Insight, 5th Edition by Peter Atkins, Loretta Jones

The equipartition theorem is a result from classical mechanics; so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant.

I know quantization describes that photons, separate "packets,"are what energy is thought to happen in a subatomic level. However, I don't how quantization is related here: because quantization is unimportant, we can use it for translational and rotational motion of molecules at room temperature and above? and also why quantization is unimportant here? I believe translational and rotational motion of molecules happen in a subatomic level. Moreover, why can't we use the equipartition theorem for vibrational motion?

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What they mean is that for translational motion the quanta are so small that using classical mechanics produces no error, i.e $k_BT$ is far far bigger than the energy gaps between translational quanta, ($k_B$ is the Boltzmann constant) the same may be true of rotational motion where quanta may be only be separated by a few wavenumbers vs $k_BT\approx 210$ wavenumbers at room temperature. It is generally true for rotational motion but not all e.g. H$_2$. Vibrations have quanta of at least hundreds of wavenumbers and $v=1 , v=2$ etc. are hardly populated at room temperature.

We can only use the equipartition theorem if $k_BT \gg \Delta E$ where $\Delta E$ s the gap between energy levels, i.e the fact that there are discrete levels is no longer important. This can occur if the energy gaps are v small or the temperature very high or both.

(Technically the 'equipartition theorem' of classical statistical mechanics states that the mean (average) value of each independent quadratic term in the energy is equal to $k_BT/2$)

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