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My question is: Why at low temperatures does a diatomic molecule result in the same specific heat capacity as a monoatomic at a constant volume?

My understanding is that at very high temperature there are two additional degrees of freedom from the vibration between the atoms, this gives the specific heat capacity of $C_v=\frac 7 2 R$ at room temperature. Vibrational degree of freedom does not occur, so we are only left with translational and rotational.

But a very low temperature it seems that rotational motion does not occur. But why?

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    $\begingroup$ For the same reason that the vibrational degrees of freedom aren't active at low temperature: there is an energy gap between the ground rotational state and first excited rotational state, so as long as $k_{\textrm{B}}T \ll $ that energy gap, the rotational degrees of freedom are "frozen out". $\endgroup$ – march Mar 25 at 20:49
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    $\begingroup$ Also, at room temperature, the correct equation is Cv=2.5R. $\endgroup$ – Chet Miller Mar 25 at 22:28

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