Rotational and Nuclear Partition Function

Finally, the rotational partition function of a diatomic molecule follows from the quantum mechanical energy level scheme  

The final expression is the classical limit, valid above a certain critical temperature, which, however, in practical cases is low (i.e. 85 K for H2, 3 K for CO). For a homonuclear or a symmetric linear molecule, the factor a equals 2, while for a het-eronuclear molecule cr=l (Tab. 3.1). This symmetry factor stems from the indistinguishable permutations the molecule may undergo due to the rotation and actually also involves the nuclear partition function. The symmetry factor can be estimated directly from the symmetry of the molecule. 

The average energy of a rotating molecule follows immediately by logarithmic differentiation  

For completeness, we also give the general formula for a larger molecule with moments of inertia /a, Ib, and Ic along the three principal axes  

As an exercise we leave the reader to show that the average energy of a diatomic molecule in the gas phase at temperatures where only the vibrational ground state is populated equals Sk T/l. What is it at high temperatures  

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