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Molecules and Crystals

Problem 6.175

The natural vibration frequency of a hydrogen molecule is equal to \(8.25 \cdot 10^{14} \mathrm{~s}^{-1},\) the distance between the nuclei is \(74 \mathrm{pm}\) Find the ratio of the number of these molecules at the first excited vibrational level \((v=1)\) to the number of molecules at the first excited rotational level \((J=1)\) at a temperature \(T=875 \mathrm{~K} .\) It should be remembered that the degeneracy of rotational levels is equal to \(2 J+1\).

Reveal Answer
 6.175. No ib/Nrot =1/3e−ℏ(ω−2B)/kT=3.1⋅10−4, where B==ℏ/2I,I is the moment of inertia of the molecule. \begin{aligned} &\text { 6.175. } N_{\text {o } i b} / N_{\text {rot }}=1 /{ }_{3} \mathrm{e}^{-\hbar(\omega-2 B) / \mathrm{k} T}=3.1 \cdot 10^{-4}, \quad \text { where } \quad B=\\ &=\hbar / 2 I, I \text { is the moment of inertia of the molecule. } \end{aligned}
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