The low and high temperature limits of vibrational partition function are $(\theta=\mathrm{h} v / \mathrm{k})$ :
(a)$\mathrm{e}^{-\frac{\theta}{\mathrm{T}}}$ and $\frac{\mathrm{T}}{\theta} \mathrm{e}^{-\frac{\theta}{\mathrm{T}}}$
(b)$\mathrm{e}^{-\frac{\theta}{2 \mathrm{T}}}$ and $\frac{\mathrm{T}}{\theta} \mathrm{e}^{-\frac{\theta}{2 \mathrm{T}}}$
(c)$\mathrm{e}^{-\frac{\theta}{2 \mathrm{T}}}$ and $\frac{\mathrm{T}}{\theta} \mathrm{e}^{-\frac{\theta}{\mathrm{T}}}$
(d)$\mathrm{e}^{-\frac{\theta}{2 \mathrm{T}}}$ and $\frac{\theta}{\mathrm{T}} \mathrm{e}^{-\frac{\theta}{2 \mathrm{T}}}$
Answer
Answer: B ✓ checked by 4AB · confidence medium
The source book printed A; on checking, B is correct — see the explanation.
Explanation
With energy measured from the potential minimum, q_v = e^(−θ/2T)/(1 − e^(−θ/T)). At low T this tends to e^(−θ/2T); at high T it tends to (T/θ)e^(−θ/2T) (b).