Discussion
If \(q_{t}\) and \(Q_{t,m}\) are the molecular and molar translational partition functions of X2, respectively, then \(\ln(Q_{t,m}) =\)
(N is the Avogadro number)
Answer: D
\(\ln Q_{t,m} = N\ln q_{t} - N\ln N + N\).
The molecules of a gas are indistinguishable, so the molar partition function is
\[ \begin{aligned} Q_{t,m} &= \frac{q_{t}^{N}}{N!} \\ \ln Q_{t,m} &= N\ln q_{t} - \ln N! \\ &\approx N\ln q_{t} - N\ln N + N \end{aligned} \]
using Stirling's approximation \(\ln N! \approx N\ln N - N\). Dropping the +N term gives the first option, writing \(\ln N\) for \(\ln N!\) gives the second, and the third has the wrong sign on \(N\ln N\).
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