Discussion
The translational, vibrational, and rotational molecular partition functions for a system containing ideal diatomic gas molecules in the canonical ensemble (N, V, T) are written as, \(q_{trans}\), \(q_{vib}\), and \(q_{rot}\), respectively. The option that correctly defines their thermodynamic variable(s) dependency is
Answer: B
The answer is \(q_{trans}(T, V)\), \(q_{vib}(T)\), \(q_{rot}(T)\).
Translational energy levels are those of a particle in a box, so they depend on the size of the container:
\[ \begin{aligned} q_{trans} &= \frac{V}{\Lambda^3} \\ \Lambda &= \frac{h}{\sqrt{2\pi m k_B T}} \end{aligned} \]
Rotational levels depend on the moment of inertia and vibrational levels on the bond's frequency. Both are properties of the molecule alone, so \(q_{rot}\) and \(q_{vib}\) depend only on T.
The other options make vibration or rotation depend on volume, which an ideal gas molecule's internal motion does not, or drop the volume from \(q_{trans}\).
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