Discussion
The molecular partition function of a system is given by
\(q(T) = \left(\dfrac{k_{B}T}{hc}\right)^{\frac{3}{2}}\)\(\left(\dfrac{8\pi^{3}mk_{B}T}{h^{2}}\right)^{\frac{3}{2}}\), where the symbols have their usual meanings.
The heat capacity at constant volume for this system is
Answer: A
The heat capacity is 3R per mole.
Only the temperature dependence of q matters. Collecting the powers of T: \(q \propto T^{3/2} \cdot T^{3/2} = T^{3}\). The molar internal energy and heat capacity follow from \(\ln q = 3\ln T + \text{const}\):
\[ \begin{aligned} U &= RT^{2}\frac{d\ln q}{dT} = RT^{2} \cdot \frac{3}{T} = 3RT \\ C_{V} &= \frac{dU}{dT} = 3R \end{aligned} \]
The first factor looks like a rotational term and the second like a translational one, each contributing 3R/2. 3R/2 counts only one of them, 9R/2 adds an extra 3R/2 that is not there, and 6R doubles the result.
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