= Solution
Apply part (b) to each supplied rotational sum. Multiplying the odd sum by its constant nuclear-spin factor three does not change $q'/q$ or $q''/q$. At the specified <temperature> the dimensionless rotational contributions are
$$
\begin{aligned}
\frac{C_{V,m,p}^{\rm rot}}R
&=\frac{22500(3.58\times10^{-5})}{1.165}
-22500\left(\frac{3.76\times10^{-3}}{1.165}\right)^2
+\frac{300(3.76\times10^{-3})}{1.165}
\simeq1.4253,\\
\frac{C_{V,m,o}^{\rm rot}}R
&=\frac{22500(-3.08\times10^{-5})}{0.970}
-22500\left(\frac{7.64\times10^{-3}}{0.970}\right)^2
+\frac{300(7.64\times10^{-3})}{0.970}
\simeq0.25265.
\end{aligned}
$$
Thus the \b[rotational <heat capacities> are $1.425R$ for <parahydrogen> and $0.253R$ for <orthohydrogen>]. For the total gas <heat capacities>, including translation and neglecting vibration,
$$
\boxed{C_{V,m,p}\simeq2.925R,\qquad C_{V,m,o}\simeq1.753R.}
$$
The difference is substantial because the permitted rotational ladders differ. The para ladder starts at $J=0$ and next reaches $J=2$, separated by $6k_{\rm B}\theta_{\rm rot}$; its excited population is changing appreciably here. The ortho ladder starts at $J=1$ and next reaches $J=3$, a larger separation of $10k_{\rm B}\theta_{\rm rot}$, so much of the ortho population remains in its lowest rotational manifold. A rotational contribution greater than $R$ is possible in the intermediate quantum regime and is not a violation of high-temperature <equipartition>.
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