 Solution

Apply part (b) to each supplied rotational sum. Multiplying the odd sum by its constant nuclear-spin factor three does not change $q^{\prime }/q$ or $q^{\prime \prime }/q$. At the specified temperature the dimensionless rotational contributions areThus the rotational heat capacities are $1.425R$ for parahydrogen and $0.253R$ for orthohydrogen. For the total gas heat capacities, including translation and neglecting vibration,The difference is substantial because the permitted rotational ladders differ. The para ladder starts at $J=0$ and next reaches $J=2$, separated by $6k_{\mathrm{B}}{\theta }_{\mathrm{rot}}$; its excited population is changing appreciably here. The ortho ladder starts at $J=1$ and next reaches $J=3$, a larger separation of $10k_{\mathrm{B}}{\theta }_{\mathrm{rot}}$, 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.