= Hydrogen rotational heat capacity
{title2=$c_{\rm rot}/k_{\rm B}=\operatorname{Var}(\epsilon)/t^2$}
Let $u_s=\langle\epsilon\rangle_s$ and $v_s=\operatorname{Var}_s(\epsilon)$ for the conditional distributions of <parahydrogen> and <orthohydrogen>, with $s=\mathrm p,\mathrm o$. Differentiating the normalized <Boltzmann factors> gives $du_s/dt=v_s/t^2$. Thus the rotational <heat capacity at constant volume> per <molecule> of a composition frozen at $x$ is
$$
\frac{c_{\rm fr}}{k_{\rm B}}=\frac{(1-x)v_{\rm p}+xv_{\rm o}}{t^2}.
$$
At conversion equilibrium, the <law of total variance> adds a nonnegative contribution from the changing species populations:
$$
\frac{c_{\rm eq}}{k_{\rm B}}=
\frac{(1-x_{\rm o}^{\rm eq})v_{\rm p}+x_{\rm o}^{\rm eq}v_{\rm o}
+x_{\rm o}^{\rm eq}(1-x_{\rm o}^{\rm eq})(u_{\rm o}-u_{\rm p})^2}{t^2}.
$$
The last term is the contribution from conversion. Compare the first two terms with a frozen sample at the same instantaneous composition; a frozen $3:1$ sample generally has different weights too. This formula follows directly from the <variance> of a mixture: subtract the overall mean inside each species, expand the square, and the cross terms vanish because each conditional deviation has zero mean. It also proves nonnegative rotational <heat capacity> in either model.
\Image[/hydrogen-equilibrium-and-frozen-rotational-heat-capacity.png]
{title=<Hydrogen rotational heat capacity>: equilibrating and frozen spin populations}
{id=image-hydrogen-equilibrium-and-frozen-rotational-heat-capacity}
{description=Original numerical sums of the stated <linear rigid rotor> model through $J=100$. The upper panel compares the equilibrium <orthohydrogen> fraction with frozen <normal hydrogen>; the lower panel also shows pure <parahydrogen>. These are dimensionless predictions, not experimental curves. Only internal <molecular rotation> contributes: <translation>, <molecular vibration>, interactions between <molecules>, and external-field splittings are omitted.}
For an original worked example at $t=1/2$, summing the allowed ladders gives $q_{\rm p}\approx1.00003072$, $q_{\rm o}\approx0.05494692$, and $x_{\rm o}^{\rm eq}\approx0.14150982$. The calculated rotational <heat capacities> are $c_{\rm eq}/k_{\rm B}\approx1.94719633$ and $c_{\rm fr}/k_{\rm B}\approx0.00110733$ for frozen <normal hydrogen>. The contrast is large because changing species accesses the $J=0$ to $J=1$ separation, whereas fixed species must use the larger $J=0$ to $2$ or $J=1$ to $3$ separations. These are model numbers at a chosen dimensionless <temperature>, not measured hydrogen data.
Both rotational <heat capacities> approach zero as $t\to0$, although frozen <normal hydrogen> retains nonzero rotational <internal energy>. Both approach $k_{\rm B}$ as $t\to\infty$ within the rotor model. The high-<temperature> limit is a formal limit with all other excitations suppressed; actual <molecular hydrogen> eventually needs <molecular vibration>, nonrigidity, and dissociation. Multiplying the per-<molecule> result by the <Avogadro constant> gives rotational <molar heat capacity> in units of the <molar gas constant> $R$. A dilute classical <ideal gas> additionally contributes $3R/2$ from <translation> to its <heat capacity at constant volume>.
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