= Rotational partition functions of hydrogen
{title2=$q_{\rm eq}=q_{\rm p}+3q_{\rm o}$}
Write the <linear rigid rotor> <energy eigenvalues> as $E_J=k_{\rm B}\theta_r J(J+1)$, where $\theta_r=\hbar^2/(2Ik_{\rm B})=hB/k_{\rm B}$ is the rotational <temperature> scale, $I$ the <moment of inertia>, $B$ the frequency-valued <rotational constant>, and $k_{\rm B}$ the <Boltzmann constant>. Put $t=T/\theta_r$ and $\epsilon_J=J(J+1)$. Define sums that include rotational multiplicity but exclude nuclear-spin multiplicity:
$$
q_{\rm p}(t)=\sum_{J\ \mathrm{even}}(2J+1)e^{-\epsilon_J/t},\qquad
q_{\rm o}(t)=\sum_{J\ \mathrm{odd}}(2J+1)e^{-\epsilon_J/t}.
$$
The <canonical partition function> for one equilibrating internal rotor is $q_{\rm eq}=q_{\rm p}+3q_{\rm o}$, so its equilibrium <orthohydrogen> fraction is
$$
x_{\rm o}^{\rm eq}=\frac{3q_{\rm o}}{q_{\rm p}+3q_{\rm o}},\qquad
\frac{x_{\rm o}^{\rm eq}}{1-x_{\rm o}^{\rm eq}}=\frac{3q_{\rm o}}{q_{\rm p}}.
$$
For $t\ll1$, the lowest allowed levels give $q_{\rm p}\simeq1$ and $q_{\rm o}\simeq3e^{-2/t}$, hence $x_{\rm o}^{\rm eq}\simeq9e^{-2/t}/(1+9e^{-2/t})\to0$. For $t\gg1$, the densely populated even and odd ladders have equal leading sums, $q_{\rm p}\sim q_{\rm o}\sim t/2$, hence $x_{\rm o}^{\rm eq}\to3/4$. The factor nine belongs to the low-<temperature> two-level approximation, and the factor three to the high-<temperature> spin multiplicities.
A frozen mixture requires two separately normalized <probability distributions>,
$$
p(J\mid\mathrm p)=\frac{(2J+1)e^{-\epsilon_J/t}}{q_{\rm p}}\quad(J\ \mathrm{even}),\qquad
p(J\mid\mathrm o)=\frac{(2J+1)e^{-\epsilon_J/t}}{q_{\rm o}}\quad(J\ \mathrm{odd}).
$$
Multiply these by the prescribed fractions $1-x$ and $x$, respectively. The nuclear-spin factor three cancels from the conditional <orthohydrogen> distribution. For fixed counts $N_{\rm o}$ and $N_{\rm p}$, the internal part of the many-<molecule> <partition function> is proportional to $q_{\rm p}^{N_{\rm p}}(3q_{\rm o})^{N_{\rm o}}$, with combinatorial factors independent of <temperature>. Its logarithm yields the two conditional means weighted by the fixed counts. Using $q_{\rm p}+3q_{\rm o}$ instead would silently permit conversion and change the physical constraint.
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