= Heat released by hydrogen spin conversion
{title2=$Q_{\rm out}/N\longrightarrow2xk_{\rm B}\theta_r$}
As $t\to0$, the conditional mean rotational <energies> approach $U_{\rm p}/N=0$ and $U_{\rm o}/N=2k_{\rm B}\theta_r$, using the para $J=0$ level as the common zero. A frozen mixture with ortho fraction $x$ therefore retains $U_{\rm fr}/N\to2xk_{\rm B}\theta_r$. If a bath keeps the sample at low <temperature> while a <catalyst> permits conversion to almost pure <parahydrogen>, the rotational <heat> delivered to the bath tends to $Q_{\rm out}/N=2xk_{\rm B}\theta_r$, or $3k_{\rm B}\theta_r/2$ for <normal hydrogen>.
One may shift the ortho ladder to its own lowest level for computing its conditional <heat capacity>, because a constant <energy> offset disappears on differentiation at fixed composition. That offset must be restored before comparing species or calculating conversion <heat>. Otherwise the calculation loses precisely the stored rotational <energy> that conversion releases. The expression above accounts for the internal rotor contribution at fixed bath <temperature>; it does not model liquid intermolecular <energies>, <latent heat>, or an adiabatic temperature rise.
A <catalyst> changes the conversion rate and allows heat removal during cooling; it does not change the free-gas equilibrium weights at a specified <temperature>. This connects the microscopic exchange rule to refrigeration: delayed conversion supplies a continuing internal <heat> source. <Rotational Raman spectroscopy> offers a complementary population probe because its $\Delta J=\pm2$ transitions stay within the even or odd ladder; line intensities require the appropriate species populations and transition response, not a bare $3:1$ intensity rule.
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