Heat released by hydrogen spin conversion ($Q_{\mathrm{out}}/N⟶2x{k}_{\mathrm{B}}{\theta }_r$)

As $t\to 0$, the conditional mean rotational energies approach $U_{\mathrm{p}}/N=0$ and $U_{\mathrm{o}}/N=2k_{\mathrm{B}}{\theta }_r$, using the para $J=0$ level as the common zero. A frozen mixture with ortho fraction $x$ therefore retains $U_{\mathrm{fr}}/N\to 2x{k}_{\mathrm{B}}{\theta }_r$. 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 $Q_{\mathrm{out}}/N=2x{k}_{\mathrm{B}}{\theta }_r$, or $3k_{\mathrm{B}}{\theta }_r/2$ 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 $\Delta J=\pm 2$ 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.