 Solution

Assume a dilute, nondegenerate, nonrelativistic ideal gas in local thermodynamic equilibrium, with ground-state atomic hydrogen, protons and electrons, and neglect excited atomic levels and interactions. Let the conserved number of hydrogen nuclei per volume be $n_0$, so $n_e=n_p=\zeta n_0$ and $n_H=(1-\zeta )n_0$. Equality of chemical potentials for $H\rightleftharpoons p+e$ gives the Saha ionization equationThe factor two is electron spin; $g_p$ and $g_H$ must be defined consistently with nuclear spin. The small difference between the atomic and proton translational partition functions has been neglected. ThereforeFor the ground-state convention $g_p=1$, $g_H=2$, the statistical prefactor $2g_p/g_H$ is one. At fixed volume $n_0$ is constant, whereas an isobaric heating process would not have a constant $A$.