= 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 equation>
$$
\frac{n_en_p}{n_H}=\frac{2g_p}{g_H}\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}e^{-E_I/(k_BT)}.
$$
The 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. Therefore
$$
\boxed{\frac{\zeta^2}{1-\zeta}=AT^{3/2}e^{-E_I/(k_BT)},\quad A=\frac{2g_p}{g_Hn_0}\left(\frac{2\pi m_ek_B}{h^2}\right)^{3/2}.}
$$
For 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 $A$.
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