= Solution
Chemical equilibrium for $p+e\leftrightarrow H$ gives
$$
\mu_H=\mu_p+\mu_e,
$$
and <charge neutrality> gives $n_p=n_e$. Insert the nonrelativistic Maxwell-Boltzmann number densities, use $m_Hc^2=m_pc^2+m_ec^2-E_{\rm bind}$, and neglect the proton correction in the reduced mass. The spin degeneracies cancel in the ground-state approximation. One obtains <Saha ionization equation>[Saha equation]
$$
\boxed{
\frac{n_H}{n_e^2}
=\left(\frac{2\pi\hbar^2}{m_ek_BT}\right)^{3/2}
\exp\!\left(\frac{E_{\rm bind}}{k_BT}\right)}.
$$
Equivalently, its reciprocal is the ionization ratio. The exponential favors neutral hydrogen as the temperature falls, while the translational phase-space factor favors ionization at high temperature.
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