= Solution
<Chemical equilibrium> for $p+e^-\rightleftharpoons H+\gamma$ gives $\mu_p+\mu_e=\mu_H$, because the <photon chemical potential> vanishes. Applying the <Maxwell-Boltzmann distribution>[nonrelativistic number-density formula] to all three massive species and taking the ratio gives
$$
\frac{n_H}{n_pn_e}
=\frac{g_H}{g_pg_e}
\left(\frac{2\pi\hbar^2}{k_BT}\right)^{3/2}
\left(\frac{m_H}{m_pm_e}\right)^{3/2}
\exp\!\left(\frac{m_pc^2+m_ec^2-m_Hc^2}{k_BT}\right).
$$
Use <charge neutrality> $n_p=n_e$, the <Hydrogen binding energy> $E_{\rm bind}=m_pc^2+m_ec^2-m_Hc^2$, $m_H\simeq m_p$, and the ground-state statistical weights $g_H=g_pg_e$. Then
$$
\boxed{n_H\simeq n_e^2
\left(\frac{2\pi\hbar^2}{m_ek_BT}\right)^{3/2}
e^{E_{\rm bind}/k_BT}}.
$$
The approximations are a dilute, nondegenerate <ideal gas> in thermal and chemical equilibrium, nonrelativistic electrons, protons and hydrogen, negligible excited hydrogen states, and negligible corrections from the proton-to-hydrogen mass ratio. This relation is the <Saha ionization equation>.
Back to article page
