 Solution

Assume a dilute nondegenerate ideal mixture in thermal and chemical equilibrium, electrical neutrality $N_e=N_p$, and ground-state hydrogen only. Choose the zero of internal energy as a separated proton and electron, so the atom has energy $-R$. Its single-particle partition function contains $e^{R/(k_{\mathrm{B}}T)}$:Their ratio has one remaining power of $V$. Neglecting binding mass in $m_H\simeq m_p$ makes the translational mass factor $(m_e{m}_p/m_H{)}^{3/2}\simeq m_e^{3/2}$. For unresolved ground-state spin states, $g_e{g}_p/g_H=1$: including proton spin gives $2\times 2/4$, or omitting it consistently gives $2\times 1/2$. The given law of mass action therefore yieldsThis is the Saha equation with the degeneracy convention used by the source. Equivalently $n_e^2/n_H=(2\pi m_e{k}_{\mathrm{B}}T/h^2{)}^{3/2}{e}^{-R/(k_{\mathrm{B}}T)}$.