= Solution
For $p+e\rightleftharpoons H+\gamma$, <chemical equilibrium> and the vanishing <photon chemical potential> imply $\mu_H=\mu_p+\mu_e$. Taking the ratio of the three <nonrelativistic Maxwell--Boltzmann number densities> cancels these <chemical potentials>:
$$
\frac{n_H}{n_pn_e}=\frac{g_H}{g_pg_e}\left(\frac{2\pi M_H}{M_pM_eT}\right)^{3/2}\exp\!\left(\frac{M_p+M_e-M_H}{T}\right).
$$
The ground-state <hydrogen atom> has four combined <electron> and <proton> <spin> states if the small hyperfine splitting is ignored, whereas $g_p=g_e=2$. Hence $g_H/(g_pg_e)=1$. With <Hydrogen binding energy> $B=M_p+M_e-M_H$ and $M_H\simeq M_p$, the <Saha ionization equation> gives
$$
\boxed{\frac{n_H}{n_pn_e}\simeq\left(\frac{2\pi}{M_eT}\right)^{3/2}e^{B/T}.}
$$
The approximation in the prefactor neglects relative corrections of order $(M_e+B)/M_p$; the <Hydrogen binding energy> must be retained in the exponential.
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