Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 9A / Solution

Chemical equilibrium for $p+e\leftrightarrow H$ givesand charge neutrality gives $n_p=n_e$. Insert the nonrelativistic Maxwell-Boltzmann number densities, use $m_H{c}^2=m_p{c}^2+m_e{c}^2-E_{\mathrm{bind}}$, and neglect the proton correction in the reduced mass. The spin degeneracies cancel in the ground-state approximation. One obtains Saha equationEquivalently, 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.