Consider the formation of atomic hydrogen in the reaction c + H+ = H, where e is an electron, as the adsorption of an electron on a proton H+.
(a) Show that the equilibrium concentrations of the reactants satisfy the relation

[e][H+]/[H] ≈ nQ exp(–I/τ),

Where I is the energy required to ionize atomic hydrogen, and nQ ≡ (mτ/2πh2)3/2 refers to the electron. Neglect the spins of the particles; this assumption does not affect the final result. The result is known as the Saha equation. If all the electrons and protons arise from the ionization of hydrogen atoms, then the concentration of protons is equal to that of the electrons, and the electron concentration is given by

[e] = [H]1/2nQ1/2 exp(– ½τ).

A similar problem arises in semiconductor physics in connection with the thermal ionization of impurity atoms that are donors of electrons.
Notice that:
(1) The exponent involves 1/2I and not I, which shows that this is not a simple “Boltzmann factor” problem. Here I is the ionization energy.
(2) The electron concentration is proportional to the square root of the hydrogen atom concentration.
(3) If we add excess electrons to the system, then the concentration of protons will decrease.
(b) Let [H(exe)] denote the equilibrium concentration of H atoms in the first excited electronic state, which is 3/4I above the ground state. Compare [H(exe)] with [e] for conditions at the surface of the Sun, with [H] ≈ 1023 cm-3 and T ≈ 5000K.