Consider an optically thick hydrogen gas in statistical equilibrium at temperature T. (“Optically thick” means that photons can travel only a small distance compared to the size of the system before being absorbed, so they are confined by the hydrogen and kept in statistical equilibrium with it.) Among the reactions that are in statistical equilibrium are H + γ ↔ e + p (ionization and recombination of H, with the H in its ground state) and e + p ↔ e + p + γ (emission and absorption of photons by “bremsstrahlung,” i.e., by the Coulomb-force-induced acceleration of electrons as they fly past protons). Let μ̃_γ , μ̃_H, μ̃_e, and μ̃_p be the chemical potentials including rest masses; let m_H, m_e, and m_p be the rest masses; denote by I (≡13.6 eV) the ionization energy of H, so that m_Hc² = m_ec² + m_pc² − I ; denote μj ≡ μ̃j − mjc²; and assume that T≪ m_ec²/k_B ≈ 6 × 10⁹ K and that the density is low enough that the electrons, protons, and H atoms can be regarded as nondegenerate (i.e., as distinguishable, classical particles).

(a) What relationships hold between the chemical potentials μ̃_γ , μ̃_H, μ̃_e, and μ̃_p?

(b) What are the number densities n_H, n_e, and n_p expressed in terms of T , μ̃_H, μ̃_e, and μ̃_p—taking account of the fact that the electron and proton both have spin 1/2 , and including for H all possible electron and nuclear spin states?

(c) Derive the Saha equation for ionization equilibrium:

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nen p nH = (2π m¸k µT)³/²-1/(kgT)¸ h3 (5.68)