# Consider an optically thick hydrogen gas in statistical equilibrium at temperature T. (Optically thick means that photons

## Question:

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_{H}c^{2} = m_{e}c^{2} + m_{p}c^{2} − I ; denote μj ≡ μ̃j − mjc^{2}; and assume that T≪ m_{e}c^{2}/k_{B }≈ 6 × 10^{9} 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:

## Step by Step Answer:

**Related Book For**

## Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics

**ISBN:** 9780691159027

1st Edition

**Authors:** Kip S. Thorne, Roger D. Blandford