Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 2 / a

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- Solution a

Solution

Chemical equilibrium for

e^{-} + p \leftrightarrow H + γ

requires

μ_{e} + μ_{p} = μ_{H}

, since the photon chemical potential vanishes. Inserting the nonrelativistic Maxwell-Boltzmann distribution and using charge neutrality

n_{p} = n_{e}

gives

\frac{n _{H}}{n _{e}^{2}} = (\frac{2 π}{m _{e} T})^{3/2} exp (\frac{m _{e} + m _{p} - m _{H}}{T}),

where the proton-to-hydrogen mass ratio and the stated degeneracy factors have been approximated by one. Thus

(\frac{n _{H}}{n _{e}^{2}})_{eq} = (\frac{2 π}{m _{e} T})^{3/2} e^{B_{H} / T} .

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