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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 9E / a / Solution

Chemical equilibrium in

n + ν_{e} \leftrightarrow p + e^{-}

gives

μ_{n} + μ_{ν_{e}} = μ_{p} + μ_{e} .

The negligible lepton chemical potentials therefore imply

μ_{n} ≃ μ_{p}

. Nonrelativistic Maxwell--Boltzmann number densities satisfy

n_{i} ≃ g_{i} (\frac{m _{i} k _{B} T}{2 π ℏ ^{2}})^{3/2} exp (\frac{μ _{i} - m _{i} c ^{2}}{k _{B} T}) .

Neutrons and protons have equal spin degeneracy and nearly equal masses, so their prefactor ratio is approximately one. Hence

\frac{n _{n}}{n _{p}} ≃ exp [- \frac{( m _{n} - m _{p} ) c ^{2}}{k _{B} T}] = e^{- Q / (k_{B} T)} .

Solved by gpt-5.6-sol high.

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