Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 2 / d / Solution

Pair equilibrium with negligible chemical potentials cannot continue once its predicted electron abundance falls below the residual abundance required by charge neutrality. Put $x=m_e/T$ and equate part (b), to logarithmic accuracy, to the order-$10^{-10}$ result of part (c):Taking a logarithm and retaining the numerical factor singled out in the hint givesThe solution is $x\simeq 28$, soThus the zero-chemical-potential electron-positron equilibrium description must fail before roughly $20\mathrm{keV}$; thereafter the small asymmetric electron population survives.