Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S16 / 1 / Solution

Assume a dilute nondegenerate hydrogen gas in thermal and chemical equilibrium, with Maxwell-Boltzmann translational partition functions, ground-state neutral atoms, single ionization and charge neutrality $n_p=n_e$. For $\mathrm{H}\rightleftharpoons p+e$, chemical equilibrium gives ${\mu }_H={\mu }_p+{\mu }_e$. The ideal-gas relation ${\mu }_i=E_{i0}+k_{B}T\log (n_i{\lambda }_i^3/g_i)$, with $E_{H0}=-R$, givesNeglecting the electron mass in $m_H/m_p$ and using the ground-state electronic statistical weights, $g_e=2$ and $g_H=2$ with cancelling nuclear-spin factors, this becomesExcited neutral states have an energy gap of at least $3R/4$ above the ground state. At the temperatures relevant to the plotted maximum their Boltzmann factors are very small despite the finite degeneracy increase. Highly excited levels also require a physical plasma cutoff; the untruncated infinite isolated-atom partition sum is not a valid finite-density model. Ground-state dominance is an approximation in this regime, not a theorem at all temperatures.

Take the heating process to keep the total hydrogen-nucleus density fixed, so $n_H+n_e=n_0$. This fixed-density assumption is needed to determine $n_0$ from the curve; freely expanding gas would require its density history. The excited-to-ground ratio is $n_2/n_H\simeq 4e^{-3R/(4k_{B}T)}$. ThereforeAt low temperature excitation is suppressed. At higher temperature excitation grows, but eventually ionization removes neutral atoms and the excited-neutral abundance decreases. To locate the maximum quantitatively, write $u=R/(k_{B}T)$. From $S/n_0=x^2/(1-x)$,Setting $d\log f/d\log T=0$ givesReading the original graph gives $T_{\max }\simeq (7.9\text{–}8.0)\times 10^3$ K and $f_{\max }\simeq 2.3\times 10^{-7}$. With $R=13.6$ eV, this implies $u\simeq 20$ and $\boxed{x_{\max }\simeq 0.82}$. Substitution intogives approximately $10^{18}{\mathrm{m}}^{-3}$. The result is particularly temperature-sensitive because of the Saha exponential; using $7900$ K gives about $9\times 10^{17}{\mathrm{m}}^{-3}$, while the coarse $8000$ K reading gives about $1.2\times 10^{18}{\mathrm{m}}^{-3}$. The plotted height is consistent with this range and should not be used to claim more precision than the source graph provides.