= Solution
Let $N=n_H+n_p$ be the conserved number density of hydrogen nuclei, whether ionized or bound, and $x=n_p/N=n_e/N$ the <ionization fraction>. The ideal <gas pressure> counts neutral atoms, free protons and free electrons:
$$
P=Nk_BT(1+x),\qquad \frac{x^2}{1-x}=\frac{S(T)}N,\quad S(T)=\frac{(2\pi m_ek_BT)^{3/2}}{h^3}e^{-R/(k_BT)}.
$$
At low temperature $x\simeq0$ and $P\simeq Nk_BT$. Heating through the ionization region increases the particle count as well as their thermal energy, giving a steeper rise. At high temperature $x\simeq1$, so $P\simeq2Nk_BT$ and the asymptotic slope is twice the low-temperature slope. There is a smooth ionization crossover, not a pressure discontinuity.
The held graph is legible. Representative readings are $P(5000\,\mathrm K)\simeq6.8\,\mathrm{Pa}$ and $P(15000\,\mathrm K)\simeq41.4\,\mathrm{Pa}$, with readout uncertainties of roughly $0.3$–$0.5\,\mathrm{Pa}$ and about $100\,\mathrm K$. They give $N\simeq P/(k_BT)$ on the neutral branch and $N\simeq P/(2k_BT)$ on the ionized branch, respectively. Both are consistent with
$$
\boxed{N\simeq(1.0\pm0.05)\times10^{20}\,\mathrm{m^{-3}}.}
$$
The error is a conservative graph-reading estimate, not a fitted experimental confidence interval.
For that density, the <Saha equation> gives $x\simeq0.50$ at $9000\,\mathrm K$, $x\simeq0.81$ at $10000\,\mathrm K$, and $x\simeq0.9992$ at $15000\,\mathrm K$. Differentiating the ionization relation gives
$$
x'=\frac{x(1-x)}{2-x}\left(\frac{3}{2T}+\frac{R}{k_BT^2}\right),\qquad P'=Nk_B(1+x+Tx').
$$
The calculated maximum slope occurs near $9.25\times10^3\,\mathrm K$, consistent with the source's visibly steepest region at roughly $(9.3\pm0.4)\times10^3\,\mathrm K$. This checks the actual slope rather than identifying it without justification with exactly half ionization.
Taking the neutral ground-state gas as the zero of internal binding energy, the matter <energy density> is translational kinetic energy plus ionization energy:
$$
u=\frac32Nk_BT(1+x)+NxR.
$$
At $15000\,\mathrm K$, with $R=13.6\,\mathrm{eV}$, these terms are about $62$ and $218\,\mathrm{J\,m^{-3}}$, giving
$$
\boxed{u\simeq2.8\times10^2\,\mathrm{J\,m^{-3}}.}
$$
Rest energies are excluded. A separate equilibrium radiation field, if the gas were in a blackbody cavity, would add its own radiation energy density; it is not part of this gas-only estimate.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2015-s16-saha-pressure.png]
{title=Calculated hydrogen pressure and ionization crossover at two conserved nucleus densities}
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