= Solution
At fixed <volume>, logarithmic <differentiation> of the <Saha ionization equation> gives
$$
\left(\frac2\zeta+\frac1{1-\zeta}\right)\zeta'=\frac1T\left(\frac32+\frac{E_I}{k_BT}\right),\quad
T\zeta'=\frac{\zeta(1-\zeta)}{2-\zeta}\left(\frac32+\frac{E_I}{k_BT}\right).
$$
<Differentiation> the <internal energy> then yields
$$
\boxed{C_V=\frac32R(1+\zeta)+R\frac{\zeta(1-\zeta)}{2-\zeta}\left(\frac32+\frac{E_I}{k_BT}\right)^2.}
$$
At low <temperature> $\zeta\to0$ and $C_V\to3R/2$. At high nonrelativistic <temperature> the gas is fully ionized, $\zeta\to1$, and $C_V\to3R$. The large intermediate excess reflects <energy> spent increasing <ionization>, as well as heating and creating additional translational particles. “Very high” here remains below regimes where relativistic, radiative or pair-creation contributions invalidate the model.
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