= Solution
{c}
A convenient parameter is the <water vapor> <mole fraction> itself. The <mass conservation> calculation yields
$$
P(x)=P_d+\frac{\epsilon\Delta p\,x}{1+ax},\qquad E(x)=xP(x).
$$
Both functions increase for $0\leq x\leq1$. At the endpoint,
$$
P(1)=E(1)=P_d+\Delta p.
$$
Therefore the physically admissible <hydrogen>–<water vapor> mixture requires
$$
\boxed{E(T_s)<P_d+\Delta p.}
$$
Beyond this bound, the formal positive <quadratic equation> <root> requires $x>1$, equivalently a negative <hydrogen> <partial pressure>. It is the physical constraints, rather than loss of real <roots>, that end the solution branch. For a concrete example, put $P_d=\Delta p=1$ in consistent <pressure> units, $\epsilon=9$, and $E=3$. The positive <root> of $P^2+23P-51=0$ is about $2.037$, giving $x\simeq1.473$, which is impossible.
The limiting <temperature> is obtained from $E(T_*)=P_d+\Delta p$:
$$
\frac1{T_*}=\frac1{T_t}-\frac{R_v}{L_v}\ln\!\frac{P_d+\Delta p}{E_t}.
$$
This finite limit exists above $T_t$ if $E_t<P_d+\Delta p<E_t\exp[L_v/(R_vT_t)]$. The upper qualification matters mathematically: the stipulated constant-<latent heat> formula has a finite limiting <saturation pressure> as $T\to\infty$, so for an arbitrarily large unspecified $P_d+\Delta p$ the printed assertion of a finite breakdown <temperature> is not unconditional. For a concrete formal counterexample, choose $P_d+\Delta p=10^{15}\,\mathrm{Pa}$: with the usual triple-point constants the model has $E(\infty)\simeq2.5\times10^{11}\,\mathrm{Pa}$, so it never reaches the proposed inventory limit. This is a counterexample inside the stipulated equations, not a physically credible high-<pressure> <ideal gas> ocean. Actual <liquid water> also ceases to exist above its <critical point>, before such an extrapolation would make sense.
Physically, the imposed finite-<pressure>-thickness layer can hold no more than $\Delta p/g$ of <water vapor> per unit area. As $x\to1$, it expels all its <hydrogen> into the dry upper layer. A still larger surface <saturation pressure> cannot coexist with this geometry. The layer must deepen, the <temperature> profile must change, or the assumed <phase equilibrium> must fail. If $P_d<\Delta p$, the additional restriction $P>\Delta p$ can also exclude the low-<temperature> end of the branch.
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