= Solution
Take the <convection> to maintain a dry, nearly reversible <adiabat>, with <specific gas constant> $R$. <Hydrostatic equilibrium> and an ideal-gas <equation of state> will be used separately to convert <pressure> to height. For an adiabatic displacement, the <first law of thermodynamics> in <enthalpy> form gives
$$
c_p(T)\,dT-\frac{dp}{\rho}=0,\qquad
\frac{A+BT}{T}\,dT=R\frac{dp}{p}.
$$
The <adiabat with linear heat capacity> follows by integrating from the surface: this gives the monotone implicit <temperature>–<pressure> relation
$$
\boxed{A\ln\frac{T}{T_s}+B(T-T_s)=R\ln\frac p{p_s}.}
$$
An explicit form, using the positive real Lambert W branch, is
$$
\boxed{T(p)=\frac AB W_0\left[\frac{BT_s}{A}e^{BT_s/A}
\left(\frac p{p_s}\right)^{R/A}\right]\quad(B>0).}
$$
As $B\to0$, $T=T_s(p/p_s)^{R/A}$. At fixed $p<p_s$,
$$
\left.\frac{\partial T}{\partial B}\right|_p
=\frac{T_s-T}{A/T+B}>0.
$$
Larger <heat capacity> therefore keeps the upper atmosphere warmer at the same <pressure>. Equivalently $d\ln T/d\ln p=R/(A+BT)$ becomes smaller. The original normalized sketch below shows the resulting family of profiles; no particular gas constants or surface <temperature> are assumed there.
The height equations are
$$
\boxed{\frac{dp}{dz}=-\rho g=-\frac{pg}{RT},\qquad
\frac{dT}{dz}=-\frac g{A+BT}.}
$$
Integrating the latter gives
$$
gz=A(T_s-T)+\frac B2(T_s^2-T^2),\qquad
T(z)=\frac{\sqrt{(A+BT_s)^2-2Bgz}-A}{B}.
$$
The physically selected root starts at $T_s$ and decreases. Its formal upper boundary is
$$
\boxed{z_* =\frac{AT_s+BT_s^2/2}{g}.}
$$
At that height $T\to0$ and $p=p_s(T/T_s)^{A/R}e^{B(T-T_s)/R}\to0$. The $B=0$ limit is $z_*=AT_s/g$. This finite-height zero is a property of the assumed polytropic-like model, not a claim that a real atmosphere reaches <absolute zero>: molecular heat capacities, composition, transport and the continuum ideal-gas description cease to support that extrapolation.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2018-variable-heat-capacity.png]
{title=Adiabats with increasing heat capacity and the limited-temperature latent heat inferred from an Antoine fit}
{description=Original calculations. Adiabats use dimensionless parameters $R/A=2/7$ and $b=BT_s/A$; the latent-heat plot distinguishes the stated fit range from an invalid extrapolation toward the known critical point.}
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