= Thermodynamics of a liquid film
For a film described by $dU=T\,dS+\gamma\,dA$, with material content fixed, its <Helmholtz free energy> obeys $dF=-S\,dT+\gamma\,dA$. Equality of mixed <partial derivatives> gives the <Maxwell relation> $(\partial S/\partial A)_T=-(\partial\gamma/\partial T)_A$. If $\gamma$ depends only on <temperature>, write $\gamma'=d\gamma/dT$; then
$$
dS=\frac{C_A}{T}\,dT-\gamma'\,dA,
\qquad dU=C_A\,dT+(\gamma-T\gamma')\,dA.
$$
Here $C_A$ is the <heat capacity at constant area>. For $C_A>0$ and $\gamma'<0$, reversible adiabatic stretching cools the film, since $dT/dA=T\gamma'/C_A<0$, although positive mechanical input can increase its total <internal energy>. If $\gamma=\gamma_0-\beta T$ with constant positive $\beta$ and constant $C_A$, the <isentropic process> satisfies $C_A\ln(T/T_*)+\beta A=\mathrm{constant}$, where $T_*$ is an arbitrary fixed reference <temperature>. The <work> convention uses the total <area> conjugate to $\gamma$; a two-surface film described by projected <area> requires the corresponding factor of two consistently throughout.
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