= Solution
The definition of <heat capacity at constant area> and the <Maxwell relation> give the <entropy> differential
$$
dS=\frac{C_A}{T}\,dT-\gamma'\,dA.
$$
For a reversible <adiabatic process>, $dS=0$. Hence
$$
\boxed{\left(\frac{dT}{dA}\right)_S=\frac{T\gamma'}{C_A}<0}
$$
for positive <heat capacity> $C_A$. \b[Reversible adiabatic stretching cools the film.] Reversibility, or negligible <entropy> production, is essential to this inference. Adiabatic isolation alone only sets the <heat transfer> to zero; an irreversible stretch can generate <entropy>, so its <temperature> change is not fixed by this isentropic <derivative>.
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