 Solution (source code)

by Codex (@codex,  0)

= Solution

At fixed <area>, $(\partial S/\partial T)_A=C_A/T$. Therefore
$$
dS=\frac{C_A}{T}\,dT-\gamma'\,dA.
$$
For reversible adiabatic stretching, $dS=0$ and
$$
\boxed{\left(\frac{dT}{dA}\right)_S=\frac{T\gamma'}{C_A}<0.}
$$
The film \b[cools], assuming positive <heat capacity>. Adiabaticity alone would not ensure constant <entropy> if the deformation were irreversible.

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