= Solution
Treat the film at fixed material content, with <area> $A$ the coordinate conjugate to the <surface tension> in the stated <work> term. The <Helmholtz free energy> is a <Legendre transform> of the <internal energy>:
$$
F=U-TS,\qquad dF=dU-T\,dS-S\,dT=-S\,dT+\gamma\,dA.
$$
Its natural independent variables are therefore $T,A$, so
$$
S=-\left(\frac{\partial F}{\partial T}\right)_A,
\qquad \gamma=\left(\frac{\partial F}{\partial A}\right)_T.
$$
Equality of mixed second <partial derivatives> gives the <Maxwell relation> for the <thermodynamics of a liquid film>:
$$
\boxed{\left(\frac{\partial\gamma}{\partial T}\right)_A
=-\left(\frac{\partial S}{\partial A}\right)_T.}
$$
The <area> convention matters: the supplied fundamental relation already specifies the coefficient of $dA$. There is no additional factor of two unless one changes from total interfacial <area> to a projected film <area> and changes the <work> term consistently.
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