A system consists of a thin film of surface area A, of internal energy U(S, A), where
dU = T dS + γ d A
Hence, the surface tension is given by
γ (S, A) = \frac{∂U(S, A)}{∂A}.
Express the heat Q_{if} to provide to the film for a variation ΔA_{if }= A_f − A_i of the surface of the film through an isothermal process at temperature T, that brings the film from an initial state i to a final state f, in terms of γ (T, A) and its partial derivatives.
Perform a Legendre transformation on the internal energy U(S, A) with respect to the entropy to define the free energy and derive its differential,
dF (T, A) = −S (T, A) dT + γ (T, A) dA
where
γ (T, A) = \frac{\partial F(T,A)}{\partial A} and S (T, A) = -\frac{\partial F(T,A)}{\partial T} .
For an isothermal process, we can compute the heat Q_{if} as,
Q_{if} = TΔS_{if} = T \frac{\partial S(T,A)}{\partial A} ΔA_{if}.
The Schwarz theorem applied to free energy F (T, A) yields,
\frac{\partial }{\partial A} \Bigl(\frac{\partial F}{\partial T} \Bigr) = \frac{\partial }{\partial T} \Bigl(\frac{\partial F}{\partial A} \Bigr).
which leads to the Maxwell relation,
\frac{\partial S(T, A)}{\partial A} = – \frac{∂γ (T, A)}{∂T}.
Hence, the heat is given by,
Q_{if} = −T \frac{∂γ (T, A)}{∂T}ΔA_{if} .