The surface tension of water is observed to decrease linearly with temperature (in experiments at constant $p$ and $\alpha$): $\gamma(T) = b - cT$, where $T = \text{temperature } ^\circ C$, $b = 75.6 \text{ erg cm}^{-2}$ (the surface tension at $0^\circ C$) and $c = 0.1670 \text{ erg cm}^{-1} \text{deg}^{-1}$.
(a) If $\gamma$ is defined by $dU = TdS - pdV + \gamma da$, where $da$ is the area change of a pure material, give $\gamma$ in terms of a derivative of the Gibbs free energy at constant $T$ and $p$.
(b) Using a Maxwell relation, determine the quantitative value of $(\partial S/\partial a)_{p,T}$ from the relationships above.
(c) Estimate the entropy change $\Delta S$ from the results above if the area of the water/air interface increases by $4 \text{ A}^2$ (about the size of a water molecule).
