The total surface energy of liquids

Let a liquid surface of area a be increased by da under reversible conditions. Assume that there is no mass transfer from the liquid to the vapour in this process, so that the liquid (bulk plus surfaces) may be treated as a closed system. The work done on the liquid during the increase in area is 7da so that, if volume changes are negligible, the first law of thermodynamics may be written  

It may readily be shown (see Thermal Physics by C.B.P. Finn in this series) that one of the Maxwell relations for a Uquid surface is  

KdS/dT Q is the heat capacity Cqj of the liquid when the surface area is held constant, so that  

The Dupre experiment shows that y does not depend on a so that 7 is a function of T only. Therefore  

In a reversible, isothermal change of surface area at constant volume dr = 0 and  

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