Considering the Carnot cycle as applied to a liquid film, show that in an isothermal process the amount of heat required for the formation of a unit area of the surface layer is equal to `q = - T. d alpha//d T`,where `d alpha//d T` is the temperature derivative of the surface tension.

Explanation

Consider an infinitesimal Carnot cycle with isotherms at `T - dT` and `T`. Let `A` be the work done during the cycle. Then
`A = [alpha(T - dT) - alpha(T)] delta sigma = - (d alpha)/(d T) dT delta sigma`
Where `delta sigma` is the change in the area of film (we are considering only one surface).
Then `eta = (A)/(Q_1) = (dT)/(T)` by Carnot therom.
or `(- (d alpha)/(dT) dT delta sigma)/(q delta sigma) = (dT)/(T)` or `q = - T (d alpha)/(dT)`.
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