Past exam of the physics B course of the University of Cambridge / 2018 / Paper 2 / Section C / C10 / Free-energy identity

Table of contents
- Solution Free-energy identity

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, d F = d U - T d S - S d T = - S d T + γ d A .

Its natural independent variables are therefore

T, A

, so

S = - (\frac{\partial F}{\partial T})_{A}, γ = (\frac{\partial F}{\partial A})_{T} .

Equality of mixed second partial derivatives gives the Maxwell relation for the thermodynamics of a liquid film:

(\frac{\partial γ}{\partial T})_{A} = - (\frac{\partial S}{\partial A})_{T} .

The area convention matters: the supplied fundamental relation already specifies the coefficient of

d A

. 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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