Past exam of the physics B course of the University of Cambridge / 2014 / Paper 2 / Section C / C11

With $F=U-TS$, the thermodynamics of a liquid film givesEquality of mixed derivatives of $F(T,A)$ gives the Maxwell relationUse the area convention already specified by $\gamma dA$; do not insert an additional factor for two film faces.

Since ${\gamma }^{\prime }=d\gamma /dT<0$, the Maxwell relation gives $(\partial S/\partial A{)}_T=-{\gamma }^{\prime }>0$. Thus isothermal stretching increases the film's entropy.

At fixed area, $(\partial S/\partial T{)}_A=C_A/T$. ThereforeFor reversible adiabatic stretching, $dS=0$ andThe film cools, assuming positive heat capacity. Adiabaticity alone would not ensure constant entropy if the deformation were irreversible.

Substitute the entropy differential into the given first law:

Reading the approximately straight trend in the supplied graph gives $\gamma (330\mathrm{K})\simeq 22\mathrm{mN}{\mathrm{m}}^{-1}$ and ${\gamma }^{\prime }\simeq -0.13\mathrm{mN}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. These are graph-read estimates, not more precise material constants. For $\Delta A=10^{-4}{\mathrm{m}}^2$, reversible isothermal stretching givesBoth are positive: stretching does mechanical work on the film and requires heat input to prevent cooling. Its internal energy increases by about $6.5\mu \mathrm{J}$.

Write $\gamma =\alpha -\beta T$, with $\beta =-{\gamma }^{\prime }\simeq 1.3\times 10^{-4}\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Constant $C_A$ then makes the reversible adiabatic conditionwhere $T_{\ast }$ is an arbitrary fixed reference temperature; absorbing it into the constant gives the requested logarithm convention. HenceHere $C_A=0.070\mathrm{J}{\mathrm{K}}^{-1}$. Since $\gamma -T{\gamma }^{\prime }=\alpha$, integration gives $\Delta U=C_A(T_f-T_i)+\alpha \Delta A$. To first order this isThere is no heat exchange on this reversible adiabat. Cooling and increasing internal energy are compatible: positive area work more than offsets the decrease of the temperature-dependent energy term.