Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2 / b / Solution

For $a<0$ and $b>0$, planar coexistence occurs at $\varphi =\pm {\varphi }_B$ with ${\varphi }_B=\sqrt{-a/b}$. Put $\alpha =f^{\prime \prime }({\varphi }_B)=a+3b{\varphi }_B^2=-2a>0$. A common small shift $\delta$ gives equal chemical potentials to first order,The thermodynamic pressure is $P=\varphi \mu -f$. Its difference between the positive interior and negative exterior is thereforeFor a sphere the Young–Laplace equation gives $P_{+}-P_{-}=2\sigma /R$. Consequently the Gibbs--Thomson relation is