Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 3 / a / Solution

The Young–Laplace equation gives a pressure excess $2\sigma /R$ inside a spherical droplet. Near coexistence, the common chemical-potential shift is $\mu \simeq \alpha \delta$, where $\alpha =f^{\prime \prime }({\varphi }_B)$, while changing phase changes the composition by approximately $2{\varphi }_B$. Balancing the capillary pressure against this thermodynamic shift gives the Gibbs--Thomson relationHence the compositions immediately outside and inside arewith