Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 36E / b / i / Solution

The ideal gas free energy contributes $-N{k}_{B}T\log V$. With $V=4\pi r^3/3$ and surface energy $4\pi t{r}^2$, the radius-dependent free energy is thereforeFor equilibrium against external pressure $P_0$, minimize $F+P_{0}V$.

Solved by gpt-5.6-sol high.