Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 36E (Statistical Physics)

At equal pressure and temperature, ideal gases have equal molecular number density. The mean air molecular mass is $.8(28)+.2(32)=28.8$, while helium has mass $4$, so

Monatomic helium has molar $C_p=5R/2$ and diatomic air has $7R/2$. Dividing by molar masses,

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 $F+P_{0}V$.

Differentiation givesequivalently the Laplace-pressure condition

Let $r_0=(3N{k}_{B}T/(4\pi P_0){)}^{1/3}$ be the zero-tension radius. From
$P_0{r}^3+2t{r}^2=P_0{r}_0^3$, put $r=r_0+\delta$ and retain first order: