Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 35B (Statistical Physics)

At a first-order transition the Gibbs free energy is continuous but a first derivative such as entropy or volume jumps; there is latent heat. At a second-order transition the first derivatives are continuous while a second derivative, such as heat capacity or susceptibility, is discontinuous or divergent, and there is no latent heat.

Equality of chemical potentials gives $d{\mu }_{\mathrm{l}}=d{\mu }_{\mathrm{g}}$ along coexistence. Sinceone obtainsUsing $L=T(s_{\mathrm{g}}-s_{\mathrm{l}})$ gives the Clausius-Clapeyron relation

With $v_{\mathrm{g}}\gg v_{\mathrm{l}}$ and $v_{\mathrm{g}}=k_{B}T/p$, the coexistence equation becomesFor constant $L$ this integrates towhere $p_0$ is constant. For $N$ gas particles, $V_{\mathrm{g}}=N{k}_{B}T/p(T)$, and therefore

In the Dieterici equation, $b$ is the excluded volume per particle caused by short-range repulsion. The positive constant $a$ measures cohesive attraction, which lowers the pressure through the Boltzmann factor.

At low density,Hence the second virial coefficient is

At the critical point, $(\partial p/\partial v{)}_T=({\partial }^{2}p/\partial v^2{)}_T=0$. The first logarithmic derivative givesAt a stationary point the second-derivative condition reduces toCombining the equations gives $v_c=2b$, and substitution then gives