Past exam of the physics B course of the University of Cambridge / 2015 / Paper 2 / Section C / C10

A liquid-gas thermodynamic critical point is the endpoint of the coexistence curve where distinct liquid and gas densities merge. In the van der Waals illustration, write the original constants as $a_v,b_v$ to distinguish them from the later polynomial coefficients. The critical isotherm has a horizontal inflection in $p(V)$; setting its first two volume derivatives to zero givesBelow this temperature the homogeneous van der Waals loop contains an unstable region; physical equilibrium instead separates into liquid and gas.^[ref]

For the critical point of a cubic density equation of state, assume $b,c>0$. Its density derivatives areBecause $n=N/V$ is a smooth monotone function of $V$ at finite positive density, a stationary inflection obeys $p_n=p_{nn}=0$ as well. Solving these equations and substituting back givesThe symbols $b,c$ here have different units from the van der Waals excluded-volume parameter. With $x=n/n_c$, $t=T/T_c$, the isotherms becomeThe spinodal densities of a cubic gas delimit the two-extrema regime. The curves are monotone for $t>1$, have a stationary inflection at $(x,p/p_c)=(1,1)$ for $t=1$, and have two extrema $x=1\pm \sqrt{1-t}$ for $0<t<1$.

For the critical-density compressibility of a cubic gas, since ${\kappa }_T=(1/n)(\partial n/\partial p{)}_T$, evaluation at the critical density givesIt diverges as $T\to T_c^{+}$ in this mean-field model. For $T<T_c$ it is negative on the homogeneous branch at $n_c$, violating mechanical stability: a pressure increase would expand rather than compress that branch.

The actual equilibrium below criticality is coexistence of two phases, with equal pressure and chemical potential and different densities. A sample's mean density can change by changing phase proportions. The unstable homogeneous branch is not a realizable equilibrium isotherm. The Maxwell construction in density coordinates requires care: the equal-area construction is performed in $p$ versus volume, not by using unweighted area in $p$ versus density. The truncated polynomial is an approximate mean-field equation, so its predictions need not give quantitative critical behaviour for a real gas.