UPSC Combined Geo-Scientist, Mains · 2020 · Mains Paper II

Q1(a). Critical-point inflection

Question

At the critical point on the p–V isotherm of a real gas, (∂ p/∂ V)_(n,T_c)=0 and (∂²p/∂ V²)_(n,T_c)=0. Explain the causes.

The critical isotherm has a stationary inflection where the liquid and vapour branches merge and their density difference vanishes.

Below the critical temperature, liquid and vapour can coexist with distinct densities. At the critical point these phases become indistinguishable.
In a smooth equation-of-state description, the two turning points delimiting the unstable loop coalesce.
The zero slope corresponds to divergent isothermal compressibility, κ_T=-V⁻¹(∂ V/∂ p)_T, in the thermodynamic limit.
These conditions determine p_c,V_c,T_c for an analytic model such as van der Waals; critical fluctuations require more refined treatment for quantitative real-fluid behaviour.

The full model answer

As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.

CoversCritical-point inflectionReal gases