Critical Point and Supercritical Fluids

Core Idea

The critical point is the endpoint of the liquid-vapor boundary on a phase diagram (T_c, P_c). Above the critical temperature, liquid and gas become indistinguishable (supercritical fluid). The critical point is characterized by (∂P/∂V)_T = 0 and (∂²P/∂V²)_T = 0.

Explainer

From phase diagrams and the Clausius-Clapeyron equation, you know that the liquid-vapor boundary is a curve in the P-T plane along which both phases coexist in equilibrium. If you follow this boundary upward — increasing both temperature and pressure — what happens? The density of the vapor increases (it becomes more compressed), while the density of the liquid decreases (thermal expansion). At some point, the two densities must converge. The critical point (T_c, P_c) is exactly where they meet: above this temperature, there is no longer a meaningful distinction between liquid and gas.

The mathematical signature of the critical point is that the P-V isotherm develops an inflection point with zero slope: (∂P/∂V)_T = 0 and (∂²P/∂V²)_T = 0 simultaneously. On the van der Waals equation, these two conditions uniquely determine T_c = 8a/27Rb and P_c = a/27b², giving the critical point in terms of the intermolecular interaction parameters a and b. Below T_c, isotherms have a "swaybacked" region where the van der Waals equation predicts (∂P/∂V)_T > 0 — a mechanically unstable region that resolves into the coexisting liquid and vapor phases via the Maxwell construction. Above T_c, isotherms are monotonically decreasing and no phase separation occurs.

A supercritical fluid exists above T_c and P_c. Because liquid and gas become indistinguishable at the critical point, you can continuously transform liquid into gas above T_c without ever crossing a phase boundary — by going around the critical point. A supercritical fluid shares properties of both phases: it has the density of a liquid but the diffusivity and viscosity of a gas, making it an excellent solvent and transport medium. Supercritical CO₂ (T_c = 304 K, P_c = 73 atm) is industrially important for decaffeination, pharmaceutical extraction, and dry cleaning precisely because it has liquid-like solvating power with gas-like penetration into porous materials.

Near the critical point, something physically dramatic occurs: critical opalescence. Density fluctuations grow over length scales comparable to the wavelength of visible light, scattering it strongly and making the fluid appear milky. This is a signature that the system has no preferred length scale — fluctuations occur at all scales simultaneously. The compressibility (∂V/∂P)_T diverges as T → T_c because the usual resistance to compression vanishes: at the critical point, it costs almost no energy to rearrange matter between liquid-like and gas-like density. These diverging fluctuations near the critical point are the entry point to the much deeper subject of critical phenomena and universal scaling.