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At the critical point (T_c, P_c), a pure substance exhibits a second-order phase transition where the liquid-vapor distinction vanishes and the density, properties, and fluctuations diverge according to power laws. Above T_c, the substance cannot be liquefied regardless of applied pressure; at T_c, the isotherm has zero slope and inflection point simultaneously. Near the critical point, properties vary dramatically with small changes in conditions, and the distinction between liquid and gas becomes ill-defined, leading to exotic states like supercritical fluids.
Examine the P-V diagram near the critical point. Observe how the liquid-gas interface disappears above T_c. Study density and property divergences.
The van der Waals equation of state (P + a/V²)(V − b) = RT captures the idea that real gases have attractive interactions (the a/V² correction reduces effective pressure) and finite molecular volume (the b correction). On a P-V diagram, this equation predicts a family of isotherms with qualitatively different shapes depending on temperature. Below a certain temperature T_c, the isotherm develops a region with negative slope (∂P/∂V > 0), which is mechanically unstable. The Maxwell construction replaces this unphysical portion with a horizontal line at the vapor pressure, representing liquid-vapor coexistence. As temperature increases, the coexistence region shrinks — the gap between liquid and gas molar volumes narrows — until at T_c the two phases have exactly the same density and the distinction between them disappears.
At the critical point (T_c, P_c, V_c), the isotherm satisfies two simultaneous conditions: (∂P/∂V)_T = 0 and (∂²P/∂V²)_T = 0. The first says the isotherm has zero slope — the liquid and gas are equally compressible. The second says it is an inflection point — there is no longer any sense in which one phase is denser than the other. For the van der Waals equation, these conditions give T_c = 8a/27Rb, P_c = a/27b², V_c = 3b, and the dimensionless critical compression factor P_cV_c/(RT_c) = 3/8. The fact that this ratio is a universal constant for van der Waals fluids (regardless of the values of a and b) is the first hint of universality: properties near the critical point are, to a surprising degree, independent of molecular details.
Near the critical point, fluctuations in density become enormous. Locally, regions of higher and lower density spontaneously form and dissolve on all length scales simultaneously — a phenomenon called critical opalescence, where the fluid scatters light strongly and becomes milky-white. This is because density fluctuations at wavelengths comparable to visible light are thermally excited when the compressibility diverges. The correlation length ξ — the spatial range over which density fluctuations are correlated — diverges as T → T_c from above: ξ ∝ |T − T_c|^{−ν}. All other singular properties follow power laws with their own critical exponents: the specific heat diverges as |T − T_c|^{−α}, the order parameter (density difference) vanishes as |T − T_c|^β below T_c, and the compressibility diverges as |T − T_c|^{−γ}.
Above T_c, no amount of pressure can liquefy the fluid — you can compress it arbitrarily without crossing a phase boundary. The result is a supercritical fluid that has liquid-like densities but gas-like transport properties (high diffusivity, low viscosity). Supercritical CO₂ (T_c = 31°C, P_c = 74 bar) is used industrially as a solvent in coffee decaffeination and pharmaceutical extraction precisely because its properties can be tuned continuously by adjusting temperature and pressure. This tunability is a direct consequence of the absence of a phase boundary — properties change smoothly, not discontinuously, as you vary the conditions above T_c.