Critical Point

A critical point has distinct meanings in physical science and mathematics. In thermodynamics, the best-known example is the liquid–vapor critical point, where the distinction between liquid and gas disappears. More generally, in statistical mechanics, critical points are associated with continuous phase transitions and fluctuations extending over increasingly large distances. In calculus, a critical point is a point in a function’s domain where its derivative vanishes or, under a common convention, does not exist. These definitions describe different concepts rather than a single shared criterion. (goldbook.iupac.org)

Liquid–vapor critical point

For a pure substance, the liquid–vapor coexistence curve ends at a particular critical temperature $T_c$ and critical pressure $p_c$. Below this endpoint, liquid and vapor can coexist in thermodynamic equilibrium with different densities. As the endpoint is approached along the coexistence curve, those densities converge; at the critical point, the two phases become indistinguishable. This is not an ordinary boiling point, at which distinct liquid and vapor phases remain present. (goldbook.iupac.org)

The density difference can serve as an order parameter measuring the distinction between the phases. It tends continuously to zero at the critical point. The latent heat of vaporization also vanishes, and the liquid–vapor interface loses its surface tension. Above the critical temperature, compression alone cannot produce a separate liquid phase through liquid–vapor condensation. Within the fluid region, liquid-like and gas-like states can instead be connected along a path around the endpoint without crossing the coexistence curve. (damtp.cam.ac.uk)

For ordinary water, the accepted critical parameters are $T_c=647.096\ \mathrm{K}$, equivalent to $373.946^\circ\mathrm{C}$, $p_c=22.064\ \mathrm{MPa}$, and a critical density of $322\ \mathrm{kg\,m^{-3}}$. A supercritical fluid is a fluid above both its critical temperature and critical pressure. The critical point is a particular state, whereas “supercritical” identifies a region of conditions. (iapws.org)

Equations of state

A classical description uses an equation of state relating pressure, temperature, and volume. For a smooth model expressed as $p(T,v)$, where $v$ is molar volume, the critical isotherm has a horizontal inflection at $v_c$:

These conditions locate the critical point in the van der Waals equation,

giving $v_c=3b$, $T_c=8a/(27Rb)$, and $p_c=a/(27b^2)$. Here $R$ is the molar gas constant, while $a$ and $b$ represent attractive interactions and excluded volume. (damtp.cam.ac.uk)

The model captures the qualitative existence of a liquid–vapor endpoint but does not accurately describe the asymptotic behavior of real fluids arbitrarily close to it. Its treatment replaces fluctuating local conditions with average quantities, missing the long-range fluctuations that dominate the critical region. (damtp.cam.ac.uk)

Fluctuations and scaling

Near a continuous critical point, the correlation length $\xi$—the characteristic distance over which fluctuations remain correlated—grows substantially. In an ideal infinite system, it diverges at criticality. Density fluctuations can consequently occur on length scales comparable to visible wavelengths, producing strong light scattering. The resulting cloudiness is called critical opalescence. (damtp.cam.ac.uk)

Many critical properties follow a power law sufficiently close to the transition. With reduced temperature $t=(T-T_c)/T_c$, representative relations are

Here $m$ denotes an order parameter below the transition and $\chi$ a corresponding response, such as magnetic susceptibility. The quantities $\nu,\beta,\gamma$ are critical exponents. A heat capacity may also display a singular dependence on temperature, although its precise form depends on the system and thermodynamic conditions. (damtp.cam.ac.uk)

True divergences require the thermodynamic limit, in which system size becomes infinite at fixed density. Finite samples round off singularities and limit the growth of correlations. Thus an experimentally observed peak need not be infinite to reflect critical behavior. (damtp.cam.ac.uk)

Universality and theoretical description

Different substances can share critical exponents despite having different microscopic constituents. Systems displaying the same asymptotic scaling belong to a universality class. Relevant distinctions include spatial dimensionality, order-parameter symmetry, and interaction range. Ordinary liquid–vapor criticality and short-range scalar magnetic ordering in three dimensions are described by the same universality class. (damtp.cam.ac.uk)

The Ising model provides a simple representation of such collective ordering. Ginzburg–Landau theory describes an order parameter varying through space, rather than only its uniform average. The renormalization group explains universality by examining how descriptions change when short-distance details are removed: different microscopic models can approach the same scale-invariant fixed point. (damtp.cam.ac.uk)

Mathematical meaning

For a real-valued function $f(x)$, introductory calculus commonly calls an interior domain point $c$ critical when its derivative satisfies $f'(c)=0$ or does not exist. Critical points are candidates for local extrema, not guarantees of them. For example, $f(x)=x^3$ has $f'(0)=0$ but no local maximum or minimum there; $f(x)=|x|$ has a minimum at zero despite lacking a derivative there. (openstax.org)

For a differentiable function of several variables, the corresponding condition is a zero gradient, $\nabla f=0$. Such a point may be a minimum, maximum, or saddle point. Second derivatives help classify it, but degenerate cases require further analysis. In mathematical optimization, boundary points must also be examined: an absolute extremum on a restricted domain need not occur at an interior critical point. (openstax.org)