Critical point (thermodynamics)

In thermodynamics, a critical point is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas comes into a supercritical phase, and so cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature ''T_c and a critical pressure p''_c, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures, and the ferromagnet–paramagnet transition in the absence of an external magnetic field.

Liquid–vapor critical point

Overview

The liquid–vapor critical point was the first critical point to be discovered, and it remains the best known and most studied one.
The figure shows the schematic P-T diagram of a pure substance. The commonly known phases solid, liquid and vapor are separated by phase boundaries, i.e. pressure–temperature combinations where two phases can coexist. At the triple point, all three phases can coexist. However, the liquid–vapor boundary terminates in an endpoint at some critical temperature ''T_c and critical pressure p''_c. This is the critical point.
The critical point of water occurs at and.
In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming even more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, all these properties change into the exact opposite: water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and mixes more readily with nonpolar gases and organic molecules.
At the critical point, only one phase exists. The heat of vaporization is zero. There is a stationary inflection point in the constant-temperature line on a PV diagram. This means that at the critical point:
Above the critical point there exists a state of matter that is continuously connected with both the liquid and the gaseous state. It is called supercritical fluid. The common textbook knowledge that all distinction between liquid and vapor disappears beyond the critical point has been challenged by Fisher and Widom, who identified a p–''T'' line that separates states with different asymptotic statistical properties. Note that even at temperatures above the critical point temperature, sufficient pressure can still potentially compress the matter into a solid.
Sometimes the critical point does not manifest in most thermodynamic or mechanical properties, but is "hidden" and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets, and a sudden enhancement in defect pair concentration.

History

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822 and named by Dmitri Mendeleev in 1860 and Thomas Andrews in 1869. Cagniard showed that CO₂ could be liquefied at 31 °C at a pressure of 73 atm, but not at a slightly higher temperature, even under pressures as high as 3000 atm.

Theory

Solving the above condition for the van der Waals equation, one can compute the critical point as
However, the van der Waals equation, based on a mean-field theory, does not hold near the critical point. In particular, it predicts wrong scaling laws.
To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties
The principle of corresponding states indicates that substances at equal reduced pressures and temperatures have equal reduced volumes. This relationship is approximately true for many substances, but becomes increasingly inaccurate for large values of p_r.
For some gases, there is an additional correction factor, called Newton's correction, added to the critical temperature and critical pressure calculated in this manner. These are empirically derived values and vary with the pressure range of interest.

Table of liquid–vapor critical temperature and pressure for selected substances

Substance	Critical temperature	Critical pressure
Argon
Ammonia
R-134a
R-410A
Bromine
Caesium
Chlorine
Ethane
Ethanol
Fluorine
Helium
Hydrogen
Krypton
Methane
Neon
Nitrogen
Oxygen
Carbon dioxide
Nitrous oxide
Sulfuric acid
Xenon
Lithium
Mercury
Sulfur
Iron
Gold
Aluminium
Water

Mixtures: liquid–liquid critical point

The liquid–liquid critical point of a solution, which occurs at the critical solution temperature, occurs at the limit of the two-phase region of the phase diagram. In other words, it is the point at which an infinitesimal change in some thermodynamic variable leads to separation of the mixture into two distinct liquid phases, as shown in the polymer–solvent phase diagram to the right. Two types of liquid–liquid critical points are the upper critical solution temperature, which is the hottest point at which cooling induces phase separation, and the lower critical solution temperature, which is the coldest point at which heating induces phase separation.

Mathematical definition

From a theoretical standpoint, the liquid–liquid critical point represents the temperature–concentration extremum of the spinodal curve. Thus, the liquid–liquid critical point in a two-component system must satisfy two conditions: the condition of the spinodal curve, and the extremum condition.