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Carbon dioxide exuding fog while cooling from supercritical to critical temperature

In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also known as a critical state, occurs under conditions (such as specific values of temperature, pressure or composition) at which no phase boundaries exist. There are multiple types of critical points, including vapor–liquid critical points and liquid–liquid critical points.

Pure substances: vapor–liquid critical point

1. Subcritical ethane, liquid and gas phase coexist
2. Critical point, opalescence
3. Supercritical ethane, fluid

The vapor–liquid critical point in a pressure–temperature phase diagram is at the high-temperature extreme of the liquid–gas phase boundary. The dotted green line shows the anomalous behavior of water.

The term critical point is sometimes used to specifically denote the vapor–liquid critical point of a material, above which distinct liquid and gas phases do not exist. As shown in the phase diagram to the right, this is the point at which the phase boundary between liquid and gas terminates. In water, the critical point occurs at around Template:Convert and 22.064 MPa (3200 PSIA or 218 atm).

As the substance approaches critical temperature, the properties of its gas and liquid phases converge, resulting in only one phase at the critical point: a homogeneous supercritical fluid. The heat of vaporization is zero at and beyond this critical point, and so no distinction exists between the two phases. On the PT diagram, the point at which critical temperature and critical pressure meet is called the critical point of the substance. Above the critical temperature, a liquid cannot be formed by an increase in pressure, even though a solid may be formed under sufficient pressure. The critical pressure is the vapor pressure at the critical temperature. The critical molar volume is the volume of one mole of material at the critical temperature and pressure.

Critical properties vary from material to material, and for many pure substances are readily available in the literature. Nonetheless, obtaining critical properties for mixtures is more challenging.

Mathematical definition

The critical isotherm with the critical point K

For pure substances, there is an inflection point in the critical isotherm (constant temperature line) on a PV diagram. This means that at the critical point:

$\left({\frac {\partial p}{\partial V}}\right)_{T}=\left({\frac {\partial ^{2}p}{\partial V^{2}}}\right)_{T}=0$

That is, the first and second partial derivatives of the pressure p with respect to the volume V are both zero, with the partial derivatives evaluated at constant temperature T. This relation can be used to evaluate two parameters for an equation of state in terms of the critical properties, such as the parameters a and b in the van der Waals equation.

Sometimes a set of reduced properties is defined in terms of the critical properties, i.e.:

$T_{r}=T/T_{c}\,$

$p_{r}=p/p_{c}\,$

$V_{r}={\frac {V}{RT_{c}/p_{c}}}\,$

$T_{c}=8a/27Rb,$

$V_{c}=3nb\,$

$P_{c}=a/27b^{2},$

where $T_{r}$ is the reduced temperature, $p_{r}$ is the reduced pressure, $V_{r}$ is the reduced volume, and $R$ is the universal gas constant.

Principle of corresponding states

Critical variables are useful for writing a varied equation of state that applies to all materials, similar to normalization. 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.

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

Substance	Critical temperature	Critical pressure (absolute)
Argon	Template:Sort	Template:Sort
Ammonia	Template:Sort	Template:Sort
Bromine	Template:Sort	Template:Sort
Caesium	Template:Sort	Template:Sort
Chlorine	Template:Sort	Template:Sort
Ethanol	Template:Sort	Template:Sort
Fluorine	Template:Sort	Template:Sort
Helium	Template:Sort	Template:Sort
Hydrogen	Template:Sort	Template:Sort
Krypton	Template:Sort	Template:Sort
CH4 (Methane)	Template:Sort	Template:Sort
Neon	Template:Sort	Template:Sort
Nitrogen	Template:Sort	Template:Sort
Oxygen	Template:Sort	Template:Sort
CO2	Template:Sort	Template:Sort
N2O	Template:Sort	Template:Sort
H2SO4	Template:Sort	Template:Sort
Xenon	Template:Sort	Template:Sort
Lithium	Template:Sort	Template:Sort
Mercury	Template:Sort	Template:Sort
Sulfur	Template:Sort	Template:Sort
Iron	Template:Sort
Gold	Template:Sort	Template:Sort
Aluminium	Template:Sort
Water	Template:Sort	Template:Sort

History

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822 and named by Thomas Andrews in 1869. He 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 3,000 atm.

Mixtures: liquid–liquid critical point

A plot of typical polymer solution phase behavior including two critical points: an LCST and a UCST.

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 (such as temperature or pressure) will lead 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 (UCST), which is the hottest point at which cooling will induce phase separation, and the lower critical solution temperature(LCST), which is the coldest point at which heating will induce phase separation.

Mathematical definition

From a theoretical standpoint, the liquid–liquid critical point represents the temperature-concentration extremum of the spinodal curve (as can be seen in the figure to the right). Thus, the liquid–liquid critical point in a two-component system must satisfy two conditions: the condition of the spinodal curve (the second derivative of the free energy with respect to concentration must equal zero), and the extremum condition (the third derivative of the free energy with respect to concentration must also equal zero or the derivative of the spinodal temperature with respect to concentration must equal zero).

In renormalization group theory

The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space. This effect is the cause of the critical opalescence that can be observed as binary fluid mixture approaches its liquid–liquid critical point.

In systems in equilibrium, the critical point is reached only by precisely tuning a control parameter. However, in some non-equilibrium systems, the critical point is an attractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as self-organized criticality.

See also

• Conformal field theory
• Critical exponents
• Critical phenomena
• Joback method, Klincewicz method, Lydersen method (Estimation of critical temperature, pressure, and volume from molecular structure)
• Lower critical solution temperature
• Percolation thresholds
• Phase transition
• Rushbrooke inequality
• Scale invariance
• Self-organized criticality
• Supercritical fluid, Supercritical drying, Supercritical water oxidation, Supercritical fluid extraction
• Tricritical point
• Triple point
• Upper critical solution temperature
• Widom scaling

Footnotes

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References

• Template:Cite web

External links

• Template:Cite web
• Template:Cite web
• Hagen Kleinert and Verena Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapur, 2001); Paperback ISBN 981-02-4658-7 (readable online here)