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# Critical point (thermodynamics)

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### Critical point (thermodynamics)

1. Subcritical ethane, liquid and gas phase coexist
2. Critical point (32.17 °C, 48.72 bar), opalescence
3. Supercritical ethane, fluid[1]

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent 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 the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures.

## Contents

• Liquid-vapor critical point 1
• Overview 1.1
• History 1.2
• Theory 1.3
• Table of liquid–vapor critical temperature and pressure for selected substances 1.4
• Mixtures: liquid–liquid critical point 2
• Mathematical definition 2.1
• In renormalization group theory 3
• Footnotes 5
• References 6

## Liquid-vapor critical point

### Overview

The liquid-vapor 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.

For simplicity and clarity, the generic notion of critical point is best introduced by discussing a specific example, the liquid-vapor critical point. This was historically the first critical point to be discovered, and it is still the best known and most studied one.

The figure to the right shows the schematic PT diagram of a pure substance (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). 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, even all three phases coexist. However, the liquid-vapor boundary terminates in an endpoint at some critical temperature Tc and critical pressure pc. This is the critical point.

In water, the critical point occurs at around 647 K (374 °C; 705 °F) and 22.064 MPa (3200 PSIA or 218 atm).[2]

In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming ever 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 prefers to mix with nonpolar gases and organic molecules.[3]

At the critical point, only one phase exists. The heat of vaporization is zero. There is an inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point:[4][5][6]

\left(\frac{\partial p}{\partial V}\right)_T = \left(\frac{\partial^2p}{\partial V^2}\right)_T = 0
The critical isotherm with the critical point K

Above the critical point one has a state of matter that is continuously connected with (can be transformed without phase transition into) 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[7] who identified a p,T-line that separates states with different asymptotic statistical properties (Fisher-Widom line).

### History

Carbon dioxide exuding fog while cooling from supercritical to critical temperature

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822[8] [9] and named by Dmitri Mendeleev in 1860[10] and Thomas Andrews in 1869.[11] 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 3,000 atm.

### Theory

Solving the above condition (\partial p / \partial V)_T=0 for the van der Waals equation, one can compute the critical point as

T_c = 8a/27Rb, V_c = 3nb, p_c = a/27b^2.

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[12]

T_r = T/T_c, p_r = p/p_c, V_r = \frac{V}{RT_c/p_c}.

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 pr.

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

Substance[13][14] Critical temperature Critical pressure (absolute)
Argon −122.4 °C (150.8 K) 48.1 atm (4,870 kPa)
Ammonia[15] 132.4 °C (405.5 K) 111.3 atm (11,280 kPa)
Bromine 310.8 °C (584.0 K) 102 atm (10,300 kPa)
Caesium 1,664.85 °C (1,938.00 K) 94 atm (9,500 kPa)
Chlorine 143.8 °C (417.0 K) 76.0 atm (7,700 kPa)
Ethanol 241 °C (514 K) 62.18 atm (6,300 kPa)
Fluorine −128.85 °C (144.30 K) 51.5 atm (5,220 kPa)
Helium −267.96 °C (5.19 K) 2.24 atm (227 kPa)
Hydrogen −239.95 °C (33.20 K) 12.8 atm (1,300 kPa)
Krypton −63.8 °C (209.4 K) 54.3 atm (5,500 kPa)
CH4 (methane) −82.3 °C (190.9 K) 45.79 atm (4,640 kPa)
Neon −228.75 °C (44.40 K) 27.2 atm (2,760 kPa)
Nitrogen −146.9 °C (126.2 K) 33.5 atm (3,390 kPa)
Oxygen −118.6 °C (154.6 K) 49.8 atm (5,050 kPa)
CO2 31.04 °C (304.19 K) 72.8 atm (7,380 kPa)
N2O 36.4 °C (309.5 K) 71.5 atm (7,240 kPa)
H2SO4 654 °C (927 K) 45.4 atm (4,600 kPa)
Xenon 16.6 °C (289.8 K) 57.6 atm (5,840 kPa)
Lithium 2,950 °C (3,220 K) 652 atm (66,100 kPa)
Mercury 1,476.9 °C (1,750.1 K) 1,720 atm (174,000 kPa)
Sulfur 1,040.85 °C (1,314.00 K) 207 atm (21,000 kPa)
Iron 8,227 °C (8,500 K)
Gold 6,977 °C (7,250 K) 5,000 atm (510,000 kPa)
Aluminium 7,577 °C (7,850 K)
Water[2][16] 373.946 °C (647.096 K) 217.7 atm (22.06 MPa)

## 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

• "Critical points for some common solvents". ProSciTech. Archived from the original on 2008-01-31.
• "Critical Temperature and Pressure". Department of Chemistry. Purdue University. Retrieved 2006-12-03.