Critical point (thermodynamics)

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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 higher temperatures, the gas 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.

Video Critical point (thermodynamics)

Liquid-vapor critical point

Overview

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 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, 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.

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

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.

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 (critical isotherm) on a PV diagram. This means that at the critical point:

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

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

Above the critical point there exists 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 who identified a p,T-line that separates states with different asymptotic statistical properties (Fisher-Widom line).

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 3,000 atm.

Theory

Solving the above condition ${\displaystyle (\partial p/\partial V)_{T}=0}$ for the van der Waals equation, one can compute the critical point as

${\displaystyle T_{c}=8a/(27Rb)}$, ${\displaystyle V_{c}=3nb}$, ${\displaystyle p_{c}=a/(27b^{2})}$.

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

${\displaystyle T_{r}=T/T_{c}}$, ${\displaystyle p_{r}=p/p_{c}}$, ${\displaystyle V_{r}={\frac {V}{RT_{c}/p_{c}}}}$.

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 p_r.

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

Maps Critical point (thermodynamics)

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 (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).

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See also

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Footnotes

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References

• "Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam" (PDF). International Association for the Properties of Water and Steam. August 2007. Retrieved 2009-06-09. 

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External links

• "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. 

Source of article : Wikipedia