View source for Critical point (thermodynamics)

{{Short description|Temperature and pressure point where phase boundaries disappear}}
[[File:CriticalPointMeasurementEthane.jpg|thumb|upright=1.5|
{{ordered list
|Subcritical [[ethane]], liquid and gas phase coexist.
|Critical point (32.17&nbsp;°C, 48.72&nbsp;bar), opalescence.
|Supercritical ethane, [[fluid]].<ref>{{cite thesis |first=Sven |last=Horstmann |title=Theoretische und experimentelle Untersuchungen zum Hochdruckphasengleichgewichtsverhalten fluider Stoffgemische für die Erweiterung der PSRK-Gruppenbeitragszustandsgleichung |language=de |trans-title=Theoretical and experimental investigations of the high-pressure phase equilibrium behavior of fluid mixtures for the expansion of the [[PSRK]] group contribution equation of state |type=Ph.D. |location=Oldenburg, Germany |publisher=[[University of Oldenburg|Carl-von-Ossietzky Universität Oldenburg]] |year=2000 |isbn=3-8265-7829-5|oclc=76176158}}</ref>
}}]]

In [[thermodynamics]], a '''critical point''' (or '''critical state''') is the end point of a phase [[Equilibrium (thermodynamics)|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 cannot be liquefied by pressure alone. At the critical point, defined by a ''critical temperature'' ''T''<sub>c</sub> and a ''critical pressure'' ''p''<sub>c</sub>, [[phase (matter)|phase]] boundaries vanish. Other examples include the [[Upper critical solution temperature|liquid–liquid critical points in mixtures]], and the ferromagnet–paramagnet transition ([[Curie temperature]]) in the absence of an external magnetic field.<ref>{{Cite book|last=Stanley|first=H. Eugene|url=https://www.worldcat.org/oclc/15696711|title=Introduction to phase transitions and critical phenomena|date=1987|publisher=Oxford University Press|isbn=0-19-505316-8|location=New York|oclc=15696711}}</ref>

== Liquid–vapor critical point ==

=== Overview ===

[[File:phase-diag2.svg|thumb|upright=1.5|The liquid–vapor critical point in a pressure–temperature [[phase diagram]] is at the high-temperature extreme of the liquid–gas phase boundary. The dashed 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 vapor–liquid 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|''P-T'' diagram]] of a ''pure substance'' (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). The commonly known [[phase (matter)|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''<sub>c</sub> and ''critical pressure'' ''p''<sub>c</sub>. This is the ''critical point''.

The critical point of water occurs at {{convert|647.096|K|C F}} and {{convert|22.064|MPa|psi atm bar}}.<ref name=IAPWS95>{{cite journal |last1=Wagner |first1=W. |last2=Pruß |first2=A. |title=The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use |journal=Journal of Physical and Chemical Reference Data |date=June 2002 |volume=31 |issue=2 |page=398 |doi=10.1063/1.1461829}}</ref>

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.<ref>Anisimov, [[Jan Sengers|Sengers]], [[Anneke Levelt Sengers|Levelt Sengers]] (2004):
Near-critical behavior of aqueous systems.
Chapter 2 in
Aqueous System at Elevated Temperatures and Pressures
Palmer et al., eds.
Elsevier.</ref>

''At'' the critical point, only one phase exists. The [[heat of vaporization]] is zero. There is a [[stationary point|stationary]] [[inflection point]] in the constant-temperature line (''critical isotherm'') on a [[PV diagram]]. This means that at the critical point:<ref name=Atkins>P. Atkins and J. de Paula, Physical Chemistry, 8th ed. (W. H. Freeman 2006), p. 21.</ref><ref>K. J. Laidler and J. H. Meiser, Physical Chemistry (Benjamin/Cummings 1982), p. 27.</ref><ref>P. A. Rock, Chemical Thermodynamics (MacMillan 1969), p. 123.</ref>

: <math>\left(\frac{\partial p}{\partial V}\right)_T = 0,</math>
: <math>\left(\frac{\partial^2p}{\partial V^2}\right)_T = 0.</math>

[[File:Real Gas Isotherms.svg|thumb|upright=1.5|Isotherms of a gas. The red line is the critical isotherm, with critical point&nbsp;K. The dashed lines represent parts of isotherms which are forbidden since the gradient would be positive, giving the gas in this region a [[Compressibility#Negative compressibility|negative compressibility]].]]

''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 [[Michael Fisher|Fisher]] and [[Benjamin Widom|Widom]],<ref>{{cite journal |last1=Fisher |first1=Michael E. |last2=Widom |first2=B. |title=Decay of Correlations in Linear Systems |journal=Journal of Chemical Physics |date=1969 |volume=50 |issue=9 |page=3756 |doi=10.1063/1.1671624 |bibcode=1969JChPh..50.3756F |url=https://aip.scitation.org/doi/abs/10.1063/1.1671624 |access-date=9 January 2023}}</ref> who identified a ''p''–''T'' line that separates states with different asymptotic statistical properties ([[Fisher–Widom line]]).

Sometimes{{Ambiguous|date=February 2021}} 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.<ref>{{cite journal |last1=Das |first1=Tamoghna |last2=Ganguly |first2=Saswati |last3=Sengupta |first3=Surajit |last4=Rao |first4=Madan |title=Pre-Yield Non-Affine Fluctuations and A Hidden Critical Point in Strained Crystals |journal=Scientific Reports |date=3 June 2015 |volume=5 |issue=1 |pages=10644 |doi=10.1038/srep10644 |pmid=26039380 |pmc=4454149 |bibcode=2015NatSR...510644D |doi-access=free }}</ref>

=== History ===
[[File:Critical carbon dioxide.jpg|thumb|Critical [[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<ref>{{cite journal |author=Charles Cagniard de la Tour |date=1822 |url=https://books.google.com/books?id=rzNCAAAAcAAJ&q=Cagniard&pg=PA127 |title=Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'essence de pétrole rectifiée |trans-title=Presentation of some results obtained by the combined action of heat and compression on certain liquids, such as water, alcohol, sulfuric ether (i.e., diethyl ether), and distilled petroleum spirit |journal=Annales de Chimie et de Physique |volume=21 |pages=127–132 |language=fr}}</ref><ref>Berche, B., Henkel, M., Kenna, R (2009) Critical phenomena: 150 years since Cagniard de la Tour. Journal of Physical Studies 13 (3), pp. 3001-1–3001-4.</ref> and named by [[Dmitri Mendeleev]] in 1860<ref>Mendeleev called the critical point the "absolute temperature of boiling" ({{lang-ru|абсолютная температура кипения}}; {{lang-de|absolute Siedetemperatur}}).
* {{cite journal |last1=Менделеев |first1=Д. |title=О расширении жидкостей от нагревания выше температуры кипения |journal=Горный Журнал [Mining Journal] |date=1861 |volume=4 |pages=141–152 |trans-title=On the expansion of liquids from heating above the temperature of boiling |language=ru}}  The "absolute temperature of boiling" is defined on p. 151.  Available at [https://upload.wikimedia.org/wikipedia/commons/e/e6/%D0%93%D0%BE%D1%80%D0%BD%D1%8B%D0%B9_%D0%B6%D1%83%D1%80%D0%BD%D0%B0%D0%BB%2C_1861%2C_%E2%84%9604_%28%D0%B0%D0%BF%D1%80%D0%B5%D0%BB%D1%8C%29.pdf Wikimedia]
* German translation: {{cite journal |last1=Mendelejeff |first1=D. |title=Ueber die Ausdehnung der Flüssigkeiten beim Erwärmen über ihren Siedepunkt |journal=Annalen der Chemie und Pharmacie |date=1861 |volume=119 |pages=1–11 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.c036497486;view=1up;seq=13 |trans-title=On the expansion of fluids during heating above their boiling point |language=de |doi=10.1002/jlac.18611190102 }} The "absolute temperature of boiling" is defined on p. 11: "{{lang|de|2=Als absolute Siedetemperatur müssen wir den Punkt betrachten, bei welchem 1) die Cohäsion der Flüssigkeit = 0° ist und a<sup>2</sup> = 0, bei welcher 2) die latente Verdamfungswärme auch = 0 ist und bei welcher sich 3) die Flüssigkeit in Dampf verwandelt, unabhängig von Druck und Volum."}} (As the "absolute temperature of boiling" we must regard the point at which (1) the cohesion of the liquid equals 0° and ''a''<sup>2</sup> = 0 [where ''a''<sup>2</sup> is the coefficient of capillarity, p. 6], at which (2) the latent heat of vaporization also equals zero, and at which (3) the liquid is transformed into vapor, independently of the pressure and the volume.)
* In 1870, Mendeleev asserted, against Thomas Andrews, his priority regarding the definition of the critical point: {{cite journal |last1=Mendelejeff |first1=D. |title=Bemerkungen zu den Untersuchungen von Andrews über die Compressibilität der Kohlensäure |journal=Annalen der Physik |date=1870 |volume=141 |pages=618–626 |doi=10.1002/andp.18702171218 |url=https://babel.hathitrust.org/cgi/pt?id=wu.89048352249;view=1up;seq=648 |series=2nd series |issue=12 |bibcode=1870AnP...217..618M |trans-title=Comments on Andrews' investigations into the compressibility of carbon dioxide |language=de}}</ref><ref>Landau, Lifshitz, Theoretical Physics, Vol. V: Statistical Physics, Ch. 83 [German edition 1984].</ref> and [[Thomas Andrews (scientist)|Thomas Andrews]] in 1869.<ref>{{cite journal |author=Andrews, Thomas |date=1869 |url=http://rstl.royalsocietypublishing.org/content/159/575.full.pdf+html |title=The Bakerian lecture: On the continuity of the gaseous and liquid states of matter |journal=Philosophical Transactions of the Royal Society |location=London |volume=159 |pages=575–590 |doi=10.1098/rstl.1869.0021 |doi-access= |s2cid=96898875 }} The term "critical point" appears on page 588.</ref> Cagniard showed that CO<sub>2</sub> could be liquefied at 31&nbsp;°C at a pressure of 73&nbsp;atm, but not at a slightly higher temperature, even under pressures as high as 3000&nbsp;atm.

=== Theory ===

Solving the above condition <math>(\partial p / \partial V)_T = 0</math> for the [[van der Waals equation]], one can compute the critical point as<ref name=Atkins/> 
: <math>T_\text{c} = \frac{8a}{27Rb},
  \quad V_\text{c} = 3nb,
  \quad p_\text{c} = \frac{a}{27b^2}.</math>
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 law]]s.

To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties<ref>{{Cite book  | last1 = Cengel | first1 = Yunus A. | last2 = Boles | first2 = Michael A. | title = Thermodynamics: an engineering approach | year = 2002 | publisher = McGraw-Hill | location = Boston  | isbn = 978-0-07-121688-3 | pages =  91–93}}</ref>

: <math>T_\text{r} = \frac{T}{T_\text{c}},
  \quad p_\text{r} = \frac{p}{p_\text{c}},
  \quad V_\text{r} = \frac{V}{RT_\text{c} / p_\text{c}}.</math>

The [[theorem of corresponding states|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''<sub>r</sub>.

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.<ref>{{cite journal |title= Compressibility Chart for Hydrogen and Inert Gases |first1= Frank D. |last1= Maslan |first2= Theodore M. |last2= Littman |journal= Ind. Eng. Chem. |year= 1953 |volume= 45 |issue= 7 |pages= 1566–1568 |doi= 10.1021/ie50523a054 }}</ref>

=== Table of liquid–vapor critical temperature and pressure for selected substances ===
{{See also|Critical points of the elements (data page)}}
{| class="wikitable sortable" style="margin:1em auto; text-align: center;"
! Substance<ref>{{cite book |last= Emsley |first= John |title= The Elements |edition= Second |publisher= [[Oxford University Press]] |year= 1991 |isbn= 978-0-19-855818-7 }}</ref><ref>{{cite book |first1= Yunus A. |last1= Cengel |first2= Michael A. |last2= Boles |title= Thermodynamics: An Engineering Approach |pages= [https://archive.org/details/thermodynamicsen00ceng_0/page/824 824] |edition= Fourth |publisher= [[McGraw-Hill]] |year= 2002 |isbn= 978-0-07-238332-4 |url-access= registration |url= https://archive.org/details/thermodynamicsen00ceng_0/page/824 }}</ref>
! Critical temperature
! Critical pressure (absolute)
|-
| [[Argon]]
| {{sort|0150.8|{{convert|-122.4|C|K}}}}
| {{sort|0048.1|{{convert|48.1|atm|kPa|abbr=on}}}}
|-
| [[Ammonia]] (NH<sub>3</sub>)<ref>{{Cite web|url=http://www.engineeringtoolbox.com/ammonia-d_971.html|title=Ammonia – NH3 – Thermodynamic Properties|website=www.engineeringtoolbox.com|access-date=2017-04-07}}</ref>
| {{sort|0405.6|{{convert|132.4|C|K}}}}
| {{sort|0111.3|{{convert|111.3|atm|kPa|abbr=on}}}}
|-

| [[R-134a]]
| {{sort|0374|{{convert|101.06|C|K}}}}
| {{sort|0040|{{convert|40.06|atm|kPa|abbr=on}}}}
|-

| [[R-410A]]
| {{sort|0346|{{convert|72.8|C|K}}}}
| {{sort|0047|{{convert|47.08|atm|kPa|abbr=on}}}}
|-

| [[Bromine]]
| {{sort|0584.0|{{convert|310.8|C|K}}}}
| {{sort|0102|{{convert|102|atm|kPa|abbr=on}}}}
|-
| [[Caesium]]
| {{sort|1938.00|{{convert|1664.85|C|K}}}}
| {{sort|0094|{{convert|94|atm|kPa|abbr=on}}}}
|-
| [[Chlorine]]
| {{sort|0417.0|{{convert|143.8|C|K}}}}
| {{sort|0076.0|{{convert|76.0|atm|kPa|abbr=on}}}}
|-
| [[Ethane]] ({{chem|C<sub>2</sub>H<sub>6</sub>}})
| {{sort|0305.2|{{convert|31.17|C|K}}}}
| {{sort|0048.077|{{convert|48.077|atm|kPa|abbr=on}}}}
|-
| [[Ethanol]] ({{chem|C<sub>2</sub>H<sub>5</sub>OH}})
| {{sort|0514.0|{{convert|241|C|K}}}}
| {{sort|0062.2|{{convert|62.18|atm|kPa|abbr=on}}}}
|-
| [[Fluorine]]
| {{sort|0144.30|{{convert|-128.85|C|K}}}}
| {{sort|0051.5|{{convert|51.5|atm|kPa|abbr=on}}}}
|-
| [[Helium]]
| {{sort|0005.19|{{convert|-267.96|C|K}}}}
| {{sort|0002.24|{{convert|2.24|atm|kPa|abbr=on}}}}
|-
| [[Hydrogen]]
| {{sort|0033.20|{{convert|-239.95|C|K}}}}
| {{sort|0012.8|{{convert|12.8|atm|kPa|abbr=on}}}}
|-
| [[Krypton]]
| {{sort|0209.4|{{convert|-63.8|C|K}}}}
| {{sort|0054.3|{{convert|54.3|atm|kPa|abbr=on}}}}
|-
| [[Methane]] (CH<sub>4</sub>)
| {{sort|0190.8|{{convert|-82.3|C|K}}}}
| {{sort|0045.79|{{convert|45.79|atm|kPa|abbr=on}}}}
|-
| [[Neon]]
| {{sort|0044.40|{{convert|-228.75|C|K}}}}
| {{sort|0027.2|{{convert|27.2|atm|kPa|abbr=on}}}}
|-
| [[Nitrogen]]
| {{sort|0126.3|{{convert|-146.9|C|K}}}}
| {{sort|0033.5|{{convert|33.5|atm|kPa|abbr=on}}}}
|-
| [[Oxygen]] (O<sub>2</sub>)
| {{sort|0154.6|{{convert|-118.6|C|K}}}}
| {{sort|0049.8|{{convert|49.8|atm|kPa|abbr=on}}}}
|-
| [[Carbon dioxide]] (CO<sub>2</sub>)
| {{sort|0304.19|{{convert|31.04|C|K}}}}
| {{sort|0072.8|{{convert|72.8|atm|kPa|abbr=on}}}}
|-
| [[Nitrous oxide]] (N<sub>2</sub>O)
| {{sort|0304.19|{{convert|36.4|C|K}}}}
| {{sort|0072.8|{{convert|71.5|atm|kPa|abbr=on}}}}
|-
| [[Sulfuric acid]] (H<sub>2</sub>SO<sub>4</sub>)
| {{sort|0927|{{convert|654|C|K}}}}
| {{sort|0045.4|{{convert|45.4|atm|kPa|abbr=on}}}}
|-
| [[Xenon]]
| {{sort|0289.8|{{convert|16.6|C|K}}}}
| {{sort|0057.6|{{convert|57.6|atm|kPa|abbr=on}}}}
|-
| [[Lithium]]
| {{sort|3223|{{convert|2950|C|K}}}}
| {{sort|0652|{{convert|652|atm|kPa|abbr=on}}}}
|-
| [[Mercury (element)|Mercury]]
| {{sort|1750.1|{{convert|1476.9|C|K}}}}
| {{sort|1720|{{convert|1720|atm|kPa|abbr=on}}}}
|-
| [[Sulfur]]
| {{sort|1314.00|{{convert|1040.85|C|K}}}}
| {{sort|0207|{{convert|207|atm|kPa|abbr=on}}}}
|-
| [[Iron]]
| {{sort|8500|{{convert|8227|C|K}}}}
|
|-
| [[Gold]]
| {{sort|7250|{{convert|6977|C|K}}}}
| {{sort|5000|{{convert|5000|atm|kPa|abbr=on}}}}
|-
| [[Aluminium]]
| {{sort|7850|{{convert|7577|C|K}}}}
|
|-
| [[Water]] (H<sub>2</sub>O)<ref name="IAPWS95" /><ref>{{cite web | title = Critical Temperature and Pressure | publisher = Purdue University | url = http://www.chem.purdue.edu/gchelp/liquids/critical.html | access-date = 2006-12-19 }}</ref>
| {{sort|0647.096|{{convert|373.946|C|K}}}}
| {{sort|0217.7|{{convert|217.7|atm|kPa|abbr=on}}}}
|}

== Mixtures: liquid–liquid critical point ==

[[File:LCST-UCST plot.svg|thumb|upright=1.5|A plot of typical polymer solution phase behavior including two critical points: a [[LCST]] and an [[Upper critical solution temperature|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) 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]] (UCST), which is the hottest point at which cooling induces phase separation, and the [[lower critical solution temperature]] (LCST), 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 (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 [[Gibbs free energy|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).

== See also ==

{{div col}}
* [[Conformal field theory]]
* [[Critical exponent]]
* [[Critical phenomena]] (more advanced article)
* [[Critical points of the elements (data page)]]
* [[Curie point]]
* [[Joback method]], [[Klincewicz method]], [[Lydersen method]] (estimation of critical temperature, pressure, and volume from molecular structure)
* [[Liquid–liquid critical point]]
* [[Lower critical solution temperature]]
* [[Néel point]]
* [[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]]
{{div col end}}

== References ==
{{Reflist|38em}}

== Further reading ==
*{{cite web | title = Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam | publisher = International Association for the Properties of Water and Steam | date = August 2007 | url = http://www.iapws.org/relguide/IF97-Rev.pdf | access-date = 2009-06-09 }}

* {{cite web |title=Critical points for some common solvents |url=http://www.proscitech.com.au/catalogue/notes/cpd.htm |archive-url=https://web.archive.org/web/20080131081956/http://www.proscitech.com.au/catalogue/notes/cpd.htm |publisher=ProSciTech |archive-date=2008-01-31}}
*{{cite web | title = Critical Temperature and Pressure | work = Department of Chemistry
| publisher = Purdue University | url = http://www.chem.purdue.edu/gchelp/liquids/critical.html | access-date = 2006-12-03 }}

{{Phase of matter}}
{{Authority control}}

[[Category:Conformal field theory]]
[[Category:Critical phenomena]]
[[Category:Phase transitions]]
[[Category:Renormalization group]]
[[Category:Threshold temperatures]]
[[Category:Gases]]