View source for Physics:Theorem of corresponding states

{{Thermodynamics|cTopic=[[Physics:List of thermodynamic properties|System properties]]}}
According to van der Waals, the '''theorem of corresponding states''' (or '''principle/law of corresponding states''') indicates that all [[Physics:Fluid|fluid]]s, when compared at the same reduced temperature and reduced pressure, have approximately the same [[Physics:Compressibility factor|compressibility factor]] and all deviate from ideal gas behavior to about the same degree.<ref>{{cite book |author1=Tester, Jefferson W.  |author2=Modell, Michael  |name-list-style=amp |title=Thermodynamics and its applications |publisher=Prentice Hall |year=1997 |isbn=0-13-915356-X}}</ref><ref>{{cite book|author1=Çengel Y.A. |author2=Boles M.A. |title=Thermodynamics: An Engineering Approach|edition=Sixth|publisher=McGraw Hill|year= 2007|isbn=9780071257718}} page 141</ref>

Material constants that vary for each type of material are eliminated, in a recast reduced form of a [[Physics:Constitutive equation|constitutive equation]]. The reduced variables are defined in terms of critical variables.

The principle originated with the work of [[Biography:Johannes Diderik van der Waals|Johannes Diderik van der Waals]] in about 1873<ref>[http://digital.library.okstate.edu/oas/oas_pdf/v56/p125_132.pdf ''A Four-Parameter Corresponding States Correlation for Fluid Compressibility Factors''] {{Webarchive|url=https://web.archive.org/web/20070317215431/http://digital.library.okstate.edu/oas/oas_pdf/v56/p125_132.pdf |date=2007-03-17 }} by Walter M. Kalback and Kenneth E. Starling, Chemical Engineering Department, [[Organization:University of Oklahoma|University of Oklahoma]].</ref> when he used the [[Physics:Critical temperature|critical temperature]] and critical pressure to derive a universal property of all fluids that follow the [[Physics:Van der Waals equation|van der Waals equation]] of state. It predicts a value of <math>3/8 = 0.375</math> that is found to be an overestimate when compared to real gases.  

[[Biography:Edward A. Guggenheim|Edward A. Guggenheim]] used the phrase "Principle of Corresponding States" in an opt-cited paper to describe the phenomenon where different systems have very similar behaviors when near a [[Critical point (thermodynamics)|critical point]].<ref>{{Cite journal |last=Guggenheim |first=E. A. |date=1945-07-01 |title=The Principle of Corresponding States |url=https://pubs.aip.org/jcp/article/13/7/253/186058/The-Principle-of-Corresponding-States |journal=The Journal of Chemical Physics |language=en |volume=13 |issue=7 |pages=253–261 |doi=10.1063/1.1724033 |issn=0021-9606}}</ref> 

There are many examples of non-ideal gas models which satisfy this theorem, such as the van der Waals model, the Dieterici model, and so on, that can be found on [[Physics:Real gas|the page on real gases]]. 

==Compressibility factor at the critical point==

The compressibility factor at the critical point, which is defined as <math>Z_c=\frac{P_c v_c \mu}{R T_c}</math>, where the subscript <math>c</math> indicates physical quantities measured at the [[Critical point (thermodynamics)|critical point]], is predicted to be a constant independent of substance by many equations of state.

The table below for a selection of gases uses the following conventions:
* <math>T_c</math>: [[Physics:Critical temperature|critical temperature]] [K]
* <math>P_c</math>: critical pressure [Pa]
* <math>v_c</math>: [[Critical point (thermodynamics)|critical specific volume]] [m<sup>3</sup>⋅kg<sup>−1</sup>]
* <math>R</math>: [[Physics:Gas constant|gas constant]] (8.314 [[Joule|J]]⋅[[Kelvin|K]]<sup>−1</sup>⋅[[Mole (unit)|mol]]<sup>−1</sup>)
* <math>\mu</math>: [[Physics:Molar mass|Molar mass]] [kg⋅mol<sup>−1</sup>]
{| class="wikitable sortable" border="1"
|-
! Substance
!<math>P_c</math> [Pa]
!<math>T_c</math> [K]
!<math>v_c</math> [m<sup>3</sup>/kg]
!<math>Z_c</math>
|-
| [[Chemistry:Properties of water|H<sub>2</sub>O]]
|{{val|21.817|e=6}}
|647.3
|{{val|3.154|e=-3}}
| 0.23<ref name="Goodstein">{{cite book|last=Goodstein|first=David|title=States of Matter|edition=1st|orig-year=1975|year=1985|publisher=General Publishing Company, Ltd.|location=Toronto, Ontario, {{wipe|Canada}}|isbn=0-486-64927-X|page=[https://archive.org/details/statesofmatter0000good/page/452 452]|chapter=6|trans-chapter=Critical Phenomena and Phase Transitions|url=https://archive.org/details/statesofmatter0000good/page/452}}</ref>
|-
| [[Physics:Helium-4|<sup>4</sup>He]]
|{{val|0.226|e=6}}
|5.2
|{{val|14.43|e=-3}}
| 0.31<ref name="Goodstein"/>
|-
|[[Chemistry:Helium|He]]
|{{val|0.226|e=6}}
|5.2
|{{val|14.43|e=-3}}
| 0.30<ref name="Boer">{{cite journal| last        = de Boer| first       = J.|date=April 1948| title       = Quantum theory of condensed permanent gases I the law of corresponding states| journal     = Physica| volume      = 14| issue = 2–3| pages       = 139–148| publisher   = [[Company:Elsevier|Elsevier]]| location    = Utrecht, Netherlands| doi         =10.1016/0031-8914(48)90032-9 |bibcode = 1948Phy....14..139D }}</ref>
|-
|[[Chemistry:Hydrogen|H<sub>2</sub>]]
|{{val|1.279|e=6}}
|33.2
|{{val|32.3|e=-3}}
|0.30<ref name="Boer"/>
|-
|[[Chemistry:Neon|Ne]]
|{{val|2.73|e=6}}
|44.5
|{{val|2.066|e=-3}}
|0.29<ref name="Boer"/>
|-
|[[Chemistry:Nitrogen|N<sub>2</sub>]]
|{{val|3.354|e=6}}
|126.2
|{{val|3.2154|e=-3}}
|0.29<ref name="Boer"/>
|-
|[[Chemistry:Argon|Ar]]
|{{val|4.861|e=6}}
|150.7
|{{val|1.883|e=-3}}
|0.29<ref name="Boer"/>
|-
|[[Chemistry:Xenon|Xe]]
|{{val|5.87|e=6}}
|289.7
|{{val|0.9049|e=-3}}
|0.29
|-
|[[Chemistry:Oxygen|O<sub>2</sub>]]
|{{val|5.014|e=6}}
|154.8
|{{val|2.33|e=-3}}
|0.291
|-
|[[Chemistry:Carbon dioxide|CO<sub>2</sub>]]
|{{val|7.290|e=6}}
|304.2
|{{val|2.17|e=-3}}
|0.275
|-
|[[Chemistry:Sulfur dioxide|SO<sub>2</sub>]]
|{{val|7.88|e=6}}
|430.0
|{{val|1.900|e=-3}}
|0.275
|-
|[[Chemistry:Methane|CH<sub>4</sub>]]
|{{val|4.58|e=6}}
|190.7
|{{val|6.17|e=-3}}
|0.285
|-
|[[Chemistry:Propane|C<sub>3</sub>H<sub>8</sub>]]
|{{val|4.21|e=6}}
|370.0
|{{val|4.425|e=-3}}
|0.267
|}

==See also==
*[[Physics:Van der Waals equation|Van der Waals equation]]
*[[Physics:Equation of state|Equation of state]]
*[[Physics:Compressibility factor|Compressibility factor]]s
*[[Biography:Johannes Diderik van der Waals|Johannes Diderik van der Waals]] equation
*Noro-Frenkel law of corresponding states

==References==
{{reflist}}

==External links==
* [https://web.archive.org/web/20110206055206/http://iptibm1.ipt.ntnu.no/~jsg/undervisning/naturgass/parlaktuna/Chap3.pdf Properties of Natural Gases]. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document)
*[http://www.sklogwiki.org/SklogWiki/index.php/Law_of_corresponding_states Theorem of corresponding states] on [http://www.sklogwiki.org/ SklogWiki].

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[[Category:Laws of thermodynamics]]
[[Category:Engineering thermodynamics]]
[[Category:Continuum mechanics]]



{{Sourceattribution|Theorem of corresponding states}}