Editing Equation of state (section)

== Physically based equations of state ==
There is a large number of physically based equations of state available today.<ref>{{Cite journal| last1=Kontogeorgis|first1=Georgios M.|last2=Michelsen|first2=Michael L.|last3=Folas|first3=Georgios K.|last4=Derawi|first4=Samer| last5=von Solms|first5=Nicolas|last6=Stenby|first6=Erling H.|date=2006-07-01|title=Ten Years with the CPA (Cubic-Plus-Association) Equation of State. Part 1. Pure Compounds and Self-Associating Systems|url=https://pubs.acs.org/doi/10.1021/ie051305v|journal=Industrial & Engineering Chemistry Research|language=en|volume=45|issue=14|pages=4855–4868| doi=10.1021/ie051305v|issn=0888-5885|url-access=subscription}}</ref><ref>{{Cite journal|last1=Kontogeorgis|first1=Georgios M.|last2=Voutsas| first2=Epaminondas C.| last3=Yakoumis|first3=Iakovos V.|last4=Tassios|first4=Dimitrios P.|date=1996-01-01|title=An Equation of State for Associating Fluids|url=https://pubs.acs.org/doi/10.1021/ie9600203|journal=Industrial & Engineering Chemistry Research| language=en|volume=35|issue=11|pages=4310–4318|doi=10.1021/ie9600203|issn=0888-5885|url-access=subscription}}</ref><ref>{{Cite journal| last1=Cotterman| first1=R. L.|author2-link=John Prausnitz|last2=Prausnitz|first2=J. M.|date=November 1986|title=Molecular thermodynamics for fluids at low and high densities. Part II: Phase equilibria for mixtures containing components with large differences in molecular size or potential energy| url=https://onlinelibrary.wiley.com/doi/10.1002/aic.690321105|journal=AIChE Journal|language=en|volume=32|issue=11| pages=1799–1812|doi=10.1002/aic.690321105| bibcode=1986AIChE..32.1799C| s2cid=96417239|issn=0001-1541|url-access=subscription}}</ref><ref>{{Cite journal|last1=Weingerl|first1=Ulrike| last2=Wendland| first2=Martin|last3=Fischer|first3=Johann|last4=Müller|first4=Andreas|last5=Winkelmann|first5=Jochen|date=March 2001| title=Backone family of equations of state: 2. Nonpolar and polar fluid mixtures|url=https://onlinelibrary.wiley.com/doi/10.1002/aic.690470317|journal=AIChE Journal|language=en|volume=47|issue=3|pages=705–717|doi=10.1002/aic.690470317|bibcode=2001AIChE..47..705W |url-access=subscription}}</ref><ref>{{Cite journal|last1=Müller|first1=Andreas|last2=Winkelmann|first2=Jochen|last3=Fischer|first3=Johann|date=April 1996|title=Backone family of equations of state: 1. Nonpolar and polar pure fluids|url=https://onlinelibrary.wiley.com/doi/10.1002/aic.690420423| journal=AIChE Journal|language=en|volume=42|issue=4|pages=1116–1126|doi=10.1002/aic.690420423|bibcode=1996AIChE..42.1116M |issn=0001-1541|url-access=subscription}}</ref><ref name="ChapmanGubbins1989" /><ref>{{Cite journal|last1=Gross|first1=Joachim|last2=Sadowski|first2=Gabriele| year=2002| title=Application of the Perturbed-Chain SAFT Equation of State to Associating Systems|journal=Industrial & Engineering Chemistry Research|volume=41|issue=22|pages=5510–5515|doi=10.1021/ie010954d}}</ref><ref>{{Cite journal|last1=Saajanlehto|first1=Meri| last2=Uusi-Kyyny|first2=Petri|last3=Alopaeus|first3=Ville|year=2014|title=A modified continuous flow apparatus for gas solubility measurements at high pressure and temperature with camera system|journal=Fluid Phase Equilibria|volume=382|pages=150–157| doi=10.1016/j.fluid.2014.08.035}}</ref> Most of those are formulated in the [[Helmholtz free energy]] as a function of temperature, density (and for mixtures additionally the composition). The Helmholtz energy is formulated as a sum of multiple terms modelling different types of molecular interaction or molecular structures, e.g. the formation of chains or [[Dipole-dipole|dipolar interactions]]. Hence, physically based equations of state model the effect of molecular size, attraction and shape as well as hydrogen bonding and polar interactions of fluids. In general, physically based equations of state give more accurate results than traditional cubic equations of state, especially for systems containing liquids or solids. Most physically based equations of state are built on monomer term describing the [[Lennard-Jones potential|Lennard-Jones fluid]]  or the [[Mie potential|Mie fluid]].

=== Perturbation theory-based models ===
[[Perturbation theory]] is frequently used for modelling dispersive interactions in an equation of state. There is a large number of perturbation theory based equations of state available today,<ref>{{Cite journal|last1=Betancourt-Cárdenas|first1=F.F.| last2=Galicia-Luna|first2=L.A.|last3=Sandler|first3=S.I.|date=March 2008|title=Equation of state for the Lennard–Jones fluid based on the perturbation theory|url=https://linkinghub.elsevier.com/retrieve/pii/S0378381207007030|journal=Fluid Phase Equilibria| language=en|volume=264|issue=1–2|pages=174–183|doi=10.1016/j.fluid.2007.11.015|url-access=subscription}}</ref><ref>{{Cite journal| last1=Levesque|first1=Dominique|last2=Verlet|first2=Loup|date=1969-06-05|title=Perturbation Theory and Equation of State for Fluids | url=https://link.aps.org/doi/10.1103/PhysRev.182.307|journal=Physical Review| language=en|volume=182| issue=1| pages=307–316|doi=10.1103/PhysRev.182.307|bibcode=1969PhRv..182..307L|issn=0031-899X|url-access=subscription}}</ref> e.g. for the classical Lennard-Jones fluid.<ref name="Stephan 112772"/><ref>{{Cite journal |last1=Lenhard |first1=Johannes |last2=Stephan |first2=Simon |last3=Hasse |first3=Hans |date=February 2024 |title=A child of prediction. On the History, Ontology, and Computation of the Lennard-Jonesium |url=https://linkinghub.elsevier.com/retrieve/pii/S0039368123001668 |journal=Studies in History and Philosophy of Science |language=en |volume=103 |pages=105–113 |doi=10.1016/j.shpsa.2023.11.007|url-access=subscription }}</ref> The two most important theories used for these types of equations of state are the Barker-Henderson perturbation theory<ref>{{Cite journal|last1=Barker|first1=J. A.|last2=Henderson|first2=D.|date=December 1967| title=Perturbation Theory and Equation of State for Fluids. II. A Successful Theory of Liquids|url=http://aip.scitation.org/doi/10.1063/1.1701689|journal=The Journal of Chemical Physics|language=en|volume=47|issue=11|pages=4714–4721| doi=10.1063/1.1701689|bibcode=1967JChPh..47.4714B|issn=0021-9606|url-access=subscription}}</ref> and the Weeks–Chandler–Andersen perturbation theory.<ref>{{Cite journal|last1=Weeks|first1=John D.|last2=Chandler|first2=David|last3=Andersen|first3=Hans C.| date=1971-06-15| title=Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids|url=http://aip.scitation.org/doi/10.1063/1.1674820| journal=The Journal of Chemical Physics|language=en|volume=54| issue=12|pages=5237–5247| doi=10.1063/1.1674820|bibcode=1971JChPh..54.5237W|issn=0021-9606|url-access=subscription}}</ref>

=== Statistical associating fluid theory (SAFT) ===
An important contribution for physically based equations of state is the [[statistical associating fluid theory]] (SAFT) that contributes the [[Helmholtz free energy|Helmholtz energy]] that describes the association (a.k.a. [[hydrogen bond]]ing) in fluids, which can also be applied for modelling chain formation (in the limit of infinite association strength). The SAFT equation of state was developed using [[Statistical mechanics|statistical mechanical]] methods (in particular the [[perturbation theory]] of Wertheim<ref>{{Cite journal|last=Wertheim|first=M. S.|date=April 1984|title=Fluids with highly directional attractive forces. I. Statistical thermodynamics|url=http://dx.doi.org/10.1007/bf01017362|journal=Journal of Statistical Physics|volume=35| issue=1–2|pages=19–34|doi=10.1007/bf01017362|bibcode=1984JSP....35...19W|s2cid=121383911|issn=0022-4715|url-access=subscription}}</ref>) to describe the interactions between molecules in a system.<ref name="ChapmanGubbins1989">{{cite journal|last1=Chapman|first1=Walter G.|last2=Gubbins|first2=K.E.|last3=Jackson|first3=G.| last4=Radosz|first4=M.| title=SAFT: Equation-of-state solution model for associating fluids|journal=Fluid Phase Equilibria|date=1 December 1989|volume=52|pages=31–38|doi=10.1016/0378-3812(89)80308-5|s2cid=53310898 | language=en|issn=0378-3812}}</ref><ref name="Chapman1988">{{cite journal|last1=Chapman|first1=Walter G.|date=1988| title=Theory and Simulation of Associating Liquid Mixtures|journal=Doctoral Dissertation, Cornell University|language=en}}</ref><ref name="ChapmanGubbins1988">{{cite journal|last1=Chapman|first1=Walter G.|last2=Jackson|first2=G.| last3=Gubbins| first3=K.E.| date=11 July 1988|title=Phase equilibria of associating fluids: Chain molecules with multiple bonding sites| journal=Molecular Physics|language=en|volume=65|pages=1057–1079|doi=10.1080/00268978800101601}}</ref> The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989.<ref name="ChapmanGubbins1989" /><ref name="Chapman1988" /><ref name="ChapmanGubbins1988" />  Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.<ref name="Chapman1988" /><ref name="ChapmanGubbins1990">{{cite journal| last1=Chapman|first1=Walter G.|last2=Gubbins|first2=K.E.|last3=Jackson|first3=G.|last4=Radosz|first4=M.|title=New Reference Equation of State for Associating Liquids|journal=Ind. Eng. Chem. Res.|date=1 August 1990|volume=29|issue=8| pages=1709–1721| doi=10.1021/ie00104a021|language=en }}</ref><ref>{{Cite journal | doi=10.1063/1.473101|title = Statistical associating fluid theory for chain molecules with attractive potentials of variable range| journal=The Journal of Chemical Physics| volume=106| issue=10| pages=4168–4186|year = 1997|last1 = Gil-Villegas|first1 = Alejandro| last2=Galindo| first2=Amparo| last3=Whitehead| first3=Paul J.| last4=Mills| first4=Stuart J.| last5=Jackson| first5=George| last6=Burgess| first6=Andrew N.|bibcode = 1997JChPh.106.4168G}}</ref>