Editing Equation of state

{{Short description|Equation describing a state of matter under a given set of conditions}}
{{About||the use of this in cosmology|Equation of state (cosmology)| the use of this concept in optimal control theory|Optimal control#General method}}

In [[physics]] and [[chemistry]], an '''equation of state''' is a [[thermodynamic equations|thermodynamic equation]] relating [[state variable]]s, which describe the state of matter under a given set of physical conditions, such as [[pressure]], [[Volume (thermodynamics)|volume]], [[temperature]], or [[internal energy]].<ref name="Perrot" >{{cite book | author=Perrot, Pierre | title=A to Z of Thermodynamics | publisher=Oxford University Press | year=1998 | isbn=978-0-19-856552-9}}</ref><ref>{{Cite book |last=Eliezer |first=S. |title=An Introduction to Equation of State: Theory and Applications |last2=Gahatak |first2=A. |last3=Hora |first3=H. |author-link3=Heinrich_Hora |publisher=Cambridge University Press |year=1986 |isbn=0521303893}}</ref> Most modern equations of state are formulated in the [[Helmholtz free energy]]. Equations of state are useful in describing the properties of pure substances and mixtures in liquids, gases, and [[solid]] states as well as the state of matter in the interior of [[star]]s.<ref>{{Cite book |last=Horedt |first=G.P. |title=Polytropes. Applications in Astrophysics and Related Fields |publisher=Dordrecht: Kluwer |year=2004 |isbn=1-4020-2350-2}}</ref> Though there are many equations of state, none accurately predicts properties of substances under all conditions. The quest for a universal equation of state has spanned three centuries.<ref>{{cite journal |last1=Kontogeorgis |first1=Georgios M. |last2=Liang |first2=Xiaodong |last3=Arya |first3=Alay |last4=Tsivintzelis |first4=Ioannis |title=Equations of state in three centuries. Are we closer to arriving to a single model for all applications? |journal=Chemical Engineering Science: X |date=May 2020 |volume=7 |pages=100060 |doi=10.1016/j.cesx.2020.100060|doi-access=free }}</ref>      
{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
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==Overview==
At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions. An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the [[ideal gas law]], which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.

The general form of an equation of state may be written as
<math display="block">f(p, V, T) = 0</math>

where <math>p</math> is the pressure, <math>V</math> the volume, and <math>T</math> the temperature of the system. Yet also other variables may be used in that form. It is directly related to [[Phase rule|Gibbs phase rule]], that is, the number of independent variables depends on the number of substances and phases in the system.

An equation used to model this relationship is called an equation of state. In most cases this model will comprise some empirical parameters that are usually adjusted to measurement data. Equations of state can also describe solids, including the transition of solids from one crystalline state to another. Equations of state are also used for the modeling of the state of matter in the interior of stars, including [[neutron star]]s, dense matter ([[quark–gluon plasma]]s) and radiation fields. A related concept is the [[perfect fluid]] [[equation of state (cosmology)|equation of state used in cosmology]].

Equations of state are applied in many fields such as [[process engineering]] and petroleum industry as well as pharmaceutical industry.

Any consistent set of units may be used, although [[International System of Units|SI]] units are preferred. [[Thermodynamic temperature|Absolute temperature]] refers to the use of the [[Kelvin]] (K), with zero being absolute zero.

*<math>n</math>, number of [[mole (unit)|moles]] of a substance
*<math>V_m</math>, <math>\frac{V}{n}</math>, [[molar volume]], the volume of 1 mole of gas or liquid
*<math>R</math>, [[gas constant|ideal gas constant]] ≈ 8.3144621{{nbsp}}J/mol·K
*<math>p_c</math>, pressure at the critical point
*<math>V_c</math>, molar volume at the critical point
*<math>T_c</math>, absolute temperature at the critical point

== Historical background ==
Equations of state essentially begin three centuries ago with the history of the [[ideal gas law]]:<ref>{{cite journal |last1=Kontogeorgis |first1=Georgios M. |last2=Liang |first2=Xiaodong |last3=Arya |first3=Alay |last4=Tsivintzelis |first4=Ioannis |title=Equations of state in three centuries. Are we closer to arriving to a single model for all applications? |journal=Chemical Engineering Science: X |date=May 2020 |volume=7 |pages=100060 |doi=10.1016/j.cesx.2020.100060|doi-access=free }}</ref>

<math display="block">pV = nRT</math>

[[Boyle's law]] was one of the earliest formulation of an equation of state. In 1662, the Irish physicist and chemist [[Robert Boyle]] performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. [[Mercury (element)|Mercury]] was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:<math display="block"> pV = \mathrm{constant}.</math>The above relationship has also been attributed to [[Edme Mariotte]] and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.

In 1787 the French physicist [[Jacques Charles]] found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. This is known today as [[Charles's law]]. Later, in 1802, [[Joseph Louis Gay-Lussac]] published results of similar experiments, indicating a linear relationship between volume and temperature:<math display="block">\frac{V_1}{T_1} = \frac{V_2}{T_2}.</math>[[Dalton's law]] (1801) of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.

Mathematically, this can be represented for <math>n</math> species as:<math display="block">p_\text{total} = p_1 + p_2 + \cdots + p_n = \sum_{i=1}^n p_i.</math>In 1834, [[Émile Clapeyron]] combined Boyle's law and Charles' law into the first statement of the ''[[ideal gas law]]''. Initially, the law was formulated as ''pV<sub>m</sub>'' = ''R''(''T<sub>C</sub>'' + 267) (with temperature expressed in degrees Celsius), where ''R'' is the [[gas constant]]. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with <math>0~^{\circ}\mathrm{C} = 273.15~\mathrm{K}</math>, giving:<math display="block">pV_m = R \left(T_C + 273.15\ {}^\circ\text{C}\right).</math>In 1873, [[J. D. van der Waals]] introduced the first [[van der Waals equation|equation of state]] derived by the assumption of a finite volume occupied by the constituent molecules.<ref name="van der Waals">{{cite book |author1=van der Waals |author2=J. D. | title=On the Continuity of the Gaseous and Liquid States (doctoral dissertation) | publisher=Universiteit Leiden | year=1873}}</ref>  His new formula revolutionized the study of equations of state, and was the starting point of [[cubic equations of state]], which most famously continued via the [[Redlich–Kwong equation of state]]<ref name=":1">{{Cite journal|last1=Redlich|first1=Otto.|last2=Kwong|first2=J. N. S.|date=1949-02-01|title=On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions.|journal=Chemical Reviews|volume=44|issue=1|pages=233–244|doi=10.1021/cr60137a013|pmid=18125401|issn=0009-2665}}</ref> and the [[#Soave modification of Redlich-Kwong|Soave modification of Redlich-Kwong]].<ref name="Soave modification of Redlich-Kwong">{{cite journal |last1=Soave |first1=Giorgio |title=Equilibrium constants from a modified Redlich-Kwong equation of state |journal=Chemical Engineering Science |date=1972 |volume=27 |issue=6 |pages=1197–1203 |doi=10.1016/0009-2509(72)80096-4 |bibcode=1972ChEnS..27.1197S }}</ref>

The van der Waals equation of state can be written as

<math display="block">\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T</math>
where <math>a</math> is a parameter describing the attractive energy between particles and <math>b</math> is a parameter describing the volume of the particles.

== Ideal gas law ==

=== Classical ideal gas law ===
The classical [[ideal gas law]] may be written
<math display="block">pV = nRT.</math>

In the form shown above, the equation of state is thus
<math display="block">f(p, V, T) = pV - nRT = 0.</math>

If the [[Perfect gas|calorically perfect]] gas approximation is used, then the ideal gas law may also be expressed as follows
<math display="block">p = \rho(\gamma - 1)e</math>
where <math>\rho</math> is the [[number density]] of the gas (number of atoms/molecules per unit volume), <math>\gamma = C_p/C_v</math> is the (constant) adiabatic index ([[heat capacity ratio|ratio of specific heats]]), <math>e = C_v T</math> is the internal energy per unit mass (the "specific internal energy"), <math>C_v</math> is the specific heat capacity at constant volume, and <math>C_p</math> is the specific heat capacity at constant pressure.

=== Quantum ideal gas law ===
Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass <math>m</math> and spin <math>s</math> that takes into account quantum effects. In the following, the upper sign will always correspond to [[Fermi–Dirac statistics]] and the lower sign to [[Bose–Einstein statistics]]. The equation of state of such gases with <math>N</math> particles occupying a volume <math>V</math> with temperature <math>T</math> and pressure <math>p</math> is given by<ref>Landau, L. D., Lifshitz, E. M. (1980). Statistical physics: Part I (Vol. 5). page 162-166.</ref>

<math display="block">p= \frac{(2s+1)\sqrt{2m^3k_\text{B}^5T^5}}{3\pi^2\hbar^3}\int_0^\infty\frac{z^{3/2}\,\mathrm{d}z}{e^{z-\mu/(k_\text{B} T)}\pm 1}</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]] and <math>\mu(T,N/V)</math> the [[chemical potential]] is given by the following implicit function
<math display="block">\frac{N}{V}=\frac{(2s+1)(m k_\text{B}T)^{3/2}}{\sqrt 2\pi^2\hbar^3}\int_0^\infty\frac{z^{1/2}\,\mathrm{d}z}{e^{z-\mu / (k_\text{B} T)}\pm 1}.</math>

In the limiting case where <math>e^{\mu / (k_\text{B} T)}\ll 1</math>, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit <math>e^{\mu/(k_\text{B} T)}\ll 1</math> reduces to

<math display="block">pV = N k_\text{B} T\left[1\pm\frac{\pi^{3/2}}{2(2s+1)} \frac{N\hbar^3}{V(m k_\text{B} T)^{3/2}}+\cdots\right]</math>

With a fixed number density <math>N/V</math>, decreasing the temperature causes in [[Fermi gas]], an increase in  the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in [[Bose gas]], a decrease in pressure from its classical value implying an effective attraction. The quantum nature of this equation is in it dependence on s and '''ħ'''.

== Cubic equations of state ==
{{Main|Cubic equations of state}}

[[Cubic equations of state]] are called such because they can be rewritten as a [[cubic function]] of <math>V_m</math>. Cubic equations of state originated from the van der Waals equation of state. Hence, all cubic equations of state can be considered 'modified van der Waals equation of state'. There is a very large number of such cubic equations of state. For process engineering, cubic equations of state are today still highly relevant, e.g. the Peng Robinson equation of state or the Soave Redlich Kwong equation of state.

== Virial equations of state ==

=== Virial equation of state ===
{{Main|Virial expansion}}
<math display="block">\frac{pV_m}{RT} = A + \frac{B}{V_m} + \frac{C}{V_m^2} + \frac{D}{V_m^3} + \cdots</math>

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from [[statistical mechanics]]. This equation is also called the [[Heike Kamerlingh Onnes|Kamerlingh Onnes]] equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the [[virial coefficient|coefficients]]. ''A'' is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient ''B'' corresponds to interactions between pairs of molecules, ''C'' to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms.  The coefficients ''B'', ''C'', ''D'', etc. are functions of temperature only.

=== The BWR equation of state ===
{{Main|Benedict–Webb–Rubin equation}}
<math display="block">
\begin{align}
  p = \rho RT &+
      \left(B_0 RT - A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 +
      \left(bRT - a - \frac{d}{T}\right) \rho^3 \\[2pt] &+
      \alpha\left(a + \frac{d}{T}\right) \rho^6 +
      \frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)
\end{align}
</math>

where
*<math>p</math> is pressure
*<math>\rho</math> is molar density

Values of the various parameters can be found in reference materials.<ref>{{cite book |author=K.E. Starling |year=1973 |title=Fluid Properties for Light Petroleum Systems |publisher=[[Gulf Publishing Company]] |isbn=087201293X |lccn=70184683 |oclc=947455}}</ref> The BWR equation of state has also frequently been used for the modelling of the [[Lennard-Jones fluid]].<ref name="Stephan 112772">{{Cite journal|last1=Stephan|first1=Simon|last2=Staubach|first2=Jens|last3=Hasse|first3=Hans|date=November 2020|title=Review and comparison of equations of state for the Lennard-Jones fluid|url=https://linkinghub.elsevier.com/retrieve/pii/S0378381220303204|journal=Fluid Phase Equilibria|language=en|volume=523|pages=112772|doi=10.1016/j.fluid.2020.112772| s2cid=224844789}}</ref><ref>{{Cite journal|last1=Nicolas|first1=J.J.|last2=Gubbins|first2=K.E.|last3=Streett|first3=W.B.| last4=Tildesley|first4=D.J.| date=May 1979|title=Equation of state for the Lennard-Jones fluid|url=https://www.tandfonline.com/doi/full/10.1080/00268977900101051|journal=Molecular Physics|language=en|volume=37|issue=5|pages=1429–1454| doi=10.1080/00268977900101051|bibcode=1979MolPh..37.1429N|issn=0026-8976|url-access=subscription}}</ref> There are several extensions and modifications of the classical BWR equation of state available.

The Benedict–Webb–Rubin–Starling<ref>{{Cite book|last=Starling|first=Kenneth E.|title=Fluid Properties for Light Petroleum Systems | publisher=Gulf Publishing Company|year=1973|page=270}}</ref> equation of state is a modified BWR equation of state and can be written as 
<math display="block">\begin{align}
p = \rho RT &+ \left(B_0 RT-A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 \\[2pt]
            &+ \left(bRT-a-\frac{d}{T} + \frac{c}{T^2}\right) \rho^3 + \alpha\left(a+\frac{d}{T}\right) \rho^6
\end{align} </math>

Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.

The Lee–Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.<ref>{{Cite journal|last1=Lee|first1=Byung Ik|last2=Kesler|first2=Michael G.|date=1975|title=A generalized thermodynamic correlation based on three-parameter corresponding states|journal=AIChE Journal|language=fr|volume=21| issue=3| pages=510–527| doi=10.1002/aic.690210313|bibcode=1975AIChE..21..510L |issn=1547-5905}}</ref>

<math display="block">
p = \frac{RT}{V} \left( 1 + \frac{B}{V_r} + \frac{C}{V_r^2} + \frac{D}{V_r^5} + \frac{c_4}{T_r^3 V_r^2}  \left( \beta + \frac{\gamma}{V_r^2} \right) \exp \left( -\frac{\gamma}{V_r^2} \right) \right)
</math>

== 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>

== Multiparameter equations of state ==
Multiparameter equations of state are empirical equations of state that can be used to represent pure fluids with high accuracy. Multiparameter equations of state are empirical correlations of experimental data and are usually formulated in the Helmholtz free energy. The functional form of these models is in most parts not physically motivated. They can be usually applied in both liquid and gaseous states. Empirical multiparameter equations of state represent the Helmholtz energy of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in temperature and density:
<math display="block">\frac{a(T, \rho)}{RT} = 
  \frac{a^\mathrm{ideal\,gas}(\tau, \delta) + a^\textrm{residual}(\tau, \delta)}{RT}</math>
with <math display="block">\tau = \frac{T_r}{T}, \delta = \frac{\rho}{\rho_r}</math>

The reduced density <math>\rho_r</math> and reduced temperature <math>T_r</math> are in most cases the critical values for the pure fluid. Because integration of the multiparameter equations of state is not required and thermodynamic properties can be determined using classical thermodynamic relations, there are few restrictions as to the functional form of the ideal or residual terms.<ref name=":2">{{Cite journal|last1=Span|first1=R.|last2=Wagner|first2=W.|date=2003|title=Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids|url=http://link.springer.com/10.1023/A:1022390430888|journal=International Journal of Thermophysics|volume=24|issue=1| pages=1–39|doi=10.1023/A:1022390430888| s2cid=116961558|url-access=subscription}}</ref><ref name=":3">{{Cite journal|last1=Span|first1=Roland|last2=Lemmon|first2=Eric W.|last3=Jacobsen| first3=Richard T|last4=Wagner|first4=Wolfgang|last5=Yokozeki|first5=Akimichi|date=November 2000|title=A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa| url=http://aip.scitation.org/doi/10.1063/1.1349047|journal=Journal of Physical and Chemical Reference Data| language=en| volume=29|issue=6|pages=1361–1433|doi=10.1063/1.1349047|bibcode=2000JPCRD..29.1361S|issn=0047-2689|url-access=subscription}}</ref> Typical multiparameter equations of state use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. Multiparameter equations of state are available currently for about 50 of the most common industrial fluids including refrigerants. The IAPWS95 reference equation of state for water is also a multiparameter equations of state.<ref name=":4">{{Cite journal|last1=Wagner|first1=W.|last2=Pruß|first2=A.|date=June 2002|title=The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use|url=http://aip.scitation.org/doi/10.1063/1.1461829| journal=Journal of Physical and Chemical Reference Data|language=en|volume=31|issue=2|pages=387–535| doi=10.1063/1.1461829| issn=0047-2689|url-access=subscription}}</ref> Mixture models for multiparameter equations of state exist, as well. Yet, multiparameter equations of state applied to mixtures are known to exhibit artifacts at times.<ref>{{Cite journal|last1=Deiters|first1=Ulrich K.| last2=Bell|first2=Ian H.|date=December 2020|title=Unphysical Critical Curves of Binary Mixtures Predicted with GERG Models|journal=International Journal of Thermophysics|language=en|volume=41|issue=12|pages=169|doi=10.1007/s10765-020-02743-3| issn=0195-928X|pmc=8191392|pmid=34121788|bibcode=2020IJT....41..169D}}</ref><ref>{{Cite journal|last1=Shi|first1=Lanlan|last2=Mao|first2=Shide|date=2012-01-01|title=Applications of the IAPWS-95 formulation in fluid inclusion and mineral-fluid phase equilibria|journal=Geoscience Frontiers|language=en|volume=3|issue=1|pages=51–58|doi=10.1016/j.gsf.2011.08.002|bibcode=2012GeoFr...3...51S |issn=1674-9871|doi-access=free}}</ref>

One example of such an equation of state is the form proposed by Span and Wagner.<ref name=":2" />

<math display="block">
\begin{align}
a^\mathrm{residual} ={}& \sum_{i=1}^8 \sum_{j=-8}^{12} n_{i,j} \delta^i \tau^{j/8}
+ \sum_{i=1}^5 \sum_{j=-8}^{24} n_{i,j} \delta^i \tau^{j/8} \exp\left(-\delta\right) \\
&+ \sum_{i=1}^5 \sum_{j=16}^{56} n_{i,j} \delta^i \tau^{j/8} \exp\left(-\delta^2\right)
+ \sum_{i=2}^4 \sum_{j=24}^{38} n_{i,j} \delta^i \tau^{j/2} \exp\left(-\delta^3\right)
\end{align}
</math>

This is a somewhat simpler form that is intended to be used more in technical applications.<ref name=":2" /> Equations of state that require a higher accuracy use a more complicated form with more terms.<ref name=":4" /><ref name=":3" />

== List of further equations of state ==

=== Stiffened equation of state ===

When considering water under very high pressures, in situations such as [[Underwater explosion|underwater nuclear explosions]], [[Extracorporeal shock wave lithotripsy|sonic shock lithotripsy]], and [[sonoluminescence]], the stiffened equation of state<ref>{{Cite journal|last1=Le Métayer|first1=O|last2=Massoni|first2=J|last3=Saurel|first3=R|date=2004-03-01|title=Élaboration des lois d'état d'un liquide et de sa vapeur pour les modèles d'écoulements diphasiques|url=https://www.sciencedirect.com/science/article/pii/S1290072903001443|journal=International Journal of Thermal Sciences|language=fr|volume=43|issue=3| pages=265–276| doi=10.1016/j.ijthermalsci.2003.09.002|issn=1290-0729|url-access=subscription}}</ref> is often used:

<math display="block">p = \rho(\gamma - 1)e - \gamma p^0 \,</math>

where <math>e</math> is the internal energy per unit mass, <math>\gamma</math> is an empirically determined constant typically taken to be about 6.1, and <math>p^0</math> is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).

The equation is stated in this form because the speed of sound in water is given by <math>c^2 = \gamma\left(p + p^0\right)/\rho</math>.

Thus water behaves as though it is an ideal gas that is ''already'' under about 20,000 atmospheres (2&nbsp;GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100&nbsp;kPa to 200&nbsp;kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1&nbsp;MPa to 2000.2&nbsp;MPa).

This equation mispredicts the [[specific heat capacity]] of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.

=== Morse oscillator equation of state ===
An equation of state of Morse oscillator has been derived,<ref>{{Cite journal |last=Al-Raeei |first=Marwan |date=2022-04-01 |title=Morse oscillator equation of state: An integral equation theory based with virial expansion and compressibility terms |journal=Heliyon |volume=8 |issue=4 |pages=e09328 |doi=10.1016/j.heliyon.2022.e09328 |doi-access=free |pmid=35520603 |issn=2405-8440|pmc=9062208 |bibcode=2022Heliy...809328A }}</ref> and it has the following form:

<math display="block">p = \Gamma_1 \nu + \Gamma_2 \nu^2</math>

Where <math>\Gamma_1</math> is the first order virial parameter and it depends on the temperature, <math>\Gamma_2</math> is the second order virial parameter of Morse oscillator and it depends on the  parameters of Morse oscillator in addition to the absolute temperature. <math>\nu</math> is the fractional volume of the system.

=== Ultrarelativistic equation of state ===

An [[ultrarelativistic fluid]] has equation of state
<math display="block">p = \rho_m c_s^2</math>
where <math>p</math> is the pressure, <math>\rho_m</math> is the mass density, and <math>c_s</math> is the [[speed of sound]].

=== Ideal Bose equation of state ===

The equation of state for an ideal [[Bose gas]] is

<math display="block">p V_m =
  RT~\frac{\operatorname{Li}_{\alpha+1}(z)}{\zeta(\alpha)}
  \left(\frac{T}{T_c}\right)^\alpha
</math>

where ''α'' is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), ''z'' is exp(''μ''/''k''<sub>B</sub>''T'') where ''μ'' is the [[chemical potential]], Li is the [[polylogarithm]], ζ is the [[Riemann zeta function]], and ''T''<sub>''c''</sub> is the critical temperature at which a [[Bose–Einstein condensate]] begins to form.

=== Jones–Wilkins–Lee equation of state for explosives (JWL equation) ===

The equation of state from Jones–Wilkins–Lee is used to describe the detonation products of explosives.
<math display="block">p = A \left( 1 - \frac{\omega}{R_1 V} \right) \exp(-R_1 V) + B \left( 1 - \frac{\omega}{R_2 V} \right) \exp\left(-R_2 V\right) + \frac{\omega e_0}{V}</math>

The ratio <math> V = \rho_e / \rho </math> is defined by using <math> \rho_e </math>, which is the density of the explosive (solid part) and <math> \rho </math>, which is the density of the detonation products. The parameters <math> A </math>, <math> B </math>, <math> R_1 </math>, <math> R_2 </math> and <math> \omega </math> are given by several references.<ref name="Dobratz">{{Cite journal |author1=B. M. Dobratz |author2=P. C. Crawford | year=1985 | title=LLNL Explosives Handbook: Properties of Chemical Explosives and Explosive Simulants|journal=Ucrl-52997 |url=https://ci.nii.ac.jp/naid/10012469501/|access-date = 31 August 2018}}</ref> In addition, the initial density (solid part) <math> \rho_0 </math>, speed of detonation <math> V_D </math>, Chapman–Jouguet pressure <math> P_{CJ} </math> and the chemical energy per unit volume of the explosive <math> e_0 </math> are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results.  Typical parameters for some explosives are listed in the table below.
{| class="wikitable centre"
!  Material !! <math>\rho_e\,</math> (g/cm<sup>3</sup>) !! <math>v_D\,</math> (m/s) !! <math>p_{CJ}\,</math> (GPa) !! <math>A\,</math> (GPa) !! <math>B\,</math> (GPa) !! <math>R_1</math> !! <math>R_2</math> !! <math>\omega</math> !! <math>e_0\,</math> (GPa)
|- 
|[[Trinitrotoluene|TNT]] || 1.630 || 6930 || 21.0 || 373.8 || 3.747 || 4.15 || 0.90 || 0.35 || 6.00 
|-
|[[Composition B]] || 1.717 || 7980 || 29.5 || 524.2 || 7.678 || 4.20 || 1.10 || 0.35 || 8.50
|-
| [[Polymer-bonded explosive|PBX 9501]]<ref name="Wilkins">{{Citation|last=Wilkins|first=Mark L.|title=Computer Simulation of Dynamic Phenomena| publisher=Springer|year=1999|page=80|url=https://books.google.com/books?id=b3npCAAAQBAJ&pg=PA1|access-date = 31 August 2018 |isbn=9783662038857}}</ref> || 1.844 || || 36.3 || 852.4 || 18.02 || 4.55 || 1.3 || 0.38 || 10.2
|}

=== Others ===
* [[Tait equation]] for water and other liquids.  Several equations are referred to as the '''Tait equation'''.
* [[Murnaghan equation of state]]
* [[Birch–Murnaghan equation of state]]
* Stacey–Brennan–Irvine equation of state<ref name="StaceyBrennan1981">{{Cite journal |url= https://espace.library.uq.edu.au/view/UQ:399907 |title= Finite strain theories and comparisons with seismological data |last1= Stacey |first1=F.D. |journal= Surveys in Geophysics |volume=4 |issue=3 |pages=189–232 |doi= 10.1007/BF01449185 |access-date= 31 August 2018 |last2= Brennan |first2= B. J. |last3= Irvine |first3= R.D. |year= 1981 |bibcode= 1981GeoSu...4..189S|s2cid= 129899060 |url-access= subscription }}</ref>
* Modified Rydberg equation of state<ref>{{Cite book|title = "Equations of states and scaling rules for molecular solids under strong compression" in "Molecular systems under high pressure" ed. R. Pucci and G. Piccino |last= Holzapfel |first= W.B. |publisher= Elsevier |year= 1991 |location= North-Holland |pages= 61–68}}</ref><ref>{{Cite journal |title= Equations of state for solids under strong compression |last= Holzapfel |first= W.B. |date=1991 |journal= High Press. Res. |volume= 7 |pages= 290–293 |doi= 10.1080/08957959108245571|orig-year= 1991}}</ref><ref name="Holzapfel1996">{{Cite journal |title= Physics of solids under strong compression |last= Holzapfel |first= Wi.B. |journal= Rep. Prog. Phys. |volume=59 |pages=29–90 |doi= 10.1088/0034-4885/59/1/002 |issue=1 |year= 1996 |bibcode= 1996RPPh...59...29H |s2cid= 250909120 |issn= 0034-4885}}</ref>
* Adapted polynomial equation of state<ref name="Holzapfel1998">{{Cite journal |title= Equation of state for solids under strong compression |last= Holzapfel |first= W.B. |journal= High Press. Res. |volume=16 |issue=2 |pages=81–126 |issn=0895-7959 |doi= 10.1080/08957959808200283 |year= 1998 |bibcode= 1998HPR....16...81H}}</ref>
* [[Johnson–Holmquist damage model|Johnson–Holmquist equation of state]]
* [[Mie–Grüneisen equation of state]]<ref>{{cite book |last1=Holzapfel |first1=Wilfried B. |editor1-last=Katrusiak |editor1-first=A. |editor2-last=McMillan |editor2-first=P. |title=High-Pressure Crystallography |date=2004 |publisher=Kluver Academic |location=Dordrecht, Netherlands |doi= 10.1007/978-1-4020-2102-2_14 |isbn= 978-1-4020-1954-8 |pages=217–236 |chapter-url=https://physik.uni-paderborn.de/fileadmin/physik/Alumni/Holzapfel_LOP/ldv-238.pdf |access-date=31 August 2018 |language=en |chapter=Equations of State and Thermophysical Properties of Solids Under Pressure|volume=140 |series=NATO Science Series}}</ref><ref name="Benjelloun">S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [https://arxiv.org/abs/2105.04845 Link to Archiv e-print] [https://hal.archives-ouvertes.fr/hal-03216379/ Link to Hal e-print]</ref>
* [[Anton-Schmidt equation of state]]
* [[State-transition equation]]

== See also ==
* [[Gas laws]]
* [[Departure function]]
* [[Table of thermodynamic equations]]
* [[Real gas]]
* [[Cluster expansion]]
* [[Polytrope]]

==References==
{{Reflist}}

== External links==
{{Topics in continuum mechanics}}
{{States of matter}}
{{Statistical mechanics topics}}
{{Authority control}}

[[Category:Equations of physics]]
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[[Category:Fluid mechanics]]
[[Category:Equations of state| ]]
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