Editing Equation of state (section)

== 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>