Editing Equation of state (section)

== 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}}</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}}</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}}</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" />