Real gas
{{#invoke:Sidebar collapsible  bodyclass = plainlist  titlestyle = paddingbottom:0.3em;borderbottom:1px solid #aaa;  title = Thermodynamics  imagestyle = display:block;margin:0.3em 0 0.4em;  image =  caption = The classical Carnot heat engine  listtitlestyle = background:#ddf;textalign:center;  expanded = Systems
 list1name = branches  list1title = Branches  list1 = Template:Startflatlist
 list2name = laws  list2title = Laws  list2 = Template:Startflatlist
 list3name = systems  list3title = Systems  list3 =
State 

Processes 
Cycles 
 list4name = sysprop  list4title = System properties
 list4 =
Functions of state 

Process functions 
 list5name = material  list5title = Material properties  list5 =
 list6name = equations  list6title = Equations  list6 = Template:Startflatlist
 Maxwell relations
 Onsager reciprocal relations
 Bridgman's equations
 Table of thermodynamic equations
 list7name = potentials  list7title = Potentials  list7 = Template:Startflatlist
 list8name = hist/cult  list8title = Template:Hlist  list8 =
History 

Philosophy 
Theories 

Key publications 
Timelines 
Template:Hlist 
 list9name = scientists  list9title = Scientists  list9 = Template:Startflatlist
 Bernoulli
 Carnot
 Clapeyron
 Clausius
 Carathéodory
 Duhem
 Gibbs
 von Helmholtz
 Joule
 Maxwell
 von Mayer
 Onsager
 Rankine
 Smeaton
 Stahl
 Thompson
 Thomson
 Waterston
 below = Book:Thermodynamics
}} Real gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behaviour of real gases, the following must be taken into account:
 compressibility effects;
 variable specific heat capacity;
 van der Waals forces;
 nonequilibrium thermodynamic effects;
 issues with molecular dissociation and elementary reactions with variable composition.
For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, realgas models have to be used near the condensation point of gases, near critical points, at very high pressures, to explain the Joule–Thomson effect and in other less usual cases.
Models
{{#invoke:mainmain}}
van der Waals model
{{#invoke:mainmain}}
Real gases are often modeled by taking into account their molar weight and molar volume
Where P is the pressure, T is the temperature, R the ideal gas constant, and V_{m} the molar volume. a and b are parameters that are determined empirically for each gas, but are sometimes estimated from their critical temperature (T_{c}) and critical pressure (P_{c}) using these relations:
Redlich–Kwong model
The Redlich–Kwong equation is another twoparameter equation that is used to model real gases. It is almost always more accurate than the van der Waals equation, and often more accurate than some equations with more than two parameters. The equation is
where a and b two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:
Berthelot and modified Berthelot model
The Berthelot equation (named after D. Berthelot^{[1]} is very rarely used,
but the modified version is somewhat more accurate
Dieterici model
This model (named after C. Dieterici^{[2]}) fell out of usage in recent years
Clausius model
The Clausius equation (named after Rudolf Clausius) is a very simple threeparameter equation used to model gases.
where
where V_{c} is critical volume.
Virial model
The Virial equation derives from a perturbative treatment of statistical mechanics.
or alternatively
where A, B, C, A′, B′, and C′ are temperature dependent constants.
Peng–Robinson model
Peng–Robinson equation of state (named after D.Y. Peng and D. B. Robinson^{[3]}) has the interesting property being useful in modeling some liquids as well as real gases.
Wohl model
The Wohl equation (named after A. Wohl^{[4]}) is formulated in terms of critical values, making it useful when real gas constants are not available.
where
Beattie–Bridgman model
^{[5]}
This equation is based on five experimentally determined constants. It is expressed as
where
This equation is known to be reasonably accurate for densities up to about 0.8 ρ_{cr}, where ρ_{cr} is the density of the substance at its critical point. The constants appearing in the above equation are available in following table when P is in KPa, v is in , T is in K and R=8.314^{[6]}
Gas  A_{0}  a  B_{0}  b  c 

Air  131.8441  0.01931  0.04611  0.001101  4.34×10^4 
Argon, Ar  130.7802  0.02328  0.03931  0.0  5.99×10^4 
Carbon Dioxide, CO_{2}  507.2836  0.07132  0.10476  0.07235  6.60×10^5 
Helium, He  2.1886  0.05984  0.01400  0.0  40 
Hydrogen, H_{2}  20.0117  0.00506  0.02096  0.04359  504 
Nitrogen, N_{2}  136.2315  0.02617  0.05046  0.00691  4.20×10^4 
Oxygen, O_{2}  151.0857  0.02562  0.04624  0.004208  4.80×10^4 
Benedict–Webb–Rubin model
{{#invoke:mainmain}}
The BWR equation, sometimes referred to as the BWRS equation,
where d is the molar density and where a, b, c, A, B, C, α, and γ are empirical constants. Note that the γ constant is a derivative of constant α and therefore almost identical to 1.
See also
References
 ↑ D. Berthelot in Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII (Paris: GauthierVillars, 1907)
 ↑ C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)
 ↑ {{#invoke:Citation/CS1citation CitationClass=journal }}
 ↑ A. Wohl "Investigation of the condition equation", Zeitschrift für Physikalische Chemie (Leipzig) 87 pp. 1–39 (1914)
 ↑ Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach 7th Edition, McGrawHill, 2010, ISBN 007352932X
 ↑ Gordan J. Van Wylen and Richard E. Sonntage, Fundamental of Classical Thermodynamics, 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3
 Dilip Kondepudi, Ilya Prigogine, Modern Thermodynamics, John Wiley & Sons, 1998, ISBN 0471973939
 Hsieh, Jui Sheng, Engineering Thermodynamics, PrenticeHall Inc., Englewood Cliffs, New Jersey 07632, 1993. ISBN 0132757028
 Stanley M. Walas, Phase Equilibria in Chemical Engineering, Butterworth Publishers, 1985. ISBN 0409951625
 M. Aznar, and A. Silva Telles, A Data Bank of Parameters for the Attractive Coefficient of the Peng–Robinson Equation of State, Braz. J. Chem. Eng. vol. 14 no. 1 São Paulo Mar. 1997, ISSN 01046632
 An introduction to thermodynamics by Y. V. C. Rao
 The correspondingstates principle and its practice: thermodynamic, transport and surface properties of fluids by Hong Wei Xiang