Editing Equation of state (section)

== 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.