Equation of state
In physics, equations of state attempt to describe the relationship between temperature, pressure, and volume for a given substance or mixture of substances. The ideal gas law, shown below, is one of the simplest equations of state. Although reasonably accurate for gases at low pressures and high temperatures, it becomes increasingly inaccurate at higher pressures and lower temperatures.
Despite its shortcomings, the ideal gas law is used extensively in many fields of science and engineering. Due to its simple form, straightforward solutions to a number of problems involving the equation of state can be obtained if the system of interrest can be assumed to behave as an ideal gas. The solutions become much more complicated and difficult to use for the cases where more accurate (and complicated) equations of state must be used.
Using statistical mechanics, the ideal gas law can be derived by assuming that a gas is composed of a large number of small molecules, with no attractive or repulsive forces. In reality gas molecules do interact with attractive and repulsive forces. In fact it is these forces that result in the formation of liquids.
A major weakness of the ideal gas law is its failure to predict the formation of liquid. Most other equations of state do predict the formation of a liquid phase. Usually these equations are cubic in volume and when solved will have either one or three real roots. When there is one real root, there is no liquid phase and the solution corresponds to the volume of the gas phase. When three real roots exist, one solution corresponds to the gas phase and one to the liquid phase. The intermediate root is an artefact and has no real meaning.
In the following equations the variables are defined as follows, any consistent set of units can be used although SI units are preferred:
Examples of Equations of State
Ideal Gas Law
Van der Waals equation
Where a, b and R are constants that depend on the specific material. They can be calculated from the critical properties as:
The Virial Equation
Although usually not the most convenient equation of state, the Virial Equation is important because it can be derived directly from statistical mechanics. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. In this case B corresponds to interactions between pairs of molecules, C to triplets, and so on.
The Redlich-Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure is less than about one-half of the ratio of the temperature to the critical temperature.
for hydrogen:
The Peng-Robinson Equation was developed in 1976 in order to satisfy the following goals:
K.E. Starling, Fluid Properties for Light Petroleum Systems. Gulf Publishing Company (1973).
Boyle's Law was perhaps the first expression of an equation of state. In 1662 Robert Boyle, an Irishman, performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. 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 carefully 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:
In 1787 the French physist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 degree interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
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=R(T+267) (with temperature expressed in degrees celsius). However, later work revealed that the number should actually be 273.2, giving:
Redlich-Kwong Equation of State
Introduced in 1949 the Redlich-Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor-liquid equilibria. Although, it can be used in conjunction with separate liquid-phase correlations for this purpose.
The Soave Equation
Where ω is the acentric factor for the species.
In 1972 Soave replaced the a/√(T) term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor. The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
The Peng-Robinson Equation of State
Where ω is the acentric factor for the species.
For the most part the Peng-Robinson Equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones.The BWRS Equation of State
Values of the various parameters for 15 substances can be found in: 
History
Boyle's law (1662)
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.Charles' law (1787)
Dalton's law of partial pressures (1801)
The Ideal gas law (1834)
van der Waals Equation of State (1873)
