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:

*P*= Pressure*V*= Molar volume, the volume of 1 mole of gas or liquid*T*= Temperature (K)

*R*= Ideal Gas Constant (8.31451 J/mol·K)

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:

- The parameters should be expressible in terms of the critical properties and the acentric factor.
- The model should provide reasonable accuracy near the critical point, particularly for calculations of the Compressibility factor and liquid density.
- The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
- The equation should be applicable to all calculations of all fluid properties in natural gas processes.

K.E. Starling, *Fluid Properties for Light Petroleum Systems.* Gulf Publishing Company (1973).

When considering water under very high pressures (typical applications are underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence) the stiffened equation of state is often used:

The equation is stated in this form because the speed of sound in water is given by .

Thus water behaves as though it is an ideal gas that is *already* under about 20000 atmospheres pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres, the water behaves as an ideal gas would do when changing from 20001 to 20002 atmospheres.

This equation mispredicts the specific heat capacity of water but few alternatives are available for severely nonisentropic processes such as strong shocks.

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:

- PV = constant

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:

- V
_{1}/T_{1}= V_{2}/T_{2}

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:

- PV=R(T+273.2)