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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 interest 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:
The ideal gas law may also be expressed as follows
where is the density, the adiabatic index, and e the internal energy. This form is purely in terms of intensive quantities and is useful when simulating the Euler equations because it expresses the relationship between internal energy and other forms of energy (such as kinetic), thus allowing simulations to obey the First Law.
Where a, b and R are constants that depend on the specific material. They can be calculated from the critical properties as:
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation a is called the attraction parameter and b the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.
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.