What Is an Equation of State?

In thermodynamics, an equation of state (EOS) is the mathematical expression that describes the interconnection between state variables — generally macroscopically observable and measurable properties — for a particular state. That state may be solid, liquid, gas or plasma. Observables or properties used in an equation of state may be varied by the theorist, but generally they completely describe the state. For example, the equation of state for “n” moles of an ideal gas can be completely described using the equation PV=nRT, where P=pressure, V=volume, R=the ideal gas constant and T=temperature. Note that an EOS is intended to describe not more than one state, whether that state is solid, liquid or gas.

So that an equation of state can more closely approximate real behavior, parameters such as the three listed above are modified by additional empirical — experimental — and even computational terms. Among these terms are atomic volume, which subtracts from total volume, and intermolecular force, which affects the distance between particles. Even these adjustments may not suffice. To reconcile the equation with the measured data it is intended to explain, virial mathematical terms and iterative computational methods may be required. Such terms obscure intellectual interpretation, but they do improve practical application.

An acceptable equation of state can be difficult to derive for liquid systems, because they experience a much greater degree of molecular interaction resulting from molecules being much closer together than for gases. Liquids are categorized based upon the magnitude of such interactions as either non-associating or as associating. Most London dispersion forces are quite weak and if they are the only intermolecular forces present, the liquid — perhaps an oil or other hydrocarbon — is non-associating. If, however, the joining of molecules is stronger, as it is for hydrogen-bonded molecules, the liquid is associating. The stronger the forces, the more complex the mathematical modeling and corresponding equation of state.

For the development of an acceptable equation, associating liquids may be considered to more closely resemble solids than non-associating liquids. Some scientists use a model incorporating a two-dimensional lattice, suggesting associating liquids possess at least some solid characteristics. A lattice that is two-dimensional rather than three-dimensional indicates the solid behavior component is limited. Since some of the particles are not considered to be part of the lattice, the name assigned to this model for fluids — whether gas or liquid — is “lattice-gas” theory. The mathematics of lattice-gas liquid equations of state can become counter-intuitive and complex, as is well illustrated by polymer-in-solvent systems.