The mechanical state of a substance, when the pressure, temperature and volume are fixed.
Thermodynamic properties of substances are obtained
experimentally. In order to make use of this data in a general way
(varying pressure, temperature and volume independently), these
properties are usually related in an equation of state (EOS). That
is, for a thermodynamic system characterized by n independent variables,
the EOS is defined as an algebraic expression connecting n+1
state variables.
The material behavior of real systems is generally
too complex to describe over wide ranges of the state variables.
For instance, the most well known EOS, the Ideal Gas Law only holds
true for gases at relatively low pressures, since it is an expression
describing model gases made up of molecules which have no volume, and
exert no forces on one another (valid for real gases when P->0).
Equations of state are commonly classified as either volume-explicit:
V = f1(T, P, x)
or pressure-explicit:
P = f2(T, V, x)
where P= pressure, V= volume, T= absolute temperature, x = composition (for mixtures).
From the beginning, it was realized that the ideal gas law was only
a rough approximation of true behavior. Deviations from ideality were
attributed to the finite volumes occupied by the molecules, and
attractive/repulsive forces between the molecules. Both these
factors were taken into account by van der
Waals in an equation that is the basis for many modern equations
of state. A work for which he received the Nobel
Prize in 1910.
Hundreds of other equations describing PVT behavior of gases
have been proposed, oftentimes associated with famous physicists
(Rankine, Clausius, Boltzmann, Planck).
Continued studies of forces between molecules led to a statistical mechanical approach in the form of a virial
equation of state.
Important equations of state
Several equations of state are simply (empirical or arbitrary)
adjustments of already existing equations. Some versions have as many as
thirty adjustable parameters. The choice of a suitable EOS is therefore
dependent on:
- Availability of data and fitting parameters.
- Suitability of the EOS for the desired state or phase transition.
- Suitability of the EOS with respect to chemical properties.
- Applicability over the desired range of pressures, temperatures and compositions.
- Computational requirements.
sources: my three thermo-books that I always struggle with: "Smith, Van Ness, Abbott", "Stanley Walas", "Abbott, Van Ness". | 
