[4] At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions.
An example of an equation of state correlates densities of gases and liquids to temperatures and pressures, known as the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures.
This equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid.
In most cases this model will comprise some empirical parameters that are usually adjusted to measurement data.
Equations of state essentially begin three centuries ago with the history of the ideal gas law:[5]
In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end.
Through these experiments, Boyle noted that the gas volume varied inversely with the pressure.
In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval.
Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
Initially, the law was formulated as pVm = R(TC + 267) (with temperature expressed in degrees Celsius), where R is the gas constant.
However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with
In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules.
Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass
, decreasing the temperature causes in Fermi gas, an increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in Bose gas, a decrease in pressure from its classical value implying an effective attraction.
If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients.
A is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases.
[15][16][17][18][19][20][21][22] Most of those are formulated in the Helmholtz free energy as a function of temperature, density (and for mixtures additionally the composition).
Hence, physically based equations of state model the effect of molecular size, attraction and shape as well as hydrogen bonding and polar interactions of fluids.
Perturbation theory is frequently used for modelling dispersive interactions in an equation of state.
There is a large number of perturbation theory based equations of state available today,[23][24] e.g. for the classical Lennard-Jones fluid.
hydrogen bonding) in fluids, which can also be applied for modelling chain formation (in the limit of infinite association strength).
The SAFT equation of state was developed using statistical mechanical methods (in particular the perturbation theory of Wertheim[28]) to describe the interactions between molecules in a system.
[20][29][30] Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.[29][31][32] Multiparameter equations of state are empirical equations of state that can be used to represent pure fluids with high accuracy.
Multiparameter equations of state are empirical correlations of experimental data and are usually formulated in the Helmholtz free energy.
Empirical multiparameter equations of state represent the Helmholtz energy of the fluid as the sum of ideal gas and residual terms.
Multiparameter equations of state are available currently for about 50 of the most common industrial fluids including refrigerants.
Yet, multiparameter equations of state applied to mixtures are known to exhibit artifacts at times.
[35][34] When considering water under very high pressures, in situations such as underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence, the stiffened equation of state[38] is often used:
This equation mispredicts the specific heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.
where α is an exponent specific to the system (e.g. in the absence of a potential field, α = 3/2), z is exp(μ/kBT) where μ is the chemical potential, Li is the polylogarithm, ζ is the Riemann zeta function, and Tc is the critical temperature at which a Bose–Einstein condensate begins to form.