Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.
Overview
The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of the simplest equations of state for this purpose is the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of more accurate equations of state have been developed for gases and liquids. At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions.
Equation of state (cosmology)
In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number 
, equal to the ratio of its pressure 
to its energy density 
:
It is closely related to the thermodynamic equation of state and ideal gas law.
The equation
The perfect gas equation of state may be written as
where 
is the mass density, 
is the particular gas constant, 
is the temperature and 
is a characteristic thermal speed of the molecules. Thus
where 
and 
for a "cold" gas, 
= speed of light.
FLRW equations and the equation of state
The equation of state may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of an isotropic universe filled with a perfect fluid. If 
is the scale factor then
If the fluid is the dominant form of matter in a flat universe, then
where 
is the proper time.
In general the Friedmann acceleration equation is
where 
is the cosmological constant and 
is Newton's constant, and 
is the second proper time derivative of the scale factor.
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