In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, and
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, an equation of state is a
thermodynamic equation relating
state variables, which describe the state of matter under a given set of physical conditions, such as
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
,
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
,
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, or
internal energy.
Most modern equations of state are formulated in the Helmholtz free energy. Equations of state are useful in describing the properties of pure substances and mixtures in liquids, gases, and
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...
states as well as the state of matter in the interior of
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s.
Overview
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
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
, 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.
The general form of an equation of state may be written as
where
is the pressure,
the volume, and
the temperature of the system. Yet also other variables may be used in that form. It is directly related to
Gibbs phase rule, that is, the number of independent variables depends on the number of substances and phases in the system.
An equation used to model this relationship is called an equation of state. In most cases this model will comprise some empirical parameters that are usually adjusted to measurement data. Equations of state can also describe solids, including the transition of solids from one crystalline state to another. Equations of state are also used for the modeling of the state of matter in the interior of stars, including
neutron stars, dense matter (
quark–gluon plasmas) and radiation fields. A related concept is the
perfect fluid equation of state used in cosmology.
Equations of state are applied in many fields such as
process engineering and petroleum industry as well as pharmaceutical industry.
Any consistent set of units may be used, although
SI units are preferred.
Absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
refers to the use of the
Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
(K), with zero being absolute zero.
*
, number of
moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
*The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
* Abraham Moles, French engin ...
of a substance
*
,
,
molar volume, the volume of 1 mole of gas or liquid
*
,
ideal gas constant ≈ 8.3144621J/mol·K
*
, pressure at the critical point
*
, molar volume at the critical point
*
, absolute temperature at the critical point
Historical background
Boyle's law was one of the earliest formulation of an equation of state. In 1662, the Irish physicist and chemist
Robert Boyle
Robert Boyle (; 25 January 1627 – 31 December 1691) was an Anglo-Irish natural philosopher, chemist, physicist, alchemist and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the founders ...
performed a series of experiments employing a J-shaped glass tube, which was sealed on one end.
Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:
The above relationship has also been attributed to
Edme Mariotte
Edme Mariotte (; ; c. 162012 May 1684) was a French physicist and priest (abbé). He is particularly well known for formulating Boyle's law independently of Robert Boyle. Mariotte is also credited with designing the first Newton's cradle.
Biogr ...
and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
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. This is known today as
Charles's law. Later, in 1802,
Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
Dalton's law (1801) of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.
Mathematically, this can be represented for
species as:
In 1834,
Émile Clapeyron combined Boyle's law and Charles' law into the first statement of the ''
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
''. Initially, the law was formulated as ''pV
m'' = ''R''(''T
C'' + 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
, giving:
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.
His new formula revolutionized the study of equations of state, and was the starting point of
cubic equations of state Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume.
Equations of state are generally ...
, which most famously continued via the
Redlich–Kwong equation of state and the
Soave modification of Redlich-Kwong.
The van der Waals equation of state can be written as
:
where
is a parameter describing the attractive energy between particles and
is a parameter describing the volume of the particles.
Ideal gas law
Classical ideal gas law
The classical
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
may be written
In the form shown above, the equation of state is thus
If the
calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows
where
is the density,
is the (constant) adiabatic index (
ratio of specific heats),
is the internal energy per unit mass (the "specific internal energy"),
is the constant specific heat at constant volume, and
is the constant specific heat at constant pressure.
Quantum ideal gas law
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
and spin
that takes into account of quantum effects. In the following, the upper sign will always correspond to
Fermi–Dirac statistics and the lower sign to
Bose–Einstein statistics. The equation of state of such gases with
particles occupying a volume
with temperature
and pressure
is given by
where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
and
the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
is given by the following implicit function
In the limiting case where
, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit
reduces to