In
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
and
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ...
, an equation of state is a
thermodynamic equation relating
state variable
A state variable is one of the set of Variable (mathematics), variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behavi ...
s, 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 and eve ...
,
volume
Volume is a measure of regions in 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 quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
, or
internal energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
.
Most modern equations of state are formulated in the
Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature ( isothermal). The change in the Helmholtz ene ...
. Equations of state are useful in describing the properties of pure substances and mixtures in liquids, gases, and
solid
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
states as well as the state of matter in the interior of
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s. Though there are many equations of state, none accurately predicts properties of substances under all conditions. The quest for a universal equation of state has spanned three centuries.
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 stat ...
, 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 star
A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s, dense matter (
quark–gluon plasma
Quark–gluon plasma (QGP or quark soup) is an interacting localized assembly of quarks and gluons at Thermodynamic equilibrium#Local and global equilibrium, thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasm ...
s) 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, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion.
Thermodynamic temperature is typically expres ...
refers to the use of the
Kelvin
The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
(K), with zero being absolute zero.
*
, number of
moles of a substance
*
,
,
molar volume
In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume (''V'') occupied by a substance to the amount of substance (''n''), usually at a given temperature and pressure. It is also eq ...
, 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
Equations of state essentially begin three centuries ago with the history 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 stat ...
:
Boyle's law
Boyle's law, also referred to as the Boyle–Mariotte law or Mariotte's law (especially in France), is an empirical gas laws, gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as:
...
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, Alchemy, alchemist and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the foun ...
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.
Biog ...
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
Jacques Alexandre César Charles (12 November 1746 – 7 April 1823) was a French people, French inventor, scientist, mathematician, and balloonist.
Charles wrote almost nothing about mathematics, and most of what has been credited to him was due ...
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
Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is:
When the pressure on a sample of a dry gas is held constant, the Kelvin ...
. 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 stat ...
''. 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
In physics and chemistry, 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, volume, temperature, or internal energy. Most mo ...
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 Equation of state, 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 ...
, 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 stat ...
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
number density of the gas (number of atoms/molecules per unit volume),
is the (constant) adiabatic index (
ratio of specific heats),
is the internal energy per unit mass (the "specific internal energy"),
is the specific heat capacity at constant volume, and
is the specific heat capacity 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 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 thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
and
the
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, 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 potent ...
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