Equation of state
Encyclopedia
In physics
and thermodynamics
, an equation of state is a relation between state variable
s. 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 function
s associated with the matter, such as its temperature
, pressure
, volume, or internal energy
. Equations of state are useful in describing the properties of fluid
s, mixtures of fluids, solid
s, and even the interior of star
s.
, which is roughly accurate for 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 much 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.
In addition to predicting the behavior of gases and liquids, there are also equations of state for predicting the volume of solid
s, including the transition of solids from one crystalline state to another. There are equations that model the interior of star
s, including neutron star
s. A related concept is the perfect fluid
equation of state used in cosmology
...
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 carefully 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
and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to 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:
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 n species as:
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, an equation of state is a relation between state variable
State function
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...
s. More specifically, an equation of state is a thermodynamic equation
Thermodynamic equations
Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process...
describing the state of matter under a given set of physical conditions. It is a constitutive equation
Constitutive equation
In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...
which provides a mathematical relationship between two or more state function
State function
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...
s associated with the matter, such as its temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, volume, or internal energy
Internal energy
In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal...
. Equations of state are useful in describing the properties of fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s, mixtures of fluids, solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s, and even the interior of star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s.
Overview
The most prominent use of an equation of state is to predict the state of gases and liquids. One of the simplest equations of state for this purpose is the ideal gas lawIdeal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
, which is roughly accurate for 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 much 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.
In addition to predicting the behavior of gases and liquids, there are also equations of state for predicting the volume of solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s, including the transition of solids from one crystalline state to another. There are equations that model the interior of star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s, including neutron star
Neutron star
A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...
s. A related concept is the perfect fluid
Perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p....
equation of state used in cosmology
Equation of state (cosmology)
In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number \! w, equal to the ratio of its pressure \! p to its energy density \! \rho: \! w=p/\rho...
...
Boyle's law (1662)
Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and chemist Robert BoyleRobert Boyle
Robert Boyle FRS was a 17th century natural philosopher, chemist, physicist, and inventor, also noted for his writings in theology. He has been variously described as English, Irish, or Anglo-Irish, his father having come to Ireland from England during the time of the English plantations of...
performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury
Mercury (element)
Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...
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 carefully 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
Edme Mariotte was a French physicist and priest.- Biography :Edme Mariotte was the youngest son of Simon Mariotte, administrator at the district Til-Châtel , and Catherine Denisot . His parents lived in Til-Châtel and had 4 other children: Jean, Denise, Claude, and Catharine...
and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
Charles's law or Law of Charles and Gay-Lussac (1787)
In 1787 the French physicist Jacques CharlesJacques Charles
Jacques Alexandre César Charles was a French inventor, scientist, mathematician, and balloonist.Charles and the Robert brothers launched the world's first hydrogen-filled balloon in August 1783, then in December 1783, Charles and his co-pilot Nicolas-Louis Robert ascended to a height of about...
found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac
Joseph Louis Gay-Lussac
- External links :* from the American Chemical Society* from the Encyclopædia Britannica, 10th Edition * , Paris...
published results of similar experiments, indicating a linear relationship between volume and temperature:
Dalton's law of partial pressures (1801)
Dalton's LawDalton's law
In chemistry and physics, Dalton's law states that the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual component in a gas mixture...
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 n species as:
-
The ideal gas law (1834)
In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas lawIdeal gas lawThe ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
. Initially the law was formulated as pVm = R(TC + 267) (with temperature expressed in degrees Celsius), where R is the gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:
Van der Waals equation of state (1873)
In 1873, J. D. van der Waals introduced the first equation of stateVan der Waals equationThe van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...
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 most famously continued via the Redlich–Kwong equation of stateRedlich–Kwong equation of stateIn physics and thermodynamics, the Redlich–Kwong equation of state is an equation that is derived from the van der Waals equation. It is generally more accurate than the van der Waals equation and the ideal gas equation, but not used as frequently because the increased difficulty in its derivatives...
and the Soave modification of Redlich-Kwong.
Major equations of state
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
In the following equations the variables are defined as follows. Any consistent set of units may be used, although SIInternational System of UnitsThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
units are preferred. Absolute temperatureThermodynamic temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
= pressure (absolute)
= volume
= number of moles of a substance
= 
= molar volumeMolar volumeThe molar volume, symbol Vm, is the volume occupied by one mole of a substance at a given temperature and pressure. It is equal to the molar mass divided by the mass density...
, the volume of 1 mole of gas or liquid
= absolute temperatureThermodynamic temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
= ideal gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
(8.314472 J/(mol·K))
= pressure at the critical point
= molar volume at the critical point
= absolute temperature at the critical point
Classical ideal gas law
The classical ideal gas lawIdeal gas lawThe ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
may be written:
The ideal gas law may also be expressed as follows
where
is the density, 
is the adiabatic index (ratio of specific heatsHeat capacity ratioThe heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume . It is sometimes also known as the isentropic expansion factor and is denoted by \gamma or \kappa . The latter symbol kappa is...
),
is the internal energy per unit mass (the "specific internal energy"), 
is the specific heat at constant volume, and 
is the specific heat at constant pressure.
Van der Waals equation of state
The Van der Waals equationVan der Waals equationThe van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...
of state may be written:
where
is molar volume, and 
and 
are substance-specific constants. They can be calculated from the critical properties 
and 
(noting that 
is the molar volume at the critical point) as:
Also written as
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation
is called the attraction parameter and 
the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.
The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons:- Molecules are thought as particles with volume, not material points. Thus
cannot be too little, less than some constant. So we get (
) instead of 
. - While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (
something) instead of 
. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~
, the force acting on the whole element is ~
~
.
With the reduced state variables, i.e. Vr=Vm/Vc, Pr=P/Pc and Tr=T/Tc, the reduced form of the Van der Waals equation can be formulated:
The benefit of this form is that for given Tr and Pr, the reduced volume of the liquid and gas can be calculated directly using Cardono's methodCubic functionIn mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...
for the reduced cubic form:
For Pr<1 and Tr<1, the system is in a state of vapor-liquid equilibrium. The reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.
Redlich–Kwong equation of state
Introduced in 1949 the Redlich–Kwong equation of stateRedlich–Kwong equation of stateIn physics and thermodynamics, the Redlich–Kwong equation of state is an equation that is derived from the van der Waals equation. It is generally more accurate than the van der Waals equation and the ideal gas equation, but not used as frequently because the increased difficulty in its derivatives...
was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor-liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.
The Redlich–Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):
Soave modification of Redlich-Kwong
Where ω is the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
for the species.
This formulation for
is due to Graboski and Daubert. The original formulation from Soave is:
for hydrogen:
In 1972 G. Soave replaced the 1/√(T) term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
(the resulting equation is also known as the Soave–Redlich–Kwong equation). The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
Note especially that this replacement changes the definition of a slightly, as the
is now to the second power.
Peng–Robinson equation of state
In polynomial form:
where,
is the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
of the species,
is the universal gas constant and Z=PV/(RT) is compressibility factorCompressibility factorThe compressibility factor , also known as the compression factor, is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase change, the lower the...
.
The Peng–Robinson equation was developed in 1976 in order to satisfy the following goals:- The parameters should be expressible in terms of the critical properties and the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
. - The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factorCompressibility factorThe compressibility factor , also known as the compression factor, is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase change, the lower the...
and liquid density. - The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
- The equation should be applicable to all calculations of all fluid properties in natural gas processes.
For the most part the Peng–Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones. The departure functionDeparture functionIn thermodynamics, a departure function is defined for any thermodynamic property as the difference between the property as computed for an ideal gas and the property of the species as it exists in the real world, for a specified temperature T and pressure P...
s of the Peng–Robinson equation are given on a separate article.
PRSV1
A modification to the attraction term in the Peng-Robinson equation of state published by Stryjek and Vera in 1986 (PRSV) significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
.
The modification is:
where
is an adjustable pure component parameter. Stryjek and Vera published pure component parameters for many compounds of industrial interest in their original journal article.
PRSV2
A subsequent modification published in 1986 (PRSV2) further improved the model's accuracy by introducing two additional pure component parameters to the previous attraction term modification.
The modification is:
where
, 
, and 
are adjustable pure component parameters.
PRSV2 is particularly advantageous for VLEVapor-liquid equilibriumVapor–liquid equilibrium is a condition where a liquid and its vapor are in equilibrium with each other, a condition or state where the rate of evaporation equals the rate of condensation on a molecular level such that there is no net vapor-liquid interconversion...
calculations. While PRSV1 does offer an advantage over the Peng-Robinson model for describing thermodynamic behavior, it is still not accurate enough, in general, for phase equilibrium calculations. The highly non-linear behavior of phase-equilibrium calculation methods tends to amplify what would otherwise be acceptably small errors. It is therefore recommended that PRSV2 be used for equilibrium calculations when applying these models to a design. However, once the equilibrium state has been determined, the phase specific thermodynamic values at equilibrium may be determined by one of several simpler models with a reasonable degree of accuracy.
Elliott, Suresh, Donohue equation of state
The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990. The equation seeks to correct a shortcoming in the Peng–Robinson EOS in that there was an inaccuracy in the van der Waals repulsive term. The EOS accounts for the effect of the shape of a non-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding.
where:
and
is a "shape factor", with 
for spherical molecules
- For non-spherical molecules, the following relation is suggested:
where 
is the acentric factorAcentric factorThe acentric factor \omega is a conceptual number introduced by Pitzer in 1955, proven to be very useful in the description of matter. It has become a standard for the phase characterization of single & pure components...
- The reduced number density
is defined as 
where
is the characteristic size parameter
is the number of molecules
is the volume of the container
The characteristic size parameter is related to the shape parameter
through
where
and 
is Boltzmann's constant.
Noting the relationships between Boltzmann's constant and the Universal gas constant, and observing that the number of molecules can be expressed in terms of Avogadro's numberAvogadro's numberIn chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...
and the molar massMolar massMolar mass, symbol M, is a physical property of a given substance , namely its mass per amount of substance. The base SI unit for mass is the kilogram and that for amount of substance is the mole. Thus, the derived unit for molar mass is kg/mol...
, the reduced number density
can be expressed in terms of the molar volume as
The shape parameter
appearing in the Attraction term and the term 
are given by
(and is hence also equal to 1 for spherical molecules).
where
is the depth of the square-well potential and is given by
, 
, 
and 
are constants in the equation of state:
for spherical molecules (c=1)
for spherical molecules (c=1)
for spherical molecules (c=1)
The model can be extended to associating components and mixtures of nonassociating components. Details are in the paper by J.R. Elliott Jr et al. (1990).
Dieterici equation of state
where a is associated with the interaction between molecules and b takes into account the finite size of the molecules, similarly to the Van der Waals equation.
The reduced coordinates are:
Virial equation of state
Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanicsStatistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. This equation is also called the Kamerlingh Onnes equation. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficientsVirial coefficientVirial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law...
. In this case B corresponds to interactions between pairs of molecules, C to triplets, and so on. Accuracy can be increased indefinitely by considering higher order terms. The coefficients B, C, D, etc. are functions of temperature only.
It can also be used to work out the Boyle Temperature (the temperature at which B = 0 and ideal gas laws apply) from a and b from the Van der Waals equation of state, if you use the value for B shown below:
The BWR equation of state
where- p = pressure
- ρ = the molar density
Values of the various parameters for 15 substances can be found in
Helmholtz Function form
Multiparameter equations of state (MEOS) can be used to represent pure fluids with high accuracy, in both the liquid and gaseous states. MEOS's represent the Helmholtz function of the fluid as the sum of ideal gas and residual terms. Both terms are explicit in reduced temperature and reduced density - thus:
Where:
The reduced density and temperature are typically, though not always, the critical values for the pure fluid.
Other thermodynamic functions can be derived from the MEOS by using appropriate derivatives of the Helmholtz function; hence, because integration of the MEOS is not required, there are few restrictions as to the functional form of the ideal or residual terms.
Typical MEOS use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. MEOS are available currently for about 50 of the most common industrial fluids including refrigerants. Mixture models also exist.
Stiffened equation of state
When considering water under very high pressures (typical applications are underwater nuclear explosionsUnderwater explosionAn underwater explosion, also known as an UNDEX, is an explosion beneath the surface of water. The type of explosion may be chemical or nuclear...
, sonic shock lithotripsy, and sonoluminescenceSonoluminescenceSonoluminescence is the emission of short bursts of light from imploding bubbles in a liquid when excited by sound.-History:The effect was first discovered at the University of Cologne in 1934 as a result of work on sonar. H. Frenzel and H. Schultes put an ultrasound transducer in a tank of...
) the stiffened equation of state is often used:
where
is the internal energy per unit mass, 
is an empirically determined constant typically taken to be about 6.1, and 
is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20,000 atmospheres).
The equation is stated in this form because the speed of sound in water is given by
.
Thus water behaves as though it is an ideal gas that is already under about 20,000 atmospheres (2 GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1 MPa to 2000.2 MPa).
This equation mispredicts the specific heat capacity of water but few simple alternatives are available for severely nonisentropic processes such as strong shocks.
Ultrarelativistic equation of state
An ultrarelativistic fluid has equation of state
where
is the pressure, 
is the mass density, and 
is the speed of soundSpeed of soundThe speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
.
Ideal Bose equation of state
The equation of state for an ideal Bose gasBose gasAn ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
is
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/kT) where μ is the chemical potentialChemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
, Li is the
polylogarithm, ζ is the Riemann zeta function, and Tc is the
critical temperature at which a Bose–Einstein condensateBose–Einstein condensateA Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near absolute zero . Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at...
begins to form.
Equations of state for solids
- Rose–Vinet equation of state
- Birch–Murnaghan equation of state
- Johnson–Holmquist damage modelJohnson–Holmquist damage modelIn solid mechanics, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged brittle materials, such as ceramics, rocks, and concrete, over a range of strain rates...
- Mie-Gruneisen equation of stateMie-Gruneisen equation of stateThe Mie-Gruneisen equation of state is a relation between the pressure and the volume of a solid at a given temperature. It is often used to determine the pressure in a shock-compressed solid...
See also
- Gas lawsGas lawsThe early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...
- Departure functionDeparture functionIn thermodynamics, a departure function is defined for any thermodynamic property as the difference between the property as computed for an ideal gas and the property of the species as it exists in the real world, for a specified temperature T and pressure P...
- Table of thermodynamic equationsTable of thermodynamic equationsThe following page is a concise list of common thermodynamic equations and quantities:-Entropy:*~ S = k_B ~, where k_B is the Boltzmann constant, and \Omega denotes the volume of macrostate in the phase space....
- Real gasReal gasReal gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behaviour of real gases, the following must be taken into account:* compressibility effects;...
- Cluster ExpansionCluster expansionIn statistical mechanics, the cluster expansion is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of...
- Molecules are thought as particles with volume, not material points. Thus