In physics and thermodynamics, an equation of state is a relation between state variables.^[1] 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. This is a discussion of a present category of science. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... A state variable is any variable which represents the state of an object. ... In thermodynamics, there are a large number of equations relating the various thermodynamic quantities. ... In structural analysis, constitutive relations connect applied stresses or forces to strains or deformations. ... In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... For other uses, see Temperature (disambiguation). ... This article is about pressure in the physical sciences. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... For other uses, see Solid (disambiguation). ... STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ...

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 law, which is roughly accurate for gases at low pressures and high 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. Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ...

In addition to predicting the behavior of gases and liquids, there are also equations of state for predicting the volume of solids, including the transition of solids from one crystalline state to another. There are equations that model the interior of stars, including neutron stars. A related concept is the perfect fluid equation of state used in cosmology. For other uses, see Solid (disambiguation). ... STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... For the Hugo Award-winning story by Larry Niven, see Neutron Star (story). ... In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p. ... 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 ρ: . It is closely related to the thermodynamic equation of state and ideal gas law. ...

Historical

Boyle's law (1662)

Boyle's Law was perhaps the first expression of an equation of state. In 1662 Robert Boyle, an Irishman, 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: Robert Boyle (25 January 1627 – 30 December 1691) was an Irish natural philosopher, chemist, physicist, inventor, and early gentleman scientist, noted for his work in physics and chemistry. ... General Name, Symbol, Number mercury, Hg, 80 Chemical series transition metals Group, Period, Block 12, 6, d Appearance silvery Standard atomic weight 200. ...

PV = constant

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. Edme Mariotte (c. ... Events January 29 - Feodor III becomes Tsar of Russia First measurement of the speed of light, by Ole Rømer Bacons Rebellion Russo-Turkish Wars commence. ...

Charles's law or Law of Charles and Gay-Lussac (1787)

In 1787 the French physicist Jacques Charles 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: Jacques Alexandre César Charles, 1820. ... --69. ... Joseph Louis Gay-Lussac. ...

V₁/T₁ = V₂/T₂

Dalton's law of partial pressures (1801)

Dalton's Law of Partial Pressure: The pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone. In chemistry and physics, Daltons law (also called Daltons law of partial pressures) 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. ...

Mathematically, this can be represented for n species as:

Pressure_total = Pressure₁ + Pressure₂ + ... + Pressure_n

The ideal gas law (1834)

In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially the law was formulated as PV_m=R(T_C+267) (with temperature expressed in degrees Celsius). 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: Emile_Clapeyron Benoit Paul Émile Clapeyron (February 26, 1799 - January 28, 1864) was an French engineer and physicist, considered as one of the founders of thermodynamics. ... Celsius is, or relates to, the Celsius temperature scale (previously known as the centigrade scale). ...

PV_m=R(T_C+273.15)

van der Waals equation of state

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.^[2] His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich-Kwong equation of state and the Soave modification of Redlich-Kwong. The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the van der Waals force. ...

Major Equations of State

In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero. Cover of brochure The International System of Units. ...

P = pressure

V = volume

n = number of moles of a substance

V_m = V/n = molar volume, the volume of 1 mole of gas or liquid

T = absolute temperature

R = ideal gas constant (8.314472 J/(mol·K))

P_c = pressure at the critical point

V_c = molar volume at the critical point

T_c = absolute temperature at the critical point

In chemistry, the molar volume of a substance is the ratio of the volume of a sample of that substance to the amount of substance (usually in mole) in the sample. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ...

Classical ideal gas law

The classical ideal gas law may be written: Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ...

The ideal gas law may also be expressed as follows

where ρ is the density, γ = C_p / C_v is the adiabatic index (ratio of specific heats), e = C_vT is the internal energy per unit mass (the "specific internal energy"), C_v is the specific heat at constant volume, and C_p is the specific heat at constant pressure.

Cubic Equations of State

van der Waals equation of state

The Van der Waals equation of state may be written: The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the van der Waals force. ...

, note that V_m is molar volume.

Where a and b are constants that depend on the specific material. They can be calculated from the critical properties as: The critical temperature of a material is the temperature above which unique liquid and gas phases do not exist. ...

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 a is called the attraction parameter and b 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:

1. Molecules are thought as particles with volume, not material points. Thus V cannot be too little, less than some constant. So we get (V − b) instead of V.
2. 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 (P + something) instead of P. 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 ~ρ, the force acting on the whole element is ~ρ²~.

Redlich-Kwong equation of state

Introduced in 1949 the Redlich-Kwong equation of state 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. Vapor-liquid equilibrium, abbreviated as VLE by some, is a condition where a liquid and its vapor (gas phase) are in equilibrium with each other, a condition or state where the rate of evaporation (liquid changing to vapor) equals the rate of condensation (vapor changing to liquid) on a molecular...

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): The critical temperature of a material is the temperature above which unique liquid and gas phases do not exist. ... The critical temperature of a material is the temperature above which unique liquid and gas phases do not exist. ...

Soave modification of Redlich-Kwong

Where ω is the acentric factor for the species. In thermodynamics, the acentric factor is a factor originally used by Pitzer as an expression in an equation for the compressibility factor. ...

for hydrogen:

In 1972 Soave replaced the a/√(T) term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor. The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials. In thermodynamics, the acentric factor is a factor originally used by Pitzer as an expression in an equation for the compressibility factor. ...

Note especially that this replacement changes the definition of a slightly, as the T_c is now to the second power.

Peng-Robinson equation of state

where, ω is the acentric factor of the species and R is the universal gas constant. In thermodynamics, the acentric factor is a factor originally used by Pitzer as an expression in an equation for the compressibility factor. ... Molar gas constant (also known as universal gas constant, usually denoted by symbol R) is the constant occurring in the universal gas equation, i. ...

The Peng-Robinson equation was developed in 1976 in order to satisfy the following goals:^[3]

1. The parameters should be expressible in terms of the critical properties and the acentric factor.
2. The model should provide reasonable accuracy near the critical point, particularly for calculations of the Compressibility factor and liquid density.
3. The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
4. 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 functions of the Peng-Robinson equation are given on a separate article. The critical temperature of a material is the temperature above which unique liquid and gas phases do not exist. ... In thermodynamics, the acentric factor is a factor originally used by Pitzer as an expression in an equation for the compressibility factor. ... Fluid Dynamics Compressibility (physics) is a measure of the relative volume change of fluid or solid as a response to a pressure (or mean stress) change: . For a gas the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal, while this difference is small in... In 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. Common departure functions include those for...

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:

c = a “shape factor”

Non-cubic Equations of State

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 Equations of State

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 mechanics. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. 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. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Virial coefficients appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density. ...

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 BWRS equation of state

Main article: Benedict-Webb-Rubin

ρ = the molar density

Values of the various parameters for 15 substances can be found in: The Benedict-Webb-Rubin equation (BWR) is an equation of state used in fluid dynamics. ...

K.E. Starling, Fluid Properties for Light Petroleum Systems. Gulf Publishing Company (1973).

Other Equations of State of Interest

Stiffened equation of state

When considering water under very high pressures (typical applications are underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence) the stiffened equation of state is often used: An underwater explosion, also known as an UNDEX, is an explosion beneath the surface of water. ... Long exposure image of multi-bubble sonoluminescence created by a high intensity ultrasonic horn immersed in a beaker of liquid. ...

where e is the internal energy per unit mass, γ is an empirically determined constant typically taken to be about 6.1, and p⁰ is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 2 gigapascals (20000 atmospheres).

The equation is stated in this form because the speed of sound in water is given by c² = γ(p + p⁰) / ρ.

Thus water behaves as though it is an ideal gas that is already under about 20000 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 do when changing from 20001 to 20002 atmospheres (2000.1 MPa to 2000.2 MPa).

This equation mispredicts the specific heat capacity of water but few alternatives are available for severely nonisentropic processes such as strong shocks. Specific heat capacity, also known simply as specific heat (Symbol: C or c) is the measure of the heat energy required to raise the temperature of a given amount of a substance by one degree. ...

Ultrarelativistic equation of state

An ultrarelativistic fluid has equation of state

where p is the pressure, μ is the energy density, and c_s is a constant referred to as the speed of sound. Sound is a vibration that travels through an elastic medium as a wave. ...

Ideal Bose equation of state

The equation of state for an ideal Bose gas is An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ...

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 potential, Li is the polylogarithm, ζ is the Riemann zeta function, and T_c is the critical temperature at which a Bose-Einstein condensate begins to form. In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... The polylogarithm (also known as de Jonquières function) is a special function Lis(z) that is defined by the sum The above definition is valid for all complex numbers s and z where |z|< 1. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... A Bose–Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero (0 kelvins or -273. ...

Equations of state for solids

• Johnson Holmquist Equation of State

In solid mechanics, the Johnson-Holmquist Equation of state is used to model the behaviour of ceramics. ...

See also

• Gas Laws
• Departure function
• Table of thermodynamic equations

The gas laws are a set of laws that describe the relationship between thermodynamic temperature (T), pressure (P) and volume (V) of gases. ... In 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. Common departure functions include those for... For more elaboration on these equations see: thermodynamic equations. ...

Bibliography

• Elliot & Lira, (1999). Introductory Chemical Engineering Thermodynamics, Prentice Hall.

References

1. ^ Perrot, Pierre (1998). A to Z of Thermodynamics. Oxford University Press. ISBN 0-19-856552-6. 
2. ^ van der Waals, J. D. (1873). On the Continuity of the Gaseous and Liquid States (doctoral dissertation). Universiteit Leiden. 
3. ^ Peng, DY, and Robinson, DB. A New Two-Constant Equation of State. Industrial and Engineering Chemistry: Fundamentals. Vol. 15 (1976) pp. 59-64.