Equation of state

For the use of this in cosmology, see Equation of state (cosmology). For the use of this concept in optimal control theory, see Optimal control § General method.

In physics, chemistry, and thermodynamics, 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. Equations of state are useful in describing the properties of pure substances and mixtures of liquids, gases, and solids, and the state of matter in the interior of stars.

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, 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

$f(p,V,T)=0$

where $\ p$ is the pressure, $\ V$ the volume, and $\ T$ 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 measurment 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 may fields such as process engineering and petroleum industry as well as pharmacutical industry.

Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to the use of the Kelvin (K), with zero being absolute zero.

$\ n$, number of moles of a substance

$\ V_{m}$, ${\frac {V}{n}}$, molar volume, the volume of 1 mole of gas or liquid

$\ R$, ideal gas constant ≈ 8.3144621 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

Historical

Boyle's law (1662)

Boyle's Law was perhaps the first expression of an equation of state. In 1662, the Irish physicist and chemist Robert Boyle 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:

$pV=\mathrm {constant} .\,\!$

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.

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 roughly 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 (Charles's Law):

${\frac {V_{1}}{T_{1}}}={\frac {V_{2}}{T_{2}}}.$

Dalton's law of partial pressures (1801)

Dalton's Law 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:

$p_{\text{total}}=p_{1}+p_{2}+\cdots +p_{n}=\sum _{i=1}^{n}p_{i}.$

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), 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 0 °C = 273.15 K, giving:

$pV_{m}=R\left(T_{C}+273.15\ {}^{\circ }{\text{C}}\right).$

Van der Waals equation of state (1873)

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 most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.

Classical ideal gas law

The classical ideal gas law may be written

$pV=nRT.$

In the form shown above, the equation of state is thus

$f(p,V,T)=pV-nRT=0.$

If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows

$p=\rho (\gamma -1)e$

where $\rho$ is the density, $\gamma =C_{p}/C_{v}$ is the (constant) adiabatic index (ratio of specific heats), $e=C_{v}T$ is the internal energy per unit mass (the "specific internal energy"), $C_{v}$ is the constant specific heat at constant volume, and $C_{p}$ 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 $m$ and spin $s$ 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 $N$ particles occupying a volume $V$ with temperature $T$ and pressure $p$ is given by

$p={\frac {(2s+1){\sqrt {2m^{3}k_{B}^{5}T^{5}}}}{3\pi ^{2}\hbar ^{3}}}\int _{0}^{\infty }{\frac {z^{3/2}\,\mathrm {d} z}{e^{z-\mu /(k_{B}T)}\pm 1}}$

where $k_{B}$ is the Boltzmann constant and $\mu (T,N/V)$ the chemical potential is given by the following implicit function

${\frac {N}{V}}={\frac {(2s+1)(mk_{B}T)^{3/2}}{{\sqrt {2}}\pi ^{2}\hbar ^{3}}}\int _{0}^{\infty }{\frac {z^{1/2}\,\mathrm {d} z}{e^{z-\mu /(k_{B}T)}\pm 1}}.$

In the limiting case where $e^{\mu /(k_{B}T)}\ll 1$, 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 $e^{\mu /(k_{B}T)}\ll 1$ reduces to

$pV=Nk_{B}T\left[1\pm {\frac {\pi ^{3/2}}{2(2s+1)}}{\frac {N\hbar ^{3}}{V(mk_{B}T)^{3/2}}}+\cdots \right]$

With a fixed number density $N/V$, decreasing the temperature causes in Fermi gas, an increase in the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in Bose gas, a decrease in pressure from its classical value implying an effective attraction.

Cubic equations of state

Cubic equations of state are called such because they can be rewritten as a cubic function of $V_{m}$. Cubic equations of state originated from van der Waals equation of state. Hence, all cubic equations of state can be considered 'modified van der Waals equation of state'. There is a very large number of such cubic equations of state. For process engineering, cubic equations of state are today still highly relevant, e.g. the Peng Robinson equation of state or the Soave Redlich Kwong equation of state.

Non-cubic equations of state

Dieterici equation of state

$p(V-b)=RTe^{-{\frac {a}{RTV}}}$

where a is associated with the interaction between molecules and b takes into account the finite size of the molecules, similar to the Van der Waals equation.

The reduced coordinates are:

$T_{c}={\frac {a}{4Rb}},\ p_{c}={\frac {a}{4b^{2}e^{2}}},\ V_{c}=2b.$

Virial equations of state

Virial equation of state

Main article: Virial expansion

${\frac {pV_{m}}{RT}}=A+{\frac {B}{V_{m}}}+{\frac {C}{V_{m}^{2}}}+{\frac {D}{V_{m}^{3}}}+\cdots$

Although usually not the most convenient equation of state, the virial equation is important because it can be derived directly from statistical mechanics. 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 coefficients. A is the first virial coefficient, which has a constant value of 1 and makes the statement that when volume is large, all fluids behave like ideal gases. The second virial coefficient 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.

One of the most accurate equations of state is that from Benedict-Webb-Rubin-Starling shown next. It was very close to a virial equation of state. If the exponential term in it is expanded to two Taylor terms, a virial equation can be derived:

$p=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}+{\frac {c}{T^{2}}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}$

Note that in this virial equation, the fourth and fifth virial terms are zero. The second virial coefficient is monotonically decreasing as temperature is lowered. The third virial coefficient is monotonically increasing as temperature is lowered.

The BWR equation of state

Main article: Benedict–Webb–Rubin equation

$p=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)$

where

p is pressure

ρ is molar density

Values of the various parameters can be found in reference materials.

Lee-Kesler equation of state

The Lee-Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.

$P={\frac {RT}{V}}\left(1+{\frac {B}{V_{r}}}+{\frac {C}{V_{r}^{2}}}+{\frac {D}{V_{r}^{5}}}+{\frac {c_{4}}{T_{r}^{3}V_{r}^{2}}}\left(\beta +{\frac {\gamma }{V_{r}^{2}}}\right)\exp \left({\frac {-\gamma }{V_{r}^{2}}}\right)\right)$

SAFT equations of state

Statistical associating fluid theory (SAFT) equations of state predict the effect of molecular size and shape and hydrogen bonding on fluid properties and phase behavior. The SAFT equation of state was developed using statistical mechanical methods (in particular perturbation theory) to describe the interactions between molecules in a system. The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989. Many different versions of the SAFT equation of state have been proposed, but all use the same chain and association terms derived by Chapman. SAFT equations of state represent molecules as chains of typically spherical particles that interact with one another through short range repulsion, long range attraction, and hydrogen bonding between specific sites. One popular version of the SAFT equation of state includes the effect of chain length on the shielding of the dispersion interactions between molecules (PC-SAFT). In general, SAFT equations give more accurate results than traditional cubic equations of state, especially for systems containing liquids or solids.

Multiparameter equations of state

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:

${\frac {a(T,\rho )}{RT}}={\frac {a^{o}(T,\rho )+a^{r}(T,\rho )}{RT}}=\alpha ^{o}(\tau ,\delta )+\alpha ^{r}(\tau ,\delta )$

where:

$\tau ={\frac {T_{r}}{T}},\delta ={\frac {\rho }{\rho _{r}}}$

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. The IAPWS95 reference equation of state for water is also an MEOS. Mixture models for MEOS exist, as well.

One example of such an equation of state is the form proposed by Span and Wagner.

$a^{res}=\sum _{i=1}^{8}\sum _{j=-8}^{12}n_{i,j}\delta ^{i}\tau ^{j/8}+\sum _{i=1}^{5}\sum _{j=-8}^{24}n_{i,j}\delta ^{i}\tau ^{j/8}\exp \left(-\delta \right)+\sum _{i=1}^{5}\sum _{j=16}^{56}n_{i,j}\delta ^{i}\tau ^{j/8}\exp \left(-\delta ^{2}\right)+\sum _{i=2}^{4}\sum _{j=24}^{38}n_{i,j}\delta ^{i}\tau ^{j/2}\exp \left(-\delta ^{3}\right)$

This is a somewhat simpler form that is intended to be used more in technical applications. Reference equations of state require a higher accuracy and use a more complicated form with more terms.

Other equations of state of interest

Stiffened equation of state

When considering water under very high pressures, in situations such as underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence, the stiffened equation of state is often used:

$p=\rho (\gamma -1)e-\gamma p^{0}\,$

where $e$ is the internal energy per unit mass, $\gamma$ is an empirically determined constant typically taken to be about 6.1, and $p^{0}$ 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 $c^{2}=\gamma \left(p+p^{0}\right)/\rho$.

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

$p=\rho _{m}c_{s}^{2}$

where $p$ is the pressure, $\rho _{m}$ is the mass density, and $c_{s}$ is the speed of sound.

Ideal Bose equation of state

The equation of state for an ideal Bose gas is

$pV_{m}=RT~{\frac {{\text{Li}}_{\alpha +1}(z)}{\zeta (\alpha )}}\left({\frac {T}{T_{c}}}\right)^{\alpha }$

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.

Jones–Wilkins–Lee equation of state for explosives (JWL equation)

The equation of state from Jones–Wilkins–Lee is used to describe the detonation products of explosives.

$p=A\left(1-{\frac {\omega }{R_{1}V}}\right)\exp(-R_{1}V)+B\left(1-{\frac {\omega }{R_{2}V}}\right)\exp \left(-R_{2}V\right)+{\frac {\omega e_{0}}{V}}$

The ratio $V=\rho _{e}/\rho$ is defined by using $\rho _{e}$ = density of the explosive (solid part) and $\rho$ = density of the detonation products. The parameters $A$, $B$, $R_{1}$, $R_{2}$ and $\omega$ are given by several references. In addition, the initial density (solid part) $\rho _{0}$, speed of detonation $V_{D}$, Chapman–Jouguet pressure $P_{CJ}$ and the chemical energy of the explosive $e_{0}$ are given in such references. These parameters are obtained by fitting the JWL-EOS to experimental results. Typical parameters for some explosives are listed in the table below.

Material	$\rho _{0}\,$ (g/cm3)	$v_{D}\,$ (m/s)	$p_{CJ}\,$ (GPa)	$A\,$ (GPa)	$B\,$ (GPa)	$R_{1}\,$	$R_{2}\,$	$\omega \,$	$e_{0}\,$ (GPa)
TNT	1.630	6930	21.0	373.8	3.747	4.15	0.90	0.35	6.00
Composition B	1.717	7980	29.5	524.2	7.678	4.20	1.10	0.35	8.50
PBX 9501	1.844		36.3	852.4	18.02	4.55	1.3	0.38	10.2

Equations of state for solids and liquids

Common abbreviations: $\eta =\left({\frac {V}{V_{0}}}\right)^{\frac {1}{3}}~,~~K_{0}^{\prime }={\frac {dK_{0}}{dp}}$

• Tait equation for water and other liquids. Several equations are referred to as the Tait equation.
• Murnaghan equation of state

$p(V)={\frac {K_{0}}{K_{0}'}}\left[\eta ^{-3K_{0}'}-1\right]$

• Birch–Murnaghan equation of state

$p(V)={\frac {3K_{0}}{2}}\left({\frac {1-\eta ^{2}}{\eta ^{7}}}\right)\left\{1+{\frac {3}{4}}\left(K_{0}'-4\right)\left({\frac {1-\eta ^{2}}{\eta ^{2}}}\right)\right\}$

• Stacey-Brennan-Irvine equation of state (falsely often refer to Rose-Vinet equation of state)

$p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{2}}}\right)\exp \left[{\frac {3}{2}}\left(K_{0}'-1\right)(1-\eta )\right]$

• Modified Rydberg equation of state (more reasonable form for strong compression)

$p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[{\frac {3}{2}}\left(K_{0}'-3\right)(1-\eta )\right]$

• Adapted Polynomial equation of state (second order form = AP2, adapted for extreme compression)

$p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[c_{0}(1-\eta )\right]\left\{1+c_{2}\eta (1-\eta )\right\}$

with

$c_{0}=-\ln \left({\frac {3K_{0}}{p_{\text{FG0}}}}\right)~,~~p_{\text{FG0}}=a_{0}\left({\frac {Z}{V_{0}}}\right)^{\frac {5}{3}}~,~~c_{2}={\frac {3}{2}}\left(K_{0}'-3\right)-c_{0}$

where $a_{0}$ = 0.02337 GPa.nm⁵. The total number of electrons $Z$ in the initial volume $V_{0}$ determines the Fermi gas pressure $p_{\text{FG0}}$, which provides for the correct behavior at extreme compression. So far there are no known "simple" solids that require higher order terms.

• Adapted polynomial equation of state (third order form = AP3)

$p(V)=3K_{0}\left({\frac {1-\eta }{\eta ^{5}}}\right)\exp \left[c_{0}(1-\eta )\right]\left\{1+c_{2}\eta (1-\eta )+c_{3}\eta (1-\eta )^{2}\right\}$

• Johnson–Holmquist equation of state

$p(V)={\begin{cases}k_{1}~\xi +k_{2}~\xi ^{2}+k_{3}~\xi ^{3}+\Delta p&\qquad {\text{Compression}}\\k_{1}~\xi &\qquad {\text{Tension}}\end{cases}}~;~~\xi :={\cfrac {V_{0}}{V}}-1$

• Mie–Grüneisen equation of state (for a more detailed discussion see refs.)

$p(V)-p_{0}={\frac {\Gamma }{V}}\left(e-e_{0}\right)$

• Anton-Schmidt equation of state

$p(V)=-\beta \left({\frac {V}{V_{0}}}\right)^{n}\ln \left({\frac {V}{V_{0}}}\right)$

where $\beta =K_{0}$ is the bulk modulus at equilibrium volume $V_{0}$ and $n=-{\frac {K'_{0}}{2}}$ typically about −2 is often related to the Grüneisen parameter by $n=-{\frac {1}{6}}-\gamma _{G}$

See also

• Gas laws
• Departure function
• Table of thermodynamic equations
• Real gas
• Cluster Expansion

External links

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