Classical ideal gas law

The classical ideal gas law may be written

$${\displaystyle pV=nRT.}$$

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

$${\displaystyle f(p,V,T)=pV-nRT=0.}$$

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

$${\displaystyle p=\rho (\gamma -1)e}$$

where ${\displaystyle \rho }$ is the density, ${\displaystyle \gamma =C_{p}/C_{v}}$ is the (constant) adiabatic index (ratio of specific heats), ${\displaystyle e=C_{v}T}$ is the internal energy per unit mass (the "specific internal energy"), ${\displaystyle C_{v}}$ is the specific heat capacity at constant volume, and ${\displaystyle C_{p}}$ 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 ${\displaystyle m}$ and spin ${\displaystyle s}$ 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 ${\displaystyle N}$ particles occupying a volume ${\displaystyle V}$ with temperature ${\displaystyle T}$ and pressure ${\displaystyle p}$ is given by^[5]

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

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

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

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

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

With a fixed number density ${\displaystyle N/V}$, 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. The quantum nature of this equation is in it dependence on s and ħ.

Virial equation of state

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

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.

The BWR equation of state

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

where

• ${\displaystyle p}$ is pressure
• ${\displaystyle \rho }$ is molar density

Values of the various parameters can be found in reference materials.^[6] The BWR equation of state has also frequently been used for the modelling of the Lennard-Jones fluid.^[7][8] There are several extensions and modifications of the classical BWR equation of state available.

The Benedict–Webb–Rubin–Starling^[9] equation of state is a modified BWR equation of state and can be written as

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

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 Lee–Kesler equation of state is based on the corresponding states principle, and is a modification of the BWR equation of state.^[10]

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

Perturbation theory-based models

Perturbation theory is frequently used for modelling dispersive interactions in an equation of state. There is a large number of perturbation theory based equations of state available today,^[19][20] e.g. for the classical Lennard-Jones fluid.^[7][21] The two most important theories used for these types of equations of state are the Barker-Henderson perturbation theory^[22] and the Weeks–Chandler–Andersen perturbation theory.^[23]

Statistical associating fluid theory (SAFT)

An important contribution for physically based equations of state is the statistical associating fluid theory (SAFT) that contributes the Helmholtz energy that describes the association (a.k.a. hydrogen bonding) in fluids, which can also be applied for modelling chain formation (in the limit of infinite association strength). The SAFT equation of state was developed using statistical mechanical methods (in particular the perturbation theory of Wertheim^[24]) to describe the interactions between molecules in a system.^[16][25][26] The idea of a SAFT equation of state was first proposed by Chapman et al. in 1988 and 1989.^[16][25][26] Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.^[25][27][28]

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^[34] is often used:

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

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

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.

Morse oscillator equation of state

An equation of state of Morse oscillator has been derived,^[35] and it has the following form:

${\displaystyle p=\Gamma _{1}\nu +\Gamma _{2}\nu ^{2}}$

Where ${\displaystyle \Gamma _{1}}$ is the first order virial parameter and it depends on the temperature, ${\displaystyle \Gamma _{2}}$ is the second order virial parameter of Morse oscillator and it depends on the parameters of Morse oscillator in addition to the absolute temperature. ${\displaystyle \nu }$ is the fractional volume of the system.

Ultrarelativistic equation of state

An ultrarelativistic fluid has equation of state

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

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

Ideal Bose equation of state

The equation of state for an ideal Bose gas is

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

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/k_BT) 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.

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

The ratio ${\displaystyle V=\rho _{e}/\rho }$ is defined by using ${\displaystyle \rho _{e}}$, which is the density of the explosive (solid part) and ${\displaystyle \rho }$, which is the density of the detonation products. The parameters ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$ and ${\displaystyle \omega }$ are given by several references.^[36] In addition, the initial density (solid part) ${\displaystyle \rho _{0}}$, speed of detonation ${\displaystyle V_{D}}$, Chapman–Jouguet pressure ${\displaystyle P_{CJ}}$ and the chemical energy per unit volume of the explosive ${\displaystyle e_{0}}$ 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	${\displaystyle \rho _{e}\,}$ (g/cm3)	${\displaystyle v_{D}\,}$ (m/s)	${\displaystyle p_{CJ}\,}$ (GPa)	${\displaystyle A\,}$ (GPa)	${\displaystyle B\,}$ (GPa)	${\displaystyle R_{1}}$	${\displaystyle R_{2}}$	${\displaystyle \omega }$	${\displaystyle e_{0}\,}$ (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[37]	1.844		36.3	852.4	18.02	4.55	1.3	0.38	10.2

Others

• Tait equation for water and other liquids. Several equations are referred to as the Tait equation.
• Murnaghan equation of state
• Birch–Murnaghan equation of state
• Stacey–Brennan–Irvine equation of state^[38]
• Modified Rydberg equation of state^[39][40][41]
• Adapted polynomial equation of state^[42]
• Johnson–Holmquist equation of state
• Mie–Grüneisen equation of state^[43][44]
• Anton-Schmidt equation of state
• State-transition equation