Editing Equation of state (section)

== List of further equations of state ==

=== Stiffened equation of state ===

When considering water under very high pressures, in situations such as [[Underwater explosion|underwater nuclear explosions]], [[Extracorporeal shock wave lithotripsy|sonic shock lithotripsy]], and [[sonoluminescence]], the stiffened equation of state<ref>{{Cite journal|last1=Le Métayer|first1=O|last2=Massoni|first2=J|last3=Saurel|first3=R|date=2004-03-01|title=Élaboration des lois d'état d'un liquide et de sa vapeur pour les modèles d'écoulements diphasiques|url=https://www.sciencedirect.com/science/article/pii/S1290072903001443|journal=International Journal of Thermal Sciences|language=fr|volume=43|issue=3| pages=265–276| doi=10.1016/j.ijthermalsci.2003.09.002|issn=1290-0729|url-access=subscription}}</ref> is often used:

<math display="block">p = \rho(\gamma - 1)e - \gamma p^0 \,</math>

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

Thus water behaves as though it is an ideal gas that is ''already'' under about 20,000 atmospheres (2&nbsp;GPa) pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100&nbsp;kPa to 200&nbsp;kPa), the water behaves as an ideal gas would when changing from 20,001 to 20,002 atmospheres (2000.1&nbsp;MPa to 2000.2&nbsp;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,<ref>{{Cite journal |last=Al-Raeei |first=Marwan |date=2022-04-01 |title=Morse oscillator equation of state: An integral equation theory based with virial expansion and compressibility terms |journal=Heliyon |volume=8 |issue=4 |pages=e09328 |doi=10.1016/j.heliyon.2022.e09328 |doi-access=free |pmid=35520603 |issn=2405-8440|pmc=9062208 |bibcode=2022Heliy...809328A }}</ref> and it has the following form:

<math display="block">p = \Gamma_1 \nu + \Gamma_2 \nu^2</math>

Where <math>\Gamma_1</math> is the first order virial parameter and it depends on the temperature, <math>\Gamma_2</math> is the second order virial parameter of Morse oscillator and it depends on the  parameters of Morse oscillator in addition to the absolute temperature. <math>\nu</math> is the fractional volume of the system.

=== Ultrarelativistic equation of state ===

An [[ultrarelativistic fluid]] has equation of state
<math display="block">p = \rho_m c_s^2</math>
where <math>p</math> is the pressure, <math>\rho_m</math> is the mass density, and <math>c_s</math> is the [[speed of sound]].

=== Ideal Bose equation of state ===

The equation of state for an ideal [[Bose gas]] is

<math display="block">p V_m =
  RT~\frac{\operatorname{Li}_{\alpha+1}(z)}{\zeta(\alpha)}
  \left(\frac{T}{T_c}\right)^\alpha
</math>

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''<sub>B</sub>''T'') where ''μ'' is the [[chemical potential]], Li is the [[polylogarithm]], ζ is the [[Riemann zeta function]], and ''T''<sub>''c''</sub> 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.
<math display="block">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}</math>

The ratio <math> V = \rho_e / \rho </math> is defined by using <math> \rho_e </math>, which is the density of the explosive (solid part) and <math> \rho </math>, which is the density of the detonation products. The parameters <math> A </math>, <math> B </math>, <math> R_1 </math>, <math> R_2 </math> and <math> \omega </math> are given by several references.<ref name="Dobratz">{{Cite journal |author1=B. M. Dobratz |author2=P. C. Crawford | year=1985 | title=LLNL Explosives Handbook: Properties of Chemical Explosives and Explosive Simulants|journal=Ucrl-52997 |url=https://ci.nii.ac.jp/naid/10012469501/|access-date = 31 August 2018}}</ref> In addition, the initial density (solid part) <math> \rho_0 </math>, speed of detonation <math> V_D </math>, Chapman–Jouguet pressure <math> P_{CJ} </math> and the chemical energy per unit volume of the explosive <math> e_0 </math> 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.
{| class="wikitable centre"
!  Material !! <math>\rho_e\,</math> (g/cm<sup>3</sup>) !! <math>v_D\,</math> (m/s) !! <math>p_{CJ}\,</math> (GPa) !! <math>A\,</math> (GPa) !! <math>B\,</math> (GPa) !! <math>R_1</math> !! <math>R_2</math> !! <math>\omega</math> !! <math>e_0\,</math> (GPa)
|- 
|[[Trinitrotoluene|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
|-
| [[Polymer-bonded explosive|PBX 9501]]<ref name="Wilkins">{{Citation|last=Wilkins|first=Mark L.|title=Computer Simulation of Dynamic Phenomena| publisher=Springer|year=1999|page=80|url=https://books.google.com/books?id=b3npCAAAQBAJ&pg=PA1|access-date = 31 August 2018 |isbn=9783662038857}}</ref> || 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<ref name="StaceyBrennan1981">{{Cite journal |url= https://espace.library.uq.edu.au/view/UQ:399907 |title= Finite strain theories and comparisons with seismological data |last1= Stacey |first1=F.D. |journal= Surveys in Geophysics |volume=4 |issue=3 |pages=189–232 |doi= 10.1007/BF01449185 |access-date= 31 August 2018 |last2= Brennan |first2= B. J. |last3= Irvine |first3= R.D. |year= 1981 |bibcode= 1981GeoSu...4..189S|s2cid= 129899060 |url-access= subscription }}</ref>
* Modified Rydberg equation of state<ref>{{Cite book|title = "Equations of states and scaling rules for molecular solids under strong compression" in "Molecular systems under high pressure" ed. R. Pucci and G. Piccino |last= Holzapfel |first= W.B. |publisher= Elsevier |year= 1991 |location= North-Holland |pages= 61–68}}</ref><ref>{{Cite journal |title= Equations of state for solids under strong compression |last= Holzapfel |first= W.B. |date=1991 |journal= High Press. Res. |volume= 7 |pages= 290–293 |doi= 10.1080/08957959108245571|orig-year= 1991}}</ref><ref name="Holzapfel1996">{{Cite journal |title= Physics of solids under strong compression |last= Holzapfel |first= Wi.B. |journal= Rep. Prog. Phys. |volume=59 |pages=29–90 |doi= 10.1088/0034-4885/59/1/002 |issue=1 |year= 1996 |bibcode= 1996RPPh...59...29H |s2cid= 250909120 |issn= 0034-4885}}</ref>
* Adapted polynomial equation of state<ref name="Holzapfel1998">{{Cite journal |title= Equation of state for solids under strong compression |last= Holzapfel |first= W.B. |journal= High Press. Res. |volume=16 |issue=2 |pages=81–126 |issn=0895-7959 |doi= 10.1080/08957959808200283 |year= 1998 |bibcode= 1998HPR....16...81H}}</ref>
* [[Johnson–Holmquist damage model|Johnson–Holmquist equation of state]]
* [[Mie–Grüneisen equation of state]]<ref>{{cite book |last1=Holzapfel |first1=Wilfried B. |editor1-last=Katrusiak |editor1-first=A. |editor2-last=McMillan |editor2-first=P. |title=High-Pressure Crystallography |date=2004 |publisher=Kluver Academic |location=Dordrecht, Netherlands |doi= 10.1007/978-1-4020-2102-2_14 |isbn= 978-1-4020-1954-8 |pages=217–236 |chapter-url=https://physik.uni-paderborn.de/fileadmin/physik/Alumni/Holzapfel_LOP/ldv-238.pdf |access-date=31 August 2018 |language=en |chapter=Equations of State and Thermophysical Properties of Solids Under Pressure|volume=140 |series=NATO Science Series}}</ref><ref name="Benjelloun">S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [https://arxiv.org/abs/2105.04845 Link to Archiv e-print] [https://hal.archives-ouvertes.fr/hal-03216379/ Link to Hal e-print]</ref>
* [[Anton-Schmidt equation of state]]
* [[State-transition equation]]