SklogWiki - User contributions [en]

Thiele hard sphere equation of state

2024-06-12T22:20:17Z

Proxima Centari: added full name

{{Stub-general}}
The '''Thiele hard sphere equation of state''' is an [[equations of state|equation of state]] for modeling a [[hard sphere model|hard sphere]] fluid developed by Everett Thiele in 1963
<ref>[https://doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics '''39''' 474 (1963)]</ref>.
The equation provides a better approximation of the repulsive forces between molecules than the [[Van der Waals equation of state|Van der Waals repulsive term]]. The equation is given below:

<math> Z_{hs} = \frac{p_{hs}V_m}{RT} = \frac{1 - \eta^3}{(1-\eta)^4} = \frac{1 + \eta + \eta^2}{(1-\eta)^3} </math>,

where:
:<math>Z_{hs}</math> is the [[compressibility factor]] of the hard sphere fluid;
:<math>p_{hs}</math> is the [[pressure]] of the fluid;
:<math>V_m</math> is the molar volume of the fluid;
:<math>T</math> is the absolute [[temperature]] of the fluid;
:<math>R</math> is the [[molar gas constant]]; and
:<math>\eta</math> is the [[packing fraction]] of the fluid.

In terms of accuracy, the Thiele equation is superseded by the [[Carnahan-Starling equation of state]]
[[category: Equations of state]]
[[category: hard sphere]]

==References==
<references/>

Thiele hard sphere equation of state

2024-06-12T22:18:38Z

Proxima Centari: added wikilinks

{{Stub-general}}
The '''Thiele hard sphere equation of state''' is an [[equations of state|equation of state]] for modeling a [[hard sphere model|hard sphere]] fluid developed by Thiele in 1963
<ref>[https://doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics '''39''' 474 (1963)]</ref>.
The equation provides a better approximation of the repulsive forces between molecules than the [[Van der Waals equation of state|Van der Waals repulsive term]]. The equation is given below:

<math> Z_{hs} = \frac{p_{hs}V_m}{RT} = \frac{1 - \eta^3}{(1-\eta)^4} = \frac{1 + \eta + \eta^2}{(1-\eta)^3} </math>,

where:
:<math>Z_{hs}</math> is the [[compressibility factor]] of the hard sphere fluid;
:<math>p_{hs}</math> is the [[pressure]] of the fluid;
:<math>V_m</math> is the molar volume of the fluid;
:<math>T</math> is the absolute [[temperature]] of the fluid;
:<math>R</math> is the [[molar gas constant]]; and
:<math>\eta</math> is the [[packing fraction]] of the fluid.

In terms of accuracy, the Thiele equation is superseded by the [[Carnahan-Starling equation of state]]
[[category: Equations of state]]
[[category: hard sphere]]

==References==
<references/>

Molar gas constant

2024-06-12T22:17:22Z

Proxima Centari: added stub template

{{Stub-general}}
The '''molar gas constant''', <math>R</math>, is given by

:8.3144621 J mol-1 K-1

It is the product of the [[Avogadro constant]] and the [[Boltzmann constant]], i.e.

:<math>R = N_A k_B</math>
==References==
#[http://physics.nist.gov/cgi-bin/cuu/Value?r|search_for=molar CODATA Internationally recommended values of the Fundamental Physical Constants]
[[category: physical constants]]

Molar gas constant

2024-06-12T22:15:52Z

Proxima Centari: simplified wording

The '''molar gas constant''', <math>R</math>, is given by

:8.3144621 J mol-1 K-1

It is the product of the [[Avogadro constant]] and the [[Boltzmann constant]], i.e.

:<math>R = N_A k_B</math>
==References==
#[http://physics.nist.gov/cgi-bin/cuu/Value?r|search_for=molar CODATA Internationally recommended values of the Fundamental Physical Constants]
[[category: physical constants]]

Thiele hard sphere equation of state

2024-06-11T20:14:45Z

Proxima Centari: Wording

{{Stub-general}}
The '''Thiele hard sphere equation of state''' is an [[equations of state|equation of state]] for modeling a [[hard sphere model|hard sphere]] fluid developed by Thiele in 1963
<ref>[https://doi.org/10.1063/1.1734272 Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics '''39''' 474 (1963)]</ref>.
The equation provides a better approximation of the repulsive forces between molecules than the [[Van der Waals equation of state|Van der Waals repulsive term]]. The equation is given below:

<math> Z_{hs} = \frac{p_{hs}V_m}{RT} = \frac{1 - \eta^3}{(1-\eta)^4} = \frac{1 + \eta + \eta^2}{(1-\eta)^3} </math>,

where:
:<math>Z_{hs}</math> is the compressibility factor of the hard sphere fluid;
:<math>p_{hs}</math> is the pressure of the fluid;
:<math>V_m</math> is the molar volume of the fluid;
:<math>T</math> is the absolute temperature of the fluid;
:<math>R</math> is the gas constant; and
:<math>\eta</math> is the [[packing fraction]] of the fluid.

In terms of accuracy, the Thiele equation is superseded by the [[Carnahan-Starling equation of state]]
[[category: Equations of state]]
[[category: hard sphere]]

==References==
<references/>

Twu-Sim-Tassone equation of state

2024-04-09T03:06:37Z

Proxima Centari: Added the exact values for Z_c and the parameters a & b

Twu, Sim and Tassone presented a cubic [[Equations of state|equation of state]] for accurate representation of hydrocarbons that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>[http://dx.doi.org/10.1016/S0378-3812(01)00663-X Chorng H. Twu, Wayne D. Sim and Vince Tassone "A versatile liquid activity model for SRK, PR and a new cubic equation-of-state TST", Fluid Phase Equilibria '''194-197''' pp. 385-399 (2002)]</ref>. With a critical [[compressibility factor]] of (Eq. 5)

:<math>Z_c = \frac{p_cv_c}{RT_c} = \frac{8}{27} </math>

it better represents the compressibility than many of than the [[Redlich-Kwong equation of state]]s, including the [[Redlich-Kwong equation of state#Soave modification | Soave modified version]], and the [[Peng and Robinson equation of state]].

The equation follows the general cubic form resulting in the equation (Eq. 2):

:<math>p=\frac{RT}{V_m-b}-\frac{a}{(V_m-0.5b)(V_m+3b)}</math>

where <math>V_m</math> is the molar volume, <math>R</math> is the [[molar gas constant]], <math>T</math> is the [[temperature]], <math>p</math> is the [[pressure]], and <math>a</math> and <math>b</math> are the attractive and repulsive parameters akin to those of the [[Van der Waals equation of state]]. Relations exists between <math>a</math> and <math>b</math> and the critical parameters <math>T_c</math> and <math>p_c</math> in the forms (Eqs. 3 and 4):

:<math>a_c= Z_c \frac{343}{216}\frac{R^2T_c^2}{p_c}</math>

:<math>b_c= Z_c \frac{RT_c}{4p_c}</math>

==References==
<references/>
;Related reading
*[http://www.aiche.org/uploadedFiles/CEP/Issues/110258.pdf Chorng H. Twu, Wayne D. Sim and Vince Tassone "Getting a Handle on Advanced Cubic Equation of State", Chemical Engineering Progress '''November''' pp. 58-65 (2002)]
[[category: equations of state]]

Equations of state

2021-06-01T00:34:37Z

Proxima Centari: /* Semi-empirical equations of state */ fixed several grammatical errors

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally, there is the [[Equation of State: Ideal Gas|ideal gas equation]]. However, one of the first to describe realistic substances was the famous [[van der Waals equation of state]]. Since then, many semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to give better estimations of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state |Amagat]]
*[[Antoine equation of state |Antoine]]
*[[Baonza equation of state |Baonza]]
*[[BACK equation of state |BACK]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state |Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb, and Rubin]]
*[[Berthelot equation of state |Berthelot]]
*[[Birch-Murnaghan equation of state |Birch-Murnaghan]]
*[[Boltzmann equation|Boltzmann]]
*[[Boynton and Bramley equation of state |Boynton and Bramley]]
*[[Brillouin equation of state |Brillouin]]
*[[Clausius equation of state |Clausius]]
*[[Cole equation of state |Cole]]
*[[Dalton's law|Dalton]]
*[[BACK equation of state |DIBACK]]
*[[Dieterici equation of state |Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state |Fouché]]
*[[Goebel equation of state |Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Holzapfel equation of state |Holzapfel]]
*[[Jäger equation of state |Jäger]]
*[[Kam equation of state |Kam]]
*[[Kumari-Dass equation of state |Kumari-Dass]]
*[[Lagrange equation of state |Lagrange]]
*[[Leduc equation of state |Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie potential|Mie]]
*[[BACK equation of state |MOBACK]]
*[[Mohsen-Nia, Modarress and Mansoori equation of state |Mohsen-Nia, Modarress, and Mansoori]]
*[[Murnaghan equation of state |Murnaghan]]
*[[Natanson equation of state |Natanson]]
*[[BACK equation of state |NIK]]
*[[Onnes equation of state |Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state |Porter]]
*[[BACK equation of state |QUABACK]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state |Recknagel]]
*[[Redlich-Kwong equation of state |Redlich-Kwong]]
*[[Reinganum equation of state |Reinganum]]
*[[Rose-Vinet (Universal) equation of state |Rose-Vinet]]
*[[Sarrau equation of state |Sarrau]]
*[[Schiller equation of state |Schiller]]
*[[Schrieber equation of state |Schrieber]]
*[[Smoluchowski equation|Smoluchowski]]
*[[Starkweather equation of state |Starkweather]]
*[[Stiffened equation of state |Stiffened]]
*[[Tait equation of state |Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tillotson equation of state |Tillotson]]
*[[Tumlirz equation of state |Tumlirz]]
*[[Twu-Sim-Tassone equation of state |Twu-Sim-Tassone]]
*[[van der Waals equation of state |van der Waals]]
*[[Walter equation of state |Walter]]
*[[Wohl equation of state |Wohl]]
*[[Phase diagrams of water |Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==See also==
*[[Pair stress approximation]]
*[[Scaled-particle theory]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Equations of state

2021-06-01T00:16:50Z

Proxima Centari: /* Semi-empirical equations of state */ fixed 2 red-links

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally there is the [[Equation of State: Ideal Gas|ideal gas equation of state]]. However, one of the first steps towards a description of realistic substances was the famous [[van der Waals equation of state]]. Since then a plethora of semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to better reproduce the foibles of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state |Amagat]]
*[[Antoine equation of state |Antoine]]
*[[Baonza equation of state |Baonza]]
*[[BACK equation of state |BACK]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state |Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb, and Rubin]]
*[[Berthelot equation of state |Berthelot]]
*[[Birch-Murnaghan equation of state |Birch-Murnaghan]]
*[[Boltzmann equation|Boltzmann]]
*[[Boynton and Bramley equation of state |Boynton and Bramley]]
*[[Brillouin equation of state |Brillouin]]
*[[Clausius equation of state |Clausius]]
*[[Cole equation of state |Cole]]
*[[Dalton's law|Dalton]]
*[[BACK equation of state |DIBACK]]
*[[Dieterici equation of state |Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state |Fouché]]
*[[Goebel equation of state |Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Holzapfel equation of state |Holzapfel]]
*[[Jäger equation of state |Jäger]]
*[[Kam equation of state |Kam]]
*[[Kumari-Dass equation of state |Kumari-Dass]]
*[[Lagrange equation of state |Lagrange]]
*[[Leduc equation of state |Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie potential|Mie]]
*[[BACK equation of state |MOBACK]]
*[[Mohsen-Nia, Modarress and Mansoori equation of state |Mohsen-Nia, Modarress and Mansoori]]
*[[Murnaghan equation of state |Murnaghan]]
*[[Natanson equation of state |Natanson]]
*[[BACK equation of state |NIK]]
*[[Onnes equation of state |Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state |Porter]]
*[[BACK equation of state |QUABACK]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state |Recknagel]]
*[[Redlich-Kwong equation of state |Redlich-Kwong]]
*[[Reinganum equation of state |Reinganum]]
*[[Rose-Vinet (Universal) equation of state |Rose-Vinet]]
*[[Sarrau equation of state |Sarrau]]
*[[Schiller equation of state |Schiller]]
*[[Schrieber equation of state |Schrieber]]
*[[Smoluchowski equation|Smoluchowski]]
*[[Starkweather equation of state |Starkweather ]]
*[[Stiffened equation of state |Stiffened]]
*[[Tait equation of state |Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tillotson equation of state |Tillotson]]
*[[Tumlirz equation of state |Tumlirz]]
*[[Twu-Sim-Tassone equation of state |Twu-Sim-Tassone]]
*[[van der Waals equation of state |van der Waals]]
*[[Walter equation of state |Walter]]
*[[Wohl equation of state |Wohl]]
*[[Phase diagrams of water |Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==See also==
*[[Pair stress approximation]]
*[[Scaled-particle theory]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Equations of state

2021-06-01T00:01:01Z

Proxima Centari: /* Semi-empirical equations of state */ fixed Boltzmann red-link

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally there is the [[Equation of State: Ideal Gas|ideal gas equation of state]]. However, one of the first steps towards a description of realistic substances was the famous [[van der Waals equation of state]]. Since then a plethora of semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to better reproduce the foibles of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state |Amagat]]
*[[Antoine equation of state |Antoine]]
*[[Baonza equation of state |Baonza]]
*[[BACK equation of state |BACK]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state |Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb, and Rubin]]
*[[Berthelot equation of state |Berthelot]]
*[[Birch-Murnaghan equation of state |Birch-Murnaghan]]
*[[Boltzmann equation|Boltzmann]]
*[[Boynton and Bramley equation of state |Boynton and Bramley]]
*[[Brillouin equation of state |Brillouin]]
*[[Clausius equation of state |Clausius]]
*[[Cole equation of state |Cole]]
*[[Dalton's law|Dalton]]
*[[BACK equation of state |DIBACK]]
*[[Dieterici equation of state |Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state |Fouché]]
*[[Goebel equation of state |Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Holzapfel equation of state |Holzapfel]]
*[[Jäger equation of state |Jäger]]
*[[Kam equation of state |Kam]]
*[[Kumari-Dass equation of state |Kumari-Dass]]
*[[Lagrange equation of state |Lagrange]]
*[[Leduc equation of state |Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie equation of state |Mie]]
*[[BACK equation of state |MOBACK]]
*[[Mohsen-Nia, Modarress and Mansoori equation of state |Mohsen-Nia, Modarress and Mansoori]]
*[[Murnaghan equation of state |Murnaghan]]
*[[Natanson equation of state |Natanson]]
*[[BACK equation of state |NIK]]
*[[Onnes equation of state |Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state |Porter]]
*[[BACK equation of state |QUABACK]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state |Recknagel]]
*[[Redlich-Kwong equation of state |Redlich-Kwong]]
*[[Reinganum equation of state |Reinganum]]
*[[Rose-Vinet (Universal) equation of state |Rose-Vinet]]
*[[Sarrau equation of state |Sarrau]]
*[[Schiller equation of state |Schiller]]
*[[Schrieber equation of state |Schrieber]]
*[[Smoluchowski equation of state |Smoluchowski]]
*[[Starkweather equation of state |Starkweather ]]
*[[Stiffened equation of state |Stiffened]]
*[[Tait equation of state |Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tillotson equation of state |Tillotson]]
*[[Tumlirz equation of state |Tumlirz]]
*[[Twu-Sim-Tassone equation of state |Twu-Sim-Tassone]]
*[[van der Waals equation of state |van der Waals]]
*[[Walter equation of state |Walter]]
*[[Wohl equation of state |Wohl]]
*[[Phase diagrams of water |Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==See also==
*[[Pair stress approximation]]
*[[Scaled-particle theory]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Equations of state

2021-05-31T23:55:13Z

Proxima Centari: /* Semi-empirical equations of state */ added Dalton's law

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally there is the [[Equation of State: Ideal Gas|ideal gas equation of state]]. However, one of the first steps towards a description of realistic substances was the famous [[van der Waals equation of state]]. Since then a plethora of semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to better reproduce the foibles of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state |Amagat]]
*[[Antoine equation of state |Antoine]]
*[[Baonza equation of state |Baonza]]
*[[BACK equation of state |BACK]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state |Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb, and Rubin]]
*[[Berthelot equation of state |Berthelot]]
*[[Birch-Murnaghan equation of state |Birch-Murnaghan]]
*[[Boltzmann equation of state |Boltzmann]]
*[[Boynton and Bramley equation of state |Boynton and Bramley]]
*[[Brillouin equation of state |Brillouin]]
*[[Clausius equation of state |Clausius]]
*[[Cole equation of state |Cole]]
*[[Dalton's law|Dalton]]
*[[BACK equation of state |DIBACK]]
*[[Dieterici equation of state |Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state |Fouché]]
*[[Goebel equation of state |Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Holzapfel equation of state |Holzapfel]]
*[[Jäger equation of state |Jäger]]
*[[Kam equation of state |Kam]]
*[[Kumari-Dass equation of state |Kumari-Dass]]
*[[Lagrange equation of state |Lagrange]]
*[[Leduc equation of state |Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie equation of state |Mie]]
*[[BACK equation of state |MOBACK]]
*[[Mohsen-Nia, Modarress and Mansoori equation of state |Mohsen-Nia, Modarress and Mansoori]]
*[[Murnaghan equation of state |Murnaghan]]
*[[Natanson equation of state |Natanson]]
*[[BACK equation of state |NIK]]
*[[Onnes equation of state |Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state |Porter]]
*[[BACK equation of state |QUABACK]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state |Recknagel]]
*[[Redlich-Kwong equation of state |Redlich-Kwong]]
*[[Reinganum equation of state |Reinganum]]
*[[Rose-Vinet (Universal) equation of state |Rose-Vinet]]
*[[Sarrau equation of state |Sarrau]]
*[[Schiller equation of state |Schiller]]
*[[Schrieber equation of state |Schrieber]]
*[[Smoluchowski equation of state |Smoluchowski]]
*[[Starkweather equation of state |Starkweather ]]
*[[Stiffened equation of state |Stiffened]]
*[[Tait equation of state |Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tillotson equation of state |Tillotson]]
*[[Tumlirz equation of state |Tumlirz]]
*[[Twu-Sim-Tassone equation of state |Twu-Sim-Tassone]]
*[[van der Waals equation of state |van der Waals]]
*[[Walter equation of state |Walter]]
*[[Wohl equation of state |Wohl]]
*[[Phase diagrams of water |Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==See also==
*[[Pair stress approximation]]
*[[Scaled-particle theory]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Amagat equation of state

2021-05-31T23:32:29Z

Proxima Centari: Tweaked last sentence and added equation

The '''Amagat equation of state''' or '''Amagat's law''' <ref>[http://gallica.bnf.fr/ark:/12148/bpt6k34859d/f343.image.langEN Émile Hilaire Amagat Emile Hilaire Amagat, "Memoire sur la compressibilite des gaz a des pressions elevees," Annales de Chimie et de Physique '''19''' pp. 345-385 (1880)]</ref>, applies to [[mixtures]] where each component is an [[ideal gas]]. It states that the volume of the mixture is the sum of the volumes of each component.

For <math>n</math> components,

<math>V_{total} = \sum_{i=1}^n V_i = V_1 + V_2 + ... + V_n</math>

==References==
<references/>

[[category: equations of state]]
[[category: ideal gas]]
[[category: mixtures]]

Universality classes

2021-04-07T03:45:00Z

Proxima Centari: Tweaked style of fractions for consistency/uniformity

'''Universality classes''' are groups of [[Idealised models | models]] that have the same set of [[critical exponents]]

:{| border="1"
|-
| dimension ||<math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || <math>\delta</math> ||<math>\nu</math> || <math>\eta</math> || class
|-
| || || || || || || || 3-state Potts
|-
| || || || || || || ||Ashkin-Teller
|-
| || || || || || || ||Chiral
|-
| || || || || || || ||Directed percolation
|-
| 2 || 0 || 1/8 || 7/4 || || 1 || 1/4 || 2D Ising
|-
| 3 || 0.1096(5) || 0.32653(10) || 1.2373(2) || 4.7893(8) || 0.63012(16) || 0.03639(15) || 3D Ising
|-
| || || || || || || ||Local linear interface
|-
| any || 0 || 1/2 || 1 || 3 || 1/2 || 0 || Mean-field
|-
| || || || || || || ||Molecular beam epitaxy
|-
| || || || || || || ||Random-field
|-
| 3 || −0.0146(8) || 0.3485(2) || 1.3177(5) || 4.780(2) ||0.67155(27) || 0.0380(4) || XY
|}

where
*<math>\alpha</math> is known as the [[Critical exponents#Heat capacity exponent| heat capacity exponent]]
*<math>\beta</math> is known as the [[Critical exponents#Magnetic order parameter exponent | magnetic order parameter exponent]]
*<math>\gamma</math> is known as the [[Critical exponents#Susceptibility exponent |susceptibility exponent ]]
*<math>\delta</math> is known as the [[Critical exponents#Equation of state exponent |equation of state exponent ]]
*<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length exponent]]
*<math>\eta</math> is known as...
=Derivations=
==3-state Potts==
[[Potts model]]
==Ashkin-Teller==
[[Ashkin-Teller model]]
==Chiral==
==Directed percolation==
==Ising==
The [[Hamiltonian]] of the [[Ising model]] is

<math>
H=\sum_{<i,j>}S_i S_j
</math>

where <math>S_i=\pm 1</math> and the summation runs over the lattice sites.

The [[Order parameters | order parameter]] is
<math>
m=\sum_i S_i
</math>

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are

:<math>
\alpha=0
</math>

(In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]])

<math>
\beta=\frac{1}{8}
</math>

<math>
\gamma=\frac{7}{4}
</math>

<math>
\delta=15
</math>

along with <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>:

:<math>
\nu=1
</math>

:<math>
\eta = 1/4
</math>

In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and [[Renormalisation group]] analysis provide accurate estimates <ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref>:

:<math>
\nu=0.63012(16)
</math>

:<math>
\alpha=0.1096(5)
</math>

:<math>
\beta= 0.32653(10)
</math>

:<math>
\gamma=1.2373(2)
</math>

:<math>
\delta=4.7893(8)
</math>

:<math>
\eta =0.03639(15)
</math>

with a critical temperature of <math>k_BT_c = 4.51152786~S </math><ref>[http://dx.doi.org/10.1088/0305-4470/29/17/042 A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General '''29''' pp. 5727-5733 (1996)]</ref>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

==Local linear interface==
==Mean-field==
The [[critical exponents]] of are derived as follows <ref>Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4 </ref>:
====Heat capacity exponent: <math>\alpha</math>====
(final result: <math>\alpha=0</math>)
====Magnetic order parameter exponent: <math>\beta</math>====
(final result: <math>\beta=1/2</math>)
====Susceptibility exponent: <math>\gamma</math>====
(final result: <math>\gamma=1</math>)
====Equation of state exponent: <math>\delta</math>====
(final result: <math>\delta=3</math>)
====Correlation length exponent: <math>\nu</math>====
(final result: <math>\nu=1/2</math>)
====Correlation function exponent: <math>\eta</math>====
(final result: <math>\eta=0</math>)
==Molecular beam epitaxy==
==Random-field==
==XY==
For the three dimensional [[XY model]] one has the following [[critical exponents]]<ref name="Campostrini2001" >[http://dx.doi.org/10.1103/PhysRevB.63.214503 Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B '''63''' 214503 (2001)]</ref>:

:<math>
\nu=0.67155(27)
</math>

:<math>\alpha = -0.0146(8)</math>

:<math>
\beta= 0.3485(2)
</math>

:<math>
\gamma=1.3177(5)
</math>

:<math>
\delta=4.780(2)
</math>

:<math>
\eta =0.0380(4)
</math>
=References=
<references/>
[[category: Renormalisation group]]

Equations of state for hard spheres

2021-04-07T03:34:28Z

Proxima Centari: Added Thiele (hard sphere) equation of state

The following is a list of [[equations of state]] designed for the [[hard sphere model]]:
*[[Carnahan-Starling equation of state]]
*[[Equations of state for crystals of hard spheres]]
*[[Hamad hard sphere equation of state]]
*[[Hansen-Goos hard sphere equation of state]]
*[[Hard hypersphere equation of state |Hard hyperspheres]]
*[[Kolafa-Labík-Malijevský equation of state]]
*[[Liu hard sphere equation of state]]
*[[Santos-Lopez de Haro hard sphere equation of state]]
*[[Thiele hard sphere equation of state]]
*[[WC1 and WC2 hard sphere equations of state]]
==See also==
*[[Equations of state for hard disks]]
*[[Equations of state for hard sphere mixtures]]
*[[Exact solution of the Percus Yevick integral equation for hard spheres]]
==Related reading==
*[http://dx.doi.org/10.1007/978-3-540-78767-9_3 A. Mulero, C.A. Galán, M.I. Parra and F. Cuadros "Equations of State for Hard Spheres and Hard Disks", Lecture Notes in Physics '''753''' Chapter 3 pp.37-109 (2008)]
[[category: equations of state]]

Equations of state

2021-04-07T03:29:37Z

Proxima Centari: /* Semi-empirical equations of state */ ce

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally there is the [[Equation of State: Ideal Gas|ideal gas equation of state]]. However, one of the first steps towards a description of realistic substances was the famous [[van der Waals equation of state]]. Since then a plethora of semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to better reproduce the foibles of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state |Amagat]]
*[[Antoine equation of state |Antoine]]
*[[Baonza equation of state |Baonza]]
*[[BACK equation of state |BACK]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state |Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb, and Rubin]]
*[[Berthelot equation of state |Berthelot]]
*[[Birch-Murnaghan equation of state |Birch-Murnaghan]]
*[[Boltzmann equation of state |Boltzmann]]
*[[Boynton and Bramley equation of state |Boynton and Bramley]]
*[[Brillouin equation of state |Brillouin]]
*[[Clausius equation of state |Clausius]]
*[[Cole equation of state |Cole]]
*[[BACK equation of state |DIBACK]]
*[[Dieterici equation of state |Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state |Fouché]]
*[[Goebel equation of state |Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Holzapfel equation of state |Holzapfel]]
*[[Jäger equation of state |Jäger]]
*[[Kam equation of state |Kam]]
*[[Kumari-Dass equation of state |Kumari-Dass]]
*[[Lagrange equation of state |Lagrange]]
*[[Leduc equation of state |Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie equation of state |Mie]]
*[[BACK equation of state |MOBACK]]
*[[Mohsen-Nia, Modarress and Mansoori equation of state |Mohsen-Nia, Modarress and Mansoori]]
*[[Murnaghan equation of state |Murnaghan]]
*[[Natanson equation of state |Natanson]]
*[[BACK equation of state |NIK]]
*[[Onnes equation of state |Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state |Porter]]
*[[BACK equation of state |QUABACK]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state |Recknagel]]
*[[Redlich-Kwong equation of state |Redlich-Kwong]]
*[[Reinganum equation of state |Reinganum]]
*[[Rose-Vinet (Universal) equation of state |Rose-Vinet]]
*[[Sarrau equation of state |Sarrau]]
*[[Schiller equation of state |Schiller]]
*[[Schrieber equation of state |Schrieber]]
*[[Smoluchowski equation of state |Smoluchowski]]
*[[Starkweather equation of state |Starkweather ]]
*[[Stiffened equation of state |Stiffened]]
*[[Tait equation of state |Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tillotson equation of state |Tillotson]]
*[[Tumlirz equation of state |Tumlirz]]
*[[Twu-Sim-Tassone equation of state |Twu-Sim-Tassone]]
*[[van der Waals equation of state |van der Waals]]
*[[Walter equation of state |Walter]]
*[[Wohl equation of state |Wohl]]
*[[Phase diagrams of water |Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==See also==
*[[Pair stress approximation]]
*[[Scaled-particle theory]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Murnaghan equation of state

2021-04-07T03:20:52Z

Proxima Centari: Fixed red-link

The '''Murnaghan equation of state''' was developed by Francis D. Murnaghan of John Hopkins University. He presented an [[Equations of state |equation of state]] suitable for representing solids <ref>[http://www.jstor.org/stable/2371405 F. D. Murnaghan "Finite Deformations of an Elastic Solid", American Journal of Mathematics '''59''' pp. 235-260 (1937)]</ref><ref>[http://www.pnas.org/content/30/9/244.full.pdf+html F. D. Murnaghan. "The Compressibility of Media under Extreme Pressures", Proceedings of the National Academy of Sciences of the United States of America '''30''' pp. 244-247 (1944)]</ref>. Having high energy dependence on volume, the equation of state has found considerable use in condensed phase media.

Three [[Thermodynamic relations |derivative relations]] are utilised to lead to the formulation, namely:

:<math>p=-\left(\frac{\partial U}{\partial V}\right)_S</math>

:<math>B=-V\left(\frac{\partial p}{\partial V}\right)_T</math>

:<math>B'=\left(\frac{\partial B}{\partial p}\right)_T</math>

Where <math>p</math> is the [[pressure]], <math>U</math> is the [[internal energy]], <math>V</math> is the volume, <math>B</math> is the [[Compressibility | bulk modulus]] and <math>B'</math> is the pressure derivative of the bulk modulus at constant [[temperature]].

Making the assumption that <math>B'</math> is constant allows one to assign a linear dependence of the bulk modulus on the pressure, which allows specifying one bulk modulus and derivative represented as <math>B_0</math> and <math>B_0'</math> as the material constants. This leads to a relationship for pressure:

:<math>p=\frac{B_0}{B_0'}\left(\left(\frac{V_0}{V}\right)^{B_0'}-1\right)</math>

That can be integrated to represent the energy of the solid:

:<math>U=U_0+\frac{B_0V}{B_0'}\left(\frac{(V_0/V)^{B_0'}}{B_0'-1}+1\right)-\frac{B_0V_0}{B_0'-1}</math>

==Regions of Applicability==
Many papers have since been published on the applicability of the Murnaghan equation of state, many of which have presented alternate equation of state forms for solids. In general, it has been shown that the Murnaghan equation of state breaks down for compression ratios greater than ~0.7-0.8 times the original volume, which occurs as a consequence of the linear dependence of the bulk modulus on pressure and constant bulk modulus pressure derivative. Several popular forms presented to address this issue are the [[Birch-Murnaghan equation of state]], the [[Rose-Vinet (Universal) equation of state|Vinet (Universal) equation of state]], the [[Holzapfel equation of state]], the [[Kumari-Dass equation of state]], and the [[Baonza equation of state]].

==References==
<references/>
[[category: equations of state]]

Thiele hard sphere equation of state

2021-02-10T19:25:47Z

Proxima Centari: Mentioned the Carnahan-Starling equation

{{Stub-general}}
The '''Thiele hard sphere equation of state''' is an [[equations of state|equation of state]] for a [[hard sphere model|hard sphere]] fluid developed by Thiele in 1963. The equation provides a better approximation of the repulsive forces between molecules than the [[Van der Waals equation of state|Van der Waals repulsive term]]. The equation is given below:

<math> Z_{hs} = \frac{p_{hs}V_m}{RT} = \frac{1 - \eta^3}{(1-\eta)^4} = \frac{1 + \eta + \eta^2}{(1-\eta)^3} </math>,

where:
:<math>Z_{hs}</math> is the compressibility factor of the hard sphere fluid;
:<math>p_{hs}</math> is the pressure of the fluid;
:<math>V_m</math> is the molar volume of the fluid;
:<math>T</math> is the absolute temperature of the fluid;
:<math>R</math> is the gas constant; and
:<math>\eta</math> is the [[packing fraction]] of the fluid.

In terms of accuracy, the Thiele equation is superseded by the [[Carnahan-Starling equation of state]]
[[category: Equations of state]]
[[category: hard sphere]]

Thiele hard sphere equation of state

2021-02-10T19:07:05Z

Proxima Centari: Added the Thiele equation

{{Stub-general}}
The '''Thiele hard sphere equation of state''' is an [[equations of state|equation of state]] for a [[hard sphere model|hard sphere]] fluid developed by Thiele in 1963. The equation provides a better approximation of the repulsive forces between molecules than the [[Van der Waals equation of state|Van der Waals repulsive term]]. The equation is given below:

<math> Z_{hs} = \frac{p_{hs}V_m}{RT} = \frac{1 - \eta^3}{(1-\eta)^4} = \frac{1 + \eta + \eta^2}{(1-\eta)^3} </math>,

where:
:<math>Z_{hs}</math> is the compressibility factor of the hard sphere fluid;
:<math>p_{hs}</math> is the pressure of the fluid;
:<math>V_m</math> is the molar volume of the fluid;
:<math>T</math> is the absolute temperature of the fluid;
:<math>R</math> is the gas constant; and
:<math>\eta</math> is the [[packing fraction]] of the fluid.

[[category: Equations of state]]
[[category: hard sphere]]

Thiele hard sphere equation of state

2021-02-10T18:15:46Z

Proxima Centari: Added categories hard sphere and Equations of state

Thiele hard sphere equation of state

2021-02-10T18:12:23Z

Proxima Centari: Created page for Thiele equation

Equations of state for hard spheres

2021-02-08T17:31:20Z

Proxima Centari: Arranged list alphabetically and removed extra copy of link Equations of state for crystals of hard spheres

The following is a list of [[equations of state]] designed for the [[hard sphere model]]:
*[[Carnahan-Starling equation of state]]
*[[Equations of state for crystals of hard spheres]]
*[[Hamad hard sphere equation of state]]
*[[Hansen-Goos hard sphere equation of state]]
*[[Hard hypersphere equation of state |Hard hyperspheres]]
*[[Kolafa-Labík-Malijevský equation of state]]
*[[Liu hard sphere equation of state]]
*[[Santos-Lopez de Haro hard sphere equation of state]]
*[[WC1 and WC2 hard sphere equations of state]]
==See also==
*[[Equations of state for hard disks]]
*[[Equations of state for hard sphere mixtures]]
*[[Exact solution of the Percus Yevick integral equation for hard spheres]]
==Related reading==
*[http://dx.doi.org/10.1007/978-3-540-78767-9_3 A. Mulero, C.A. Galán, M.I. Parra and F. Cuadros "Equations of State for Hard Spheres and Hard Disks", Lecture Notes in Physics '''753''' Chapter 3 pp.37-109 (2008)]
[[category: equations of state]]

Kolafa-Labík-Malijevský equation of state

2021-02-08T17:19:58Z

Proxima Centari: Rectified a grammatical error

{{stub-general}}
The '''Kolafa-Labík-Malijevský equation of state''' is an [[equations of state |equation of state]] for the fluid phase of three-dimensional [[hard sphere model|hard spheres]].
It was generated by the appropriate fitting of very precise [[molecular dynamics]] simulation
results
<ref> [http://dx.doi.org/10.1039/b402792b Jirí Kolafa, Stanislav Labík and Anatol Malijevský, "Accurate equation of state of the hard sphere fluid in stable and metastable regions" Physical Chemistry Chemical Physics '''6''' pp. 2335- (2004)] </ref>. This equation is much more precise than the [[Carnahan-Starling equation of state|Carnahan-Starling]] equation of state.
== References ==
<references />
[[category: Equations of state]]