SklogWiki - User contributions [en]

Kumari-Dass equation of state

2011-11-13T05:18:24Z

Josephrpeterson:

Another model based on a linear bulk modulus equation, in the spirit of the [[Murnaghan equation of state]] was presented by Kumari and Dass<ref>Kumari, M. and Dass, N. An equation of state applied to 50 solids. ''Condensed Matter'', 2:3219, 1990.</ref>. The equation of state does not correctly model the bulk modulus as the pressure tends towards infinity, as it remains bounded. This is apparent in the equation relating the bulk modulus to pressure:

:<math>B=B_0+\frac{B_0'}{\lambda}\left(1-e^{-\lambda P}\right)</math>

where <math>B_0</math> is the isothermal bulk modulus, <math>B_0'</math> is the pressure derivative of the bulk modulus and <math>\lambda</math> is a softening parameter for the bulk modulus. This leads to a equation for pressure dependent on these parameters of the form:

:<math>P=\frac{1}{\lambda}\left[\frac{\lambda B_0 \left(V/V_0\right)^{-\lambda B_0 + B_0'}+B_0'}{\lambda B_0 + B_0'}\right]</math>

==References==
<references/>

[[category: equations of state]]

Kumari-Dass equation of state

2011-11-13T05:14:38Z

Josephrpeterson: Created page with "Another model based on a linear bulk modulus equation, in the spirit of the Murnaghan equation of state was presented by Kumari and Dass<ref>Kumari, M. and Dass, N. An equati..."

Baonza equation of state

2011-11-07T03:53:49Z

Josephrpeterson: Created page with "Baonza, ''et al'' formulated an equation based on a linear bulk modulus called the '''Baonza equation of state'''<ref>V.G. Baonza, M. Taravillo, M. Caceres, and J. Nunez, Univers..."

Baonza, ''et al'' formulated an equation based on a linear bulk modulus called the '''Baonza equation of state'''<ref>V.G. Baonza, M. Taravillo, M. Caceres, and J. Nunez, Universal features of the equation of state of solids from a pseudospinodal hypothesis, ''Phys. Rev. B'' 53:5252, 1996.</ref>. It has a simple analytical form but also gives similar accuracy to the [[Rose-Vinet (Universal) equation of state]]. The equation of state is:

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

where <math>B_0</math> is the isothermal bulk modulus, <math>B_0'</math> is the pressure derivative of the bulk modulus and <math>\gamma</math> relates the bulk modulus and its pressure derivative via:

:<math>B=B_0\left(1+\frac{B_0'}{B_0}P\right)^{\gamma}</math>

==References==
<references/>

[[category: equations of state]]

Rose-Vinet (Universal) equation of state

2011-11-07T03:33:38Z

Josephrpeterson:

==Vinet==

In order to rectify the excessive stiffness of the [[Murnaghan equation of state]] as well as represent the exponential dependence of the repulsion as solid undergoes infinite compression, Vinet proposed an equation of state that has widely as either the '''Vinet equation of state''' or '''Universal equation of state'''<ref>P.Vinet, J.R. Smith, J. Ferrante, and J.H. Rose. Temperature effects on the universal equation of state of solids. ''Phys. Rev. B'', 35:1945,1987</ref>. The latter due to the fact that the equation of state was formulated so that one form could represent all solids, depending only on the calibration point. Using the shorthand for the cube root specific volume:

:<math>\eta=\left(\frac{V}{V_0}\right)^{\frac{1}{3}}</math>

the equation of state is:

:<math>p=3B_0\left(\frac{1-\eta}{\eta^2}\right)e^{\frac{3}{2}(B_0'-1)(1-\eta)}</math>

==Rose-Vinet==

==References==
<references/>

[[category: equations of state]]

Rose-Vinet (Universal) equation of state

2011-11-07T03:32:37Z

Josephrpeterson: Created page with "==Vinet== In order to rectify the excessive stiffness of the Murnaghan equation of state as well as represent the exponential dependence of the repulsion as solid undergoes ..."

Twu-Sim-Tassone equation of state

2011-11-07T03:03:23Z

Josephrpeterson:

Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>Twu, C.H., et al, "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 <math>Z_c</math> of 0.2962, it better represents many the compressibility of than the [[Redlich-Kwong equation of state]], including the Soave modified version, and the [[Peng and Robinson equation of state]]. This allows it to better represent long chain hydrocarbons<ref>Twu, C.H., Sim, W.D., Tassone, V. "Getting a Handle on Advanced Cubic Equation of State." ''Measurement & Control'', pp. 58-65, (2002).</ref>.

The equation follows the general cubic form resulting in the equation:

:<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, 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:

:<math>a=\frac{0.427481R^2T_c^2}{P_c}</math>
:<math>b=\frac{0.086641RT_c}{P_c}</math>

==References==
<references/>

[[category: equations of state]]

Redlich-Kwong equation of state

2011-11-07T02:59:40Z

Josephrpeterson:

The Redlich-Kwong equation of state is<ref>[http://dx.doi.org/10.1021/cr60137a013 Otto Redlich and J. N. S. Kwong "On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions", Chemical Reviews '''44''' pp. 233 - 244 (1949)]</ref>:

:<math>\left[ p + \frac{a}{T^{1/2}v(v+b)} \right] (v-b) = RT</math>

where

:<math>a= 0.4278 \frac{R^2T_c^{2.5}}{P_c}</math>

and

:<math>b= 0.0867 \frac{RT_c}{P_c}</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the pressure at the critical point.

==Soave Modification==
A modification of the the Redlich-Kwong equation of state was presented by Giorgio Soave in order to allow better representation of non-spherical molecules<ref>[http://dx.doi.org/10.1016/0009-2509(72)80096-4 Giorgio Soave "Equilibrium constants from a modified Redlich-Kwong equation of state", Chemical Engineering Science '''27''' pp. 1197-1203 (1972)]</ref>. In order to do this, the square root temperature dependence was replaced with a temperature dependent acentricity factor:

:<math>\alpha(T)=\left(1+\left(0.48508+1.55171\omega-0.15613\omega^2\right)\left(1-\sqrt\frac{T}{T_c}\right)\right)^2 </math>

where <math>T_c</math> is the critical temperature and <math>\omega</math> is the acentric factor for the gas. This leads to an equation of state of the form:

:<math> \left[p+\frac{a\alpha(T)}{v(v+b)}\right]\left(v-b\right)=RT</math>

or equivalently:

:<math> p=\frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)}</math>

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

Redlich-Kwong equation of state

2011-11-07T02:57:53Z

Josephrpeterson:

The Redlich-Kwong equation of state is<ref>[http://dx.doi.org/10.1021/cr60137a013 Otto Redlich and J. N. S. Kwong "On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions", Chemical Reviews '''44''' pp. 233 - 244 (1949)]</ref>:

:<math>\left[ p + \frac{a}{T^{1/2}v(v+b)} \right] (v-b) = RT</math>

where

:<math>a= 0.4278 \frac{R^2T_c^{2.5}}{P_c}</math>

and

:<math>b= 0.0867 \frac{RT_c}{P_c}</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the pressure at the critical point.

==Soave Modification==
A modification of the the Redlich-Kwong equation of state was presented by Giorgio Soave in order to allow better representation of non-spherical molecules<ref>[http://dx.doi.org/10.1016/0009-2509(72)80096-4 Giorgio Soave "Equilibrium constants from a modified Redlich-Kwong equation of state", Chemical Engineering Science '''27''' pp. 1197-1203 (1972)]</ref>. In order to do this, the square root temperature dependence was replaced with a temperature dependent acentricity factor:

:<math>\alpha=\left(1+\left(0.48508+1.55171\omega-0.15613\omega^2\right)\left(1-\sqrt\frac{T}{T_c}\right)\right)^2 </math>

where <math>T_c</math> is the critical temperature and <math>\omega</math> is the acentric factor for the gas. This leads to an equation of state of the form:

:<math> \left[p+\frac{a\alpha}{v(v+b)}\right]\left(v-b\right)=RT</math>

or equivalently:

:<math> p=\frac{RT}{v-b}-\frac{a\alpha}{v(v+b)}</math>

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

Twu-Sim-Tassone equation of state

2011-11-04T22:38:37Z

Josephrpeterson:

Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>Twu, C.H., et al, "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 <math>Z_c</math> of 0.2962, it better represents many the compressibility of than the [[Redlich-Kwong equation of state]], including the Soave modified version, and the [[Peng and Robinson equation of state]]. This allows it to better represent long chain hydrocarbons<ref>Twu, C.H., Sim, W.D., Tassone, V. "Getting a Handle on Advanced Cubic Equation of State." ''Measurement & Control'', pp. 58-65, (2002).</ref>.

The equation follows the general cubic form resulting in the equation:

:<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, 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]].

==References==
<references/>

[[category: equations of state]]

Twu-Sim-Tassone equation of state

2011-11-04T22:37:54Z

Josephrpeterson:

Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>Twu, C.H., et al, "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 <math>Z_c</math> of 0.2962, it better represents many the compressibility of than the [[Redlich-Kwong equation of state]], including the Soave modified version, and the [[Peng and Robinson equation of state]] and has been shown to better represent long chain hydrocarbons<ref>Twu, C.H., Sim, W.D., Tassone, V. "Getting a Handle on Advanced Cubic Equation of State." ''Measurement & Control'', pp. 58-65, (2002).</ref>.

The equation follows the general cubic form resulting in the equation:

:<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, 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]].

==References==
<references/>

[[category: equations of state]]

Twu-Sim-Tassone equation of state

2011-11-04T22:36:38Z

Josephrpeterson:

Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>Twu, C.H., et al, "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 <math>Z_c</math> of 0.2962, it better represents many the compressibility of than the [[Redlich-Kwong equation of state]], including the Soave modified version, and the [[Peng and Robinson equation of state]] and has been shown to better represent long chain hydrocarbons<ref>Twu, C.H., Sim, W.D., Tassone, V. "Getting a Handle on Advanced Cubic Equation of State." ''Measurement & Control'', pp. 58-65, (2002).</ref>.

The equation follows the general cubic form resulting in the equation:

:<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, 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]].

==References==
<references/>

Twu-Sim-Tassone equation of state

2011-11-04T22:33:55Z

Josephrpeterson: Created page with "Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state..."

Twu, Sim and Tassone presented a cubic equation of state for accurate representation of hydrocarbon that has become known as the '''Twu-Sim-Tassone''' or '''TST equation of state'''<ref>Twu, C.H., et al, "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 <math>Z_c</math> of 0.2962, it better represents many the compressibility of than the Redlich-Kwong, including the Soave modified version, and the Peng-Robinson equation of state and has been shown to better represent long chain hydrocarbons<ref>Twu, C.H., Sim, W.D., Tassone, V. "Getting a Handle on Advanced Cubic Equation of State." ''Measurement & Control'', pp. 58-65, (2002).</ref>.

The equation follows the general cubic form resulting in the equation:

:<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, and <math>a</math> and <math>b</math> are the attractive and repulsive parameters akin to those of the [http://www.sklogwiki.org/SklogWiki/index.php/Van_der_Waals_equation_of_state van Der Waals equation of state].

==References==
<references/>

User:Josephrpeterson

2011-11-04T22:05:02Z

Josephrpeterson:

==Joseph R. Peterson==

[[File:jrp.jpeg]]

'''Institution:''' [http://www.sci.utah.edu/ Scientific Computing and Imaging Institute, University of Utah]

'''Position:''' Staff Specialist

'''Research Group:''' [http://csafe.utah.edu/ Center for Simulation of Accidental Fires and Explosions]

'''Undergraduate Institution:''' [http://www.utah.edu/ University of Utah]

'''Degree:''' Chemistry BS, Computer Science BS

'''Years:''' 2007-2012

==Publications list==

[http://csafe.utah.edu/pdf/papers/Tgrid10PAPPSv4.pdf Berzins, M.; Luitjens, J.; Meng, Q.; Harman, T.; Wight, C.; Peterson, J. "Uintah - a scalable framework for hazard analysis", Proceedings of the Teragrid 2010 Conference, No 3., Note: Awarded Best Paper in the Science Track!, 2010]

File:Jrp.jpeg

2011-11-04T22:01:45Z

Josephrpeterson: uploaded a new version of "File:Jrp.jpeg"

File:Jrp.jpeg

2011-11-04T21:59:58Z

Josephrpeterson:

User:Josephrpeterson

2011-11-04T21:57:09Z

Josephrpeterson: Created page with "==Joseph R. Peterson== File:jrp.jpeg '''Institution:''' [http://www.sci.utah.edu/ Scientific Computing and Imaging Institute, University of Utah] '''Position:''' Staff Spe..."

Birch-Murnaghan equation of state

2011-11-03T02:56:13Z

Josephrpeterson:

An extension, or rather a generalization, of the [[Murnaghan equation of state]] was presented by Albert F. Birch in 1947. <ref> [http://adsabs.harvard.edu/abs/1947PhRv...71..809B Birch, Francis (1947). "Finite Elastic Strain of Cubic Crystals". Physical Review 71 (11): 809–824.] </ref> It has become known as the '''Birch-Murnaghan equation of state'''. The generalization followed from the identification that the strain energy could be approximated as a taylor series based on the finite strain in the crystal. Common orders include first, second and third, with the first reducing to the Murnaghan equation of state.

Since finite strain is represented as:

:<math> f=\frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{2/3}-1\right]</math>

An energy for the strain is defined as a Taylor expansion:

:<math> E=a+bf+cf^2+df^3...</math>

A pressure, then is the derivative of this equation:

:<math>P=-\left(\frac{\partial E}{\partial f}\right)\left(\frac{\partial f}{\partial V}\right)</math>

The second order form is thus:

:<math> P=\frac{3K_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right] </math>

Where <math>K_0</math> is the isothermal (or calibration) bulk modulus. However, since this form is not dependent on the bulk modulus derivative, <math>K_0'</math>, it is rarely used and either the first order or third order form are used. The third order shows increased accuracy over the Murnaghan equation of state and has a relatively simple analytical form:

:<math> P=\frac{3K_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right]\left[1-\frac{3}{4}\left(K_0'-4\right)\left(\left(\frac{V_0}{V}\right)^{2/3}-1\right)\right]</math>

==References==
<references/>

Birch-Murnaghan equation of state

2011-11-03T02:55:12Z

Josephrpeterson: Created page with "An extension, or rather a generalization, of the Murnaghan equation of state was presented by Albert F. Birch in 1947. <ref> [http://adsabs.harvard.edu/abs/1947PhRv...71..809..."

An extension, or rather a generalization, of the [[Murnaghan equation of state]] was presented by Albert F. Birch in 1947. <ref> [http://adsabs.harvard.edu/abs/1947PhRv...71..809B Birch, Francis (1947). "Finite Elastic Strain of Cubic Crystals". Physical Review 71 (11): 809–824.] </ref> It has become known as the '''Birch-Murnaghan equation of state'''. The generalization followed from the identification that the strain energy could be approximated as a taylor series based on the finite strain in the crystal. Common orders include first, second and third, with the first reducing to the Murnaghan equation of state.

Since finite strain is represented as:

:<math> f=\frac{1}{2}\left[\left(\frac{V_0}{V}\right)^{2/3}-1\right]</math>

An energy for the strain is defined as a Taylor expansion:

:<math> E=a+bf+cf^2+df^3...</math>

A pressure, then is the derivative of this equation:

:<math>P=-\left(\frac{\partial E}{\partial f}\right)\left(\frac{\partial f}{\partial V}\right)</math>

The second order form is thus:

:<math> P=\frac{3K_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right] </math>

Where <math>K_0</math> is the isothermal (or calibration) bulk modulus. However, since this form is not dependent on the bulk modulus derivative, <math>K_0'</math>, it is rarely used and either the first order or third order form are used. The third order shows increased accuracy over the Murnaghan equation of state and has a relatively simple analytical form:

:<math> P=\frac{3K_0}{2}\left[\left(\frac{V_0}{V}\right)^{7/3}-\left(\frac{V_0}{V}\right)^{5/3}\right]\left[1-\frac{3}{4}\left(K_0'-4\right)\left(\frac{V}{V_0}\right)^{-2/3}-1\right]</math>

==References==
<references/>

Equations of state

2011-11-03T02:23:46Z

Josephrpeterson: /* Semi-empirical equations of state */

'''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]]
*[[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]]
*[[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]]
*[[Murnaghan equation of state | Murnaghan]]
*[[Natanson equation of state | Natanson]]
*[[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]]
*[[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 ]]
*[[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

2011-11-02T16:30:56Z

Josephrpeterson: Created page with "Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids <ref> [http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pm..."

Francis D. Murnaghan of John Hopkins University presented an equation of state suitable for representing solids <ref> [http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1078704 F.D. Murnaghan. "The Compressibility of Media under Extreme Pressures. ''Proceedings of hte National Academy of Sciences of the United States of America'', 30(9):224-247, 1944] </ref>. The equation has become known as the '''Murnaghan''' equation of state. Having high energy dependence on volume, the equation of state has found considerable use in condensed phase media.

Three derivative relations are assumed that lead to the formulation:

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

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

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

Where <math>P</math>, <math>E</math>, <math>V</math>, <math>K</math> and <math>K'</math> are the pressure, energy, volume, bulk modulus, and the pressure derivative of the bulk modulus at constant temperature.

Making the assumption that <math>K'</math> is constant allows to assign a linear dependence of the bulk modulus on the pressure, which allows specifying one bulk modulus and derivative represented as <math>K_0</math> and <math>K_0'</math> as the material constants. This leads to a relationship for pressure:

:<math>P=\frac{K_0}{K_0'}\left(\left(\frac{V_0}{V}\right)^{K_0'}-1\right)</math>

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

:<math>E=E_0+\frac{K_0V}{K_0'}\left(\frac{(V_0/V)^{K_0'}}{K_0'-1}+1\right)-\frac{K_0V_0}{K_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 Vinet (Universal) equation of state, the Holzapfel equation of state, the Kumari-Dass equation of state, and the Baonza equation of state.

==References==
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