Editing Carnahan-Starling equation of state

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[[Image:CS_EoS_plot.png|thumb|350px|right]]
The '''Carnahan-Starling equation of state'''  is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres"  Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).
The '''Carnahan-Starling equation of state'''  is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres"  Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).


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where:
where:
*<math> Z </math> is the [[compressibility factor]]
*<math> p </math> is the [[pressure]]
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> V </math> is the volume
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*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 <ref>[https://arxiv.org/abs/cond-mat/0605392 Hongqin Liu "A very accurate hard sphere equation of state over the entire stable and metstable region", arXiv:cond-mat/0605392 (2006)]</ref>.
==Virial expansion==
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"></ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0  Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0  Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
{| style="width:40%; height:100px" border="1"
|-  
|-  
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| 4 || 18.3647684 || 18
| 4 || 18.3647684 || 18
|-  
|-  
| 5 || 28.22451(26) || 28
| 5 || 28.224512 || 28
|-  
|-  
| 6 || 39.81515(93) || 40
| 6 || 39.8151475 || 40
|-
|-
| 7 || 53.3444(37) || 54
| 7 || 53.3444198 || 54
|-
|-
| 8 || 68.538(18) || 70
| 8 || 68.5375488 || 70
|-
|-
| 9 || 85.813(85) || 88
| 9 || 85.8128384 || 88
|-
|-
| 10 || 105.78(39) || 108
| 10 || 105.775104 || 108
|}
|}


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Isothermal [[compressibility]]:
Isothermal [[compressibility]]:


:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} -1 =  \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>
:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} =  \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>


where <math>\eta</math> is the [[packing fraction]].
where <math>\eta</math> is the [[packing fraction]].
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The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).
== Kolafa correction ==
Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy <ref>[http://dx.doi.org/10.1063/1.4870524 Miguel Robles, Mariano López de Haro and Andrés Santos "Note: Equation of state and the freezing point in the hard-sphere model", Journal of Chemical Physics '''140''' 136101 (2014)]</ref>:
: <math>
Z =  \frac{ 1 + \eta + \eta^2 -  \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.
</math>
== Liu correction ==
Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude <ref>[https://arxiv.org/abs/2010.14357 Hongqin Liu "Carnahan Starling type equations of state for stable hard disk and hard sphere fluids", arXiv:2010.14357]</ref>:
: <math>
Z =  \frac{ 1 + \eta + \eta^2 -  \frac{8}{13}\eta^3 - \eta^4 + \frac{1}{2}\eta^5 }{(1-\eta)^3 }.
</math>


== See also ==  
== See also ==  
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