Carnahan-Starling equation of state: Difference between revisions

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[[Image:CS_EoS_plot.png|thumb|right]]
[[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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*<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>):
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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} =  \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} -1 =  \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 ==  

Revision as of 11:15, 10 November 2020

The Carnahan-Starling equation of state is an approximate (but quite good) equation of state for the fluid phase of the hard sphere model in three dimensions. It is given by (Ref [1] Eqn. 10).

where:

The Carnahan-Starling equation of state is not applicable for packing fractions greater than 0.55 [2].

Virial expansion

It is interesting to compare the virial coefficients of the Carnahan-Starling equation of state (Eq. 7 of [1]) with the hard sphere virial coefficients in three dimensions (exact up to , and those of Clisby and McCoy [3]):

Clisby and McCoy
2 4 4
3 10 10
4 18.3647684 18
5 28.224512 28
6 39.8151475 40
7 53.3444198 54
8 68.5375488 70
9 85.8128384 88
10 105.775104 108

Thermodynamic expressions

From the Carnahan-Starling equation for the fluid phase the following thermodynamic expressions can be derived (Ref [4] Eqs. 2.6, 2.7 and 2.8)

Pressure (compressibility):


Configurational chemical potential:

Isothermal compressibility:

where is the packing fraction.

Configurational Helmholtz energy function:

The 'Percus-Yevick' derivation

It is interesting to note (Ref [5] Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution of the Percus Yevick integral equation for hard spheres via the compressibility route, to one third via the pressure route, i.e.

The reason for this seems to be a slight mystery (see discussion in Ref. [6] ).

Kolafa correction

Jiri Kolafa produced a slight correction to the C-S EOS which results in improved accuracy [7]:

Liu correction

Hongqin Liu proposed a correction to the C-S EOS which improved accuracy by almost two order of magnitude [8]:

See also

References