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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. (Eqn. 10 in Ref 1).
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).


: <math>
: <math>
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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
*<math> N </math> is the number of particles
*<math> N </math> is the number of particles
*<math> k_B  </math> is the [[Boltzmann constant]]
*<math> k_B  </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
 
*<math> T </math> is the absolute temperature
 
*<math> \eta </math> is the [[packing fraction]]:
*<math> \eta </math> is the [[packing fraction]]:


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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==
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"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.22451(26) || 28
|-
| 6 || 39.81515(93)  || 40
|-
| 7 || 53.3444(37) || 54
|-
| 8 || 68.538(18) || 70
|-
| 9 || 85.813(85) || 88
|-
| 10 || 105.78(39) || 108
|}
==Thermodynamic expressions==
==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase  
From the Carnahan-Starling equation for the fluid phase  
the following thermodynamic expressions can be derived
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref>  Eqs. 2.6, 2.7 and 2.8)
(Eq. 2.6, 2.7 and 2.8 in Ref. 2)


[[Pressure]] (compressibility):  
[[Pressure]] (compressibility):  


:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>
:<math>\frac{\beta p^{CS}}{\rho} = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>
 


Configurational [[chemical potential]]:
Configurational [[chemical potential]]:


:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>
:<math>\beta \overline{\mu }^{CS} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>


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}{kT} \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]].
Configurational [[Helmholtz energy function]]:
:<math> \frac{ A_{ex}^{CS}}{N k_B T}  = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>
==The 'Percus-Yevick' derivation==
==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048    G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics  '''54''' pp. 1523-1525 (1971)] </ref>  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.
It is interesting to note (Ref 3 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.


:<math>Z = \frac{ p V}{N k_B T} =  \frac{2}{3} \left[  \frac{(1+\eta+\eta^2)}{(1-\eta)^3}  \right] +  \frac{1}{3} \left[    \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}  \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>
:<math>Z = \frac{ p V}{N k_B T} =  \frac{2}{3} \left[  \frac{(1+\eta+\eta^2)}{(1-\eta)^3}  \right] +  \frac{1}{3} \left[    \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}  \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>


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. 4).
== 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 ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]
 
== References ==
== References ==
<references/>
#[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)]
#[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]
#[http://dx.doi.org/10.1063/1.1675048    G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics  '''54''' pp. 1523-1525 (1971)]
#[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)]
[[Category: Equations of state]]
[[Category: Equations of state]]
[[category: hard sphere]]
[[category: hard sphere]]
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