Difference between revisions of "Carnahan-Starling equation of state"

From SklogWiki
Jump to: navigation, search
m (Slight tidy)
Line 18: Line 18:
  
 
*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
 
*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
The CS eos is not applicable for packing fractions greater than 0.55 <ref>https://arxiv.org/pdf/cond-mat/0605392.pdf</ref>.
+
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"
 
|-  
 
|-  

Revision as of 14:23, 18 June 2018

CS EoS plot.png

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).


Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.

where:

 \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V}

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 B_4, and those of Clisby and McCoy [3]):

n Clisby and McCoy B_n=n^2+n-2
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):

\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}


Configurational chemical potential:

\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}

Isothermal compressibility:

\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}

where \eta is the packing fraction.

Configurational Helmholtz energy function:

 \frac{ A_{ex}^{CS}}{N k_B T}  = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}

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.

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 }

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]:


Z =  \frac{ 1 + \eta + \eta^2 -  \frac{2}{3}(1+\eta) \eta^3 }{(1-\eta)^3 }.

See also

References