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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. | |||
: <math> | : <math> | ||
Line 7: | Line 6: | ||
where: | where: | ||
*<math> p </math> is the [[pressure]] | *<math> p </math> is the [[pressure]] | ||
*<math> V </math> is the volume | *<math> V </math> is the volume | ||
Line 18: | Line 16: | ||
*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter. | *<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter. | ||
==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 | ||
( | (Eq. 2.6, 2.7 and 2.8 in Ref. 2) | ||
[[Pressure]] (compressibility): | [[Pressure]] (compressibility): | ||
:<math>\frac{p^{CS} | :<math>\frac{\beta p^{CS}}{\rho} = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math> | ||
Configurational [[chemical potential]]: | Configurational [[chemical potential]]: | ||
:<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}{ | :<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]]. | ||
==The 'Percus-Yevick' derivation== | ==The 'Percus-Yevick' derivation== | ||
It is interesting to note (Ref | 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. | The reason for this seems to be a slight mystery (see discussion in Ref. 4). | ||
== 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]] |