Editing Carnahan-Starling equation of state
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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. 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> 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. | ||
==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. | | 5 || 28.224512 || 28 | ||
|- | |- | ||
| 6 || 39. | | 6 || 39.8151475 || 40 | ||
|- | |- | ||
| 7 || 53. | | 7 || 53.3444198 || 54 | ||
|- | |- | ||
| 8 || 68. | | 8 || 68.5375488 || 70 | ||
|- | |- | ||
| 9 || 85. | | 9 || 85.8128384 || 88 | ||
|- | |- | ||
| 10 || 105. | | 10 || 105.775104 || 108 | ||
|} | |} | ||
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[[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 <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 <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. | ||
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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> ). | ||
== References == | == References == | ||
<references/> | <references/> | ||
[[Category: Equations of state]] | [[Category: Equations of state]] | ||
[[category: hard sphere]] | [[category: hard sphere]] |