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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Isothermal [[compressibility]]: | Isothermal [[compressibility]]: | ||
:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} | :<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> | ||
where <math>\eta</math> is the [[packing fraction]]. | where <math>\eta</math> is the [[packing fraction]]. | ||
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Configurational [[Helmholtz energy function]]: | Configurational [[Helmholtz energy function]]: | ||
:<math> \frac{ A_{ex}^{CS}}{ | :<math> \frac{ A_{ex}^{CS}}{k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math> | ||
==The 'Percus-Yevick' derivation== | ==The 'Percus-Yevick' derivation== | ||
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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> ). | ||
== See also == | |||
== | |||
*[[Kolafa-Labík-Malijevský equation of state]] | *[[Kolafa-Labík-Malijevský equation of state]] | ||