Carnahan-Starling equation of state: Difference between revisions
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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> [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). | ||
: <math> | : <math> | ||
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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== | |||
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 those calculated using a [[Padé approximants | Padé approximant]] by Ree and Hoover <ref>[http://dx.doi.org/10.1063/1.1725286 Francis H. Ree and William G. Hoover "Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks", Journal of Chemical Physics '''40''' pp. 939- (1964)] </ref>: | |||
{| style="width:45%; height:100px" border="1" | |||
|- | |||
| <math>B_n</math> || Ree and Hoover||<math>B_n=n^2+n-2</math> | |||
|- | |||
| 2 || 4 || 4 | |||
|- | |||
| 3 || 10 || 10 | |||
|- | |||
| 4 || 18.36 || 18 | |||
|- | |||
| 5 || 28.2 || 28 | |||
|- | |||
| 6 || 39.5 || 40 | |||
|} | |||
==Thermodynamic expressions== | ==Thermodynamic expressions== | ||
From the Carnahan-Starling equation for the fluid phase | From the Carnahan-Starling equation for the fluid phase | ||
Revision as of 10:26, 28 September 2009
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).
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }. }
where:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } is the pressure
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } is the volume
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } is the number of particles
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B } is the Boltzmann constant
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T } is the absolute temperature
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta } is the packing fraction:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma } is the hard sphere diameter.
Virial expansion
It is interesting to compare the virial coefficients of the Carnahan-Starling equation of state (Eq. 7 of [1]) with those calculated using a Padé approximant by Ree and Hoover [2]:
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n} | Ree and Hoover | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n=n^2+n-2} |
| 2 | 4 | 4 |
| 3 | 10 | 10 |
| 4 | 18.36 | 18 |
| 5 | 28.2 | 28 |
| 6 | 39.5 | 40 |
Thermodynamic expressions
From the Carnahan-Starling equation for the fluid phase the following thermodynamic expressions can be derived (Ref [3] Eqs. 2.6, 2.7 and 2.8)
Pressure (compressibility):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\beta p^{CS}}{\rho} = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}}
Configurational chemical potential:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \overline{\mu }^{CS} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}}
Isothermal compressibility:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} is the packing fraction.
The 'Percus-Yevick' derivation
It is interesting to note (Ref [4] 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.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. [5] ).
References
- ↑ 1.0 1.1 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics 51 pp. 635-636 (1969)
- ↑ Francis H. Ree and William G. Hoover "Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks", Journal of Chemical Physics 40 pp. 939- (1964)
- ↑ 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)
- ↑ 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)
- ↑ 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)