Vega equation of state for hard ellipsoids: Difference between revisions

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The '''Vega''' equation of state for hard (biaxial) [[Hard ellipsoids |ellipsoids]] is given by (Ref. 1 Eq. 20):
The '''Vega''' [[equations of state |equation of state]] for an isotropic fluid of hard (biaxial) [[Hard ellipsoid model |ellipsoids]] is given by
<ref>[http://dx.doi.org/10.1080/002689797169934 Carlos Vega "Virial coefficients and equation of state of hard ellipsoids", Molecular Physics '''92''' pp. 651-665 (1997)]</ref>
(Eq. 20):


:<math>
:<math>
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where <math>Z</math> is the [[compressibility factor]] and <math>y</math> is the [[volume fraction]], given by
where <math>Z</math> is the [[compressibility factor]] and <math>y</math> is the [[volume fraction]], given by
<math>y= \rho V</math> where <math>\rho</math> is the [[number density]].
<math>y= \rho V</math> where <math>\rho</math> is the [[number density]].
The virial coefficients are given by the fits
The [[Virial equation of state |virial coefficients]] are given by the fits


:<math>B_3^* =  10 + 13.094756 \alpha'  - 2.073909\tau' + 4.096689 \alpha'^2  
:<math>B_3^* =  10 + 13.094756 \alpha'  - 2.073909\tau' + 4.096689 \alpha'^2  
Line 27: Line 29:
</math>
</math>


where


:<math>B_n^*= B_n/V^{n-1}</math>,


where <math>B_n^*= B_n/V^{n-1}</math>,


:<math>\tau' = \frac{4 \pi R^2}{S} -1,</math>
:<math>\tau' = \frac{4 \pi R^2}{S} -1,</math>
Line 38: Line 41:


where <math>V</math> is
where <math>V</math> is
the volume, <math>S</math>, the surface area,  and <math>R</math> the mean radius of curvature.
the volume, <math>S</math>, the surface area,  and <math>R</math> the mean radius of curvature. These can be calculated
 
using this [[Mathematica]] notebook file for
For  <math>B_2</math> see [[B_2 for any hard convex body]].
[http://ender.quim.ucm.es/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid].
For  <math>B_2</math> see the page "[[Second virial coefficient]]".
==References==
==References==
#[http://dx.doi.org/10.1080/002689797169934 Carlos Vega "Virial coefficients and equation of state of hard ellipsoids", Molecular Physics '''92''' pp. 651-665 (1997)]
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba  "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45  (2007)]
[[category: equations of state]]
[[category: virial coefficients]]
{{numeric}}

Latest revision as of 12:28, 19 February 2010

The Vega equation of state for an isotropic fluid of hard (biaxial) ellipsoids is given by [1] (Eq. 20):

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 = 1+B_2^*y + B_3^*y^2 + B_4^*y^3 + B_5^*y^4 + \frac{B_2}{4} \left( \frac{1+y+y^2-y^3}{(1-y)^3} -1 -4y -10y^2 -18.3648y^3 - 28.2245y^4 \right) }

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 Z} is the compressibility factor and 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 y} is the volume fraction, given by 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 y= \rho V} where is the number density. The virial coefficients are given by the fits

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_3^* = 10 + 13.094756 \alpha' - 2.073909\tau' + 4.096689 \alpha'^2 + 2.325342\tau'^2 - 5.791266\alpha' \tau',}


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_4^* = 18.3648 + 27.714434\alpha' - 10.2046\tau' + 11.142963\alpha'^2 + 8.634491\tau'^2 - 28.279451\alpha' \tau' - 17.190946\alpha'^2 \tau' + 24.188979\alpha' \tau'^2 + 0.74674\alpha'^3 - 9.455150\tau'^3,}

and

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_5^* = 28.2245 + 21.288105\alpha' + 4.525788\tau' + 36.032793\alpha'^2 + 59.0098\tau'^2 - 118.407497\alpha' \tau' + 24.164622\alpha'^2 \tau' + 139.766174\alpha' \tau'^2 - 50.490244\alpha'^3 - 120.995139\tau'^3 + 12.624655\alpha'^3\tau', }

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 B_n^*= B_n/V^{n-1}} ,


and

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 \alpha' = \frac{RS}{3V}-1.}

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 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 S} , the surface area, and 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 R} the mean radius of curvature. These can be calculated using this Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid. For 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_2} see the page "Second virial coefficient".

References[edit]

  1. ↑ Carlos Vega "Virial coefficients and equation of state of hard ellipsoids", Molecular Physics 92 pp. 651-665 (1997)

Related reading

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