Second virial coefficient - Revision history

Carl McBride: Added 'unkown' mixing rule

2019-12-10T13:36:49Z

Added 'unkown' mixing rule

← Older revision		Revision as of 15:36, 10 December 2019
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	:<math>B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math>		:<math>B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math>
	where <math>x_i</math> and <math>x_j</math> are the mole fractions of the <math>i</math>th and <math>j</math>th component gasses of the mixture.		where <math>x_i</math> and <math>x_j</math> are the mole fractions of the <math>i</math>th and <math>j</math>th component gasses of the mixture.
			==Unknown==
			(<ref>I am not sure where this mixing rule was published</ref>)
			:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
	==See also==		==See also==
	*[[Virial equation of state]]		*[[Virial equation of state]]

Carl McBride: /* Admur and Mason mixing rule */ correction

2019-12-03T11:53:51Z

Admur and Mason mixing rule: correction

← Older revision		Revision as of 13:53, 3 December 2019
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	where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].		where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].
	==Admur and Mason mixing rule==		==Admur and Mason mixing rule==
	~~For the~~ [[second virial coefficient]] of a mixture		The [[second virial coefficient]] for a mixture of <math>n</math> components is given by (Eq. 11 in
	<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref> ~~(this reference does not say this~~)		<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref>)
	:<math>B_{~~ij} = \frac~~{\~~left(B_~~{ii}^{1/3}~~+B_~~{jj}^{1/3}~~\right)~~^3}{8}</math>		:<math>B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math>
			where <math>x_i</math> and <math>x_j</math> are the mole fractions of the <math>i</math>th and <math>j</math>th component gasses of the mixture.

	==See also==		==See also==

129.241.228.167: /* Admur and Mason mixing rule */

2019-12-03T08:33:29Z

Admur and Mason mixing rule

← Older revision		Revision as of 10:33, 3 December 2019
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	==Admur and Mason mixing rule==		==Admur and Mason mixing rule==
	For the [[second virial coefficient]] of a mixture		For the [[second virial coefficient]] of a mixture
	<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref>		<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref> (this reference does not say this)
	:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>		:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>

	==See also==		==See also==
	*[[Virial equation of state]]		*[[Virial equation of state]]

Carl McBride: /* References */ Added a recent publication

2017-12-04T10:42:27Z

References: Added a recent publication

← Older revision		Revision as of 12:42, 4 December 2017
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	*[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)]		*[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)]
	*[http://dx.doi.org/10.1080/00268976.2016.1263763 Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics '''115''' pp. 1191-1199 (2017)]		*[http://dx.doi.org/10.1080/00268976.2016.1263763 Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics '''115''' pp. 1191-1199 (2017)]
			*[https://doi.org/10.1063/1.5004687 Elisabeth Herold, Robert Hellmann, and Joachim Wagner "Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry", Journal of Chemical Physics '''147''' 204102 (2017)]

	[[Category: Virial coefficients]]		[[Category: Virial coefficients]]

Carl McBride: /* References */ Added a recent publication

2017-05-19T14:29:37Z

References: Added a recent publication

← Older revision		Revision as of 16:29, 19 May 2017
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	*[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398- (1941)]		*[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398- (1941)]
	*[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)]		*[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)]
			*[http://dx.doi.org/10.1080/00268976.2016.1263763 Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics '''115''' pp. 1191-1199 (2017)]


	[[Category: Virial coefficients]]		[[Category: Virial coefficients]]

Carl McBride: Added the Admur and Mason mixing rule

2011-11-24T15:24:49Z

Added the Admur and Mason mixing rule

← Older revision		Revision as of 17:24, 24 November 2011
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	where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].		where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]].
			==Admur and Mason mixing rule==
			For the [[second virial coefficient]] of a mixture
			<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref>
			:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
	==See also==		==See also==
	*[[Virial equation of state]]		*[[Virial equation of state]]

Carl McBride: Added some information on evaluation of the integral

2011-05-03T15:08:36Z

Added some information on evaluation of the integral

← Older revision		Revision as of 17:08, 3 May 2011
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	The second virial coefficient, in three dimensions, is given by		The second virial coefficient, in three dimensions, is given by

	:<math>B_{2}(T)= - \frac{1}{2} \int \left( ~~\left\langle~~ \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)~~\right\rangle~~ -1 \right) 4 \pi r^2 dr </math>		:<math>B_{2}(T)= - \frac{1}{2} \int \left( \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right) -1 \right) 4 \pi r^2 dr </math>

	where <math>\Phi_{12}({\mathbf r})</math> is the [[intermolecular pair potential]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. Notice that the expression within the parenthesis		where <math>\Phi_{12}({\mathbf r})</math> is the [[intermolecular pair potential]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. Notice that the expression within the parenthesis
	of the integral is the [[Mayer f-function]].		of the integral is the [[Mayer f-function]].

			In practice the integral is often ''very hard'' to integrate analytically for anything other than, say, the [[Hard sphere: virial coefficients \| hard sphere model]], thus one numerically evaluates

			:<math>B_{2}(T)= - \frac{1}{2} \int \left( \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle -1 \right) 4 \pi r^2 dr </math>

			calculating

			:<math> \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle</math>

			for each <math>r</math> using the numerical integration scheme proposed by Harold Conroy <ref>[http://dx.doi.org/10.1063/1.1701795 Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics '''47''' pp. 5307 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01597437 I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics '''39''' pp. 65-79 (1989)]</ref>.
	==Isihara-Hadwiger formula==		==Isihara-Hadwiger formula==
	The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara		The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara

Carl McBride: /* Isihara-Hadwiger formula */ Corrected journal title

2010-10-08T16:17:41Z

Isihara-Hadwiger formula: Corrected journal title

← Older revision		Revision as of 18:17, 8 October 2010
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	and the Swiss mathematician Hadwiger in 1950		and the Swiss mathematician Hadwiger in 1950
	<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>		<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>
	<ref>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" ~~Experimentia~~ '''7''' pp. 395-398 (1951)]</ref>		<ref>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia '''7''' pp. 395-398 (1951)]</ref>
	<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>		<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>
	The second virial coefficient for any hard convex body is given by the exact relation		The second virial coefficient for any hard convex body is given by the exact relation

Carl McBride: /* Isihara-Hadwiger formula */ Added a DOI

2010-10-08T16:16:24Z

Isihara-Hadwiger formula: Added a DOI

← Older revision		Revision as of 18:16, 8 October 2010
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	and the Swiss mathematician Hadwiger in 1950		and the Swiss mathematician Hadwiger in 1950
	<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>		<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>
	<ref>H. Hadwiger "" Experimentia '''7''' pp. 395- (1951)</ref>		<ref>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experimentia '''7''' pp. 395-398 (1951)]</ref>
	<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>		<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>
	The second virial coefficient for any hard convex body is given by the exact relation		The second virial coefficient for any hard convex body is given by the exact relation
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	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.

	==Hard spheres==		==Hard spheres==
	For the [[hard sphere model]] one has <ref>Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40</ref>		For the [[hard sphere model]] one has <ref>Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40</ref>

Carl McBride: Added an internal link

2010-10-07T16:23:19Z

Added an internal link

← Older revision		Revision as of 18:23, 7 October 2010
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	The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second virial coefficient represents the initial departure from [[ideal gas \|ideal-gas]] ~~behavior~~.		The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second [[Virial equation of state \|virial coefficient]] represents the initial departure from [[ideal gas \|ideal-gas]] behaviour.
	The second virial coefficient, in three dimensions, is given by		The second virial coefficient, in three dimensions, is given by