Virial equation of state - Revision history

103.221.246.79 at 13:51, 17 November 2020

2020-11-17T13:51:56Z

← Older revision		Revision as of 15:51, 17 November 2020
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	*<math> V </math> is the volume		*<math> V </math> is the volume
	*<math> N </math> is the number of molecules		*<math> N </math> is the number of molecules
	*<math>T</math> is the [[temperature]]		*<math> T </math> is the [[temperature]]
	*<math>k_B</math> is the [[Boltzmann constant]]		*<math>k_B</math> is the [[Boltzmann constant]]
	*<math> \rho \equiv \frac{N}{V} </math> is the (number) density		*<math> \rho \equiv \frac{N}{V} </math> is the (number) density

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

2018-01-09T11:15:16Z

References: Added a recent publication

← Older revision		Revision as of 13:15, 9 January 2018
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	*Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928		*Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
	*[http://dx.doi.org/10.1063/1.4929392 Nathaniel S. Barlow, Andrew J. Schultz, Steven J. Weinstein and David A. Kofke "Analytic continuation of the virial series through the critical point using parametric approximants", Journal of Chemical Physics '''143''' 071103 (2015)]		*[http://dx.doi.org/10.1063/1.4929392 Nathaniel S. Barlow, Andrew J. Schultz, Steven J. Weinstein and David A. Kofke "Analytic continuation of the virial series through the critical point using parametric approximants", Journal of Chemical Physics '''143''' 071103 (2015)]
			*[https://doi.org/10.1063/1.5016165 Harold W. Hatch, Sally Jiao, Nathan A. Mahynski, Marco A. Blanco, and Vincent K. Shen "Communication: Predicting virial coefficients and alchemical transformations by extrapolating Mayer-sampling Monte Carlo simulations", Journal of Chemical Physics '''147''' 231102 (2017)]

	[[category:equations of state]]		[[category:equations of state]]

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

2015-09-22T12:31:45Z

References: Added a recent publication

← Older revision		Revision as of 14:31, 22 September 2015
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	*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]		*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
	*Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928		*Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
			*[http://dx.doi.org/10.1063/1.4929392 Nathaniel S. Barlow, Andrew J. Schultz, Steven J. Weinstein and David A. Kofke "Analytic continuation of the virial series through the critical point using parametric approximants", Journal of Chemical Physics '''143''' 071103 (2015)]


	[[category:equations of state]]		[[category:equations of state]]

Carl McBride: Added mention of PIMC calculations

2011-04-08T09:32:22Z

Added mention of PIMC calculations

← Older revision		Revision as of 11:32, 8 April 2011
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	where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]).		where ''f'' is the [[Mayer f-function]] (see also: [[Cluster integrals]]).
	See also:		See also <ref>[http://dx.doi.org/10.1080/002689796173453 M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics '''89''' pp. 1005-1017 (1996)]</ref>
	*[http://dx.doi.org/10.1080/002689796173453 M. S. Wertheim "Fluids of hard convex molecules III. The third virial coefficient", Molecular Physics '''89''' pp. 1005-1017 (1996)]

	==Convergence==		==Convergence==
	For a commentary on the convergence of the virial equation of state see <ref>[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]</ref> and section 3 of <ref>[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]</ref>		For a commentary on the convergence of the virial equation of state see <ref>[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]</ref> and section 3 of <ref>[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]</ref>.
			==Quantum virial coefficients==
			Using the [[path integral formulation]] one can also calculate the virial coefficients of quantum systems <ref>[http://dx.doi.org/10.1063/1.3573564 Giovanni Garberoglio and Allan H. Harvey "Path-integral calculation of the third virial coefficient of quantum gases at low temperatures", Journal of Chemical Physics 134, 134106 (2011)]</ref>.
	==References==		==References==
	<references/>		<references/>

Carl McBride: Added reference by Ursell

2009-11-20T15:59:40Z

Added reference by Ursell

← Older revision		Revision as of 17:59, 20 November 2009
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	The '''virial equation of state''' is used to describe the behavior of diluted gases.		The '''virial equation of state''' is used to describe the behavior of diluted gases.
	It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the		It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the
	density or the pressure. Such an expansion was first introduced by in 1885 by Thiesen <ref>[http://dx.doi.org/10.1002/andp.18852600308 M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik '''24''' pp. 467-492 (1885)]</ref> and extensively studied by Heike Kamerlingh Onnes <ref> H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)</ref>		density or the pressure. Such an expansion was first introduced in 1885 by Thiesen <ref>[http://dx.doi.org/10.1002/andp.18852600308 M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik '''24''' pp. 467-492 (1885)]</ref> and extensively studied by Heike Kamerlingh Onnes <ref> H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)</ref>
	<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]</ref>. In the ~~first case:~~		<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]</ref>, and mathematically by Ursell <ref>[http://dx.doi.org/10.1017/S0305004100011191 H. D. Ursell "The evaluation of Gibbs' phase-integral for imperfect gases", Mathematical Proceedings of the Cambridge Philosophical Society '''23''' pp. 685-697 (1927)]</ref>. One has

	:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.		:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.

Carl McBride: Added an eary reference of Thiesen

2009-11-20T15:51:40Z

Added an eary reference of Thiesen

← Older revision		Revision as of 17:51, 20 November 2009
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	The '''virial equation of state''' is used to describe the behavior of diluted gases.		The '''virial equation of state''' is used to describe the behavior of diluted gases.
	It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the		It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the
	density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes ~~in 1901~~ <ref> H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)</ref>		density or the pressure. Such an expansion was first introduced by in 1885 by Thiesen <ref>[http://dx.doi.org/10.1002/andp.18852600308 M. Thiesen "Untersuchungen über die Zustandsgleichung", Annalen der Physik '''24''' pp. 467-492 (1885)]</ref> and extensively studied by Heike Kamerlingh Onnes <ref> H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)</ref>
	<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]</ref>. In the first case:		<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]</ref>. In the first case:

Carl McBride: Added a book reference.

2009-10-21T09:31:05Z

Added a book reference.

← Older revision		Revision as of 11:31, 21 October 2009
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	The '''virial equation of state''' is used to describe the behavior of diluted gases.		The '''virial equation of state''' is used to describe the behavior of diluted gases.
	It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the		It is usually written as an expansion of the [[compressibility factor]], <math> Z </math>, in terms of either the
	density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes in 1901 (~~Ref~~. 1 and 2). In the first case:		density or the pressure. Such an expansion was first introduced by Heike Kamerlingh Onnes in 1901 <ref> H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)</ref>
			<ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]</ref>. In the first case:

	:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.		:<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}</math>.
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	==Convergence==		==Convergence==
	For a commentary on the convergence of the virial equation of state see ~~Ref 4~~ and section 3 of ~~Ref~~. 5.		For a commentary on the convergence of the virial equation of state see <ref>[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]</ref> and section 3 of <ref>[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]</ref>
	==References==		==References==
	~~# H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)~~		<references/>
	#[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' ~~pp. 125-147 (1902)]~~		'''Related reading'''
	#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]		*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
	~~#[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz~~ and ~~O. Penrose~~ "~~Convergence~~ of ~~Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]~~		*Edward Allen Mason and Thomas Harley Spurling "The virial equation of state", Pergamon Press (1969) ISBN 0080132928
	~~#[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions~~", ~~Journal of Physics: Condensed Matter '''20''' 283102~~ (~~2008~~)]
	[[category:equations of state]]		[[category:equations of state]]

Carl McBride: /* Virial coefficients */ Added internal link.

2008-08-01T11:27:54Z

Virial coefficients: Added internal link.

← Older revision		Revision as of 13:27, 1 August 2008
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	*<math> B_k\left( T \right) </math> is called the k-th virial coefficient		*<math> B_k\left( T \right) </math> is called the k-th virial coefficient
	==Virial coefficients==		==Virial coefficients==
	The [[second virial coefficient]] represents the initial departure from ideal-gas ~~behavior~~		The [[second virial coefficient]] represents the initial departure from [[Ideal gas \|ideal-gas]] behaviour

	:<math>B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math>		:<math>B_{2}(T)= \frac{N_A}{2V} \int .... \int (1-e^{-\Phi/k_BT}) ~d\tau_1 d\tau_2</math>

Carl McBride: Rearranged references slightly.

2008-06-16T14:03:07Z

Rearranged references slightly.

← Older revision		Revision as of 16:03, 16 June 2008
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	==Convergence==		==Convergence==
	~~#[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz~~ and O. ~~Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''~~5~~''' pp~~. ~~841-847 (1964)]~~		For a commentary on the convergence of the virial equation of state see Ref 4 and section 3 of Ref. 5.
	==References==		==References==
	# H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)		# H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Communications from the Physical Laboratory of the University of Leiden '''71''' pp. 3-25 (1901)
	#[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]		#[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]
	#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]		#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
			#[http://dx.doi.org/10.1063/1.1704186 J. L. Lebowitz and O. Penrose "Convergence of Virial Expansions", Journal of Mathematical Physics '''5''' pp. 841-847 (1964)]
	#[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]		#[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]
	[[category:equations of state]]		[[category:equations of state]]

Carl McBride: /* References */ Added a new reference.

2008-06-16T13:58:58Z

References: Added a new reference.

← Older revision		Revision as of 15:58, 16 June 2008
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	#[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]		#[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=436&view=image&startrow=1 H. Kammerlingh Onnes "Expression of the equation of state of gases and liquids by means of series", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''4''' pp. 125-147 (1902)]
	#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]		#[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A Beattie and Walter H Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
			#[http://dx.doi.org/10.1088/0953-8984/20/28/283102 A. J. Masters "Virial expansions", Journal of Physics: Condensed Matter '''20''' 283102 (2008)]
	[[category:equations of state]]		[[category:equations of state]]