Cluster integrals - Revision history

Carl McBride: Added paper by Hellmann and Bich

2011-09-01T13:21:30Z

Added paper by Hellmann and Bich

← Older revision		Revision as of 15:21, 1 September 2011
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	The first three cluster integrals are (~~Ref. 1~~ Eq. 13.6)		The first three cluster integrals are (Eq. 13.6 in <ref>Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13</ref>)
	:<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>		:<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>

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	note <math>b_4 = \frac{2}{3}\beta_1^3 + \beta_1 \beta_2 + \frac{1}{4}\beta_3</math>		note <math>b_4 = \frac{2}{3}\beta_1^3 + \beta_1 \beta_2 + \frac{1}{4}\beta_3</math>
			==Hellmann and Bich diagrams==
			Hellmann and Bich have rederived the [[virial equation of state]] from the [[Grand canonical ensemble#Grand canonical partition function \| grand canonical partition function]] without restricting themselves to pairwise [[intermolecular pair potential]]s <ref>[http://dx.doi.org/10.1063/1.3626524 Robert Hellmann and Eckard Bich "A systematic formulation of the virial expansion for nonadditive interaction potentials", Journal of Chemical Physics '''135''' 084117 (2011)]</ref>. This leads to expressions for the virial coefficients that, for <math>B_6</math> and beyond, require the evaluation of far fewer diagrams when compared to the original diagrams of Mayer or to the reformulation of Ree and Hoover <ref>[http://dx.doi.org/10.1063/1.1726136 Francis H. Ree and William G. Hoover "Reformulation of the Virial Series for Classical Fluids", Journal of Chemical Physics '''41''' 1635 (1964)]</ref>.
	==See also==		==See also==
	*[[Cluster diagrams]]		*[[Cluster diagrams]]
	==References==		==References==
	~~# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.~~		<references/>
	#[http://dx.doi.org/10.1016/0003-4916(58)90058-7 Edwin E. Salpeter "On Mayer's theory of cluster expansions", Annals of Physics '''5''' pp. 183-223 (1958)]		;Related reading
			*[http://dx.doi.org/10.1016/0003-4916(58)90058-7 Edwin E. Salpeter "On Mayer's theory of cluster expansions", Annals of Physics '''5''' pp. 183-223 (1958)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

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

2008-04-28T12:37:05Z

References: Added a reference.

← Older revision		Revision as of 14:37, 28 April 2008
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	==References==		==References==
	# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.		# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.
			#[http://dx.doi.org/10.1016/0003-4916(58)90058-7 Edwin E. Salpeter "On Mayer's theory of cluster expansions", Annals of Physics '''5''' pp. 183-223 (1958)]
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 14:23, 31 July 2007

2007-07-31T14:23:50Z

← Older revision		Revision as of 16:23, 31 July 2007
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	The simplest cluster is that consisting of a single molecule, not bound to any other.		The simplest cluster is that consisting of a single molecule, not bound to any other.
	A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:		A cluster of three specified identical molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:

	[[Image:ijk.png]]		[[Image:ijk.png]]
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	The first three cluster integrals are (Ref. 1 Eq. 13.6)		The first three cluster integrals are (Ref. 1 Eq. 13.6)
	:<math>b_1 = \frac{1}{V}\int d\tau_1 =1</math>		:<math>b_1 = \frac{1}{1!V}\int d\tau_1 =1</math>

	Ref. 1 Eq. 13.7:		Ref. 1 Eq. 13.7:

	:<math>b_2 = \frac{1}{2V} \iint f(r_{12}) d\tau_2 d\tau_1 = \frac{1}{2}\int_0^\infty 4\pi r^2 f(r) dr</math>		:<math>b_2 = \frac{1}{2!V} \iint f(r_{12}) d\tau_2 d\tau_1 = \frac{1}{2}\int_0^\infty 4\pi r^2 f(r) dr</math>

	and Ref. 1 Eq. 13.8:		and Ref. 1 Eq. 13.8:

	:<math>b_3 = \frac{1}{6V} \iiint (f_{31} f_{21} + f_{32}f_{31} + f_{32}f_{21} + f_{32}f_{31}f_{21}) d\tau_3 d\tau_2 d\tau_1</math>		:<math>b_3 = \frac{1}{3!V} \iiint (f_{31} f_{21} + f_{32}f_{31} + f_{32}f_{21} + f_{32}f_{31}f_{21}) d\tau_3 d\tau_2 d\tau_1</math>

	using the [[Mayer f-function]] notation.		using the [[Mayer f-function]] notation.
			==Irreducible clusters==
			Irreducible clusters are denoted by <math>\beta_k</math>

			:<math>\beta_1 = \int f_{31} d\tau_3 = \frac{1}{V}\iint f_{12}d\tau_1 d\tau_2 =\int_0^\infty 4 \pi r^2 f(r) dr</math>

			note <math>b_2 = \frac{1}{2}\beta_1</math>.



			:<math>\beta_2 = \frac{1}{2V}\iiint f_{32}f_{31}f_{21} d\tau_1 d\tau_2 d\tau_3 </math>

			note <math>b_3 = \frac{1}{2} \beta_1^2 + \frac{1}{3}\beta_2</math>

			:<math>\beta_3 = \frac{1}{6V}\iiiint (3f_{43}f_{32}f_{21}f_{41}+6f_{43}f_{32}f_{21}f_{41}f_{31} + f_{43}f_{32}f_{21}f_{41}f_{31}f_{42})d\tau_1 d\tau_2 d\tau_3 d\tau_4 </math>

			note <math>b_4 = \frac{2}{3}\beta_1^3 + \beta_1 \beta_2 + \frac{1}{4}\beta_3</math>
	==See also==		==See also==
	*[[Cluster diagrams]]		*[[Cluster diagrams]]

Carl McBride at 13:46, 31 July 2007

2007-07-31T13:46:14Z

@@ Line 5: / Line 5: @@
 The simplest cluster is that consisting of a single molecule, not bound to any other.
 A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:
 ==See also==
 *[[Cluster diagrams]]
 ==References==
 # Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.
 [[Category: Statistical mechanics]]

Carl McBride: /* References */

2007-07-31T13:16:09Z

References

← Older revision		Revision as of 15:16, 31 July 2007
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	*[[Cluster diagrams]]		*[[Cluster diagrams]]
	==References==		==References==
	# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940)		# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940) Chapter 13.
	[[Category: Statistical mechanics]]		[[Category: Statistical mechanics]]

Carl McBride at 13:12, 31 July 2007

2007-07-31T13:12:12Z

← Older revision		Revision as of 15:12, 31 July 2007
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			In an [[ideal gas]] there are no intermolecular interactions. However, in an imperfect or real gas, this is not so, and the
			[[second virial coefficient]] is other than zero. Mayer and Mayer developed a theoretical treatment of the
			[[Virial equation of state \|virial coefficients]] in terms of '''cluster integrals'''.

	The simplest cluster is that consisting of a single molecule, not bound to any other.		The simplest cluster is that consisting of a single molecule, not bound to any other.
	A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:		A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:

Carl McBride: New page: The simplest cluster is that consisting of a single molecule, not bound to any other. A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways: ==Se...

2007-07-31T13:00:40Z

New page: The simplest cluster is that consisting of a single molecule, not bound to any other. A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways: ==Se...

New page

The simplest cluster is that consisting of a single molecule, not bound to any other.
A cluster of three specified molecules, ''i'', ''j'' and ''k'' may be formed in any of four ways:
==See also==
*[[Mayer f-function]]
*[[Cluster diagrams]]
==References==
# Joseph Edward Mayer and Maria Goeppert Mayer "Statistical Mechanics" John Wiley and Sons (1940)
[[Category: Statistical mechanics]]