Van der Waals equation of state - Revision history

Carl McBride: /* Critical point */ Replaced P_c with p_c to be consistent

2011-11-07T15:25:42Z

Critical point: Replaced P_c with p_c to be consistent

← Older revision		Revision as of 17:25, 7 November 2011
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	:<math>a= \frac{27}{64}\frac{R^2T_c^2}{~~P_c~~}</math>		:<math>a= \frac{27}{64}\frac{R^2T_c^2}{p_c}</math>


	:<math>b= \frac{RT_c}{~~8P_c~~}</math>		:<math>b= \frac{RT_c}{8p_c}</math>

	==Virial form==		==Virial form==

Carl McBride: /* Critical exponents */ Fixed internal link

2011-07-20T13:08:31Z

Critical exponents: Fixed internal link

← Older revision		Revision as of 15:08, 20 July 2011
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	[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]		[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
	==Critical exponents==		==Critical exponents==
	The [[critical exponents]] of the Van der Waals equation of state place it in the [[Universality classes#Mean field \| mean field universality class]].		The [[critical exponents]] of the Van der Waals equation of state place it in the [[Universality classes#Mean-field \| mean field universality class]].

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

Carl McBride: /* Critical exponents */ Replaced section with a link

2011-07-20T13:02:08Z

Critical exponents: Replaced section with a link

← Older revision		Revision as of 15:02, 20 July 2011
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	[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]		[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
	==Critical exponents==		==Critical exponents==
	The [[critical exponents]] of ~~are derived as follows <ref>Linda E. Reichl "A Modern Course~~ in ~~Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9~~.~~4 </ref>:~~		The [[critical exponents]] of the Van der Waals equation of state place it in the [[Universality classes#Mean field \| mean field universality class]].

	~~====Heat capacity exponent: <math>\alpha</math>====~~
	~~(final result: <math>\alpha=0</math>)~~
	~~====Magnetic order parameter exponent: <math>\beta</math>====~~
	~~(final result: <math>\beta=1/2</math>)~~
	~~====Susceptibility exponent: <math>\gamma</math>====~~
	~~(final result: <math>\gamma=1</math>)~~

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

Carl McBride: /* Critical exponents */ Added final results for the moment

2011-07-15T15:52:04Z

Critical exponents: Added final results for the moment

← Older revision		Revision as of 17:52, 15 July 2011
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	====Heat capacity exponent: <math>\alpha</math>====		====Heat capacity exponent: <math>\alpha</math>====
			(final result: <math>\alpha=0</math>)
	====Magnetic order parameter exponent: <math>\beta</math>====		====Magnetic order parameter exponent: <math>\beta</math>====
			(final result: <math>\beta=1/2</math>)
	====Susceptibility exponent: <math>\gamma</math>====		====Susceptibility exponent: <math>\gamma</math>====
			(final result: <math>\gamma=1</math>)

	==See also==		==See also==
	*[[Zeno line#Batchinsky law \| Batchinsky law]]		*[[Zeno line#Batchinsky law \| Batchinsky law]]

Carl McBride: Started section on Critical exponents

2011-07-15T15:46:08Z

Started section on Critical exponents

← Older revision		Revision as of 17:46, 15 July 2011
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	The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:		The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
	[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]		[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
			==Critical exponents==
			The [[critical exponents]] of are derived as follows <ref>Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4 </ref>:

			====Heat capacity exponent: <math>\alpha</math>====
			====Magnetic order parameter exponent: <math>\beta</math>====
			====Susceptibility exponent: <math>\gamma</math>====
	==See also==		==See also==
	*[[Zeno line#Batchinsky law \| Batchinsky law]]		*[[Zeno line#Batchinsky law \| Batchinsky law]]

Carl McBride: /* Critical point */ Slight tidy.

2010-10-25T09:14:41Z

Critical point: Slight tidy.

← Older revision		Revision as of 11:14, 25 October 2010
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	:<math>\left.~~V_c~~\right.=3b</math>.		:<math>\left.v_c\right.=3b</math>


	~~and~~		along with a critical point [[compressibility factor]] of


	:<math>\frac{~~p_cV_c~~}{~~T_c~~}= \frac{3R}{8}</math>		:<math>\frac{p_c v_c}{RT_c}= \frac{3}{8} = 0.375</math>


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	:<math>b= \frac{RT_c}{8P_c}</math>		:<math>b= \frac{RT_c}{8P_c}</math>

	==Virial form==		==Virial form==
	One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:		One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:

Carl McBride: Added a See also section

2010-10-07T10:09:35Z

Added a See also section

← Older revision		Revision as of 12:09, 7 October 2010
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	The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:		The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
	[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]		[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
			==See also==
			*[[Zeno line#Batchinsky law \| Batchinsky law]]
	==References==		==References==
	<references/>		<references/>

Carl McBride: /* Virial form */ trivial tidy

2010-10-06T15:18:02Z

Virial form: trivial tidy

← Older revision		Revision as of 17:18, 6 October 2010
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	One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:		One can re-write the van der Waals equation given above as a [[virial equation of state]] as follows:

	:<math>Z := \frac{pV}{nRT} = \frac{1}{1- \frac{nb}{V}} - \frac{an}{~~VRT~~} </math>		:<math>Z := \frac{pV}{nRT} = \frac{1}{1- \frac{bn}{V}} - \frac{an}{RTV} </math>

	Using the well known [http://mathworld.wolfram.com/SeriesExpansion.html series expansion] <math>(1-x)^{-1} = 1 + x + x^2 + x^3 + ...</math>		Using the well known [http://mathworld.wolfram.com/SeriesExpansion.html series expansion] <math>(1-x)^{-1} = 1 + x + x^2 + x^3 + ...</math>
	one can write the first term of the right hand side as <ref>This expansion is valid as long as <math>-1 < x < 1</math>, which is indeed the case for <math>nb/V</math> </ref>:		one can write the first term of the right hand side as <ref>This expansion is valid as long as <math>-1 < x < 1</math>, which is indeed the case for <math>bn/V</math> </ref>:

	:<math>\frac{1}{1- \frac{nb}{V}} = 1 + \frac{nb}{V} + \left( \frac{nb}{V} \right)^2 + \left( \frac{nb}{V} \right)^3 + ... </math>		:<math>\frac{1}{1- \frac{bn}{V}} = 1 + \frac{bn}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ... </math>

	Incorporating the second term of the right hand side in its due place leads to:		Incorporating the second term of the right hand side in its due place leads to:

	:<math>Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} + \left( \frac{nb}{V} \right)^2 + \left( \frac{nb}{V} \right)^3 + ...</math>.		:<math>Z = 1 + \left( b -\frac{a}{RT} \right) \frac{n}{V} + \left( \frac{bn}{V} \right)^2 + \left( \frac{bn}{V} \right)^3 + ...</math>.

	From the above one can see that the [[second virial coefficient]] corresponds to		From the above one can see that the [[second virial coefficient]] corresponds to
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	:<math>B_{3}(T)= b^2 </math>		:<math>B_{3}(T)= b^2 </math>

	==Boyle temperature==		==Boyle temperature==
	The [[Boyle temperature]] of the van der Waals equation is given by		The [[Boyle temperature]] of the van der Waals equation is given by

Carl McBride: Added virial expansion and Boyle temperature

2010-10-06T15:01:36Z

Added virial expansion and Boyle temperature

@@ Line 40: / Line 40: @@
 :<math>b= \frac{RT_c}{8P_c}</math>
 ==Dimensionless formulation==
 If one takes the following reduced quantities

Carl McBride: Changed references to Cite format

2010-09-23T11:08:57Z

Changed references to Cite format

← Older revision		Revision as of 13:08, 23 September 2010
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	The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]], takes into account two features that are absent in the [[Equation of State: Ideal Gas \| ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,		The '''van der Waals equation of state''', developed by [[ Johannes Diderik van der Waals]] <ref>J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873)</ref>
			<ref>English translation: [http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930]</ref>, takes into account two features that are absent in the [[Equation of State: Ideal Gas \| ideal gas]] equation of state; the parameter <math> b </math> introduces somehow the repulsive behavior between pairs of molecules at short distances,
	it represents the minimum molar volume of the system, whereas <math> a </math> measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.		it represents the minimum molar volume of the system, whereas <math> a </math> measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.
	==Equation of state==		==Equation of state==
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	The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:		The following image is a plot of the isotherms <math>T/T_c</math> = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:
	[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]		[[Image:vdW_isotherms.png\|center\|Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the van der Waals equation of state]]
	==~~Interesting reading~~==		==References==
			<references/>
			'''Related reading'''
	*[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910]		*[http://nobelprize.org/nobel_prizes/physics/laureates/1910/waals-lecture.pdf Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910]
	*Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de van der Waals", Anales de la Real Sociedad Española de Química '''101''' #1 pp. 19-30 (2005)		*Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de van der Waals", Anales de la Real Sociedad Española de Química '''101''' #1 pp. 19-30 (2005)
	~~==References==~~
	*J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).
	~~English translation:~~
	*[http://store.doverpublications.com/0486495930.html J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930]
	[[Category: equations of state]]		[[Category: equations of state]]