Berthelot equation of state - Revision history

Carl McBride at 13:31, 8 January 2014

2014-01-08T13:31:24Z

← Older revision		Revision as of 15:31, 8 January 2014
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	:<math>\frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </math>		:<math>Z_c = \frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </math>

Carl McBride: /* Low pressure variant */ References needed

2010-09-29T15:41:07Z

Low pressure variant: References needed

← Older revision		Revision as of 17:41, 29 September 2010
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	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point.
	==Low pressure variant==		==Low pressure variant==
	Berthelot also proposed an [[Equations of state \|equation of state]] for use at low pressures:		Berthelot also proposed an [[Equations of state \|equation of state]] for use at low pressures{{reference needed}}:


	:<math>p = \frac{RT}{v} \left( 1 + \frac{9}{128} \frac{pT_c}{p_c T} \left( 1- \frac{6T_c^2}{T^2} \right) \right)</math>		:<math>p = \frac{RT}{v} \left( 1 + \frac{9}{128} \frac{pT_c}{p_c T} \left( 1- \frac{6T_c^2}{T^2} \right) \right)</math>

	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride at 14:16, 22 September 2010

2010-09-22T14:16:37Z

← Older revision		Revision as of 16:16, 22 September 2010
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	:<math>b= \frac{v_c}{3}</math>		:<math>b= \frac{v_c}{3}</math>

	and ~~the~~ critical [[compressibility factor]] is		and giving a critical [[compressibility factor]] of


	:<math>\frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </math>		:<math>\frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </math>

Carl McBride at 14:09, 22 September 2010

2010-09-22T14:09:16Z

← Older revision		Revision as of 16:09, 22 September 2010
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	:<math>b= \frac{v_c}{3}</math>		:<math>b= \frac{v_c}{3}</math>

	and		and the critical [[compressibility factor]] is

	:<math>\frac{~~RT_c~~}{~~p_cv_c~~} = \frac{8}{3} ~~\approx 2~~.~~667~~ </math>		:<math>\frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </math>

Carl McBride: Added low pressure EOS

2010-09-22T09:02:13Z

Added low pressure EOS

← Older revision		Revision as of 11:02, 22 September 2010
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	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point.
			==Low pressure variant==
			Berthelot also proposed an [[Equations of state \|equation of state]] for use at low pressures:


			:<math>p = \frac{RT}{v} \left( 1 + \frac{9}{128} \frac{pT_c}{p_c T} \left( 1- \frac{6T_c^2}{T^2} \right) \right)</math>
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride at 12:49, 21 September 2010

2010-09-21T12:49:42Z

← Older revision		Revision as of 14:49, 21 September 2010
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	At the [[critical points \| critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right\|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right\|_{T=T_c}=0 </math>,		At the [[critical points \| critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right\|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right\|_{T=T_c}=0 </math>,

	which leads to (Eqs. 4.1 - 4.3 <ref>[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]</ref><ref> [http://www.ucm.es/info/molecsim/Berthelot_EOS.sws SAGE Notebook Worksheet] for use in the mathematics ~~program~~ [http://www.sagemath.org/ SAGE]</ref>)		which leads to (Eqs. 4.1 - 4.3 <ref>[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]</ref><ref> [http://www.ucm.es/info/molecsim/Berthelot_EOS.sws SAGE Notebook Worksheet] for use in the open-source mathematics software [http://www.sagemath.org/ SAGE]</ref>)

Carl McBride: Added 1907 reference

2010-09-21T12:40:13Z

Added 1907 reference

← Older revision		Revision as of 14:40, 21 September 2010
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	The '''Berthelot equation of state''' <ref>[http://dx.doi.org/10.1051/jphystap:018990080026300 D. J. Berthelot "Sur Une Méthode Purement Physique Pour La Détermination des Poids Moléculaires des Gaz et des Poids Atomiques de Leurs Éléments", J. Phys., '''8''' pp. 263-274 (1899)]</ref> can be written as		The '''Berthelot equation of state''' <ref>[http://dx.doi.org/10.1051/jphystap:018990080026300 D. J. Berthelot "Sur Une Méthode Purement Physique Pour La Détermination des Poids Moléculaires des Gaz et des Poids Atomiques de Leurs Éléments", J. Phys., '''8''' pp. 263-274 (1899)]</ref><ref>D. Berthelot "", Travaux et Mémoires du Bureau international des Poids et Mesures '''Tome XIII''' (Paris: Gauthier-Villars, 1907)</ref>
			can be written as

	:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.		:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.
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	:<math>a = ~~3T_cP_cV_c~~^2</math>		:<math>a = 3 T_c p_c v_c^2</math>


	:<math>b= \frac{~~V_c~~}{3}</math>		:<math>b= \frac{v_c}{3}</math>

	and		and

	:<math>R = \frac{~~8P_cV_c~~}{~~3T_c~~}</math>		:<math>\frac{RT_c}{p_cv_c} = \frac{8}{3} \approx 2.667 </math>


	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>~~P_c~~</math> is the pressure and <math>~~V_c~~</math> is the volume at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point.
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride: Slight tidy

2010-09-21T12:00:13Z

Slight tidy

← Older revision		Revision as of 14:00, 21 September 2010
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	:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.		:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.

	At the [[critical points \| critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right\|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right\|_{T=T_c}=0 </math>, which leads to (Eqs. 4.1 - 4.3 <ref>[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]</ref><ref> [http://www.ucm.es/info/molecsim/Berthelot_EOS.sws SAGE Notebook Worksheet] for use in the mathematics program [http://www.sagemath.org/ SAGE]</ref>)		At the [[critical points \| critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right\|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right\|_{T=T_c}=0 </math>,

	:<~~math~~>~~a = 3T_cp_cv_c^2~~</~~math~~>		which leads to (Eqs. 4.1 - 4.3 <ref>[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]</ref><ref> [http://www.ucm.es/info/molecsim/Berthelot_EOS.sws SAGE Notebook Worksheet] for use in the mathematics program [http://www.sagemath.org/ SAGE]</ref>)


	:<math>b= \frac{~~v_c~~}{3}</math>		:<math>a = 3T_cP_cV_c^2</math>


			:<math>b= \frac{V_c}{3}</math>

	and		and

	:<math>R = \frac{~~8p_cv_c~~}{3T_c}</math>		:<math>R = \frac{8P_cV_c}{3T_c}</math>


	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature ~~and~~ <math>P_c</math> is the pressure at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature, <math>P_c</math> is the pressure and <math>V_c</math> is the volume at the critical point.
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride: Recalculated a, b, and R

2010-09-21T11:47:59Z

Recalculated a, b, and R

← Older revision		Revision as of 13:47, 21 September 2010
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	The '''Berthelot equation of state''' <ref>[http://dx.doi.org/10.1051/jphystap:018990080026300 D. J. Berthelot "Sur Une Méthode Purement Physique Pour La Détermination des Poids Moléculaires des Gaz et des Poids Atomiques de Leurs Éléments", J. Phys., '''8''' pp. 263-274 (1899)]</ref> can be written as		The '''Berthelot equation of state''' <ref>[http://dx.doi.org/10.1051/jphystap:018990080026300 D. J. Berthelot "Sur Une Méthode Purement Physique Pour La Détermination des Poids Moléculaires des Gaz et des Poids Atomiques de Leurs Éléments", J. Phys., '''8''' pp. 263-274 (1899)]</ref> can be written as

	:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>		:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.

	~~where~~		At the [[critical points \| critical point]] one has <math>\left.\frac{\partial p}{\partial v}\right\|_{T=T_c}=0 </math>, and <math>\left.\frac{\partial^2 p}{\partial v^2}\right\|_{T=T_c}=0 </math>, which leads to (Eqs. 4.1 - 4.3 <ref>[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]</ref><ref> [http://www.ucm.es/info/molecsim/Berthelot_EOS.sws SAGE Notebook Worksheet] for use in the mathematics program [http://www.sagemath.org/ SAGE]</ref>)

	:<math>a= ~~\frac{27}{64}R~~^2 \frac{~~T_c^3~~}{~~P_c~~}</math>		:<math>a = 3T_cp_cv_c^2</math>


			:<math>b= \frac{v_c}{3}</math>

	and		and

	:<math>b=\frac{~~RT_c~~}{~~8P_c~~}</math>		:<math>R = \frac{8p_cv_c}{3T_c}</math>


	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature and <math>P_c</math> is the pressure at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature and <math>P_c</math> is the pressure at the critical point.
	==References==		==References==
	<references/>		<references/>
	~~'''Related reading'''~~
	*[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride: Added DOI link

2010-09-17T09:00:37Z

Added DOI link

← Older revision		Revision as of 11:00, 17 September 2010
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			The '''Berthelot equation of state''' <ref>[http://dx.doi.org/10.1051/jphystap:018990080026300 D. J. Berthelot "Sur Une Méthode Purement Physique Pour La Détermination des Poids Moléculaires des Gaz et des Poids Atomiques de Leurs Éléments", J. Phys., '''8''' pp. 263-274 (1899)]</ref> can be written as

	:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>		:<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>

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	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature and <math>P_c</math> is the pressure at the critical point.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points \| critical]] temperature and <math>P_c</math> is the pressure at the critical point.
	==References==		==References==
	~~# D~~. J. Berthelot "~~", J. Phys.~~, '''8''' pp. ~~263~~- (~~1899~~)		<references/>
			'''Related reading'''
			*[http://dx.doi.org/10.1021/ed039p464 Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education '''39''' (9) pp. 464-465 (1962)]
	[[category: equations of state]]		[[category: equations of state]]