Clausius equation of state - Revision history

Carl McBride: Corrected T_c to be T_c^3 not T_c^2

2012-09-07T07:52:27Z

Corrected T_c to be T_c^3 not T_c^2

← Older revision		Revision as of 09:52, 7 September 2012
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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>, which leads to <ref>For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout] produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick].</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>, which leads to <ref>For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout] produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick].</ref>

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

	:<math>b= v_c - \frac{RT_c}{4P_c}</math>		:<math>b= v_c - \frac{RT_c}{4P_c}</math>

Carl McBride: Changed comment to be a reference.

2009-11-20T16:22:59Z

Changed comment to be a reference.

← Older revision		Revision as of 18:22, 20 November 2009
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	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.

	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		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 <ref>For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout] produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick].</ref>

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

	:<math>c= \frac{3RT_c}{8P_c}-v_c</math>		:<math>c= \frac{3RT_c}{8P_c}-v_c</math>

	~~For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout]~~
	~~produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick].~~
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride: Improved references.

2009-11-20T11:57:55Z

Improved references.

← Older revision		Revision as of 13:57, 20 November 2009
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	The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Eq. 3 <ref>[http://dx.~~doi~~.~~org~~/10.~~1021/ie50663a005 K~~. K. ~~Shah and G~~. ~~Thodos~~ "~~A Comparison of Equations of State~~", ~~Industrial & Engineering Chemistry~~ '''57''' pp. 30-37 (~~1965~~)]</ref>)		The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> <ref>[http://dx.doi.org/10.1002/andp.18812501007 R. Clausius "Ueber die theoretische Bestimmung des Dampfdruckes und der Volumina des Dampfes und der Flüssigkeit", Annalen der Physik und Chemie '''14''' pp. 279-290 (1881)]</ref> is given by (Eq. 1 <ref>[http://gallica.bnf.fr/ark:/12148/bpt6k30574.pleinepage.f941.N4 E. Sarrau "Sur la compressibilité des fluides", Comptes Rendus des Séances de l'Académie des Sciences. Paris '''101''' pp. 941-944 (1885)]</ref>)

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

Carl McBride: Removed mention of Eq. 4 of Shah and Thodos: a b anc c are mixed up.

2009-11-20T11:26:58Z

Removed mention of Eq. 4 of Shah and Thodos: a b anc c are mixed up.

← Older revision		Revision as of 13:26, 20 November 2009
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	The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (~~Equations~~ 3 ~~and 4 in~~ <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>)		The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Eq. 3 <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>)

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

Carl McBride: Corrected a b and c

2009-11-20T10:36:41Z

Corrected a b and c

← Older revision		Revision as of 12:36, 20 November 2009
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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>, which leads to		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

	:<math>a = ~~v_c -~~ \frac{~~RT_c~~}{~~4P_c~~}</math>		:<math>a = \frac{27R^2T_c^2}{64P_c}</math>

	:<math>b= \frac{~~3RT_c~~}{~~8P_c~~}~~-v_c~~</math>		:<math>b= v_c - \frac{RT_c}{4P_c}</math>

	and		and

	:<math>c= \frac{~~27R^2T_c^2~~}{~~64P_c~~}</math>		:<math>c= \frac{3RT_c}{8P_c}-v_c</math>

			For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout]
			produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick].
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Nice and Tidy: Slight tidy.

2009-10-20T13:51:42Z

Slight tidy.

← Older revision		Revision as of 15:51, 20 October 2009
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	:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.</math>		:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.</math>

			where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.

	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		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
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	:<math>c= \frac{27R^2T_c^2}{64P_c}</math>		:<math>c= \frac{27R^2T_c^2}{64P_c}</math>

	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.
	==References==		==References==
	<references/>		<references/>
	[[category: equations of state]]		[[category: equations of state]]

Carl McBride: Rewording + internal link

2009-10-20T12:23:36Z

Rewording + internal link

← Older revision		Revision as of 14:23, 20 October 2009
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	:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.</math>		:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.</math>

	~~In the case of~~ the critical ~~isotherm~~ 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>, ~~leading~~ to		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

	:<math>a = v_c - \frac{RT_c}{4P_c}</math>		:<math>a = v_c - \frac{RT_c}{4P_c}</math>

Carl McBride: Added comment on derivation of a, b, and c

2009-10-20T10:14:31Z

Added comment on derivation of a, b, and c

← Older revision		Revision as of 12:14, 20 October 2009
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	The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Equations 3 and 4 in <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>)		The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Equations 3 and 4 in <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>)

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

	~~where~~		In the case of the critical isotherm 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>, leading to

	:<math>a = v_c - \frac{RT_c}{4P_c}</math>		:<math>a = v_c - \frac{RT_c}{4P_c}</math>

Carl McBride: Added source of the equations given in this page.

2009-10-20T10:01:22Z

Added source of the equations given in this page.

← Older revision		Revision as of 12:01, 20 October 2009
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	The '''Clausius''' [[Equations of state \| equation of state]] ~~is given~~ by <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref>		The '''Clausius''' [[Equations of state \| equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Equations 3 and 4 in <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>)

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

Carl McBride: Undo revision 9115 by 140.136.232.232 (Talk) I think things were correct as they were.

2009-10-20T09:56:21Z

Undo revision 9115 by 140.136.232.232 (Talk) I think things were correct as they were.

← Older revision		Revision as of 11:56, 20 October 2009
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	where		where

	:<math>a = \frac{~~27R^2T_c^2~~}{~~64P_c~~}</math>		:<math>a = v_c - \frac{RT_c}{4P_c}</math>

	:<math>b= ~~v_c -~~ \frac{~~RT_c~~}{~~4P_c~~}</math>		:<math>b= \frac{3RT_c}{8P_c}-v_c</math>

	and		and

	:<math>c= \frac{~~3RT_c~~}{~~8P_c~~}~~-v_c~~</math>		:<math>c= \frac{27R^2T_c^2}{64P_c}</math>

	where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.		where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, 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, and <math> v_c </math> is the critical volume per mol.