Clausius equation of state: Difference between revisions
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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 ( | 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> | ||
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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 = | :<math>a = \frac{27R^2T_c^3}{64P_c}</math> | ||
:<math>b= \frac{ | :<math>b= v_c - \frac{RT_c}{4P_c}</math> | ||
and | and | ||
:<math>c= \frac{ | :<math>c= \frac{3RT_c}{8P_c}-v_c</math> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: equations of state]] | [[category: equations of state]] | ||
Latest revision as of 09:52, 7 September 2012
The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius [1] [2] is given by (Eq. 1 [3])
=RT.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f02e21ed2347a97172552cb26efb9eeb9920888)
where is the pressure, is the temperature, is the volume per mol, and is the molar gas constant. is the critical temperature and is the pressure at the critical point, and is the critical volume per mol.
At the critical point one has , and , which leads to [4]
and
References[edit]
- ↑ R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie 9 pp. 337-357 (1880)
- ↑ 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)
- ↑ E. Sarrau "Sur la compressibilité des fluides", Comptes Rendus des Séances de l'Académie des Sciences. Paris 101 pp. 941-944 (1885)
- ↑ For details see the Mathematica printout produced by Dr. John L. Hardwick.