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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> <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>) | | The '''Clausius equation of state''' is given by |
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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> |
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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 |
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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>
| | :<math>a = v_c - \frac{RT_c}{4P_c}</math> |
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| :<math>a = \frac{27R^2T_c^3}{64P_c}</math> | | :<math>b= \frac{3RT_c}{8P_c}-v_c</math> |
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| :<math>b= v_c - \frac{RT_c}{4P_c}</math>
| | and |
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| | :<math>c= \frac{27R^2T_c^2}{64P_c}</math> |
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| :<math>c= \frac{3RT_c}{8P_c}-v_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. |
| ==References== | | ==References== |
| <references/>
| | #[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Ueber das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)] |
| [[category: equations of state]] | | [[category: equations of state]] |