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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><ref>D. Berthelot "", Travaux et Mémoires du Bureau international des Poids et Mesures '''Tome XIII''' (Paris: Gauthier-Villars, 1907)</ref>
| | :<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math> |
| can be written as
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| :<math>RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)</math>.
| | 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>,
| | :<math>a= \frac{27}{64}R^2 \frac{T_c^3}{P_c}</math> |
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| 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>)
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| | :<math>b=\frac{RT_c}{8P_c}</math> |
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| :<math>a = 3 T_c p_c v_c^2</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. |
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| :<math>b= \frac{v_c}{3}</math>
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| and giving a critical [[compressibility factor]] of
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| :<math>Z_c = \frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </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, <math>p_c</math> is the pressure and <math>v_c</math> is the volume at the critical point. | |
| ==Low pressure variant==
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| Berthelot also proposed an [[Equations of state |equation of state]] for use at low pressures{{reference needed}}:
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| :<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>
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| ==References== | | ==References== |
| <references/>
| | # D. J. Berthelot "", J. Phys., '''8''' pp. 263- (1899) |
| [[category: equations of state]] | | [[category: equations of state]] |