Berthelot equation of state: Difference between revisions

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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>.


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 open-source mathematics software [http://www.sagemath.org/ SAGE]</ref>)
 
 
:<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 giving a critical [[compressibility factor]] of
 
 
:<math>Z_c = \frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 </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.
==Low pressure variant==
Berthelot also proposed an [[Equations of state |equation of state]] for use at low pressures{{reference needed}}:


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


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/>
[[category: equations of state]]
[[category: equations of state]]

Latest revision as of 14:31, 8 January 2014

The Berthelot equation of state [1][2] can be written as

.

At the critical point one has , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 } ,

which leads to (Eqs. 4.1 - 4.3 [3][4])


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 3 T_c p_c v_c^2}


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b= \frac{v_c}{3}}

and giving a critical compressibility factor of


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_c = \frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 }


where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is the pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature and is the molar gas constant. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} is the critical temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c} is the pressure and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_c} is the volume at the critical point.

Low pressure variant[edit]

Berthelot also proposed an equation of state for use at low pressures[?]


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)}

References[edit]

  1. ↑ 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)
  2. ↑ D. Berthelot "", Travaux et Mémoires du Bureau international des Poids et Mesures Tome XIII (Paris: Gauthier-Villars, 1907)
  3. ↑ Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education 39 (9) pp. 464-465 (1962)
  4. ↑ SAGE Notebook Worksheet for use in the open-source mathematics software SAGE
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