Berthelot equation of state: Difference between revisions

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


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


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


and
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>
 
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>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>


:<math>b=\frac{RT_c}{8P_c}</math>
==References==
==References==
# D. J. Berthelot "", J. Phys., '''8''' pp. 263- (1899)
<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 ,

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



and giving a critical compressibility factor of



where is the pressure, is the temperature and is the molar gas constant. is the critical temperature, is the pressure and is the volume at the critical point.

Low pressure variant[edit]

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


References[edit]