Berthelot equation of state: Difference between revisions
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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. | 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: | |||
:<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> | |||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: equations of state]] | [[category: equations of state]] | ||
Revision as of 10:02, 22 September 2010
The Berthelot equation of state [1][2] can be written as
- 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 RT = \left( p + \frac{a}{Tv^2} \right) \left( v - b\right)} .
At the critical point one has 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 p}{\partial v}\right|_{T=T_c}=0 } , and ,
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
- 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 \frac{RT_c}{p_cv_c} = \frac{8}{3} \approx 2.667 }
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 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 R}
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
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
- ↑ 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)
- ↑ D. Berthelot "", Travaux et Mémoires du Bureau international des Poids et Mesures Tome XIII (Paris: Gauthier-Villars, 1907)
- ↑ Antony F. Saturno "Daniel Berthelot's equation of state", Journal of Chemical Education 39 (9) pp. 464-465 (1962)
- ↑ SAGE Notebook Worksheet for use in the open-source mathematics software SAGE