Berthelot equation of state
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 the critical compressibility factor is
- 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{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 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