Clausius equation of state: Difference between revisions
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The '''Clausius''' [[Equations of state | equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Equations 3 and 4 in <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>) | The '''Clausius''' [[Equations of state | equation of state]], proposed in 1880 by [[Rudolf Julius Emanuel Clausius]] <ref>[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Über das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]</ref> is given by (Equations 3 and 4 in <ref>[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]</ref>) | ||
:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT</math> | :<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.</math> | ||
In the case of the critical isotherm 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>, leading to | |||
:<math>a = v_c - \frac{RT_c}{4P_c}</math> | :<math>a = v_c - \frac{RT_c}{4P_c}</math> | ||
Revision as of 11:14, 20 October 2009
The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius [1] is given by (Equations 3 and 4 in [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 \left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.}
In the case of the critical isotherm 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 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 } , leading to
- 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 = v_c - \frac{RT_c}{4P_c}}
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 c= \frac{27R^2T_c^2}{64P_c}}
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, 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 } is the volume per mol, 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 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 P_c} is the pressure at the critical point, 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 critical volume per mol.