Clausius equation of state: Difference between revisions
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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 | 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 | ||
:<math>a = | :<math>a = \frac{27R^2T_c^2}{64P_c}</math> | ||
:<math>b= \frac{ | :<math>b= v_c - \frac{RT_c}{4P_c}</math> | ||
and | and | ||
:<math>c= \frac{ | :<math>c= \frac{3RT_c}{8P_c}-v_c</math> | ||
For details see the [[Mathematica]] [http://urey.uoregon.edu/~pchemlab/CH417/Lect2009/Clausius%20equation%20of%20state%20to%20evaluate%20a%20b%20c.pdf printout] | |||
produced by [http://www.uoregon.edu/~chem/hardwick.html Dr. John L. Hardwick]. | |||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: equations of state]] | [[category: equations of state]] | ||
Revision as of 11:36, 20 November 2009
The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius [1] is given by (Equations 3 and 4 in [2])
=RT.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f02e21ed2347a97172552cb26efb9eeb9920888)
where is the pressure, is the temperature, is the volume per mol, and is the molar gas constant. 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.
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 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
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a={\frac {27R^{2}T_{c}^{2}}{64P_{c}}}}
- 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= 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{3RT_c}{8P_c}-v_c}
For details see the Mathematica printout produced by Dr. John L. Hardwick.