Second virial coefficient: Difference between revisions
Carl McBride (talk | contribs) (New page: The '''second virial coefficient''' is usually written as ''B'', or <math>B_2</math>. The second virial coefficient is given by :<math>B_{2}(T)= - \frac{1}{2} \int \left(\langle \exp(-\fr...) |
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The '''second virial coefficient''' is usually written as ''B'' | The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second virial coefficient represents the initial departure from [[ideal gas |ideal-gas]] behavior. | ||
The second virial coefficient is given by | The second virial coefficient is given by | ||
:<math>B_{2}(T)= - \frac{1}{2} \int \left(\langle \exp(-\frac{\Phi_{12}({\mathbf r})}{k_BT})\rangle -1 \right) 4 \pi r^2 dr </math> | :<math>B_{2}(T)= - \frac{1}{2} \int \left( \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle -1 \right) 4 \pi r^2 dr </math> | ||
where <math>\Phi_{12}({\mathbf r})</math> is the [[intermolecular pair potential]]. Notice that the expression within the parenthesis | where <math>\Phi_{12}({\mathbf r})</math> is the [[intermolecular pair potential]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. Notice that the expression within the parenthesis | ||
of the integral is the [[Mayer f-function]]. | of the integral is the [[Mayer f-function]]. | ||
==For any hard convex body== | ==For any hard convex body== | ||
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the volume, <math>S</math>, the surface area, and <math>R</math> the mean radius of curvature. | the volume, <math>S</math>, the surface area, and <math>R</math> the mean radius of curvature. | ||
==Hard spheres== | ==Hard spheres== | ||
For hard spheres one has | For hard spheres one has (McQuarrie, 1976, eq. 12-40) | ||
:<math>B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr | :<math>B_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr | ||
</math> | </math> | ||
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:<math>B_{2}= \frac{2\pi\sigma^3}{3}</math> | :<math>B_{2}= \frac{2\pi\sigma^3}{3}</math> | ||
Note that <math>B_{2}</math> for the [[hard sphere model| hard sphere]] is independent of temperature. | Note that <math>B_{2}</math> for the [[hard sphere model| hard sphere]] is independent of [[temperature]]. | ||
==See also== | |||
*[[Boyle temperature]] | |||
==References== | ==References== | ||
[[Category: Virial coefficients]] | [[Category: Virial coefficients]] | ||
Revision as of 10:11, 12 July 2007
The second virial coefficient is usually written as B or 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 B_2} . The second virial coefficient represents the initial departure from ideal-gas behavior. The second virial coefficient is given by
- 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_{2}(T)= - \frac{1}{2} \int \left( \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle -1 \right) 4 \pi r^2 dr }
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 \Phi_{12}({\mathbf r})} is the intermolecular pair potential, 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 k_B} is the Boltzmann constant. Notice that the expression within the parenthesis of the integral is the Mayer f-function.
For any hard convex body
The second virial coefficient for any hard convex body is given by the exact relation
- 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{B_2}{V}=1+3 \alpha}
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 \alpha = \frac{RS}{3V}}
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 V} is the volume, 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 S} , the surface area, 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} the mean radius of curvature.
Hard spheres
For hard spheres one has (McQuarrie, 1976, eq. 12-40)
- 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_{2}(T)= - \frac{1}{2} \int_0^\sigma \left(\langle 0\rangle -1 \right) 4 \pi r^2 dr }
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 B_{2}= \frac{2\pi\sigma^3}{3}}
Note that 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_{2}} for the hard sphere is independent of temperature.