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The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second | 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, in three dimensions, is given by | The second virial coefficient, in three dimensions, is given by | ||
:<math>B_{2}(T)= - \frac{1}{2} \int \left( | :<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]], ''T'' is the [[temperature]] and <math>k_B</math> is the [[Boltzmann constant]]. 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]]. | ||
==Isihara-Hadwiger formula== | ==Isihara-Hadwiger formula== | ||
The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara | The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara and the Swiss mathematician Hadwiger in 1950. | ||
and the Swiss mathematician Hadwiger in 1950 | |||
The second virial coefficient for any hard convex body is given by the exact relation | The second virial coefficient for any hard convex body is given by the exact relation | ||
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where <math>V</math> is | where <math>V</math> is | ||
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. | ||
====References==== | |||
#[http://dx.doi.org/10.1063/1.1747510 A. Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics '''18''' pp. 1446-1449 (1950)] | |||
#[http://dx.doi.org/10.1143/JPSJ.6.40 Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. I. Second Virial Coefficient for Rigid Ovaloids Model", Journal of the Physical Society of Japan '''6''' pp. 40-45 (1951)] | |||
#[http://dx.doi.org/10.1143/JPSJ.6.46 Akira Isihara and Tsuyoshi Hayashida "Theory of High Polymer Solutions. II. Special Forms of Second Osmotic Coefficient", Journal of the Physical Society of Japan '''6''' pp. 46-50 (1951)] | |||
#H. Hadwiger "" Mh. Math. '''54''' pp. 345- (1950) | |||
#H. Hadwiger "" Experimentia '''7''' pp. 395- (1951) | |||
#H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955) | |||
==Hard spheres== | ==Hard spheres== | ||
For the [[hard sphere model]] one has | For the [[hard sphere model]] 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 is independent of [[temperature | Note that <math>B_{2}</math> for the hard sphere is independent of [[temperature]]. | ||
==Excluded volume== | ==Excluded volume== | ||
The second virial coefficient can be computed from the expression | The second virial coefficient can be computed from the expression | ||
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where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]]. | where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]]. | ||
==See also== | ==See also== | ||
*[[Virial equation of state]] | *[[Virial equation of state]] | ||
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*[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] | *[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] | ||
==References== | ==References== | ||
#Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) (Re-published) ISBN 978-1-891389-15-3 | |||
#[http://dx.doi.org/10.1063/1.1750922 W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics '''9''' pp. 398- (1941)] | |||
#[http://dx.doi.org/10.1063/1.481106 G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics '''112''' pp. 5364-5369 (2000)] | |||
[[Category: Virial coefficients]] | [[Category: Virial coefficients]] | ||