Second virial coefficient: Difference between revisions

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m (→‎Hard spheres: Added an internal link.)
(Added B2 for vdW and changed references to Cite format)
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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 and the Swiss mathematician Hadwiger in 1950.
The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara
<ref>[http://dx.doi.org/10.1063/1.1747510 Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics '''18''' pp. 1446-1449 (1950)]</ref>
<ref>[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)]</ref>
<ref>[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)]</ref>
and the Swiss mathematician Hadwiger in 1950
<ref>H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. '''54''' pp. 345- (1950)</ref>
<ref>H. Hadwiger "" Experimentia '''7''' pp. 395- (1951)</ref>
<ref>H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)</ref>
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====
==Hard spheres==
#[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)]
For the [[hard sphere model]]  one has <ref>Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3  Eq. 12-40</ref>
#[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==
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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Note that <math>B_{2}</math> for the hard sphere is independent of [[temperature]]. See also: [[Hard sphere: virial coefficients]].
Note that <math>B_{2}</math> for the hard sphere is independent of [[temperature]]. See also: [[Hard sphere: virial coefficients]].
==Van der Waals equation of state==
For the [[Van der Waals equation of state]] one has:
:<math>B_{2}(T)=  b -\frac{a}{RT} </math>


For the derivation [[Van der Waals equation of state#Virial form | click here]].
==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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*[[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
<references/>
#[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)]
'''Related reading'''
#[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)]
*[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]]

Revision as of 16:28, 7 October 2010

The second virial coefficient is usually written as B or as . The second virial coefficient represents the initial departure from ideal-gas behavior. The second virial coefficient, in three dimensions, is given by

where is the intermolecular pair potential, T is the temperature and is the Boltzmann constant. Notice that the expression within the parenthesis of the integral is the Mayer f-function.

Isihara-Hadwiger formula

The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara [1] [2] [3] and the Swiss mathematician Hadwiger in 1950 [4] [5] [6] The second virial coefficient for any hard convex body is given by the exact relation

or

where

where is the volume, , the surface area, and the mean radius of curvature.

Hard spheres

For the hard sphere model one has [7]

leading to

Note that for the hard sphere is independent of temperature. See also: Hard sphere: virial coefficients.

Van der Waals equation of state

For the Van der Waals equation of state one has:

For the derivation click here.

Excluded volume

The second virial coefficient can be computed from the expression

where is the excluded volume.

See also

References

  1. Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics 18 pp. 1446-1449 (1950)
  2. 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)
  3. 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)
  4. H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. 54 pp. 345- (1950)
  5. H. Hadwiger "" Experimentia 7 pp. 395- (1951)
  6. H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)
  7. Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40

Related reading