Second virial coefficient: Difference between revisions

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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( \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle -1 \right) 4 \pi r^2 dr </math>  
:<math>B_{2}(T)= - \frac{1}{2} \int \left( \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right) -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]].
In practice  the integral is often ''very hard'' to integrate analytically for anything other than, say, the [[Hard sphere: virial coefficients | hard sphere model]], thus one numerically evaluates
:<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>
calculating
:<math> \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle</math>
for each <math>r</math> using the numerical integration scheme proposed by Harold Conroy <ref>[http://dx.doi.org/10.1063/1.1701795 Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics '''47''' pp. 5307 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01597437 I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics '''39''' pp. 65-79 (1989)]</ref>.
==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

Revision as of 16:08, 3 May 2011

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

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Phi _{12}({\mathbf {r} })} 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.

In practice the integral is often very hard to integrate analytically for anything other than, say, the hard sphere model, thus one numerically evaluates

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B_{2}(T)=-{\frac {1}{2}}\int \left(\left\langle \exp \left(-{\frac {\Phi _{12}({\mathbf {r} })}{k_{B}T}}\right)\right\rangle -1\right)4\pi r^{2}dr}

calculating

for each using the numerical integration scheme proposed by Harold Conroy [1][2].

Isihara-Hadwiger formula

The Isihara-Hadwiger formula was discovered simultaneously and independently by Isihara [3] [4] [5] and the Swiss mathematician Hadwiger in 1950 [6] [7] [8] The second virial coefficient for any hard convex body is given by the exact relation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B_{2}=RS+V}

or

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 the hard sphere model one has [9]

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. See also: Hard sphere: virial coefficients.

Van der Waals equation of state

For the Van der Waals equation of state 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 B_{2}(T)= b -\frac{a}{RT} }

For the derivation click here.

Excluded volume

The second virial coefficient can be computed from the expression

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{1}{2} \iint v_{\mathrm {excluded}} (\Omega,\Omega') f(\Omega) f(\Omega')~ {\mathrm d}\Omega {\mathrm d}\Omega'}

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_{\mathrm {excluded}}} is the excluded volume.

See also

References

  1. ↑ Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics 47 pp. 5307 (1967)
  2. ↑ I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics 39 pp. 65-79 (1989)
  3. ↑ Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics 18 pp. 1446-1449 (1950)
  4. ↑ 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)
  5. ↑ 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)
  6. ↑ H. Hadwiger "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. 54 pp. 345- (1950)
  7. ↑ H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia 7 pp. 395-398 (1951)
  8. ↑ H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)
  9. ↑ Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40

Related reading

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