Second virial coefficient
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 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
calculating
for each using the numerical integration scheme proposed by Harold Conroy [1][2].
Isihara-Hadwiger formula[edit]
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
or
where
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 the mean radius of curvature.
Hard spheres[edit]
For the hard sphere model one has [9]
leading to
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[edit]
For the Van der Waals equation of state one has:
For the derivation click here.
Excluded volume[edit]
The second virial coefficient can be computed from the expression
where is the excluded volume.
Admur and Mason mixing rule[edit]
The second virial coefficient for a mixture of components is given by (Eq. 11 in [10])
where 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 x_j} are the mole fractions of the th and th component gasses of the mixture.
Unknown[edit]
([11])
See also[edit]
References[edit]
- ↑ Harold Conroy "Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals", Journal of Chemical Physics 47 pp. 5307 (1967)
- ↑ 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)
- ↑ Akira Isihara "Determination of Molecular Shape by Osmotic Measurement", Journal of Chemical Physics 18 pp. 1446-1449 (1950)
- ↑ 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)
- ↑ 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 "Einige Anwendungen eines Funkticnalsatzes fur konvexe Körper in der räumichen Integralgeometrie" Mh. Math. 54 pp. 345- (1950)
- ↑ H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia 7 pp. 395-398 (1951)
- ↑ H. Hadwiger "Altes und Neues über Konvexe Körper" Birkäuser Verlag (1955)
- ↑ Donald A. McQuarrie "Statistical Mechanics", University Science Books (2000) ISBN 978-1-891389-15-3 Eq. 12-40
- ↑ I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids 1 pp. 370-383 (1958)
- ↑ I am not sure where this mixing rule was published
Related reading
- W. H. Stockmayer "Second Virial Coefficients of Polar Gases", Journal of Chemical Physics 9 pp. 398- (1941)
- 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)
- Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics 115 pp. 1191-1199 (2017)
- Elisabeth Herold, Robert Hellmann, and Joachim Wagner "Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry", Journal of Chemical Physics 147 204102 (2017)