Second virial coefficient: Difference between revisions
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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]] | The '''second virial coefficient''' is usually written as ''B'' or as <math>B_2</math>. The second [[Virial equation of state |virial coefficient]] represents the initial departure from [[ideal gas |ideal-gas]] behaviour. | ||
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( \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== | |||
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>[http://dx.doi.org/10.1007/BF02168922 H. Hadwiger "Der kinetische Radius nichtkugelförmiger Moleküle" Experientia '''7''' pp. 395-398 (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 | ||
:<math>B_2=RS+V</math> | |||
or | |||
:<math>\frac{B_2}{V}=1+3 \alpha</math> | :<math>\frac{B_2}{V}=1+3 \alpha</math> | ||
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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. | ||
==Hard spheres== | ==Hard spheres== | ||
For hard | 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> | ||
:<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 [[ | 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== | |||
The second virial coefficient can be computed from the expression | |||
:<math>B_{2}= \frac{1}{2} \iint v_{\mathrm {excluded}} (\Omega,\Omega') f(\Omega) f(\Omega')~ {\mathrm d}\Omega {\mathrm d}\Omega'</math> | |||
where <math>v_{\mathrm {excluded}}</math> is the [[excluded volume]]. | |||
==Admur and Mason mixing rule== | |||
The [[second virial coefficient]] for a mixture of <math>n</math> components is given by (Eq. 11 in | |||
<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures", Physics of Fluids '''1''' pp. 370-383 (1958)]</ref>) | |||
:<math>B_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j</math> | |||
where <math>x_i</math> and <math>x_j</math> are the mole fractions of the <math>i</math>th and <math>j</math>th component gasses of the mixture. | |||
==Unknown== | |||
(<ref>I am not sure where this mixing rule was published</ref>) | |||
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math> | |||
==See also== | ==See also== | ||
*[[Virial equation of state]] | *[[Virial equation of state]] | ||
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*[[Boyle temperature]] | *[[Boyle temperature]] | ||
*[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] | *[[Joule-Thomson effect#Joule-Thomson coefficient | Joule-Thomson coefficient]] | ||
==References== | |||
<references/> | |||
'''Related reading''' | |||
*[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)] | |||
*[http://dx.doi.org/10.1080/00268976.2016.1263763 Michael Rouha and Ivo Nezbeda "Second virial coefficients: a route to combining rules?", Molecular Physics '''115''' pp. 1191-1199 (2017)] | |||
*[https://doi.org/10.1063/1.5004687 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)] | |||
[[Category: Virial coefficients]] | [[Category: Virial coefficients]] | ||
Latest revision as of 14:36, 10 December 2019
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 behaviour. The second virial coefficient, in three dimensions, 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( \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right) -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.
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 (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 }
calculating
- 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 \left\langle \exp\left(-\frac{\Phi_{12}({\mathbf r})}{k_BT}\right)\right\rangle}
for each 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} 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
- 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=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[edit]
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[edit]
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[edit]
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.
Admur and Mason mixing rule[edit]
The second virial coefficient for a mixture of 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 n} components is given by (Eq. 11 in [10])
- 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_{ {\mathrm {mix}} } = \sum_{i=1}^{n} \sum_{j=1}^{n} B_{ij} x_i x_j}
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 x_i} 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 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 j} th component gasses of the mixture.
Unknown[edit]
([11])
- 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_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}}
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)