Clausius–Mossotti relation

The Clausius–Mossotti relation expresses the dielectric constant (relative permittivity, ε_r) of a material in terms of the atomic polarizability, α, of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It is named after Ottaviano-Fabrizio Mossotti and Rudolf Clausius. It is equivalent to the Lorentz–Lorenz equation. It may be expressed as:[1][2]

{\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}={\frac {N\alpha }{3\varepsilon _{0}}}

where

$\varepsilon _{r}=\epsilon /\epsilon _{0}$ is the dielectric constant of the material, which for non-magnetic materials is equal to $n^{2}$ where $n$ is the refractive index
$\varepsilon _{0}$ is the permittivity of free space
$N$ is the number density of the molecules (number per cubic meter), and
$\alpha$ is the molecular polarizability in SI-units (C·m²/V).

In the case that the material consists of a mixture of two or more species, the right hand side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by i in the following form:[3]

{\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}=\sum _{i}{\frac {N_{i}\alpha _{i}}{3\varepsilon _{0}}}

In the CGS system of units the Clausius–Mossotti relation is typically rewritten to show the molecular polarizability volume $\alpha '=\alpha /(4\pi \varepsilon _{0})$ which has units of volume (m³).[2] Confusion may arise from the practice of using the shorter name "molecular polarizability" for both $\alpha$ and $\alpha '$ within literature intended for the respective unit system.

Lorentz–Lorenz equation

The Lorentz–Lorenz equation is similar to the Clausius–Mossotti relation, except that it relates the refractive index (rather than the dielectric constant) of a substance to its polarizability. The Lorentz–Lorenz equation is named after the Danish mathematician and scientist Ludvig Lorenz, who published it in 1869, and the Dutch physicist Hendrik Lorentz, who discovered it independently in 1878.

The most general form of the Lorentz–Lorenz equation is (in CGS units)

{\frac {n^{2}-1}{n^{2}+2}}={\frac {4\pi }{3}}N\alpha _{\mathrm {m} }

where $n$ is the refractive index, $N$ is the number of molecules per unit volume, and $\alpha _{\mathrm {m} }$ is the mean polarizability. This equation is approximately valid for homogeneous solids as well as liquids and gases.

When the square of the refractive index is $n^{2}\approx 1$ , as it is for many gases, the equation reduces to:

n^{2}-1\approx 4\pi N\alpha _{\mathrm {m} }

or simply

n-1\approx 2\pi N\alpha _{\mathrm {m} }

This applies to gases at ordinary pressures. The refractive index $n$ of the gas can then be expressed in terms of the molar refractivity $A$ as:

n\approx {\sqrt {1+{\frac {3Ap}{RT}}}}

where $p$ is the pressure of the gas, $R$ is the universal gas constant, and $T$ is the (absolute) temperature, which together determine the number density $N$ .

Accordingly $N=N_{\mathrm {A} }\cdot c$ holds, with $c$ the molar concentration. If one replaces $n$ with the complex refractive index $m=n+ik$ , with the absorption index $k$ , it follows that:

m\approx 1+c{\frac {N_{\mathrm {A} }\cdot \alpha }{2\varepsilon _{0}}}

Therefore the imaginary part, the absorption index, is proportional to the molar concentration

k\approx c{\frac {N_{\mathrm {A} }\cdot \alpha ''}{2\varepsilon _{0}}}

and, therefore, to the absorbance. Accordingly, Beer's law can be derived from the Lorentz-Lorenz relation.[4] The change of the real refractive index in diluted solutions is therefore also approximately linearly depending on the molar concentration.[5]

References

Rysselberghe, P. V. (January 1932). "Remarks concerning the Clausius–Mossotti Law". J. Phys. Chem. 36 (4): 1152–1155. doi:10.1021/j150334a007.
Atkins, Peter; de Paula, Julio (2010). "Chapter 17". Atkins' Physical Chemistry. Oxford University Press. pp. 622–629. ISBN 978-0-19-954337-3.
Corson, Dale R; Lorrain, Paul (1962). Introduction to electromagnetic fields and waves. San Francisco: W.H. Freeman. p. 116. OCLC 398313.
Thomas Günter Mayerhöfer, Jürgen Popp (2020-05-12), "Beyond Beer's law: Revisiting the Lorentz-Lorenz equation", ChemPhysChem (in German), n/a (n/a), pp. 1218–1223, doi:10.1002/cphc.202000301, ISSN 1439-4235, PMC 7317954, PMID 32394615
Thomas G. Mayerhöfer, Alicja Dabrowska, Andreas Schwaighofer, Bernhard Lendl, Jürgen Popp (2020-04-20), "Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration", ChemPhysChem (in German), 21 (8), pp. 707–711, doi:10.1002/cphc.202000018, ISSN 1439-4235, PMC 7216834, PMID 32074389CS1 maint: multiple names: authors list (link)

Bibliography

Lakhtakia, A (1996). Selected papers on linear optical composite materials. Bellingham, Wash., USA: SPIE Optical Engineering Press. ISBN 978-0-8194-2152-4. OCLC 34046175.
Böttcher, C.J.F. (1973). Theory of Electric Polarization (2nd ed.). Elsevier. doi:10.1016/c2009-0-15579-4. ISBN 978-0-444-41019-1.
Clausius, R. (1879). Die Mechanische Behandlung der Electricität. Wiesbaden: Vieweg+Teubner Verlag. doi:10.1007/978-3-663-20232-5. ISBN 978-3-663-19891-8.
Born, Max; Wolf, Emil (1999). "section 2.3.3". Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.). Cambridge New York: Cambridge University Press. ISBN 0-521-64222-1. OCLC 40200160.
Lorenz, Ludvig, "Experimentale og theoretiske Undersogelser over Legemernes Brydningsforhold", Vidensk Slsk. Sckrifter 8,205 (1870) https://www.biodiversitylibrary.org/item/48423#page/5/mode/1up
Lorenz, L. (1880). "Ueber die Refractionsconstante". Annalen der Physik und Chemie (in German). Wiley. 247 (9): 70–103. Bibcode:1880AnP...247...70L. doi:10.1002/andp.18802470905. ISSN 0003-3804.
Lorentz, H. A. (1881). "Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase". Annalen der Physik (in German). Wiley. 248 (1): 127–136. Bibcode:1881AnP...248..127L. doi:10.1002/andp.18812480110. ISSN 0003-3804.
O. F. Mossotti, Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso, Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena, vol. 24, p. 49-74 (1850).

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[1] Rysselberghe, P. V. (January 1932). "Remarks concerning the Clausius–Mossotti Law". J. Phys. Chem. 36 (4): 1152–1155. doi:10.1021/j150334a007.

[Atkins-2] Atkins, Peter; de Paula, Julio (2010). "Chapter 17". Atkins' Physical Chemistry. Oxford University Press. pp. 622–629. ISBN 978-0-19-954337-3.

[3] Corson, Dale R; Lorrain, Paul (1962). Introduction to electromagnetic fields and waves. San Francisco: W.H. Freeman. p. 116. OCLC 398313.

[4] Thomas Günter Mayerhöfer, Jürgen Popp (2020-05-12), "Beyond Beer's law: Revisiting the Lorentz-Lorenz equation", ChemPhysChem (in German), n/a (n/a), pp. 1218–1223, doi:10.1002/cphc.202000301, ISSN 1439-4235, PMC 7317954, PMID 32394615

[5] Thomas G. Mayerhöfer, Alicja Dabrowska, Andreas Schwaighofer, Bernhard Lendl, Jürgen Popp (2020-04-20), "Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration", ChemPhysChem (in German), 21 (8), pp. 707–711, doi:10.1002/cphc.202000018, ISSN 1439-4235, PMC 7216834, PMID 32074389CS1 maint: multiple names: authors list (link)