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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:
In the CGS system of units the Clausius-Mossotti relation is typically rewritten to show the molecular polarizability volume which has units of volume (m3). Confusion may arise from the practice of using the shorter name "molecular polarizability" for both and within literature intended for the respective unit system.
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)
where is the refractive index, is the number of molecules per unit volume, and 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 , as it is for many gases, the equation reduces to:
This applies to gases at ordinary pressures. The refractive index of the gas can then be expressed in terms of the molar refractivity as:
where is the pressure of the gas, is the universal gas constant, and is the (absolute) temperature, which together determine the number density .
Accordingly holds, with the molar concentration. If one replaces with the complex refractive index , with the absorption index , it follows that:
Therefore the imaginary part, the absorption index, is proportional to the molar concentration
and, therefore, to the absorbance. Accordingly, Beer's law can be derived from the Lorentz-Lorenz relation. The change of the real refractive index in diluted solutions is therefore also approximately linearly depending on the molar concentration.
^Rysselberghe, P. V. (January 1932). "Remarks concerning the Clausius-Mossotti Law". J. Phys. Chem. 36 (4): 1152-1155. doi:10.1021/j150334a007.
^ abAtkins, Peter; de Paula, Julio (2010). "Chapter 17". Atkins' Physical Chemistry. Oxford University Press. pp. 622-629. ISBN978-0-19-954337-3.
^Corson, Dale R; Lorrain, Paul (1962). Introduction to electromagnetic fields and waves. San Francisco: W.H. Freeman. p. 116. OCLC398313.
^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, ISSN1439-4235, PMC7216834, PMID32074389CS1 maint: multiple names: authors list (link)
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).