{{Short description|Equation for a material's dielectric constant given its atomic polarizability}}

In [[electromagnetism]], the '''Clausius–Mossotti relation''', named for [[Ottaviano-Fabrizio Mossotti|O.&nbsp;F.&nbsp;Mossotti]] and [[Rudolf Clausius]], expresses the [[dielectric constant]] (relative [[permittivity]] {{math|''ε''<sub>r</sub>}}) of a material in terms of the atomic [[polarizability]] {{mvar|α}} of the material's constituent atoms and/or molecules, or a [[homogeneous mixture]] thereof. It is equivalent to the {{pslink|Lorentz–Lorenz equation}}, which relates the [[refractive index]] (rather than the dielectric constant) of a substance to its polarizability. It may be expressed in [[SI units]] as<ref>{{cite journal |last=Rysselberghe |first=P. V. |title=Remarks concerning the Clausius–Mossotti Law |journal=J. Phys. Chem. |date=January 1932 |volume=36 |issue=4 |pages=1152–1155 |doi=10.1021/j150334a007}}</ref><ref name=Atkins>{{cite book |title=Atkins' Physical Chemistry |year=2010 |publisher=Oxford University Press |isbn=978-0-19-954337-3 |pages=622–629 |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |chapter=Chapter 17}}</ref> <math display="block"> \frac{\varepsilon_\text{r} - 1}{\varepsilon_\text{r} + 2} = \frac{N \alpha}{3\varepsilon_0}, </math> where : <math>\varepsilon_\text{r} = \varepsilon/\varepsilon_0</math> is the [[dielectric constant]] of the material, which for non-magnetic materials is equal to {{math|''n''{{sup|2}}}}, where {{mvar|n}} is the [[refractive index]]; : {{math|''ε''{{sub|0}}}} is the [[permittivity of free space]]; : {{mvar|N}} is the [[number density]] of the molecules (m<sup>−3</sup>); : {{mvar|α}} is the [[Electric susceptibility#Molecular polarizability|molecular polarizability]] ([[Coulomb|C]]·m<sup>2</sup>/V).

In the case that the material consists of a mixture of two or more species, the right side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by {{mvar|i}} in the following form:<ref>{{Cite book |title=Introduction to electromagnetic fields and waves |last1=Corson |first1=Dale R. |last2=Lorrain |first2=Paul |date=1962 |publisher=W.&nbsp;H.&nbsp;Freeman |location=San Francisco |language=en |oclc=398313 |page=116}}</ref> <math display="block"> \frac{\varepsilon_\text{r} - 1}{\varepsilon_\text{r} + 2} = \sum_i \frac{N_i \alpha_i}{3\varepsilon_0}. </math>

In the [[Centimetre–gram–second system of units|CGS system of units]] the Clausius–Mossotti relation is typically rewritten to show the [[Polarizability#Definition|molecular polarizability ''volume'']] <math>\alpha' = \frac{\alpha}{4\pi\varepsilon_0},</math> which has units of volume (cm<sup>3</sup>).<ref name=Atkins/> Confusion may arise from the practice of using the shorter name "molecular polarizability" for both <math>\alpha</math> and <math>\alpha'</math> within literature intended for the respective unit system.

The Clausius–Mossotti relation assumes only an induced dipole relevant to its polarizability and is thus inapplicable for substances with a significant permanent [[dipole]]. It is applicable to gases such as {{chem2|[[Nitrogen|N2]], [[CO2]], [[CH4]]}} and {{chem2|[[Hydrogen|H2]]}} at sufficiently low densities and pressures.<ref>{{Cite journal |last=Uhlig |first=H. H. |last2=Keyes |first2=F. G. |date=1933-02-01 |title=The Dependence of the Dielectric Constants of Gases on Temperature and Density |url=https://aip.scitation.org/doi/10.1063/1.3247827 |journal=The Journal of Chemical Physics |volume=1 |issue=2 |pages=155–159 |doi=10.1063/1.3247827 |issn=0021-9606 |url-access=subscription}}</ref> For example, the Clausius–Mossotti relation is accurate for N<sub>2</sub> gas up to 1000&nbsp;atm between 25&nbsp;°C and 125&nbsp;°C.<ref>{{Cite journal |last=Michels |first=A. |last2=Jaspers |first2=A. |last3=Sanders |first3=P. |date=1934-05-01 |title=Dielectric constant of nitrogen up to 1000 atms. Between 25&nbsp;°C and 150&nbsp;°C |url=https://www.sciencedirect.com/science/article/pii/S0031891434802509 |journal=Physica |language=en |volume=1 |issue=7 |pages=627–633 |doi=10.1016/S0031-8914(34)80250-9 |issn=0031-8914 |url-access=subscription}}</ref> Moreover, the Clausius–Mossotti relation may be applicable to substances if the applied electric field is at a sufficiently high frequencies such that any permanent dipole modes are inactive.<ref>{{Cite book |last=Böttcher |first=C. J. F. |url=https://linkinghub.elsevier.com/retrieve/pii/C20090155794 |title=Theory of Electric Polarization |date=1973 |publisher=Elsevier |isbn=978-0-444-41019-1 |language=en |doi=10.1016/c2009-0-15579-4}}</ref>

==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 [[Gaussian units|Gaussian-CGS]] units) <math display="block"> \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha_\text{m}, </math> where {{mvar|n}} is the refractive index, {{mvar|N}} is the number of molecules per unit volume, and <math>\alpha_\text{m}</math> 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 <math>n^2 \approx 1</math>, as it is for many gases, the equation reduces to <math display="block"> n^2 - 1 \approx 4 \pi N \alpha_\text{m} </math> or simply <math display="block"> n - 1 \approx 2 \pi N \alpha_\text{m}. </math>

This applies to gases at ordinary pressures. The refractive index {{mvar|n}} of the gas can then be expressed in terms of the [[molar refractivity]] {{mvar|A}} as <math display="block"> n \approx \sqrt{1 + \frac{3 A p}{R T}}, </math> where {{mvar|p}} is the pressure of the gas, {{mvar|R}} is the [[universal gas constant]], and {{mvar|T}} is the (absolute) temperature, which together determine the number density {{mvar|N}}.

==References== {{reflist}}

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