# Clausius–Mossotti relation

# Clausius–Mossotti relation

The **Clausius–Mossotti relation** expresses the dielectric constant (relative permittivity, εr) of a material in terms of the atomic polarizibility, 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]}

where

is the dielectric constant of the material

is the permittivity of free space

is the number density of the molecules (number per cubic meter), and

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: (see Lorrain and Corson - Electromagnetic Field and Waves, 1962, 2nd Edition, page 116)

`In theCGS system of unitsthe Clausius–Mossotti relation is typically rewritten to show themolecular polarizability`

*volume*which has units of volume (m3).^{[2]}Confusion may arise from the practice of using the shorter name "molecular polarizability" for bothandwithin 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)

`whereis therefractive index,is the number of molecules per unit volume, andis the meanpolarizability. 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:`

or simply

`This applies to gases at ordinary pressures. The refractive indexof the gas can then be expressed in terms of themolar refractivityas:`

`whereis the pressure of the gas,is theuniversal gas constant, andis the (absolute) temperature, which together determine the number density.`

## References

*J. Phys. Chem*.

**36**(4): 1152–1155. doi:10.1021/j150334a007.

*Atkins' Physical Chemistry*. Oxford University Press. pp. 622–629. ISBN 978-0-19-954337-3.