Derivation of Clausius-Mossotti Equation

The Clausius-Mossotti equation is a fundamental equation in physical chemistry that relates the dielectric constant of a material to its molecular polarizability. It is used to explain the behavior of substances under the influence of an external electric field.

Let a substance of molecular weight 'M', density 'd' and dielectric constant 'ϵ', be present between two plates having charges per unit area, +q and −q of a condenser.
Then the number of molecules per c.c. = N.d/M
The field of force due to charges be F which acts on the molecules of substances so as to produce an average induced moment 'μ' per molecule.
Hence, Induced moment per c.c.:
z = (μ.N.d)/M = (α.F.N.d)/M    (As μ = α.F)    ----- Equation: 1
Where, N = Avogadro Number

Now, let a thin slice of the dielectric of thickness 'x' and cross section 'a' be placed between the plates of a condenser.
The induced moment = a.x.z
But induced moment = Charge × Distance
Hence, Charge on the slice surface = ± (a.x.z)/x = ± a.z
or, Charge density = ± z

If the slice of a substance is large enough to cover the entire space between the plates then the charge on the plates = ±q and the charge on the surface of the dielectric adjoining the plates = ±z. Hence, the effective charge density = q − z.

Two forces are acting on the unit charge: the force F₁ due to the charge (q − z) on the plate charge interface and F₂ exerted by the layers of molecules surrounding it. In order to calculate F₂, let the unit charge be enclosed in a sphere of induced charges and let the surface of the surrounding sphere be divided into strips.

The area of the strip laying between the angle θ and (θ + dθ) = 2𝜋 r cosθ . r . dθ
The strip makes an angle θ to the field, then the charge density = z sinθ
Hence, the total charge on the strip = 2𝜋 r² cosθ . sinθ . dθ
Thus, force on unit charge = (1/r²) × 2𝜋 z cosθ sinθ dθ = 2𝜋 z cosθ sinθ dθ

The force exerted on the unit charge parallel to the field = 2𝜋 z sinθ cosθ dθ × sinθ
So, the force exerted by the entire spherical surface:

The force due to plate charge, F₁ = 4𝜋(q − z)
Since, F = F₁ + F₂
or, F = 4𝜋(q − z) + 4𝜋z/3
or, F = (12𝜋q − 12𝜋z + 4𝜋z)/3
or, 3F = 12𝜋q − 8𝜋z
or, 12𝜋q = 3F + 8𝜋z
or, q = (3F + 8𝜋z)/12𝜋
and 12𝜋q − 12𝜋z = 3F − 4𝜋z
or, q − z = (3F − 4𝜋z)/12𝜋

Therefore, dielectric constant:

This is called the Clausius-Mossotti Equation. Since N and α do not depend on temperature, P_m does not depend on temperature and should be determined only by the nature of molecules. Again, ϵ is dimensionless, hence the polarization is expressed in units of volume. Knowing the molecular weight and density, the molar polarization in a given field can be calculated.

Limitations of the Clausius-Mossotti Equation

The Clausius-Mossotti relation is derived based on the following assumptions:

1. Polarization is considered as proportional to the field.
2. The polarizable molecules are isotropic.
3. Absence of short-range interaction.

The above mentioned conditions are satisfied with neutral molecules having no constant dipoles (i.e. non-polar). Therefore, the equation is applicable to neutral liquids and especially to gases. The Clausius-Mossotti Equation is not applicable to strong solutions and solids.