Clausius Mossotti equations

Thus far we have assumed that the applied field or external field was the local field inside the dielectric. This is not quite true because the aligned dipoles produce a local field that adds to the applied field. We now set out to determine how this local field relates to the applied field. 

Geometry for obtaining the local field using the Clausius-Mozzotti model. 

If we cut out a small spherical cavity around the point in question, the field in the center of this sphere will be given by the sum of the applied field E, the field due to the charges on the inside surface of the sphere, the field due to the dipoles outside the sphere and the field due to the dipoles inside the sphere. For materials with cubic symmetry, the latter contribution vanishes by s)Tnmetry, as does the field from the dipoles outside of the cavity. The charge density on the surface of the cavity is given by P cos / . The field due to the surface charges inside the cavity is then given by 

Putting this back into Equation 23.42 and solving for P, 

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Another way to obtain a relative permitivity is using some simple equations that relate relative permitivity to the molecular dipole moment. These are derived from statistical mechanics. Two of the more well-known equations are the Clausius-Mossotti equation and the Kirkwood equation. These and others are discussed in the review articles referenced at the end of this chapter. The com-... 

After a little manipulation we derive the Clausius-Mossotti equation... 

The derivation of the Clausius-Mossotti equation follows as above, except that the quantity in square brackets has to replace a, and we obtain finally... 

It is also worth noting that Equation 9.1 indicates a connection between C44, hardness, and e. The dielectric constant, e depends on the polarizability, a of each alkali halide through the Clausius-Mossotti equation ... 

Clausius-Mossotti equation). In this expression, V designates the mole volume and Ae, Be, Cf,... are the first, second, third,... virial dielectric coefficients. A similar expansion exists for the refractive index, n, which is related to the (frequency dependent) dielectric constant as n2 = e (Lorentz-Lorenz equation, [87]). The second virial dielectric coefficient Be may be considered the sum of an orientational and a polarization term, Be = B0r + Bpo, arising from binary interactions, while the second virial refractive coefficient is given by just the polarization term, B = Bpo at high enough frequencies, the orientational component falls off to small values and the difference Be — B may be considered a measurement of the interaction-induced dipole moments [73],... 

Clausius-Mossotti equation). The AE, BE,. .., are the first, second,. .., dielectric virial coefficients, given by... 

Having obtained statistically converged results for the dipole polarizability of liquid argon, we now consider the resulting values for the dielectric constant. Again, we use the three theoretical models and simply obtain the dielectric constant e, from the dipole polarizability, using the Clausius-Mossotti equation [34] ... 

The average dipole moment of a water molecule is given by m = y /jocab where y is the polarizability. Using for y the Clausius—Mossotti equation,... 

Merging the previous equations, the anisotropic Clausius-Mossotti equation becomes... 

See also -> Debye-Clausius-Mossotti equation, and -> Clausius-Mossotti equation. 

Clausius-Mossotti equation — Named after Clausius and Ottaviano Fabrizio Mossotti (1791-1863). It relates the electron -> polarizability a of an individual molecule to the optical -> dielectric constant (relative permittivity) r of the bulk material. 

Debye-Clausius-Mossotti equation -> Debye expanded the - Clausius-Mossotti equation and related the molar polarization P with the -> dielectric constant er, the electron polarizability a of an individual molecule, and the - dipole moment p P = nN0 a + = f Vm... 

Vm is the molar volume of the compound). The Debye-Clausius-Mossotti equation is applicable only to nonpolar gases at moderate pressure, and to nonpolar solvents and solutes in nonpolar solvents. For polar gases and polar solvents the -> Onsager equation gives more precise data. 

This is the Clausius-Mossotti equation. The quantity Pm is called the molar polarization and has the dimensions of volume per mole. 

The classical treatment of nonpolar dielectric materials is expressed by the Clausius-Mossotti equation. Polar materials in nonpolar solvents are better handled by Debye s modification, which allows for the permanent dipole of the molecule. Onsager made the next major step by taking into account the effect of the dipole on the surrounding medium, and finally Kirkwood treated the orientation of neighboring molecules in a more nearly exact manner. (See Table 2-1.) The use of these four theoretical expressions can be quickly narrowed. Because of their limitations to nonpolar liquids or solvents, the Clausius-Mossotti and Debye equations have little application to H bonded systems. Kirkwood s equation has great potential interest, but in the present state of the theory of liquids the factor g is virtually an empirical constant. The equation has been applied in only a few cases. 

The dipole moment induced per unit of volume V is called the molar polarization Pm> and is defined by the Clausius-Mossotti equation as ... 

Table I lists the experimentally measured dielectric constants at 0% and 40% RH, polymer densities, and the weight uptake of water per 100 g film at 40% RH for all the samples investigated. The polarizability of the absorbed water in the films was calculated using the Clausius-Mossotti equation (Equation 1) which relates macroscopic dielectric constant to molecular polarizability as follows ...

The dipole moment of a selected functional group in the polymer can also be calculated using the Clausius-Mossotti equation. According to Van Krevelan, (15) the "effective polarizability" of a functional group in the polymer is calculated from the measured dielectric constant, the polymer density and the number of moles of that group in the polymer repeat unit. Using the data for the six polyamide-imides listed in Table I, the effective polarizability of the amide group can be determined from... 

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