The Clausius-Mosotti equation

In this case, the scattering serves as a means for counting the number of molecules (or particles, or objects) per unit volume (N/V). It is seen that the polarizability, a, will be greater for larger molecules, which will scatter more. If we take the Clausius-Mosotti equation [16] ... 

This result, called the Clausius-Mosotti equation, gives the relationship between the relative dielectric constant of a substance and its polarizability, and thus enables us to express the latter in terms of measurable quantities. The following additional comments will connect these ideas with the electric field associated with electromagnetic radiation ... 

The Clausius-Mosotti equation with n written for can be used to... 

The Clausius-Mosotti equation relates the polarizability of a substance to... 

Pq the dipole or orientation polarisation P itself is defined by the Clausius-Mosotti Equation... 

The quantity on the left side of the equation is called the molar polarization, and this expression is the Clausius—Mosotti equation. On the right side the quantity a is the polarizability, which measures the ease with which an induced moment is... 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

Hence there exists a relationship between AE and the dielectric constant e which is correlated with a by the Clausius-Mosotti equation ... 

Use the Clausius-Mosotti equation from physical chemistry (Atkins 1994, Chapter 22) to introduce refractive index n... 

The capacitance may change with temperature not only because the dimensions of the capacitor change but also because the permittivity of the dielectric changes. To gain some insight into the sources of the variation in permittivity with temperature, the Clausius-Mosotti equation (Eq. (2.88)) can be differentiated with respect to temperature to give... 

For a number of dielectrics with e, > 30, TC is negative and within 15% of —aLer as illustrated by the examples given in Table 5.6. Eq. (5.34) suggests that the temperature variation of polarizability is small compared with the volume expansion coefficient in these cases. Lower-permittivity oxides have positive TC s and in their case the temperature coefficient of polarizability can be assumed to exceed the volume expansion coefficient. However, the extent to which the Clausius-Mosotti equation can be applied to ionic solids is open to debate. 

The quantity sr is directly sensitive to the detailed chemical composition of the sample. However, the quantitative theory that relates the observed er to the concentrations and dipole moments of the various polar segments present has proved quite difficult to use. The simplest approach is based on the Clausius-Mosotti equation as modified for permanent moments by Debye28). The Debye approach, although overly simple, revealed that sr should decrease with increasing temperature, and should reflect changing concentrations of polar constituents during a reaction. 

By rearranging Equation 22.12 with the Clausius-Mosotti equation, TCK is negatively proportional to the thermal expansion coefficient,... 

From the Clausius-Mosotti equation (Equation 22.16), the relationship between the dielectric polarizabilities and the measured dielectric constant can be expressed by Equation 22.17. 

Finally, Shannon obtained 61 sets of ionic polarizabilities for 129 oxides and 25 fluorides using the Clausius-Mosotti equation and least square refinements, and suggested the periodic table of ionic polarizabilities. Therefore the dielectric constant of materials with compositional changes can be successfully predicted by Equation 22.17 and Equation 22.18. Erom another arrangement of Equation 22.16, the theoretical dielectric constant can be obtained from the total ionic polarizabilities in Equation 22.19. Erom Equation 22.17 through Equation 22.19, the theoretical values of dielectric constant and polarizabilities can be obtained as well as the measured values ... 

The influence of Van der Waals interactions on the polarizability of interacting molecules manifests itself in deviations from the Clausius-Mosotti equation in the Kerr effect and in collision induced li t scattering Ahhougji measurements of these effects are all performed on bulk systems in thermodynamical equilibrium and not on Van der Waals molecules per se, we will nevertheless say a few words about pair polarizabilities, because, just as in the case of the collision induced IR absorption, much can be learned about Van der Waals interactions from the comparison of experimental and computational results. 

The usual criterion is that the molar volume, V, must become equal to the molar refractivity, R = 7rNoa, where a is the gas-phase polarizability. Then the Clausius-Mosotti equation becomes... 

The molar refractivity is the volume of the substance taken up by each mole of that substance. In SI units, MR is expressed as m /mol. MR is a molecular descriptor of a liquid, which contains both information about molecular volume and polarizability, usually defined by the Lorenz-Lorentz equation [Lorentz, 1880a, 1880b] (also known as the Clausius-Mosotti equation) ... 

It remains now to relate the molecular quantities ccq and p to the macroscopic polarizability or dielectric constant, which can be measured experimentally. This is a very difficult task and will not be carried out in a rigorous fashion here. Rather, we start our discussion with an approximate equation, given by Debye, which describes the complex dielectric constant in terms of molecular properties. We rationalize the form of the equation through the Clausius-Mosotti equation and then show how e (o)) and s"((o) can be derived from this expression. Additional factors that were not included in Debye s original work, such as the effect of the reaction field and orientation correlation-which are important in condensed phases-will also be discussed... 

Our explanation of equation (7-31) makes use of the Clausius-Mosotti equation, which is derived as follows ... 

It is now easy to understand the origin of equation (7-31). One sees that it is of the form of the Clausius-Mosotti equation where the complex dielectric constant rather than % or eR values is used. The complex formulation introduces a frequency dependence, which appears in the last term of equation (7-31). One would expect the time-dependent contribution to be related to the difference between instantaneous and long-time behavior and, indeed, this is correct, because the factor multiplying the frequency dependence in equation (7-31) is merely the difference between equations (7-47) and (7-40). In fact, these two expressions may be combined with equation (7-47) to yield... 

This is the Clausius-Mosotti equation. The molar polarization P is defined by... 

Derive the Clausius—Mosotti equation using text Equations (11.3), (11.4) and (11.5). [Note derivation is not given in the answers at the end of this book.]... 

Starting from the Clausius-Mosotti equation (9.7) and assuming that it can be applied to each principal direction Ox, in an orthorhombic crystal, show that the Lorentz-Lorenz equation can be written in a form that leads to the equation , = (1 + 2x,)/(l — x,), where , is the refractive index for light polarised parallel to Ox,- and X,- = q ,-/(3 oF), with a,- equal to the polarisability per unit cell for light polarised parallel to Ox,- and V the volume of the unit cell. 

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