Macroscopic dipole moment

As the metallic particles are assumed to be sufficiently large for macroscopic dielectric theory to be applicable, we can substitute for a the expression for the polarisability of metallic particle immersed in an insulator. The dipole moment is given by the integration of the polarisation over the volume V. Thus, if the polarisation is uniform ... 

Molecules do not consist of rigid arrays of point charges, and on application of an external electrostatic field the electrons and protons will rearrange themselves until the interaction energy is a minimum. In classical electrostatics, where we deal with macroscopic samples, the phenomenon is referred to as the induced polarization. I dealt with this in Chapter 15, when we discussed the Onsager model of solvation. The nuclei and the electrons will tend to move in opposite directions when a field is applied, and so the electric dipole moment will change. Again, in classical electrostatics we study the induced dipole moment per unit volume. 

A macroscopic model for regular air/solution interfaces has been proposed by Koczorowski et al 1 The model is based on the Helmholtz formula for dipole layers using macroscopic quantities such as dielectric constants and dipole moments. The model quantitatively reproduces Ax values [Eq. (37)], but it needs improvement to account for lateral interaction effects. 

Without specifying the dimensions and spatial configuration of the solvation shell, we will treat it in terms of its macroscopic characteristic, like of other dielectric materials. First, consider a polar solution, in which the solutes possess a constant dipole moment as is shown in Fig. 4. In each solvate of a solution, the immediate surroundings are polarized due to the dipole moment, /[Pg.201]

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

After these preliminary remarks, the term polarity appears to be used loosely to express the complex interplay of all types of solute-solvent interactions, i.e. nonspecific dielectric solute-solvent interactions and possible specific interactions such as hydrogen bonding. Therefore, polarity cannot be characterized by a single parameter, although the polarity of a solvent (or a microenvironment) is often associated with the static dielectric constant e (macroscopic quantity) or the dipole moment p of the solvent molecules (microscopic quantity). Such an oversimplification is unsatisfactory. 

The BI for any binding system can be written as 0 = 6(C a), where C is the ligand concentration and a is a set of parameters that could be molecular (such as the mass or the dipole moment of the ligand) or macroscopic (such as the temperature or concentration of some solutes). 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

If the unit cell has a dipole moment, as is the case for polar crystals, the macroscopic polarization must be taken into account. At the point r, it is given by (4a /3 V) iunit r (P. Becker 1990, unpublished results), or... 

The dipole moment of adsorbed atoms can be derived by measuring the change in work function as a result of the presence of the adatoms. In macroscopic measurements, the states of the deposited adatoms are unknown. They may combine into clusters of different sizes and some of them may be absorbed into lattice steps. One can also use the field emission microscope for this purpose. However, similar uncertainties exist. To achieve a well characterized surface and a known number of individual adsorbed atoms on a surface, a combined experiment with field ion microscope observations and field emission Fowler-Nordheim (F-N) plots has been most successful.198,200... 

By comparing the slopes of F-N-plots, as shown in Fig. 4.40(e), with and without adatoms on the surface, the dipole moments of adatoms are then derived. Table 4.7 lists some of the data obtained by different investigators. These values are in fair agreement with those obtained by other macroscopic techniques.201 One must recognize, however, that values obtained with macroscopic techniques are measured without the knowledge of whether these are single adatoms, or whether these adatoms have already been combined into clusters of different sizes, while some of them may in fact have been absorbed into lattice steps. FIM experiments have... 

It seems that there is a need to reexamine, some of the basic quantities used in transport processes, like Thiele numbers, attempting to connect them to more chemical quantities. For example, the macroscopic quantity, e the dielectric constant, can be interpreted in terms of dipole moment distribution, and the dipole moment has immediate structural implications. Now to talk of a dielectric constant in the interaction of two atoms would be a rather useless exercise, since the dilectric constant is a continuous matter concept, not a discrete matter concept. In the same... 

From the molecular viewpoint, the medium acquires macroscopic polarization M because the molecules possess dipole moments pu that preferentially align along the electric field of the charged plates, as depicted schematically below ... 

Two polarization mechanisms are possible. If the molecules possess a permanent electric dipole moment pbp rm, each molecule can align its moment with the field direction by reorientation, producing a macroscopic dipole moment. Even if perm = 0 in the field-free limit, each molecule can achieve a field-dependent dipole moment pind by induction. The induced dipole moment is proportional to field strength, pind = a , where a is the electric polarizability of the molecule. In both cases, work must be performed on the system to achieve the macroscopic polarization. Molecules with large permanent dipole moments correspond to high k. 

For practical purposes the solvent is described as a continuum, so that the dimensions of the solvent molecules do not appear explicitly in the interaction energy equations the permanent dipole moments and the polarizabilities of the solvent are expressed as functions of macroscopic properties which are the dielectric constant D and the refractive index n the interaction... 

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