Debye restriction

In polyatomic molecules, however, other relaxation pathways can show up in such cases, using combination of intermode energy transfer with relaxation to circumvent the Debye restriction. Consider for example a pair of molecular modes... 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Though the applicability of Eq. (2.39) is restricted, it has certain advantages over the conventional equation of orientational diffusion proposed by Debye [1]... 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

With these assumptions, it is readily verified that each term in (301) is finite, and we may thus restrict ourselves to the lowest-order term. Taking for the Debye-Hiickel result (168)... 

Mie wrote the scattering and absorption cross sections as power series in the size parameter 0, restricting the series to the first few terms. This truncation of the series restricts the Mie theory to particles with dimensions less than the wavelength of light but, unlike the Rayleigh and Debye approximations, applies to absorbing and nonabsorbing particles. 

The part of the activity coefficients depending on k (Equation 33) can be simplified further if suitable average values are introduced. If the restriction d a is sufficiently well fulfilled, the term in y is small compared with the term in <5. The latter therefore represents the main influence caused by the presence of dipoles since the terms in are from ionic charges. They are identical with terms of corresponding order in the Debye-Hiickel theory. 

Finally, the study of the protons of the polymer chain measured by incoherent neutron scattering allows the identification of two distinct types of motion (a) a vibrational motion of the Debye-Waller type and (b) a slow jump-like diffusive motion of the whole chain confined within the volume restricted by... 

Dipole-Dipole Interaction. The first of the four terms in the total electrostatic energy depends on the permanent dipole moment of the solute molecule of radius a (assuming a spherical shape) immersed in a liquid solvent of static dielectric constant D. The function f(D) = 2(D - l)/(2D + 1) is known as the Onsager polarity function. The function used here is [f(D) — f(n2)] so that it is restricted to the orientational polarity of the solvent molecules to the exclusion of the induction polarity which depends on the polarizability as of the solvent molecules, related to the slightly different Debye polarity function q>(n2) according to... 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

For the near wall solution, from the slipping plane to 3 Debye lengths from the wall, the solution is restricted to cases where there are no inertial effects within this region. In other words the left hand side of (5) goes to zero and we are left with an equation of the same form as the DC electro-osmosis equation but now with an AC electric field. The solution to this equation is... 

Together with an appropriate closure for the pair and triplet distribution functions, one may restrict consideration in this limit to the first two equations in the hierarchy (23). Again, in this approximation, one may follow a course of the Debye-Hiickel approach to obtain the mean field potential, while image forces are accounted for. In this way, the distribution of ions in the system will be known, and interaction forces can be calculated on the basis of this distribution. 

I should note that the limit of sensitivity of these experiments restricts us to saying that the phase transitions are simultaneous only in the sense that they both occur between -1 and 0°C (in either direction). More subtle variations, of the order of 0.1°C, would not have been detectable. According to Debye-Huckel theory [3], the depression of the freezing point of pure water is 0.18°C in a 0.1 M uni-univalent electrolyte solution. We would expect the clay to cause a further small depression in the freezing point, as discussed below. Within these limits, the temperature where both the freezing transition and the gel-crystalline phase transition occur is the same in our model clay colloid system, and it can be concluded to be the ordinary freezing point of the soaking solution. 

Although this is clearly an extreme case, and mobilities are commonly much higher than this and Debye lengths smaller, nevertheless, oxide semiconductors may well be limited in practical application by carrier transport. Naturally, if the bias is such that the semiconductor is not in depletion, there will be no restriction arising from transport indeed, the semiconductor will behave like a metal under these circumstances. 

The usefulness of Frohhch s formula (2.53) is mainly restricted by our ignorance of the correlation factor g, which necessarily depends on the shapes of molecules and the disposition of the permanent dipoles within them, the anisotropy of pOlarisability and the presence of charge distributions of higher orders of symmetry. The theory gives us a good general understanding of the behaviour of polar materials, but deviations from the simple Debye equation (2.44) can often only be discussed in qualitative terms. 

Quantitative Tests of the Debye-Hiickel Limiting Equation.—Although the Debye-Hiickel equations are generally considered as applying to solutions of strong electrolytes, it is important to emphasize that they are by no means restricted to such solutions they are of general applicability and the only point that must be noted is that in the calculation of the ionic strength the actual ionic concentrations must be employed. 

Induced Interaction between Two Multipole Systems. Equation (58) defines in general form the classical electrostatic interaction of two electric systems having permanent multipoles and pj" , in conformity with the classical theory of Keesom. In the classical approach also, as shown by Debye and Falkenhagen, one has to take into consideration energies due to interactions between the permanent multipoles of the one system and electric multipoles induced in the other, and vice versa. Restricting the problem in a first approximation to the energy arising from the mutual interaction of dipoles, we can write ... 

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