Debye radius

We can see from this equation that the potential / at the point r = 0 has the value that would exist if there were at distance 1/k a point charge -zj or, if we take into account the spherical symmetry of the system, if the entire ionic atmosphere having this charge were concentrated on a spherical surface with radius 1/k around the central ion. Therefore, the parameter = 1/k with the dimensions of length is called the ejfective thickness of the ionic atmosphere or Debye radius (Debye length). This is one of the most important parameters describing the ionic atmosphere under given conditions. 

According to Eq. (7.34), the values of the Debye radius depend on the ionic strength of the solution and increase with decreasing ionic strength they are 0.3, 3, and 30nm for values of 4 of 1, 10 , and 10 " M (note that here / is given in the units mol/L). 

As expected, the D-H theory tells us that ions tend to cluster around the central ion. A fundamental property of the counterion distribution is the thickness of the ion atmosphere. This thickness is determined by the quantity Debye length or Debye radius (1/k). The magnitude of 1/k has dimension in centimeters, as follows ... 

As it was first noted by Zeldovich [33] it is not easy to distinguish experimentally between exponents 1 and 3/4 (equations (2.1.8) and (2.1.77)). The approach just presented cannot be applied to charged reactants since their electrostatic attraction cuts off spatial fluctuation spectrum at the Debye radius. 

Despite the fact that formalism of the standard chemical kinetics (Chapter 2) was widely and successfully used in interpreting actual experimental data [70], it is not well justified theoretically in fact, in its derivation the solution of a pair problem with non-screened potential U (r) = — e2/(er) is used. However, in the statistical physics of a system of charged particles the so-called Coulomb catastrophes [75] have been known for a long time and they have arisen just because of the neglect of the essentially many-particle charge screening effects. An attempt [76] to use the screened Coulomb interaction characterized by the phenomenological parameter - the Debye radius Rd [75] does not solve the problem since K(oo) has been still traditionally calculated in the same pair approximation. 

Later, considering the problem from a macroscopic point of view, H. Casi-mir and D. Polder (Netherlands, 1948), and E. M. Lifshitz (1954), obtained a different, more rapid law of interaction decay. Only recently L. P. Pitaevskii showed that the contradiction does not indicate an error Ya.B. studied an extreme case of large Debye radius, and this case is realizable in principle. 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

Two systems discussed in the present study are determined by two different volume fraction r] = (it/(])(> I )A occupied by the micellar macroions in a bulk suspension, namely, r] = 0.01 and 0.05. According to Table 1, these two volume fractions are associated with the SDS micellar solutions characterized by two different surfactant concentrations of 0.03 and 0.10 mol/L and two different values of the Debye radius h 1. To be close to the real macroion suspensions, the value of the Debye parameter, used in computer modelling, was fixed at kD = 1.5 when macroion volume fraction is rj = 0.01, and kD = 2 in the case of rj = 0.05. Afterwards, two simulated macroion system that are characterized by these two sets of parameters are referred to as the macroion suspension of low surfactant concentration and high surfactant concentrations, respectively. 

In Eqs. (13.4) and (13.5) k is rec dissociation and recombination rate constants within a contact ion pair of a radius a, Hj, is the Debye radius. For the systems with pronounced geminate recombination it is possible to fit nonlinear R OH decay to a numerical solution of a system of DSE equations [18]. Equation... 

Dimensional Analysis The constant /I in Equation 1.31 is the reciprocal Debye radius. 1116 Dimensional analysis of this term is instructive because it involves a number of often needed constants and occurs frequently in a variety of contexts. 

The recombination rate in such a microspace can be estimated according to Goselle et al. (1979). When two reactants, one of which is much smaller and more diffusive (H+) than the other (0-), are locked in a space with internal diameter R, we can regard the heavier reactant (0-) as practically immobile target in the center of a reaction sphere. The proton can diffuse through the reaction space, but wherever it penetrates the Coulomb cage of the proton emitter, protonation takes place. The rate constant of the reaction is thus controlled by two radii, the Debye radius (RD), where electrostatic interaction dominates, and the radius of the reaction sphere, out of which the proton cannot escape. 

The two electrostatic terms are functions of 5, which is the ratio between the Debye radius of the molecule i D = ZxZ ellzkT (the distance at which the electrostatic force equals to the thermal energy kT) and r. ... 

As the electrolyte concentration is low and the Debye radius many times exceeds this distance an identification of the potential in this zone with the potential of the adsorbing ions is reasonable. At high electrolyte concentration the diffuse layer thickness can be comparable to this distance. Even if the counterions are indifferent their distribution in this layer cannot be neglected because it decreases the electrostatic component of surfactant ion adsorption., i.e. enhance its adsorption. In this case we have to consider a discreteness of charges must, the formation of a counter ion atmosphere around the adsorbed ions and their overlap with the neighbour adsorbed ions. 

Figure 4. Schematic presentation of the reaction space for proton-excited pyranine anion recombination in the thin water layer between phospholipid membranes of multilamellar vesicles. The proton release is depicted at the center of the layer and diffuses in concentric shells. When the diffusion radius exceeds the distance to the membrane (dw/2), the shape of the diffusion space deviates from spherical symmetry and approaches cylindrical symmetry. R0 is the reaction radius, R is the unscreened Debye radius of pyranine (R d = 28.3 A ). in this scheme is 30 A, and the size of the water molecules is drawn to...

Space Distribution of Electron Density and Electric Field Around a Thermo-lonized Aerosol Particle. Consider the thermal ionization of a macro-particle of radius 10 p.m, work function 3 eV, and temperature 1500 K. Find the electron concentration just near the surface of the aerosol particle and corresponding value of the Debye radius, r. Calculate the total electric charge of the macro-particle, the electric field on its surface, and the electric field at a distance 3 p.m from the surface. 

The Debye radius td is a plasma parameter characterizing the quasi neutrality. It represents the characteristic size of the charge separation and plasma polarization. If the electron density is high and the Debye radius is small (ro R), then the deviation from quasi neutrality is small, electrons and ions move together, and diffusion is ambipolar. If, vice versa, electron density is relatively low and the Debye radius is large (rp > R), then the plasma is not quasi neutral, and electrons and ions move separately and diffusion is free. For calculations of the Debye radins it is convenient to use the following numerical formula ... 

The Debye radius gives the characteristic plasma size scale required for the shielding of an external electric field. The same distance is necessary to compensate the electric field of a specified charged particle in plasma. In other words, the Debye radius indicates the scale of plasma quasi-neutrality. There is the correlation between the Debye radius and plasma ideality. The non-ideality parameter F is related to the number of plasma particles in the Debye sphere, For plasma consisting of electrons and positive ions,... 

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