Charge Distribution around an Ion

The net charge density distribution of the ion cloud around a reference ion i of charge ezi follows from Equations 3.10 and 3.12 as 

As the cloud is outside the reference ion, corrections of order 1 /n, with n being the total number of ions in the solution, are ignored. As in the Debye-Hiickel theory, by expanding the exponential term and keeping up to the linear term in if, we obtain 

With the definition of the inverse Debye length k (Equation 3.16), the charge density in the cloud becomes 

The charge density around an ion has a sign opposite to that of the reference ion and is proportional to the potential in the Debye-Hiickel approximation. The 

The fraction of net charge, dezchud, between r and r q- in the cloud is Ttr Pcioud, proportional to /cr exp(-/Yr), as seen from Equations 3.39 and 3.40. Rewriting this result. 

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The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

Polarization of the electronic charge distribution around an ion including its hydration shell near a polyelectrolyte surface,... 

Relaxation effect Hindrance of the undisturbed movement of the ions in the solution, produced in an external electric field by the nonspherical symmetry of the charge distributions around the ions. 

Ionic crystals are formed between highly electropositive and highly electronegative elements when electron transfer from the former to the latter occurs, resulting in oppositely charged ions with closed shell electronic configuration. X-ray diffraction studies of ionic crystals reveal an essentially spherical charge distribution around the... 

A spectacular advance in the understanding of the distribution of charges around an ion in solution was achieved in 1923 by Debye and Hiickel. It is as significant in the understanding of ionic solutions as the Maxwell theory of the distribution of velocities is in the understanding of gases. 

At this point, the symbol k is used only to reduce the tedium of writing. It turns out later, however, that k is not only a shorthand symbol it contains information concerning several fundamental aspects of the distribution of ions around an ion in solution. In Chapter 6 it will be shown that it also contains information concerning the distribution of charges near a metal surface in contact with an ionic solution. In terms ofK, the linearized P-B expression (3.19) is... 

Once the charge distribution q (x) around an ion is known, the pressure may be calculated directly as the sum of a kinetic and an electrostatic term... 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

Thus, there is an apparent excess of ions of opposite charge around any ion. To some extent, a similar arrangement of ions is found in a dilute solution. Distribution of... 

Debye-Huckel Theory, As shown above the cations and anions in an aqueous solution are not uniformly distributed due to forces of interaction between them (ion-ion interaction). There is a statistical excess (over bulk concentration) of opposite charges around a given ion. Thus, ions in solution are surrounded by an ionic atmosphere of an opposite charge. The total charge in this ionic atmosphere is of opposite sign and equal to the charge of the particular ion. 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

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