Solution of a Symmetrical Electrolyte

The single-ion activity coefficients approach unity in the limit of infinite dilution  

In other words, we assume that in an extremely dilute electrolyte solution each individual ion behaves like a nonelectrol5Te solute species in an ideal-dilute solution. At a finite solute molality, the values of y+ and y are the ones that allow Eq. 10.1.10 to give the correct values of the quantities (/l+ — jjJ ) and (/l- — We have no way to actually measure these quantities experimentally, so we cannot evaluate either y+ or y.  

The approximations in these equations are like those in Table 9.6 for nonelectrolyte solutes they are based on the assumption that the partial molar volumes F+ and V- are independent of pressure. 

from Eq. 10.1.9, we have the following relations between the chemical potentials and molalities of the ions  

Like the values of y+ and y, values of the single-ion quantities a+, a-, F+, and F-caimot be determined by experiment. 

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Field saturation. Consider a particle occupying a convex open domain CR3 (or ft2) with a smooth boundary du>, charged to the electric potential > 0, at equilibrium with an infinite solution of a symmetric electrolyte of a given average concentration. (Properties described below are directly generalizable to an arbitrary electrolyte or electrolyte mixture.)... 

For an aqueous solution of a symmetrical electrolyte at 25°C, equation (7.6) becomes... 

CHAPTER 10 ELECTROLYTE SOLUTIONS 10.2 Solution of a Symmetrical Electrolyte... 

Keh and Chen [3] obtained an analytical expression for the diffusiophoretic velocity of a dielectric sphere surrounded by a thin but polarized electric double layer in the solution of a symmetric electrolyte. 

We shall consider in this section a diluted solution of a symmetrical, (1-1) electrolyte confined between two thick, infinite slabs (Fig. 4). Further, it is assumed that ions are point charges, i.e., in the one-body and pair electrostatic potentials there is no short-range component. Moreover, the surface charge is smeared out over the particle surface, and, therefore, it may be represented by an average density a Severity of these assumptions will be discussed later on in the text. The dielectric permittivity is assumed to preserve its constant bulk value down to the surface. 

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at separation h immersed in a solution containing a symmetrical electrolyte of valence z and bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7-9]. We denote by Ni and Zi, respectively, the density and valence of charged groups in membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. Without loss of generality we may assume that Zj > 0 and Z2 may be either positive or negative and that Eq. (13.1) holds. The Poisson-Boltzmann equations (13.2)-(13.4) for the potential distribution j/(x) are rewritten in terms of the scaled potential y = zeif/IkT as... 

The electric double layer thickness is a function of the fluid ionic strength. For aqueous solutions having a symmetric electrolyte (e.g., NaCl), the electric double layer thickness, 1/k, at 25 °C is (54)... 

To get the main idea of the charge effect on adsorption kinetics, it is sufficient to consider an aqueous solution of a symmetric (z z) ionic surfactant in the presence of an additional indifferent symmetric (z z) electrolyte. When a new interface is created or the equilibrium state of an interfacial layer disturbed a diffusion transport of surface active ions, counterions and coions sets in. This transport is affected by the electric field in the DEL. According to Borwankar and Wasan [102], the Gouy plane as the dividing surface marks the boundary between the diffuse and Stem layers (see Fig. 4.10). When we denote the surfactant ion, the counterion and the coion, respectively, with the indices / = 1, 2 and 3, the transport of the ionic species with valency Z/ and diffusion coefficient A, under the influence of electrical potential i, is described by the equation [2, 33] ... 

For the special case of a symmetric electrolyte solution (z. z = 1 1), the Poisson-Boltzmann equation reduces to... 

Gouy-Chapman solution for a symmetric electrolyte of valence V ... 

In order to obtain identical values for the second minimum in different electrolyte solutions, a specific relationship must be observed between the concentration c and the valence of the counter-ion z of a symmetrical electrolyte. This relationship, for a given surface potential, can be represented as follows ... 

If we confine ourselves to the case of a symmetrical electrolyte 1.1 in aqueous solution, it is possible to describe the solution, in normalised MKSA units, in terms of its characteristic parameters. 

In a solution of a single electrolyte solute that is not necessarily symmetrical, the ion molalities are related to the overall solute molality by... 

Let us first consider the electrostatic (double-layer) interaction between two identical charged plane-parallel surfaces across a solution of a symmetrical Z Z electrolyte. If the separation between the two planes is very large, the number concentration of both counterions and coions would be equal to its btllk value, no, in the middle of the film. However, at finite separation, h, between the surfaces, the two electric double layers overlap and the counterion and coion concentrations in the middle of the film, nio and respectively, are no longer equal. As pointed out by Langmuir [311], the electrostatic disjoining pressure, Del, can be identified with the excess osmotic pressure in the middle of the film ... 

If this equation is employed for a solution of a single valence-symmetrical electrolyte (z+ = z = z), then... 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

The question to be discussed is whether saturation of the electric field (asserted by Proposition 2.1) implies saturation of the interparticle force of interaction. Consider for definiteness repulsion between two symmetrically charged particles in a symmetric electrolyte solution. In the onedimensional case (for parallel plates) the answer is known—the force of repulsion per unit area of the plates saturates. (This follows from a direct integration of the Poisson-Boltzmann equation carried out in numerous works, primarily in the colloid stability context, e.g., [9]. Recall that again in vacuum, dielectrics, or an ionic system with a linear screening, the appropriate force grows without bound with the charging of the particles.)... 

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