Diffuse layer theory

The nonlocal diffuse-layer theory near Eam0 has been developed283 with a somewhat complicated function oLyjind of solvent structural parameters. At low concentrations,/ ) approaches unity, reaching the Gouy-Chapman Qatc- 0. At moderate concentrations, deviations from this law are described by the effective spatial correlation range A of the orientational polarization fluctuations of the solvent. 

It is assumed that the quantity Cc is not a function of the electrolyte concentration c, and changes only with the charge cr, while Cd depends both on o and on c, according to the diffuse layer theory (see below). The validity of this relationship is a necessary condition for the case where the adsorption of ions in the double layer is purely electrostatic in nature. Experiments have demonstrated that the concept of the electrical double layer without specific adsorption is applicable to a very limited number of systems. Specific adsorption apparently does not occur in LiF, NaF and KF solutions (except at high concentrations, where anomalous phenomena occur). At potentials that are appropriately more negative than Epzc, where adsorption of anions is absent, no specific adsorption occurs for the salts of... 

The charge density on the electrode a(m) is mostly found from Eq. (4.2.24) or (4.2.26) or measured directly (see Section 4.4). The differential capacity of the compact layer Cc can be calculated from Eq. (4.3.1) for known values of C and Cd. It follows from experiments that the quantity Cc for surface inactive electrolytes is a function of the potential applied to the electrode, but is not a function of the concentration of the electrolyte. Thus, if the value of Cc is known for a single concentration, it can be used to calculate the total differential capacity C at an arbitrary concentration of the surface-inactive electrolyte and the calculated values can be compared with experiment. This comparison is a test of the validity of the diffuse layer theory. Figure 4.5 provides examples of theoretical and experimental capacity curves for the non-adsorbing electrolyte NaF. Even at a concentration of 0.916 mol dm-3, the Cd value is not sufficient to permit us to set C Cc. 

According to the diffusion layer theory, for which the transport process is rate-limiting, kT kR, so that k = kT. According to the interfacial barrier theory, for which the surface reaction is rate-limiting, kR kT, so that it, = R-... 

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

The original and simplest form of the diffusion layer theory was developed by Nemst [105] and Brunner [106], who assumed that the mass flux is given by Fick s first law of diffusion. In that case,... 

Equation (43) describes the transport-controlled dissolution rate of a solid according to the diffusion layer theory in its simplest form. The mass transfer coefficient here is given by k, = kT = Dlh. 

The simplest version of the Stem theory consists in eliminating the point-charge approximation of the diffuse-layer theory. This is done in exactly the same way [Fig. 6.66(a)] as in the theory of ion-ion interactions (see Chapter 3) the ion centers are taken as not coming closer titan a certain distance a from the electrode. 

This differential equation for the space variation of the potential inside the semiconductor can be easily identified with that for the space variation of the potential inside the electrolyte in the Gouy-Chapman theory of the diffuse layer (Section 6.6.1). The solution can therefore be borrowed from the diffuse-layer theory. One has, from Eq. (6.125),... 

In a more general case where the sinusoidal perturbation is imposed to a steady-state, concentration distributions due to a DC current passage (fDC), the surface concentrations Co(0) and cR(0), are expressed as functions of the experimentally accessible variables c, cj), and E using the diffusion layer theory ... 

Substantial efforts have been made to develop physicochemical models for ion exchange based on the Gouy-Chapman diffuse-layer theory (e.g., 9, 10). This work not only has provided insight into the role of diffuse-layer sorption in the ion-exchange process but also has pointed to the need to consider other factors, especially specific sorption at the surface. Consideration of specific sorption enables description of the different tendencies of ions to... 

Several physicochemical models of ion exchange that link diffuse-layer theory and various models of surface adsorption exist (9, 10, 14, 15). The difficulty in calculating the diffuse-layer sorption in the presence of mixed electrolytes by using analytical methods, and the sometimes over simplified representation of surface sorption have hindered the development and application of these models. The advances in numerical solution techniques and representations of surface chemical reactions embodied in modem surface complexation mod-... 

In what concerns the tests of the ionic isotherm, they can be carried out only if an analytic expression of is available. That is, the present theory must be necessarily combined with a diffuse layer theory. Therefore, the validity of these isotherms depends to a certain degree on the validity of theory of the diffuse layer adopted for the calculation of For simplicity we used the Gouy-Chapman theory and the well known equation [49] ... 

Any species, the surface excesses of which are not totally described by the diffuse-layer theory, are said to be specifically adsorbed. cr the quantity of charge, that cannot be accounted for by the diffuse-layer model, is the charge at the electrode surface... 

Mainly because of the work of Dzombak and Morel (1990) in providing a consistent theory and especially a self-consistent database of equilibrium constants, most widely available modeling programs use the double diffusive layer theory adopted by them. However, there are many other alternatives. 

This theory was followed by one in which it was assumed that the excess charge on the solution side is not localized at the interfacial plane on the solution side but is diffuse due to the net effect of the electrical and thermal influences (5,7). According to this model (diffuse layer theory) there is a parabolic dependence of the capacity on charge which is found to be in agreement with experiment at low concentrations. However, at higher concentration, the predicted values of the capacity are too high. 

Surface complexation models, on the other hand, account explicitly for the electrical state of the sorbing surface (e.g., Adamson, 1976 Stumm, 1992). This class of models includes the constant capacitance, double layer, and triple layer theories (e.g., Westall and Hohl, 1980 Sverjensky, 1993). Of these, double layer theory (also known as diffuse layer theory) is most fully developed in the literature and probably the most useful in geochemical modeling (e.g., Dzombak and Morel, 1987). 

It follows from the Gouy-Chapman diffuse-layer theory that if =... 

Still more interesting conclusions may be drawn from eq. (49), as to the effect of the valency of the ions upon the flocculating concentrations. Eq. (49) contains v both explicitly and in the quantity y, implicitly. If, however, the double layer potential is sufficiently large, so that even for univalent fons (p = 1) the factor y approaches 1, the value of y will, a fortiori, be = 1. for larger valencies, and therefore practically independent of v. (For z = 8, y = (e — l)/(e -I- 1) = 0.9 and y == 0.864). In that case the concentration c is simply proportional to v. Hence, under the conditions presumed in this chapter (involving in particular the assumption that the diffuse layer theory of Gouy and Chapman may be applied), eq. (49) leads to the very important result that we must expect the quantities of 1—1 valent, 2—2 valent and 3—3 valent electrolyte, needed to flocculate a lyophobic sol or suspension, to be in a ratio... 

A comparison of the experimental data with the solution based on the diffusimi layer theory and the numerical solution presented by Coueignoux and Schuhmann [171] was made by Deslouis et al. [177]. Figure 4.19 presents the normalized complex plane plots for ferricyanide obtained at different potentials corresponding to 1/4, 1/2, and 3/4 of the limiting currents. It is obvious that the approximation using the simplified Nemst diffusion layer theory, curve (a), goes above the experimental points while the numerical solution, curve (b), is much better. 

R. De Levie, /. Electroanal. Chem., 278, 17 (1990). Notes on Gouy Diffuse-Layer Theory. 

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