Excess surface charge density

In the case of the adsorption of ions, the adsorption free energy is proportional to the excess surface charge density [37],... 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

When ions specifically adsorb at the interface, the excess surface charge density is divided into three parts, the charges in the two diffuse parts of the double layer, q and q, and the charge due to the specifically adsorbed ions, q° [25]. The electroneutrality condition of the entire interfacial region is... 

The thermodynamic relationships among surface tension, surface excess, surface charge density, and double-layer capacitance are shown in Fig. 5H, which follows from Eq. 29H. It is clear that the double-... 

FIGURE 7.3. Illustration of the instability window in electrocapiUary curve (a), excess surface charge density vs. potential curve (b) and double-layer capacitance vs. potential curve (c) in the absence (dashed lines) and the presence (solid lines) of the adsorpdon and partition of an ionic surfactant. Parameters used for calculation are the same as those in Figure 7.1. Shaded region shows the potential range where the system is thermodynamically unstable. Horizontal dashed lines in the middle of b and c represent the lines of zero surface charge and of zero capacitance, respectively. See text for parameters used for the calculation. Adapted in Figure 1 in Ref. [14]. 

In the case of the adsorption of ions, the adsorption free energy is proportional to the excess surface charge density [37], or the phase-boundary potential, AQlinear relationship between qw and Aq(/> [31]. On the other hand, the adsorption free energy is quadratically dependent on Aq(/> in the case of the adsorption of neutral compounds [38]. It seems that within the experimentally available range of the potential window, ca. 300 mV the linear term prevails in the adsorption of ionic surfactants [39,40]. Therefore,... 

When both metals. Mi and M2, are zero charged (their excess surface charge density ionic double layers are absent on their surfaces. 

Image forces play a significant role in electric double-layer effects. The excess surface charge density is... 

The distance-dependence of the surface potentials depends on a number of parameters such as the dielectric constant, the ionic strength, the net excess surface charge density, temperature etc. An indication of the order of magnitude and distances involved of this is shown in Figure 5.3. 

One additional equation that is needed to solve this problem is the Gauss theorem, which relates the excess surface-charge density to the gradient of the potential... 

This is a beautiful theory, in that it allows us to calculate the excess surface-charge density and the double-layer capacitance from well-known principles of electrostatics (the Poisson equation and the Gauss theorem) and thermodynamics (the Boltzmann equation). It has, however, one major drawback it does not predict the correct experimental results Perhaps it would be more accurate to state that agreement between theory and experiment is found only in dilute solutions and over a limited range of potentials, near the potential of zero charge, as seen in Figure 8.4. 

Several equations follow directly from this equation. The partial derivative of y with respect to potential, at constant composition of the solution, yields the excess surface charge density... 

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