Surface charge equation

The surface potential is not accessible by direct experimental measurement it can be calculated from the experimentally determined surface charge (equations 32-34) by equations 40a and 40b. The zeta potential, calculated from electrophoretic measurements, is typically lower than the surface potential, J/q, calculated from diffuse double-layer theory. The zeta potential reflects the potential difference between the plane of shear and the bulk phase. The distance between the surface and the shear plane cannot be defined rigorously. 

It is reasonable to assume that H, OH, HCOa", and C02(aq) are able to interact as potential-determining species (adsorption or desorption) with CaC03(s) and affect its surface charge (equation 44). In a system [CaC03(s), CO2, H2O] where CaC03(s) (calcite) is equilibrated with pco2 = constant, the following equilibrium is valid ... 

There is no experimental way to measure o- (As mentioned before, the zeta potential as obtained, for example, from electrophoretic measurements is smaller than i/. ) But as discussed, we can obtain the surface charge (equation 33) and then compute the surface potential J/ on the basis of the diffuse double-layer model with equation 40a equation 40a in simplified form for 25 °C is... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

Electrostatic Interaction. Similarly charged particles repel one another. The charges on a particle surface may be due to hydrolysis of surface groups or adsorption of ions from solution. The surface charge density can be converted to an effective surface potential, /, when the potential is <30 mV, using the foUowing equation, where -Np represents the Faraday constant and Ai the gas law constant. 

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

The polyelectrolyte chain is often assumed to be a rigid cylinder (at least locally) with a uniform surface charge distribution [33-36], On the basis of this assumption the non-linearized Poisson-Boltzmann (PB) equation can be used to calculate how the electrostatic potential

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The non-zero, in general, value of e Pw-e PR in Equation (7.18) implies that there are net surface charges on the gas exposed electrode surfaces. These charges (q+,q.) have to be opposite and equal as the cell is overall electrically neutral and all other charges are located at the metal-solid electrolyte interfaces to maintain their electroneutrality. The charges q+=-q. are quite small in relation to the charges, Q, stored at the metal-electrolyte interface but nevertheless the system has, due to their presence, an excess electrostatic energy ... 

This equation was hrst obtained by Gabriel Lippmann in 1875. The Lippmann equation is of basic importance for electrochemistry. It shows that surface charge can be calculated thermodynamically from data obtained when measuring ESE. The values of ESE can be measured with high accuracy on liquid metals [e.g., on mercury (tf= -39°C)] and on certain alloys of mercury, gallium, and other metals that are liquid at room temperature. 

ElectrocapiUary curves have a maximum. At this point, according to Eq. (10.32), the surface charge Qg = 0. The potential, E, of the maximum is called the point of zero charge (PZC). Knowing the charge density Qgyi, one can calculate the interfacial potential contained in Eq. (10.1). This is insufficient, however, for a calculation of the total Galvani potential, since other terms in this equation cannot be determined experimentally. 

The availability of the surface charge results in redistribution of free charge carriers in semiconductor which leads to formation of a compensating space charge and electric field E related to the value of the volume charge through the Poisson equation ... 

The above relationships were derived for low electrode coverages by the adsorbed substance, where a linear adsorption isotherm could be used. Higher electrode coverages are connected with a marked change in the surface charge. The two-parallel capacitor model proposed by Frumkin and described by the equation... 

The two terms in the potential drop are the dipole moments of free charges and the permanent dipole moments. Each may be divided into contributions of components of the two phases. Equation (6) may be rewritten in terms of the surface charge densities of Eqs. (3) and (4) ... 

This helps explain58,73 why simple variational calculations can give good work functions only the electronic tail, spilling over from the positive background, is involved in the surface contribution. Equation (32) also holds72 for the charged interface, in the form... 

Since surface charges depend on the electrostatic potential (Eq. 4.20), Eqs. 4.20-4.22 are solved in an iterative way leading to self-consistent surface charges. At the end of this procedure, surface charges and the electrostatic potential satisfy the boundary condition specified in Eq. 4.21. In practical applications, this self-consistent procedure for calculating reaction field potential is coupled to self-consistent procedure which governs solving the Kohn-Sham equations. A special case for infinite dielectric constant outside the cavity... 

Here, clayer = Ksjdlayer is the capacitance per unit area (in farads/m2) of the layer. The surface voltage Vlayer can be related to the accumulated surface charge a/ (in C/m2) by the following equation ... 

Now, at the point of zero charge, equation (2.9) implies that A = 0 i.e. that the pzc corresponds to a potential drop across the interface of zero and, from equation (2.2), that M = s. This is not found in practice owing to the layer of water molecules at the electrode surface that are present even at the pzc. These water dipoles give rise to an additional contribution to A, see Figure 2.4(a). This additional potential drop, AD, will change sign according to the orientation of the water dipoles at the electrode, and equation (2.2) can thus be re-written as ... 

Figure 2.5 (a) Schematic plot of surface tension vs. potential according to equation <2.15) and assuming that cations and anions behave in identical manners, (b) Plot of surface charge density vs. potential obtained riu the differentiation of the curve in (a), (c) Plot of differential capacitance vs. potential obtained via the differentiation of the curve in (b). 

In Reaction 10.11 the change Az in surface charge is —zq because the uncom-plexed sites Ap carry no charge. With this in mind, we can write a generalized mass action equation cast in the form of Equation 10.8,... 

The iteration step, however, is complicated by the need to account for the electrostatic state of the sorbing surface when setting values for mq. The surface potential T affects the sorption reactions, according to the mass action equation (Eqn. 10.13). In turn, according to Equation 10.5, the concentrations mq of the sorbed species control the surface charge and hence (by Eqn. 10.6) potential. Since the relationships are nonlinear, we must solve numerically (e.g., Westall, 1980) for a consistent set of values for the potential and species concentrations. 

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