Charge-parabola

It thus defines the curvature of the energy-charge parabola. Chemically, it is the inverse of the charge capacity (k) or polarizability 31... 

A further feature of the fit parameters obtained by Blundy and Wood (1994) for plagioclase and clinopyroxene was that the partitioning parabolae become tighter E increases) and displaced to lower as charge increases (Law et al. 2000 Blundy and Dalton 2001). These observations have since been confirmed by a large number of... 

The next step is obvious. Differentiate the y versus V electrocapillary curve at various values of cell potential, and plot these values of the slope (electrode charge) as a function of potential (Fig. 636). If the electrocapillary curve were a perfect parabola, then the charge (strictly, excess chaige density) on the electrode would vary linearly with the cell potential (Fig. 6.56). 

Fig. 6.56. When the electrocapillary y vs. E curve is a perfect parabola, the electrode-charge density varies linearly with potential difference. A further differentiation gives a capacity that is potential inde-pendent. However, experimental electrocapillary curves are not perfect parabolas.

Now, the cosh function gives inverted parabolas [Fig. 6.65(b)]. Hence, according to the simple diffuse-charge theory, the differential capacity of an electrified interface should not be a constant. Rather, it should show an inverted-parabola dependence on the potential across the interface. This, of course, is a welcome result because the major weakness of the Helmholtz-Perrin model is that it does not predict any variation in capacity with potential, although such a variation is found experimentally [Fig. 6.65(b)],... 

Gouy—Chapman Diffuse-Charge Model Jec0kT % ze y 2n Smh 2kT 2 nkT J kT V, = e" Pol 0 er s tial x-> It predicts that differential capacities have the shape of inverted parabolas. Ions are considered as pointcharges. Ion-ion interactions are not considered. The dielectric constant is taken as a constant. 

On this basis it seems that metal properties do affect the total capacity, C, through the changes in M- Thus, the next question seems quite obvious Would it be possible to measure and then corroborate its contribution to the total capacity of the double layer Unfortunately a direct measurement of CM is not possible because the metal will always form part of the total double layer and therefore only the total capacity can be measured. However, we may still have some weapons left. It is possible to obtain an indirect measurement of in the absence of specific adsorption. The way to proceed is as follows. From Eq. (6.124) we see that CH is independent of the concentration in solution, in contrast to the term Qji which involves the term c0 [see Eq. (6.130)]. However, CM should be independent of the concentration of the solution since it involves only the electrode properties. Thus, it is reasonable to combine the concentration-independent terms and say, for example, that the term CM is included in the CH term.48 Thus, a plot of CH vs. the charge of the electrode, qu, would give an indication of the effect of CM on the interfacial properties. Figure 6.70 shows one of those graphs. Thus, the shape of this graph, the asymmetric parabola, is most probably due to the influence of the properties of the metal in the interfacial properties. 

It will be seen that the values of the space-charge capacities are low (-0.01-1 fiF cm 2) compared with the capacities (-17 (J.F cm 2) of the region between the semiconductor surface and the OHP plane, the Helmholtz-Perrin parallel-plate region. That is why the space-charge capacities (the inverted parabolas) are noticed, for the observed capacity is given by two capacitors in series, the space charge, Csc, and Helmholtz-Perrin HP capacitors. Thus,... 

In order to make the comparison between Ep and Ep/2 measurements summarized in Table 9, the two quantities were measured in separate experiments. A recent study by Eliason and Parker has shown that this is not necessary [57]. Analysis of theoretical LSV waves by second-order linear regression showed that data in the region of Ep are very nearly parabolic. The data in Fig. 9 are for the LSV wave for Nernstian charge transfer. The circles are theoretical data and the solid line is that described by a second-order polynomial equation. It was concluded that no detectable error will be invoked in the measurement of LSV Ep and Ip by the assumption that the data fit the equation for a parabola as long as the data is restricted to about 10 mV on either side of the maximum. This was verified by experimental measurements on both a Nernstian and a kinetic system. 

THOMSON PARABOLA METHOD. The method of investigating the charge-to-mass ratio of positive ions in which the ions are acted upon by electric and magnetic fields applied in the same direction normal to the path of the ions. It can be shown that ions of a given charge-to-mass ratio but different velocities will be deflected so as to form a parabola. 

If we measure electrocapillary curves of mercury in an aqueous medium which contains KF, NaF, or CsF, then we observe that the typical parabolas become narrower with increasing concentration. Explanation With increasing salt concentration the Debye-length becomes shorter, the capacity of the double layer increases. The maximum of the electrocapillarity curve, and thus the point of zero charge (pzc), remains constant, i.e., neither the cations nor fluoride adsorb strongly to mercury. 

Table 9.4 represents the calculated AG values for the charge separation and charge recombination processes. Hereby, the charge recombination falls into the inverted regime of the Marcus parabola. With these values in hand, it was possible to place the different possible reaction pathways in a state diagram (Fig. 9.25). 

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