Capacity of the double layer

By analogy with the Helmholtz condenser formula, for small potentials the diffuse double layer can be likened to an electrical condenser of plate distance /k. For larger yo values, however, a increases more than linearly with o, and the capacity of the double layer also begins to increase. 

Fig. V-12. Variation of the integral capacity of the double layer with potential for 1 N sodium sulfate , from differential capacity measurements 0, from the electrocapillary curves O, from direct measurements. (From Ref. 113.)...

At the electrocapillary maximum, the charge density, a, is zero (point of zero charge) (Fig. A.4.5c). By definition, the differential capacity of the double layer, Cd, is equal (Second Lippmann Equation). 

As was mentioned in the introduction, Frumkin and his group [Kuchinsky et al. (107) and Frumkin et al. (10 )] proposed an electrochemical theory, according to which the adsorption of electrolytes by carbon would be determined by the electrical potential at the carbon-solution interface, and by the capacity of the double layer. At higher concentrations some physical adsorption of acid might occur in addition. 

Although an alteration in the specific inductive capacity of the double layer may be effected by the selective adsorption of the non-electrolyte yet such an explanation is evidently by no means adequate thus we find 1500 millimols. of ethyl alcohol as effective as 70 millimols. of isoamyl alcohol, whilst the dielectric capacities of water and the two alcohols stand in the ratio 81 20 8 5 7 again the anomalous behaviour of di- and tetravalent cations as noted by Efruyt and van Duin is not explicable on this hypothesis. 

What is the significance of this equation It shows that the slope of the curve of the electrode charge versus cell potential yields the value of the differential capacity of the double layer. In the case of an ideal parabolic y versus Vcurve, which yields a linear q versus Vcurve, one obtains a constant capacitance (Fig. 6.56). 

How would contribute to the total capacity of the double layer We pointed out earlier that electrons are able to jump or come out of the electrode surface, and that they are the main contributors to the surface potential (xM or gM), and therefore,... 

Fig6.69. Effectofthechargeofthemetal on the spillover electrons and on the metal capacity of the double layer. 

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. 

Fig. 8.2. An early transient. Current density is constant. Potential builds up first through charging of the double layer, but at a higher potential, electrons pass across the interface, i.e., current flows and the double layer behaves as a leaky capacitor. The very early sections of the transient (double-layer condenser not leaking) can be used to obtain the capacity of the double layer because, there, there is a negligible Faradaic current through the interfacial region and the current goes overwhelmingly to charging the double layer. C = (dq/dV) = (idt/dV).

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. 

If the cations and anions were exactly similar in their effect on the tension, and had either no tendency to be adsorbed, or the ions of each sign were equally adsorbed, we should expect the fall of tension produced by a given departure of the potential E, from that necessary to produce an uncharged surface of mercury, to produce equal decreases in surface tension whether the changes in tension were positive or negative. The curve would then be symmetrical about the maximum. It will be shown below that, if the capacity of the double layer is independent of the applied voltage, that is independent of the nature of the ions or other substances in it, an exact parabola is to be expected for the curve. A few solutions approach the pure parabolic form, but most deviate from it considerably. 

If D is the dielectric constant of the parallel plate condenser formed by these two plane layers of opposite charges, r their distance apart, or the thickness of the double layer , then the capacity of the double layer per square centimetre is Djterr if a is the amount of charge in each layer per square centimetre, the potential difference between the two sides of this idealized double layer is = terra ID. The force on the charge a in a field of potential gradient X is Xa but the resistance to motion of the layer of liquid at the distance from the solid surface where the outer side of this double layer is situated, is rj(v[r), where y is the viscosity and v the velocity with which the liquid moves. At steady motion the driving force and viscous resistance to motion... 

Earlier theories by Gouy, Chapman, and Hcrzfeld discussed the double layer as wholly of this diffuse type but Stem points out that these give far too high values for the capacity of the double layer, partly because in them the ions are supposed mathematically to be able to approach indefinitely close to the solid surface, which is impossible physically owing to the size of the ions. Stern s theory gives a complicated expression for the capacity of the double layer, but accounts reasonably well for the experimental values. Though the layer is largely diffuse in many cases, the capacity is usually of the same order as if the layer were of the plane parallel type, because most of the ions are fairly close to the fixed part of the layer. 

Another model is the Gouy-Chapman model, which assumes that the charges are in a well-defined, compact plane in the solid phase, but the distribution of the counterions is diffuse in the liquid phase. Consequently, the potential changes exponentially in the liquid. However, capacities calculated by the model are significantly different from the experimentally measured capacities of the double layers. 

Stem s Theory of the Double Layer.—The variations of capacity of the double layer with the conditions, the influence of electrolytes on the zeta-potential, and other considerations led Stern to propose a model for the double layer which combines the essential characteristics of the Helmholtz and the Gouy theories. According to Stern the double layer consists of two parts one, which is approximately of a molecular diame r in thickness, Is supposed to remain fixed to the surface, while ihe other is anlttfuse layer extending for me distance into tlie solution The fall of potential in the fixed layer is sharp while that in the diffuse layer is gradual, the decrease being exponential in nature, as required by equation (5). 

To be more precise, the capacity of the double layer around the electrodes should be added. 2Note that the term ohmic drop often also refers to the potential difference between the working electrode and the reference electrode. 

Using liquid mercury, the Cambridge workers assumed that the area of the electrocatalyst was what it seemed to be no invisible roughness. The capacity (around 20 pP cm 2) was then taken by Bowden and Rideal to be the real capacity of the double layer. (By real, they meant unaffected by invisible micro-roughness. )... 

In the oldest theory of the electrode-electrolyte interface (compact layer theory), the double layer was considered analogous to a parallel plate condenser with a plane of charges on the metal side and a second plane of opposite charges on the solution side (4,5). According to this theory, the capacity of the double layer should be independent of potential which is not observed experimentally. 

The ensemble Helmholtz layer/Gouy-Chapman layer constitutes the electrochemical double layer. Its thickness is in the order of a few tens of Angstroms. This layer is generally represented by the series combination of two capacitances relative to the diffuse and compact layers, Cdia and Qomp- The capacity of the double layer, Cd, is thus equal to ... 

Fundamental studies on such systems ate scarce, but some attention has been paid to dispersion in hydrocarbon solutions of surface-active agents for which the ionic concentrations are extremely small, e.g. 10 ° mol dm corresponding to I/k 10 psn. Since 1/k is large the capacity of the double layer is small and only a small surface charge density is necessary to obtain an appreciable surface potential Furthermore, the slow decay in potential from the surface means that the zeta potential, readily obtained from electrophoresis experiments, may be equated with considerable accuracy to the surface potential. 

The capacity of the double layer depends strongly on the solvent but there is a qualitative similarity in the shape of capacity-potential curves in all solvents including water. Early measurements seemed to indicate qualitative differences between water and such non-aqueous solvents as methanol, ethanol and ammonia which have featureless capacity curves in contrast to the characteristic humped curves found in water. However, more recent studies have shown that capacity humps occur commonly in solvents of widely differing types. They are found in solvents of high and medium dielectric constant and probably have a common origin in field reorientation of solvent dipoles. 

F. 1.1.1 A simple electronic scheme equiveilent to the electrochemical cell Ru, resistance uncompensated in the regular three-electrode system Q, differential capacity of the double layer Rf, resisttmce to faradaic current at the electrode surface Rq, solution resistemce compensated in the three-electrode system... 

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