Work terms, Double-layer effects

Although the foregoing electron-transfer theory is preoccupied with describing the electron-transfer step itself, in order to understand the kinetics of overall reactions it is clearly also important to provide satisfactory models for the effective free energy of forming the precursor and successor states from the bulk reactant and product, wv and ws, respectively. As outlined in Sect. 2.2, it is convenient to describe the influence of the precursor and successor state stabilities upon the overall activation barrier using relations such as 

The first term in brackets in eqn. (7a), wv, is that associated with the precursor stability constant [eqns. (4) and (13)], which relates koh and kel [eqn. (10)]. The second term, ocet (wa — wf), accounts for the effect of the double layer on the driving force for the electron-transfer step. Thus, when 

Taken together, these two terms, comprising the conventional doublelayer effect, can be thought of as the influence of the surface upon the transition-state stability, presuming that the reactant-surface interactions in the transition state are an approximately weighted mean of those in the adjacent precursor and successor states. (The appropriate weighting factor is the transfer coefficient aet.) This therefore constitutes the thermodynamic catalytic influence of the surface, as distinct from the intrinsic catalytic effect as defined above. The former, but not the latter, is conventionally termed the double-layer effect, even though both, in fact, involve surface environmental influences upon the transition state stability. 

The theoretical estimation of wp and wB for inner-sphere pathways is clearly difficult since the formation of chemical bonds is necessarily involved. However, at least for strongly adsorbed reactants, Kp is commonly sufficiently large so that it (and hence wp) can be obtained experimentally from measurements of Tp [eqn. (10a)], even though some extrapolation of the Kp values from potentials where the reactant is stable to those of kinetic interest is normally required [20, 57], Experimental estimates of Ks are usually more difficult to obtain, although it can sometimes be assumed that K Ks [12a, 57], 

Few outer-sphere electrode reactions have precursor-state concentrations that are measurable [21] so that it is usual to estimate wp and ws from double-layer models. The simplest, and by far the most commonly used, treatment is the Frumkin model embodied in eqns. (8) and (8a) whereby, as noted in Sect. 2.2, the sole contributor to wp and ws is presumed to be electrostatic work associated with transporting the reactant from the bulk solution to the o.H.p. at an average potential j)d. This potential is usually calculated from the Gouy-Chapman (GC) theory [58], 

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Similarly, double-layer effects on the MHL law may be expressed through the same work terms ... 

It is important to notice that the rate of a given outer sphere electrode redox reaction should be independent of the nature of the metal electrode if allowance is made for electrostatic work terms or double layer effects which will, of course, be dependent on the nature of the electrode material. Inner sphere reactions, on the other hand, are expected to be catalytic with kinetics strongly dependent on the electrode surface due to specific adsorption interactions. 

Most treatments of such double-layer effects assume that the microscopic solvation environment of the reacting species within the interfacial region is unaltered from that in the bulk solution. This seems oversimplified even for reaction sites in the vicinity of the o.H.p., especially since there is evidence that the perturbation of the local solvent structure by the metal surface [18] extends well beyond the inner layer of solvent molecules adjacent to the electrode [19]. Such solvent-structural changes can yield considerable influences upon the reactant solvation and hence in the observed kinetics via the work terms wp and wR in eqn. (7a) (Sect. 2.2). While the position of the reaction site for inner-sphere processes will be determined primarily by the stereochemistry of the reactant-electrode bond, such solvation factors can influence greatly the spatial location of the transition state for other processes. 

The broad applicability often claimed lor Eq. (j) of 12.3.7.2, therefore, needs to be tempered with an awareness of its often serious limitations . Large breakdown in the applicability of such simple relationships may result from several factors, such as nonelectrostatic contributions to the work terms, differences in between corresponding homogeneous and heterogeneous reactions, and specific solvation effects. Further measurements of electrochemical-rate parameters with due regard for double-layer effects are needed to resolve this question. 

Since Bis via Gauss s Law of electrodynamics proportional to the local excess free charge it follows that the term fjeV VGj is proportional to the net charge stored in the metal in region G. This net charge, however, was shown above to be zero, due to the electroneutrality of the backspillover-formed effective double layer at the metal/gas interface and thus Dfje w.Gj must also vanish. Consequently Eq. (5.47) takes the same form with Eq. (5.19) where, now, O stands for the average surface work function. The same holds for Eq. (5.18). 

The presence of the work terms Wp and w, in Eq. (n) of 12.3.7.2 indicates that the structure of the metal-solution double-layer region can exert an important influence upon the kinetics of electrode reactions. The effect of varying the double-layer structure upon electrode kinetics has long been an active research topic, especially at the Hg-HjO interface for which much thermodynamic information exists concerning double-layer composition and structure. ... 

Equation (a), with set equal to ( >, is surprisingly successful in describing the effect of varying the double-layer structure upon the kinetics of electrochemical reactions at Hg electrodes, at least in the absence of specific adsorption of the supporting electrolyte (i.e., when the inner-layer region adjacent to the electrode contains only solvent molecules). However, this does not necessarily imply that average electrostatic interactions provide the sole contribution to the work terms, because contributions may arise from other sources that remain constant under these conditions. In particular, inner-sphere pathways commonly involve reaction sites within the outer Helmholtz plane. Consequently, the overall work terms consist of separate contributions from transporting the reactant from the bulk solution to this outer plane and from this plane to the reaction site within the inner layer. The latter will then be independent of and, therefore, influence only k j.. in Eq. (a). 

Following the first indication in the work of Stout" that b can be independent of temperature, Bockris and Parsons, and Bockris et showed that a similar effect arose in the h.e.r. at Hg in methanolic HCl between 276 and 303 K below these temperatures, b apparently varied in the conventional way with T. However, the derived a values showed a considerable spread. Variations of the temperature effect in b were discussed in terms of the possible influence of impurities but an overall assessment of all other, more recent, observations of the dependence of 6 on T for various types of reactions leads to the conclusion that the unconventional dependence is not due to some incidental effect of impurities. In fact, in another paper, Bockris and Parsons" suggested that the temperature dependence of p for the h.e.r. at Hg arose because of expansion of the inner region of the double layer with temperature. They also noted that, formally, for b to be independent of T, the entropy of activation should be a function of electrode potential. 

In Fig. II. 1.12, cyclic voltammograms incorporating both IR drop and capacitance effects are shown. Effects for the ideal case of a potential independent working electrode capacitance give rise to an additional non-Faradaic current (Fig. II. 1.12b) that has the effect of adding a current, /capacitance = Cw x v, to both the forward and backward Faradaic current responses. Tbe capacitance, Cw, is composed of several components, e.g. double layer, diffuse layer, and stray capacitance, with the latter becoming relatively more important for small electrodes [61]. On the other hand, the presence of uncompensated resistance causes a deviation of the applied potential from the ideal value by the term R x /, where R denotes the uncompensated resistance and I the current. In Fig. II. 1.12, the shift of the peak potential, and indeed the entire curve due to the resistance, can clearly be seen. If the value of Ru is known (or can be estimated from the shape of the electrochemically reversible... 

Interestingly, the capacitance in CT-containing solutions is smaller than in SO solutions alone and smaller, also, than that for the oxide-flhn region (Pajkossy [1994]). This is not explicable (cf. Pajkossy [1994]) in terms of simple specific adsorption effects since theories predict an increase, rather than a decrease, of capacitance. In these cases, it seems that the double-layer and adsorption pseudocapacitance are coupled in an inseparable way as argued by Delahay [1966] and as pointed out in the recent studies by Germain et al. [2004] in our laboratory in work at Au (cf. Cahan et al. [1991]). 

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