Double layer correction

When the concentration of the inert electrolyte is low, the electrostatic potential at the reaction site differs from that in the bulk and changes with the applied potential. This results in two effects [4]  

The surface concentrations c x and cjed differ from those in the bulk even if the surface region and the bulk are in equilibrium. Using the same arguments as in the Gouy-Chapman theory, the surface concentration cs of a species with charge number z is  

On application of an overpotential rj, the Gibbs energy of the electron-transfer step changes by eo[r) — Afa rj), where Afa(rj) is the corresponding change in the potential fa at the reaction site. Consequently, rj must be replaced by [rj — Afa r )] in the Butler-Volmer equation (5.13). 

These modifications are known as the Frumkin double-layer corrections. They are useful when the electrolyte concentration is sufficiently low, so that fa can be calculated from Gouy-Chapman theory, and the uncertainty in the position of the reaction site is unimportant. Whenever possible, kinetic investigations should be carried out with a high concentration of supporting electrolyte, so that double-layer corrections can be avoided. 

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The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

A theoretical approach based on the electrical double layer correction has been proposed to explain the observed enhancement of the rate of ion transfer across zwitter-ionic phospholipid monolayers at ITIES [17]. If the orientation of the headgroups is such that the phosphonic group remains closer to the ITIES than the ammonium groups, the local concentration of cations is increased at the ITIES and hence the current observed due to cation transfer is larger than in the absence of phospholipids at the interface. This enhancement is evaluated from the solution of the PB equation, and calculations have been carried out for the conditions of the experiments presented in the literature. The theoretical results turn out to be in good agreement with those experimental studies, thus showing the importance of the electrostatic correction on the rate of ion transfer across an ITIES with adsorbed phospholipids. 

The double-layer correction has been applied in perchlorate, nitrate, and chloride media and the true rate constant, k°t, varies... 

The k°a values in KF, KC1, and KBr have been corrected by applying the Frumkin theory for the double-layer correction as cf>2 data were available in the literature for these media. The k°t values obtained in Cl and Br media correspond to CdCl+ and CdBr complexes, respectively, as they correlate93 with corrected values... 

FIGURE 1.23. Variations of the transfer coefficient with the electrode potential derived from convolutive cyclic voltammetry of the following systems with double layer correction, t-nitrobutane in acetonitrile ( ), r-nitrobutane in DMF ( ), nitrodurene in acetonitrile + 2%H20 (a), nitrodurene in acetonitrile ( ), nitromesitylene in acetonitrile (y). Data from reference 64 and references therein. 

The possibility of studying the heterogeneous ET to dialkyl peroxides at the mercury electrode has provided the opportunity to test the dissociative ET theory using experimental activation/driving force relationships. It was thus possible to observe parabolic patterns in agreement with the theory and to use the potential dependence of a to determine the double-layer corrected values. Thus, using the convolution analysis approach, values for E roor/rovro were determined for a number of peroxides in both acetonitrile and DMF solutions. Representative results are summarized in Table 4. 

The first exponential shows the potential dependence of the rate constant upon the measured applied metal—solution potential difference (0m — 0s)- The second exponential is the double layer correction to the rate constant and accounts for the effects of both concentration and potential at the pre-reaction plane. 

From inspection of the second exponential in eqn. (105) and Fig. 3, it appears that double layer corrections to the observed electrode kinetic paramenters are more important at low ionic concentration and high ionic charge of the reacting particle and at potentials close to the pzc. 

From eqn. (105) and eqns. (80) and (84) in Sect. 3.2, the Butler— Volmer equation can be written in terms of the double layer correction... 

A consequence of eqn. (108) is that the true transfer coefficient, at, can be calculated from the apparent transfer coefficient with the double layer correction calculated from the diffuse double layer theory [6]... 

In the presence of specific adsorption of anions other than the reactant species, the double layer correction outlined above can be extended assuming that the pre-electrode layer is still the OHP and taking into account the total electrode charge [46]. 

The second term in eqn. (110) is the double layer correction to the observed reaction order due to the changes in the interfacial potential distribution with the bulk concentration of the ionic reactant. When 9A02/9 In [O] = 0, eqn. (110) becomes identical to eqn. (89) for concentrations instead of activities. This occurs in the presence of large excess of supporting electrolyte, since the concentration of the reacting ion 02o does not determine the interfacial potential distribution and the true reaction order is obtained in eqn. (110). 

This relation emphasizes that only part of the double-layer correction upon AG arises from the formation of the precursor state [eqn. (4a)]. Since the charges of the reactant and product generally differ, normally wp = ws and so, from eqn. (9) the work-corrected activation energy, AG orr, will differ from AG. [This arises because, according to transition-state theory, the influence of the double layer upon AG equals the work required to transport the transition state, rather than the reactant, from the bulk solution to the reaction site (see Sect. 3.5.2).] Equation (9) therefore expresses the effect of the double layer upon the elementary electron-transfer step, whereas eqn. (4a) accounts for the work of forming the precursor state from the bulk reactant. These two components of the double-layer correction are given together in eqn. (7a). 

Therefore, for most experimental conditions, the transfer coefficient for the electron-transfer step, aet, is predicted to approximate 0.5 with deviations from this value at moderate driving forces expected most often for processes featuring large inner-shell structural changes [55]. By and large, these expectations are borne out by experiment work-corrected transfer coefficients in the range ca. 0.35-0.65 are commonly observed for simple one-electron redox couples, although the extraction of ae, values is often impeded by uncertainties in the double-layer corrections. 

In the context of the present discussion, it is worth noting that virtually all the experimental systems that exhibit such "anomalous temperature-dependent transfer coefficients are multistep inner-sphere processes, such as proton and oxygen reduction in aqueous media [84]. It is therefore extremely difficult to extract the theoretically relevant "true transfer coefficient for the electron-transfer step, ocet [eqn. (6)], from the observed value [eqn. (2)] besides a knowledge of the reaction mechanism, this requires information on the potential-dependent work terms for the precursor and successor state [eqn. (7b)]. Therefore the observed behavior may be accountable partly in terms of work terms that have large potential-dependent entropic components. Examinations of temperature-dependent transfer coefficients for one-electron outer-sphere reactions are unfortunately quite limited. However, most systems examined (transition-metal redox couples [2c], some post-transition metal reductions [85], and nitrobenzene reduction in non-aqueous media [86]) yield essentially temperature-independent transfer coefficients, and hence potential-independent AS orr values, within the uncertainty of the double-layer corrections. 

Variation in the metal surface composition is, then, generally expected to yield large variations in the observed rate constant for inner-sphere pathways since the reaction energetics will be sensitive to the chemical nature of the metal surface. For outer-sphere reactions, on the other hand, the rate constants are anticipated to be independent of the electrode material after correction for electrostatic work terms provided that adiabatic (or equally non-adiabatic) pathways are followed. Although a number of studies of the dependence of the rate constants for supposed outer-sphere reactions on the nature of the electrode material have been reported, relatively few refer to sufficiently well-defined conditions where double-layer corrections are small or can be applied with confidence [111-115]. Several of these studies indeed... 

Since anions and cations adsorb at oxide electrodes positive and negative to the pzc, respectively, electrostatic work terms (double layer corrections) should contribute to the activation free energy barrier for adsorbed electroactive ions depending on the position of the reaction site. Not much attention has been paid to this phenomenon yet. Trasatti and co-workers... 

Diffuse double layer correction in electrode kmetics, 205, 210... 

Diffuse-Double-Layer Corrections in Electrode Kinetics... 

These effects are quite remarkable, and the diffuse-double-layer correction cannot be neglected for such highly charged ions under any circumstances. 

The second line for Mn gives analysis for double-layer corrected kinetic data. 

The primary purpose of this review is to summarize comprehensively advances in the study of this kinetic aspect of charge transfer across ITIES since 1981, when Koryta and Vanysek gave a timely review at that early stage of the development of electrochemistry at ITIES. Reviews [5-14] and monographs [15, 16] are available of other aspects of the electrochemistry at ITIES, e.g., ion transfer facilitated by ionophores, applications to analytical purposes or to liquid extraction, and instrumentation. In a recent review on charge transfer across ITIES, Girault [14] addressed key issues regarding the mechanism of ion transfer the dependence of the rate constant of ion transfer on the applied potential, the presence of an activation barrier, the double layer correction, the effect of solvent viscosity, theoretical treatments, etc. Since the author s [14] opinions differ in several respects from ours, we have tried to review this subject as systematically and critically as possible. 

The above relations assume that the concentration of reactant, r, at the interface does not significantly depend on potential. When r is an ion, such as HaO this can be taken account of by introducing so-called double-layer corrections and/or using an excess concentration of an inert supporting electrolyte. When r is a molecule, separate experiments may be necessary to determine the potential dependence of its coverage, as also are required with respect to its bulk concentration in order to interpret the reaction order in that reactant (see Section X). 

Because the first and second bracketed terms in Eq. (n) represent the surface and bulk portions, respectively, of the thermodynamic influence upon it is convenient to define a double-layer corrected rate constant,... 

The double-layer corrected deal enthalpy of activation, (AH ), obtained from the slope of a plot of — R In k against temperature at a given nonisoAermal cell potential ( 12.3.7.1) is approximately equal to the activation enthalpy, AH, for the elementary reaction at that potential . Also, the real activation enthalpy, (AHJ), , obtained from the temperature dependence of the double-layer corrected standard rate constant, k rr> can be identified approximately with the intrinsic barrier , AG. Therefore, from Eqs. (n) and (o) ... 

Ikeda et al. later published results on the CO2 reduction at an Au electrode in phosphate buffer solutions of pH 2 to 6.8. They showed that I c, also obtained from analytical measurements, is proportional to the pressure of CO2. The Tafel slope is ca. 120 mV decade . Their potential partial current relation obtained with electrolytes at pH below 4.3 agreed well with Hori et aL s obtained with 0.5 M KHCO3 at pH 7.5 after the double layer correction due to the difference of the electrolyte concentration. This fact shows that CO formation at Au electrode does not depend on pH of the electrolyte, and that the proton donor is not H but H2O molecules. 

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