A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 32
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Although the rate constants of homogeneous reactions are not depen-3.5 Multistep Mechanisms113dent on potential, they affect the overall current-potential characteristic by their impact onthe surface concentrations of species that are active at the interface. Some of the most interesting applications of electroanalytical techniques have been aimed at unraveling thehomogeneous chemistry following the electrochemical production of reactive species,such as free radicals. Chapter 12 is devoted to these issues.(d) Chemically Reversible Processes Near EquilibriumA number of experimental methods, such as impedance spectroscopy (Chapter 10), arebased on the application of small perturbations to a system otherwise at equilibrium.These methods often provide the exchange current in a relatively direct manner, as longas the system is chemically reversible.
It is worthwhile for us to consider the exchangeproperties of a multistep process at equilibrium. The example that we will take is theoverall process О + ne ^ R, effected by the general mechanism in (3.5.8)-(3.5.10) andhaving a standard potential E® .At equilibrium, all of the steps in the mechanism are individually at equilibrium, andeach has an exchange velocity. The electron-transfer reactions have exchange velocitiesthat can be expressed as exchange currents in the manner that we have already seen.There is also an exchange velocity for the overall process that can be expressed as an exchange current.
In a serial mechanism with a single RDS, such as we are now considering,the overall exchange velocity is limited by the exchange velocity through the RDS. From(3.4.4), we can write the exchange current for the RDS as/ O r d s = FAk^s(Co,)eqe-af^-E0^(3.5.34)The overall exchange current is и-fold larger, because the pre- and postreactions contribute n' + n" additional electrons per electron exchanged in the RDS. Thus,i 0 = nFA^ s (Co0 e q e- a / ( £ e q- £ ° r d s )We can use the fact that the prereactions are at equilibrium to express (CoOeqCQ By substitution from (3.5.13),i0 = nFAk°Tds C*e-"'/№eq-£Opre)e-a/№eq-SWLet us multiply by unity in the form e(n' + a^(E(3.5.35)mterms of(3.5.36)~E * and rearrange to obtain/0 = nFAk%, г«'/(£Орге-£О')еа/(£&-£О')С *e-(«'+«)/№eq-£0'>(3.5.37)Because equilibrium is established, the Nernst equation for the overall process isapplicable. Taking it in the form of (3.5.16) and raising both sides to the power— (nr + a)/n, we haveI0 = r^AI^^'^U-^'^^Tds-E^c*[!-(„'+*)/,,] c * [<л'+«)/л](3e5e38)Note that the two exponentials are constants of the system at a given temperature andpressure.
It is convenient to combine them into an apparent standard rate constant for theoverall process, k®pp, by definingС = k°rdsen^E^-E\^E°^-*°')(3.5.39)so that the final result is reached:*о =(3.5.40)This relationship applies generally to mechanisms fitting the pattern of(3.5.8)—(3.5.10), but not to others, such as those involving purely homogeneous pre- or114Chapter 3. Kinetics of Electrode Reactionspostreactions or those involving different rate-determining steps in the forward and reverse directions. Even so, the principles that we have used here can be employed to derive an expression like (3.5.40) for any other pattern, provided that the steps arechemically reversible and equilibrium applies. It will be generally possible to expressthe overall exchange current in terms of an apparent standard rate constant and the bulkconcentrations of the various participants.
If the exchange current can be measuredvalidly for a given process, the derived relationship can provide insight into details ofthe mechanism.For example, the variation of exchange current with the concentrations of О and Rcan provide (ri + a)/n for the sequential mechanism of (3.5.8)-(3.5.10). By an approachsimilar to that in Section 3.4.4, one obtains the following from (3.5.40):dlog CZ/ct*o \ri+a(3.5.42)\d log iSince n is often independently available from coulometry or from chemical knowledge ofthe reactants and products, one can frequently calculate ri + a. From its magnitude, itmay be possible to estimate separate values for ri and a, which in turn may afford chemical insight into the participants in the RDS.
Practice in this direction is available in Problems 3.7 and 3.10.As we have seen here, the apparent standard rate constant, k®w, is usually not a simple kinetic parameter for a multistep process. Interpreting it may require detailed understanding of the mechanism, including knowledge of standard potentials or equilibriumconstants for various elementary steps.We can usefully take this discussion a little further by developing a current-overpotential relationship for a quasireversible mechanism having the pattern of(3.5.8)-(3.5.10). Beginning with (3.5.24)-(3.5.26), we multiply the first term by unityin the form of exp [—(ri + a)f(Eeq— Eeq)] and the second by unity in the form ofexp [(n" + 1 - a)f(Eeq- £ e q )].
The result is""'R(0,{n+1,(n"+1t)e "~ ~ '(3.5.43)a E-a)fE£f[n"E°post+(l-a)E%fe]{n"+\-a)f(E-Eeq-«)/ eqg/№post+(l- ) °rds\ein"+1-a)f(E-Eeeqn)Multiplication of the first term by unity in the form of exp [—(ri + a)f(E0' - E0)] andthe second by unity in the form of exp [(n" + 1 - a)f(E0' - E0)] gives- nFAk®dsCR(0,{n +1i)e "(3.5.44)where E — EQq has been recognized as 17. The first exponential in each of the two termscan be rewritten as a function of bulk concentrations by raising (3.5.16) to the appropriatepower and substituting. The result is^atfn(3.5.45)3.6 Microscopic Theories of Charge Transfer115Division by the exchange current, as given by (3.5.40), and consolidation of the bulk concentrations provides1C=Ads. O(°> 0uKCapp0Пп'Е%ге+о£%Ь-{п'+а)Е ']е-{п'+<ф,еO-a)fV* app(3.5.46)^ Rwhere we have recognized that n' + л" + 1 = л.
Substitution for ,consolidation of the exponentials leads to the final result,from (3.5.39) and(3.5.47)which is directly analogous to (3.4.10).When the current is small or mass transfer is efficient, the surface concentrations donot differ from those of the bulk, and one has(3.5.48)which is analogous to (3.4.11). At small overpotentials, this relationship can be linearizedvia the approximation ex ~ 1 + x to givei.
•Y-t•(3.5.49)which is the сои^ефай of (3.4.12). The charge-transfer resistance for this multistep system is thenRTnFi,(3.5.50)which is a generalization of (3.4.13).The arguments leading to (3.5.47)-(3.5.50) are particular to the assumed mechanisticpattern of (3.5.8)-(3.5.10), but similar results can be obtained by the same techniques forany quasireversible mechanism. In fact, (3.4.49) and (3.4.50) are general for quasireversible multistep processes, and they underlie the experimental determination of in viamethods, such as impedance spectroscopy, based on small perturbations of systems atequilibrium.3.6 MICROSCOPIC THEORIES OF CHARGE TRANSFERThe previous sections dealt with a generalized theory of heterogeneous electron-transferkinetics based on macroscopic concepts, in which the rate of the reaction was expressedin terms of the phenomenological parameters, lc° and a.
While useful in helping to organize the results of experimental studies and in providing information about reaction mechanisms, such an approach cannot be employed to predict how the kinetics are affected bysuch factors as the nature and structure of the reacting species, the solvent, the electrodematerial, and adsorbed layers on the electrode. To obtain such information, one needs amicroscopic theory that describes how molecular structure and environment affect theelectron-transfer process.A great deal of work has gone into the development of microscopic theories over thepast 45 years. The goal is to make predictions that can be tested by experiments, so that116Chapter 3.
Kinetics of Electrode Reactionsone can understand the fundamental structural and environmental factors causing reactions to be kinetically facile or sluggish. With that understanding, there would be a firmerbasis for designing superior new systems for many scientific and technological applications. Major contributions in this area have been made by Marcus (37, 38), Hush (39, 40),Levich (41), Dogonadze (42), and many others. Comprehensive reviews are available(43-50), as are extensive treatments of the broader related field of electron-transfer reactions in homogeneous solution and in biological systems (51-53). The approach taken inthis section is largely based on the Marcus model, which has been widely applied in electrochemical studies and has demonstrated the ability to make useful predictions aboutstructural effects on kinetics with minimal computation.
Marcus was recognized with theNobel Prize in Chemistry for his contributions.At the outset, it is useful to distinguish between inner-sphere and outer-sphere electron-transfer reactions at electrodes (Figure 3.6.1). This terminology was adopted fromthat used to describe electron-transfer reactions of coordination compounds (54). Theterm "outer-sphere" denotes a reaction between two species in which the original coordination spheres are maintained in the activated complex ["electron transfer from one primary bond system to another" (54)]. In contrast, "inner-sphere" reactions occur in anactivated complex where the ions share a ligand ["electron transfer within a primary bondsystem" (54)].Likewise, in an outer-sphere electrode reaction, the reactant and product do not interact strongly with the electrode surface, and they are generally at a distance of at least asolvent layer from the electrode.
A typical example is the heterogeneous reduction ofRu(NH 3 )6 + , where the reactant at the electrode surface is essentially the same as in thebulk. In an inner-sphere electrode reaction, there is a strong interaction of the reactant,intermediates, or products with the electrode; that is, such reactions involve specific adsorption of species involved in the electrode reaction. The reduction of oxygen in waterand the oxidation of hydrogen at Pt are inner-sphere reactions. Another type of innersphere reaction features a specifically adsorbed anion that serves as a ligand bridge to ametal ion (55).
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