A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 11
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At any time, the volume of the diffusion layer isCo(x =II5(r2)8(/ 3 )Figure 1.4.5 Growth of thediffusion-layer thickness with time.1.4 Introduction to Mass-Transfer-Controlled ReactionsNo convectiont35Figure 1.4.6 Current-time transient for a potential step toa stationary electrode (no convection) and to an electrode instirred solution (with convection) where a steady-state current isattained.A8o(t). The current flow causes a depletion of O, where the amount of О electrolyzedis given byMoles of О electrolyzedin diffusion layerГ (mi ^ ( Qf'idtfidt(Лл~mBy differentiation of (1.4.30) and use of (1.4.29),[Cg - Co(x = 0)] A d8(t)2dt=|nF= 0)](1.4.31)ord8(t)flfr=2D O5(0(1.4.32)Since 5(0 = 0 at Г = 0, the solution of (1.4.32) is8(t) = 2VD~t(1.4.33)and(1.4.34)This approximate treatment predicts a diffusion layer that grows with tl/2 and a currentthat decays with t~l/2.
In the absence of convection, the current continues to decay, butin a convective system, it ultimately approaches the steady-state value characterized byS(0 ~ ^o (Figure 1.4.6). Even this simplified approach approximates reality quiteclosely; equation 1.4.34 differs only by a factor of 2/тг1^2 from the rigorous descriptionof current arising from a nernstian system during a potential step (see Section 5.2.1).1.5 SEMIEMPIRICAL TREATMENT OF NERNSTIANREACTIONS WITH COUPLED CHEMICAL REACTIONSThe current-potential curves discussed so far can be used to measure concentrations,mass-transfer coefficients, and standard potentials.
Under conditions where the electrontransfer rate at the interface is rate-determining, they can be employed to measure heterogeneous kinetic parameters as well (see Chapters 3 and 9). Often, however, one isinterested in using electrochemical methods to find equilibrium constants and rate constants of homogeneous reactions that are coupled to the electron-transfer step. This section provides a brief introduction to these applications.361,5.1Chapter 1.
Introduction and Overview of Electrode ProcessesCoupled Reversible ReactionsIf a homogeneous process, fast enough to be considered always in thermodynamic equilibrium (a reversible process), is coupled to a nernstian electron-transfer reaction, then one canuse a simple extension of the steady-state treatment to derive the i-E curve.
Consider, for example, a species О involved in an equilibrium that precedes the electron-transfer reaction14A «± О + qYО + ne «± R(1.5.1)(1.5.2)For example, A could be a metal complex, MYq+; О could be the free metal ion, M n + ;and Y could be the free, neutral ligand (see Section 5.4.4). For reaction 1.5.2, the Nernstequation still applies at the electrode surface,(1.5.3)and (1.5.1) is assumed to be at equilibrium everywhere:C(1.5.4)AHenceKCA(x = 0)nF(1.5.5)Assuming (1) that at t = 0, C A = C*> CY = C*, and C R = 0 (for all JC); (2) that C* is solarge compared to C* that CY(x = 0) = C* at all times; and (3) that К «1; then atsteady statenFA- CA(x = 0)](1.5.6)(1.5.7)= 0)(1.5.8)Then, as previously,С Ax = 0) =(// ~ 0nFAmkCR(x = 0) =(1.5.9)(1.5.10)E = El/2 + (0.059/л) log -4—where14To simplify notation, charges on all species are omitted.(T = 25°)(1.5.11)1.5 Semiempirical Treatment of Nemstian Reactions with Coupled Chemical Reactions ^1 37Thus, the i-E curve, (1.5.11), has the usual nernstian shape, but Ещ is shifted in a negative direction (since К «1) from the position that would be found for process 1.5.2 unperturbed by the homogeneous equilibrium.
From the shift of Ещ with log Cy, bothq [= —(n/0.059)(dEi/2/d log C Y ] and К can be determined. Although these thermodynamic and stoichiometric quantities are available, no kinetic or mechanistic informationcan be obtained when both reactions are reversible.Coupled Irreversible Chemical ReactionsWhen an irreversible chemical reaction is coupled to a nernstian electron transfer, the i-Ecurves can be used to provide kinetic information about the reaction in solution.
Considera nernstian charge-transfer reaction with a following first-order reaction:О + ne ^± R(1.5.13)k(1.5.14)R->Tlwhere к is the rate constant (in s ) for the decomposition of R. (Note that к could be apseudo-first-order constant, such as when R reacts with protons in a buffered solution andк = к'Сц+.) As an example of this sequence, consider the oxidation of p-aminophenol inacid solution.2H + + 2e+ NH 3(1.5.15)(1.5.16)Reaction 1.5.16 does not affect the mass transfer and reduction of O, so (1.4.6) and(1.4.9) still apply (assuming CQ = CQ and C R = 0 at all x at t = 0). However, the reactioncauses R to disappear from the electrode surface at a higher rate, and this difference affects the i-E curve.In the absence of the following reaction, we think of the concentration profile for R asdecreasing linearly from a value CR(x = 0) at the surface to the point where C R = 0 at 8,the outer boundary of the Nernst diffusion layer.
The coupled reaction adds a channel fordisappearance of R, so the R profile in the presence of the reaction does not extend as farinto the solution US 8. Thus, the added reaction steepens the profile and augments masstransfer away from the electrode surface. For steady-state behavior, such as at a rotatingdisk, we assume the rate at which R disappears from the surface to be the rate of diffusionin the absence of the reaction [(mRCR(;t = 0); see (1.4.8)] plus an increment proportionalto the rate of reaction [/X£CR(JC = 0)]. Since the rate of formation of R, given by (1.4.6),equals its total rate of disappearance, we havenFA= mo\C% - Co(x = 0)] - mRCR(x = 0) + fxkCR(x = 0)(1.5.17)38Chapter 1.
Introduction and Overview of Electrode Processeswhere /UL is a proportionality constant having units of cm, so that the product \xk has dimensions of cm/s as required. In the literature (3), \x is called the reaction layer thickness. For our purpose, it is best just to think of /UL as an adjustable parameter. From(1.5.17),Substituting these values into the Nernst equation for (1.5.13) yields(1.5.20)orE = E[/2 + ° ^ iOgfc^(at25°C)(1.5.21)where•Ч^Ц^)(1.5.22)or^[^(1.5.23))where Ещ is the half-wave potential for the kinetically unperturbed reaction.Two limiting cases can be defined: (a) When fik/m^ «1, that is /лк « m R , the effect of the following reaction, (1.5.14), is negligible, and the unperturbed i-E curve results, (b) When fxk/mR »1, the following reaction dominates the behavior andт(1.5.24)^тThe effect is to shift the reduction wave in & positive direction without a change in shape.For the rotating disk electrode, where m R = 0.62DR/3<<>1/2^~1/6, (1.5.24) becomes [assumi n g ^ Ф/(со)]г,ЕШг^0.059,= El/2 + - a - logM0 6 2 D 2^-0.059 11 / 6--2Гbg со/1*1*4(1.5.25)An increase of rotation rate, со, will cause the wave to shift in a negative direction (towardthe unperturbed wave; see Figure 1.5.1).
A tenfold change in со causes a shift of 0.03/n V.A similar treatment can be given for other chemical reactions coupled to the chargetransfer reaction (4). This approach is often useful in formulating a qualitative or semiquantitative interpretation of i-E curves. Notice, however, that unless explicit expressionsfor m R and [i can be given in a particular case, the exact values of к cannot be determined.The rigorous treatment of electrode reactions with coupled homogeneous chemical reactions is discussed in Chapter 12.1.6 The Literature of Electrochemistry39Figure 1.5.1 Effect of an irreversiblefollowing homogeneous chemical reactionon nernstian i-E curves at a rotating diskelectrode.
(7) Unperturbed curve. (2) and (3)Curves with following reaction at two rotationrates, where the rotation rate for(3) is greater than for (2).1.6 THE LITERATURE OF ELECTROCHEMISTRYWe now embark on more detailed and rigorous considerations of the fundamentalprinciples of electrode reactions and the methods used to study them. At the outset,we list the general monographs and review series in which many of these topics aretreated in much greater depth. This listing is not at all comprehensive, but does represent the recent English-language sources on general electrochemical subjects.
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