A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 29
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Let us seenow if the kinetic model yields that thermodynamic relation as a special case. From equation 3.3.11 we have, at zero current,FAk°Co(0, O e - ^ e q - * * ) = FAk°CR(0, t)ea-^)f(EQq-EQl)( 3 A 1 )Since equilibrium applies, the bulk concentrations of О and R are found also at the surface; hencef(Eeq-E°')e=£R( 3 A 2 )Rwhich is simply an exponential form of the Nernst relation:(3.4.3)Thus, the theory has passed its first test of compatibility with reality.Even though the net current is zero at equilibrium, we still envision balanced faradaicactivity that can be expressed in terms of the exchange current, / 0 , which is equal in magnitude to either component current, ic or /a.
That is,OI)(3.4.4)3.4 Implications of the Butler-Volmer Model for the One-Step, One-Electron Process • 99If both sides of (3.4.2) are raised to the -a power, we obtain-af(Eeq-E0')e=(^(3.4.5)RSubstitution of (3.4.5) into (3.4.4) gives 7; _i0 -R(3.4.6)The exchange current is therefore proportional to k° and can often be substituted for k° inkinetic equations. For the particular case where CQ = CR = C,/0 = FAk°C(3.4.7)Often the exchange current is normalized to unit area to provide the exchange currentdensity, jo =3.4.2The Current-Overpotential EquationAn advantage of working with /Q rather than k° is that the current can be described interms of the deviation from the equilibrium potential, that is, the overpotential, 77, ratherthan the formal potential, E0'.
Dividing (3.3.11) by (3.4.6), we obtainC o (0,C R (0,(3.4.8)or(3.4.9)«0The ratios (CQ/C^T and (Со/С|)~ ( 1 ~ а ) are easily evaluated from equations 3.4.2 and3.4.5, and by substitution we obtain(3.4.10)where rj = E - £ e q . This equation, known as the current-overpotential equation, will beused frequently in later discussions. Note that the first term describes the cathodic component current at any potential, and the second gives the anodic contribution.8The behavior predicted by (3.4.10) is depicted in Figure 3.4.1. The solid curveshows the actual total current, which is the sum of the components ic and /a, shown asdashed traces.
For large negative overpotentials, the anodic component is negligible;hence the total current curve merges with that for /c. At large positive overpotentials, thecathodic component is negligible, and the total current is essentially the same as /a. Ingoing either direction from £ e q , the magnitude of the current rises rapidly, because theexponential factors dominate behavior, but at extreme 17, the current levels off. In these7The same equation for the exchange current can be derived from the anodic component current i,d at E = Ещ.Since double-layer effects have not been included in this treatment, k° and i0 are, in Delahay's nomenclature(8), apparent constants of the system.
Both depend on double-layer structure to some extent and are functionsof the potential at the outer Helmholtz plane, Ф2, relative to the solution bulk. This point will be discussed inmore detail in Section 13.7.8100Chapter 3. Kinetics of Electrode Reactions1.00.8—/^0.6'/~~—Total current0.44001300I200-100I|100/7Eeq/// °А,-200J-300I-400IT|, mV*--0.4--0.6--0.8-ч1.0Figure 3.4.1 Current-overpotential curves for the system О + e <=± R with a = 0.5, T = 298 K,//c = -//a = // and /0/// = 0.2. The dashed lines show the component currents ic and /a.level regions, the current is limited by mass transfer rather than heterogeneous kinetics. The exponential factors in (3.4.10) are then moderated by the factors CQ(0, 0/Q)*and CR(0, t)IC^, which manifest the reactant supply.3.4.3Approximate Forms of the I-IJ Equation(a) No Mass-Transfer EffectsIf the solution is well stirred, or currents are kept so low that the surface concentrations donot differ appreciably from the bulk values, then (3.4.10) becomes(3.4.11)which is historically known as the Butler-Volmer equation.
It is a good approximation of(3.4.10) when / is less than about 10% of the smaller limiting current, //c or //a. Equations 1.4.10 and 1.4.19 show that C o (0, i)IC% and C R (0, 0/CR will then be between 0.9and 1.1.The curves in Figure 3.4.2 show the behavior of (3.4.11) for different exchange current densities. In each case a = 0.5. Figure 3.4.3 shows the effect of a in a similar manner. There the exchange current density is 10~6 A/cm2 for each curve.
A notable featureof Figure 3.4.2 is the degree to which the inflection at Ещ depends on the exchange current density.Since mass-transfer effects are not included here, the overpotential associated withany given current serves solely to provide the activation energy required to drive the heterogeneous process at the rate reflected by the current. The lower the exchange current,the more sluggish the kinetics; hence the larger this activation overpotential must be forany particular net current.3.4 Implications of the Butler-Volmer Model for the One-Step, One-Electron Process101j, цА/ст 24b)(a)-400300100I200-100-200-300-400--2--4--6--8Figure 3.4.2 Effect of exchange current density on the activation overpotential required to delivernet current densities. (a)jo = 1 0 A/cm (curve is indistinguishable from the current axis),(b)j0 = 10 6 A/cm2, (c)j0 = 10 9 A/cm2.
For all cases the reaction is О + e : *- R with a = 0.5and Г = 298 К.If the exchange current is very large, as for case (a) in Figure 3.4.2, then the systemcan supply large currents, even the mass-transfer-limited current, with insignificant activation overpotential. In that case, any observed overpotential is associated with changingsurface concentrations of species О and R.
It is called a concentration overpotential andcan be viewed as an activation energy required to drive mass transfer at the rate needed tosupport the current. If the concentrations of О and R are comparable, then Eeq will be nearE® , and the limiting currents for both the anodic and cathodic segments will be reachedwithin a few tens of millivolts of E° .On the other hand, one might deal with a system with an exceedingly small exchangecurrent because k° is very low, as for case (c) in Figure 3.4.2. In that circumstance, no sigj, цА/ст 2/8уf-— a = 0 . 5- a = 0.75—-j6a = 0.254212001150I10015 0 ^ ^/////iiiiй#*/у'^0**-/ / '-2/-4 --1-50%1-100I-150I-200r\, mV-6 -8Figure 3.4.3 Effect of the transfer coefficient on the symmetry of the current-overpotential curvesfor О + e ?± R with T = 298 К andy 0 = Ю" 6 A/cm2.102 -•• Chapter 3. Kinetics of Electrode Reactionsnificant current flows unless a large activation overpotential is applied.
At a sufficientlyextreme potential, the heterogeneous process can be driven fast enough that mass transfercontrols the current, and a limiting plateau is reached. When mass-transfer effects start tomanifest themselves, then a concentration overpotential will also contribute, but the bulkof the overpotential is for activation of charge transfer.
In this kind of system, the reduction wave occurs at much more negative potentials than E° , and the oxidation wave liesat much more positive values.The exchange current can be viewed as a kind of "idle current" for charge exchangeacross the interface. If we want to draw a net current that is only a small fraction of thisbidirectional idle current, then only a tiny overpotential will be required to extract it.
Evenat equilibrium, the system is delivering charge across the interface at rates much greaterthan we require. The role of the slight overpotential is to unbalance the rates in the two directions to a small degree so that one of them predominates. On the other hand, if we askfor a net current that exceeds the exchange current, the job is much harder. We have todrive the system to deliver charge at the required rate, and we can only do that by applying a significant overpotential. From this perspective, we see that the exchange current isa measure of any system's ability to deliver a net current without a significant energy lossdue to activation.Exchange current densities in real systems reflect the wide range in k°.
They may exceed 10 A/cm2 or be less than pA/cm2 (8-14, 28-31).(b) Linear Characteristic at Small rjFor small values of x, the exponential ex can be approximated as 1 + JC; hence for sufficiently small 77, equation 3.4.11 can be reexpressed as(3.4.12)which shows that the net current is linearly related to overpotential in a narrow potentialrange near Eeq. The ratio — r]/i has units of resistance and is often called the charge-transfer resistance, Rct:(3.4.13)This parameter is the negative reciprocal slope of the /-77 curve where that curve passesthrough the origin (77 = 0, / = 0). It can be evaluated directly in some experiments, and itserves as a convenient index of kinetic facility. For very large &°, it approaches zero (seeFigure 3.4.2).(c) Tafel Behavior at Large 77For large values of 77 (either negative or positive), one of the bracketed terms in (3.4.11)becomes negligible.
For example, at large negative overpotentials, e x p ( - a / n ) > > exp[(l- a)/n] and (3.4.11) becomes(3.4.14)orRT,(3.4.15)3.4 Implications of the Butler-Volmer Model for the One-Step, One-Electron Process103Thus, we find that the kinetic treatment outlined above does yield a relation of the Tafelform, as required by observation, for the appropriate conditions. The empirical Tafel constants (see equation 3.2.4) can now be identified from theory as 9-23RTaF(3.4.16)The Tafel form can be expected to hold whenever the back reaction (i.e., the anodicprocess, when a net reduction is considered, and vice versa) contributes less than 1% ofthe current, ora-<*)fve< 0.01,(3.4.17)which implies that |TJ| > 118 mV at 25°C.
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