\ OX /x=o

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(5.4.1) =nFDKed[ \

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\ OX /x=o

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(5.4.2)

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In general, for a larger number of electroactive substances, the partial current densities corresponding to the individual substances are additive. It is necessary to clarify the meaning of the distance, x = 0, from the electrode. We must remember that the concentration changes resulting from transport processes appear up to distances considerably larger than the dimensions of the electrical double layer (the space charge region, i.e. the region of the diffuse electrical layer, is mostly spread out to a distance of at most several tens of nanometres). Thus 'zero' distance from the electrode corresponds to points lying just outside the diffuse electrical layer. At this distance, the concentrations of the electroactive substances are not yet affected by the space charge. The influence of the surface charge is shown, however, in the rate constants of the electrode reactions which appear in the boundary conditions (see Section 5.3.2). The current density corresponding to the electrode reaction (5.2.1) is described by Eq. (5.2.13). Combination of Eqs (5.2.13) and (5.4.2) yields

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(5.4.3)

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kacRed

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(5.4.4)

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If the rates of the electrode reactions are large and the system is fairly close to equilibrium (the electrode potential is quite close to the reversible electrode potential), then the right-hand sides of Eqs (5.4.3) and (5.4.4) correspond to the difference between two large numbers whose absolute values are much larger than those of the left-hand sides. The left-hand side can then be set approximately equal to zero, A:ccOx k3icKcd ~ 0, and in view of Eqs (5.2.14) and (5.2.17),

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r( - >F] = exp Red

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(5A5)

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which is a form of the Nernst equation. The electrode reaction thus proceeds approximately at equilibrium (this is often termed a 'reversible' electrode process). When A o or A >0, then cRed 0 or cOx *0. A limiting current is then > formed on the polarization curve that is independent of the electrode potential (see page 286). If the initial concentration, e.g. of the oxidized

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281 component, equals c , then the equations of the limiting diffusion current density yd are obtained for various types of diffusion and convective diffusion; these equations follow from Eqs (2.5.8), (2.7.12) and (2.7.17) after multiplication of the right-hand sides by nF. In the general case, the initial concentration of the oxidized component equals Cox ar*d that of the reduced component c Red . If the appropriate differential equations are used for transport of the two electroactive forms (see Eqs 2.5.3 and 2.7.16) with the corresponding diffusion coefficients, then the relationship between the concentrations of the oxidized and reduced forms at the surface of the electrode (for linear diffusion and simplified convective diffusion to a growing sphere) is given in the form (5.4.6) For the sake of simplicity, it will further be assumed that c Red = 0. For the boundary conditions (5.4.5), i.e. the prescribed ratio of the concentrations of the two forms, we may use the transformation -A(D Ox /D Red ) 1/2 cg 1 + A(DOx/DRed)l/2 Together with the boundary condition (5.4.5) and relationship (5.4.6), this yields the partial differential equation (2.5.3) for linear diffusion and Eq. (2.7.16) for convective diffusion to a growing sphere, where D = DOx and c = CoJ[l +k(DOx/DRed)1/2]. As for linear diffusion, the limiting diffusion current density is given by the Cottrell equation

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(5.4.8)

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and, for convective diffusion to a growing sphere (cf. Eq. (2.7.17)), by the Ilkovic equation of instantaneous current, jd = -nFc0Ox(7DoJ3jtt)1/2 (5.4.9)

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the current density at an arbitrary potential for a 'reversible' electrode process is

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l + A(Poxd/ >Red)1/2

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(5A10)

whence

where / and / d are the currents corresponding to the current densities / and

For j = i/d, the half-wave potential Em is obtained:

Em_E

+ l l

(5.4.12)

As the ratio of the diffusion coefficients in Eq. (5.4.12) is mostly very close to unity, Em~EOf. For a 'slow' electrode reaction (boundary conditions 5.4.3 and 5.4.4) the substitutions, Eq. (5.4.6), kJkc = X, c = [1 + A(DOx/DRed)1/2]cOx -A(DOx/Z)Red)1/2^ox and D = DOJ[1 + X(DOx/DKcd)m] convert the boundary conditions (5.4.3) and (5.4.4) to the form = 0, t > 0, dc D = kcc dx (5.4.13)

while the partial differential equations (2.5.3) and (2.7.16) preserve their original form. The solution of this case is