## B3 Diffusion Kinetic Model for a Single Potential Step

When electron transfer is not fully at equilibrium during the time scale of the experiment, the shape of the Q(t) vs. t curve is controlled by electrode kinetics and diffusion. Analysis of such data by nonlinear regression analy-

Table 12.3 Model for Reversible Electron Transfer in Double Potential Step Chronocoulometry [2]

Assumptions: Reversible electron transfer, linear and edge diffusion Regression equation:

Q(t < t) = b0 + bttm + b2t Q(t < r) = bx[tm - (t - T)m] + b2r + b,(r - t) ba = Gdi; i>] = IFADo1 Ctt'xa\ b2 = aFADnC(nr)m; = aFADRC(nr)

Qdi = double layer charge; F = Faraday's constant, A = electrode area; D0 and Dr = diffusion coefficients; C = concentration of reactant; a = constant accounting for nonlinear diffusion; and r = radius of disk or spherical electrode; model adaptable to D0 + Dr

Regression parameters: Data:

Special instructions: Logical statements required in subroutine for computing Q(t)\ IF t < r THEN (use) 0(f) = Q(t < t) IF t > t THEN (use) <2(0 = Q(t > t) Background may need subtraction or consideration in the model for solid electrodes; obtain Do from b\.

Table 12.4 Model for Diffusion-Kinetic Control in Single Potential Step Chronocoulometry [3]

Assumptions: Kinetic and linear diffusion control of Q, no adsorption Regression equation:

y = b0>'2; b0 = A; b, = g bo ff = nFAK°' exp{-a{F/RT)n(E-E°')}

A = k°'lD-£2 exp{~a(F/RT)n(E - E°')} + D~mR exp{-a(F/RT)n(E - £°')}] F = Faraday's constant, A = electrode area; DR and D0 = diffusion coefficients;

C = concentration of reactant; T = temperature in Kelvins. Regression parameters: Data:

Special instructions: Estimation of D and k°' require values of ES" and a to be known or determined from a voltammetric method; background requires subtraction; compute exp(y2) erf(y) from standard series expansion [5]

sis provides the diffusion coefficient simultaneously with the apparent standard heterogeneous rate constant for electron transfer. The model is called the diffusion-kinetic model (Table 12.4).

The diffusion-kinetic model has been applied to the oxidation of ferrocy-anide ion:

In neutral 0.1 M KC1 this redox couple showed an anodic voltammetric peak at about 0.25 V vs. SCE. Starting from initial potentials of -0.2 V, single potential step chronocoulometry was done with final potentials of 0.4, 0.5, and 0.6 V vs. SCE. These final potentials were well positive of the E°' of this redox couple as judged from the midpoint of the anodic and cathodic peaks in the cyclic voltammogram. When the reversible model in Table 12.1 was fit to the background subtracted data, nonrandom deviation plots (Figure 12.5) and a D about 15% larger than the accepted value were obtained [3],

The final potential used in these experiments should have been positive enough to ensure adherence of the data to the conditions of the reversible model. However, the model failed to fit the data. It was suspected that an inorganic polymer film formed on the electrode during the chronocoulometry may have partially blocked the surface and slowed down electron transfer [3],

Analysis of the background-subtracted ferrocyanide Q-t data with the diffusion-kinetic model in Table 12.4 showed all the characteristics of a good fit (Figure 12.6), including a random deviation plot. The average D of (6.3 ± 0.15) X 10 6 cm2 s_1 was in good agreement with the value from the literature of 6.5 X 10~6 cm2 s"1. The k°' of (6 ± 2) X 10 3 cm s"1 at pH 7 (7 X 10 3 cm s"1 was found by CV) decreased to (2.0 ± 0.9) X 10"4 cm a

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Figure 12.5 Results of regression analysis for uncorrected chronocoulometric data (Ej = -0.20 V vs. SCE; Ef = 0.40 V vs. SCE) for 0.2 mM ferrocyanide in 0.1 M KC1, pH 12: (a) experimental charge (O); line from fit onto reversible model; (b) plot of relative residuals from the analysis. (Reprinted with permission from [3], copyright by VCH.)

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Figure 12.5 Results of regression analysis for uncorrected chronocoulometric data (Ej = -0.20 V vs. SCE; Ef = 0.40 V vs. SCE) for 0.2 mM ferrocyanide in 0.1 M KC1, pH 12: (a) experimental charge (O); line from fit onto reversible model; (b) plot of relative residuals from the analysis. (Reprinted with permission from [3], copyright by VCH.)

s 1 at pH 12, in line with the hypothesis of film formation on the electrode, which is facilitated by alkaline solutions [3]. This example illustrates the importance of choosing the correct model for the system to obtain accurate parameters.