This technique has advantages in the estimation of electron transfer rate constants compared to other voltammetric methods. The signal to noise ratio of the pulse techniques is generally better than in methods in which current is measured continuously. NPV is particularly suitable for regression analysis because it has a closed form model for voltammograms under diffusion-kinetic control.

In normal pulse voltammetry, a series of periodic pulses of linearly increasing or decreasing potential are applied to the electrochemical cell (cf., Figure 11.3). All pulses begin at the same base potential, at which there is no electrolysis at the working electrode. At the end of each pulse, the potential returns to the base value and a time lag precedes the next pulse. Current measurements are made at the end of each pulse, allowing the unwanted charging current to decay relative to the analytical faradaic current. A resulting increase in faradaic to charging current ratio in the current vs. potential curve, relative to methods such as cyclic voltammetry that measure current continuously, means that the signal to noise ratio is often larger and background currents are more well behaved and easier to model or subtract.

The model for NPV has been described by Osteryoung, O'Dea, and Go [12, 13]. It assumes that the electrode reaction under consideration is

or its reverse. The model is summarized in Table 11.7.

The term x cxp(x2) erfc(x) in Table 11.7 can be computed by Oldham's approximation [14] to an accuracy of better than ±0.2%. Box 11.1 gives

Table 11.7 Model for Obtaining Electron Transfer Rate Constants from Normal Pulse Voltammetry

Assumptions: Voltammogram controlled by electrode kinetics and linear diffusion; a does not depend on E; D values of O and R are equal Regression equations:

i(t) = idTTll2x expO2) erfc(x)/(l + ff) + b5E + b6 6 = exp[nF/RT)(E - £°')] (reductions) id = nFADmCd(vmtpm) x = k(1 + 6)0~«tpm k = k°'IDm

R = gas constant, F = Faraday's constant, T = temperature in Kelvins; id = limiting current, C0 = concentration of electroactive species, a = electrochemical transfer coefficient, tp = pulsewidth, D = diffusion coefficient, and A = electrode area Regression parameters: Data:

id E°' a k b5 b6 i vs. E Special instructions: Keep (F/RT) fixed; use optimum pulsewidth range so that voltammogram is not reversible or totally irreversible; obtain D0 values from limiting currents [1, 2]; x exp(x2) erfc(x) is computed by Oldham's approximation [14]; use 6 = exp[~(nF/ RT)(E - E°')} for oxidations

Subroutine 11.1. BASIC subroutine for analysis of quasireversible NPV data with background based on Model in Table 11.7

10 THETA = EXP(38.92»(A(2,I%)-B(0))) 12 IF B(2) <0 THEN B(2) = 1 15 IF B(3)>1.02 THEN B(3) = .5

20 XXX = B(2)*(l + THETA)*THETAA(-B(3))*SQR(TPULSE)

30 PXX = .8577*(1-(.024*XXXA2)*(1-.8577*XXX/3.1415927#A2))

60 YC = B(1)*FXX/(1+THETA) - B(4)»(A(2,I%)) + B(5)

Documentation:

parameters: B(0) = E°'; B(l) = ¡a; B(2) = k; B(3) = a; B(4) = background slope;

B(5) = background intercept, line 10 - computes 8

lines 12-18 - sets limits on parameters to avoid unproductive parameter space, line 20 - computes x line 30 - 50 computes 7C1/2 x exp(x2)erfc(x) by Oldham's approximation line 60 - computes NPV current at potential E_

Box 11.1 BASIC subroutine for use in nonlinear regression analysis of quasireversible NVP data with a background based on the model in Table 11.7.

the BASIC code for computing the NPV current-potential curve employing Oldham's approximation (Table 11.7) within a nonlinear regression program.

The subroutine in Box 11.1 was used in the analysis of data for the ferri/ ferrocyanide redox couple on glassy carbon electrodes in 0.5 M KN03, which was discussed earlier for carbon microdisk electrodes. Table 11.6 shows that the results from the NPV analysis [10] agree well with those of microelectrode voltammetry and with other studies using similarly activated glassy carbon electrodes.

If we are interested mainly in k°', it is essential to obtain the kinetic parameter k with good accuracy. We must consider the influence of pulsewidth on the error in rate constant k°', which is contained in k in the model in Table 11.7. This was done by analyzing theoretical data with normally distributed noise of ±0.5% of the limiting current [10]. Because the true parameters are known for these data sets, their errors are the differences of the true values from those computed from nonlinear regression. Results show that the correct choice of pulse width in NPV is critical for accurate estimations of k°'.

As pulsewidth is increased above about 10 ms for data with k = 12, the shape of the current-potential curve eventually becomes reversible and the kinetic information in it decreases. For data with 1-2 ms pulsewidths, errors in k from the regression analysis were about ±1%. In contrast, analysis of data with a pulse width of 30 ms gave a 20% error in k. Hence, data for k— 12 and tp > 30 ms gave excellent fits to the reversible model in Table 11.2.

Data representing k = 12 and three values of a show an optimum range of pulsewidths for obtaining k with minimum errors (Figure 11.9). This optimum range does not seem to depend critically on a. Errors in k for a = 0.35 and a = 0.5 fall on nearly the same line. However, the a = 0.65 data showed an extended optimum range in the small pulsewidth region. Errors in k over this range were larger that those for a < 0.5.

Optimum pulse widths shifted to longer times as k was decreased (Figure 11.10). For k = 5, small errors in k were found in the 0.5-20 ms pulsewidth range. Data for 30-50 ms pulses gave very slow convergence of a Marquardt-Levenberg regression program, and the computations usually did not converge even after 120 iterative cycles. Data for k = 1 and 0.1 at pulse widths <5 ms showed significant errors in k. Errors in k for k - 0.1 remained relatively large (15%) even at pulsewidths of 200 ms. At this small value of k, voltammograms are apparently approaching the irreversible limit, beyond which they contain little information about electrode kinetics.

These error analyses clearly show that an optimum range of NPV pulsewidths is required to estimate k°' values reliably. If rough estimates of

0 alpha 0.5 A alpha 0.65 O alpha 0.35

0 alpha 0.5 A alpha 0.65 O alpha 0.35

Figure 11.9 Influence of pulsewidth on errors in kinetic parameter for k = 12 and three different values of a. Random absolute noise of ±0.5%id with id = 1.0 ju.A, E° = -0.3 V, 65 = 0.8 /xA V-1, and b6 = 0.1 ¡xA. (Reproduced with permission from [10], copyright by the American Chemical Society.)

pulse width, ms

Figure 11.9 Influence of pulsewidth on errors in kinetic parameter for k = 12 and three different values of a. Random absolute noise of ±0.5%id with id = 1.0 ju.A, E° = -0.3 V, 65 = 0.8 /xA V-1, and b6 = 0.1 ¡xA. (Reproduced with permission from [10], copyright by the American Chemical Society.)

k°' are obtained before the regression analysis, the appropriate pulsewidth ranges can be chosen from Figures 11.9 and 11.10.

The concept of an optimum time range in electrochemical experiments is a general one. If the time scale is too short or too long for the data to contain sufficient information about the desired kinetic constant, errors will be large. The situation in homogeneous chemical kinetics is similar.

Figure 11.10 Influence of pulsewidth on errors in kinetic parameter for k = 5 (a = 0.35), 1 (a = 0.35), and 0.1 (a = 0.5). Random noise of ±0.5%/,, with id = 1.0 ¿¿A, E° = -0.3 V, bi = 0.8 yu.A V-1, and bb = 0.1 ¡jlA. (Reproduced with permission from [10], copyright by the American Chemical Society.)

pulse width, ms

Figure 11.10 Influence of pulsewidth on errors in kinetic parameter for k = 5 (a = 0.35), 1 (a = 0.35), and 0.1 (a = 0.5). Random noise of ±0.5%/,, with id = 1.0 ¿¿A, E° = -0.3 V, bi = 0.8 yu.A V-1, and bb = 0.1 ¡jlA. (Reproduced with permission from [10], copyright by the American Chemical Society.)

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