Analysis of data reflecting the EC reaction mechanism just discussed requires double potential step chronocoulometry, because only the response during the reverse step is influenced by the chemical reaction. In many cases, however, the product of the chemical step can be electrolyzed. This leads to additional charge on the forward potential step above that expected for an EC reaction. Such an electrode reaction pathway is called an ECE mechanism (Box 12.3), in which the EC process is followed by a second electron transfer.

Although the electrode reaction mechanism of an ECE process may be complex, the data on the forward pulse of a potential step reflect the rate of the chemical reaction. This is because, when the potential step (for oxidations) is well positive of both formal potentials for A/B and C/D couples, the electrolysis of C provides additional charge over that obtained from only the first electron transfer reaction. The amount of additional

A f B + e' |
(12.7) |

B + S-V C |
(12.8) |

A f B + e" |
(12.9) | |

B C |
(rds) |
(12.10) |

C- D + e" |
E°2. |
(12.11) |

Box 12.3 ECE pathway, reversible electron transfer.

Box 12.3 ECE pathway, reversible electron transfer.

charge is proportional to the rate of transformation of B to C. Thus, single potential step chronocoulometry can be used to find the rate constant of the chemical reaction. The data analysis problem is a bit simpler than for a double potential step experiment.

We now discuss analysis of data on ECE reactions. Equation (12.10) is written as a pseudo-first-order reaction to simplify the model. In general, we would have

The usual pseudo-first-order condition for ECE reactions is [5] > 10X [A]. If this can be achieved experimentally, with kr in a range where the charge passed represents less then 2F/mol of A, then k can be estimated by nonlinear regression analysis. The second-order rate constant is found from k2 = /c/[S]. The pulse width r must be chosen correctly to achieve the above conditions [8].

As discussed earlier, if EpU|Se §> E\ the intermediate C will be oxidized as soon as it is formed. The electron can be lost either at the electrode (eq. (12.11)) or in a thermodynamically favored chemical dispro-portionation step (eq. (12.13), Box 12.4) [9]. This latter pathway is featured in the so-called DISP1 mechanism shown in box 12.4, which has eq. (12.10) as rate-determining step.

Actually, the general mechanism for such reactions includes both Boxes 12.3 and 12.4, and the possibility of eq. (12.13) as a rate-determining step (rds) [9]. However, if the second-order reaction in eq. (12.13) is the rds, the peak or half-wave potential determined by a voltammetric method (cf. Chapter 11) will vary in a predictable way with the concentration of A [9].

A ^ B+ e~ |
E°\ |
(12.9) |

BAC |
(rds) |
(12.10) |

B + C ^ A + D |
fast. |
(12.13) |

Box 12.4 DISP1 pathway, reversible electron transfer.

Box 12.4 DISP1 pathway, reversible electron transfer.

Thus, systems with second-order rate-determining steps are easily recognized in preliminary voltammetric studies. For a first-order rds, peak potential does not depend on concentration of A.

In this section we consider only the ECE and DISP1 models, in which the pseudo-first-order conversion of B to C is the rds. We then discuss distinguishing between these two pathways for the hydroxylation of an organic cation radical.

The ECE model is given in Table 12.7. As for determination of rate constants with normal pulse voltammetry, the pulse width (r) is critical for accurate estimation of k. Pointwise variance analysis showed that range 0 < log^r < 1.7 gave the smallest error in k. The use of r values giving log&Twell outside this range could give errors as large as 200% [8], Considering that modern commercial instrumentation can easily provide 20-40 data points for r = 20 ms, the upper limit for determination of k by this method is about 2.5 X 103 s"1.

Table 12.8 presents the model for the DISP1 pathway (Box 12.4). The same consideration applies as for the ECE mechanism, that the pulsewidth should obey the relation 0 < log&T< 1.7.

The models in Tables 12.7 and 12.8 were employed in a chronocoulome-tric study of the electrochemical hydroxylation of tetrahydrocarbazoles 1 and 2 [8], which are model compounds for structurally related anticancer indole alkaloids. Their anodic hydroxylation products in acetonitrile/water

Table 12.7 Model for ECE Pathway with Reversible Electron Transfer in Single Potential Step Chronocoulometry [8]

Assumptions: Reversible electron transfer, linear and edge diffusion; D0 = DR; pseudo-first-

order rds (Box 12.3) Regression equation:

Q(t) = b0 + bx\AtmvV2 - b2 -m&ri(b2t)m\ + ht bo = Qdu ¿i = FADy2C; b2 = k; b3 = aFADC/r

Qdi = double layer charge; F = Faraday's constant, A = electrode area; D = diffusion coefficient; C = concentration of reactant; a - constant accounting for nonlinear diffusion; and r = radius of disk electrode Regression parameters: Data:

Special instructions: Use simplex or steepest decent algorithms for fitting data (Marquardt algorithm gave convergence problems) [10]. Correct range of pulse widths must be used for accurate k estimates [8]; that is, -0 < log£:t < 1.7. For some solid electrodes, background may need subtraction or consideration in the model. Compute erf from (7r1/2/2)[erf(x)]/x = exp^x2)^!, 2/3; x2)

where

,Fj(a, b; x2) = 1 + {a!b)z + a(a + 1 )/[b(b + l)]z2/2! + . . . Rummer's confluent hypergeometric function, or from alternative standard series expansion [5].

Table 12.8 Model for DISPI Pathway with Reversible Electron Transfer in Single Potential Step Chronocoulometry [8]

Assumptions: Reversible electron transfer, linear and edge diffusion; D0 = DR\ pseudo-first-

order rds (Box 12.4) Regression equation:

Q(t) = b0 + ¿>,7r1/2[4r"2 - 0.667V2 - t"2 exp(62r) - 0.886 erf{(&2;)}"2] + b,t b0 = 0di; bt = FADmC; b2 = 2k; i>, = aFADC/r Qdi = double layer charge; F = Faraday's constant; A = electrode area; D = diffusion coefficient; C = concentration of reactant; a = constant accounting for nonlinear diffusion; and r = radius of disk electrode Regression parameters: Data:

Special instructions: Use simplex or steepest decent algorithms for fitting data (Marquardt algorithm gave convergence problems) [10]. Correct range of pulsewidths must be used for accurate ^estimates [8]; that is, -0 < logfcr£ 1.7. For some solid electrodes, background may need subtraction or consideration in model. Compute erf from (771/2/2)[erf(x)]/x = exp(-*2),F,(l, 2/3; x2)

where

,F,(a, b; x2) = 1 + (%)z + a(a + 1 )/[b(b + l)z2/2! + . . . Rummer's confluent hypergeometric function, or from alternative standard series expansion [5],

(>2M) [11] are 3 and 4. Carbon paste electrodes were used, which did not require background subtraction.

H COOMe l-carboxymethyltetrahydrocarbazoles

H COOMe l-carboxymethyltetrahydrocarbazoles

HO COOMe

1 -hydroxy- l-carboxymethyltetrahydrocarbazoles (products)

COOMe possible reactive intermediates in hydroxylation

COOMe

Figure 12.10 Residual plot for chronocoulometry of 0.65 mM 2 in acetonitrile with 11-1 M water and 0.2 M LiCL04 fit onto DISP1 model in Table 12.8. The solid line shows the residual (deviation) pattern for theoretical ECE data with k = 25 sfit onto DISP1 model in Table 12.8. (Reprinted with permission from [8], copyright by the American Chemical Society.)

Figure 12.10 Residual plot for chronocoulometry of 0.65 mM 2 in acetonitrile with 11-1 M water and 0.2 M LiCL04 fit onto DISP1 model in Table 12.8. The solid line shows the residual (deviation) pattern for theoretical ECE data with k = 25 sfit onto DISP1 model in Table 12.8. (Reprinted with permission from [8], copyright by the American Chemical Society.)

Single potential step Q-t data for 1 and 2 obtained in acetonitrile containing 3-17 M water were analyzed by using the models in Tables 12.7 and 12.8. Comparisons of residual plots (cf. Section 3.C.1) were used to find the model that best fit the data. A typical residual plot from a fit of data onto the DISP1 model (Figure 12.10) showed a distinct nonrandom pattern, the same as for regression of theoretical noise-free ECE data onto the DISP1 model. When the same data were analyzed with the ECE model, random deviation plots were obtained (Figure 12.11). The data for both

Figure 12.11 Residual plot for Q-t data used in Figure 12.10 analyzed by using the ECE model in Table 12.7. (Reprinted with permission from [8], copyright by the American Chemical Society.)

Figure 12.12 Experimental (O) and best fit calculated line (ECE model, Table 12.7) for chronocoulometric data used to obtain Figure 12.11. (Reprinted with permission from [8], copyright by the American Chemical Society.)

Figure 12.12 Experimental (O) and best fit calculated line (ECE model, Table 12.7) for chronocoulometric data used to obtain Figure 12.11. (Reprinted with permission from [8], copyright by the American Chemical Society.)

compounds studied were in excellent agreement with this model (Figure 12.12). Thus, the ECE model in Box 12.3 appears to be the best choice for the oxidation pathway of 1 and 2 [8].

The values of k2 computed for 1 and 2 and their deuterio derivatives increased with decreasing concentration of water. The value of In k2 of a reaction of an ion with a dipolar molecule in solution should increase approximately linearly with the inverse of the dielectric constant of the solution. This relation was obeyed rather well (Figure 12.13). Therefore, the differences in k2 at different water concentrations were explained by the influence of water on the dielectric constant of the medium.

The results of these kinetic analyses were combined with quantum mechanical studies of possible reactants and intermediates to propose a detailed mechanism for the hydroxylations [8]. The proposed reactive intermediates are the cation radicals 6 and 8.

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