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S. FELDBERC
Theory of Regenerative Second-Order Mechanisms in Chronoamperometry. The Paradox of Disproportionation1 by S. Feldberg Brookhaven National Laboratory, Upton, New York
(Received Julu IR, 1968)
11973
A regenerative second-order reaction following electron transfer at an electrode surface produces a change in the apparent number of electrons transferred in the overall reaction. The reaction is usually written as the disproportionationof an electrode product 0-
A@B k
2B+A+D
This mechanism, however, leads to an interesting paradox since it implies that the reaction e-
B e D
(1-3)
proceeds more easily than reaction 1-1. Why, then, does not the overall reaction proceed as 2e-
A*D
and how is it possible to observe second-order behavior? There are several solutions to this paradox ranging from the obvious expedient of declaring reaction 1-3 to be kinetically hindered, to some more complicated mechanisms involving additional kinetic steps, In this paper three alternatives to the disproportionation mechanism are presented and analyzed: self-dimerizationof the electrode product and the subsequent electrochemical and chemical reaction of the dimer; dimerization of the electrode product with the parent and the subsequent electrochemical and chemical reaction of the dimer; and an equilibrium reaction of the electrode product preceding the second-order chemical reaction. These alternatives can exhibit behavior quite similar to the disproportionationmechanism when studied by the chronoamperometrictechnique.
A regenerative second-order reaction following electron transfer a t an electrode surface leads to a change in the number of electrons transferred in the overall reaction. The mechanism is usually written as the disproportionation of an electrode product 0-
Mechanism 1 :
A-B
(1-1)
ki
2B-A+D
( 1-21
This mechanism, however, leads to an interesting paradox since it implies that the reaction e-
B-D
(1-3)
proceeds more easily than reaction 1. Why, then, does not the reaction proceed as 2e-
A 3 3 D
(1-4)
and how is it possible to observe second-order behavior? An obvious solution to the paradox is to declare reaction 1-3 to be kinetically hindered. The acceptability of this condition depends of course on the nature of the The Journal of Physical Chemistry
chemical species involved. There are, however, at least three other alternatives to the disproportionation mechanism which circumvent the paradox : selfdimerization of the electrode product and the subsequent electrochemical and chemical reactions of the dimer; dimerization of the electrode product with the parent and subsequent electrochemical and chemical reaction; and an equilibrium reaction of the electrode product preceding a second-order chemical reaction. The objective of this paper is to indicate the mechanistic limitations of the alternatives to the disproportionation mechanism as well as the limitations of the disproportionation mechanism itself. The currenttime-kinetic relationships for the chronoamperometric mode of investigation are compared and shown to be similar. The calculations were made using the technique of digital simulation.2 The basic assumptions are that diffusion coefficients of all species are identical, that the electrode potential is such that the rates of all heterogeneous electron transfer reactions are diffusion limited, and semiinfinite linear diffusion obtains (i.e., a planar electrode). ( 1 ) Research supported by the
U,9. Atomic Energy Commission. (2) 9. W. Feldberg, "Electroanalytical Chemistry-A Series Of Advances," Vol. 111, A. J. Bard, Ed.
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AMPEROMETRY REGENERATIVE SECOND-ORDER MECHANISMS IN CHRONO
I
I
I
I
1
I
-1
CHRONOAMPEROMETRY MECHANISM 1,4 MECHANISM 2
-
I
1
I
I
-1-1
I-
z W
2P
K,,,=Y/O; 0
2.0' 1.8.
I
Because of the ambiguity of the details of reaction 2-4, the chronoamperometric working curves have been calculated for three limiting cases: (a) k2t = k2b = 0; (b) kzf >>>1>>>k2b; (3) kzb >> 1 >> kzt. The plots Of n.,,/nl 218. log# ($ = kZtCA) for cases a, b, and c are shown in Figure 1 (Kzes = O/O, Keeq = 00, and K2eq= 0, respectively), where n2 = 2nl. If reaction 2-4 does not occur (case a ) , then kz is defined by dB/dt = -2kzB'
(3)
If, however, reaction 2-4 does proceed (case b ) , then the mechanism is virtually identical with mechanism 1, except for the additional mode of removal of species B, and dB/dt = -4kzB2
log,, 9
--
Figure 1. Chronoamperometric working ciirves: mechanism 1, I)= k l l C ~ -; mechanism 2, # 5 k z l C ~ ; - mechani.sm 3, I)= Ic3tC~; mechanism 4 (same curve as for mechanism 1) I)= K4e~htCA.
--
,
The chronoamperometric working curve calculated for mechanism 1 is shown in Figure l ? apparent n us. log$, where $ = kltCA and where CA is the bulk concentration of the parent species A, and
(4)
The Kzeq= m curve, Figure 1, is identical in form with the curve for mechanism 1 (Figure 1), but shifted by - 0.3 log unit ( -log 2) . The curves for the three cases vary in qualitatively the same manner as the analogous curves for the firstorder ECE mechanism.6 The variations, however, are less striking. Dimerization of the Electrode Product With the Parent. Adams7v8 has suggested the following mechanistic scheme as a viable alternative to self-dimerization (mechanism 2) 9110-
Mechanism 3 :
It is assumed, of course, that reaction 3 is blocked. Self-Dimerization of the Electrode Product. One alternative to mechanism 1 involves dimerization of the electrode product
A'3B
(3-1)
ks
A+B-AB n3e
(3-2)
-
A B e C
(3-3)
11.18-
Mechanism 2:
A'3R
nze-
C-D
(2-3)
Except for the second-order kinetics of reaction 2-2, the mechanism is virtually identical with the ECE mechanism described in the l i t e r a t ~ r e . ~However, ,~ as has been recently pointed out,6 one must also consider the reaction
+ 2B e 2A + D k2f
C
(for n2 = 2n )
(2-4)
kab
Reaction 2-4 proceeds as a third-order reaction or more likely as two sequential second-order reactions; thus it is probably considerably less significant than the analogous reaction in a first-order ECE mechanism.
When n1 = 1, n2 = 2, and n3 = 1, the apparent n will shift from 1 to 2 with increasing katC. Innumerable mechanistic complications arise if one considers all the possible interactions of the three redox couples. For the sake of sanity and simplicity consider only the (3) These calculations agree quite well with those of Booman and Pence for the same mechanism [ G . L. Booman and D. T. Pence, Anal. Chem., 37, 1366 (1965)l. There appear to be some small errors in their calculation, e.g., as kltCA approaches zero, the term n.,,/nl approaches 1.01 instead of 1.00. Their curve is also shifted along the abscissa by 0.3 log unit since they define the rate of disappearance of species B by the second-order reaction as dB/dt = -kB2. This notation, which has been used by nearly all authors, is incorrect and should be dB/dt = -2kB2, since two molecules of species B are removed each time the reaction occurs. (4) A. 0.Testa and W. H. Reinmuth, Anal. Chem., 33, 1320 (1960). (5) Q. 9. Alberts and I. Shain, ibid., 35, 1859 (1963). (6) M. D. Hawley and 9. W. Feldberg, J. Phys. Chem., 7 0 , 3459 (1966). (7) R. N. Adams, private communication. (8) E. T. Seo, R. F. Nelson, J. M. Fritsch, L. 8. Marcoux, D. W. Leedy, and R . N. Adams, J. Amer. Chem. Soe., 8 8 , 3498 (1966). Volume 73,Number 6 May I969
S. FELDBERG
1240 reaction
assumed that K4eq
