Anal. Chem. 1994,66, 3182-3181
Response of an ECE Reaction to a Potential Leap Sten 0. Engblom,t Jan C. Myland, and Kelth B. Oldham' Department of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8
A technique is described for analyzing the chemical step of an ECE mechanism. The current following the imposition of a large, concentration-polarizing step is semiintegrated, the semiintegralbeing then displayedas a function of the logarithm of time. A wave-shaped response is predicted, during which the semiintegral doubles in magnitude (if each E reaction involves one electron). The wave may be analyzedto find values of the equilibrium constant K and each of the rate constants kf and kb. Such a total analysis is possible only if the pK lies between -1 and +3. If pKis less than -1, kfalone is accessible, whereas if pK> 3, the product kfKis the measurableconstant. In electrochemistry, and especially in organic electrochemistry, the ECE mechanism is widespread.' A typical example is the occurrence of a protonation in solution after an electron transfer. If the product is itself electroactive, a second electron transfer is possible. The second transfer may occur at the electrode, and this is the circumstance addressed here. An alternative pathway, via a homogeneous rather than a heterogeneous reaction, is the so-called disproportionation (or DISP) route. The ECE-DISP dichotomy has been extensively analyzed and illustrated by the Saveant group.2 Because it involves a second-order reaction, the DISP parallel pathway is always disfavored by diminishing the bulk concentration of the reactant and we shall consider it no further in this article. Except for this limitation, we place no restrictions on the kinetics or thermodynamics of the charge transfers and the chemical reaction. We do, however, require the chemical reaction to be of first, or pseudo first, order. The chemical step is our main concern, and we present below a novel method for determining the rate and equilibrium constants for this reaction. There exists a fairly large body of theoretical knowledge about the ECE mechanism, as well as a voluminous literature on experimental examples. Someof the first theoretical reports in polarography were published by Koutecky3 and by Tachi and Sendaa4A theory applicable to chronopotentiometry was published by Testa and R e i n m ~ t h who , ~ also introduced the now-standard terminology to concisely describe these and other types of coupled reactions. The first theory applicable to controlled-potential voltammetry was that by Nicholson and Shaine6 Sav6ant and co-workers have developed the voltaf On leave from The Water and Environment District of Vaasa, Vasa, Finland. (1) Bard, A. J.; Faulkner, L. R. Elecfrochemical Methods: Fundamenfals and Applicafiom;John Wiley & Sons: New York, 1980; p 431. (2) Andrieux, C. P.; Savbnt, J.-M. In Investigation of Rates and Mechanisms ofReactionr, Techniques ofchemisrry; Berasconi, C. F., Ed.; John Wiley & Sons: New York, 1986; V12; pp 305-309. (3) Koutecky, J. Collecf. Czech. Chem. Commun. 1953, 18, 183. (4) Tachi, 1.; Senda, M. In Advances in Polarography; Longmuir, I. S.,Ed.; Pergamon Press: London, 1960; pp 454-464. (5) Testa, A. C.; Reinmuth. W. H. Anal. Chem. 1961, 33, 1320. (6) Nicholson,'R. S.;Shain, I. Anal. Chem. 1965, 37, 178.
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Ana~lcalChemlstry,Vol. 68, No. 19, October 7, 1994
mmetric theory of the ECE mechanism in a number of papers, (see ref 7, for example). Imbeaux and Sav6ants were the first to suggest the use of semiintegration in the voltammetric study of the ECE mechanism. Other workers who have used convolutive methods to analyze ECE cyclic voltammograms include Neudeck and D i t t r i ~ h . ~ In its most general manifestation, the ECE mechanism may have as many as 12 unknown parameters-four diffusivities, three rate constants (one homogeneous and two heterogeneous), two standard potentials, one equilibrium constant (for the C step) and two transfer coefficients. By confining our attention to the chemical step alone, we reduce this number to a manageable level. Thus, to obviate dependence on the E reactions, we address an excitation signal consisting of a single large step in potential to a region where concentration polarization is complete, Le., where the surface concentrations of the reactants are always equal to zero. The first workers to consider chronoamperometry under these conditions in the study of the ECE mechanism were Alberts and Shain,lo who derived theoretical expressions for the currents generated by both a reversible and an irreversible chemical step. Semiintegration of the ECE current, using a double-potential-step approach, was first suggested by Lawson and Maloy.'l These workers derived the expression for the semiintegral of the current for an irreversible chemical reaction. The semiintegral approach was further developed by Bess et a1.,12 also assuming an irreversible chemical step.
THEORY The reactions under consideration are coupled according to the following scheme:
All symbols here and below have their usual electrochemical meanings, but to aid the reader, a glossary is provided at the end of this article. Reduced species R is present at t = 0 with a uniform concentration equal to its bulk value cb. The other three species are produced as the experiment proceeds and their bulk concentrations are all equal to zero. We treat the case of planar, semiinfinite diffusion. For mathematical tractability, it is necessary to assume that species Q and P share the same diffusion coefficient D,and for convenience, we take D to be the diffusivity of R also. Further, to keep the (7) Nadjo, L.; Savbnt, J. M. J . Electroanal. Chem. 1973, 48, 113. (8) Imbeaux, J. C.; SavCant, J. M. J. Electroanal. Chem. 1973, 44, 169. (9) Neudeck, A.; Dittrich, J. J. Elecfroanal. Chem. 1989, 264, 91. (10) Albcrts, G. S.;Shain, 1. Anal. Chem. 1963, 35, 1859. (11) Lawson, R. J.; Maloy, J. T. Anal. Chem. 1974, 46, 559. (12) Bess, R. C.; Cranston, S.E.; Ridgway, T . H. Anal. Chem. 1976, 48, 1619.
0003-2700/94/0366-3182$04.50/0
0 1994 Amerlcan Chemlcal Soclety
expressions simple, the numbers of electrons transferred in the two electrode reactions are each taken as one. The theory can however easily be extended to cover any pair of electron numbers. As stated above, we restrict attention to a "potential leap", i.e., an experiment in which a large potential step takes the electrode from a region of total inactivity to one of complete concentration polarization for both electroreactants. Thus, the surface concentrations c; and cb both equal zero at all times t > 0 during the experiment. We treat oxidative electron transfers, with anodic current being considered positive. To find an expression for the current, we start by looking at the factors that affect the local concentrations CQ and cp of the species that undergo chemical interconversion. These concentrations change with time as a result of fickian diffusion and first-order chemical reaction according to the equations
_ 'a - D a%' -- kFQ
equal to
--
kf/kb.
1+
i2
FcbD1/2-
If this is less than unity, then
1 - K exp(-2kt) K- 1 ( a t )112
i f K < l (9) where erf( ) denotes the error function,13 k is the mean rate constant of the reaction, (kf + k b ) / 2 , and a is used as an abbreviation for 2kK2/(K2 - 1). In the special case when the two rate constants kf and k b are equal, so that the equilibrium constant equals unity, the simple result
kbCp
ax2
at
holds, whereas the relationship
and (3)
=[
. .
exp(-2kt)daw{$) ut)'/2 - daw((ot)'/')]]
K where x is the dimension measured normal to the electrode and D is the common value of the diffusivities of species P and Q. We seek to solve these equations subject to the initial conditions at t = 0 for all x > 0
cp = cQ = 0
(4)
i f K > 1 (11) applies when the equilibrium constant exceeds unity. Here daw( ) denotes Dawson's integral.13 Another simple case arises when the chemical step is irreversible, so that K is infinite; then
and the boundary condition cp = 0
at x = 0 for all t
>0
(5)
and thereby to evaluate the current density arising from the second electron transfer. This is given by i, = FD(ac,/ax)'
(6)
according to Fick's first and Faraday's laws. A superscript "s" is used, as previously, to denote conditions at the electrode surface. We could similarly write
but, of course, the current arising from the first electron transfer obeys the Cottrell equation i, = Fcb(D/at)'/'
(8)
because the electrode is completely concentration polarized with respect to the initial electroreactant R. A solution to this equation set may be obtained via Laplace transformation in the manner described in Appendix B. It transpires that the form adopted by the expression for i 2 depends on the magnitude of the equilibrium constant K ,
In a different formalism, eqs 9, 11, and 12 were derived by Alberts and Shain'O in the 1960s. These authors also presented the K = solution for the case of spherical diffusion. Of course, the individual iz and i l current densities are not independently measureable; only their sum i = il + i 2 is accessible. Information about the chemical step is present in i 2 alone but the current i l will, during the whole experiment, form an interfering cottrellian background. The study would be considerably simplified if one could replace this transient background with a time-invariant one. The remedy is semiintegration. It is well-known that the semiintegral of a cottrellian current produces a constant. Semiintegration of the current, or equivalently stated, the convolution between the current and l / ( a ~ ) l can / ~ , be performed either electronically14 or c ~ m p u t a t i o n a l l y . We ~ ~ will therefore investigate the semiintegral equivalents of eqs 9-12, using the symbol m to represent the semiintegral (with lower limit zero) of the current density i. Appendix B may again be consulted for the mathematical details. (13) Spanier, J.; Oldham, K. B. An Atlas of Functions; Hemisphere Publishing Corp. and Springer-Verlag: Washington and Berlin, 1987; Chapters 40 and 49. (14) Oldham, K. B. A n d . Chem. 1973, 45, 39. (1 5) Myland, J. C.; Oldham, K. B. Fundumentals of EIectrochemicd Science; Academic Press: San Diego, CA, 1994; pp 260, 448.
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diagram are shown examples of m versus kt curves for different values of K. The graph is plotted semilogarithmically. Notice that all curves show a steady progression, as time increases, from a value of m equal to FcbD112to twice this value. This is because, at the high initial values of the current density, the chemical step in unable to provide a significant supply of species P, whereas it is successful in converting virtually all the Q produced at the electrode into P at the lower current densities that are encountered later in the experiment. Appropriate ways of analyzing experimental m versus t data to find K and k will be the theme of the next section.
2 .o
1.8
9 c
1.6
1
E
1.4
12
1.o
ID2
10’
loo
IO’
101
102
kt Figure 1. Semlintegrai of the current density, normalized by division by Fc?D”*, plotted as a function of kf, whlch has been scaled logarithmically. The numbers on each curve indicate the value of the equilibrium constant K. Unlabeled curves have K = 1, 2, and 5.
Once again, the case when kf = kb, so the equilibrium constant is unity, is special and leads to the simple result m/FcbD’IZ= 2 - exp{-kt)[I,(kt)
+ I,(kt)]
if K = 1 (13)
where IO() and 11( ) are the hyperbolic Bessel functions of orders 0 and l . I 3 The other case that results in a simplified expression for m arises when the chemical reaction is completely irreversible. As kb becomes small, Kgrows beyond any limit and ifK=m
DISCUSSION One of the most striking features, to an electrochemist, of the curves in Figure 1 is their familiar shape. The resemblance to classical voltammetric waves is strong: each curve in Figure 1 displays a transition from a lower plateau of height FcbD112 through an escarpment region to an upper plateau of height equal to 2FcbD1f2.This is exactly what one would expect of a voltammetric wave progressing, with time, from a oneelectron to a two-electron oxidation. Notice, moreover, that the “voltammograms” display a transition in shape, with varying K, from a steep “reversible-like” wave at high K to a less steep wave as K declines. Like irreversible voltammetric waves, theshape becomesconstant at small K, so that a further decrease in Kcauses only a shift along the axis. However, the analogy must not be pushed too far. The waves do not match the shape of true voltammetric waves nor is the similarity to the reversible/irreversible transition anything but a mathematical accident. Nevertheless, another feature of the analogy is that the chemical reaction can be fully characterized (Le., both the thermodynamic Kvalue and the kinetic k value can both be measured) only in the “quasireversible region”. By analogy with voltammetric practice, we may quantify the slopes of the waves in Figure 1 by adopting the “Tomes technique” of measuring the difference between the threequarter-wave point and the one-quarter-wave point. In the present context, this means finding the difference
a result that has been presented earlier.” In the general case we obtain the complicated but exact relation
-FC~D’/’
K- 1
exp(-kt)I,(kt)
if 1 # K < m (1 5 )
The theory is now fully developed and has so far been exact. However, we would like to treat one approximation of practical importance. If K is very small compared to unity, eq 15 can be simplified (see Appendix B) to become -= m
2 - exp(2kK’t) erf~((2kK’t)’/~)
if K
