Anal. Chem. 1999, 71, 3905-3909
Correspondence
Chronoamperometry of Surface-Confined Redox Couples for Irreversible Two-Step and Three-Step Consecutive Reaction Mechanisms Vilmos Kertesz, James Q. Chambers,* and Adam N. Mullenix
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600
Chronoamperometry has been widely employed to determine kinetic rate constants for electron-transfer reactions of surface-confined redox couples. When the mechanism for the transformation is more complicated than a oneelectron transfer, deviations can be expected from an exponential decay of the current transient. Theory is presented in this paper for irreversible, two-step and three-step consecutive reaction mechanisms in the form of analytical solutions of the corresponding differential equations. The theory should find application to surface electrode reactions with two-electron “n-values”. The chronoamperometic, potential-step method has been used effectively to study the kinetics of electrode reactions in which the redox partners are confined to the electrode surface.1-4 For the process, Ox + e- f R, proceeding at an overpotential where the backward reaction is small, it is well established that the current is simply given by first-order exponential decay, I ) I0 exp(-kt), where k is the rate of the process. Thus, a simple data analysis allows extraction of the rate constant from the current decay at a given overpotential without any assumption regarding the potential dependence of the rate constant. Many electrode reactions, however, are mechanistically more complex than a first-order, one-electron-transfer reaction and usually involve a sequence of steps each of which will have a characteristic rate constant. Somewhat more complicated than the above first-order process is a consecutive, irreversible two-step reaction sequence: e-, k1
e-, k2
A 98 B 98 C
(1)
where A is converted to C via the intermediate B. To the authors’ knowledge, the solution to this case in solution kinetics,5 which has been known for more than a century, has not been used for the analysis of simple electrochemical surface reactions. However, for the thin-layer voltammetric experiment, or for reactions (1) (2) (3) (4) (5)
Chidsey, C. E. D. Science 1991, 251, 919-922. Finklea, H. O.; Hanshew, D. D. J. Am. Chem. Soc. 1992, 114, 3173-3181. Creager, S. E.; Weber, K. Langmuir 1993, 9, 844-850. Forster, R. J.; Faulkner, L. R. J. Am. Chem. Soc. 1994, 116, 5444-5452. Laidler, K. J. Chemical Kinetics, 2nd ed.; McGraw-Hill: 1965.
10.1021/ac990114x CCC: $18.00 Published on Web 07/08/1999
© 1999 American Chemical Society
confined to an electrode surface where a mass balance condition is maintained, the classical solution for the case of consecutive reaction transposes directly into the chronoamperometric theory. Another common sequence is a three-step reaction sequence: e-, k1
k2
e-, k3
A 98 B 98 C 98 D
(2)
While this sequence has been written as an ECE mechanism with a rate constant k2 for the intermediate chemical step, the treatment below for this mechanism is easily adapted to other combinations of electron-transfer and chemical steps. In fact, the result below is readily generalized to an n-step sequence. Solutions for the differential equations corresponding to these two-step and three-step sequences and the corresponding currents are given below. RESULTS Theory of the Two-Step Mechanism. The mass balance condition for the above EE reaction sequence is given by eqs 3 and 4:
ΓT ) ΓA + ΓB + ΓC
(3)
1 ) θA + θ B + θ C
(4)
or
where Γi are the individual surface coverages (in mol/cm2), θi ) Γi/ΓT are the fractional coverages, and ΓT is the total surface coverage. If the adsorption/desorption steps are slow on the chronoamperometric time scale, ΓT will be constant for a simple two-step surface redox reaction on an electrode of constant area A. For the above mechanism, it is convenient to evaluate the overall current in two parts, I1 and I2, corresponding to the two kinetically irreversible steps in the reaction sequence. Analytical Chemistry, Vol. 71, No. 17, September 1, 1999 3905
I1 ) -FA I2 ) FA
dθA dΓA ) -FAΓT ) k1FAΓA dt dt
(5)
dΓC dθC ) FAΓT ) k2FAΓB dt dt
(6)
Thus the problem reduces to the differential equations,
dθA ) -k1θA dt
(7)
dθB ) k1θA - k2θB dt
(8)
dθC ) k2θB dt
(9)
and
with the initial conditions that θA ) 1, θB ) 0, and θC ) 0 at t ) 0. By correspondence with the classical solution for a consecutive two-step reaction,5 we have
θA ) e-k1t
(10)
and
1 θC ) [k (1 - e-k1t) - k1(1 - e-k2t)] k2-k1 2
(11)
θB ) 1-θA-θC
(12)
Figure 1 shows the time dependence of the fractional surface coverages and the chronoamperometric response for a case where the first step is 2 times slower than the second. (In all the figures, the current axes have units of I/FAΓT in s-1.) The intermediate nature of species B is clearly evident; note that the dashed line corresponds to k2 ) 0, in which case
θB ) 1 - e-k1t
e-,k1
e-,k2
of the time for a A sf B sf C, EE reaction. For k1 ) 2000 and k2 ) 4000: (1) θA; (2) θB; (3) θC. For k1 ) 2000 and k2 ) 0: (4) θB. (b) Chronoamperometric curve for an EE reaction, where k1 ) 2000, and k2 ) 4000 (see panel a).
Theory of the Three-Step Mechanism. For the simple ECE reaction sequence above, the corresponding kinetic equations are
dθA ) -k1θA dt
(17)
dθB ) k1θA-k2θB dt
(18)
dθC ) k2θB-k3θC dt
(19)
dθD ) k3θC dt
(20)
and
(13)
Substitution into the above equations for I1 and I2 gives
I1 ) FAΓTk1e-k1t
Figure 1. (a) Calculated fractional surface coverages as a function
(14)
where again θA ) 1 at t ) 0. The solution for this set of differential equations, which can be verified by substitution, gives
θA ) e-k1t
and
I2 ) FAΓT
k1k2 -k1t (e - e-k2t) k2-k1
(15)
[
θC ) k1k2
The current I2, which tracks dθC/dt, exhibits a maximum given by eq 16
tmax )
ln(k1/k2) k1 - k2
(
θB ) k1
(16)
3906 Analytical Chemistry, Vol. 71, No. 17, September 1, 1999
and
(21)
e-k1t e-k2t + k2 - k1 k1 - k2
)
e-k1t e-k2t + + (k2 - k1)(k3 - k1) (k1 - k2)(k3 - k2)
(22)
]
e-k3t (23) (k1 - k3)(k2 - k3)
θD )
[
k2k3
(k2 - k1)(k3 - k1)
+
k1k2 (k2 - k3)(k1 - k3)
k1k3 (k1 - k2)(k3 - k2) -
+
k2k3e-k1t (k2 - k1)(k3 - k1)
k1k3e-k2t (k1 - k2)(k3 - k2)
-
k1k2e-k3t
-
]
(k2 - k3)(k1 - k3)
(24)
In this case, the current I2 is proportional to dθD/dt and is given by eq 25.
[
I2 ) FAΓTk1k2k3
e-k1t e-k2t + + (k2 - k1)(k3 - k1) (k1 - k2)(k3 - k2)
]
e-k3t (25) (k2 - k3)(k1 - k3)
The method described by Frost and Pearson offers an alternative approach to Eqns 21-24.6 Figure 2 shows the chronoamperometric response and the time dependence of the fractional surface coverages corresponding to eqs 21-25, when k1 ) k2/2 ) k3/3. It must be noted, that eqs 11, 15, 16, and 22-25 are undefined if some of the rate constants are equal. However, l’Hospital’s rule allows transformation of these equations into well-defined forms in this case; see Appendix section. DISCUSSION Consideration of the above theory leads to several observations that impact the chronoamperometric response of a surface layer containing electroactive sites that can undergo multielectron transformations. In the analysis of these data, a primary question often concerns the n-value, i.e., the number of electrons transferred per redox site. Experimentally, deviation of the ln(current) vs time plot from linear behavior indicates that a simple one-step, rate-controlling reaction mechanism with a unique E°′ is not operative. Interestingly, as shown in the calculated currents in Figures 1 and 2 for the two-step mechanism, when the rates of the individual steps are of the same order of magnitude, or if k2 > k1, maximums will be seen in the current-time curves. Also note that all of the above theory is well-behaved if the rate constants are equal. For example, if k1 ) k2 ) k, eq 15 becomes eq A2. Thus I2 exhibits a maximum given by eq 16 for any ratio of k1 and k2 (or by 1/k if k1 ) k2; see eq A3). The above treatment should prove useful for the analysis of data obtained in spectroelectrochemical experiments in which an intermediate in an electrode reaction is detected. The case for an EE reaction when k2 is much greater than k1 is also of interest. Here I1 and I2 are equal for nearly the complete duration of the current transient and I ) 2FAΓTk1e-k1t; i.e., a twoelectron reaction would be observed proceeding with a rate constant equal to k1. In this case, an intercept corresponding to I(t)0) ) Qk1 and a slope equal to -k1 would be seen on the ln(I) vs time plot where Q ) 2FAΓT, which of course could be (6) Frost, A. A.; Pearson, R. G. Kinetics and Mechanism, 2nd ed.; John Wiley & Sons: New York, 1961; pp 174-177.
Figure 2. (a) Calculated fractional surface coverages as a function e-,k1
k2
e-,k3
of the time for a A sf B sf C sf D, ECE reaction, where k1 ) 2000, k2 ) 4000, and k3 ) 6000: (1) θA; (2) θB; (3) θC; (4) θD. (b) Chronoamperometric curve for an ECE reaction, where k1 ) 2000, k2 ) 4000, and k3 ) 6000 (see a).
determined by integration of the I-time curve, is the charge for a two-electron transformation. Without additional evidence, such as knowledge of the molecular footprints of the redox sites, the chronoamperometric data alone would not allow the one-electron (which exhibits the same values of the intercept and the slope on the ln(I) vs time curve) and the two-electron reactions to be distinguished. There is an other interesting case. When k1 ) 2k2, I ) 2FAΓTk2e-k2t. This is equivalent to a two-electron process with a rate constant equal to k2. Here the intercept of the ln(I) vs time plot is I(t)0) ) Qk2 and the slope is -k2. This is very similar to the k2 . k1 case described above; in this case as well, the chronoamperometric curve does not allow one to distinguish between this case and a simple one-electron process. The equivalent situation in homogeneous solution kinetics is well-known.7 In Figure 3 some ln(I) vs time plots are shown for different ratios of the k1 and k2 values. The ln plot for the k1 ) 2k2 case is shown in panel b. The above treatment applies to redox transitions in which the components are confined to the electrode surface, for example, by adsorption or in a thin layer of solution under the usual conditions of thin-layer electrochemistry. Prominent candidates for application of the above theory are SAMs containing quinonoid species, which are known to exhibit kinetic complications in their voltammetric behavior8-11 or the reduction of surface metal oxide films. (7) Espenson, J. H. Chemical kinetics and reaction mechanisms, 2nd ed.; McGraw-Hill: New York, 1995. (8) Finklea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubenstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19, pp 109-335. (9) Bravo, B. G.; Mebrahtu, T.; Soriaga, M.; Zapien, D. C.; Hubbard, A. T.; Stickney, J. L. Langmuir 1987, 3, 595. (10) Zhang, L.; Lu, T.; Gokel, G. W.; Kaifer, A. E. Langmuir 1993, 9, 786.
Analytical Chemistry, Vol. 71, No. 17, September 1, 1999
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Figure 3. The ln(current) vs time curves of an EE reaction for different ratios of k1 and k2. Values of k2/k1: (a) 0.1, (b) 0.5, (c) 1, (d) 2, (e) 5, (f) 10, (g) 20, (h) 50, and (i) 100.
Some limitations of the above theory can be noted. Inherent in the treatment is the assumption that the molecular areas occupied by the redox states, i.e., species A-D, are equal and that the total surface coverage, ΓT is constant during the redox transformation. Another limitation is that only totally irreversible steps are considered and the backward reactions are not included in the mechanisms. For electron-transfer steps, this limitation can usually be overcome experimentally by application of an electrode potential in a region where the overpotential is large. For coupled chemical steps, i.e., an ECE mechanism, this implies the assumption that the kinetic state is “frozen” at t ) 0. This condition can usually be met by application of a suitable potential to a cell that is at open circuit for t < 0. Another caveat concerns the observation of nonlinear ln(I) vs time plots for real data that contain a double-layer charging component. When the cell time constant is significant compared to the time constant for the Faradaic reaction, the chronoamperometric transient will exhibit
a maximum and nonlinear ln(I) vs time curves will be observed.12 Also, dispersion of E°′ values for an one-electron step will give rise to nonlinear plots.13
(11) Whittemore, N. A.; Mullenix, A. N.; Inamati, G. B.; Manoharan, M.; Cook, P. D.; Tuinman, A. A.; Baker, D. C.; Chambers, J. Q. Bioconjugate Chem. 1999, 10, 261-270. (12) Finklea, H. O. in: ref 7, p 287. (13) Tender, L.; Carter, M. T.; Murray, R. W. Anal. Chem. 1994, 66, 31733181. (14) Stewart, J. Calculus, 2nd ed.; Wadsworth: Belmont, CA, 1991.
3908 Analytical Chemistry, Vol. 71, No. 17, September 1, 1999
ACKNOWLEDGMENT This research was supported by the National Science Foundation (NSF Grant CHE-9616994) and The University of Tennessee, Knoxville. APPENDIX L’Hospital’s rule14 was used to determine the form of the corresponding equations for the cases when some of the rate constants were equal. If k1 ) k2 ) k, then eqs 11, 15, 16, and 22 become, respectively
θC ) 1 - e-kt - kte-kt
(A1)
I2 ) FAΓTk2te-kt
(A2)
tmax ) 1/k
(A3)
θB ) kte-kt
(A4)
If k1 ) k2 ) k * k3, then eqs 23-25 become
[e-kt(k3t - kt - 1) + e-k3t]
θC ) k2
(A5)
(k3 - k)2 [e-kt(k3t - kt - 1) + e-k3t]
θD ) 1 - e-kt - kte-kt - k2
(k3 - k)2
[e-kt(k3t - kt - 1) + e-k3t] I2 ) FAΓTk3k2 (k3 - k)2
θD ) 1 - e-k1t - k1
[e-kt(k1t - kt - 1) + e-k1t] kk1 (A12) (k1 - k)2
(A6)
(A7)
and
[e-kt(k1t - kt - 1) + e-k1t] I2 ) FAΓTk1k2 (k1 - k)2
If k1 ) k3 ) k * k2, then eqs 23-25 become
θC ) kk2
[e-kt(k2t - kt - 1) + e-k2t]
(A13)
(A8)
(k2 - k)2
If k1 ) k2 ) k3 ) k, then eqs 23-25 become
[e-kt - e-k2t] θD ) 1 - e-kt - k k2 - k [e kk2
[e-kt - e-k1t] k1 - k
-kt
(k2t - kt - 1) + e (k2 - k)
-k2t
2
]
(A9)
θC ) (k2t2/2)e-kt
(A14)
θD ) 1 - (1 + kt + (k2t2/2))e-kt
(A15)
I2 ) FAΓT(k3t2/2)e-kt
(A16)
and
and
[e-kt(k2t - kt - 1) + e-k2t]
2
I2 ) FAΓTk2k
(k2 - k)
2
(A10)
If k2 ) k3 ) k * k1, then eqs 23-25 become
θC ) kk1
[e-kt(k1t - kt - 1) + e-k1t] (k1 - k)2
Received for review February 3, 1999. Accepted May 24, 1999.
(A11) AC990114X
Analytical Chemistry, Vol. 71, No. 17, September 1, 1999
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