1480
Langmuir 1999, 15, 1480-1490
Voltammetry of Surface Electrodimerization Processes. Application to the Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury Juan Jose´ Calvente, Marı´a Luisa Gil, Rafael Andreu,* Emilio Rolda´n, and Manuel Dominguez Department of Physical Chemistry, University of Sevilla, 41012 Sevilla, Spain Received August 3, 1998 The theory for surface dimerization reactions under linear voltammetric conditions, and in the absence of mass transport complications, has been extended to remove the kinetic constraints of previous treatments. The solution of the initial value problem is shown to be identical when reactant and products remain in the adsorbed state and when the adsorbed reactant is allowed to equilibrate with its bulk concentration at any potential. Analytical expressions for the voltammetric waves are derived for some kinetic limits of interest, and empirical equations are developed to relate the peak coordinates and wave width with experimental variables when only numerical solutions are available. Theoretical predictions are applied to the analysis of the oxidation of adsorbed 2-mercaptoethyl ether on mercury, which is shown to consist of a monoelectronic exchange followed by a fast reversible dimerization.
Introduction Coupling between homogeneous dimerization reactions and electrode/solution charge transfer processes has been extensively studied in the electrochemical literature.1 Comparatively, much less attention has been paid to surface dimerization processes. Examples of electrochemically triggered surface dimerizations include the reduction of a series of organomercuric derivatives2 and that of p-diacetylbenzene3 on mercury. On the other hand, second (or higher)-order surface reactions are likely to be involved in the formation of long range ordered interfacial structures known as self-assembled monolayers4 (SAMs). Among several types of SAMs that have been described so far, alkanethiol monolayers deposited on gold, silver, and mercury have deserved preferential attention.5 The first stage of their formation, either under open circuit conditions6 or under potentiostatic control,7 implies the oxidative adsorption of the thiol group on the metallic substrate. Then, chemisorbed molecules interact with each other to build closed packed interfacial structures. Recent experimental evidence6,8-14 indicates the presence of (1) Galus, Z. Fundamentals of Electroanalytical Analysis; Ellis Horwood: Chichester, 1976. (2) Laviron, E.; Roullier, L. Electrochim. Acta 1972, 18, 237. (3) Meunier-Prest, R.; Gaspard, C.; Laviron, E. J. Electroanal. Chem. 1996, 410, 145. (4) Ulman, A. Introduction to Ultrathin Organic Films from Langmuir-Blodgett to Self-Assembly; Academic Press: San Diego, CA, 1991. (5) Finklea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19. (6) Schlenoff, J. B.; Li, M.; Ly, H. J. Am. Chem. Soc. 1995, 117, 12528. (7) Weisshaar, D. E.; Lamp, B. D.; Porter, M. D. J. Am. Chem. Soc. 1992, 114, 5860. (8) Nuzzo, R. G.; Zegarski, B. R.; Dubois, L. H. J. Am. Chem. Soc. 1987, 109, 733. (9) Fenter, P.; Eberhardt, A.; Eisenberger, P. Science 1994, 266, 1216. (10) Yeganeh, M. S.; Dougal, S. M.; Polizzoti, R. S.; Rabinowitz, P. Phys. Rev. Lett. 1995, 74, 1811. (11) Nishida, N.; Hara, M.; Sasabe, H.; Knoll, W. Jpn. J. Appl. Phys., Part 2 1996, 35, L799. (12) Nishida, N.; Hara, M.; Sasabe, H.; Knoll, W. Jpn. J. Appl. Phys., Part 1 1996, 35, 5866. (13) Nishida, N.; Hara, M.; Sasabe, H.; Knoll, W. Jpn. J. Appl. Phys., Part 1 1997, 36, 2379. (14) Voets, J.; Gerritsen, J. W.; Grimbergen, R. F. P.; van Kempen, H. Surf. Sci. 1998, 399, 316.
adsorbed dimers in alkanethiol monolayers deposited on Au(111), thus pointing to the presence of surface dimerization steps in the formation of these structures. A surface electrodimerization process can be described in terms of three elementary steps: reactant adsorption, electron transfer, and subsequent dimerization. Up to now, the only theoretical treatment for surface dimerization reactions under linear voltammetric conditions has been developed by Laviron.16 He solved the appropriate initial value problem by including the following assumptions: (a) all the species intervening in the electrochemical process (reactant, monomer, and dimer) remain attached to the electrode surface and, from the point of view of mass transfer, are effectively isolated from the solution, (b) adsorption is described according to a Langmuir isotherm, (c) electron transfer is reversible and fast enough to stay in equilibrium at any potential, and (d) dimerization is irreversible. He also developed diagnostic criteria to distinguish between surface and volume dimerization reaction pathways.17 The main aim of this work is to extend the theoretical description of surface dimerization reactions along two lines: (i) Two Limiting Cases in the Absence of Mass Transport Effects. The influence of reactant diffusion on the voltammetric response can be neglected either when the experimental time scale is short and the reactant concentration low enough that no significant mass transport takes place during the experiment (as discussed by Laviron18) or when the reactant concentration is high and the scan rate low enough so that the flux demand to form a monolayer of adsorbed products is too low to entail significant changes in the reactant concentration profile (as discussed by White et al.19). In both cases, a substantial mathematical simplification is achieved. In a recent work20 (15) Sawaguchi, T.; Mizutani, F.; Taniguchi, Y. Langmuir 1998, 14, 3565. (16) Laviron, E. Electrochim. Acta 1971, 16, 409. (17) Meunier-Prest, R.; Laviron, E. J. Electroanal. Chem. 1997, 437, 61. (18) Laviron, E. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1982; Vol. 12. (19) Hatchett, D. W.; Stevenson, K. J.; Lacy, W. B.; Harris, J. M.; White, H. S. J. Am. Chem. Soc. 1997, 119, 6596.
10.1021/la980961f CCC: $18.00 © 1999 American Chemical Society Published on Web 01/06/1999
Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury
Langmuir, Vol. 15, No. 4, 1999 1481
the transition between these two limiting cases has been described in the absence of kinetic complications. Here we will show that they share a common mathematical solution when appropriate correspondences between dimensionless variables are observed. (ii) Bidirectional Dimerization Kinetics. By analogy with homogeneous dimerizations, where fast reversible or quasireversible dimerizations have been observed,21,22 it seems desirable to explore the expected voltammetric response in the presence of a bidirectional surface dimerization reaction. Implicit analytical solutions will be obtained for the irreversible and fast reversible limits, and they will be shown to transform into simple analytical expressions under appropriate conditions. The more general quasireversible case will be solved numerically, and empirical relationships will be developed to describe the peak parameters and wave width behavior as a function of characteristic parameters. Theoretical predictions will be applied to the analysis of 2-mercaptoethyl ether (MEE) oxidative adsorption on mercury. Oxidation of thiols on mercury leads to the formation of organomercuric salts.23-26 There is some controversy regarding the oxidation state of mercury in these salts.27-30 Analysis of the bulk electrolysis products23,24,31 of hydrophilic thiols indicates the formation of a soluble Hg(RS)2 species, though Hg(I) may still play a role as a reaction intermediate. These results are in agreement with the expected displacement of the Hg(I) disproportionation equilibrium toward the formation of Hg(II) species in the presence of thiols32 and indicate the likely occurrence of dimerization steps in the overall oxidation mechanism. As compared to compounds in previous studies, MEE offers the possiblity of observing a reversible thiol oxidation under submonolayer conditions, which are more amenable to comparison with theoretical results.
Differential equations were integrated employing the DDEBDF subroutine from the SLATEC program library. The code was written in FORTRAN and implemented on a 166 MHz Pentium PC computer. The program was checked by performing calculations under conditions where analytical solutions were available.
Experimental Section Solutions of 0.5 M AcOH/NaAcO at pH ) 4 were prepared from Merck (p.a.) reagents and water purified with a Millipore Milli-Q system. 2-Mercaptoethyl ether ((HSCH2CH2)2O) was purchased from Aldrich and used without further purification. A sodium-saturated calomel electrode and a platinum foil (1 cm2 of area) were used as reference and auxiliary electrodes, respectively. The working electrode was a static drop mercury electrode (EGG PAR 303A). The drop area (0.0106 cm2) was determined by weighing 3 sets of 10 drops. Voltammetric experiments were performed with a homemade generator/potentiostat. Voltammograms were recorded and stored with a Nicolet 410 Oscilloscope. Solutions were deareated with a presaturated nitrogen stream prior to admitting mercury into the cell. All measurements were carried out at 25 ( 0.2 °C. (20) Calvente, J. J.; Gil, M. A.; Andreu, R.; Rolda´n, E.; Dominguez, M. Langmuir, in press. (21) Evans, D. H.; Jimenez, P. J.; Kelly, M. J. J. Electroanal. Chem. 1984, 163, 145. (22) Rueda, M.; Compton, R. G.; Alden, J. A.; Prieto, F. J. Electroanal. Chem. 1998, 443, 227. (23) Stankovich, M.; Bard, A. J. J. Electroanal. Chem. 1977, 75, 487. (24) Casassas, E.; Arin˜o, C.; Esteban, M.; Mu¨ller, C. Anal. Chim. Acta 1988, 206, 65. (25) Muskal, N.; Turyan, Y.; Mandler, D. J. Electroanal. Chem. 1996, 409, 131. (26) Heyrovsky, M.; Mader, P.; Vavricka, S.; Vesela´, V.; Fedurco, M. J. Electroanal. Chem. 1997, 430, 103. (27) Kolthoff, I. M.; Barnum, C. J. Am. Chem. Soc. 1940, 62, 3061. (28) Miller, I. R.; Teva, J. J. Electroanal. Chem. 1972, 36, 157. (29) Chen, S.; Abrun˜a, H. D. J. Phys. Chem. B 1997, 101, 167. (30) Stevenson, K. J.; Mitchell, M.; White, H. S. J. Phys. Chem. B 1998, 102, 1235. (31) Calvente, J. J.; Gil, M. A.; Andreu, R. To be submitted. (32) Brodersen, K.; Hummel, H. In Comprehensive Coordination Chemistry; Wilkinson, G., Ed.; Pergamon: Oxford, 1987; Vol. 5.
Theory (a) Initial Value Problem in the Absence of a Reactant Concentration Gradient (High Reactant Concentration and/or Low Scan Rate). Let us consider the following mechanism scheme: KaR
MS + Rsol {\} MR + Ssol Es
MR {\} MO + ne-
(I)
kd
MO + MO {\ } M2O2 (Z) k -d
where a reduced species R displaces an adsorbed solvent molecule (or a cluster of solvent molecules) S from the electrode surface M and undergoes a surface redox process. The resulting oxidized species O does not desorb, but it may be involved in a subsequent surface dimerization reaction. Thus, the final product Z, is a dimer, which may include two substrate atoms. The voltammetric behavior to be expected from Scheme I will be described under the following restrictions: (i) Though species R and S are freely exchanged between the adsorbed state and the solution bulk, we will neglect any change in their concentration during the voltammetric experiment; that is, we will exclude the presence of any concentration gradient. This will be the case whenever the oxidized products are formed from a preadsorbed monolayer of R or when the scan rate is low enough and the reactant (and solvent) concentration is high enough so that flux requirements are fulfilled without significant perturbation of the concentration profiles. (ii) Adsorption and electron transfer steps will be assumed to remain at equilibrium at any potential, and adsorption of R will be described by a Langmuir isotherm with a potential-independent adsorption coefficient KaR. Adsorption and electron-transfer equilibria will be expressed as
KaR )
ΓRcS ΓScR
(1)
ΓO nF (E - Es) ) ξ ) exp ΓR RT
(
)
(2)
where Γi and ci stand for the surface and bulk concentrations of the i species, respectively, and Es is the formal potential of the surface redox process. On the other hand, the dimerization rate is given by
dΓZ ) kdΓO2 - k-dΓZ dt
(3)
with the following initial conditions:
ΓR(0) ) ΓR* )
KaRcRΓS ; ΓO(0) ) ΓZ(0) ) 0 cS
(4)
Equations 1-4 are to be solved to obtain the surface concentrations and their time derivatives, which are closely related to the faradaic current if in a voltammetric experiment,
1482 Langmuir, Vol. 15, No. 4, 1999
if ) nFAv
Calvente et al.
(
)
dΓZ dΓO +2 dt dt
where A is the electrode surface area and v is the scan rate. It is convenient to reformulate the initial value problem in terms of dimensionless variables. Therefore, we define
cR Γi ;θ ) cS i Γi,m
(6a)
2kdΓR,m k-d ;k h -d ) a a
(6b)
T ) at; cjR ) k hd )
θR θO ) (ξθR*) θR*
(5) and
θZ ) 1 -
(7)
Es* ) Es -
θR
(8)
(1 - θR - θO - θZ)cjR
(9)
dθZ )k h dθ2O - k h -dθZ dT
(10)
KaRcjR ; θ (0) ) θZ(0) ) 0 (11) KaRcjR + 1 O
and the dimensionless current I is also defined as
I)
d(θO + θZ) if ) (ξ nFAΓR,ma dξ
(12)
{
λ
(13a)
λ
i
s
λ
λ
(13b)
where Ei is the initial potential and Tλ is the dimensionless time corresponding to the beginning of the reverse scan. Then, the plus sign in eq 12 applies to the forward scan and the minus sign to the reverse scan. More simplified expressions are obtained by noting that eqs 8 and 11 relate θO + θZ to the θR/θR* ratio in a straightforward way:
θ O + θZ ) 1 -
θR θR*
Then, from eqs 14 and 9 it follows that
(18)
d(θR/θR*) dξ*
I ) -ξ*
(19)
Generally speaking, no analytical solution for I is available, and the differential equation (10) has to be solved numerically. However, as we will see below, some limiting cases of eq 10 are amenable to more or less explicit analytical solutions. But first, we will see how an analogous set of equations describes a closely related physical situation. (b) Initial Value Problem for Surface-Isolated Reactant and Products (Low Reactant Concentration and/or High Scan Rate). Let us consider the following simplified version of Scheme I:
kd
MO + MO {\ } M2O2 (Z) k -d
nF 0eTeT (E - E )), (RT nF exp(-T + 2T ) exp( (E - E )), T e T e 2T RT s
ξ* ) ξθR*
Es
ξ) i
(17)
MR {\} MO + ne-
where it has been assumed that ΓS,m ) ΓO,m ) ΓR,m ) 2ΓZ,m, in agreement with Scheme I and the dimeric character of Z. In addition, the time derivative has been replaced by a derivative with respect to ξ, since for a cyclic voltammetric experiment, in which oxidation takes place in the forward scan, ξ is given by
exp(T) exp
RT ln θR* nF
so that the dimensionless current may be expressed as
θO )ξ θR
θR* ) θR(0) )
(16)
and to define
nFv RT
Now eqs 1-4 can be rewritten as follows:
KaR )
θR (1 - ξθR*) θR*
Therefore, all the influence of cR and KaR on the voltammetric wave can be expressed in terms of the initial coverage of R. Moreover, from the above relationships it seems more convenient to refer the potential scale to a concentration-dependent formal potential defined as
where Γi,m is the maximum surface concentration of species i and
a)
(15)
(14)
(II)
where only the adsorbed R species undergoes an oxidation on the electrode to produce either adsorbed O or its dimer Z. The voltammetric response corresponding to Scheme II is obtained in the absence of diffusional complications by assuming that the experimental time scale is short enough, and/or the reactant bulk concentration is small enough, so that diffusional mass transport is negligible along the voltammetric wave and the adsorbed reactant and products are effectively isolated from the solution. By recalling our previous definitions, we can write for Scheme II:
θO θZ θR + + )1 θR* θR* θR*
(20)
θO/θR* )ξ θR/θR*
(21)
( )
θO d(θZ/θR*) ) (k h dθR*) dT θR*
2
θZ -k h -d θR*
θO(0) θZ(0) θR(0) ) 1; ) )0 θR* θR* θR* and the dimensionless current is given by
(22) (23)
Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury
I θR*
) (ξ
d((θO + θZ)/θR*) dξ
(24)
so that the I/θR* ratio scales the faradaic current with ΓR(0) rather than with ΓR,m, as in the definition of I (see eq 12). Equations 20-24 represent the mathematical formulation of the initial value problem corresponding to Scheme II, and they can be shown to be identical, except for the definition of the dimensionless variables involved, to eqs 8-12, which define the initial value problem for Scheme I. To facilitate this comparison, eqs 8 and 9 can be rewritten in the following way:
θR + θO + θZ ) 1 (≡eq 8) θR*
(25)
θO ) ξ* (≡eq 9) θR/θR*
(26)
When eqs 20-24 are compared with eqs 25, 26, and 10-12, it turns out that both sets of equations represent the same mathematical problem. In the next sections we will present some analytical and numerical solutions corresponding to Scheme I; they can be easily converted into solutions of Scheme II by observing the following correspondences: (i) The variables θO and θZ in Scheme I should be replaced by θO/θR* and θΖ/θR*, respectively, in Scheme II. (ii) The dimensionless dimerization rate constant k h d and the equilibrium constant K h d in Scheme I should be h dθR*, respectively, in Scheme substituted by k h dθR* and K II. (iii) The potential dependence is expressed in terms of ξ in Scheme II instead of ξ* in Scheme I. Note that ξ is independent of the reactant concentration. (iv) The dimensionless current I in Scheme I should be replaced by I/θR* in Scheme II, so that it is scaled now with ΓR(0) instead of ΓR,m. Two limits of surface dimerization kinetics, which lead to analytical expressions for the voltammetric wave, will be examined next. From now on, only Scheme I will be considered explicitely, but expressions applying to Scheme II are easily derived by applying the above correspondences, and their use will be exemplified in the analysis of the experimental results. (c) Irreversible Surface Dimerization Reaction. In this case k h -d ) 0 and the dimerization rate is
dθZ )k h dθO2 dT
( )
d(θR/θR*) θR/θR* θR ξ* + +k hd dξ* ξ* + 1 ξ* + 1 θR*
2
)0
(28)
the solution being
1 (29) -ξ* + ln(1 + ξ*) (1 + ξ*) 1+k hd ξ* + 1
[
(
where the initial potential has been assumed to be negative enough so that (ξ*)T)0 , 1. From eqs 29 and 19 the anodic dimensionless current is obtained:
Iir a )
[1 + k h d ln(1 + ξ*)] ξ* 2 -ξ* (1 + ξ*) 1 + k + ln(1 + ξ*) hd ξ* + 1
)]
[
(
)]
2
(30)
A similar procedure leads to the following expression for the dimensionless cathodic current corresponding to the reverse scan:
Iir c )
-ξ* (1 + ξ*)2
[
1-
[
k hd ln(ξ* + 1) Bλ
]
k h d -ξ* + ln(ξ* + 1) Bλ 1 Bλ ξ* + 1
(
]
)
2
(31)
where Bλ is related to the choice of the switching potential (ξλ*) and is given by
[
hd Bλ ) 1 + 2k
]
-ξλ* + ln(ξλ* + 1) ξλ* + 1
(32)
(c.1) Forward Scan for Irreversible Dimerization with a High Value of the Dimerization Rate Constant (k hd g 3000). For high k h d values the voltammetric wave will show at potentials negative of Es*, so that ξ* , 1 will be satisfied, and eq 30 is transformed into
≈ Iir,∞ a
k h d(ξ*)2 [2k h d(ξ*)2 + 1]2
(33)
the peak coordinates being
(ξa,p*)ir,∞ )
x
(34)
1 2
(35)
2 h hd
and
Iir,∞ a,p )
The voltammetric wave is symmetrical with respect to the peak potential, and the half peak current is reached at ) Ea,p + Ea,p/2
RT ln(3 - 2x2)1/2 ) nF Ea,p - 22.6/n mV (at 25 °C) (36)
+ ) Ea,p + Ea,p/2
RT ln(3 + 2x2)1/2 ) nF Ea,p + 22.6/n mV (at 25 °C) (37)
(27)
The corresponding initial value problem for Scheme II has been solved by Laviron.16 Adapting his approach to Scheme I, eqs 9, 14, and 27 were rearranged to give a Bernouilli’s ordinary differential equation of the form
θR ) θR*
Langmuir, Vol. 15, No. 4, 1999 1483
so that the half-height semiwidths wa ) Ea,p - Ea,p/2 and + + wa ) Ea,p/2 - Ea,p are identical to those expected for a simple 2n-electron surface redox process.18 Therefore, the anodic wave has the characteristics typical of a 2n-electron surface process and shifts toward more negative potentials as k h d increases. (c.2) Reverse Scan for Irreversible Dimerization with a Low Value of the Dimerization Rate Constant (k h d e 1). When the switching potential is kept positive enough, so that Eλ - Es* g (100/n) mV, and provided that the
1484 Langmuir, Vol. 15, No. 4, 1999
Calvente et al.
dimerization rate constant is small, eq 31 can be approximated by
-ξ* (ξ* + 1)2(1 + k h dQλ)
Iir,0 ≈ c
(38)
is high enough, the voltammetric wave will develop in a potential region negative of Es*, so that the condition ξ* , 1 will be fulfilled at any potential of interest; under this restriction, eqs 46 and 47 can be simplified to give
Irev,∞ ) -Irev,∞ ≈ a c
where Qλ is determined by the choice of the switching potential:
Qλ ) 2
[
-ξλ* + ln(1 + ξλ*) 1 + ξλ*
]
(39)
(ξc,p*)
)1
Iir,0 c,p
-1 (1 + k h dQλ)-1 ) 4
ir,0
) (Ec,p/2
ir,0
) Eir,0 c,p +
RT ln(3 + 2x2) ) nF Eir,0 c,p + 45.3/n mV (at 25 °C) (42)
) Eir,0 c,p +
RT ln(3 - 2x2) ) nF Eir,0 c,p - 45.3/n mV (at 25 °C) (43)
(d) Fast Dimerization in Equilibrium. In this case: h -d . 1, so that the dimerization reaction remains k h d and k under equilibrium conditions, and eq 10 transforms into
K hd )
k hd k h -d
)
θZ
(44)
θO2
where K h d is the dimensionless equilibrium constant for the dimerization reaction. By combining eqs 9, 14, and 44 the following quadratic equation results:
( )
θR K h d ξ* θR*
2
θR -1)0 + (ξ* + 1) θR*
(45)
the solution being
-(ξ* + 1) + x(ξ* + 1)2 + 4K h d(ξ*)2 θR ) θR* 2K h d(ξ*)2
(46)
By differentiating eq 45 with respect to ξ* and recalling eq 19, the dimensionless current is obtained in an implicit form
( ( (
) ))
θR θR ξ* 1 + 2K h dξ* θ * θ R R* rev Irev a ) -Ic ) θR 1 + ξ* 1 + 2K h dξ* θR*
(47)
(d.1) Reversible Dimerization with a High Value of the hd Dimerization Equilibrium Constant (K h d g 500). When K
(48)
( ) 1 + x2 2K hd
1/2
(49)
and rev,∞ ) 6 - 4x2 ≈ 0.343 Irev,∞ a,p ) -Ic,p
(41)
and eventually disappears. In the meantime, the wave remains symmetrical with respect to Es*, and it displays the characteristic shape of a simple surface redox process,18 with + ) (Ec,p/2
(ξp*)rev,∞ )
(40)
the peak current decreases on increasing k h d according to
2K h d(ξ*)2 x4K h d(ξ*)2 + 1
which leads to the following values for the coordinates of the voltammetric peak:
Whereas the voltammetric peak potential remains at ir,0
[x4K h d(ξ*)2 + 1 - 1]2
(50)
It may also be interesting to note that the half-height potentials will occur at
(Ep/2)
rev,∞
) Ep +
RT ln 0.331 788 ) nF Ep - 28.3/n mV (at 25 °C) (51)
(E+ p/2)
rev,∞
) Ep +
RT ln 4.262 40 ) nF Ep + 37.2/n mV (at 25 °C) (52)
Therefore, the voltammetric wave is not symmetrical with respect to the peak potential and displays a half-height width wa ) wc ) 65.5/n mV. Results and Discussion (a) General Behavior. Analytical solutions for Schemes I and II were derived in the previous section under either irreversible or fast reversible surface dimerization conditions. In some limiting cases, the voltammetric wave was shown to obey simple analytical expressions. It is also interesting to explore how the characteristics of the voltammetric wave evolve from those corresponding to a simple n-electron surface redox process + (ξp,a* ) ξp,c* ) 1, Ip,a ) Ip,c ) 0.25, and w+ c ) wc ) wa ) wa ) 90.5/n mV at 25 °C) as the dimerization reaction starts to operate under irreversible (Figure 1a), fast reversible (Figure 1c), or quasi-reversible (Figure 1b) kinetic conditions. From a qualitative point of view, anodic waves are expected to become higher and narrower and to shift toward more negative potentials as k h d increases. However, the specific features of this transition depend on the reversibility of the dimerization reaction, that is, on the k h -d value. When k-d ≈ 0 (irreversible dimerization, Figure 1a), anodic waves remain symmetrical with respect to their peak potential. Upon increasing k h d, their shape changes gradually from that typical of an n-electrons surface wave to that of a 2n-electron surface wave (Ip,a ) 0.5 and wa ) 90.5/2n mV) and they shift toward more negative potentials. On the other hand, cathodic waves do not shift to a significant extent and remain centered around Ep,c ) Es*, but their height decreases abruptly as the dimerization process starts to operate. h -d . 1 (fast reversible dimerization, When k h d and k Figure 1c), cyclic voltammograms remain symmetrical with respect to the potential axis, but they may lose their symmetry with respect to the axis defined by the two peak
Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury
Langmuir, Vol. 15, No. 4, 1999 1485
(b) Peak Coordinates and Wave Width. Voltammetric data are usually collected as a function of reactant concentration and scan rate, but from a theoretical point of view, it is more convenient to consider their influence in terms of θR*, k h d, and k h -d. All the scan rate dependence is included in the dimensionless dimerization rate conh -d; whereas the role of cR (in the absence stants k h d and k of diffusional effects) is more indirect, it determines the initial θR* value in Schemes I and II and shifts the adsorption equilibrium of the reactant in Scheme I. It should be noted also that, when voltammograms for h dθR*, Scheme II are computed, k h d has to be replaced by k which is now a function of both scan rate and reactant concentration. (b.1) Dependence on θR*. Formulation of the initial value problem for Scheme I in terms of dimensionless variables shows that the normalized anodic wave Ia ) f(ξ*) is only a function of the k h d and k h -d values. Therefore, neither the peak current nor the wave shape will depend on θR* or the reactant concentration. On the other hand, peak potentials will obey an expression of the type
ξa,p* ) ξa,pθR* ) f(k h d,k h -d)
(53)
which can be rearranged to give
Ea,p ) Es -
Figure 1. Normalized voltammograms corresponding to Scheme I: (a) irreversible dimerization, k h d ) 4 × 10-16 (1), 0.2 (2), 40 (3), and 4000 (4); (b) quasi-reversible dimerization, k hd ) 4 × 103 and k h -d ) 2.5 × 103 (1), 25 (2), 1.3 (3), and 0.13 (4); (c) fast and reversible dimerization, k h d ≡ 1.6 × 10-19 (1), 4 (2), 160 (3), and 16 000 (4). θR* ) 1.
potentials. Both waves shift toward more negative poh -d ratio, and eventually, tentials upon increasing the k h d/ k the dimensionless peak currents reach a limiting value equal to 0.343. Then, the negative and positive semiwidths + differ and become equal to wc ) wa ) 28.3/n mV and wc + ) wa ) 37.2/n mV, respectively. The transition between the reversible and irreversible cases may be observed by lowering the k h -d value, at a given k h d (Figure 1b). As long as k h -d . 1, voltammetric waves reproduce the reversible shift of Figure 1c (curves 1 and 2 in Figure 1b). But as soon as k h -d e 1, anodic waves approach the symmetric shape characteristic of the irreversible dimerization case, while cathodic waves become broader and tend to adopt the sigmoidal shape which is typical of a kinetically controlled CE process. No attempt will be made to describe quantitatively all of the intermediate voltammetric shapes. However, useful diagnostic criteria can still be obtained from the dependence of peak parameters on scan rate and reactant bulk concentration. In the absence of analytical solutions, empirical expressions will be developed to reproduce accurately the behavior of peak currents, peak potentials, and wave widths, as predicted from the numerical solution of the initial value problem. This is the subject of the following section.
KaRcjR RT RT ln + ln f(k h d,k h -d) (54) nF KaRcjR + 1 nF
Then Ea,p is expected to be concentration-independent in the presence of strong reactant adsorption (KaRcjR . 1) and to vary linearly with the logarithm of the reactant concentration when the reactant is weakly adsorbed (KaRcjR , 1). Similar arguments can be applied to anodic waves corresponding to Scheme II. By recalling the correspondence between the dimensionless variables that describe Schemes I and II, it can be shown that
h dθR*,k h -d) ξa,p ) f(k
(55)
so that the Ea,p dependence on θR*, or the reactant concentration, has to be traced back to that of k h dθR* (see below). Analogously, wave widths will also be affected by changes in k h dθR*. A more simple dependence is expected for anodic peak currents, which are now directly proportional to θR*. Analysis of the cathodic peak parameters and the wave width is further complicated by the choice of the switching potential and will be considered later in connection with h -d on our numerical results. the influence of k h d and k (b.2) Dependence on k h d and k h -d. As indicated previously, the normalized anodic waves for Scheme I are uniquely h -d, or by k h dθR* and k h -d for Scheme II. defined by k h d and k h d (or Figure 2a illustrates the variation of Ea,p with ln k k h dθR*) for several values of k-d. Except for the irreversible limit (k-d ) 0, curve 1), a sigmoidal transition between two plateaus is obtained. The high-k h d plateau corresponds to the reversible limit (k h -d, and k h -d . 1), and its value is h d plateau corresponds to a short a function of K h d. The low-k experimental time scale, so that the redox process is not perturbed by the dimerization reaction. It was found that this type of functionality could be reproduced satisfactorily by the following expression:
Ea,p ) Es* -
(
)
k hd + k h -d + 2 RT ln 2nF k h -d + 2
(56)
1486 Langmuir, Vol. 15, No. 4, 1999
Calvente et al. rev,∞ where Iir,∞ a,p ) 0.5 (see eq 35), Ia,p ) 0.343 (see eq 50), and E Ia,p ) 0.25 is the dimensionless peak current corresponding to a simple surface redox process. The dotted lines in Figure 1b were computed from eq 58. When k h -d ) 0, Ia,p ≡ Iir a,p, and eq 58 simplifies to
(
)
ir Iir,∞ a,p - Ia,p
ln
Iir a,p
-
IEa,p
x2 + kh d
) 0.75 + ln
(59)
k hd
h -d . 1, Ia,p ≡ Irev When k h d and k a,p , and eq 58 simplifies to
ln
(
)
rev Irev,∞ a,p - Ia,p
Irev a,p
-
IEa,p
x1 + Kh d
) 0.3 + ln
(60)
K hd
Equation 59 describes a kinetic transition from k hd ) 0 is a function of scan rate, whereas eq to k h d f ∞, and Iir a,p 60 describes a thermodynamic transition from K h d ) 0 to K h d f ∞, and Irev a,p does not depend on scan rate. Analysis of the half-height peak width w as a function of scan rate is commonly used as a suitable diagnostic criterion to distinguish among different mechanisms. In agreement with the fact that the charge under the h -d values, voltammetric wave is independent of the k h d and k the behavior of wa was found to parallel that of Ia,p, so that an increase in the peak current was accompanied by a decrease in the wave width (see Figure 2c). The following expression was found to reproduce accurately the behavior h d and k h -d values: of wa for a wide range of k
wa ) 0.33 wir,∞ k hd a e rev,∞ 0.48 (wir,∞ )e k h -dβaw + 2 a /wa
Figure 2. Peak potential (a), current (b), and width (c) of the anodic wave. Symbols are values obtained from the numerical solution of the initial value problem. Dotted lines were derived from eq 56 (a), eq 58 (b), and eq 61 (c), respectively. These plots also represent the solution of Scheme II when Es* is replaced by Es, Ia,p by Ia,p/θR*, and k h d by k h dθR*.
Differences between numerically calculated Ea,p values and those estimated from eq 56 are within (2 mV, that is, well within the range of experimental error for a wide h -d values (see Figure 2a). range of k h d and k The dependence of Ia,p (or Ia,p/θR*) on the logarithm of h dθR*) is depicted in Figure 2b. The full lines, k h d (or k corresponding to the irreversible limiting case (top) and to the absence of a follow-up chemical reaction (bottom), embrace all the intermediate kinetic possibilities. The most striking feature of these plots is the development of a maximum, whose abscissa (k h d)max satisfies
(k h d)max ) e
0.65
(1 + K h d)
0.65
Ia,p )
+
IEa,p
(
)
k hd + k h -d + 2
rev,∞ 0.25 (Iir,∞ k h -d + 2 2k h -d + a,p /Ia,p )e -0.4 k hd + k h -d + 2 1/2 e k hd + e0.25k h -d + 2 2k h -d + 2
(
e0.48k h -dβa,w + 2
)
2
1/2
(58)
+
(
(
)
k hd + k h -d + 2 2k h -d + 2
)
k hd + k h -d + 2 2k h -d + 2
1/2
1/2
(61)
) 45.2/n mV (see eqs 36 and 37), wrev,∞ ) where wir,∞ a a 65.5/n mV (see eqs 51 and 52), wEa ) 90.6/n mV (see eqs 42 and 43), and
βa,w )
k h -d
(62)
1 k h -d + 1+k h -d
The dotted lines in Figure 1c were estimated from eq 61. As in the case of the peak current, simplified expressions can be derived from eq 61. When k h -d ) 0 and wa ≡ wir a , the wave width is given by
ln
(57)
The following empirical expression gives an accurate h d and description (within (1%) of the Ia,p dependence on k k h -d -0.4 k hd Iir,∞ a,p e
e0.33k hd
+ wEa
(
)
wir,∞ - wir a a wir a
-
wEa
) ln
k hd
x2 + kh d
(63)
h -d . 1 and wa ≡ wrev and when k h d and k a , the wave width is given by
ln
(
)
wrev,∞ - wrev a a wrev a
-
wEa
) 0.2 + ln
K hd
x1 + Kh d
(64)
As long as the switching potential is kept positive enough, that is, Eλ - Es* g 200/n mV, the cathodic peak potentials are independent of the choice of the switching potential and can be approximated by
Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury
Langmuir, Vol. 15, No. 4, 1999 1487
Figure 3. Peak potentials of the cathodic wave. Symbols are values obtained from the numerical solution of the initial value problem. Dotted lines were derived from eq 65. These plots also represent the solution of Scheme II when Es* is replaced by Es and k h d by k h dθR*.
(
Ec,p ) Es* -
(
RT × 2nF
)
k h d + 1.9 × 106
)
1.6
(k h -d(k h d)1.3 + 0.026) 1.9 × 106 ln k h d + 1.9 × 106 1.6 (1 + K h d)-1 k h -d(k h d)1.3 + 0.026 1.9 × 106 (65)
(
)
As in the case of the anodic peak potentials, two plateaus (corresponding to high and low scan rates) can also be observed for the variation of the cathodic peak potential with k h d in Figure 3. The transition between the two h -d decreases. plateaus takes place at higher k h d values as k The dotted lines in Figure 3 were derived from eq 65; they can be seen to reproduce the numerical estimates of Ec,p within (4 mV. Cathodic peak currents retain some dependence on the choice of the switching potential even when Eλ - Es* g 200/n mV. Figure 4a shows how under irreversible h -d increases, dimerization conditions -0.25 e Ic,p e 0. As k ) Ic,p develops a maximum and approaches the Irev,∞ c,p -0.343 value in the high-k h d limit. Eventually, for high enough k h -d values the dimerization equilibrium is shifted toward the monomers to such an extent that dimerization does not affect Ic,p any more (curve 9 in Figure 4a). The following empirical expression reproduces within (0.005 units the values of Ic,p: Ic,p ) Iir,∞ a,p h dβc,I Iir,∞ a,p k rev,∞ 2.25 (Iir,∞ k h -d/(Iir,∞ a,p e a,p + Ia,p )) + 2
k h dβc,I k h -de2.25 + 2
+
(
( )
E + (Iir,∞ a,p + Ia,p)
k h -d + k hd + 2 2k h -d + 2
)
k h -d + k hd + 2 2k h -d + 2
0.5
0.5
(66) where
Figure 4. (a) Peak currents of the cathodic wave for Eλ - Es* ) 200 mV. k-d ) 0 (1), 0.01 (2), 0.1 (3), 1 (4), 10 (5), 50 (6), 500 (7), 2500 (8), and 106 (9) s-1. (b) Influence of the switching potential on the cathodic peak currents for Eλ - Es* ) 200 (‚‚‚O‚‚‚), 300 (s), and 400 (‚‚‚4‚‚‚) mV and k-d ) 0.01 (1), 0.1 (2) and 50. (3) s-1. These plots also represent the solution of Scheme II when Ic,p is replaced by Ic,p/θR* and k h d by k h dθR*. Table 1. Values of ωI,1 and ωI,2 in Eq 67 for Different Switching Potentials Eλ - Es*/V
ωI,1
ωI,2
0.2 0.3 0.4
3.35 3.79 4.09
0.70 0.49 0.42
βc,I )
h -d + ωI,2) eωI,1(k eωI,1-1.6k h -d + ωI,2
(67)
ωI,1 and ωI,2 depend on the switching potential and have been computed for some Eλ - Es* values (see Table 1). Figure 4b illustrates this influence of Eλ - Es* on Ic,p. It can be observed that Ic,p becomes independent of the switching potential when k h d > 1 and that the influence of h -d. When Ic,p is dependent on Eλ increases on decreasing k Eλ, larger scan amplitudes result in a lowering of the absolute value of the peak current. Finally, cathodic wave widths display a qualitative behavior similar to that of Ic,p, as indicated in Figure 5, though they do not vary with the switching potential for Eλ - Es* g 200/n mV. It should be noted that wc f ∞, as Ic,p f 0. Therefore, we have restricted our analysis to measurable peaks with wc e 120 mV. The following expression reproduces the numerical wc values within (4 mV:
1488 Langmuir, Vol. 15, No. 4, 1999
Calvente et al.
Figure 5. Width of the cathodic wave. k-d ) 0.1 (1), 1 (2), 10 (3), 50 (4), 500 (5), and 2500 (6) s-1. These plots also represent the solution of Scheme II when k h d is replaced by k h dθR*.
wc ) w
(
E
k h d + xK h d + 0.35
(
xKh d + 0.35
)( ) 2.5
h d + xK h d + 0.35 wE k βc,w xKh d + 0.35
e1.1k h -d +
2.5
e1.1k h -d +
)
(k h d + (k h d + 1)-1) k hd + 1
(k h d + (k h d + 1)-1) k hd + 1 (68)
where
wExK hd + 1 + βc,w )
xKh d + 1 +
wrev,∞ e0.2K h d(k h -d + 1) a k h -d + (k h -d + 1)-1 e0.2K h d(k h -d + 1)
(69)
k h -d + (k h -d + 1)-1
(c) Comparison with Experiment. In order to verify experimentally some of the previous theoretical predictions, a voltammetric study of the 2-mercaptoethyl ether (MEE) oxidative adsorption on mercury was performed. As compared to other short-chain thiols, MEE adsorbs rather strongly on mercury in acidic media. Experiments were restricted to low bulk reactant concentrations (757.5 µM) to minimize mass-transfer effects, so that experimental conditions approached the requirements of Scheme II. Voltammograms were recorded in the 0.1-500 Vs-1 scan rate range, where three types of voltammetric behavior can be observed (see Figure 6). At low scan rates (v < 10 V s-1), anodic waves have a typical diffusion-controlled shape, whereas cathodic waves are narrower and more symmetrical, reflecting the electrochemical stripping of an adsorbed product. Under these conditions, |Ic,p| > Ia,p and Ea,p > Ec,p, as expected for a diffusion-controlled surface voltammetric wave.20 Peak splitting results from the enhancement of the reactant local concentration in the vicinity of the electrode during the desorption process. The characteristics of a reversible surface process, Ia,p and Ic,p ∝ v, Ia,p/|Ic,p| ≈ 1, and Ea,p ≈ Ec,p * f(v), are approached as the scan rate increases. However, at high enough scan rates (v > 100 V s-1), peak potentials start to depart from each other, indicating the onset of kinetic control of the electrochemical process. Therefore, the 10-100 V s-1 scan
Figure 6. (a) Evolution of experimental voltammetric shape with scan rate. (b) Dependence of peak current on scan rate: (b) anodic scan; (2) cathodic scan. (c) Dependence of peak potential on scan rate: (b) anodic scan; (2) cathodic scan. The MEE concentration is 10 µM.
rate interval seems to be the most appropriate for comparison with the theoretical expressions derived before. The presence of a 10-20 mV shift between the anodic and cathodic peak potentials in this interval, however, should be noted. This shift is scan-rateindependent but tends to increase with reactant concentration (or the amount of adsorbed reactant). Similar splittings have been observed also for other thiol- and sulfur-containing compounds,19,29,33-35 and it seems to be a common feature of surface redox processes involving strong interaction with the substrate. The MEE bulk concentration determines the initial thiol coverage (θR*), as long as enough time is allowed to establish equilibrium conditions at the beginning of the experiment. The influence of reactant concentration, or θR*, is illustrated in Figure 7, which shows a typical set of voltammograms recorded at 25 V s-1, after approximately 1 min of equilibration at the initial potential in a stirred cell solution. In the presence of low reactant coverages, the observed current merges with the capacitative current of the supporting electrolyte on both sides of the voltammetric peak. When the reactant coverage increases, a small (but significant) faradaic current is observed at potentials positive of the anodic peak. Numerical simulations, including mass transport from the solution, also predict some current tailing due to reactant diffusion.36 But a detailed comparison shows that diffusion cannot account quantitatively for the observed effect. On the other hand, bulk electrolysis experiments,31 performed at more positive potentials and in the presence of higher reactant concentrations, indicate that two electrons are exchanged (33) Hulbert, H. M.; Shain, I. Anal. Chem. 1970, 42, 162. (34) Hatchett, D. W.; Gao, X.; Catron, S. W.; White, H. S. J. Phys. Chem. 1996, 100, 331. (35) Kontrec, J.; Svetlicic. Electrochim. Acta 1998, 43, 589. (36) Numerical simulations including reactant diffusion from the solution were performed by the orthogonal collocation technique as described in ref 20.
Oxidation of Adsorbed 2-Mercaptoethyl Ether on Mercury
Langmuir, Vol. 15, No. 4, 1999 1489
Figure 7. Cyclic voltammograms recorded at 25 V s-1 for the following MEE concentrations: (O) 0, (4) 7.5, (0) 10, (3) 25, (]) 50, and (b) 75 µM.
Figure 9. Anodic peak potentials (a) and wave widths (b) as a function of MEE surface concentration; (O) ΓR(0) estimated directly from the peak current; (b) ΓR(0) estimated from eqs 73 and 74; (‚‚‚) theoretical prediction for Kd ) 0; (- - -) theoretical prediction for Kd f ∞; (s) optimized fit for Kd ) 1011 mol-1 cm2.
Irev,∞ i ) rev,∞ ) ip I p
1
x
[ 2(1 + x2)(ξ/ξp)2 + 1 - 1]2
x
0.343(1 + x2) (ξ/ξ )2 2(1 + x2)(ξ/ξ )2 + 1 p p Figure 8. Comparison between theoretical and experimental normalized voltammograms. (- - -) n ) 1 simple redox process; (‚‚‚) n ) 2 simple redox process; (s) n ) 1 with reversible dimerization according to eq 70; (4) cMEE ) 25 µM and v ) 25 V s-1; (0) cMEE ) 50 µM and v ) 25 V s-1; (]) cMEE ) 75 µM and v ) 25 V s-1.
per MEE molecule. As we will see below, only one electron per MEE molecule is likely to be exchanged along most of the anodic surface wave. Therefore the tailing effect may tentatively be ascribed to the kinetically-controlled oxidation of the second thiol group of adsorbed MEE molecules. Supporting electrolyte voltammograms were subtracted from those recorded in the presence of MEE. A comparison between normalized background-corrected waves and theoretical surface waves is displayed in Figure 8. It may be observed that experimental waves cannot be described by expressions derived for simple surface redox processes18 with n ) 1 or 2, even after consideration of interaction coefficients between adsorbed molecules18 (not shown in Figure 8). However, it can be noted that the cathodic side of the experimental waves approaches that of a simple redox process involving two electrons (n ) 2), whereas the anodic side is better described by a one-electron redox process. This is exactly the behavior to be expected when charge transfer is perturbed by the presence of a fast reversible dimerization with a high value for its equilibrium constant, which corresponds to eqs 48-52 of Scheme I. Therefore, an improved description of the wave shape is obtained from
(70)
where
ξ/ξp ) exp
nF (E - E )) (RT p
(71)
which is the analogous result for Scheme II of the ratio between our previous eqs 48 and 50 (i.e. assuming that K h dθR* > 500) and is uniquely determined by the choice of n when potentials are referred to the E - Ep coordinates. The full lines in Figure 8 were derived from eq 70 with n ) 1; the reasonable fit to the experimental wave shapes suggests that oxidation of adsorbed MEE molecules takes place through a single electron exchange followed by a fast reversible dimerization step. In order to estimate Es, Kd, and ΓR(0), peak parameters and wave widths were analyzed as a function of reactant concentration. To avoid interferences from the tailing currents, quantitative estimates were restricted to the anodic waves. Initial values of ΓR(0) were obtained from the peak currents according to
ΓR(0) )
ia,pRT 0.343n2F2Av
(72)
which is the analogue of eq 50 for Scheme II. Then, variation of the peak potentials and wave widths with reactant surface concentration is close to that expected by assuming K h dθR* > 500 (broken lines and open symbols in Figure 9). However, quantitative fits require a somewhat lower value of the surface dimerization equilibrium
1490 Langmuir, Vol. 15, No. 4, 1999
Calvente et al.
considered in this work, MEE adsorption can be described by a Henry isotherm with KaRΓs/cs ) 1.5 × 10-3 cm (see Figure 10b). The presence of a following dimerization reaction in the mechanism of MEE oxidation is consistent with previous observations on the oxidation of other thiols on mercury.24,26 Though the nature of the dimer remains unsolved, the most likely mechanism involves the formation of a Hg(I) derivative according to
HSRORSHads + Hg T HSRORSHgads + e- + H+ (75) followed by its dimerization:
2HSRORSHgads T HSRORSHgHgSRORSHads (76) The dimeric product of eq 76 may eventually disproportionate to give
HSRORSHgHgSRORSHads T (HSRORS)2Hgads + Hg (77) The presence of the above disproportionation equilibrium would not change our previous analysis, but now Kd should be reinterpreted in terms of the two chemical equilibria. Conclusions Figure 10. (a) Optimized fits of anodic waves for the parameter values indicated in the text: (]) cMEE ) 75 µM, (3) 50 µM, (4) 25 µM, (0) 10 µM, and (O) 7.5 µM and v ) 25 V s-1. (b) MEE surface concentrations derived from eqs 73 and 74 as a function of MEE concentration.
constant, which in turn slightly modifies the initial estimate of ΓR(0). Model parameters can be determined in a consistent way by fitting simultaneously peak potentials and currents to the following two equations:
Ea,p ) Es and
(
0.343 -
ln
RT ln(2KdΓR(0) + 1) 2nF
ia,p nFAΓR(0)a
ia,p nFAΓR(0)a
- 0.25
)
) 0.3 + ln
(73)
x1 + 2KdΓR(0) 2KdΓR(0)
(74)
which are the analogues of eqs 56 and 60 for Scheme II under fast reversible dimerization conditions. Best fits (full lines in Figure 9) were obtained for Kd ) 1 ( 0.5 × 1011 mol-1 cm2, Es ) -0.39 ( 0.01 V, and the ΓR(0) values depicted in Figure 10b. Figure 10a shows how these parameter values are able to reproduce quantitatively the observed anodic waves, except for the tailing contribution indicated above. For the low thiol concentrations
A theoretical analysis of voltammetric waves perturbed by a following surface dimerization process has been carried out in the absence of diffusional effects. The previous work of Laviron has been extended to avoid kinetic restrictions on the dimerization step. It has been shown that transition from reversible to irreversible dimerization kinetics involves significant changes in wave shape. In particular, the wave width and peak current corresponding to a fast reversible dimerization are similar to those expected for a simple suface redox process exchanging 1.5n electrons. Rather simple analytical expressions have been derived for some limiting kinetic cases. Empirical equations, relating peak coordinates and wave widths with scan rate and reactant concentration, have been developed to allow a straightforward voltammetric diagnosis in the presence of a bidirectional quasireversible dimerization process. From the point of view of diffusional flux requirements, it has been shown that two extreme situations, where adsorbed reactant molecules are either isolated from or equilibrated with the solution bulk, share a common mathematical solution of the initial value problem. Theoretical predictions were applied to the interpretation of the voltammetric features of 2-mercaptoethyl ether oxidation on mercury. A satisfactory description of the voltammograms was obtained by assuming the presence of a fast reversible dimerization reaction following a single electron exchange. Acknowledgment. This work was supported by the Spanish DGICYT under Grant PB095-0537. LA980961F
