Langmuir 2002, 18, 2765-2770
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Method of Separation and Determination of the Characteristics of the Adsorbed and Nonadsorbed States of Electroactive Substances on Electrodes M. Łobacz, M. Orlik, J. Stroka, and Z. Galus* University of Warsaw, Department of Chemistry, Pasteura 1, 02093 Warsaw, Poland Received July 9, 2001. In Final Form: October 26, 2001
The electrochemical analysis of the properties of the adsorbed reactant state for the case when it is energetically close to the nonadsorbed state of the electroactive species is presented. The method of separation of the corresponding adsorption and adsorption-free contributions to the overall faradaic response, which may also supply the voltammetric properties of adsorbed state, is given. It is based on recording, for a given solution, two normal pulse, charge-potential (voltocoulometric) curves. The first curve is obtained by integration of the faradaic current over the entire duration of the potential pulse t (Qt-E curve). For the second curve the charge is measured with the exclusion of the first millisecond after the application of the potential pulse (Qt-1 vs E curve). The numerical model of such dependencies proves that the shape of their difference ∆Q ) Qt - Qt-1 as a function of the electrode potential satisfactorily approximates the electrochemical properties of the adsorption state. Such an approach is recommended as a simple way of analysis of the properties of the adsorption state of electroactive substances which do not yield a distinct separate faradaic response from the adsorption state.
Introduction Many reactions in heterogeneous systems occur with the participation of the adsorbed state which can be formed at the gas/solid, gas/liquid, liquid/liquid, or liquid/solid interfaces.1 In some cases (typical, e.g., of many gas-phase reactions in contact with a solid catalyst), the processes occurring from the adsorbed state may be the only observable ones, since the noncatalyzed route appears to be too slow. In such a case the kinetic data exhibit a direct correlation with the properties of the adsorbed state, the characteristics of which are then relatively easy to determine. However, when the rates of reactions from the adsorbed and nonadsorbed states are comparable, one can get only common combined kinetic information from both reaction routes. Under such circumstances, it would be useful to have a simple way to extract the properties of adsorbed reactant contribution to the overall kinetic response. The method of such separation depends on the particular kind of kinetic information which can be obtained for a given system. In the present paper we consider the case of the reactants which exhibit the surface affinity to a metallic surface and are also electroactive, and as a consequence they can be studied by electrochemical methods. To yield the faradaic response in the form of only one common current (or charge) vs potential dependence for the adsorbed and nonadsorbed species,2-3 reactants should exhibit relatively small free energy of adsorption (as, e.g., for the ligand induced adsorption4-8). (1) Laviron, E. In Electroanalytical Chemistry; Bard, A. J., Ed.; Dekker: M.: New York, 1982; Vol. 12, p 63. (2) Osteryoung, R. A.; Christie, J. H. J. Phys. Chem. 1967, 71, 1348. (3) Anson, F. C.; Barclay, D. J. Electroanal. Chem. 1970, 28, 71. (4) Lovricˇ, M.; Komorski-Lovricˇ, Sˇ . Langmuir 1995, 11, 1784. (5) Lovricˇ, M. J. Electroanal. Chem. 1985, 183, 107. (6) Galus, Z. Fundamentals of Electrochemical Analysis, 2nd ed.; Horwood, E., Ed.; PWN: New York; Warsaw, 1994. (7) Xuan, H. T.; Maksymiuk, K.; Stroka, J.; Galus, Z. Electroanalysis 1996, 8, 34. (8) Bogucka, A.; Gromulska, A.; Stroka, J.; Galus, Z. Polish J. Chem. 2001, 75, 93.
These composite signals are different from the classical voltammetric dependencies influenced by strong adsorption, when distinct prewaves or postwaves (so-called Brdicˇka waves), originating from the electrode processes of adsorbed states only, were observed. Our approach is described here for the liquid/liquid, i.e., mercury electrode/aqueous solution interface. The mercury electrode was chosen because it has a well-defined surface and thus easily yields reproducible results, which is particularly useful for the verification of the validity of the new method of analysis that we propose. Once verified, it can be also applied for solid electrodes. For the mercury electrode one can find a number of electrochemical systems exhibiting a relatively weak adsorption of the reactant and therefore forming one common signal. Curves 1 in Figure 1 show selected examples. For the present studies we chose the process illustrated by Figure 1C: the electroreduction of the cryptate-Tl+ (1:1) complex, Tl(2,2,2)+, to the thallium amalgam. In this case the metal ion is interacting with the electrode via adsorbed ligand11 and the charge-transfer step is followed by the release of a free ligand molecule. Our previous studies of this process12 showed that electroreduction of Tl(2,2,2)+ occurs simultaneously from the bulk and from the adsorption states, so it meets the principal assumption of our present analysis. Before12 we focused on the kinetic parameters of the redox couple and on the thermodynamic properties of the Tl(2,2,2)+ complex, largely free of adsorption effects. Now we present the way to separate its adsorption characteristics from the analysis of the charge-potential relationships, such as those shown in Figure 1. We verify our approach by the numerical simulation of the studied process. (9) Gromulska, A.; Stroka, J.; Galus, Z., unpublished results. (10) Z˙ mudzin, U.; Łobacz, M.; Stroka, J.; Galus, Z., unpublished results. (11) Carla, M.; Gambi, C. M. C.; Bagolini, P. J. Phys. Chem. 1996, 100, 11 067. (12) Łobacz, M.; Orlik, M., Stroka, J.; Galus, Z. Electroanalysis, in press.
10.1021/la011046o CCC: $22.00 © 2002 American Chemical Society Published on Web 03/03/2002
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Łobacz et al. tion over sampling time, at a given electrode potential. The latter method consists of constructing the chronocoulometric response vs the electrode potential, which is changed as a function of time as in the normal pulse polarography. The resulting Q-E relationship is further called the voltocoulometric curve. Chronocoulometric experiments were carried out with a CH model 660 analyzer. For the voltocoulometry the homemade apparatus, with the current integration times ranging from 4 to 100 ms, with or without the exclusion of the first millisecond, was applied, and the Q-E curves were recorded with an X-Y recorder LY 14100-II (Linseis).
Results
Figure 1. Exemplary experimental voltocoulometric Qt-E (1) and Qt-1-E (2) curves for (A) the 1-nitrodecane with relative concentration c/cs (cs, concentration of saturated solution) equal to 0.80, in aqueous solution containing 0.50 M NaClO4 and 0.05 M HClO4. Pulse times tp ) 64 ms.9 (B) The Cd(His-)2 complex recorded in aqueous solution at pH ) 10.6, containing 0.50 M NaClO4, 2.5 × 10-3 M L-histidine, and Britton-Robinson buffer. Pulse times tp ) 81 ms.10 (C) The Tl(2,2,2)+ complex in aqueous solution containing 5 × 10-4 M Tl+, 2.5 × 10-4 M cryptand, and 0.10 M TEAP. Pulse times tp ) 25 ms.
Experimental Section Chemicals. Fluka manufactured thallium(I) nitrate, tetraethylammonium perchlorate (TEAP), and cryptand(2,2,2) from Merck were used without further purification. All chemicals used were p.a. grade. Solutions used were prepared with triply distilled water, purified in a final step using Millipore filters. Apparatus and Experimental Methods. The electrochemical studies were carried out in a three-electrode cell with a water jacket at 25 ( 0.1 °C. A static mercury drop electrode (Laboratorni Prˇistroje, Prague) was used as a working electrode, while the platinum foil (2 cm2) served as a counter electrode. All potentials were measured against a calomel electrode (SSCE) with saturated sodium chloride. The electrode processes and adsorption of reactant were investigated by using single-step chronocoulometry and voltocoulometry. The former technique involves the current integra-
Voltocoulometric Experiments and the Concept of Analysis. Two sets of experiments were performed. In the first set the current was integrated during total pulse time (curves denoted further as Qt vs E), while in the second set the integration started with 1-ms delay after the pulse application (curves Qt-1 vs E). To investigate the induced adsorption of the complex, such chargepotential dependencies were recorded for different thallium(I) and cryptand concentrations in aqueous solutions. All solutions contained 0.10 M TEAP. Tl+ concentration was varied from 5 × 10-4 to 5 × 10-3 M, and the cryptand(2,2,2) concentration was changed in the range from 5 × 10-3 to 2.5 × 10-2 M. The voltocoulometric cathodic Qt-E and Qt-1-E curves were recorded in the potential range from -0.50 to -1.20 V. A pair of typical curves Qt-E and Qt-1-E is given in Figure 1C. On the cathodic Qt-1 vs E curves a maximum was always observed. The formation of this maximum was generally explained as originating from the potential-dependent electroreduction of adsorbed reactant that was not electrolyzed within the first millisecond after the application of the potential pulse.12 This maximum should be particularly noticeable for the irreversible electrode processes (cf. Figure 1A), when the electrolysis of the adsorption layer occurs in relatively large time, 1 or 2 orders of magnitude greater than the first millisecond of delay in current integration. Thus, despite this delay quite a large amount of adsorbed species remains ready for further electrolysis at this potential. But it is noteworthy that in the opposite limiting case, i.e., for the reversible redox processes Ox + ne ) Red, coupled also with quickly establishing adsorption equilibrium Ox(bulk) ) Ox(ads), this maximum, though relatively weak, is still observed, since then to the diffusion layer a large amount of Red, originating both from bulk and adsorbed Ox, is injected. According to Nernst equation, for a given electrode potential, the local pre-electrode high concentration of Red maintains enhanced concentrations of Ox, which prevents instant electroreduction of Ox down to the equilibrium surface concentration Γox at less negative potentials. This Γox is attained only asymptotically in a time scale typical of diffusion of Red toward the bulk of the solution; therefore, within the first millisecond only a small portion of adsorbed Ox species can be electrolyzed. However, at far negative potentials, both bulk and adsorbed portions of Ox are electrolyzed within times shorter than 1 ms, so the charge recorded with this delay does not contain any detectable adsorption contribution. Thus it becomes clear that we discuss here quite a general phenomenon that is observed for the parallel electrode processes of the reactant located in the bulk solution phase and in the adsorbed state (cf. Figure 1). Since in our particular case the Tl(2,2,2)+ electroreduction is quasi-reversible, both limiting mechanisms, described above, interplay in the formation of a maximum shown in Figure 1C. Now, let us consider more quantitatively the contribution of adsorption to the recorded voltocoulometric curves.
Adsorbed and Nonadsorbed States on Electrodes
Langmuir, Vol. 18, No. 7, 2002 2767 Table 1. Charges and Surface Concentrations of Adsorbed Tl(2,2,2)+ Cryptate in Solutions Containing 2.5 × 10-2 M (2,2,2), 0.10 M TEAP, and Different Concentrations of Tl+ Cationsa [Tl+], M
QAds, µC cm-2
5 × 10-4 1 × 10-3 2 × 10-3 5 × 10-3
8.6b 11.9b 9.0b 10.8b
Γ of Tl(2,2,2)+, mol cm-2 8.9 × 10-11 b 1.23 × 10-10 b 9.3 × 10-11 b 1.12 × 10-10 b
8.3c 9.0c 8.5c
8.6 × 10-11 c 9.3 × 10-11 c 8.8 × 10-11 c
a Initial potential E ) -0.50 V. b Data calculated from extrapoi lation of QLt vs tp1/2 dependencies (from ref 12). c Data calculated from differences (QLt - QLt-1) (present work).
Figure 2. (A) Experimental dependence of ∆Q on E for Tl(2,2,2)+ cryptate electroreduction in aqueous solution containing 1 × 10-3 M Tl+, 2.5 × 10-2 M (2,2,2), and 0.10 M TEAP. tp ) 49 ms. (B) Simulated dependencies obtained for the parameters specified in part A and the caption to Figure 3. Curves: (1) ∆Q-E relationship, (2) pure adsorption contribution QAds-E calculated within the first 1 ms of the potential pulse, and (3) pure adsorption contribution QAds-E calculated within 49 ms.
An usual way of extraction of such an adsorption contribution to the faradaic charge is based on the dependence of the limiting charge QLt vs tp1/2 which we used previously.12 This approach is, however, limited to the range of potentials, where the contribution from the reactant electroreduction from the solution is controlled by the rate of transport only, when the potential of the electrode is sufficiently negative. As a result we obtain the charge related only to the total amount of the electroactive species, adsorbed at the initial potential, in time preceding the potential step for electrolysis. For determination of the adsorption parameters, this information is of course valuable, but it was our aim to go into more profound analysis of the adsorbed state and to find a relatively simple approach that would allow us to obtain the characteristics of this state in the entire range of the potentials of the formation of the voltocoulometric wave. In other words, we wanted to use electrochemical methods not only for the separation of the signal of the adsorbed state from the total signal but also to get its full electrochemical characteristics. The method which we propose in this paper is based on the analysis of the differences ∆Q ) (Qt - Qt-1) as a function of the electrode potential. Such an experimentally determined dependence is given as an illustration in Figure 2A. The difference of the limiting charges, ∆QL, of such two dependencies is equal to the charge collected during the first millisecond of electrolysis and is given by the equation
∆QL ) QLt - QLt-1 ) Qdl + QL1 + nFΓox
reaction potential and at the initial potential and nFΓox and QL1 stand for the charge due to the electrode reaction of adsorbed reactant and the limiting charge of the diffusing reactant, during the first millisecond after the pulse application, respectively. QL1 may be easily calculated or determined from the QL vs tp1/2 plots and Qdl at least estimated from the independent experiment (blank probe). If QL1 and Qdl are relatively low, the ∆QL values are expected to be close to adsorption charges determined previously12 from the conventional QLt vs t1/2 relationship. For quantitative verification of the present analysis, these two sets of values are collected and compared in Table 1. The data obtained, using both methods, are in satisfactory agreement. Thereby, the values of QAds appear to be practically independent of the pulse time, when calculated from the QLt - QLt-1 differences recorded at different tp. On the basis of the above results, valid for limiting charges, one can expect that also charge values calculated for less negative potentials, down to the foot of the Q-E wave, will also approximate quite well actual adsorption contributions. However, then, according to the mechanism of the formation of maximum, described above, not the entire surface excess Γox, as in (1), but only its part, ∆Γox, that underwent electrolysis within the first millisecond is included in this relation:
∆Q(E) ) Qt - Qt-1 ) Qdl + Q1 + nF∆Γox
To check these considerations quantitatively we made appropriate numerical calculations, using the model process Tl(2,2,2)+ + e f Tl(Hg). Construction of the Numerical Model. The electrode reactions of reactant from the solution (Oxd) and from the surface (OxAds) occur in two parallel processes: ks,1
Oxd + ne {\}Red
(3)
ks,2
OxAds + ne {\}Red
(4)
where ks,1 and ks,2 denote the appropriate apparent standard rate constants. Adsorbed and bulk species remain in fast local surface equilibrium Oxd a OxAds, described by the appropriate adsorption isotherm. The faradaic current calculated according to the following relationship (for mathematical details see Appendix)
(
If ) nFA fox(0,t) +
)
∆Γox ∆t
(5)
was numerically integrated over sampling time tp, to yield a faradaic charge:
(1)
where Qdl is the difference of double layer charge at the
(2)
Q(tp) )
t ∫0+t p
del
If(t) dt = ∆t
∑j If(j)
(6)
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Figure 3. Voltocoulometric Qt-1-E curves for the solution containing 5 × 10-4 M Tl+, 5 × 10-3 M cryptand, and 0.10 M TEAP. Pulse times tp ) 49 ms. Curves: (O) experimental; (-) simulated. In simulation the parameters E0f, Dox, Dred, c0ox, Γox∞, and A given in the text were used and additionally ks,1 ) 8 × 10-3 cm s-1, Rn ) 0.2, βox ) 4.4 × 106 cm3 mol-1, and a ) 2.5 were assumed. The complex kinetic parameter ks,3 that includes the standard rate constant from the adsorbed state (cf. Appendix) was equal to 0.27 cm s-1.
In this integration formula the contribution of current within the first tdel period could be included or neglected. The obtained Q vs E relationship forms a desired normal pulse (Qt vs E or Qt-1 vs E) voltocoulometric curve. For thallium(I) electroreduction, considered here, n ) 1, and the following input parameters were chosen on the basis of our experimental studies:12 Ef0 ) -0.725 V, Dox ) 5.2 × 10-6 cm2 s-1, Dred ) 9.9 × 10-6 cm2 s-1 (ref 13), cox0 ) 5 × 10-7 mol cm-3, Γox∞ ) 4.2 × 10-10 mol cm-2, electrode surface area A ) 0.0106 cm2. The best concordance between the experimental and simulated Qt-1 vs E curves was further found for the following adjustable parameters: βox ) 4.4 × 106 cm3 mol-1, ks,1 ) 8 × 10-3 cm s-1, ks,3 ) 0.27 cm s-1, Rn ) 0.2, a ) 2.5 (see Appendix for the explanation of symbols). The comparison of the fitted and experimental Qt-1-E curves, for the exemplary integration time tp ) 49 ms, is shown in Figure 3. A quite close course of both curves suggests that the simulated Qt-1-E and thus also Qt-E dependencies can be used further for the verification of the adsorption analysis, based on the ∆Q-E relationship. Curve 1 in Figure 2B shows the ∆Q-E dependence (cf. eq 2), simulated for the parameters listed above and for the integration time tp ) 49 ms. The striking similarity of the shape of this curve to the experimental one (from Figure 2A) confirms that the model used for simulation is correct. Slight differences between those curves may be, at least partly, due to the fact that the relatively small effect of double-layer charging Qdl was not included in the simulations. Furthermore, the simulation procedure allows comparison to the model ∆Q vs E curve with the pure model adsorption contribution to the overall faradaic charge, recorded within the first millisecond of electrolysis (curve 2 in Figure 2B). Since curves 1 and 2 are rather close, it appears that indeed such a simple approach, consisting in the analysis of the ∆Q ) f(E) relationship, allows one to extract quite well the characteristics of the electrode process of the adsorbed state of the reactant. For comparison, curve 3 in Figure 2B shows the simulated pure adsorption contribution that is hidden in overall faradaic responses recorded for the entire sampling time of 49 ms. (13) Bellavance, M. I.; Miller, B. In Encyclopedia of Electrochemistry of the Elements; Bard, A. J., Ed.; Dekker: New York, 1975; Vol. 4, p 179.
Łobacz et al.
Figure 4. Influence of the delay time of charge integration on simulated voltocoulometric dependencies. Langmuir isotherm, a ) 0. Integration delay times indicated in the Figure, Rn ) βn ) 0.5, tp ) 100 ms. All other parameters as in the caption to Figure 3.
The shapes of these dependencies are concordant with the above-given idea of the prolonged electrolysis of the adsorbed layer. Discussion It was shown that the easily available dependence of ∆Q on E approximates quite well a voltocoulometric characteristics of the adsorbed reactant. One may explore what experimental factors could be adjusted to make this approach more precise, flexible, and applicable to other redox systems characterized by different electrochemical and adsorption characteristics. It appears that such an important and easily adjustable parameter is the delay in the charge sampling (current integration) time after potential pulse application, since various electrochemical systems, in dependence on, e.g., degree of reversibility of electrode processes and magnitude of the diffusion coefficient, may be characterized by different time scales of the electrolysis of adsorbed reactant. Evidently, for a given system, the shorter this delay time is, the relatively stronger adsorption effects should be exposed on the voltocoulometric curves and ∆Q vs E relationships. Quantitative illustration of the importance of the delay time on the shape of the simulated voltocoulometric curves, for the simplified case of a Langmuir isotherm, is shown in Figure 4. Charge maximum decreases with increasing delay time, and simultaneously the maximum is shifted to less negative potential values. The maximum on the curves is particularly large for delay times not exceeding 1 ms. On the other hand, for delay times longer than 10 ms, the maximum practically disappears. Let us consider how this delay time influences the shapes of the ∆Q-E relationships. One may suppose that for sufficiently short delay times the ∆Q vs E relationship, after subtraction of the double-layer charge Qdl, should give practically pure characteristics of the adsorption wave that would be recorded for the integration time equal to delay time. Figure 5 shows the ∆Q vs E exemplary relationships simulated for various delay times after the pulse application with the simulated pure adsorption contribution integrated within the time equal to delay time. For delay times longer than 1 ms (Figure 5C), the contribution of the diffusion charge recorded within this time to the full curve introduces a substantial difference between the considered curves. However, one can note that for 0.1 ms (Figure 5A) both relationships,
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Langmuir, Vol. 18, No. 7, 2002 2769
Figure 6. Simulated voltocoulometric curves for different standard electroreduction rate constants from the adsorbed state ks,3. Langmuir isotherm, βox ) 4.4 × 106 cm 3 mol-1, ks,1 ) 8 × 10-3 cm s-1 in bulk, and tp ) 100 ms without first 1 ms. All other parameters as in the caption to Figure 3.
Figure 7. Simulated voltocoulometric curves Qt-E (---) and Qt-1-E (-) for different adsorption constants βox of the reactant in adsorbed state. Langmuir isotherm, ks,1 ) 8 × 10-3 cm s-1 in bulk, ks,3 ) 0.27 cm s-1 in adsorbed state, and tp ) 100 ms without first 1 ms.
Figure 5. Comparison of the ∆Q vs E relationships (0) simulated for various delay times after the pulse application with the simulated pure adsorption contribution (-) for the current integration time equal to delay time: (A) 0.1; (B) 1; (C) 4 ms. Parameters of the model correspond to the Tl(2,2,2)+ electroreduction (cf. Figures 2 and 3 and text).
which form in this case rather drawn-out, apparently irreversible waves, practically overlap with each other. In the considered case, both the kinetics of the chargetransfer step from the adsorbed state and the transport properties of the Red species are operating, since the Tl(2,2,2)+ electroreduction is quasi-reversible.12 For another, highly irreversible electrode process, such a relationship as in Figure 5A would constitute clear kinetic characteristics of the electroreduction from the adsorbed state. Finally, for a more extensive analysis of the effect of the adsorbed state on the shape of recorded voltammetric dependencies, using our numerical model we have also checked the influence of the standard rate constant ks,3 from the adsorbed state (Figure 6) and of the adsorption constant βox (Figure 7) on the shape and the size of the maximum on the simulated Qt-1-E voltocoulometric curves. Acceleration of the electrode process from adsorbed state, similarly to the decrease of the adsorption energy, caused the decrease of the charge maximum and its shift to less negative potentials. This sensitivity of the size and
the shape of the maximum to the kinetic and thermodynamic parameters of the adsorbed state of the experimental system makes possible their determination from the voltocoulometric dependencies, according to a quite general methodology, described in this paper. Conclusions We presented the novel way of studying the electrode processes complicated by adsorption of electroactive species, for the frequently occurring case when both forms of reactant, in the solution and in the adsorbed state, are energetically close. The method is based on recording of a faradaic charge with some delay (1 ms or less) (Qt-del) and without a delay (Qt) after the application of the potential pulse, to get the ∆Q ) Qt - Qt-del difference. Using a digital simulation model we showed that the ∆Q vs E dependence gives voltocoulometric characteristics of the reactant in the adsorbed state. This approach provides a simple method for studying the properties of adsorbed reactants. Though the cryptate complex of Tl(I) was considered as an experimental system, the phenomena and their analysis, described in this paper, are quite common for electrode processes complicated by adsorption of electroactive species. Appendix The basis of the Numerical Model. It is assumed that the electroreduction proceeds according to reactions
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3 and 4. OxAds and Oxd are always in a local surface equilibrium, described by the Frumkin isotherm:
βoxcox(0,t) )
Γox Γox∞ - Γox
( )
exp
aΓox Γox∞
(7)
where cox(0,t) is the surface concentration of Ox in the solution at the reaction site and a is the coefficient of lateral interactions in the adsorbed layer, positive for repulsive forces. Despite the sphericity of the mercury electrode, due to relatively short sampling times used in our experiments (e100 ms) it was sufficient to apply the linear diffusion approximation for the transport of electroactive species between the bulk and the reaction site:
∂2cox ∂cred ∂2cred ∂cox ) Dox 2 ) Dred 2 ∂t ∂t ∂x ∂x
(
[
- fox(0,t) 1 +
]
kf′∆x kb′∆x + - kf′cox(1,t) + kb′cred(1,t) 2Dox 2Dred kb′∆x 1+ 2Dred (10)
( )
where ∆x is the resolution of the spatial grid in the discretization of the diffusion layer, and the complex rate constants kf′ and kb′ are defined by
[
{
exp
[
kb′ ) kf′ exp
)
(14) Feldberg, S. W. In Electrochemistry. Calculations, Instrumentation; Mattson, J. S., Mark, H. B., Macdonald, H. C., Eds.; Dekker: New York, 1972; Vol. II, pp 185-215.
×
Γox∞
-RnF(E - Efd0) RT
]
nF(E - Efd0) RT
]
(11)
(12)
Thereby Efd0 is a formal potential of the Ox/Red couple in the solution, ks,3 ) ks,2βox((n-Rn)/n)βred((Rn)/n) is treated as a single (adjustable) parameter, since individual values of βRed and ks,2 are not known a priori and as is the surface activity for the Frumkin isotherm:15
∆Γox ∆Γox ∆Γred + (9) =∆t ∆t ∆t
where fox(0,t) and fred(0,t) are the diffusion fluxes of Ox and Red, respectively, at the electrode surface. At every time step ∆t of the simulation of the faradaic response to the single normal pulse of the potential, the actual change of the surface excess of Ox was calculated by using the relationship
( )]} [ Γox
kf′ ) as ks,1 + ks,3 exp (λ - 1)a
(8)
As the numerical procedure, the finite-differences algorithm, based on Feldberg’s approach for cyclic voltammetry,14 generalized for an n-electron process and adapted for the conditions of our voltocoulometric technique, was applied. In terms of this approach, where the adsorption equilibria of only Ox species are considered (βox . βred), the following boundary condition for the surface fluxes of Ox and Red species is met:
fox(0,t) + fred(0,t) ) -
∆Γox ) ∆t
as )
(
)
Γox∞ - Γox Γox exp - λa ∞ ∞ Γox Γox
(13)
with a Temkin “coverage parameter” 0 < λ < 1,15 here assumed to be equal to 0.5. One should also note that eq 10 is valid if pairs of transfer coefficients (Rn, βn) are assumed to be identical for the electroreduction from the solution and from the adsorbed state, which significantly simplifies mathematical derivations. LA011046O (15) Mohilner, D. M. J. Phys. Chem. 1969, 73, 2652.
