Reverse Pulse Voltammetry and Double Potential Step

Nov 6, 1998 - Much more significant and useful are reverse pulse polarography ..... are not well defined and therefore it is difficult to measure the ...
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Anal. Chem. 1998, 70, 5237-5243

Reverse Pulse Voltammetry and Double Potential Step Chronoamperometry as Useful Tools for Characterization of Electroactive Systems under the Conditions of Mixed Diffusional and Migrational Transport Wojciech Hyk and Zbigniew Stojek*

Department of Chemistry, University of Warsaw, ul. Pasteura 1, PL-02-093, Warsaw, Poland

A numerical model for reverse pulse voltammetry and double potential step chronoamperometry is presented for the systems with mixed diffusional and migrational transport. The calculations that have been done using this model indicate that under the conditions of moderate deficiency or absence of supporting electrolyte the ratio of the plateau current magnitudes, predicted for reverse pulse voltammetry, depends on the type of electrode reaction. For each type of electrode process, this ratio is also a function of support ratio (the ratio of bulk concentrations of supporting electrolyte and analyte). It has been found that for any concentration of supporting electrolyte the ratio of the magnitudes of the plateau currents changes linearly with the reciprocal of the square root of the pulse time. Such dependencies can be used to detect the contribution of migration to the current measured and to determine the diffusion coefficient of the primary substrate. The determination of these quantities requires the registration of only one double potential step chronoamperometric curve or several reverse pulse voltammograms. The reported possibilities should be especially useful for the cases where the support ratio is unknown or an addition of supporting electrolyte is impossible, e.g., for solid-state electrochemistry. For the purpose of validation of the model presented, an expression for the diffusional reverse pulse current at a hemispherical microelectrode has been derived analytically, and experimental measurements for a model redox system (ferrocene) have been performed. The use of microelectrodes makes it possible to extend electrochemical studies to highly resistive media, including solutions containing no deliberately added supporting electrolyte and generally solutions of low support ratio (the ratio of bulk concentrations of supporting electrolyte and substrate).1 In such systems, in addition to diffusion, migration contributes substantially to the transport of species. Several experimental2-6 and * Corresponding author: (e-mail) [email protected]. (1) Wightman, R. M.; Wipf, D. O. Voltammetry at Ultramicroelectrodes. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1988; Vol. 15. 10.1021/ac980748o CCC: $15.00 Published on Web 11/06/1998

© 1998 American Chemical Society

theoretical papers7-12 addressed the important issues related to this problem. To find out how migration contributes to the current measured for a given system, one might wish to do an extra experiment with excess supporting electrolyte. This is not always possible. High concentration of the analyte and limited solubility of supporting electrolyte may be one of the problems, solid state of the system examined may be another. The application of a reversal method, which is reported in this paper, may allow one to skip the addition of excess supporting electrolyte, since the substrate and product are of different charge, and therefore, the forward and reverse currents will be affected by migration differently. The simplest reversal method based on the control of potential is double potential step chronoamperometry (DPSC), in which the first step is used to generate some species, and the second step is used to reverse the effect of the initial step. The DPSC wave form is shown in Figure 1a. Though this technique is not of great analytical importance, most of the other reversal methods are based on it. Much more significant and useful are reverse pulse polarography (RPP) and voltammetry (RPV).13 These electroanalytical techniques provide valuable electrochemical information, including kinetic and mechanistic parameters. The introduction of a waiting period, or waiting time, tW, to the reverse pulse voltammetric wave form, as shown in Figure 1b, simplifies both the mathematical description and interpretation of the (2) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 172, 299. (3) Malmsten, R. A.; Smith, C. P.; White, H. S. J. Electroanal. Chem. 1986, 215, 223. (4) Ciszkowska, M.; Stojek, Z. J. Electroanal. Chem. 1986, 213, 189. (5) Cooper, J. B.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1992, 331, 877. (6) Jaworski, A.; Donten, M.; Stojek, Z. Anal. Chim. Acta 1995, 305, 106. (7) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Electroanal. Chem. 1984, 172, 11. (8) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1987, 225, 49. (9) Amatore, C.; Bartlet, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (10) Oldham, K. B. J. Electroanal. Chem. 1988, 250, 1. (11) Baker, D. R.; Verbrugge, M. W.; Newman, J. J. Electroanal. Chem. 1986, 314, 23. (12) Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (13) Osteryoung, J.; Kirova-Eisner, E. Anal. Chem. 1980, 52, 62.

Analytical Chemistry, Vol. 70, No. 24, December 15, 1998 5237

Figure 1. Potential wave forms for double potential step chronoamperometry (a) and reverse pulse voltammetry with waiting period (b). Notation used: t0, initial time; tG, generating time; tP, pulse time; tW, waiting period (waiting time). Open and filled circles in (b) denote currents measured at the end of generating time (IG) and pulse time (IRPV), respectively. E ) 0 marks the standard potential. At E ) EG, the product is generated. Before applying the potential EG, the electrode is kept at a neutral potential, which is usually equal to Efinal.

experimental waves obtained at solid electrodes,14 since during this period the initial concentrations in the system studied are being restored. Reverse pulse voltammetry with waiting period is given as RPV-WP further throughout the paper. The restoration of the initial concentrations in RPP is done by simply dislodging the Hg drop at the end of the pulse time. All of the theoretical models published on double potential step chronoamperometry and reverse pulse voltammetries are restricted to the cases where the transport of species is achieved only by diffusion and therefore to the conditions of excess supporting electrolyte.15 The aim of this work is to extend the theoretical descriptions of reversal methods, such as DPSC and RPV-WP, for the purpose of diagnosis of electrode processes occurring under the conditions of mixed diffusional and migrational transport and, particularly, for the systems where the concentration of supporting electrolyte cannot be changed. The results obtained with the numerical model derived are compared with experimental data. To validate partially the model proposed, we have derived the theoretical relations for DPSC and RPV-WP at hemispherical and spherical microelectrodes and for diffusional transport. EXPERIMENTAL SECTION Double potential step chronoamperometry and reverse pulse voltammetry were performed using a PARC, model 273A, poten(14) Stojek, Z.; Jaworski, A. Electroanalysis 1992, 4, 317. (15) Galus, Z. Fundamentals of Electrochemical Analysis; Ellis Horwood, New York, 1994.

5238 Analytical Chemistry, Vol. 70, No. 24, December 15, 1998

tiostat which was controlled via software. Since a convenient setup for reverse pulse voltammetry is not provided by the manufacturer, this technique was constructed as a set of appropriate doublestep chronoamperograms. The corresponding potential wave forms and the notation used in this paper for DPSC and RPVWP are presented in Figure 1a and b, respectively. All electrodes used were platinum. A platinum wire was the counter electrode, and another wire was the quasi-reference electrode, just to eliminate a possible leak of electrolyte from the bridge. The repeatability of the formal potential measured as the half-wave potential for excess supporting electrolyte was good. The platinum working microdisk of 5.5 µm in radius (Project Ltd., Warsaw) was initially polished with 0.1-µm diamond suspension (Buehler) on a wet pad. The electrode was rinsed with a direct stream of ultrapure water (Milli-Q, Millipore). Water was removed from the electrode surface using a dry tissue and then the electrode was held in pure acetonitrile for a couple of minutes. After the measurements, the electrode surface was inspected optically with an Olympus, model PME 3, inverted metallurgical microscope. The experiments were performed at 20 °C. The cell was enclosed in a solid aluminum Faraday cage. The uncertainty related to the fact that disk microelectrodes are used to test the theory derived for hemispherical microelectrodes will be discussed in the section reporting on the comparison with experiment. Ferrocene ((C5H5)2Fe) from Merck was used without purification. Acetonitrile was of AnalaR grade. The manufacturer’s specification is ∼0.1% water in this solvent. Tetrabutylammonium perchlorate (TBAP) purchased from Fluka, is known to completely dissociate in acetonitrile and therefore it was used as the supporting electrolyte. NUMERICAL INFORMATION Theoretical and Numerical Bases. The considered electrode process is Nernstian and proceeds at a hemispherical microelectrode. It can be written as

SZs S PZp ( ne (for primary and reverse processes)

(1)

where zS, zP are charges of substrate (S) and product (P), respectively; n is the number of electrons transferred per molecule (n ) zP - zS). In addition to electroactive species, the system also contains a specified amount of a binary supporting electrolyte, which consists of univalent cation (C) and anion (A). It is also assumed that the substrate, if ionic, is introduced into the solution as a salt; thus one molecule of it is accompanied by an appropriate number of oppositely charged univalent ions (X). Under the conditions of small support ratio, the mass flow occurs by diffusion and migration. Changes in the concentration of any species (ci) at time t are ruled by the transport equation which, in the spherical coordinates (r), has the form

(

)

(

)

∂ci ∂2ci 2 ∂ci ziF ∂2Φ ∂ci ∂Φ 2ci ∂Φ + - Di 2 + - Di c + )0 ∂t r ∂r RT i ∂r2 ∂r ∂r r ∂r ∂r

i ) S, P, A, C, and X

(2)

where Di and zi are the diffusion coefficient and the charge of ith

species, respectively, Φ is the electrostatic potential existing in the solution, treated as a macroscopically observed ohmic drop, and F, R, and T have their usual meanings. In our considerations we assume that the microelectrode is not too small and pulse time not too short, so that the doublelayer thickness is an insignificant fraction of the depletion layer thickness and the electroneutrality principle holds. To validate this assumption we have used the criteria proposed by Norton et al.16 For 1:1 electrolyte, in a solvent with relative dielectric permittivity in the range from 10 to 50, the approximate values of the thickness of the double layer, for the lowest concentration of the electrolyte considered (5 × 10-6 M), are 13.6 and 30.4 nm, respectively (T ) 20 °C), while the approximated thickness of the depletion layer (δ ) (πDt)1/2) is, e.g., 6.8 µm for a diffusion coefficient of 2.1 × 10-9 m2/s (ferrocene in acetonitrile) and 7 ms after pulse application (the shortest pulse time considered in this work). This means that the electric field within this layer has an insignificant effect on the transport of ions to the microelectrode surface for almost the entire duration of a 7-ms pulse. The existence of a potential gradient in the transport equation might indicate that there is a separation of charges. However, the magnitude of this separation is so small that it does not have any significant influence on the concentration profiles.17 More details on the methodology of numerical solving of the equation set 2 can be found in our previous papers.18,19 Generally, the simulation scheme for the reverse pulse voltammetry with waiting period involves the following operations for the jth time step: (1) calculation of resistance and ohmic drop using the concentration profiles obtained in the previous time step, (2) setting the new boundary conditions, and (3) evaluation of the transport and construction of new concentration profiles. The variables and parameters of equations taken to the simulation have been made dimensionless using the following rules:

E° is the standard potential of redox system, and IGss is the steadystate generating current. All numerical data presented in this paper have been obtained for the following dimensionless simulation parameters: jrmax ) 180 (space size), ∆rj ) 0.005 (distance increment), htG ) 100 (generating time), ∆thG ) 0.0005 (generating time increment), htP ) 0.5, 1.0, ..., 10.0 (pulse times), ∆thP ) 0.0001 (pulse time increment), h ) 1 (potential increment), nP E h G ) 20 (generating potential), ∆E ) 40 (number of pulses). Although the simulated data have been obtained for equal diffusion coefficients and one-electron processes, the simulation scheme derived in this paper is also stable for unequal diffusivities and higher than one-electron reactions. Validation. The simulation procedure and the appropriate program were validated by comparing the simulation results with analytical equations that were derived by us for purely diffusional transport to a hemispherical electrode. Equality of diffusion coefficients of both reactants, and equality of numbers of electrons transferred in primary and reverse steps, were assumed. The equation for dimensionless reverse current in double potential step chronoamperometry is

hI R(th) )

1

(

1

1

-

xπ xht

)

where ht is dimensionless time measured from the moment of imposition of the second (reverse) pulse; see also Figure 1a. It is worth noting that eq 3 is identical to the corresponding expression for linear diffusion.20,21 The equation for reverse pulse voltammetry with waiting period has the form

hI RPV(E h) )

(π(thG + ht P))-1/2 + 1 1 + exp[-nE h]

-

ht P-1/2 - (thG + ht P)-1/2 π1/2 (1 + exp[nE h ])

jci ) ci/cSb (dimensionless concentration of ith species) ht ) tDS/re2 (dimensionless time, including tP and tG) jr ) r/re (dimensionless distance) D h i ) Di/DS (dimensionless diffusion coefficient of ith species) b

Ψ ) (F/RT)(Φ - Φ ) (dimensionless electrostatic potential) 0

E h ) (F/RT)(E - E ) (dimensionless electrode potential) hI )

I IGss

)

I (dimensionless current intensity) 2πnFDScSbre

where cSb and DS are bulk concentration and diffusion coefficient of primary substrate, respectively, re is the radius of the microelectrode, Φb is the electrostatic potential in bulk of the solution, (16) Norton, J. D., White, H. S., Feldberg, S. W. J. Phys. Chem., 1990, 94, 6772. (17) Jaworski, A.; Stojek, Z.; Osteryoung, J. G. Anal. Chem. 1995, 67, 3349. (18) Hyk, W.; Palys, M.; Stojek, Z. J. Electroanal. Chem. 1996, 415, 13. (19) Hyk, W.; Stojek, Z. J. Electroanal. Chem. 1997, 439, 81.

(3)

xht + ht G

(4) where n and E h denote the number of electrons exchanged in the primary step of electrode process and the actual reverse pulse potential, respectively. In the derivation of eq 4, it was assumed that the generating potential is sufficiently high, and as a result the surface concentration of the primary substrate is practically equal to 0. The parameter htW does not appear in eq 4 because we have assumed that this time is sufficiently long to restore the initial concentration profiles after each reverse pulse. The graphical representations of eqs 3 and 4 are given in Figure 2. An excellent agreement with expressions given above was obtained when migrational terms were neglected in our simulation scheme. The differences between the predicted and simulated results were less than 0.1%. Software. The simulation programs were written in ANSI C to ensure their portability. The programs were equipped with a graphics interface. They were compiled and tested with an IBM Pentium and a Cray YMP 4E computer. The output data were (20) Kambara, T. Bull. Chem. Soc. Jpn. 1954, 27, 523. (21) Colyer, C. L.; Hempstead, M. R.; Oldham, K. B. J. Electroanal. Chem. 1987, 218, 15.

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The appropriate expression for the conditions of linear diffusion (reverse pulse polarography) was derived by Osteryoung and Kirova-Eisner.13 The formula found by these authors has the following form

| |

IRP ) (1 + tG/tP)1/2 - 1 IDC

(6)

It is seen that expression 6 is equal to the numerator of the formula found for the conditions of spherical diffusion (eq 5). By expanding the function given by eq 5 in series for htp f 0 (which means that pulse time is significantly shorter than generating time), one can find a linear relationship between IRP/ IDC and reciprocal of the square root of the pulse time. This linear relation has the general form

|IRP/IDC| ) aj ht P-1/2 + b Figure 2. Theoretical diffusional reverse pulse voltammograms calculated for two values of dimensionless pulse time: 0.5 (a) and 10 (b). The inset shows the theoretical diffusional reverse current in double-step chronoamperometry. Currents were made dimensionless with respect to steady-state generating current.

processed using either commercially available or homemade programs. RESULTS AND DISCUSSION We have considered the following four classes of typical Nernstian electrode processes:

I S0 ) P+/- ( e II S+/- ) P0 - e III S+/- ) P2+/2- ( e IV S2+/2- ) P+/- - e

The above equations are written for both oxidation and reduction processes. For example, case IV is either the reduction of (+2) substrate to (+1) product or the oxidation of (-2) substrate to (-1) product. Excess Supporting Electrolyte. For excess supporting electrolyte, i.e., under the conditions of purely diffusional transport, the values of IRP/IDC (representations of these quantities are given graphically in Figure 2) and E1/2RP (half reverse pulse wave potential) are the same for all types of electrode processes discussed in this work. The value of IRP/IDC depends on pulse time width, while that of E1/2RP is equal to the standard potential of the redox system examined. Under the conditions of spherical diffusion, for RPV-WP at a microelectrode, the analytical formula for the ratio IRP/IDC can be found after some rearrangements of eq 4. The expression found is the following:

| |

IRP (1 + ht G/thP)1/2 - 1 ) IDC [π(thG + ht P)]1/2 + 1

(5)

5240 Analytical Chemistry, Vol. 70, No. 24, December 15, 1998

(7)

According to the expanding done, coefficient aj has the following physical representation: htG1/2[1 + (πthG)1/2]-1. Coefficient b is equal to -[1 + (πthG)1/2]-1. For a sufficiently long generating time, i.e., when steady state is reached during the primary step, coefficients aj and b approach π-1/2 (or re(πDS)-1/2, in the dimensional form) and 0, respectively. Thus, the value of the slope of the linear dependence of IRP/IDC vs tP-1/2 can be used for the determination of primary substrate diffusion coefficient, interestingly, without the necessity of knowledge of the substrate concentration. Deficiency or Absence of Supporting Electrolyte. The results presented in this and the following paragraphs have been obtained for reverse pulse voltammetry with waiting period. Figure 3a-d presents the dependencies of IRP/IDC vs dimensionless pulse time for selected values of support ratio, and for the four schemes of electrode reactions considered (I-IV, respectively). We have chosen the number 10-4 as the lower limit of support ratio in our calculations, just because, in many real systems with the analyte at a level of 10-3 M or less, the concentration of unwanted ionic impurities, including those produced in the process of dissociation of the solvent, exceeds 10-7 M.22 Therefore, the case of support ratio equal to 10-4 can be treated as the system in which no ionic support was added externally. For low support ratio, or in the absence of deliberately added supporting electrolyte, the values of either IRP/IDC or E1/2RP differ from the appropriate values found for purely diffusional transport. In other words, under such conditions, the reverse pulse voltammograms exhibit a sensitivity with respect to both the amount of inert electrolyte in the system and the type of electrode process that takes place at the microelectrode. The specific results obtained for different classes of the electrode processes are discussed below. Class I. The changes in IRP/IDC with pulse time for various support ratios are presented qualitatively in Figure 3a. It is seen that for class I this ratio remains practically unchanged for a wide range of supporting electrolyte concentration. Under the conditions of low support ratio, the transport of uncharged substrate is not directly influenced by migration18 in the primary process. The (22) Hyk, W.; Stojek, Z. J. Electroanal. Chem. 1997, 422, 179.

Figure 3. Theoretical dependencies of reverse-pulse currents ratio, |IRP/IDC|, with respect to dimensionless pulse time for various support ratios: (1) 0.0001, (2) 0.001, (3) 0.01, (4) 0.1, (5) 1, (6) 10, (7) 100. (a-d) refer to electrode processes of classes I-IV, respectively. Note: for class I (a), numbers 1-6 refer to support ratios equal to 0.005, 0.05, 0.5, 5, 50, and 500, respectively.

steady-state, or rather quasi-steady-state, generating current, as well as the concentration profiles of electroactive species and conductivity and ohmic drop in the depletion layer, has values similar to those for excess supporting electrolyte. Therefore, the transport of the charged product, which becomes the substrate in the reverse step, is not influenced by migration. However, in the reverse step, the direction of the counterion flux has to be changed. This flux reorientation is the source of an extra potential drop, particularly when the supporting ions are present at low level in the bulk of the solution. Therefore, a shift in E1/2RP is observed. Obviously, this shift increases with a decrease in electrolyte concentration and for the lowest support ratio considered (0.005) is approximately equal to 3RT/F vs the standard potential. For a small support ratio and short pulse times, the reverse pulse voltammograms (RPV-WP) are not well defined and therefore it is difficult to measure the wave height. We attribute this to slow transport of the counterion and the existence of the corresponding ohmic drop.18 Therefore, more reasonable parameters are obtained from reverse pulse voltammograms calculated or recorded for longer pulse times (greater than 2re2/DS, i.e., 29 ms if re ) 5.5 µm and DS ) 2.1 × 10-9 m2/s).

Classes II, III, and IV. The changes of IRP/IDC with pulse time for various support ratios are presented graphically in Figure 3b-d. It is seen that for a given pulse time this ratio changes with respect to concentration of supporting electrolyte. The shorter pulse time and the smaller support ratio the more significant differences are observed, and thus one can easier distinguish these classes of electrode reactions. For instance, for pulse time equal to 0.5re2/DS (i.e., 7 ms if re ) 5.5 µm and DS ) 2.1 × 10-9 m2/s) and for the smallest support ratio considered (1 × 10-4), the ratio IRP/IDC is 0.932 (class II), 0.515 (class III), and 0.881 (class IV). We want to point out here that for very short pulse times the assumption of neglecting the double-layer effects is rather poor. Under the conditions of low support ratio, including the absence of supporting electrolyte, the transport of a charged substrate is strongly influenced by migration12,19 in the primary process. The steady-state, or rather quasi-steady-state, generating currents, and thus the concentration profiles of electroactive species, are different compared to the excess electrolyte case. Depending on the charge of the substrate and the sign of the potential applied to the electrode, and thus the nature of the electrode process, these currents (the fluxes) are either larger Analytical Chemistry, Vol. 70, No. 24, December 15, 1998

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Table 1. Coefficients of Linear Regression (see Eq 7) for Electrode Processes of Class I

a

support ratio

aj

b

0.005a 0.05 0.5 5 50 500

0.472 0.514 0.524 0.530 0.531 0.532

-0.040 -0.048 -0.051 -0.050 -0.050 -0.050

Reverse pulse voltammetric wave is not well defined.

Table 2. Coefficients of Linear Regression (See Eq 7) for Electrode Processes of Classes II, III, and IV type of electrode process II

III

IV

support ratio

aj

b

aj

b

aj

b

0.0001 0.001 0.01 0.1 1 10 100

0.707 0.708 0.718 0.727 0.628 0.547 0.533

-0.068 -0.068 -0.069 -0.070 -0.060 -0.052 -0.051

0.390 0.390 0.391 0.400 0.452 0.515 0.530

-0.036 -0.036 -0.036 -0.037 -0.043 -0.049 -0.050

0.667 0.667 0.667 0.662 0.629 0.559 0.535

-0.062 -0.062 -0.061 -0.061 -0.059 -0.053 -0.051

or smaller than those for excess supporting electrolyte. Such a situation influences the transport of the product, which becomes the substrate in the reverse step. Therefore, the electrochemical response in this step, i.e., the magnitude of the reverse pulse current, is different for various types of electrode processes. For classes II, III, and IV, no shift in E1/2RP is observed because, even if no supporting electrolyte was added, the substrate is ionic, and therefore the conductivity of the system is sufficiently large. The value of E1/2RP is equal to E°. Generalizations. It has been found that, for any support ratio and for any type of electrode process, the IRP/IDC parameter changes linearly with the reciprocal of the square root of the pulse time according to eq 7. The obtained coefficients (aj , b) from the fitting procedure for the types of processes considered are given in Tables 1 and 2. The correlation coefficients departed from 1 by less than 0.0001. The results for the first type of electrode processes are presented in a separate table because, in contrast to other types, the calculations in this case were performed for different values of support ratio. For class I, it is seen that coefficients aj and b are practically constant. Therefore, for any support ratio, the value of diffusion coefficient of uncharged primary substrate can be extracted from the value of aj , analogously to the excess supporting electrolyte case. For classes II, III, and IV coefficients aj and b are functions of the support ratio. Thus, in these cases, the value of aj gives information on the contribution of migration (and on support ratio) to the transport of the reactants. If the determined value of coefficient aj , for sufficiently long generating time, is greater (less) than π-1/2, the primary substrate is attracted to (repelled from) the electrode surface, or in other words, the total flux of the substrate is increased (decreased). Comparison with Experiment. To confirm our predictions, we did some experimental measurements. The process that we chose for experiments was the oxidation of ferrocene, Fc/Fc+. 5242 Analytical Chemistry, Vol. 70, No. 24, December 15, 1998

Figure 4. Experimental (open circles) and calculated (solid lines) dimensionless reverse pulse voltammograms for two values of support ratio: 0.002 (a) and 20 (b). Primary electrode process is oxidation of ferrocene (class I). Generating time 3 s (EG ) 0.8 V), pulse time 50 ms, waiting time 2 s (Efinal ) -0.2 V), potential step 0.02 V, bulk concentration of ferrocene 2.5 mM, and radius of disk microelectrode 5.5 µm.

Ferrocene gives a one-electron Nernstian anodic wave at a platinum microelectrode and, therefore, can be treated as a model system for class I considered in this work. Since disk microelectrodes were used in the measurements, and the model was developed for hemispherical geometry, to compare the data, we employed in the calculations a hemisphere radius equal to 2rdisk/ π. The factor 2/π is absolutely correct for steady state. To verify its correctness for transient processes, we have compared the current-time expression derived for a microdisk electrode by Shoup and Szabo23 (I ) 4nFDcbrdisk[0.7854 + 0.8862τ-1/2 + 0.2146 exp(-0.7823τ-1/2)]), where τ ) 4Dt/(rdisk)2, with the appropriate expression for a hemispherical electrode15 (I ) 2πnFDcbrhemisph[1 + rhemisph/(πDt)1/2]). For the pulse time of 50 ms that has been used in the experiments, the calculated ratio of the currents flowing at appropriate disk and hemisphere microelectrodes departs from the factor 2/π by only 0.37%. This comparison has been done for diffusional transport; however, the corresponding number for mixed diffusional and migrational transport is similar.22 A full verification of the model presented with other redox systems is the subject of our next paper. Figure 4 presents experimental (open circles) and calculated (solid lines) dimensionless reverse pulse voltammograms (RPVWP) for two extreme values of support ratio, i.e., 0.002 (a) and 20 (b). It is seen that the experimental and calculated waves fit each other very well. The differences between the values of IRP/IDC obtained experimentally and theoretically do not exceed 1%. Applying the procedure for the determination of primary substrate diffusion coefficient described above, we have found the value of the ferrocene diffusion coefficient in acetonitrile; it equals 2.11 × 10-9 m2/s (T ) 20 °C). This value is in a very good agreement with the literature value24 (2.17 × 10-9 m2/s, T ) 25 °C, csupp.el.(TBAP) ) 0.2 M). (23) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237. (24) Baur, J. E.; Wightman, R. M. J. Electroanal. Chem. 1991, 305, 73.

CONCLUSIONS It is apparent that reverse pulse voltammetry with waiting period and double potential step chronoamperometry can be used for the quantitative determination of the migrational contribution to the current measured in the redox systems studied with microelectrodes. After detection of such contribution, it would be apparently incorrect to use the diffusional equations for the calculation of the diffusion coefficients, which is sometimes done in solid-state electrochemistry. The determination of migrational contribution to the reactants’ total transport, and the identification of the type of the electrode process studied, can be done by measuring the IRP/IDC ratio in a single experiment and then comparing it with the tabulated theoretical data. According to the numerical model that was presented in this paper, the choice of appropriate values for the parameters that characterize the reversal techniques (tG, tW, tP), is essential for making reliable and accurate conclusions. It has been found that the generating time should be long enough to reach the nearly steady state in the primary step. This means that this parameter should not be less than 100re2/DS (i.e., 1.5 s if re ) 5.5 µm and DS ) 2.1 × 10-9 m2/s). The length of the waiting period (rest time) can be less than that of the generating time. Regarding the choice of the pulse time length: there are two opposite tendencies which have to be taken into account. In reverse pulse voltammograms, on one hand, the value of the ratio IRP/IDC obtained for a very short pulse time is more sensitive to migration, but on the other hand, the influence of the effects related to the double layer becomes larger. For the first class of electrode processes, in addition to the tendencies mentioned above, another tendency is observed: for very short pulse times, the reverse pulse wave is not well defined, apparently due to insufficient transport of counterions to the electrode surface. Thus, the optimal values of

pulse time can be something in the range of 1- and 2.5re2/DS (i.e., 14.5 and 36 ms, if re ) 5.5 µm and DS ) 2.1 × 10-9 m2/s) for classes II, III, and IV and class I, respectively. For all classes of electrode processes and for any amount of supporting electrolyte in the system, the changes of the ratio IRP/ IDC vs the reciprocal of the square root of the pulse time are linear. This is a very valuable feature of RPV-WP. The value of the slope of such dependencies informs us about the migrational contribution to the reactants’ total transport and indirectly about the type of electrode reaction. For charged primary substrate, this value is either greater or less than that derived for purely diffusional transport, while for an uncharged primary substrate, these values are practically the same even for a very small support ratio. For an uncharged primary substrate, no matter what the value of the support ratio is, the diffusion coefficient can be easily determined by comparing the slopes of the IRP/IDC vs tP-1/2 dependencies obtained experimentally and derived theoretically. The model for migrational reverse pulse voltammetry and the procedure for the determination of primary substrate diffusivity have been compared and tested with a real redox system. It has been shown that the agreement is very good. ACKNOWLEDGMENT This work was supported by a grant from the University of Warsaw (BST 562/5/97). The computation time provided by the Interdisciplinary Centre of Modelling and the Department of Mathematics and Informatics of the University of Warsaw is gratefully acknowledged. Received for review July 9, 1998. Accepted September 17, 1998. AC980748O

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