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Determination of the chemical and electrochemical parameters for a CE system by methods of convolution electrochemistry
Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 563 (2004) 283–290 www.elsevier.com/locate/jelechem
Determination of the chemical and electrochemical parameters for a CE system by methods of convolution electrochemistry Y.I. Moharram
*
Chemistry Department, Faculty of Science, Tanta University, 31527 Tanta, Egypt Received 7 July 2003; accepted 6 September 2003
Abstract For a rapid electron transfer process which follows a reversible chemical step (CE), an analysis procedure is presented for the determination of chemical and electrochemical parameters. It is based on methods of convolution electrochemistry and can be applied to chronoamperometric curves. The main equations of the reported procedure are very simple and are easily analyzed. The analysis procedure has been applied to both simulated and experimental data. As an experimental example, the electroreduction of Cdþ2 in DMF containing 0.5 M TEAP at 25 °C was investigated. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Cadmium ion; Digital simulation; Voltammetry; Convolution; Rate constants
1. Introduction In the literature, the rapid electron transfer behaviour of Cdþ2 on a mercury cathode has been studied to examine new electroanalytical methods or equipment [1–4]. Several authors [5–8] supposed the existence of a chemical step (CE) preceding electron transfer for the electroreduction of Cdþ2 on a mercury cathode in aqueous media. In nonaqueous media, Bielger et al. [9] determined the kinetic parameters of the Cdþ2 /Cd(Hg) reaction in pure and mixed solvents e.g. (methanol + acetonitrile, and N-methylformamide + N ; N ;dimethylformamide). The rate determining step was shown to be the reduction of Cdþ2 to Cdþ2 and the rate constant for the system Cdþ2 /Cd was measured and ranged from 0.01 to 0.45 cm s1 . These authors suggested that there was no correlation between the rate constant and the available real Gibbs energies of solvation. They explained that this relation could be obscured by the absence of a Frumkin correction. Hills and Peter [10] studied the cyclic voltammetric measurements of the reduction of cadmium ion in dimethylsulphoxide. They *
Present address: Teacher College, P.O. 2375, Dammam 31451, Saudi Arabia. E-mail address: [email protected] (Y.I. Moharram). 0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.07.044
showed that the electrode process was controlled by a first order homogeneous reaction which preceded charge transfer. Fawcett and Lasia [11] measured the standard rate constant and apparent transfer coefficients for the electroreduction of Cdþ2 at Hg in DMF containing various concentrations of tetraalkylammonium salts as base electrolytes. They concluded that the model for amalgam formation in which ion transfer in the double layer is the slow step. Brisard and Lasia [12] studied the electroreduction of Cdþ2 in dimethylsulphoxide containing TEAP as the supporting electrolyte. The reduction mechanism revealed the presence of a homogeneous chemical reaction prior to the electron transfer. The kinetic parameters were determined using cyclic voltammetry with convolution and chronoamperometry. Recently, in our laboratories, El-Hallag et al. [13] studied the electrode reduction of Cdþ2 in aqueous NaNO3 solution as the supporting electrolyte. They applied convolution/deconvolution voltammetry combined with a digital simulation technique for determination of the relevant chemical and electrochemical parameters. Diagnostic criteria and methods of analysis were presented. The agreement between experimental and theoretical data indicated that the reduction of Cdþ2 at the mercury electrode proceeds via the CE mechanism [13,14]. Also, in our laboratories, a convolution method was applied to
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cyclic voltammetric data analysis for Cdþ2 /Cd(Hg) in dimethylsulphoxide at a mercury cathode [15]. The involvement of a CE preceding the electron transfer (CE) was examined via digital simulation studies. The homogenous and heterogeneous rate constants of the electroreduction of Cdþ2 /Cd(Hg) were estimated. The aim of the present paper is to determine chemical and electrochemical rate constants by convolution cyclic voltammetry combined with digital simulation. The methods of convolution electrochemistry have also been applied to chronoamperometric data. The application of the methods provides the opportunity to obtain accurate kinetic parameters in case of a rapid electron transfer process following a reversible CE. Experimental examples are presented using the electroreduction of Cdþ2 at the HMDE electrode in 0.5 M TEAP as supporting electrolyte in dimethylformamide.
2. Theory The objective of this section is to describe the treatment of electrochemical data for a reversible chemical reaction prior to an electron transfer step, CE using the convolution transform of the current [16–19]. This method is generally applicable to cyclic voltammetry and chronoamperometry. It was shown [16] that under diffusion control the convolution current can be given as Z t 1 IðuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; I1 ¼ pffiffiffi ð1Þ p 0 ðt uÞ where IðuÞ is the experimental current, u is a dummy variable, and t is the time calculated from the start of the current flow. It was demonstrated [20] that the kinetic convolution I2 can be expressed as Z t IðuÞekc ðtuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; I2 ¼ ð2Þ pðt uÞ 0 where kc is the homogenous chemical rate constant. The deconvoluted current data, dI1 /dt, are obtained by the action of a semidifferentiation operator on the convoluted current I1 . The form of the deconvoluted current–potential curves is predicted to display a minimum epc and a maximum epa corresponding to the cathodic and anodic waves, respectively. For a reversible electron transfer process epc and are equal epa and occur in potentials each of which equals the polarographic half-wave potential, E1=2 [21]. The values of the half-peak width ðWp ¼ 3:526RT =nF Þ ¼ 90:53=n mV at 25 °C [21]. A schematic illustration of the selected CE mechanism is given in Eqs. (3) and (4) as follows k1
X O; k1
ð3Þ
ks ;E0
O þ ne R;
ð4Þ
kc ¼ k1þ k1 ;
ð5Þ
K ¼ k1 =k1 ;
ð6Þ
DO ;DR
where ks is the heterogeneous rate constant, E0 is the standard electrode potential, kc is the homogeneous chemical rate constant, and K is the equilibrium constant. DO ; DR are the diffusion coefficients of the oxidized and reduced species, respectively, which are assumed to be equal for simplicity. It was shown [22] that the appropriate solution of diffusion equations via Laplace transformation in terms of I1 and I2 convolution of the current, gives the concentrations of each species X, O, and R at the surface which can be expressed as cXð0;tÞ ¼ cbulk X
I1 I2 ; nFSD1=2 ð1 þ KÞ
ð7Þ
cOð0;tÞ ¼ cbulk O
KI1 þ I2 ; nFSD1=2 ð1 þ KÞ
ð8Þ
I1 ; ð9Þ nFSD1=2 where n is the number of electrons in the electrochemical step, F is the Faraday constant, S is the electrode area and cð0;tÞ ; cbulk are the concentrations of the species at the surface and in the bulk of the solution, respectively. At the electrode surface, the Butler–Volmer relationship applies to the oxidized and the reduced species of the electrochemical step represented in Eq. (4) and can be given as cRð0;tÞ ¼ cbulk þ R
I ¼ nFSðkhf cOð0;tÞ khb cRð0;tÞ Þ:
ð10Þ
Hence, substituting Eqs. (8) and (9) for the surface concentrations of O and R in the above equation, we have ID1=2 KI þ I2 þ en ðIlim ¼ IOlim 1 R I1 Þ; khf 1þK
ð11Þ
where khf and khb are the forward and backward heter1=2 ogeneous rate constants, respectively, IOlim ¼ ncbulk O FSD lim bulk 1=2 and IR ¼ ncR FSD are the limiting values of the convolution current of the oxidized and reduced species, respectively, n ¼ ðE E0 ÞnF =RT , and the remaining symbols have their usual significance. When the potential is driven to a sufficiently extreme value past the wave, the value of en equals zero, khf tends to infinity and Eq. (11) reduces to IOlim ¼
KI1 þ I2 : 1þK
ð12Þ
For a CE mechanism, IOlim cannot be readily obtained since a rising plateau value or sweep rate dependence of the limiting value of the convolution current occurs.
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In chronoamperometry, the potential is kept constant. As a result Eq. (11) can be rewritten in the following form khf KI þ I2 khb I ¼ 1=2 IOlim 1 ð13Þ þ 1=2 ðIRlim I1 Þ: D 1þK D In the limiting case (at a sufficiently large overpotential, when khf > khb the second term of Eq. (13) can be neglected and then rearrangement of Eq. (13) gives the following linear relationship between the current I, and CE convolution current ðICE Þ
285
particular potential dependence of the electron transfer rate constant. In this respect, the technique lends itself to the study of such potential dependences and provides a means of assessing adherence to a kinetic model such that of the Butler–Volmer model. A second advantage lies in the fact that after a brief initial period there is no contribution to the observed current from the charging of the electrical double layer at the electrode surface.
3. Experimental
khf khf K ICE ; I ¼ 1=2 IOlim 1=2 D D 1þK
ð14Þ
3.1. Electrode, electrochemical cell, and instrumentation
ICE ¼ I1 þ I2 ;
ð15Þ
I ¼ a bICE :
ð16Þ
Electrochemical experiments were performed using a hanging mercury drop electrode (area ¼ 2.61 106 m2 ) model 303A connected to a Model 362 potentiostat/ galvanostat (from EG&G). Background data were stored and subtracted from the experimental data set, minimizing side effects such as double layer charging current. AgjAgCljKClsat: was used as the reference electrode and Pt-wire as the counter electrode. The stability of the reference electrode, 3 mV was checked periodically against a 1 mM ferrocene solution in the solvent of interest by measuring the peak potential, Ep , of Fcþ /Fc after the current–potential measurements for the compound under investigation were completed. Repeated experiments over a period of one week gave an Ep of Fcþ /Fc in DMF equal to 0.581 0.003 V vs. AgjAgCl. We chose the Fcþ /Fc system because it is of great interest in electrochemistry, since it is often chosen as a model system in electrochemical studies [24,25]. Internal resistance ohmic drop distortions were minimized by applying suitable positive feedback compensation. The electrochemical data were processed on a PC computer using the Condecon 300 package (from EG&G). The capture facilities of Condecon 300 software are such that compensation has to be made after data collection. This allows criteria for correct compensation to be assessed. Accordingly, software compensation was used throughout. All measurements were performed at a temperature of 25 °C. The same temperature was considered in the calculations. The scan rate was in the range from 0.02 to 5 V/s for all data reported. Chronoamperometric experiments were performed under the same conditions as the cyclic voltammetry. The duration of polarization, t, was 0.2 s and its lower limit was dictated by the time required to collect 500 data points. Its upper limit was imposed by the increasing contribution of convection within the electrolyte to the mass transport at long times. The current was filtered prior to capture to reduce the charging current associated with the capacitance of the electrical double layer in order to improve the signal to noise ratio. The theoretical methods of convolution electrochemistry
The simplicity of this relationship (Eq. (16)) makes this a very attractive route for data analysis. Here a (intercept), and b (gradient) are coefficients obtained by plotting the voltammetric current, I, against the summed term I1 þ I2 CE convolution current ICE a¼
khf lim I ; D1=2 O
ð17Þ
b¼
khf K : D1=2 1 þ K
ð18Þ
For a reversible CE (K ¼ 1) preceding an electron transfer process khf : ð19Þ D1=2 Eq. (14) indicates that in the presence of chemical complications, as in the CE reaction, a linear fit for the graph of I vs. ICE is obtained. This agrees well with the linearization procedure of Oldham [23] for a simple electron transfer. It then requires only an assessment of the diffusion coefficient to be made using the following equation
b ¼ 1=2
a ¼ 1=2IOlim ; b rffiffiffiffiffiffi a FS: D ¼ 2 ncbulk b O
ð20Þ ð21Þ
Eq. (21) is used successfully to calculate the value of the diffusion coefficient. Thus, the values of khf can be calculated directly from Eq. (19). Furthermore, the variation of the gradient (b) with overpotential is a route to ks and a and hence a test for the Butler–Volmer relationship. Thus, the application of convolution electrochemistry to chronoamperometric curves possesses some attractive aspects in the field of mechanistic analysis. It is simple to perform, and, because of the fixed potential at which the system is studied, there is no need to presuppose a
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were applied to the chronoamperometric curves using the EG & G Condecon software package. 3.2. Chemicals and solvents Reagent grade chemicals and thrice distilled mercury were employed in all cases. Solutions of Cdþ2 (0.1 mol m3 ) in 0.5 M TEAP as supporting electrolyte in dimethylformamide were investigated. Working solutions were degassed thoroughly with oxygen-free nitrogen and a nitrogen atmosphere was maintained above the solution throughout the course of the measurements. For several experiments, extra care was taken to exclude residual water from the DMF solvent by using methods of purification of the solvent [26].
4. Results and discussion 4.1. Direct comparison of simulation/experimental voltammograms A cyclic voltammogram of the electroreduction of Cdþ2 at the HMDE in DMF + 0.5 M TEAP electrolyte solution at a temperature of 25 °C and a scan rate of 1 V/s, after applying background subtraction and correction for uncompensated resistance is shown in Fig. 1. Superimposed on this is a digital simulation for a CE reaction. At this scan rate the following parameters were used to produce the best fit. The standard electrode potential for the redox couple, E° ¼ 1:03 V, the area of the electrode ¼ 2.61 106 m2 , the diffusion coefficient of the species DðDO ; DR Þ ¼ 3 1010 m2 /s, the number of electrons in
Fig. 1. Theoretical (dotted), and experimental (solid) lines for CV scans of Cdþ2 in DMF containing 0.5 M TEAP at HMDE ðv ¼ 1 V/s). Simulation parameters employed were: E° ¼ 1:03 V, S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
the electrochemical step, n ¼ 2, and the heterogeneous rate constant, ks ¼ 7 103 m/s. Simulating this situation (a CE reaction) gave for the homogeneous chemical rate constant, kc ¼ 6 s1 , an almost perfect fit which is a direct validation of parameters, noting at present that the electron transfer is reversible. The heterogeneous rate constant, ks was determined from a table of the peak to peak separation ðDEp Þ between the forward and backward scans generated by simulation. The shape of the voltammogram is quite insensitive to the value of the transfer coefficient a as reported in the literature [27] if the value of ks is greater than 5 103 m/s so the correct value of a cannot be estimated via simulation. Fig. 2 shows a plot of Ip =ðvÞ1=2 vs. scan rate, v, for experimental and simulated data. The values of Ip =ðvÞ1=2 decrease with increasing sweep rate as reported in Table 1. Also the cathodic to anodic peak current ratio, Ipc =Ipa changes with variation of the scan rate and is not equal to unity, which confirms the CE reaction. The values of DEp (Table 1) indicates that the electroreduction of Cdþ2 ions occurs through a two electron reversible change. 4.2. Convolution and deconvolution voltammetry The convolution transform [16–21] has been shown to be effective in testing the adherence of a system to a chosen electron transfer model and can lead to adequate parameter assessment from experimental data. In this system the following are noted: (a) The I1 convolution does not maintain a constant potential past the wave but returns to zero on the reverse scan as cRð0;tÞ is proportional to I1 (see Eq. (9)). Fig. 3 shows that the rate of
Fig. 2. Plot of Ip =ðvÞ1=2 vs. scan rate, v for experimental (.......) and simulated (––––) data.
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287
Table 1 Comparison between the experimental and the simulated peak characteristics at different scan rates v/V s1
0.02 0.05 0.1 0.2 0.5 1
Ip =A
Ip v1=2 /AV1=2 s1=2
DEp /mV
Wp /mV
epc =epa
CV
Sim
CV
Sim
CV
Sim
CV
Sim
CV
Sim
0.201 0.289 0.358 0.448 0.658 0.899
0.211 0.287 0.361 0.437 0.645 0.890
1.421 1.290 1.133 1.002 0.930 0.899
1.492 1.281 1.142 0.978 0.912 0.890
31 31 31 31 32 32
32 32 30 31 33 32
46 47 47 47 46 46
47 48 46 46 47 47
0.56 0.59 0.63 0.67 0.70 0.77
0.55 0.53 0.61 0.67 0.71 0.76
Simulation parameters employed were: E° ¼ 1:03 V, S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
change of I1 with potential reaches a maximum in the region of E ¼ E1=2 but falls as the rate of production of O becomes the rate-determining step. The I1 convolution will not reach a plateau value however, until cXð0;tÞ þ cOð0;tÞ ¼ 0. This confirms the CE reaction.
Fig. 3. I1 convolution of a 0.1 mol m3 solution of Cdþ2 ions in DMF containing 0.5 M TEAP at HMDE (v ¼ 1 V/s).
(b) Fig. 3 also shows the close overlay on the plot for the two sweep halves, which indicates reversible behaviour of the electron transfer process. The logarithmic analysis
Fig. 4. Deconvoluted current data dI1 =dt of the investigated 0.1 mol m3 solution of Cdþ2 ions in DMF containing 0.5 M TEAP at HMDE ðv ¼ 1 V/s).
Table 2 Values of I1B I1A , I2A I2B , K, and kc of experimental and simulated results kc /s1
102 ðI1B I1A Þ/A s1=2
102 ðI2A I2B Þ/A s1=2
K
CV
CV
Sim
CV
Sim
CV
Sim
(a) At scan rates 1 and 0.5 V/s 0.1 4.682 5.127 1.0 3.0 6.0 8.0 10
6.063 5.624 5.243 4.692 4.387 4.176
6.859 6.409 5.889 5.292 4.974 4.682
1.2950 1.2012 1.1198 1.0021 0.9369 0.8919
1.3380 1.2501 1.1483 1.0322 0.9702 0.9132
3.20
3.00
(b) At scan rates 0.1 and 0.05 V/s 0.1 3.179 3.754 1.0 3.0 6.0 8.0 10
4.266 4.071 3.656 3.192 2.845 2.709
5.152 4.884 4.467 3.829 3.499 3.349
1.3420 1.2807 1.1500 1.0041 0.8950 0.8520
1.3723 1.3010 1.1900 1.0201 0.9321 0.8921
3.15
3.00
Sim
1010 D/m 2 s1
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of the convoluted data is tested for different reaction schemes (Erev , EC, CE. . .Þ [22]. It is found that the data for CV waves follow the following Eq. (22) satisfactorily for a CE reaction RT RT I lim I1 ln k1 þ ln : ð22Þ 2nF nF I lim Linearity of the logarithmic plot and coincidence of the forward and reverse sweeps over a wide range of potential sweep rates provide a rigorous test method E ¼ E0
of the adherence of a system to a CE scheme. The plot of the relation lnðI lim I1 Þ=I lim vs. E is significant in that all the available data obtained experimentally are tested. Good straight lines are obtained for the two sweep halves with a gradient equal to 13 mV, which indicates that the electroreduction of Cd2þ ions is a reversible two electron transfer process. From the intercept of Eq. (22), a value of E1=2 of )1.03 0.02 V was found. (c) Deconvoluted current data, dI1 =dt in Fig. 4, show the expected deviation in the cathodic to anodic deconvoluted current ratios epc =epa in a manner which varies with scan rate. Table 2 also shows that, with increasing scan rate, the epc =epa ratios increase which is entirely consistent with a CE reaction. The values of the half-peak width ðWp Þ of the deconvoluted current data, dI1 =dt (Table 2) give the expected theoretical value for a reversible two electron transfer process (45.27 mV at 25 °C) [21]. The cathodic deconvoluted current peak potential is located at the same potential as the anodic peak. This potential is E1=2 [21]. The value of E1=2 ()1.03 V) obtained by this method is in good agreement with those values obtained from digital simulation. This observation confirms the rapid electron transfer process. 4.3. Determination of kc and K and D for a CE mechanism Considering the two sets of data at different scan rates, the relationship (12) applies to both data sets. If we use the same test values of the homogeneous chemical rate constant, kc to evaluate the kinetic convolution current, I2 , the value of the equilibrium constant K
Fig. 5. Plots of K vs. kc (test) for: (a) experimental (.......) and (b) simulated (––––) data for a CE system obtained for Cdþ2 in DMF containing 0.5 M TEAP at HMDE at scan rates: (i) 1 and 0.5 V/s and (ii) at scan rates 0.1 and 0.05 V/s Simulation parameters employed were: S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
Fig. 6. Dependence of experimental (a) and simulated (b) voltammetric current I and the CE convolution current, ICE according to Eq. (16) in chronoamperometry at E ¼ 0:88 V for Cdþ2 in DMF containing 0.5 M TEAP at the HMDE. Simulation parameters employed were: S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 6:8 103 m/s, kc ¼ 6 s1 and t ¼ 0:2 s.
Y.I. Moharram / Journal of Electroanalytical Chemistry 563 (2004) 283–290
289
Table 3 Analysis of the current I and the CE convolution current, ICE according to Eq. (16) )E/V
0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80
)ln khf
105 khf /m s1
1010 D/m2 s1
106 intercept/A
Gradient/s1=2
CV
Sim
CV
Sim
CV
Sim
CV
Sim
CV
Sim
1.037 0.771 0.574 0.427 0.317 0.236 0.176 0.126 0.097
1.253 0.866 0.647 0.479 0.355 0.277 0.209 0.152 0.115
2.231 1.659 1.234 0.917 0.683 0.508 0.378 0.271 0.208
2.882 1.990 1.488 1.103 0.817 0.637 0.481 0.348 0.263
8.26 6.15 4.57 3.39 2.53 1.88 1.39 1.00 0.77
9.98 6.89 5.15 3.82 2.83 2.21 1.67 1.21 0.91
9.40 9.69 9.99 10.29 10.59 10.88 11.18 11.51 11.77
9.21 9.58 9.87 10.17 10.47 10.72 11.00 11.32 11.60
3.43
3.00
Simulation parameters employed were: D ¼ 3:00 1010 m2 /s, ks ¼ 7 103 m s1 , n ¼ 2; t ¼ 0:2 s, and c ¼ 0:1 mol m3 .
should also be the same. As IOlim is a constant, the value of K corresponding to this particular test value kc can be obtained via K ¼ I2A IB2 = I1B IA ð23Þ 1 : Thus, by varying the test value kc , a series of K can be calculated. A similar treatment can be performed on another pair of data sets under different scan rates. It is proved that the plots of K vs. kc should cross exactly at the point where the true values of K and kc are located (Fig. 5). The intersections yield the true values of K ¼ 1 and kc ¼ 6 s1 , which are exactly the same as those entered for the simulation (Fig. 5). This in turns leads to the knowledge of IOlim via Eq. (12) and subsequently the diffusion coefficient of species O (Table 2). The values of I2A IB2 ; I1B IA 1 , K, D and kc of two different sets of experimental and simulated data at scan rates 1 and 0.5 and 0.5 and 0.1 mV/s are reported in Table 2.
4.4. Method of convolution electrochemistry applied to chronoamperometric curves Chronoamperometric experiments were performed on a HMDE (electrode area, S ¼ 2:61 106 m2 ) in DMF + 0.5 M TEAP electrolyte solution at a temperature of 25 °C. The concentration of electroactive species was 0.1 mol m3 and the duration of polarization, t, was 0.2 s. Data capture and subsequent treatment were facilitated by the EG&G Condecon 300 software. The characteristic linear relationship between the current I, and the CE convolution current, ICE Eq. (16) is evident in Fig. 6. Table 3, lists the intercepts (a) and the gradients (b) at various overpotentials. The diffusion coefficient has been calculated successfully from Eq. (21). The values of diffusion coefficients (Table 3) are in good agreement with those values previously obtained from convolution voltammetry. The potential dependent rate constants khf are calculated from Eq. (19). Volmer behaviour was proved by analyzing the ln khf vs. E dependence within the potential range studied, as is shown in Fig. 7. Good linearity is observed, and from the gradient and intercept, the values a ¼ 0:35 0:01 and ks ¼ 6:8 103 0:3 m s1 were determined, respectively. The value of the heterogeneous rate constant, ks obtained, agrees very well with the value used in the simulation.
5. Conclusion
Fig. 7. Plot of ln khf vs. E for experimental (a) and simulated (b) data for the reduction of Cdþ2 in DMF containing 0.5 M TEAP at the HMDE. The khf values employed are those obtained from Table 3.
Our study allows the extraction of chemical and electrochemical parameters in the case of a CE mechanism via convolution electrochemistry. Digital simulation is found to give good results confirming the accuracy of the proposed method. The convolution procedure can also be applied also to chronoamperometric curves. A linear relationship between the current I, and CE convolution current (ICE Þ is evident. The simplicity of this relationship makes this a very attractive route for data analysis. Experimental verification on
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a real system was carried out with the reduction of Cdþ2 in DMF containing 0.5 M TEAP at the HMDE.
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Electroanalytical Chemistry Journal of Electroanalytical Chemistry 563 (2004) 283–290 www.elsevier.com/locate/jelechem
Determination of the chemical and electrochemical parameters for a CE system by methods of convolution electrochemistry Y.I. Moharram
*
Chemistry Department, Faculty of Science, Tanta University, 31527 Tanta, Egypt Received 7 July 2003; accepted 6 September 2003
Abstract For a rapid electron transfer process which follows a reversible chemical step (CE), an analysis procedure is presented for the determination of chemical and electrochemical parameters. It is based on methods of convolution electrochemistry and can be applied to chronoamperometric curves. The main equations of the reported procedure are very simple and are easily analyzed. The analysis procedure has been applied to both simulated and experimental data. As an experimental example, the electroreduction of Cdþ2 in DMF containing 0.5 M TEAP at 25 °C was investigated. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Cadmium ion; Digital simulation; Voltammetry; Convolution; Rate constants
1. Introduction In the literature, the rapid electron transfer behaviour of Cdþ2 on a mercury cathode has been studied to examine new electroanalytical methods or equipment [1–4]. Several authors [5–8] supposed the existence of a chemical step (CE) preceding electron transfer for the electroreduction of Cdþ2 on a mercury cathode in aqueous media. In nonaqueous media, Bielger et al. [9] determined the kinetic parameters of the Cdþ2 /Cd(Hg) reaction in pure and mixed solvents e.g. (methanol + acetonitrile, and N-methylformamide + N ; N ;dimethylformamide). The rate determining step was shown to be the reduction of Cdþ2 to Cdþ2 and the rate constant for the system Cdþ2 /Cd was measured and ranged from 0.01 to 0.45 cm s1 . These authors suggested that there was no correlation between the rate constant and the available real Gibbs energies of solvation. They explained that this relation could be obscured by the absence of a Frumkin correction. Hills and Peter [10] studied the cyclic voltammetric measurements of the reduction of cadmium ion in dimethylsulphoxide. They *
Present address: Teacher College, P.O. 2375, Dammam 31451, Saudi Arabia. E-mail address: [email protected] (Y.I. Moharram). 0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.07.044
showed that the electrode process was controlled by a first order homogeneous reaction which preceded charge transfer. Fawcett and Lasia [11] measured the standard rate constant and apparent transfer coefficients for the electroreduction of Cdþ2 at Hg in DMF containing various concentrations of tetraalkylammonium salts as base electrolytes. They concluded that the model for amalgam formation in which ion transfer in the double layer is the slow step. Brisard and Lasia [12] studied the electroreduction of Cdþ2 in dimethylsulphoxide containing TEAP as the supporting electrolyte. The reduction mechanism revealed the presence of a homogeneous chemical reaction prior to the electron transfer. The kinetic parameters were determined using cyclic voltammetry with convolution and chronoamperometry. Recently, in our laboratories, El-Hallag et al. [13] studied the electrode reduction of Cdþ2 in aqueous NaNO3 solution as the supporting electrolyte. They applied convolution/deconvolution voltammetry combined with a digital simulation technique for determination of the relevant chemical and electrochemical parameters. Diagnostic criteria and methods of analysis were presented. The agreement between experimental and theoretical data indicated that the reduction of Cdþ2 at the mercury electrode proceeds via the CE mechanism [13,14]. Also, in our laboratories, a convolution method was applied to
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cyclic voltammetric data analysis for Cdþ2 /Cd(Hg) in dimethylsulphoxide at a mercury cathode [15]. The involvement of a CE preceding the electron transfer (CE) was examined via digital simulation studies. The homogenous and heterogeneous rate constants of the electroreduction of Cdþ2 /Cd(Hg) were estimated. The aim of the present paper is to determine chemical and electrochemical rate constants by convolution cyclic voltammetry combined with digital simulation. The methods of convolution electrochemistry have also been applied to chronoamperometric data. The application of the methods provides the opportunity to obtain accurate kinetic parameters in case of a rapid electron transfer process following a reversible CE. Experimental examples are presented using the electroreduction of Cdþ2 at the HMDE electrode in 0.5 M TEAP as supporting electrolyte in dimethylformamide.
2. Theory The objective of this section is to describe the treatment of electrochemical data for a reversible chemical reaction prior to an electron transfer step, CE using the convolution transform of the current [16–19]. This method is generally applicable to cyclic voltammetry and chronoamperometry. It was shown [16] that under diffusion control the convolution current can be given as Z t 1 IðuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; I1 ¼ pffiffiffi ð1Þ p 0 ðt uÞ where IðuÞ is the experimental current, u is a dummy variable, and t is the time calculated from the start of the current flow. It was demonstrated [20] that the kinetic convolution I2 can be expressed as Z t IðuÞekc ðtuÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; I2 ¼ ð2Þ pðt uÞ 0 where kc is the homogenous chemical rate constant. The deconvoluted current data, dI1 /dt, are obtained by the action of a semidifferentiation operator on the convoluted current I1 . The form of the deconvoluted current–potential curves is predicted to display a minimum epc and a maximum epa corresponding to the cathodic and anodic waves, respectively. For a reversible electron transfer process epc and are equal epa and occur in potentials each of which equals the polarographic half-wave potential, E1=2 [21]. The values of the half-peak width ðWp ¼ 3:526RT =nF Þ ¼ 90:53=n mV at 25 °C [21]. A schematic illustration of the selected CE mechanism is given in Eqs. (3) and (4) as follows k1
X O; k1
ð3Þ
ks ;E0
O þ ne R;
ð4Þ
kc ¼ k1þ k1 ;
ð5Þ
K ¼ k1 =k1 ;
ð6Þ
DO ;DR
where ks is the heterogeneous rate constant, E0 is the standard electrode potential, kc is the homogeneous chemical rate constant, and K is the equilibrium constant. DO ; DR are the diffusion coefficients of the oxidized and reduced species, respectively, which are assumed to be equal for simplicity. It was shown [22] that the appropriate solution of diffusion equations via Laplace transformation in terms of I1 and I2 convolution of the current, gives the concentrations of each species X, O, and R at the surface which can be expressed as cXð0;tÞ ¼ cbulk X
I1 I2 ; nFSD1=2 ð1 þ KÞ
ð7Þ
cOð0;tÞ ¼ cbulk O
KI1 þ I2 ; nFSD1=2 ð1 þ KÞ
ð8Þ
I1 ; ð9Þ nFSD1=2 where n is the number of electrons in the electrochemical step, F is the Faraday constant, S is the electrode area and cð0;tÞ ; cbulk are the concentrations of the species at the surface and in the bulk of the solution, respectively. At the electrode surface, the Butler–Volmer relationship applies to the oxidized and the reduced species of the electrochemical step represented in Eq. (4) and can be given as cRð0;tÞ ¼ cbulk þ R
I ¼ nFSðkhf cOð0;tÞ khb cRð0;tÞ Þ:
ð10Þ
Hence, substituting Eqs. (8) and (9) for the surface concentrations of O and R in the above equation, we have ID1=2 KI þ I2 þ en ðIlim ¼ IOlim 1 R I1 Þ; khf 1þK
ð11Þ
where khf and khb are the forward and backward heter1=2 ogeneous rate constants, respectively, IOlim ¼ ncbulk O FSD lim bulk 1=2 and IR ¼ ncR FSD are the limiting values of the convolution current of the oxidized and reduced species, respectively, n ¼ ðE E0 ÞnF =RT , and the remaining symbols have their usual significance. When the potential is driven to a sufficiently extreme value past the wave, the value of en equals zero, khf tends to infinity and Eq. (11) reduces to IOlim ¼
KI1 þ I2 : 1þK
ð12Þ
For a CE mechanism, IOlim cannot be readily obtained since a rising plateau value or sweep rate dependence of the limiting value of the convolution current occurs.
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In chronoamperometry, the potential is kept constant. As a result Eq. (11) can be rewritten in the following form khf KI þ I2 khb I ¼ 1=2 IOlim 1 ð13Þ þ 1=2 ðIRlim I1 Þ: D 1þK D In the limiting case (at a sufficiently large overpotential, when khf > khb the second term of Eq. (13) can be neglected and then rearrangement of Eq. (13) gives the following linear relationship between the current I, and CE convolution current ðICE Þ
285
particular potential dependence of the electron transfer rate constant. In this respect, the technique lends itself to the study of such potential dependences and provides a means of assessing adherence to a kinetic model such that of the Butler–Volmer model. A second advantage lies in the fact that after a brief initial period there is no contribution to the observed current from the charging of the electrical double layer at the electrode surface.
3. Experimental
khf khf K ICE ; I ¼ 1=2 IOlim 1=2 D D 1þK
ð14Þ
3.1. Electrode, electrochemical cell, and instrumentation
ICE ¼ I1 þ I2 ;
ð15Þ
I ¼ a bICE :
ð16Þ
Electrochemical experiments were performed using a hanging mercury drop electrode (area ¼ 2.61 106 m2 ) model 303A connected to a Model 362 potentiostat/ galvanostat (from EG&G). Background data were stored and subtracted from the experimental data set, minimizing side effects such as double layer charging current. AgjAgCljKClsat: was used as the reference electrode and Pt-wire as the counter electrode. The stability of the reference electrode, 3 mV was checked periodically against a 1 mM ferrocene solution in the solvent of interest by measuring the peak potential, Ep , of Fcþ /Fc after the current–potential measurements for the compound under investigation were completed. Repeated experiments over a period of one week gave an Ep of Fcþ /Fc in DMF equal to 0.581 0.003 V vs. AgjAgCl. We chose the Fcþ /Fc system because it is of great interest in electrochemistry, since it is often chosen as a model system in electrochemical studies [24,25]. Internal resistance ohmic drop distortions were minimized by applying suitable positive feedback compensation. The electrochemical data were processed on a PC computer using the Condecon 300 package (from EG&G). The capture facilities of Condecon 300 software are such that compensation has to be made after data collection. This allows criteria for correct compensation to be assessed. Accordingly, software compensation was used throughout. All measurements were performed at a temperature of 25 °C. The same temperature was considered in the calculations. The scan rate was in the range from 0.02 to 5 V/s for all data reported. Chronoamperometric experiments were performed under the same conditions as the cyclic voltammetry. The duration of polarization, t, was 0.2 s and its lower limit was dictated by the time required to collect 500 data points. Its upper limit was imposed by the increasing contribution of convection within the electrolyte to the mass transport at long times. The current was filtered prior to capture to reduce the charging current associated with the capacitance of the electrical double layer in order to improve the signal to noise ratio. The theoretical methods of convolution electrochemistry
The simplicity of this relationship (Eq. (16)) makes this a very attractive route for data analysis. Here a (intercept), and b (gradient) are coefficients obtained by plotting the voltammetric current, I, against the summed term I1 þ I2 CE convolution current ICE a¼
khf lim I ; D1=2 O
ð17Þ
b¼
khf K : D1=2 1 þ K
ð18Þ
For a reversible CE (K ¼ 1) preceding an electron transfer process khf : ð19Þ D1=2 Eq. (14) indicates that in the presence of chemical complications, as in the CE reaction, a linear fit for the graph of I vs. ICE is obtained. This agrees well with the linearization procedure of Oldham [23] for a simple electron transfer. It then requires only an assessment of the diffusion coefficient to be made using the following equation
b ¼ 1=2
a ¼ 1=2IOlim ; b rffiffiffiffiffiffi a FS: D ¼ 2 ncbulk b O
ð20Þ ð21Þ
Eq. (21) is used successfully to calculate the value of the diffusion coefficient. Thus, the values of khf can be calculated directly from Eq. (19). Furthermore, the variation of the gradient (b) with overpotential is a route to ks and a and hence a test for the Butler–Volmer relationship. Thus, the application of convolution electrochemistry to chronoamperometric curves possesses some attractive aspects in the field of mechanistic analysis. It is simple to perform, and, because of the fixed potential at which the system is studied, there is no need to presuppose a
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were applied to the chronoamperometric curves using the EG & G Condecon software package. 3.2. Chemicals and solvents Reagent grade chemicals and thrice distilled mercury were employed in all cases. Solutions of Cdþ2 (0.1 mol m3 ) in 0.5 M TEAP as supporting electrolyte in dimethylformamide were investigated. Working solutions were degassed thoroughly with oxygen-free nitrogen and a nitrogen atmosphere was maintained above the solution throughout the course of the measurements. For several experiments, extra care was taken to exclude residual water from the DMF solvent by using methods of purification of the solvent [26].
4. Results and discussion 4.1. Direct comparison of simulation/experimental voltammograms A cyclic voltammogram of the electroreduction of Cdþ2 at the HMDE in DMF + 0.5 M TEAP electrolyte solution at a temperature of 25 °C and a scan rate of 1 V/s, after applying background subtraction and correction for uncompensated resistance is shown in Fig. 1. Superimposed on this is a digital simulation for a CE reaction. At this scan rate the following parameters were used to produce the best fit. The standard electrode potential for the redox couple, E° ¼ 1:03 V, the area of the electrode ¼ 2.61 106 m2 , the diffusion coefficient of the species DðDO ; DR Þ ¼ 3 1010 m2 /s, the number of electrons in
Fig. 1. Theoretical (dotted), and experimental (solid) lines for CV scans of Cdþ2 in DMF containing 0.5 M TEAP at HMDE ðv ¼ 1 V/s). Simulation parameters employed were: E° ¼ 1:03 V, S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
the electrochemical step, n ¼ 2, and the heterogeneous rate constant, ks ¼ 7 103 m/s. Simulating this situation (a CE reaction) gave for the homogeneous chemical rate constant, kc ¼ 6 s1 , an almost perfect fit which is a direct validation of parameters, noting at present that the electron transfer is reversible. The heterogeneous rate constant, ks was determined from a table of the peak to peak separation ðDEp Þ between the forward and backward scans generated by simulation. The shape of the voltammogram is quite insensitive to the value of the transfer coefficient a as reported in the literature [27] if the value of ks is greater than 5 103 m/s so the correct value of a cannot be estimated via simulation. Fig. 2 shows a plot of Ip =ðvÞ1=2 vs. scan rate, v, for experimental and simulated data. The values of Ip =ðvÞ1=2 decrease with increasing sweep rate as reported in Table 1. Also the cathodic to anodic peak current ratio, Ipc =Ipa changes with variation of the scan rate and is not equal to unity, which confirms the CE reaction. The values of DEp (Table 1) indicates that the electroreduction of Cdþ2 ions occurs through a two electron reversible change. 4.2. Convolution and deconvolution voltammetry The convolution transform [16–21] has been shown to be effective in testing the adherence of a system to a chosen electron transfer model and can lead to adequate parameter assessment from experimental data. In this system the following are noted: (a) The I1 convolution does not maintain a constant potential past the wave but returns to zero on the reverse scan as cRð0;tÞ is proportional to I1 (see Eq. (9)). Fig. 3 shows that the rate of
Fig. 2. Plot of Ip =ðvÞ1=2 vs. scan rate, v for experimental (.......) and simulated (––––) data.
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287
Table 1 Comparison between the experimental and the simulated peak characteristics at different scan rates v/V s1
0.02 0.05 0.1 0.2 0.5 1
Ip =A
Ip v1=2 /AV1=2 s1=2
DEp /mV
Wp /mV
epc =epa
CV
Sim
CV
Sim
CV
Sim
CV
Sim
CV
Sim
0.201 0.289 0.358 0.448 0.658 0.899
0.211 0.287 0.361 0.437 0.645 0.890
1.421 1.290 1.133 1.002 0.930 0.899
1.492 1.281 1.142 0.978 0.912 0.890
31 31 31 31 32 32
32 32 30 31 33 32
46 47 47 47 46 46
47 48 46 46 47 47
0.56 0.59 0.63 0.67 0.70 0.77
0.55 0.53 0.61 0.67 0.71 0.76
Simulation parameters employed were: E° ¼ 1:03 V, S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
change of I1 with potential reaches a maximum in the region of E ¼ E1=2 but falls as the rate of production of O becomes the rate-determining step. The I1 convolution will not reach a plateau value however, until cXð0;tÞ þ cOð0;tÞ ¼ 0. This confirms the CE reaction.
Fig. 3. I1 convolution of a 0.1 mol m3 solution of Cdþ2 ions in DMF containing 0.5 M TEAP at HMDE (v ¼ 1 V/s).
(b) Fig. 3 also shows the close overlay on the plot for the two sweep halves, which indicates reversible behaviour of the electron transfer process. The logarithmic analysis
Fig. 4. Deconvoluted current data dI1 =dt of the investigated 0.1 mol m3 solution of Cdþ2 ions in DMF containing 0.5 M TEAP at HMDE ðv ¼ 1 V/s).
Table 2 Values of I1B I1A , I2A I2B , K, and kc of experimental and simulated results kc /s1
102 ðI1B I1A Þ/A s1=2
102 ðI2A I2B Þ/A s1=2
K
CV
CV
Sim
CV
Sim
CV
Sim
(a) At scan rates 1 and 0.5 V/s 0.1 4.682 5.127 1.0 3.0 6.0 8.0 10
6.063 5.624 5.243 4.692 4.387 4.176
6.859 6.409 5.889 5.292 4.974 4.682
1.2950 1.2012 1.1198 1.0021 0.9369 0.8919
1.3380 1.2501 1.1483 1.0322 0.9702 0.9132
3.20
3.00
(b) At scan rates 0.1 and 0.05 V/s 0.1 3.179 3.754 1.0 3.0 6.0 8.0 10
4.266 4.071 3.656 3.192 2.845 2.709
5.152 4.884 4.467 3.829 3.499 3.349
1.3420 1.2807 1.1500 1.0041 0.8950 0.8520
1.3723 1.3010 1.1900 1.0201 0.9321 0.8921
3.15
3.00
Sim
1010 D/m 2 s1
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of the convoluted data is tested for different reaction schemes (Erev , EC, CE. . .Þ [22]. It is found that the data for CV waves follow the following Eq. (22) satisfactorily for a CE reaction RT RT I lim I1 ln k1 þ ln : ð22Þ 2nF nF I lim Linearity of the logarithmic plot and coincidence of the forward and reverse sweeps over a wide range of potential sweep rates provide a rigorous test method E ¼ E0
of the adherence of a system to a CE scheme. The plot of the relation lnðI lim I1 Þ=I lim vs. E is significant in that all the available data obtained experimentally are tested. Good straight lines are obtained for the two sweep halves with a gradient equal to 13 mV, which indicates that the electroreduction of Cd2þ ions is a reversible two electron transfer process. From the intercept of Eq. (22), a value of E1=2 of )1.03 0.02 V was found. (c) Deconvoluted current data, dI1 =dt in Fig. 4, show the expected deviation in the cathodic to anodic deconvoluted current ratios epc =epa in a manner which varies with scan rate. Table 2 also shows that, with increasing scan rate, the epc =epa ratios increase which is entirely consistent with a CE reaction. The values of the half-peak width ðWp Þ of the deconvoluted current data, dI1 =dt (Table 2) give the expected theoretical value for a reversible two electron transfer process (45.27 mV at 25 °C) [21]. The cathodic deconvoluted current peak potential is located at the same potential as the anodic peak. This potential is E1=2 [21]. The value of E1=2 ()1.03 V) obtained by this method is in good agreement with those values obtained from digital simulation. This observation confirms the rapid electron transfer process. 4.3. Determination of kc and K and D for a CE mechanism Considering the two sets of data at different scan rates, the relationship (12) applies to both data sets. If we use the same test values of the homogeneous chemical rate constant, kc to evaluate the kinetic convolution current, I2 , the value of the equilibrium constant K
Fig. 5. Plots of K vs. kc (test) for: (a) experimental (.......) and (b) simulated (––––) data for a CE system obtained for Cdþ2 in DMF containing 0.5 M TEAP at HMDE at scan rates: (i) 1 and 0.5 V/s and (ii) at scan rates 0.1 and 0.05 V/s Simulation parameters employed were: S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 7 103 m/s, kc ¼ 6 s1 .
Fig. 6. Dependence of experimental (a) and simulated (b) voltammetric current I and the CE convolution current, ICE according to Eq. (16) in chronoamperometry at E ¼ 0:88 V for Cdþ2 in DMF containing 0.5 M TEAP at the HMDE. Simulation parameters employed were: S ¼ 2:61 106 m2 , DðDO ; DR Þ ¼ 3 1010 m2 /s, n ¼ 2; ks ¼ 6:8 103 m/s, kc ¼ 6 s1 and t ¼ 0:2 s.
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289
Table 3 Analysis of the current I and the CE convolution current, ICE according to Eq. (16) )E/V
0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80
)ln khf
105 khf /m s1
1010 D/m2 s1
106 intercept/A
Gradient/s1=2
CV
Sim
CV
Sim
CV
Sim
CV
Sim
CV
Sim
1.037 0.771 0.574 0.427 0.317 0.236 0.176 0.126 0.097
1.253 0.866 0.647 0.479 0.355 0.277 0.209 0.152 0.115
2.231 1.659 1.234 0.917 0.683 0.508 0.378 0.271 0.208
2.882 1.990 1.488 1.103 0.817 0.637 0.481 0.348 0.263
8.26 6.15 4.57 3.39 2.53 1.88 1.39 1.00 0.77
9.98 6.89 5.15 3.82 2.83 2.21 1.67 1.21 0.91
9.40 9.69 9.99 10.29 10.59 10.88 11.18 11.51 11.77
9.21 9.58 9.87 10.17 10.47 10.72 11.00 11.32 11.60
3.43
3.00
Simulation parameters employed were: D ¼ 3:00 1010 m2 /s, ks ¼ 7 103 m s1 , n ¼ 2; t ¼ 0:2 s, and c ¼ 0:1 mol m3 .
should also be the same. As IOlim is a constant, the value of K corresponding to this particular test value kc can be obtained via K ¼ I2A IB2 = I1B IA ð23Þ 1 : Thus, by varying the test value kc , a series of K can be calculated. A similar treatment can be performed on another pair of data sets under different scan rates. It is proved that the plots of K vs. kc should cross exactly at the point where the true values of K and kc are located (Fig. 5). The intersections yield the true values of K ¼ 1 and kc ¼ 6 s1 , which are exactly the same as those entered for the simulation (Fig. 5). This in turns leads to the knowledge of IOlim via Eq. (12) and subsequently the diffusion coefficient of species O (Table 2). The values of I2A IB2 ; I1B IA 1 , K, D and kc of two different sets of experimental and simulated data at scan rates 1 and 0.5 and 0.5 and 0.1 mV/s are reported in Table 2.
4.4. Method of convolution electrochemistry applied to chronoamperometric curves Chronoamperometric experiments were performed on a HMDE (electrode area, S ¼ 2:61 106 m2 ) in DMF + 0.5 M TEAP electrolyte solution at a temperature of 25 °C. The concentration of electroactive species was 0.1 mol m3 and the duration of polarization, t, was 0.2 s. Data capture and subsequent treatment were facilitated by the EG&G Condecon 300 software. The characteristic linear relationship between the current I, and the CE convolution current, ICE Eq. (16) is evident in Fig. 6. Table 3, lists the intercepts (a) and the gradients (b) at various overpotentials. The diffusion coefficient has been calculated successfully from Eq. (21). The values of diffusion coefficients (Table 3) are in good agreement with those values previously obtained from convolution voltammetry. The potential dependent rate constants khf are calculated from Eq. (19). Volmer behaviour was proved by analyzing the ln khf vs. E dependence within the potential range studied, as is shown in Fig. 7. Good linearity is observed, and from the gradient and intercept, the values a ¼ 0:35 0:01 and ks ¼ 6:8 103 0:3 m s1 were determined, respectively. The value of the heterogeneous rate constant, ks obtained, agrees very well with the value used in the simulation.
5. Conclusion
Fig. 7. Plot of ln khf vs. E for experimental (a) and simulated (b) data for the reduction of Cdþ2 in DMF containing 0.5 M TEAP at the HMDE. The khf values employed are those obtained from Table 3.
Our study allows the extraction of chemical and electrochemical parameters in the case of a CE mechanism via convolution electrochemistry. Digital simulation is found to give good results confirming the accuracy of the proposed method. The convolution procedure can also be applied also to chronoamperometric curves. A linear relationship between the current I, and CE convolution current (ICE Þ is evident. The simplicity of this relationship makes this a very attractive route for data analysis. Experimental verification on
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a real system was carried out with the reduction of Cdþ2 in DMF containing 0.5 M TEAP at the HMDE.
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