6578 E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585chloride dihydrate (EuCl3 , Aldrich, 99.99%), 3-nitrophthalicacid (Ph(COOH)2 NO2 , Aldrich, 95%), 3-nitrophenol (Ph(OH)NO2 ,Aldrich, 99%), mercury(I) nitrate dihydrate (Hg2 (NO3 )2 , Aldrich,>97%), acetonitrile (MeCN, Fischer Scientiﬁc, dried and distilled,99%), dimethylsulfoxide (DMSO, Aldrich, anhydrous, 99.9%),tetra-n-butylammonium perchlorate (TBAP, Fluka, Puriss electro-chemical grade, 99%), sodium perchlorate (NaClO4 , Sigma–Aldrich,98+%), tetra-n-butylammonium hexaﬂuorophosphate (Bu4 NPF6 ,Aldrich, 98%), potassium nitrate (KNO3 , Aldrich, 99+% ACSreagent, 0.10 M), tetra-n-butylammonium hydroxide (Bu4 NOH,Sigma–Aldrich, 40 wt.% in water), perchloric acid (HClO4 , Aldrich,70%) and nitric acid (HNO3 , Fisher Scientiﬁc, 70%, 0.15 M) were allused as received without further puriﬁcation. For DPV and RPV experiments a 1 mM solution of the oxidizedspecies of the corresponding redox couple was prepared. The pHof Eu3+ solutions was adjusted to 4.5 by addition of HClO4. The3-nitrophenolate monoanion and 3-nitrophthalate dianion solu-tions were prepared in the electrochemical cell by adding anexcess of base (Bu4 NOH) to the corresponding 3-nitrophenol and3-nitrophthalic acid solutions .2.2. Working electrode Mercury was electrodeposited by holding the potential at−0.245 V on Pt disc electrodes with r0 = 25 m from a 10 mMsolution of Hg2 (NO3 )2 with 0.1 M KNO3 as supporting electrolyte Fig. 1. Differential Pulse Voltammetry: (a) potential-time function; (b) DPV signal. Reverse Pulse Voltammetry: (c) potential–time function; (d) RPV signal.(acidiﬁed with 0.5% of HNO3 ), as described in literature [16,17]. The platinum electrode radius was calibrated by analyzing thesteady-state voltammetry of a 2 mM solution of Fc in acetonitrile ing the uncompensated solution resistance (Ru ) , its effect oncontaining 0.1 M TBAP, adopting a value of diffusion coefﬁcient of the DPV and RPV signals was studied by numerical simulation (seeferrocene in MeCN of D = 2.3 × 10−9 m2 s−1 at 25 ◦ C . Appendix A), demonstrating that this is negligible for our working Before the mercury deposition the platinum electrode was pol- conditions. At least three experiments were performed with eachished using 3.0, 1.0, and 0.1 m diamond spray on soft lapping system, a good reproducibility of results being found.pads (Kemet, U.K.). The mercury nitrate solution was bubbled withnitrogen gas (BOC, Guildford, Surrey, U.K.) for 30 min to remove 3. Results and discussionatmospheric oxygen, and nitrogen atmosphere was maintainedduring the deposition. 3.1. Theory The size of the hemisphere was controlled by means of theamount of charge transferred, which for a 25 m radius electrode 3.1.1. Differential Pulse Voltammetry (DPV)corresponds to 213.7 C. Next, it was conﬁrmed electrochemically In the DPV technique successive double potential pulses arefrom the steady-state voltammetry of reduction of [Ru(NH3 )6 ]3+ , applied, with a constant pulse height ( E) and the second pulsedemonstrating that the ratio of limiting steady-state currents being much shorter than the ﬁrst one (t2 t1 , see the schemebefore and after deposition was close to /2 . shown in Fig. 1a). The current is measured at the end of each poten- tial pulse, (I1 (t1 )) and (I2 (t2 )), and the DPV signal is a plot of the2.3. Instrumentation difference between the two current samples ( I = I2 (t2 ) − I1 (t1 )) vs. the arithmetic average of both potential values (E1,2 = (E1 + E2 )/2, A computer-controlled -Autolab potentiostat Type II (Eco- see Fig. 1b). The initial equilibrium conditions are restored beforeChemie, Netherlands) was used to undertake the mercury the application of each double pulse, by renewal of the electrodedeposition, the calibration of the mercury hemispherical electrodes (mercury drop electrode), by open circuiting the working electrodeas well as the DPV and RPV experiments.  or by setting the applied potential at the adequate value . For DPV and RPV experiments a three-electrode set-up was An explicit analytical solution for the DPV response of a chargeemployed, with a 25 m radius Hg electrode as working electrode transfer processes with ﬁnite electrode kinetics has been recentlyand a coiled Pt wire as the counter electrode. For measurements deduced , valid for spherical electrodes of any size (includingin aqueous solution a Saturated Calomel Electrode (SCE) was planar electrodes and hemispherical ultramicroelectrodes):employed as a reference electrode; for experiments in DMSO, a ksmicro · K −˛ · f ( I 2 1 /2) micro −˛ 2commercial ‘no leak’ reference electrode comprising Ag/Ag+ in a = 1 + ks · K2 · (1 + K2 ) · F Id (∞) 1 + ksmicro K −˛ (1 + K ) 2PEEK barrel ﬁtted with a membrane junction (66-EE009, Cypress 2 2 (1)electrodes) was used. Solutions were bubbled with N2 before the experiments and where all the variables and functions are given in Table 1 andpositive pressure of N2 was maintained throughout. Before the Appendix B. This expression is fully applicable under the typicalapplication of each double pulse, equilibrium conditions were conditions of the DPV technique (t1 ≥ 50·t2 ).restored by open circuiting the working electrode for 10 s. Chronoamperograms were recorded under identical conditions 3.1.2. Reverse Pulse Voltammetryof DPV and RPV experiments in the absence of the electroactive According to the RPV waveform (Fig. 1c), the applied poten-species, conﬁrming in all cases that the background currents were tial is set at a value corresponding to diffusion limiting conditionsnegligible in the scales of potential and time of interest. Regard- (E1 → −∞ for a reduction process) in the interval 0 ≤ t ≤ t1 and
E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585 6579Table 1Deﬁnitions. r0 Radius of the electrode tj , j ≡ 1, 2 Duration of the jth potential pulse D Diffusion coefﬁcient of the electroactive species ci∗ , i ≡ O, R Bulk concentration of species i c∗ = R c∗ Ratio of the bulk concentrations of the electroactive O species (j) ci , i ≡ O, R; j ≡ 1, 2 Concentration proﬁle of species i for the jth potential pulse kf , kb Heterogeneous rate constants of reduction, oxidation processes k0 Standard heterogeneous rate constant k0 ·r0 micro ks = D Dimensionless rate constant at microelectrodes under steady-state conditions ˛ Electron transfer coefﬁcient E0 Formal potential of the electroactive couple Ej , j ≡ 1, 2 Potential applied during the jth pulse E1 +E2 E1,2 = 2 Abscissa potentialstepped from E1 to E2 for t ≥ t1 . The current is measured at the endof the second pulse (t = t1 + t2 ) and plotted vs. the second potentialvalue (see Fig. 1d). As in DPV, after each double pulse the initialequilibrium conditions are re-established. The analytical solution for this technique at spherical electrodesfor processes of any reversibility degree is given by : p Id (t2 )IRPV = Id,DC + micro K −˛ (1 + K ) 1 + ks 2 2 √ micro −˛ Z 2 − ˇ1/2 + ks K2 (1 + K2 ) 2 [Z · F( 2) + Y · H( 2 , ˇ)] (2) Fig. 2. (a) Experimental DPV curves obtained for 3-nitrophenolate−/2− • (᭹), 3- nitrophthalate2−/3− • ( ) and Eu3+/2+ ( ); t1 = 0.5 s, t1 /t2 = 50, E = −50 mV. (b)with all the variables and functions given in Table 1 and Appendix Experimental RPV curves for the above systems; t1 = 0.5 s, t1 /t2 = 20. a vs. SCE refer-B. ence electrode for Eu 3+/2+ ; vs. Ag/Ag reference electrode for 3-nitrophenolate−/2− • + and 3-nitrophthalate2−/3− • . Solutions: 1 mM 3-nitrophenolate− , 0.1 M Bu4 NPF6 in This solution is rigorously valid whatever the lengths of either DMSO; 1 mM 3-nitrophthalate2− , 0.1 M Bu4 NPF6 in DMSO; 1 mM Eu3+ , 0.1 M NaClO4potential pulse (t1 , t2 ), which is signiﬁcant since the RPV problem is in H2 O (pH = 4.5).usually tackled by assuming a much shorter second pulse (t2 t1 ),otherwise it presents a considerable difﬁculty even under lineardiffusion conditions . will be seen, from the direct visual inspection of the shape of the experimental responses, qualitative kinetic information about the3.2. Experimental results systems can be easily extracted (Figs. 2–4). Also quantitative data were determined by means of the theory [7,8] from single-point The experimental veriﬁcation of the theory as well as the ﬁtting of the experimental results to theoretical working curvesdemonstration of the value of the DPV and RPV techniques in (Figs. 6 and 7), the values obtained for the kinetic parameters andkinetic studies is carried out from three one-electron reduction the formal potential being indicated in Tables 2 and 3.processes for which very different electrochemical reversibility has In Fig. 2 the experimental DPV and RPV curves for all the redoxbeen reported [10–14]: system are plotted. In Fig. 2a, the effect of the electrode kinetics on the position and shape of DPV curves is clearly observed. The- 3-nitrophenolate−/2−• position of the peak with respect to the formal potential (E0 ) is greatly affected by the electrochemical reversibility of the system − k0 ,˛ 2− • [Ph(O− )NO2 ] + e− [Ph(O− )NO2 ] (I) so that the more irreversible the electrochemical process is, the greater the shift of the peak potential with respect to the formal- 3-nitrophthalate2−/3−• potential . Thus, in the case of Eu3+/2+ (grey points) there is a sig- 2− k0 ,˛ 3− • niﬁcant shift of the peak potential towards more negative values, [Ph(COO− )2 NO2 ] + e− [Ph(COO− )2 NO2 ] (II) related with the small heterogeneous rate constant of the sys-- Eu3+/2+ tem (see Tables 2 and 3). The DPV curve of 3-nitrophthalate2−/3−• (white points) is slightly shifted with respect to E0 whereas for k0 ,˛ Eu3+ + e− Eu2+ (III) 3-nitrophenolate−/2−• (black points) the peak potential and the for- mal potential coincide. So, from the position of the DPV response The experimental application of the DPV and RPV techniques the reversibility of the redox systems is qualitative determined,to the kinetic study of these three redox systems was performed ﬁnding that this decreases in the order: 3-nitrophenolate−/2−• > 3-on mercury hemispherical microelectrodes of ca. 25 m radius. As nitrophthalate2−/3−• > Eu3+/2+ .
6580 E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585Fig. 3. Experimental DPV curves with positive ( E = +50 mV, triangularpoints) and negative ( E = −50 mV, circular points) pulse height values for p Fig. 4. Experimental Id,RPV /Id (t2 ) vs. t2 /(t1 + t2 ) curves obtained for 3-3-nitrophenolate−/2− • (black points), 3-nitrophthalate2−/3− • (white points) and −/2− •Eu3+/2+ (grey points). t1 = 1 s, t1 /t2 = 100. a vs. SCE reference electrode for Eu3+/2+ ; vs. nitrophenolate (᭹), 3-nitrophthalate2−/3− • ( ) and Eu3+/2+ ( ). Straight linesAg/Ag+ reference electrode for 3-nitrophenolate−/2− • and 3-nitrophthalate2−/3− • . corresponding to the linear regression of the experimental results are also plottedOther experimental conditions as in Fig. 2. for each case, the values of the correlation coefﬁcient (R), the slope and the intercept given in the table imbedded. t1 = 0.5 s. Other experimental conditions as in Fig. 2. Information about the electrode kinetics can also be inferred of the former having a greater heterogeneous rate constant (seefrom the shape of DPV curves. As is well known , for given Fig. 6).˛ value and experimental conditions (r0 , t2 and t1 values) the According to the above discussed, the DPV technique is a goodDPV peak becomes smaller and broader as the heterogeneous rate method to determine the kinetic parameters by single-point ﬁt ofconstant decreases. Thus, in the case of 3-nitrophenolate−/2−• a the peak current and the peak potential, which are strongly depen-much greater and sharper DPV peak is obtained, which points dent on the kinetics of the electrode process.out the greater reversible behaviour of this system. On the other In Fig. 2b the experimental RPV curves obtained are shown. Thehand, the DPV curves of 3-nitrophthalate2−/3−• and Eu3+/2+ show effect of the electrode kinetics on the shape of the RPV curves isbroader and smaller peaks, characteristic of more sluggish charge very signiﬁcant so that the morphology of the responses enables thetransfer processes. The peak of 3-nitrophthalate2−/3−• is espe- estimation of the electrode kinetics from simple visual inspection.cially interesting (see graph imbedded in Fig. 2a) since it shows Thus, a gradual split of the RPV curve into two waves is found as thea remarkable asymmetry with a ‘tail’ at the more negative poten- system becomes more irreversible. Thus, for 3-nitrophenolate−/2−•tials. This feature of the DPV curve is indicative of a small value of (black points) a well-shaped sigmoidal signal is obtained, charac-the electron transfer coefﬁcient (˛). The smaller ˛ value together teristic of fast charge transfer. On the other hand, the RPV curvewith the smaller diffusion coefﬁcient (therefore, smaller spheric- obtained for Eu3+/2+ (grey points) clearly shows a split of the RPVity) of 3-nitrophthalate also explains that the peak height obtained curve into a cathodic and an anodic wave, showing that the chargefor 3-nitrophthalate2−/3−• is smaller than that of Eu3+/2+ in spite transfer process is much more sluggish. An intermediate behaviourTable 2Heterogeneous rate constant (k0 ), electron transfer coefﬁcient (˛) and formal potential (E0 ) corresponding to the best ﬁt of theoretical working curves (Eq. (1)) to the DPVexperimental results (Fig. 6). Redox couple Mediuma D × 106 /cm2 s−1 b k0 /cm s−1 ˛ E0 /V 3- Nitrophenolate−/2− • 0.1 M Bu4 NPF6 , DMSO 3.30 ± 0.03 1.4 × 10−2 0.45 −1.415c 3-Nitrophthalate2−/3− • 0.1 M Bu4 NPF6 , DMSO 1.88 ± 0.06 1.5 × 10−3 0.37 −1.503c Eu3+/2+ 0.1 M NaClO4 , H2 O (pH = 4.5) 6.11 ± 0.05 1.7 × 10−4 0.69 −0.643d , e a Concentration substrate: 1 mM of 3-nitrophenolate− , 3-nitrophthalate2− , Eu3+ . b Calculated from several repeat potential step chronoamperograms. c vs. Ag/Ag+ reference electrode. d vs. SCE reference electrode. e Value taken from the data determined from RPV experiments (Table 3).Table 3Heterogeneous rate constant (k0 ), electron transfer coefﬁcient (˛) and formal potential (E0 ) corresponding to the best ﬁt of theoretical working curves (Eq. (2)) to the RPVexperimental results (Fig. 7). Redox couple Mediuma k0 /cm s−1 ˛ E0 /V −/2− • −2 3-Nitrophenolate 0.1 M Bu4 NPF6 , DMSO 1.6 × 10 0.46 b −1.410c , b 3-Nitrophthalate2−/3− • 0.1 M Bu4 NPF6 , DMSO 1.7 × 10−3 0.40 −1.515c Eu3+/2+ 0.1 M NaClO4 , H2 O (pH = 4.5) 1.4 × 10−4 0.72 −0.643d a Concentration substrate: 1 mM of 3- nitrophenolate− , 3-nitrophthalate2− , Eu3+ . b Values calculated from the ﬁt of the complete RPV curve. c vs. Ag/Ag+ reference electrode. d vs. SCE reference electrode.
E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585 6581 p Fig. 6. Experimental values of (a) Ip /Id (t2 ) and (b) Ep vs. log( 2) for 3-Fig. 5. Typical shape of the signals recorded in (a) DPV and (b) RPV techniques −/2− • 2−/3− • nitrophenolate (᭹), 3-nitrophthalate ( ) and Eu3+/2+ ( ) and theoreticalindicating the points of the curves from which they can be easily characterized: working curves (from Eq. (1), solid lines) corresponding to the best ﬁt to experimen-(a) DPV: peak current ( Ip ) and peak potential (Ep ); (b) RPV: cathodic and anodic tal results. t1 /t2 = 50. a vs. SCE reference electrode for Eu3+/2+ ; vs. Ag/Ag+ referencelimiting currents (Id,DC and Id,RPV , respectively) and cathodic and anodic half-wave electrode for 3-nitrophenolate−/2− • and 3-nitrophthalate2−/3− • . The value of the cat anpotentials (E1/2 and E1/2 , respectively). sum of the squared residuals (S) obtained from the ﬁtting is indicated on each 2 curve: S = (experimental value − theoretical value) . Other experimental con- ditions as in Fig. 2.is observed for the RPV curve obtained with 3-nitrophthalate2−/3−•(white points), in which the split starts to appear but it is not The RPV technique is also very useful for the full analysis offully developed. It can be inferred that the RPV technique is very the electrode reaction since the anodic branch of the RPV curve isappropriate for the study of the product of the electrode reac- strongly affected by the behaviour of the reaction product. In partic-tion in comparison with single-pulse techniques (like Normal Pulse ular, from the anodic limiting current of the RPV curve (Id,RPV = IRPVVoltammetry) where obtaining two separate irreversible waves is (E2 → +∞)) the existence of any process involving the reactiononly possible when both electroactive species comprising the redox product can be inferred such as, for example, a coupled subse-couple are initially present. quent chemical reaction (EC mechanism). These complications can In DPV, an alternative simple way for the elucidation of electrode be discarded by taking into account that for a simple charge trans-kinetics is by doing DPV experiments with positive and negative fer process (E mechanism) the expression of Id,RPV when only the E values, and studying the symmetry of the peaks obtained. The oxidized species is initially present ( = 0) is :asymmetry between both peaks increases as the system becomesmore irreversible . Id,RPV t2 p = −1 (3) In Fig. 3, the experimental DPV curves obtained with Id (t2 ) t1 + t2 E = −50 mV (normal mode) and E = +50 mV (reverse mode) areplotted for the three redox systems considered. The asymmetry of which is valid for any electrode radius and for any reversibil- pthe peaks corresponding to positive and negative E values is very ity degree. As a result, if the plot Id,RPV /Id (t2 ) vs. t2 /(t1 + t2 ) isapparent in the case of Eu3+/2+ reduction, for which a notably larger a straight line with slope = 1 and intercept = −1 the system underpeak is obtained when E < 0. On the other hand, the peaks corre- study corresponds to an E mechanism, since if the reduced speciessponding to 3-nitrophenolate−/2−• are almost symmetrical, which is involved in a following chemical reaction deviation from theagain points out a more rapid charge transfer process. An interme- above linear relationship is obtained as well as smaller anodicdiate situation is found for 3-nitrophthalate2−/3−• , a slightly larger currents.peak for E < 0 being obtained. So, this type of analysis also allows In Fig. 4, this simple diagnostic test is applied to the experi-us to easily classify the reversibility degree of the electrode pro- mental systems here studied. In all cases, very good correlationcesses. coefﬁcients (R) are obtained for the linear regression of the plots
6582 E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585Fig. 7. Experimental values of (a) E1/2 (= E1/2 − E1/2 ), and (b) and (c) an cat E1/2 (= E1/2 + E1/2 ) vs. log( an cat 2) for 3-nitrophenolate−/2− • (᭹), 3-nitrophthalate2−/3− • ( ) and Eu3+/2+( ) and theoretical working curves (from Eq. (2), solid lines) corresponding to the best ﬁt to experimental results. t1 /t2 = 20. a vs. SCE reference electrode for Eu3+/2+ ; vs.Ag/Ag+ reference electrode for 3-nitrophenolate−/2− • and 3-nitrophthalate2−/3− • . The value of the sum of the squared residuals (S) obtained from the ﬁtting is indicated on 2each curve: S = (experimental value − theoretical value) . Other experimental conditions as in Fig. 2. p is directly related to the duration of the double pulse so that theId,RPV /Id (t2 ) vs. t2 /(t1 + t2 ), with slope and intercepts values veryclose to 1 and −1, respectively, as required for an E mechanism. greater 2 , the longer t1 + t2 . DPV and RPV curves can be described by the characteristic As can be observed in Fig. 6b, the duration of the doubleparameters shown in Fig. 5. Thus, DPV curves can be character- pulse slightly affects the position of the DPV peaks of 3-ized by the peak current ( Ip ) and the peak potential (Ep ) (see nitrophenolate−/2−• , which is characteristic of rapid electrode pro-Fig. 5a), and RPV curves by the cathodic and anodic limiting cur- cesses . On the other hand, for Eu3+/2+ and 3-nitrophthalate2−/3−•rents (Id,DC and Id,RPV , respectively) and the cathodic and anodic the peak potential shifts towards less negative values when the cat anhalf-wave potentials (E1/2 and E1/2 , respectively) (see Fig. 5b). double pulse duration increases (i.e., when 2 increases), as cor- This allows a straightforward way of analysis of the experimen- responds to slower charge transfer reactions . In all cases, the ptal response by single-point ﬁt to theoretical results, instead of the normalized peak current Ip /Id (t2 ) increases with 2 (see Fig. 6a).more complex ﬁt of the whole curve. Thus, from the experimental The solid lines in Fig. 6 are the theoretical working curves cor-values of the above parameters for different lengths of the double responding to the best ﬁt to the experimental results. Thus, frompulse (t1 + t2 ) (keeping the ratio t1 /t2 constant), the quantitative the simultaneous ﬁtting of the variation of the peak current and thecharacterization of the electrode kinetics can be performed [7,8]. peak potential, the values of the heterogeneous rate constant, theFor this purpose, general working curves can be easily obtained transfer coefﬁcient and the formal potential are determined (valuesfrom Eqs. (1) and (2): Ip and Ep vs. log( 2 ) for DPV (Fig. 6), and given in Table 2). The ﬁtting was done by computerized minimum an cat E1/2 (= E1/2 − E1/2 ) and an cat E1/2 (= E1/2 + E1/2 ) vs. log( 2 ) for RPV least square procedure of theoretical and experimental results. The value of the sum of the squared residuals (S) is shown on the curves(Fig. 7). as an indicator of the goodness of the ﬁtting. Regarding DPV technique, in Fig. 6 the variation of the normal- p For very slow charge transfer processes, as is the case of Eu3+/2+ized peak current ( Ip /Id (t2 ), Fig. 6a) and the peak potential (Ep , redox system, the peak current is not sensitive to k0 value . Nev-Fig. 6b) obtained for each system are plotted vs. log( 2 ). The exper- ertheless, if the formal potential of the system is known, the kineticimental points correspond to DPV experiments undertaken with parameters can be extracted from the working curves considereddifferent t1 + t2 values, being t1 /t2 = 50 in all cases. Note that given in Fig. 6. Thus, from the variation of the normalized peak currentthat the ratio t1 /t2 is kept constant, the parameter 2 (= 2 Dt2 /r0 ) p ( Ip /Id (t2 )) with log( 2 ) the ˛ value is determined and, once ˛ and
E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585 6583Table 4Data reported in the literature for the diffusion coefﬁcient (D), heterogeneous rate constant (k0 ) and electron transfer coefﬁcient (˛) of the systems under study at mercuryelectrodes. Redox couple Medium D × 106 /cm2 s−1 k0 /cm s−1 ˛ Methoda Reference −/2− • −2 3-Nitrophenolate 0.1 M Bu4 NPF6 , DMSO 4.0 2.1 × 10 0.43 DCV  3-Nitrophthalate2−/3− • 0.1 M Bu4 NPF6 , DMSO 1.3 1.1 × 10−3 0.34 DCV  Eu3+/2+ 1 M NaClO4 , H2 O (pH = 3.0) 7.0 2.5 × 10−4 0.60 d.c. polarography  0.5 M NaClO4 , H2 O ∼7.5 9.3 × 10−5 0.68 Chrono-coulometry  1 M KCl, H2 O – 2.6 × 10−4 0.54 DPP  0.1 M NaClO4 , H2 O (pH = 2.0) – 10−3 0.76 SWV  1 M NaClO4 , H2 O 5.7 – – Chrono-potentiometry  a DCV: Derivative Cyclic Voltammetry; DPP: Differential Pulse Polarography; SWV: Square Wave Voltammetry.E0 values are known, the heterogeneous rate constant is obtained behaviours of E1/2 with 2 are obtained for ˛ > 0.5 (Eu3+/2+ ) andfrom the plot of the peak potential (Ep ) vs. log( 2 ). ˛ < 0.5 (3-nitrophthalate2−/3−• ) (see Fig. 7b). Thus, from the curves In Fig. 7 the results obtained from RPV experiments for t1 /t2 = 20 E1/2 vs. log( 2 ) the ˛ value is immediately obtained and onceare shown. The parameter E1/2 is related to the separation of this is determined, the value of the heterogeneous rate constant isthe cathodic and anodic branches of RPV curves and so with the extracted from the curve E1/2 vs. log( 2 ) (Fig. 7a). Finally, oncereversibility of the system. Hence, the greater the E1/2 value, the ˛ and k0 values are known, the value of the formal potential isthe more irreversible the process, so that the greatest E1/2 determined from E1/2 vs. log( 2 ) curves.values are obtained for Eu3+/2+ and the smallest ones for 3- From the comparison of the ﬁtting procedures carried out innitrophenolate−/2−• (see Fig. 7a). Figs. 6 and 7 we deduce that, although both DPV and RPV techniques The parameter E1/2 informs about the relative symmetry are useful for the study of the electrode kinetics, the RPV tech-of the cathodic and anodic branches and it is strongly dependent nique is more powerful for the case of irreversible systems sinceon the value of the electron transfer coefﬁcient. Thus, contrasting the separation of the cathodic and anodic branches (i.e., E1/2 ) is always sensitive to k0 value , E1/2 increasing as k0 decreases. On the other hand, as was previously indicated, the peak current in DPV becomes independent of k0 value for very slow charge transfer processes . In Fig. 8 the experimental and theoretical DPV and RPV curves for t1 = 0.5 s are plotted. In all cases, the agreement between theory and experiments is quite satisfactory which validates the analy- sis of the curves by single-point ﬁt, much easier than the ﬁt of the whole curve, as well as the theory for double potential pulse recently developed [7,8]. Moreover, the data of the kinetic parame- ters here obtained from DPV and RPV experiments show a good or reasonable concordance with those reported in the literature with different electrochemical techniques (see Table 4), which validates the applicability of both double pulse techniques for kinetic studies. 4. Conclusions The value of Differential Pulse Voltammetry and Reverse Pulse Voltammetry for the characterization of the electrode kinetics of the electrode process has been experimentally demonstrated by studying three redox systems with notably different electrode kinetics at mercury hemispherical microelectrodes. Simultane- ously, the applicability of a recent theory for double pulse at spherical electrodes has been validated. The qualitative analysis of the electrochemical reversibility of these systems has been performed by visual analysis of DPV and RPV responses, as well as from the symmetry of DPV peaks obtained with positive and negative pulse heights. In addition, sim- ple methodology for the detection of following coupled chemical reactions from the time behaviour of the anodic limiting current of the RPV curve is described. Finally, the kinetic parameters and the formal potential of the redox couples under study are determined by simple ﬁt of exper- imental responses obtained with different values of the length of the double potential pulse.Fig. 8. Fit of the experimental (points) and theoretical (solid line) (a) DPV and (b) RPVcurves for 3-nitrophenolate−/2− • (᭹), 3-nitrophthalate2−/3− • ( ) and Eu3+/2+ ( ). AcknowledgementsThe theoretical curves correspond to the k0 , ˛ and E0 values obtained from the ﬁt toworking curves shown in Figs. 6 and 7 (values given in Tables 2 and 3). a vs. SCE refer-ence electrode for Eu3+/2+ ; vs. Ag/Ag+ reference electrode for 3-nitrophenolate−/2− • A.M., F.M.-O. and E.L. greatly appreciate the ﬁnancial supportand 3-nitrophthalate2−/3− • . Other experimental conditions as in Fig. 2. provided by the Dirección General de Investigación (MEC) (Project
6584 E. Laborda et al. / Electrochimica Acta 55 (2010) 6577–6585Number CTQ2009-13023) and by the Fundación SENECA (Project the relevant dimensionless parameter to take into account iR-dropNumber 08813/PI/08). Also, E.L. thanks the Ministerio de Ciencia e effect isInnovacion for the grant received. ∗ F 2 r0 cO D1/2 ς= (A2) RT t 1/2Appendix A. 1 The evaluation of the effects of the uncompensated solution In Fig. A.1 DPV and RPV curves for different values of ς areresistance (Ru ) on the DPV and RPV responses has been performed plotted considering a hemispherical electrode with r0 = 25 m. Theby digital simulation with a homemade program. response expected with no iR-drop effects is also shown (circle In this program, we have used an exponentially expanding grid points).with high expansion factors and four-point formulae for the dis- It is clearly demonstrated that for ς ≤ 2 × 10−2 (– – – –) the DPVcretisation of spatial derivatives (as shown in Ref. ). Under these and RPV responses are almost indistinguishable from that corre-conditions, a Thomas-like algorithm is feasible and the u–v method sponding to no iR-drop effects. So, under the working conditions for boundary conditions fully applicable. In this way, the sur- of this paper (ς < 5 × 10−3 ) the effect of the solution resistance isface concentration of both species and their surface gradient are negligible and it can be ignored.very simply related, and the implicit calculation of surface con- It is worth to highlight the tolerance of the RPV technique to thecentrations with the inclusion of iR-drop effects is reduced to the solution resistance so that the signal is only signiﬁcantly affectedsolution of one-unknown equation. For time-integration we have for large ς values (see Fig. A.1b).used EXTRAP4 algorithm taking 5-time intervals  in each poten-tial step (which gives rise to very fast calculations). Appendix B. Being the resistivity of the solution, with the assumption thatfor a hemispherical electrode uncompensated resistance can be ∗ FAcO Devaluated as  Id (∞) = (B1) r0Ru = (A1) 2 r0 p ∗ D Id (t) = FAcO (B2) t r0 Id,DC = Id (∞) 1+ (B3) D(t1 + t2 ) F Kj = exp (E − E 0 ) j ≡ 1, 2 (B4) RT j 2 Dt2 2 = (B5) r0 t2 ˇ= (B6) t1 + t2 t2 0 −˛ 2 = 2 +2 k K2 (1 + K2 ) (B7) D 2 (t1 + t2 ) 0 −˛ 1 = +2 k K1 (1 + K1 ) (B8) ˇ D 1+kmicro K1−␣ (1 + ) s 2 Z=− (B9) 1 + kmicro K−˛ (1 + K2 ) s 2 2 ␤ Y=− (B10) 2 ˛ micro K −˛ (1 + K ) 1 + ks K2 2 2 f (x) = 1 − K2 − (1 − K1 ) K1 micro K −˛ (1 + K ) 1 + ks 1 1 micro K −˛ (1 + K ) 1 + ks 2 2 + 1− · F(x) (B11) micro K −˛ (1 + K ) 1 + ks 1 1Fig. A.1. Inﬂuence of the uncompensated solution resistance (Ru ) on the (a) DPVand (b) RPV responses. Several values of ς parameter are considered: ς = 2 × 10−3 ∞ √(working conditions, —), ς = 2 × 10−2 (– – – –), ς = 0.1 (· · ·) and ς = 0.2 (– · · – · · – ·). (−1)i xi+1 x 2 xCircle points (᭹) correspond to the response obtained with no solution resistance. F(x) = = x · exp · erfc (B12) i p 2 2 2r0 = 25 m, t1 = 2.5 s, t1 /t2 = 100, D = 10−5 cm2 s−1 ; (a) E = −50 mV. i=0 l=0 l
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