Reciprocal Derivative Chronopotentiometry with Programmed Current: Influence of the Reversibility

Reciprocal Derivative Chronopotentiometry with ProgrammedCurrent: Influence of the ReversibilityAngela Molina*, Joaquin Gonzalez and Marien M. Moreno

Departamento de QuÌmica FÌsica. Facultad de QuÌmica. Universidad de Murcia. Espinardo, 30100. Murcia, Spaine-mail: [email protected]

Received: March 9, 2001Final version: May 23, 2001

AbstractThe behavior of the system Cd�2/Cd(Hg) in the absence and presence of an aliphatic alcohol in noncomplexing mediais analyzed in reciprocal derivative and double derivative chronopotentiometry with programmed current (RDCP andRDDCP, respectively). These electrochemical methods have been compared with derivative and double derivativevoltammetry (DV and DDV, respectively). When the above system behaves as non reversible, it is not possible toestablish a total analogy between chronopotentiometric and voltammetric responses, contrary to the case of areversible process.The peaks obtained in RDCP are better resolved than those obtained in DVand the different operational principles ofRDCP compared to those of DV can reduce its susceptibility to ohmic drop effects. Moreover, RDCP and RDDCPare very versatile in the determination of kinetic parameters of electrode processes since, by varying the currentamplitude and/or the power of time u in the programmed current, it is easy to influence the reversibility of the process.

Keywords: Reciprocal derivative chronopotentiometry, Programmed currents, Cadmium reduction, Quasireversibleprocesses

1. Introduction

In a previous article we compared the responses obtained inderivative and differential voltammetric techniques andthose corresponding to reciprocal derivative chronopoten-tiometry with a programmed current of the form I(t)� I0tu,with u��1/2 (dtu�1/2/dE�E and d2tu�1/2/dE2�E responses)for reversible processes [1]. We demonstrated that it ispossible to establish a total analogy between both potentio-static and chronopotentiometric responses [1, 2].In this work we analyze the influence of the reversibility

on the response corresponding to reciprocal derivativechronopotentiomentry (RDCP) and reciprocal double de-rivative chronopotentiomentry (RDDCP) with pro-grammed current for which the first (dtu�1�2N /dE) and second(d2tu�1�2N /dE2) reciprocal derivatives of the normalized E/tu�1�2N curves are obtained. We have compared thesederivatives with those obtained in derivative voltammetry(DV) and double derivative voltammetry (DDV) (first (dIN/dE) and second (d2IN/dE2) derivatives of the IN/E curve).In contrast to what occurs for reversible processes, in the

case of quasirreversible and totally irreversible processes,normalized chronopotentiometric and voltammetric re-sponses can never be identical due to the fact that theirtemporal dependence cannot be eliminated, as it can be inreversible processes [1]. However, the dtu�1�2N /dE�E(RDCP) and d2tu�1�2N /dE2�E (RDDCP) curves for u�0present a parallel behavior to that of the first and secondderivatives of the normalized IN/E curve, except for theparticular case u� 0 (constant current, I(t)� I0), for whichthe dtu�1�2N /dE±E curve does not present a peak when atotally irreversible process is analyzed. This behavior

enables us to distinguish a totally irreversible process fromanother quasirreversible one, although to determine kineticparameters of the charge transfer reaction� and k�0 it will benecessary to work with programmed current instead ofconstant current (i.e., values of u different from zero) forwhich analytical expressions of the peak height and peakpotentials as functions of � and k�0 are given.The use of RDCP with programmed current proposed in

this article improves the usually used constant current in thedetermination of thermodynamic and kinetic parameters ofelectrode processes. This is because the possibility of beingable to choose different values of u leads to the following,among others, advantages:

± Amore suitable selection of the transition times in awiderrange of concentration than does a constant current [1].

± An easy analysis of the reversibility of the process [3]since, a quasirreversible process can be treated asreversible (by increasing the value of u and/or decreasingthe current amplitude I0) or as totally irreversible (bydecreasing the value of u and/or increasing I0).

± An immediate distinction between a reversible and aquasirreversible process since the normalized RDCP(dtu�1�2N /dE ±E) andRDDCP(d2tu�1�2N /dE2) responses varywith u for quasirreversible and totally irreversible proc-esses, whereas these responses remain independent of ufor reversible ones.

If we compareRDCPwith potentiostatic techniques, we canstate that:

± The signal obtained with this technique presents moreanalytical and kinetic advantages than those correspond-

281

Electroanalysis 2002, 14, No. 4 ¹ WILEY-VCH Verlag GmbH, 69469 Weinheim, Germany, 2002 1040-0397/02/0402-0281 $ 17.50+.50/0

ing to derivative voltammetry (DV) since, in the firstplace, these signals are less noise influenced than thosecorresponding to DV [1] and, in the second place, thepeaks obtained in RDCP are narrower than thoseobtained in DV in the case of quasirreversible or totallyirreversible electrode processes.

± The use of this technique in the determination of kineticparameters of electrode processes is also more advanta-geous than the use of cyclic voltammetry (CV) since, inorder to increase the sensitivity of the response in CV it isconvenient to increase the sweep rate, which causes asizeable distortion on the I/E curve due to the increase ofthe ohmic drop, a feature which can be confused with thatcaused by a slow charge transfer reaction [4, 5]. However,in order to increase the sensitivity of theRDCP technique(i. e., to increase the peak height), it is necessary todecrease the applied current, whichmeans that the ohmicdrop effects become less relevant [6, 7].

It is also of interest to point out that the peaks obtained inRDCP and RDDCP are scarcely affected by double layereffects on account of the fact that their having been obtainedfrom the central zone of the E vs. tu�1/2 curve [3].We have carried out the experimental study of the Cd�2�

NaClO4 ¥ 0.5 M system with different concentrations of analiphatic alcohol,n-pentanol, which inhibits the reduction ofthe cadmium [8 ± 12]. This system is of great interest sincethe diffusion of a reactant from the bulk the solution to theelectrode surface through an adsorbed layer that decreasethe rate of charge transfer is one of the models used in thestudy of membranes or layers that separate two phases orsolutions of different chemical compositions [10]. We haveused RDCP and RDDCP techniques to characterize thebehavior of the cadmium reduction in solutions withdifferent concentrations of n-pentanol which gives rise todifferent values of the kinetic constant of this process [10]and in each case the values of the kinetic and thermody-namic parameters of the charge transfer are determined.Wehave compared the response of these techniques with thatcorresponding to DV, with a better resolution beingobtained in RDCP. Thus, for example, for the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in a saturated solution ofn-pentanol, RDCP curves with peak half widths of 93 mVare obtained for the less favorable case of the application ofa current ramp, as opposed to the 120 mVobtained in DV.

2. Experimental

2.1. Apparatus and Electrodes

Computer driven potentiostat-galvanostat was designedand constructed by Quiceltron (Spain).Pulse andwaveform generation and data acquisition were

performed using i-SBXDD4 and DAS16-330i (Computer-Boards, USA) boards, respectively. All computer programswere written in our laboratory. See [1] and [3] for details ofthe experimental set-up.

A three electrode cell was employed in the experiments.A static mercury drop electrode (SMDE, electrode radii r0� 0.0487 and 0.0463 cm), served as working electrode. TheSMDE was constructed using a DME, EA 1019-1 (Met-rohm) to which a homemade valve was sealed. Theelectrode radius of the SMDE was determined by weighinga large number of drops. The counter electrode was a Pt foiland the reference electrode was a Ag �AgCl �KCl 1.0 Melectrode.In the case of the SMDE, the errors due to the influence of

the electrode sphericity and of the cadmium amalgamationin the measurement of the peak currents and peak heightswere less than 4%. In order to evaluate these errors we havecompared Equations 12 and 17 for an SMDE consideringspherical diffusion and amalgam formation of references[13, 14] corresponding to the i vs. E and E vs. t curves,respectively, with Equations 1 and 17 in this article. Theinfluence of the electrode sphericity on the peak potentialswas negligible.

2.2. Signal Processing

In the experimentalmeasurements of the potential time andcurrent potential curves we have used different digital noisefilters of the instrument supported software. The experi-mental I/E and E/t curves were then smoothed by applyingthe moving average smoothing procedure proposed bySavitzky and Golay [15], and transformed to the corre-sponding IN/E and t

u�1�2N E curves, respectively (with IN� I/Id

and tN� t/�). The curves obtained for the solutions ofcadmium in absence of n-pentanol and cadmium in thepresence of n-pentanol 0.1 M were fitted with a sigmoidalregression. In the case of the curves corresponding to thesaturated n-pentanol solution, we programmed theoreticalequations for the totally irreversible E ± t curve to carry outthe regressions. In both cases we used the programSigmaPlot [16], and the fitted curves obtained were numeri-cally differentiated by using a finite differences formula offifth degree [17] to give the dtu�1/2/dE and dI/dE curves. Thesecond order differentiated curves d2tu�1/2/dE2 and d2I/dE2,were obtained by differentiating the first order differ-entiated curves.In the determination of the kinetic parameters of the

quasireversible and totally irreversible systems under study,wehave considered a similar experimental time scale in bothchronopotentiometric and voltammetric techniques due tothe fact that the temporal dependence of the responsescannot be eliminated in this case.Peak potentials and peak heights in DVand RDCP were

measured from the fitted differentiated curves withoutfurther analysis. Cross potentials (i. e., potentials at whichthe current/signal measured in DDVor in RDDCP is equalto zero) were obtained from a linear interpolation of thecentral zone of the derivative and differential curves.

282 A. Molina et al.

Electroanalysis 2002, 14, No. 4

2.3. Chemical Reagents

n-Pentanol, NaClO4 and CdSO4 were of Aldrich reagentgrade. These chemical reagents were used without furtherpurification.Waterwas bidistilled and nitrogen gaswas usedfor deaeration.Saturated solutions of n-pentanol were prepared by

mechanical stirring (over 3 days) of a solvent with an excessof the aliphatic alcohol.Freshly prepared solutions were used in all experiments.The diffusion coefficients of cadmium were determined

by chronoamperometricmeasurments with the obtention ofthe following values:DCd�2� (7.20� 0.05)� 10�6 cm2 s�1 andDCd(Hg)�(1.65� 0.50)� 10�5 cm2 s�1, which are in goodagreement with those reported in the literature [14, 18 ± 20].

3. Theory

3.1. Reciprocal Derivative Chronopotentiometry (RDCP)and Reciprocal Double DerivativeChronopotentiometry (RDDCP) with ProgrammedCurrent

When a programmed current of the form I(t)� I0tu, with u�� 1/2 is applied to a plane electrode, the potential time curvecorresponding to a quasirreversible charge transfer reactionis given by [3, 21 ± 22],

��DA

�p2u�1

2k�0�1�2e��tuN � 1� tu�1�2N � �e�tu�1�2N 1

with k�0 and� being the heterogeneous rate constant and thecharge transfer coefficient of the electrode process, respec-tively, and tN a normalized (dimensionless) time given by,

tN � t�

2

with being the transition time of the experiment,

� � nFAc�A��DA

�p2u�1

2I0

� �2� 2u�1 3

Moreover,

� � nFRT

E t � E�0� �� 4

� ��DA

DB

�5

p2u�1 �2� 4u� 3 �2

� u� 1 6

In Equations 3 ± 6 E(t) is the time dependent measuredpotential, E�0 is the formal potential of the electroactivecouple, c�

A is the bulk concentration of the oxidized speciesA, n is the number of transferred electrons,Di (i�A orB) isthe diffussion coefficient of the i species, � is the EulerGamma function and R, Tand F have their usual meanings.In the case of the potential time response given from

Equation 1, it can be observed that the potential � dependson the time through the dimensionless variable tu�1�2N (i. e.,(t/�)u�1/2). This fact has suggested our obtaining the first andsecond reciprocal derivatives of the theoretical E vs. tu�1�2N

curve, which are the responses obtained in RDCP andRDDCP, respectively. Sowe first obtain the tu�1�2N vs.E curvefrom Equation 1 by applying any numerical method (forexample the bisection method), and then we calculatenumerically the first and second derivative by using a finitedifferences formula of fifth degree [17].By making k�0� 1 cm s�1 in Equation 1, we easily deduce

the expressions for the peak potentials, Ep, cross potential,Ec, and peak heights, yp, corresponding to a totally irrever-sible process as functions of the kinetic parameters of thecharge transfer reaction� andk�0 and also of the exponent ofthe power current u, which are shown in Table 1. Table 2shows the values of Ep, Ec and yp for two usual values of u(u�1/2 and u�1).Moreover, we have also shown in Table 3 the values of the

above parameters Ep, Ec and yp, obtained in DVand DDVfor comparison.

4. Results and Discussion

For the deduction of the RDCP (dtu�1�2N /dE vs. E) andRDDCP (d2tu�1�2N /dE2 vs. E) curves we have proceeded asfollows:Once we have obtained the E/t response we calculate the

transition time of the experiment � and obtain the normal-ized E/tu�1�2N (with tN� t/�) curve instead the usual chrono-potentiogram (i. e., we plot the E/t1�2N when a constantcurrent, u� 0, is applied, the E/tN curve when a current thatvarieswith the square root of time,u� 1/2, is applied and theE/t3�2N curve when a current ramp, u� 1, is applied). Fromthese curves we have numerically obtained the first recip-rocal derivative (RDCP) or the second reciprocal derivative(RDDCP) by following the procedure indicated in theExperimental section.Figure 1 shows the normalized RDCP responses (dtu�1�2N /

dE±E curves) obtained for the application of a currentramp (I(t)� I0t, Figure 1a), of a current that varies with thesquare root of time (I(t)� I0t1/2, Figure 1b), and of a currentstep (I(t)� I0, Figure 1c), to the Cd�2 1.6 mM�NaClO4 ¥0.5 M system in absence of n-pentanol (curves 1), in asolution 0.1 M of n-pentanol (curves 2) and in a saturatedsolution of n-pentanol (curves 3).As can be observed in this Figure, the behavior of this

system in RDCP is strongly affected by the presence of n-pentanol such that as the peak potentials shift towardsmorenegative values, the peak half widths increase and the peak

283Reciprocal Derivative Chronopotentiometry with Programmed Current


heights decrease as the concentration of n-pentanol in-creases.These results indicate to us that the charge transfer

process becomes more irreversible the higher the concen-tration of n-pentanol. The reason for this lies in the fact thatthe presence of adsorbed alcohol inhibits the charge transfer

and causes a decrease in the reversibility of the process [8,10].The behavior observed when a programmed current (u�

0, see Figures 1a and 1b) is used is qualitatively similar tothat obtained in derivative voltammetry [23] and in differ-ential pulse voltammetry [30]. When a constant current is

Table 1. Totally irreversible processes. Theoretical expressions for the peak potentials, cross potentials and peak heights correspondingto reciprocal derivative chronopotentiometry with programmed current (RDCP, dtu�1�2N /dE vs. E curves, with tN� t/�) and to reciprocaldouble derivative chronopotentiometry with programmed current (RDDCP, d2tu�1�2N /dE2 vs. E curves), see Equations 1 ± 3. I(t)� I0tu,u�0. When u� 0, peaks are not observed.Technique Peak potentials Ep Cross potential Ec Peak heights Peak half widths

RDCPdtu�1�2N /dEvs. E

Ep � E�0 � RT�nF

ln k�0��

DA

�� ± yRDCPp

�� nF 2u� 1 RT

� WRDCP1�2 � RT

�nFx

� RT�nF

ln2 1�Au

p2u�1Au�u�1�2u

1 �Au 1�Au 2u�Au

2 � 1� T2 T2u� 2u�1 1

1� T1 T2u� 2u�1 2

3

RDDCPd2tu�1�2N /dEvs. E

Eminp � E�0 � RT�nF

ln k�0��

DA

�� Ec � E�0 � RT

�nFln k�0

��

DA

�� yRDDCPmin

�� RT�nF

ln F B1 4 � RT�nF

ln2 1�Au

p2u�1Au� u�1�2 u

7 � �nFRT

2u� 1 � �2

G B1 8

Emaxp � E�0 � RT�nF

ln k�0��

DA

�� yRDDCPmax

�� RT�nF

ln F B2 5 � �nFRT

2u� 1 � �2

G B2 9

Eminp � Emaxp

�� RT�nF

lnF B1 F B2 6 yRDDCPmin

yRDDCPmax

�� G B1

G B2 10

With yRDCPp � dtu�1�2N /dE)p, yRDDCPmax �d2tu�1�2N /dE2)max, yRDDCPmin � d2tu�1�2N /dE)min, Au��2u� 1 2u � 2u, F(Bi)� 2(1�Bi)/(p2u�1 B

u�u�1�2i , G(Bi)� (Bi�B2i )

(B2i � 4uBi�2u)/(2u�Bi)3, with i� 1, 2, and B1 and B2 being the greatest and lowest positive roots, respectively, of the equation B4� 8uB3� 24u2B2� (24u2

� 4u)B� 4u2� 0 in the interval 0�B� 1. T1 � 1� Cu ��Cu � 1 2�8uCu

� ��2� T2 � 1� Cu �

��Cu � 1 2�8uCu

� ��2 and Cu � 0�5 1� 4u�

2��2u 2u� 1 �.

Table 2. Totally irreversible processes. theoretical expressions for the peak potentials, cross potentials and peak heights corresponding to RDCP,dtu�1�2N /dE vs. E curves and RDDCP, d2tu�1�2N /dE2 vs. E curves, for power currents of the form I(t)� I0t1/2 and I(t)� I0t

Tech-nique

u Peak potentials Ep Cross potential Ec Peak heights Peak halfwidths

RDCP 1/2 Ep � E�0 � RT�nF

ln k�0 ��DA

� �� 0�215

�1 ± yRDCPp

�� 0�343�nFRT

2 2�35RT�nF

3

1 Ep � E�0 � RT�nF

ln k�0 ��DA

� �� 0�348

�4 ± yRDCPp

�� 0�303�nFRT

5 2�51RT�nF

3

RDDCP Eminp � E�0 � RT�nF

ln k�0 ��DA

� �� 1�172

�7 Ec � E�0 � RT

�nF� yRDDCPmin

�� 0�152 �nFRT

� �2

1/2 Emaxp � E�0 � RT�nF

ln k�0 ��DA

� �� 0�704

�8 � ln k�0

��

DA

�� 0�215

� �10 yRDDCPmax

�� 0�227 �nFRT

� �2±

Eminp � Emaxp

�� 1�87 RT�nF

9

�� yRDDCPmin �yRDDCPmax

�� 0�668 13


ln k�0 ��DA

� �� 1�438

�14 Ec � E�0 � RT

�nF� yRDDCPmin

�� 0�127 �nFRT

� �2 ±

1 Emaxp � E0 � RT�nF

ln k�0 ��DA

� �� 0�717

�15 � ln k�0

��

DA

�� 0�348

� �17 yRDDCPmax

�� 0�160 �nFRT

� �2

Eminp � Emaxp

�� 2�156 RT�nF

16

�� yRDDCPmin �yRDDCPmax

�� 0�791 20



used, a totally irreversible process is characterized by theabsence of peak in the response, as is shown in curve 3 ofFigure 1c.Starting from the peak parameter measurements and

from the study of the influence of the power of time value uon the RDCP curves we have deduced that the behavior ofthe Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system in absence of n-pentanol can be considered as totally reversible foru�1 andu�1/2, whereas when the n-pentanol concentration is 0.2 M(saturation concentration), the behavior of the chargetransfer is totally irreversible for the three values of thepower of time in the applied current, u, considered inFigure 1. For an intermediate n-pentanol concentration(0.1 M), the system behaves as quasirrersible. Belowwewillcomment in greater detail on each of these situations.

4.1. Reversible Behavior

In the first place we have used RDCP and RDDCP toanalyze the behavior of the Cd�2 1.6 mM�NaClO2 ¥ 0.5 Msystem in absence of n-pentanol. Figure 2 shows theexperimental normalized RDCP (dtu�1�2N /dE±E) andRDDCP (d2tu�1�2N /dE2 ±E) curves (Figures 2a and 2b, re-spectively), corresponding to the application of a pro-grammed current of the form I(t)� I0tu for two differentvalues of the power of time u (1 and 1/2) and a currentamplitude I0� 30 �A s�u (with the transition times being ��1.071 s and 0.942 s, respectively). We have also included inthese Figures the experimental DV (dIN/dE±E) and DDV(d2IN/dE2 ±E) curves (see curves with white circles inFigures 2a and 2b, respectively) of this experimental systemcorresponding to a time pulse t�0.10 s.

From the curves shown in Figure 2a we can see that theRDCP responses are practically superimposable and, there-fore, are not affected by the u value, with their peakpotentials and heights remaining practically constant (seeTable 4). This behavior is characteristic of a reversibleelectrode process as has been indicated in reference [1].Thus, in these conditions the peak potential is equal to thereversible half wave potential of the system, Er

1�2, and thepeak height and peak half width take the values 19.49 V�1

(�nF/(4RT)) and 0.045 V (�3.53RT/(nF)), respectively(see [1, 23]).From these Figures the total agreement is also clear

between the voltammetric and chronopotentiometric de-rivative normalized curves (see curves with white circles inFigure 2a and 2b), which is also typical of reversiblebehavior [1], although the widest discrepancies with respectto the theoretical values of the peak height and peak halfwidths are obtained in DV (see Table 4).We have measured the reversible half-wave potential of

this system from the peak potentials of the dtu�1�2N /dE±Ecurves corresponding to a series of five essays with aprogrammed current of the form I0t, u�1, and a currentamplitude I0� 30 �A s�1, obtaining Er

1�2�E�0 ± (RT/nF)�ln �� (� 0.553�0.001) V. The potential is the mean of thefive experimental values and the error corresponds to thestandard deviation. From this result we have obtained thefollowing value for the formal potential E�0� (� 0.558�0.001) V, which is in good agreement with the reporteddata in the literature [8, 10, 14].From the RDDCP curves shown in Figure 2b we can

observe that, in the same way as that previously discussedfor the RDCP curves, the second derivatives are alsosuperimposable and are practically unaffected by the value

Table 3. Totally irreversible processes. Theoretical expressions for the peak potentials, cross potentials and peak currents correspondingto derivative voltammetry (DV) and double derivative voltammetry (DDV) (See [23 ± 27]). IM� I/Id with Id being the limiting diffusioncurrent for a plane electrode (Cottrell×s current)

Tech-nique

Peak potentials Ep Cross potential Ec Peak currents Peak half widths

DVdIN /dEvs. E

Ep � E�0 � RT�nF

ln k�0��tDA

�� 1 ± dIN�dEp � 2 W1�2 � 2�80

RT�nF

3

�0�591 RT�nF

� 0�319�nFRT

DDV>d2IN /dE2

vs. E


ln k�0��tDA

�� 4 Ec � E�0 � RT

�nFln k�0

��tDA

�� d2IN�dE

2minp

�� ±

�0�463 RT�nF

�0�591 RT�nF

7 � 0�134 �nFRT

� �28

Emaxp � E�0 � RT�nF

ln k�0��tDA

�� 5 d2IN�dE

2maxp

�� 1�661RT

�nF � 0�192 �nFRT

� �29

Eminp � Emaxp

�� 2�124 RT�nF

6 d2IN�dE2minp

d2IN�dE2maxp

�� 0�700 10



of u. Therefore, the peak and cross potentials remainindependent of the power of time u, with the differencebetween the maximum and the minimum potentials, �Ep,given in Table 4 being very close to 34 mV (�Ep� 68/nmVfor a reversible system according to Equation 4 of Table 2 in[1]), and with errors lower than 3 mV in all cases.The ratio between the maximum and the minimum peak

heights of the RDDCP response, � yRDDCPmin /yRDDCPmax � , for a

reversible process must be the unit, independently of u (seeEquation 6 of Table 2 in reference [1]). From Figure 2b wecan see that whereas the right branch of the RDDCP curvesare perfectly superimposable, slight discrepancies areobserved in the region corresponding to the maximum inthe left branch of these Figures. Thus, we obtain � yRDDCPmin /yRDDCPmax �� 1.04 for u�1 (current ramp) and � yRDDCPmin /yRDDCPmax �� 1.06 for u�1/2.

4.2. Irreversible Behavior

Figure 3 shows the experimental normalized RDCP andRDDCP curves (Figures 3a and 3b, respectively), corre-sponding to the application of a programmed current of theform I(t)� I0tu for three values of the u exponent (1, 1/2 and

Fig. 1. Experimental RDCP curves (dtu�1�2N /dE vs. E, with tN� t/�) obtained for the Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system inabsence of n-pentanol (curves 1), in a solution of n-pentanol 0.1 M(curves 2) and in a saturated solution of n-pentanol (0.2 M, curves3) on a SMDE. I(t)� I0tu. The values of the power of time in theapplied current u, the current amplitude I0 (in �A s�u) and �u�1/2

(in su�1/2) are: a) u�1, 60, 0.4744; b) u�1/2, 50, 0.4832 and c) u�0,40, 0.4743. A� 0.0270 cm2, n�2, DCd�2� 7.2� 10�6 cm2 s�1, ��0.66, T�297.7 K.

Fig. 2. (–) Experimental RDCP (dtu�1�2N /dE vs. E, with tN� t/�,Figure 2a) and RDDCP (d2tu�1�2N /dE2 vs. E, Figure 2b) curvesobtained for the Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system in absenceof n-pentanol on a SMDE. I(t)� I0tu. The values of the power oftime u in the applied current are on the curves. I0� 30 �A s�u. (�)Experimental DV (dIN/dE vs. E, with IN� I/Id, Figure 2a) andDDV (d2IN/dE2 vs. E, Figure 2b) curves corresponding to the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in absence of n-pentanol on aSMDE. Time pulse t�0.10 s, Id� 42.95 �A. A� 0.0299 cm2. Otherconditions as in Figure 1.



0), to the Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system with aconcentration of n-pentanol 0.2 M, which corresponds to asaturated solution of this alcohol, together with the DVandDDV curves (see curves with white circles in Figures 3a and3b, respectively), for the same system.From Figure 3 it can be deduced immediately that the

normalized curves obtained in RDCP (see Figure 3a) and inRDDCP (see Figure 3b) are strongly dependent on thevalue of the exponent u of the programmed current applied,a fact which shows that the behavior of this system is notreversible. Furthermore, we can see that the RDCP curvecorresponding to a current step presents no peak (see curvewith u�0 in Figure 3a). The absence of peak in the RDCPresponse for the application of a constant current is a typicalfeature of a totally irreversible process, which means that inthese conditions the process behaves as totally irreversiblefor u�0 (constant current).In Figure 4 we have represented the variation of the peak

potentials Ep with ln �1/2 corresponding to the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in a saturated solution ofn-pentanol for four different current amplitudes I0 whenu�1 (circles) and u�1/2 (squares). As can be observed, thisvariation is linear with slope (RT/�nF) independently of thevalue of u. This behavior shows that this system also behavesas totally irreversible when u�0 (see Equations 1 and 10 ofTable 2).From Figure 3a it can also be deduced that for u�0, the

peak potentials in RDCP curves are shifted towards morenegative values and the peak heights increase as u dimin-ishes. These results are in agreement with those shown inTables 1 and 2.As the peak heights of the normalized RDCP curves are

only dependent on the value of u and the charge transfercoefficient� (seeEquation 2 in Table 1), we have calculatedthe value of � from the experimental RDCP peak heights ofcurves in Figure 3a and obtained practically identical valuesfor different u powers (see Table 5). Likewise, by applyingEquation 1 in Table 1 we have calculated k�0 from the peakpotentials of the RDCP curves and have again obtainedvalues that are in agreement, when different values of thepower of time u are used (see Table 5).It is important to point out the improvement in the peak

resolution that RDCP possesses when we compare it with

Table 4. Experimental values of the peak and cross potentials, normalized peak heights and normalized peak currents in RDCP,RDDCP, DV and DDV techniques for the Cd2� 1.6 mM�NaClO4 ¥ 0.5 M system in absence of n-pentanol on a SMDE (see Figure 2).The current applied in RDCP and RDDCP hast the form I(t)� I0tu with I0� 30 �A s�u. The time pulse for DV and DDV has beent� 0.10 s, n� 2, A� 0.0299 cm2, DCd�2� 7.2� 10�6 cm2 s�1, �� 0.66, T� 297.7��

RDCP RDDCP

I(t)� I0tu, u Ep (V) yRDCPp (V�1) WRDCP1�2 (V) Eminp (V) Emaxp (V) Ecross (V) Eminp � Emaxp

�� V I0t, u� 1 �0.553 19.410 0.046 �0.570 �0.534 �0.553 0.036I0t1/2, u� 1/2 �0.553 19.714 0.045 �0.570 �0.535 �0.553 0.035

DV DDV

Ep (V) � dIN/dEp� (V�1) WDV1�2 (V) Eminp (V) Emaxp (V) Ecross (V) Eminp � Emaxp

�� V �0.553 19.847 0.048 �0.570 �0.534 � 0.552 0.036

Fig. 3. (–) Experimental RDCP (dtu�1�2N /dE vs. E, with tN� t/�,Figure 3a) and RDDCP (d2tu�1�2N /dE2 vs. E, Figure 3b) curvesobtained for the Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system in asaturated solution of n-pentanol (0.2 M) on a SMDE. I(t)� I0tu.The values of the power of time u in the applied current are on thecurves. The values of the current amplitude I0 (in �A s�u) and �u�1/2

(in su�1/2) are: 30, 0.958; 50, 0.483 and 40, 0.474. (�) ExperimentalDV (dIN/dE vs. E, with IN� I/Id, Figure 3a) and DDV (d2IN/dE2 vs.E, Figure 3b) curves corresponding to the Cd�2 1.6 mM�NaClO4

¥ 0.5 M system in a saturated solution of n-pentanol (0.2 M) on aSMDE. Time pulse t�0.50 s, Id� 18.38 �A. Other conditions as inFigure 1.



the signal obtained in derivative voltammetry. From Fig-ure 3a it is evident that the peaks obtained in RDCP arebetter resolved than those obtained in DV [28, 29]. So, forthe least favorable case of a current ramp, u�1 (see curvewith u�1 in Figure 3a), we obtain a value of WRDCP

1�2 �93 mV, as opposed to the 120 mV corresponding to theDV curve (see curve with white circles in Figure 3a).Moreover, it can be also seen in Figures 3a and 3b that the

RDCP andRDDCP curves do not coincide with theDVandDDVcurves, respectively (see curvewithwhite circles). Thereason for this is that the temporal dependence of bothresponses cannot be eliminated as it can in the case of atotally reversible process (see Figure 2 for comparison).According toEquations 1, 3 ± 4, 10 and 12 ± 13 inTable 2 andEquations 1, 5 and 6 in Table 3, the DV and DDV peaksappear at more positive potentials than those of the RDCPcurves when a similar time scale is used in both techniques.The peak currents in the DV and DDV curves have an

intermediate value between that obtained in RDCP andRDDCP, respectively, for u�1 and u�1/2 (see Equation 2of Table 3 for the peak current and Equations 2 and 11 ofTable 2 for the peak heights yp).FromFigure 3b for RDDCP, it can be observed that when

u diminishes both peak potentials are shifted towards morenegative values and �Ep and Ry diminish (see Table 5).From the values of�Ep of curves of Figure 3b it is possible

to determine the value of � by using Equation 5 of Table 1.We have recalculated � and obtained the values whichappear in Table 5. It should be noted that both the value ofthe peak height yRDCPp in the RDCP curves of Figure 3a, andthe difference between the peak potentials �Ep in theRDDCP curves of Figure 3b are independent of theduration time of the experiment (see Equations 2 and 5,respectively, of Table 1).In all the cases the values of� and k�0 obtained are in good

agreement with the data reported for this system in theliterature [8, 14, 31].

4.3. Quasireversible Behavior

In Section 4.1. it has been demonstrated that the Cd�2

1.6 mM�NaClO4 0.5 M system in absence of n-pentanolpresents a totally reversible behavior when a current timefunction I(t)� I0tu with u�1 and 1/2 and I0� 30�As�u isapplied. However, when a constant current or even when apower of time u�1/5 of higher current amplitude I0 is used,the peak potentials obtained in RDCP are shifted towardsmore negative values as u diminishes or I0 increases. Thisbehavior indicates that the process becomes more irrever-sible as u decreases, something which is in agreement withthe theoretical predictions [3, 14].For quasireversible processes it is not possible to use any

simplification in the theoretical equations for the potentialtime response and its derivatives for the characterization ofthe peak potentials, cross potential and peak heights of theRDCP and RDDCP responses.In order to obtain the values of the heterogeneous rate

constant k�0 and the charge transfer coefficient � of the Cd�2

Fig. 4. Dependence of the experimental peak potential Ep of theRDCP curves (dtu�1�2N /dE vs. E, with tN� t/�) with the logarithm of�1/2 corresponding to the Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system ina saturated solution of n-pentanol (0.2 M) on a SMDE. I(t)� I0tu.The values of the power of time u are: (�) u�1; (�) u�1/2. Thevalues of I0 (in �A s�u) are in the curves. T�297.5 K. Otherconditions as in Figure 1.

Table 5. Experimental values of the peak and cross potentials and normalized peak heights in RDCP and RDDCP techniques for theCd�2 1.6 mM�NaClO4 ¥ 0.5 M system in a saturated solution of n-pentanols (0.2 M) on a SMDE (see Figure 3). I(t)� I0tu. The values ofthe current amplitude I0 (in �A s�u) and �u�1/2 in su�1/2) are: u� 1, 30, 0.959 and u� 1/2, 50, 0.483. n� 2, A� 0.0270 cm2, DCd�2� 7.2� 10�6cm2 s�1, �� 0.66, T� 297.7 ��

RDCP

I(t)� I0tu, u Ep (V) yRDCPp

�� V�1 � log (k×0 cm s�1))

I0t, u� 1 �0.610 9.452 0.40� 0.02 � 3.34� 0.03I0t1/2, u� 1/2 �0.626 10.429 0.39� 0.02 � 3.32� 0.02

RDDCP

I(t)� I0tu, u Eminp (V) Emaxp (V) Eminp � Emaxp

�� V yRDDCPmin �yRDDCPmax

��

I0t, u� 1 �0.649 �0.577 0.072 0.784 0.38� 0.02I0t1/2, u� 1/2 �0.648 �0.589 0.059 0.640 0.40� 0.02



1.6 mM�NaClO4 ¥ 0.5 M system in these conditions, poten-tial time curves have been calculated numerically by usinggeneralEquation 1 for different values ofk�0 and�, and theirreciprocal derivatives have been obtained. By comparingthe experimental curves obtained for a value of u�1/10 in arange of transition times �� 0.5 ± 1.0 s with the calculatedtheoretical RDCP curves (see Figures 5a and 5b whichcorrespond to two values of the current amplitude in theapplied current, I0� 30 and 40 �A s�1/10, respectively), wehave obtained that the best fittings correspond to thefollowing values:

log k�0� (�1.22� 0.02) cm s�1 and �� (0.50� 0.02)(Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system in absence ofn-pentanol)

with errors in the peak and cross potentials being lowerthan 1 mVand differences in the peak heights below 4%.These values are in good agreement with previous resultsin the literature [10, 11].Figure 6 shows the RDCP and RDDCP curves (Figur-

es 6a and 6b, respectively), for the system Cd�2 1.6 mM�NaClO4 ¥ 0.5 M in a nonsaturated solution of n-pentanol(0.1 M). From these figures is clear that these curves are notcoincident for the different values of u and, therefore, thissystem is not reversible. However, unlike the curves for theCd�2 1.6 mM�NaClO4 ¥ 0.5 M system in a saturated solu-tion of inhibitor (Figure 3), the RDCP curve correspondingto the application of a constant current (curve with u�0 inFigure 6a) presents a peak and we can, therefore, conclude

Fig. 5. Comparison between theoretical (�) and experimental(–) RDCP curves (dtu�1�2N /dE vs. E, with tN� t/�) for the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in absence of n-pentanol on aSMDE when a power current I(t)� I0t1/10, u�1/10 is applied(Quasireversible behavior). The values of I0 (in �A s�0.1) and of �0.6

(in s0.6) are: a) 30, 0.7837 and b) 40, 0.5878. A� 0.0299 cm2.Theoretical RDCP curves have been calculated from Equa-tions 1 ± 3 and 12 with n�2, DCd�2� 7.2� 10�6 cm2 s�1, �� 0.66, k�0� 0.06 cm s�1, �� 0.50 and E�0�� 0.558 V. The values of �0.6 (ins0.6) calculated from Equation 3 are: a) 0.7837 and b) 0.5878.

Fig. 6. Quasireversible behavior of experimental RDCP (dtu�1�2N /dE vs. E, with tN� t/�, Figure 6a) and RDDCP (d2tu�1�2N /dE2 vs. E,Figure 6b) curves (–) obtained for the Cd�2 1.6 mM�NaClO4 ¥0.5 M system in a solution of n-pentanol 0.1 M on a SMDE when apower current I(t)� I0tu with u�0, 1/2 and 1 is applied. The valuesof the current amplitude I0 (in �A s�u) and �u�1/2 (in su�1/2) are:u�1, 30, 0.954; u�1/2, 50, 0.483 and u�0, 40, 0.474. (�)Experimental DV (dIN/dE vs. E, with IN� I/Id, Figure 6a) andDDV (d2IN/dE2 vs. E, Figure 6b) curves corresponding to the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in a solution 0.1 M of n-pentanolon a SMDE. Time pulse t�0.50 s, Id� 18.05 �A. Other conditionsas in Figure 1.



that under these conditions the behavior of the system isquasirreversible for u�0.We have also confirmed that the variation of the peak

potential of the RDCP curves obtained for different valuesof the current amplitude I0 for u�1 and u�1/2, versus theln �1/2 is not linear, which points out that the behavior of thissystem is also quasirreversible for u greater than zero (seeFigure 4 for a totally irreversible process for comparison).Moreover, it is also observed in Figure 6a that, as was

previously described in Figure 3a for a totally irreversibleprocess, the half widths of the RDCP curves decrease with ufrom a value of 75 mV for the case of u�1 to a value of52 mV foru�0 and are, in all cases, lower than that obtained

in DV, which has a value of 81 mV (see curve with whitecircles in Figure 6a).As in the previously discussed case, in order to determine

the kinetic parameters of this system we must use generalEquation 1 for the potential time curve and generatederivative curves for different values of k�0 and � in orderto obtain a good agreement between the experimentalvalues and the theoretical curves.By comparing the experimental RDCP curves obtained

for values of u�1 and 1/2 in a range of transition times ��0.5 ± 1.0 s with the theoretical curves (see Figure 7 obtainedfor u�1, Figure 7a, and u�1/2, Figure 7b with I0� 30 �As�u), we have obtained the following values of the logarithmof the rate constant and the charge transfer coefficient:log k�0� (� 2.94� 0.03) cm s�1 and �� 0.50� 0.01(Cd�2 1.6 mM�NaClO4 ¥ 0.5 M system in a solution 0.1 Mof n-pentanol)

with errors in the peak and cross potentials being lower than1 mVand differences in the peak heights below 4%. Thesevalues are in good agreement with those obtained in theliterature [14].

5. Acknowledgements

The authors greatly appreciate the financial support pro-vided by the Direccio¬ n General de Investigacio¬ n CientÌficay Te¬cnica (Project Number PB96-1095) and to the Funda-cio¬ n SENECA (Expedient number 00696/CV/99). JG andMMM express their thanks to CajaMurcia and to theFundacion SENECA, respectively, for the grants received.

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Fig. 7. Comparison between theoretical (�) and experimental(–) RDCP curves (dtu�1�2N /dE vs. E, with tN� t/�) for the Cd�2

1.6 mM�NaClO4 ¥ 0.5 M system in a solution of n-pentanol 0.1 Mon a SMDE when a power current I(t)� I0tu is applied. I0�30.0 �A s�u, A� 0.0270 cm2. The values of the power of time inthe applied current u and �u�1/2 (in su�1/2) are: a) u�1, 0.9487 andb) u�1/2, 0.8053. Theoretical RDCP curves (dtu�1�2N /dE vs. E)have been calculated from Equations 1 ± 3 and 12 with n�2, DCd�2

� 7.2� 10�6 cm2 s�1, �� 0.66, k�0� 1.5� 10�3 cm s�1, �� 0.50, E�0�� 0.558 V. The values of �u�1/2 (in su�1/2) calculated fromEquation 3 are: a) 0.9487 and b) 0.8053.



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