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Journal of Electroanalytical Chemistry 648 (2010) 67–77
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Journal of Electroanalytical Chemistry
journal homepage: www.elsevier .com/locate / je lechem
Analytical solution for Reverse Pulse Voltammetry at spherical electrodes:A remarkably sensitive method for the characterization of electrochemicalreversibility and electrode kinetics
Ángela Molina a,*, Francisco Martínez-Ortiz a, Eduardo Laborda a, Richard G. Compton b,**
a Departamento de Química Física, Universidad de Murcia, Espinardo, 30100 Murcia, Spainb Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom
a r t i c l e i n f o
Article history:Received 12 May 2010Received in revised form 22 June 2010Accepted 25 June 2010Available online 1 July 2010
Keywords:Reverse Pulse VoltammetrySlow charge transfer processHeterogeneous rate constantTransfer coefficientMicroelectrodes
1572-6657/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.jelechem.2010.06.019
* Corresponding author. Tel.: +34 68 307100; fax: +** Corresponding author.
E-mail addresses: [email protected] (Á. Molina), rich(R.G. Compton).
a b s t r a c t
Reverse Pulse Voltammetry (RPV) is a powerful double pulse technique for kinetic studies. The theory ofthis technique for slow charge transfer processes at spherical electrodes is developed. An explicit analyt-ical solution is deduced applicable to electrodes of any size, whatever the reversibility of the system andthe length of the two potential pulses. From this solution we examine the influence on the RPV responseof the electrode sphericity and the kinetics of the electrode process. It is shown that visual inspection ofthe curves provides immediate valuable information on electrochemical kinetics. Moreover, simple diag-nosis criteria are established and general working curves derived for the quantitative extraction of thekinetic parameters and the formal potential. Some particularities in the morphology of the anodic branchof RPV curves for slow charge transfer processes when the second potential pulse is sufficiently large arealso described. The merits of RPV as an exquisitely sensitive measure of electrochemical reversibility andelectrode kinetics are advocated.
� 2010 Elsevier B.V. All rights reserved.
1. Introduction
This paper seeks to advocate Reverse Pulse Voltammetry (RPV)as a very powerful double pulse technique for the characterizationof electrode reactions [1,2] and of electrochemical reversibility andelectrode kinetics in particular. In the method, during the first stepthe potential is set at diffusion limiting conditions and subse-quently the electrogenerated species is electrochemically exam-ined by scanning the second applied potential in the potentialregion of interest. The current is sampled at the end of the secondpulse and RPV curve is a plot of the sampled current vs. the value ofthe second potential.
RPV technique offers several advantages against other commonelectrochemical techniques such as cyclic voltammetry, permittinga higher control of the reaction conditions, minimization of charg-ing current effects and it is more convenient for quantitative anal-ysis [3]. In addition, the mathematical treatment needed to tacklethis technique is considerably less complex.
Among other applications, this technique has been employedfor evaluation of the electrochemical reversibility of electrode
ll rights reserved.
34 68 364148.
processes [3,4], in the study of electrogenerated products [3] andreaction mechanisms [5–8] and for determination of diffusioncoefficients [9], being particularly valuable when unstable prod-ucts are involved.
According to the potential program of RPV technique, beforethe application of each double pulse, equilibrium conditionsare recovered [5,8,10]. So, from a theoretical point of view, theRPV problem is equivalent to that of the application of a doublepotential pulse for which the first step is controlled by diffusion.The theory of RPV technique for charge transfer processes withfinite kinetics has been only developed for planar diffusion[11–15], but no analytical solution is available for electrodes ofspherical geometry. The aim of this paper is to fill this gap bypresenting an easily manageable rigorous and explicit analyticalsolution for the current–potential–time response under RPV con-ditions, valid for spherical electrodes of any size, for any lengthof the two potential pulses and whatever the reversibility of thecharge transfer process. We solve the problem by means of amathematical procedure based on an extension of Koutecky’sdimensionless parameter method detailed in Appendix A.
Having at our disposal an explicit analytical solution is a greatadvance with respect to numerical calculations since from the ana-lytical equation the variables determining the system response canbe entirely examined and simplified expressions for particularcases of interest can be derived. Thus, from the solution obtainedwe analyze the effect of the electrode sphericity as well as the
68 Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77
value of the RPV for the determination of the electrode kinetics. Inthis paper it is shown that the electron transfer coefficient and theelectrochemical reversibility can be easily estimate by analysis ofRPV curves. Indeed, at spherical electrodes the width of RPV curvesstrongly depends on the heterogeneous rate constant and the sym-metry on the transfer coefficient, in such a way that simple visualinspection of RPV curves provides an assessment of the kineticcharacter of the charge transfer reaction. An additional insight ofthis paper is that single point measurements in RPV curves (thehalf wave potentials of the anodic and cathodic branches) are goodenough to achieve the complete characterization of the systemfrom RPV experiments performed with different double pulsedurations, making unnecessary the use of complex fitting proce-dures. For this purpose, working curves for the extraction of the ki-netic parameters and the formal potential are given.
The case at which both electroactive species are initially presentin solution is also covered by the theory here developed. This as-pect is of great interest since there exist several situations at whichthe presence of both members of the redox couple is expected,such as for the study of metal species [16] or of some ions whichdisproportionate into two electroactive species [17].
It is worth highlighting that these theoretical results are validfor any value of the duration of either potential pulses, which isespecially valuable when studying charge transfer processes withfinite kinetics. The appearance of a maximum in the anodic branchof the RPV curve is described for charge transfer processes with fi-nite kinetics when the duration of the second pulse is similar tothat of the first one. This behaviour is analyzed and the conditionsunder which it is most obvious are pointed out.
The clear merits of RPV for quantitative voltammetric measure-ments of electrode processes are evident.
2. Theory
We will study the case of a charge transfer reaction taking placeat a stationary spherical electrode:
Oþ e� ��! ��kf
kb
R ð1Þ
After applying an electrochemical perturbation, the diffusion masstransport of the electroactive species is described by the followingdifferential equation system:
dcðjÞO ðr; tÞ ¼ dcðjÞR ðr; tÞ ¼ 0; j � 1;2 ð2Þ
where the superscript refers to the jth pulse and d is the diffusionoperator for spherical geometry:
(a)
Fig. 1. Reverse Pulse Voltammetry technique: (
d ¼ @
@t� D
@2
@r2 þ2r@
@r
!ð3Þ
We assume that both electroactive species have the same value ofthe diffusion coefficient (DO = DR = D).
The RPV waveform consists of successive double potentialpulses, the first potential value being set at diffusion limiting con-ditions (E1 ? �1, 0 6 t 6 t1) and the value of the second one vary-ing towards anodic potentials (E2, t1 6 t 6 t2) [1,2,11]. The currentis sampled at the end of the second pulse (I2(t2)) and plotted vs.E2 value (see Fig. 1).
Given that the initial equilibrium conditions are re-establishedbefore each double potential pulse, the theory for RPV techniquematches that for the application of two potential steps withE1 ? �1. By dividing the double pulse time into two periods(0 6 t 6 t1 and t1 6 t 6 t2), the solution for the first step corre-sponds to the well-known solution for a charge transfer processat a spherical electrode under limiting conditions, the current–time expression being [1,2]:
I1dðtÞ ¼ Idð1Þ 1þ r0ffiffiffiffiffiffiffiffiffipDtp
� �ð4Þ
For t P t1 the applied potential is stepped to E2 and the bound-ary value problem is given by:
s2 � 0; r !1s2 ¼ 0; r � r0
�cð2ÞO ¼ cð1ÞO ; cð2ÞR ¼ cð1ÞR ð5Þ
s2 > 0; r ¼ r0 :@cð2ÞO ðr; tÞ
@r
!r¼r0
¼ � @cð2ÞR ðr; tÞ@r
!r¼r0
ð6Þ
D@cð2ÞO
@r
!r¼r0
¼ k0K�a2 cð2ÞO ðr0; tÞ � k0K1�a
2 cð2ÞR ðr0; tÞ ð7Þ
In Eq. (7) we assume that the transition state does not changewith potential and so the same value of a applies to the forwardand reverse processes.
We introduce the following variable change:
uðjÞi ðr; tÞ ¼cðjÞi ðr; tÞ � r
c�O � r0;
i � O;Rj � 1;2
ð8Þ
so that the differential equation system becomes equivalent to thatof linear diffusion:
@uðjÞi ðr; tÞ@t
� D@2uðjÞi ðr; tÞ
@r2 ¼ 0;i � O;R
j � 1;2ð9Þ
(b)
a) potential–time program; (b) RPV curve.
Table 1Notation and definitions.
r0 Radius of the electrodetj, j � 1, 2 Duration of the jth potential pulseD Diffusion coefficient of the electroactive speciesc�i ; i � O;R Bulk concentration of species i
cðjÞi ; i � O;Rj � 1;2
Concentration profile of species i for the jth potential pulse
kf, kb Heterogeneous rate constants of reduction, oxidationprocesses
k0 Standard heterogeneous rate constant
kmicros ¼ k0 � r0=D
a Electron transfer coefficientE0 Formal potential of the electroactive coupleEj, j � 1, 2 Potential applied during the jth pulse
Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77 69
Given that the diffusion operator is linear, the solutions of thedifferential equation system can be supposed as linear combina-tions of solutions:
uð2ÞO ðr; tÞ ¼ uð1ÞO ðr; tÞ þ ~uð2ÞO ðr; tÞ ð10Þuð2ÞR ðr; tÞ ¼ uð1ÞR ðr; tÞ þ ~uð2ÞR ðr; tÞ ð11Þ
where uð1Þi ; i � O;R are the solutions corresponding to the first po-tential step and ~uð2Þi ðr; tÞ are the unknown partial solutions.
Considering the RPV waveform, the problem notably simplifiesas a consequence of the condition E1 ? �1. Thus, the solutions forthe first pulse fulfil the condition that the surface concentration ofboth species are constant [18]:
uð1ÞO ðr0; tÞ ¼ 0; uð1ÞR ðr0; tÞ ¼ 1þ l ð12Þ
and the derivative of the function uð1ÞO ðr; tÞ at r = r0 is given by:
@uð1ÞO ðr; tÞ@r
!r¼r0
¼ 1ffiffiffiffiffiffiffiffiffipDtp þ 1
r0ð13Þ
Taking into account the above considerations, the boundary va-lue problem turns into:
s2 � 0; r !1s2 ¼ 0; r � r0
�~uð2ÞO ¼ 0; ~uð2ÞR ¼ 0 ð14Þ
s2 > 0; r ¼ r0 :@~uð2ÞO ðr; tÞ
@r
!r¼r0
�~uð2ÞO ðr0; tÞ
r0
¼ � @~uð2ÞR ðr; tÞ@r
!r¼r0
�~uð2ÞR ðr0; tÞ
r0
24
35 ð15Þ
@~uð2ÞO
@r
!r¼r0
�~uð2ÞO ðr0; tÞ
r0� k0K�a
2
D~uð2ÞO ðr0; tÞ � K2~uð2ÞR ðr0; tÞ� �
¼ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipDðt1 þ t2Þ
p þ 1r0
!� k0K1�a
2
Dð1þ lÞ ð16Þ
Next, we introduce the dimensionless variables:
s2 ¼r � r0
2ffiffiffiffiffiffiffiffiDt2p ð17Þ
v2 ¼2ffiffiffiffiffiffiffiffiDt2p
r0þ 2
ffiffiffiffit2
D
rk0K�a
2 ð1þ K2Þ ð18Þ
b ¼ t2
t1 þ t2ð19Þ
and by following the mathematical procedure detailed in AppendixA, we obtain the expression for the current–potential–time re-sponse at the second potential step under RPV conditions:
IRPV ¼ I1dðt1 þ t2Þ þIpdðt2Þ
1þ kmicros K�a
2 ð1þ K2Þ
�ffiffiffiffipp
2Zv2 � b1=2 þ kmicro
s K�a2 ð1þ K2Þ Z � Fðv2Þ þ Y � Hðv2; bÞ
� �ð20Þ
with all the variables and functions given in Table 1 and AppendixB.
The validity of the above analytical solution has been studied bycomparison with numerical calculations [19]. An excellent agree-ment between analytical and numerical results was obtained forany electrode size, for any length of the potential pulses and what-ever the reversibility of the electrode process.
2.1. Particular cases
From the above expression (Eq. (20)), others for some particularcases of interest can be deduced:
– Planar electrodes (r0 ?1) [11]:
IpRPV ¼ Ip
dðt1 þ t2Þ þ Ipdðt2Þ½Zp � Fðvp
2Þ þ Y � Hðvp2;bÞ� ð21Þ
– Ultramicroelectrodes (UME, r0 ffiffiffiffiffiffiffiffiffiffiffipDt2p
) [20]:
IUMERPV ¼ Idð1Þ � kmicro
s � K�a2 ð1� lK2Þ
1þ kmicros K�a
2 ð1þ K2Þð22Þ
– Reversible process (k0 ?1) [9,21]:
IrevRPV ¼ I1dðt1 þ t2Þ � Ip
dðt2Þð1þ lÞ K2
1þ K2
� �1þ
ffiffiffiffiffiffiffiffiffiffiffipDt2p
r0
� �ð23Þ
– Double Potential Step Chronoamperometry with diffusion limitingcurrent potentials (E1 ? �1, E2 ? +1) [21]:
Id;RPV ¼ IRPVðK2 !1Þ ¼ I1dðt1 þ t2Þ � Ipdðt2Þð1þ lÞ 1þ
ffiffiffiffiffiffiffiffiffiffiffipDt2p
r0
� �ð24Þ
3. Results and discussion
In Fig. 2 we study the influence of the kinetics of the chargetransfer process on RPV curves at a typical spherical microelec-trode (r0 = 50 lm) by plotting the IRPV vs. E2 curves for differentvalues of the heterogeneous rate constant (k0, Fig. 2a) and thetransfer coefficient (a, Fig. 2b).
As can be seen in Fig. 2a, the value of k0 affects the shape of RPVcurves so that when k0 diminishes there is a gradual split into twowaves: a cathodic wave corresponding to the reduction of speciesO and an anodic one related to the oxidation of the species R elec-trogenerated during the first pulse. This split starts appearing forquasi-reversible systems and it is fully developed for totally irre-versible ones. Thus, in the latter case two well-defined waves areobtained which separates as k0 value diminishes.
Regarding the transfer coefficient (a), Fig. 2b shows the influ-ence of this parameter on the RPV curves. It is observed that thecathodic wave shifts towards more negative potentials (i.e., greateroverpotential) as the a value diminishes, whereas the anodic oneshifts towards more positive potentials when the a value increases.Thus, the response in RPV reflects the symmetry of the electrode
(b)
E2 - E0' / V-0.4 -0.2 0.0 0.2 0.4
Ι RPV
/ Ι d
p (t 2)
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
(a)
E2 - E0' / V-0.4 -0.2 0.0 0.2 0.4
Ι RPV
/ Ι d
p (t 2)
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0 5(cm / s) 10k −=
310−
1
0.5
0.3α =
0.7
410−
Fig. 2. (a) Influence of the heterogeneous rate constant (k0) on RPV curves (Eq. (20)); k0 values marked on the curves, a = 0.5. (b) Influence of the electron transfer coefficient(a) on RPV curves (Eq. (20)); a values marked on the curves, k0 = 10�3 cm/s. r0 = 50 lm, t1 = 1 s, t1/t2 = 20, D = 10�5 cm2/s, l = 0.
70 Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77
reaction so that for low values of the transfer coefficient a ‘‘facili-tated” anodic process is obtained, and vice versa for high a values.
The above features of the shape of RPV curves give rise to simpleand direct qualitative criteria for the investigation of the kineticcharacteristics of the electrode process; in particular, from visualinspection of the RPV response one can estimate the reversibilityof the system as well as whether the transfer coefficient is higher,equal or smaller than 0.5.
As expected, neither the cathodic nor the anodic limiting cur-rents are affected by the kinetic parameters (see Eq. (24)), thesemagnitudes being useful for the determination of bulk concentra-tions, electrode radius or diffusion coefficients [9,21].
Eq. (20) is valid for spherical electrodes of any size, which en-ables us to analyze the effect of the electrode sphericity on RPV re-sponse. In Fig. 3 this influence is plotted for the three general cases:reversible (33), quasi-reversible (� � �� � �� � �� � �� � �) and totally irre-versible (– – – –) systems, considering that both electroactive spe-cies are initially present (l = 1).
We can observe that the ratio |Id,RPV/Id,DC| decreases as theelectrode radius diminishes. At ultramicroelectrodes (Fig. 3c)
the steady state is reached and the second potential pulse doesnot ‘‘remember” the first one. As a consequence, the RPV re-sponse tends to that corresponding to Normal Pulse Voltamme-try I2 vs. E2 (grey lines in Fig. 3c) [9] and the anodic–cathodiccurve obtained under these conditions fulfils that |Id,RPV/Id,DC| ? 1.
The analytical solution herein presented (Eq. (20)) is totallygeneral and rigorous for RPV conditions (E1 ? �1) and so thecomplete analysis of the influence of the different experimentalvariables can be carried out with no limitations. In particular, forthe study of slow charge transfer processes the effect of the dura-tion of the second potential pulse (t2) is very interesting.
In Fig. 4, RPV curves for different values of t2 are plotted forreversible (Fig. 4a), quasi-reversible (Fig. 4b) and irreversible(Fig. 4c) systems. We can observe the time evolution of the shapeof the RPV curve, which shows the transition towards more revers-ible behaviour as time passes. Moreover it is clear the rapid de-crease of the anodic current with time in relation to the cathodicone so that for t2 = 2 � t1 the relative magnitude of the anodicbranch is almost null in all the cases.
(a)
E2 - E0' / V-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Ι RPV
/ Ι d
p (t 2)
-1.5
-1.0
-0.5
0.0
(b)
E2 - E0' / V-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Ι RPV
/ Ι d
p (t 2)
-2
-1
0
1
(c)
E2 - E0' / V-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
Ι RPV
/ Ι d
p (t 2) ;
Ι NPV
/ Ι d
p (t)
-20
-10
0
10
20
Fig. 3. Influence of electrode sphericity on RPV curves for reversible k0 = 1 cm/s (33), quasi-reversible k0 = 10�3 cm/s (� � �� � �� � �� � �� � �) and irreversible k0 = 10�5 cm/s systems(– – – –), when both electroactive species are initially present with l = 1. Three electrode sizes are considered: (a) planar electrode (Eq. (21), n2 ? 0), (b) microelectrode (Eq.(20), n2 = 0.5) and (c) ultramicroelectrode (Eq. (22), n2 = 20); grey lines correspond to NPV curves with t = 0.05s. Other conditions as in Fig. 2.
Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77 71
Fig. 4. Variation of the RPV curve with t2 for: (a) reversible k0 = 1 cm/s, (b) quasi-reversible k0 = 10�3 cm/s and (c) irreversible k0 = 10�5 cm/s systems at a sphericalmicroelectrode (r0 = 50 lm, Eq. (20)). Four t2 values are considered: t2 = 10 ms (33), t2 = 50 ms , (– – – –) t2 = 200 ms (–�–�–�–) and t2 = 2s (� � �� � �� � �� � �� � �). Other conditions asin Fig. 2. For each case the double step chronoamperograms corresponding to four different regions of the RPV curve are shown.
72 Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77
Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77 73
For a simple charge transfer process (E mechanism), when onlythe oxidized species is initially present (i.e., l = 0) and the diffusioncoefficients of both electroactive species have the same value(DO = DR = D) the anodic current density (Id;RPV=Ip
dðt2Þ) is indepen-dent of the electrode size and of the reversibility of the electrodeprocess, being given by [21]:
Id;RPVðl ¼ 0ÞIp
dðt2Þ¼ b1=2 � 1 ð25Þ
In the case of planar electrodes and ultramicroelectrodes,Id;RPVðl ¼ 0Þ=Ip
dðt2Þ is also independent of the value of the relationbetween the diffusion coefficients [9].
So, recording Id,RPV for different t2 values and checking the lineardependence Id;RPV=Ip
dðt2Þ vs. b1/2 is an easy experimental procedurewith which to discard the possibility of complications subsequentto the electrode reaction with the advantage of not being affectedby any uncertainty in the electrode radius.
By extending the study of the t2 influence to larger electrodes, astriking maximum is found in the anodic wave of the RPV curve for
Fig. 5. (a) Variation of RPV curve with t2 for an irreversible system (k0 = 10�5 cm/s) at a sI2 � t2 chronoamperograms for two values of the second potential pulse: E2 � E0 = + 300 mof k0 (indicated on the graph) at a planar electrode (Eq. (21)) with t2 = 2 � t1; (d) RPV curvsystem (k0 = 10�5 cm/s) with t2 = 2 � t1 (Eq. (20)). Other conditions as in Fig. 2. (For interpweb version of this article.)
quasi-reversible and irreversible systems when t2 is sufficientlylarge (see Fig. 5a).
This atypical anodic peak is characteristic, and so indicative, ofslow charge transfer processes (k0 < 10�3 cm/s) not being found forreversible ones as can be seen in Fig. 5c. In this figure we also ob-serve that the maximum increases as k0 diminishes up to a limitsituation corresponding to totally irreversible systems for whichthe magnitude of the peak is independent of k0 whereas its posi-tion shifts towards more positive potentials as k0 value diminishes.
Note that the appearance of the anodic peak implies that, be-yond the peak potential, smaller anodic currents are obtained atmore anodic potentials. This situation is shown in Fig. 5b wherewe can see that at a given time there is a crossing of the chronoam-perograms so that the current corresponding to the maximum(E2 � E0 = + 300 mV, black curve) becomes greater than the anodiclimiting current (E2 � E0 = + 700 mV, blue curve). The appearanceof a maximum in the anodic branch of RPV and Normal Pulse Vol-tammetry curves has also been described in amalgam systemswhen small sized electrodes are used [18,22], although in that casethe phenomenon is related with the depletion of species R whereas
pherical electrode with r0 = 200 lm (Eq. (20)). t2 Values indicated on the graph; (b)V (black line) and E2 � E0 = + 700 mV (blue line); (c) RPV curves for different values
es for different electrode radius (r0 values indicated on the graph) for an irreversibleretation of the references to colour in this figure legend, the reader is referred to the
(a)
Log (ksmicro)
-4 -3 -2 -1 0 1 2 3 4
Δ E1/
2 / m
V
0
200
400
600
800
1000
1200
α = 0.5
0.3
0.7
Log (ksmicro)
-4 -3 -2 -1 0 1 2 3 4
Σ E1/
2 - 2
E0 / m
V
-400
-200
0
200
400
α = 0.5
0.3
0.7(b)
Fig. 6. Variation of: (a) DE1/2 and (b) RE1/2 � 2E0 with kmicros for different values of the transfer coefficient (indicated on the curves). n2 = 0.45, t1/t2 = 20.
74 Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77
in the present case it is related with the reversibility of the elec-trode process.
This unusual feature in RPV response is the more apparent thelonger the second pulse duration (see Fig. 5a) and it is promotedby large electrodes, so that the greatest peak is obtained at planarelectrodes whereas it is not observed at microelectrodes (seeFig. 5d).
3.1. Determination of kinetic parameters of the charge transfer process
As was discussed from the results obtained in Fig. 2, the shapeand symmetry of the RPV curve gives information about the kinet-ics of the electrode process. To take advantage of this feature forthe extraction of kinetic parameters we will define the followingparameters:
DE1=2 ¼ Ean1=2 � Ecat
1=2 ð26ÞRE1=2 ¼ Ean
1=2 þ Ecat1=2 ð27Þ
where Ecat1=2 is the potential at which the current takes the half value
of the cathodic limit current (i.e., IRPVðEcat1=2Þ ¼ Id;DC=2) and Ean
1=2 the
potential at which the current takes the half value of the anodic lim-it current (i.e., IRPVðEan
1=2Þ ¼ Id;RPV=2).These two magnitudes are directly related with the shape of
RPV curves and so they are very useful for the quantitative studyof the system kinetics. Thus, the parameter DE1/2 is related to theseparation of the cathodic and anodic branches of RPV curve sothat its value increases as the system behaviour is more irrevers-ible (see Figs. 6a and 7a). On the other hand, the parameter RE1/2
refers to the relative symmetry of the cathodic and the anodicbranches of RPV curves, which is very dependent on the value ofthe transfer coefficient (see Figs. 6b and 7b).
The variation of DE1/2 and RE1/2 with n2 ¼ 2ffiffiffiffiffiffiffiffiDt2p
=r0�
is studiedin Fig. 8 with the aim of establishing a procedure for the extractionof the kinetic parameters, k0 and a, with RPV technique. Note thatthese curves are completely general, since they are given in func-tion of the determining dimensionless parameters kmicro
s and n2,and therefore they can be used as working curves for kineticstudies.
In Fig. 8a, the curves DE1/2 vs. Log(n2) are plotted for a widerange of kmicro
s , from totally irreversible to reversible systems. Notethat given a kmicro
s value, the variation of the sphericity (n2) is
(b)
Log (t2 / s)-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Σ E1/
2 -
2 E0
/ mV
-400
-200
0
200
400
0.2
0.3
0.4
α = 0.5
0.6
0.7
0.8
(a)
Log (t2 / s)-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Δ E1/
2/ m
V
200
300
400
500
0.3
α = 0.5
0.7
Fig. 7. Variation of: (a) DE1/2 and (b) RE1/2 � 2E0 with the duration of the second pulse (t2) for different values of the transfer coefficient (indicated on the curves).k0 = 10�3 cm/s, r0 = 50 lm, t1 = 1s.
Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77 75
equivalent to the variation of the double pulse duration. Fromthese working curves the heterogeneous rate constant of the elec-trode process can be determined by means of RPV experimentswith different values of the double pulse duration, keeping con-stant the ratio t1/t2.
As has been previously discussed, the RE1/2 parameter isgreatly affected by the value of the transfer coefficient. InFig. 8b the RE1/2 � 2E0 vs. Log(n2) curves are plotted for severala values. We can observe that there exists a linear region theslope of which notably depends on the value of the transfercoefficient. Moreover, the slope is found to be almost indepen-dent of t1/t2 and kmicro
s , existing a direct correspondence betweenits value and a. In the table embedded in Fig. 8b the slopes forseveral a values are presented from which this kinetic parametercan be easily determined.
Once k0 and a parameters are known, the formal potential canalso be extracted from Fig. 8b from the difference between the va-
lue of the intercept of the curve RE1/2 vs. Log(n2) obtained experi-mentally and the intercept corresponding to the theoretical curveRE1/2 � 2E0 vs. Log(n2).
So, we have demonstrated that from the simultaneous analysisof the variation of DE1/2 and RE1/2 with the double pulse duration,the complete characterization of the redox system is feasible withRPV technique, obtaining the values of the heterogeneous rate con-stant, the electron transfer coefficient and the formal potential.
4. Conclusions
A rigorous analytical solution is obtained for charge transferprocesses with finite kinetics in Reverse Pulse Voltammetry atspherical electrodes. This is valid whatever the value of the elec-trode radius, the reversibility degree of the electrode process orthe length of the potential pulses.
(a)
Log (ξ2)-3 -2 -1 0 1
Δ E1/
2 / m
V
0
200
400
600
800
1000
ksmicro =
1
0.1
10
0.5
50
2.5
0.25
0.05
0.010.025
Log (ξ2)-3 -2 -1 0 1
Σ E1/
2 -
2E0 /
mV
-600
-450
-300
-150
0
150
300
450
600
α = 0.5
0.7
0.4
0.6
0.8
0.3
0.2
α - value Slope / mV
0.2 +220.2
0.3 + 112.6
0.4 + 49.6
0.5 0.15
0.6 -49.5
0.7 -112.9
0.8 -220.8
(b)
Fig. 8. (a) DE1/2 vs. Log(n2) working curves for different values of kmicros (values indicated on the curves), a = 0.5, t1/t2 = 20; (b) RE1/2 � 2E0 vs. Log(n2) working curves for
different values of a (values indicated on the curves), kmicros ¼ 1, t1/t2 = 20; the slope of the linear region is given in the table embedded for all the a values considered.
76 Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77
The effects of the electrode sphericity on the response as well asof the system reversibility are examined, and it is pointed out thatfrom direct analysis of RPV curves it is possible to estimate thereversibility of the system.
The influence of the duration of the second pulse is also studied.Thus, the appearance of an anomalous maximum in the anodicbranch of the RPV curve is described for sluggish charge transferprocesses when the second potential pulse is long enough. Thisatypical behaviour is more apparent at large electrodes.
Finally, working curves are given for the determination of thekinetic parameters and the formal potential of the system fromRPV experiments performed with different lengths of the doublepotential pulse.
Acknowledgements
A.M., F.M.-O. and E.L. greatly appreciate the financial supportprovided by the Dirección General de Investigación (MEC) (Project
Number CTQ2009-13023) and by the Fundación SENECA (ProjectNumber 08813/PI/08). Also, E.L. thanks the Ministerio de Cienciae Innovacion for the grant received.
Appendix A
To solve the problem corresponding to the second potential stepwe apply Koutecky’s dimensionless parameter method [23,24] bysupposing that the solutions of the differential equation system(9) are functional series of the dimensionless variables v2 and b:
~uð2ÞO ðr; tÞ ¼ ~uð2ÞO ðs2;v2;bÞ ¼X
i;j
ri;jðs2Þvi2b
j=2 ðA1Þ
~uð2ÞR ðr; tÞ ¼ ~uð2ÞR ðs2;v2;bÞ ¼X
i;j
di;jðs2Þvi2b
j=2 ðA2Þ
Taking into account the definition of s2, v2 and b the differentialequation system and the boundary value problem turn into:
Á. Molina et al. / Journal of Electroanalytical Chemistry 648 (2010) 67–77 77
@2 ~uð2ÞO
@s22þ 2s2
@~uð2ÞO
@s2� 2v2
@~uð2ÞO
@v2� 4bð1� bÞ @~uð2Þ
O@b ¼ 0
@2 ~uð2ÞR
@s22þ 2s2
@~uð2ÞR
@s2� 2v2
@~uð2ÞR
@v2� 4bð1� bÞ @~uð2Þ
R@b ¼ 0
9>=>; ðA3Þ
s2 ?1
~uð2ÞO ð1Þ ¼ 0; ~uð2ÞR ð1Þ ¼ 0 ðA4Þ
s2 = 0
@~uð2ÞO
@s2
!s2¼0
� v2
h2~uð2ÞO ð0Þ ¼ �
@~uð2ÞR
@s2
!s2¼0
� v2
h2~uð2ÞR ð0Þ
24
35 ðA5Þ
@~uð2ÞO
@s2
!r¼r0
� v2
h2~uð2ÞO ð0Þ �
v2
h2kmicro
s K�a2 ~uð2ÞO ð0Þ � K2~uð2ÞR ð0Þ� �
¼ � 2ffiffiffiffipp b1=2 þ Zv2 ðA6Þ
and the expression for the current is given by:
I2
nFADc�O¼ I1dðt1 þ t2Þ þ
12ffiffiffiffiffiffiffiffiDt2p @~uð2ÞO
@s2
!s2¼0
�v2
h2~uð2ÞO ð0Þ
24
35
¼ I1dðt1 þ t2Þ þ1
2ffiffiffiffiffiffiffiffiDt2p
Xi;j
r0i;jð0Þvi2b
j=2 � 1h2
Xi;j
ri;jð0Þviþ12 bj=2
" #
ðA7Þ
where h2 ¼ 1þ kmicros K�a
2 ð1þ K2Þ.By introducing the expressions (A1) and (A2) into Eqs. (A3)–
(A6), the differential equation system and the boundary valueproblem become:
r00i;jðs2Þ þ 2s2r0i;jðs2Þ � 2ðiþ jÞri;jðs2Þ ¼ �2ðj� 2Þri;j�2ðs2Þd00i;jðs2Þ þ 2s2d
0i;jðs2Þ � 2ðiþ jÞdi;jðs2Þ ¼ �2ðj� 2Þdi;j�2ðs2Þ
)ðA8Þ
s2 ?1
ri;jð1Þ ¼ 0 di;jð1Þ ¼ 0 i; j � 0 ðA9Þ
s2 = 0
r0i;jð0Þ �ri�1;jð0Þ
h2¼ �d0i;jð0Þ þ
di�1;jð0Þh2
ðA10Þ
r0i;jð0Þ �ri�1;jð0Þ
h2� kmicro
s K�a2
h2ðri�1;jð0Þ � K2di�1;jð0ÞÞ ¼
� 2ffiffiffipp for i ¼ 0; j ¼ 1
Z for i ¼ 1; j ¼ 00 otherwise
8><>:
ðA11Þ
By applying Eqs. (A9)–(A11), we obtain the following relationships:
r00;1ð0Þ ¼ �2ffiffiffiffipp ðA12Þ
r00;jð0Þ ¼ 0 for j–1 ðA13Þr01;0ð0Þ ¼ Z ðA14Þr0i;jð0Þ ¼ ri�1;jð0Þ for i � 1; j odd ðA15Þr0i;jð0Þ ¼ ri;jð0Þ ¼ 0 for i � 1 ; j even ðA16Þ
From these results, and taking into account Eq. (A7), the expressionfor the RPV current is deduced (Eq. (20)).
Appendix B
Functions and variables
l ¼ c�Rc�O
ðB1Þ
Idð1Þ ¼FAc�OD
r0ðB2Þ
IpdðtÞ ¼ FAc�O
ffiffiffiffiffiffiDpt
rðB3Þ
Id;DC ¼ I1dðt1 þ t2Þ ¼ Idð1Þ 1þ r0ffiffiffiffiffiffiffipDp
ðt1 þ t2Þ
!ðB4Þ
gj ¼F
RTðEj � E0Þ j � 1;2 ðB5Þ
Kj ¼ expðgjÞ j � 1;2 ðB6Þ
n2 ¼2ffiffiffiffiffiffiffiffiDt2p
r0ðB7Þ
vp2 ¼ 2
ffiffiffiffit2
D
rk0K�a
2 ð1þ K2Þ ¼ ðv2Þr0!1 ðB8Þ
Z ¼ � 1þ kmicros K1�a
2 ð1þ lÞ1þ kmicro
s K�a2 ð1þ K2Þ
ðB9Þ
Zp ¼ �K2ð1þ lÞ1þ K2
¼ ðZÞr0!1 ðB10Þ
Y ¼ � 2v2
ffiffiffiffibp
rðB11Þ
Fðv2Þ ¼X1i¼0
ð�1Þiviþ12Qi
l¼0pl
¼ffiffiffiffipp
2v2 � expðv2=2Þ2 � erfcðv2=2Þ ðB12Þ
Hðv2;bÞ ¼P1i¼0
ð�1Þiviþ12Qi
l¼0
pl
1þP1k¼1
ð2k�1Þ!ibk
22k�1k!ðk�1Þ!ðiþ2kÞ
� �for v2 < 10
Hðv2;bÞ ¼ffiffiffiffiffiffiffiffiffiffiffi1� b
pþ
ffiffiffiffipp P1
k¼1
ð�1Þk�1ð2k�1Þ!bk
ðk�1Þ!v2k�12
þP1i¼1
ð�1Þi2ð2i�1Þ!ði�1Þ!v2i
21�
P1k¼1
21�2kð2k�1Þ!ð2iþ1Þbk
k!ðk�1Þ!ð2k�2i�1Þ
� �for v2 > 10
ðB13Þ
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