Sd Article 7

Journal of Electroanalytical Chemistry 505 (2001) 100–108www.elsevier.nl/locate/jelechem

Adsorptive square wave voltammetry of metal complexes.Effect of ligand concentration

Part I. Theory

Fernando Garay *INFIQC, Departamento de Fısico Quımica, Facultad de Ciencias Quımicas, Uni�ersidad Nacional de Cordoba, Pabellon Argentina, Ala 1, 2°

piso, Ciudad Uni�ersitaria, 5000 Cordoba, Argentina

Received 8 December 2000; received in revised form 19 February 2001; accepted 1 March 2001

Abstract

The electrochemical behaviour of non-labile metallic complexes under square wave voltammetry (SWV) conditions is analysedtheoretically, considering the influence of ligand adsorption–desorption processes as well as the ligand concentration on thequasi-reversible redox reaction mechanism. The dependence of current and peak potentials on the transfer reaction rate and oncomplex stoichiometry is considered. Voltammetric responses for processes in which the ligand is desorbed or remains adsorbedafter the electroreduction process are compared. Diagnosis criteria for the selection of optimum SWV parameters in view ofanalytical applications are presented for each case. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Adsorptive square wave voltammetry; Mechanistic studies; Mathematical models

1. Introduction

The adsorption of electroactive analytes on the elec-trode surface is used commonly as a preconcentrationstep in electroanalytic procedures. This step improvesthe sensitivity of the analytical technique, allowingtrace level determinations [1–6]. Square wave voltam-metry (SWV) is one of the electrochemical techniquesmore widely applied in quantitative analysis, due to itshigh sensitivity, which is mainly the consequence of therejection of most of the capacitive currents [7]. Thealternating application of oxidation and reductionpulses, characteristic of SWV, allows the concentrationgradients of reagents and products close to the surfaceto be rebuilt in each wave. This fact allows the determi-nation of both species almost simultaneously. The dif-ferential peak current (��p) and the peak potential (Ep)are the most useful parameters of SWV from the ana-lytical point of view. Nevertheless, ��p values alsohave been employed in some kinetic analyses, since aplot of ��p versus the charge transfer rate constant (ks)

depicts a maximum for quasi-reversible redox reactionswith adsorbed species; this ‘quasi-reversible maximum’can be employed to obtain ks [8–10].

One of the modeling advantages is the optimisationof the experimental conditions for quantitative determi-nations of metallic complexes at trace levels. Manystudies in the literature take into account adsorbedspecies involved in charge transfers of different ratesand examine also the effect of irreversible chemicalreactions coupled to the electrochemical step [9–22].Models that simulate the voltammetric response for theadsorptive accumulation of metallic cations have alsobeen developed. In all cases, the strong and labilecomplexation forms of the metallic cations by organicand inorganic ligands have been considered, respec-tively, under conditions where the ligand is in greatexcess with regard to the metal concentration [12–16].

Simulation of mechanisms involving adsorbed speciesrequires the relationship between bulk concentrationsof soluble species and their surface excesses at a giventemperature. One of the simplest models considers alinear relationship between bulk and surface concentra-tions. This type of isotherm is applicable at low surfacecoverages, which are found frequently in trace analysis

* Tel.: +54-351-4334169; fax: +54-351-4334188.E-mail address: [email protected] (F. Garay).

0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 -0728 (01 )00459 -4

F. Garay / Journal of Electroanalytical Chemistry 505 (2001) 100–108 101

[12]. Nevertheless, a great excess of organic ligand isusually added to the analytical solution in the adsorp-tive quantification of metallic cations. This excess doesnot necessarily increase the sensitivity of the response,but it invalidates the applicability of the linearisotherm.

In the present work, a theoretical model for a reac-tion mechanism considering diffusion effects and theadsorption of reagents is proposed. The model assumesinert complexes with stability constants high enough toconsider that no free metal will be detected after theformation of the complex. The concentration and ad-sorption effects of the ligand on the voltammometricresponse are analysed. This concentration was alwayskept above the assumed trace amounts of metal. Theligand excess range with regard to the total metallic ionconcentration was studied from 10−5 to 104. In thisway, the evaluation of experimental cases under linearadsorption conditions can be accomplished.

2. Mathematical model

The formation in the bulk solution of only onechemically stable complex species is assumed:

M+(sol)+uL−

(sol) ��Kst,o

MLu(sol) (1)

Here, the u value can be 1 or 2, depending on thecomplex stoichiometry. For simplicity, the charge onthe oxidised complex is omitted, all Mn+ are symbol-ised by M+ and the suffix u in the stability (Kst,o) andadsorption (Kad) constants are also omitted; c* standsfor bulk concentrations. It is considered that cL* �cM+*and the distribution of these ionic species is definedaccording to:

cM+ (ini)* =cMLu (eq)* (2)

cL (eq)* =cL (ini)* −ucM+ (ini)* (3)

where the subscripts (ini) and (eq) indicate the speciesconcentration before the ligand is added and after thecomplexing equilibrium is reached, respectively. Theligand–complex concentration ratio is defined as:

RLM=cL* (cMLu* )−1 (4)

Considering the above conditions, two reactionschemes were analysed. The first one assumes that theproduct of the adsorbed complex reduction is the lig-and in solution, Eq. (5), while in the second scheme theadsorbed ligand remains at equilibrium with its solublefraction, Eq. (6).

MLu(sol) ��Kad,ML

MLu(ad) ��ks

ne−M(Hg)(sol)+uL(sol) (5)

MLu(sol) ��Kad,ML

MLu(ad) ��ks

ne−M(Hg)(sol)

+uL(ad) ��Kad,L

uL(sol) (6)

where Kad,ML and Kad,L are the adsorption constants forthe oxidised metal complex and the free ligand, respec-tively. As the adsorption constants are defined for theforward reactions of Eqs. (5) and (6) their dimensionsare in centimetres. A complete list of symbols is sum-marised in the nomenclature.

As stated before, adsorption isotherms for the metal-lic complex and for the free ligand are assumed to belinear. Also, no redox reactions involving the ligand aresupposed to occur within the working potential range.Considering one-dimensional diffusion, Eqs. (5) and (6)can be evaluated mathematically with the followingdifferential equations:

�cMLu/�t=D(�2cMLu

/�x2) (7)

�cM(Hg)/�t=D(�2cM(Hg)/�x2) (8)

�cL/�t=D(�2cL/�x2) (9)

For simplicity, a common value of the diffusion coeffi-cient, D=1×10−5 cm2 s−1 was assumed for all diffus-ing species. The following initial and boundaryconditions are considered.

t=0, x�0:

cMLu=cMLu

* ; �MLu

ini =Kad,MLcMLu* (10)

cM(Hg)=0; cL=cL* (11)

�Lini=Kad,LcL* (12)

t�0, x��:

cMLu�cMLu

* ; cL�cL* ; cM(Hg)�0 (13)

x=0:

�MLu=Kad,MLcMLu

(14)

D(�cMLu/�x)x=0=��MLu

/�t+I/nFA (15)

D(�cM(Hg)/�x)x=0= −I/nFA (16)

D(�cL/�x)x=0= −I/nFA (17)

�L=Kad,LcL (18)

D(�cL/�x)x=0=��L/�t−I/nFA (19)

Boundary conditions (12), (18) and (19) are appliedonly to Eq. (6) whereas condition (17) is operative onlyfor Eq. (5). All the other conditions apply to bothreaction schemes. Provided that the rate of the electro-chemical step is considerably faster than the complexdissociation, the reduced complex species could be as-sumed to be an intermediate species in equilibrium withthe dissociated products:

Kst, r=�M°Lu(cM(Hg)cL

u )−1 (20)

F. Garay / Journal of Electroanalytical Chemistry 505 (2001) 100–108102

where Kst,r describes the chemical equilibrium betweenthe adsorbed reduced complex and its dissociated spe-cies, M° (in the amalgam) and L (in solution, close tothe electrode surface). The rate of the charge transferreaction is given by the well-known equation:

I(t)/nFA=ks exp[−��(t)]{�MLu−�M°Lu

exp[�(t)]}(21)

where the symbols have their usual meaning and �(t) isa function of the reaction scheme considered, accordingto the following expressions [11,13]:

�(t)Eq. (5)=nF [E(t)−E°]/RT+ ln(Kad,ML)

+ ln(Kst,o/Kst,r) (22)

�(t)Eq. (6)=nF [E(t)−E°]/RT+ ln(Kad,ML/Kad,L)

+ ln(Kst,o/Kst, r) (23)

where E(t) is the square wave (SW) potential functionand E° is the standard potential for the simple redoxreaction involving soluble free metal species.

In order to simulate the reactions described in Eqs.(5) and (6), the current is normalised according to�(t)=I(t)/(nFAf�MLu

ini ) where f is the SW frequency. Anumerical integration method is employed in the resolu-tion of differential equations (7), (8) and (9) under therelevant boundary conditions, Eqs. (10)– (19) [23]. Thefollowing results were obtained.

Reaction scheme of Eq. (5), assuming u=1:

0=� (m)2 +�(m){Za[T(m)�(m)+1]+2�a(m)}

+ZaT(m)[�b(m)− f −1]+�a(m)[Za+�a(m)] (24)


0=� (m)3 +� (m)

2 {Za+3�a(m)}

+�(m){(Za/2)2[T(m)�(m)+1]

+�a(m)(2Za+3�a(m))}

+ (Za/2)2[T(m)(�b(m)− f −1)+�a(m)]

+ (�a(m))2[Za+�a(m)] (25)


0=� (m)2 +�(m){Zb[T(m)�(m)+1]+�a(m)+�c(m)}

+ZbT(m)[�b(m)− f −1]+�a(m)[Zb+�c(m)] (26)


0=� (m)3 +� (m)

2 {Zb+2�c(m)+�a(m)}

+�(m){(Zb/2)2[T(m)�(m)+1]+�c(m)(Zb+�c(m))

+�a(m)(Zb+2�c(m))}

+ (Zb/2)2[T(m)(�b(m)− f −1)+�a(m)]

+�c(m)�a(m)[Zb+�c(m)] (27)

where �a(m)=�m−1j=1 �( j )S(i ); �b(m)=�m−1

j=1 �( j )Q(i );�c(m)P(1)=�m−1

j=1 �( j )P(i ); �(m)=exp[��(m)]k s−1+Q(1);

S(i )= (i )1/2− (i−1)1/2; i=m− j+1; Q(i )={�S(i )+Y1(i )}a1

−1; P(i )={�S(i )+Y2(i )}a2−1; Yy(i )=

{exp[ay2(�i )] erfc[ay(�i )1/2]−exp[ay

2(�(i−1))] erfc[ay(�-(i−1))1/2]}ay

−1. In this latter equation, y subscripts canbe either 1 or 2 for the complex or ligand parameters,respectively. Accordingly, a1=DML

1/2 Kad,ML−1 ; a2=DL

1/2-Kad,L

−1 ; Za=RLMKad,MLDL1/2[ f� ]−1; Zb=RLM-

Kad,ML[Kad,L fP(1)]−1; T(m)=DM(Hg)1/2 {�rs exp[�(m)]}−1.

The constant rs=1 cm determines the surface and bulkstandard concentration relationship [22]. The parameter�=2(�/�)1/2 depends upon the period �= (qf )−1 in-volved in each numerical integration step, where qstands for the number of subintervals considered ineach wave and a value of q=40 was employed.

3. Results and discussion

Fig. 1 shows the theoretical voltammograms for bothreaction schemes, Eqs. (5) and (6). Direct (�f) andreverse (�b) normalised currents as a function of poten-tial at 100 Hz are presented for different values of ks

(curves a– f). The first scheme, Eq. (5), is analysed foru=1 and 2, whereas Eq. (6) is analysed for u=2. Thenormalised currents present asymmetric bell-shapedprofiles, with marked differences, depending on ks. Thesweeps start at the positive potential limit and thereducing current, �f, is considered to be negative.Curves calculated with Eqs. (24), (25) and (27) usingRLM=10 are arranged in Fig. 1A, B and C, respec-tively. Almost identical �–E profiles were obtained forthe lowest ks value irrespective of the reaction schemeand the complex stoichiometry, as is shown in curves a,Fig. 1A, B and C. This is a clear indication that neitherthe chemical state of the reduced products nor thestoichiometry of the complex has any influence on thecurrent–potential profiles, since ks is the only parame-ter controlling the oxidation step. As ks is increased, thequasi-reversible (curves b–e), and the reversible voltam-metric responses (curves f) are sensitive to the chemicalproperties of the complex, depending on both the uvalue and the final state of the ligand. The voltammet-ric profiles presented in Fig. 1C are independent of RLM

values for RLM�0.1. This indicates that (�cL/�x)x=0�0 for all t. In this way, further increments of cL* will notaffect the current profile, therefore Eqs. (4), (20) and(21) allow the same expression found by other authorswhen cL* �cM* to be obtained [11,20]:

I(t)/nFA=ks exp(−��(t))

[�MLp−rscM(Hg),(x=0) exp(�(t))] (28)


Curves b– f in Fig. 1A and B show wider and lowerpeaks than those of Fig. 1C. Accordingly, the RLM

value of 10 is not high enough to discard the liganddiffusion effect, which is enhanced for u=2, Fig. 1B.The scheme of Eq. (6) for u=1 gave a voltammometricresponse that was indistinguishable from that of Fig.1C.

Fig. 2 shows a theoretical analysis of the dependencesof ��p (Fig. 2A) and Ep (Fig. 2B) with log(ks) atdifferent RLM values. In both figures, full lines corre-spond to the scheme of Eq. (5), in which a liganddesorption step during the complex reduction is consid-ered, whereas dashed lines correspond to the limitingcase given by Eq. (28). The symbols indicate the differ-ent values of RLM ranging from 10 to 104. Open sym-bols stand for 1:1 complexes, Eq. (24), whereas the fullones refer to 1:2 stoichiometry, Eq. (25). The resolutionof Eq. (28) depicts the typical ��p maximum forquasi-reversible reactions involving adsorbed reactants,which is clearly seen in Fig. 2A. Nevertheless, if theligand desorption takes place during the reduction pro-cess, this quasi-reversible maximum gradually decreaseswith RLM, until it virtually disappears for RLM�101

and �102 for u=1 and u=2, respectively. Both com-plex stoichiometries analysed present very dissimilarcurrent–potential dependences when RLM�100; thesedifferences can be employed to determine the u valuefor a given reaction. The differences at the voltammet-ric profiles for both stoichiometries decrease graduallyas RLM is increased, becoming non-detectable forRLM�104. For this concentration ratio, Eqs. (24) and(25) tend to Eq. (28). The same behaviour is observed

for reaction schemes where the ligand remained ad-sorbed, Eq. (6), indicating that Eqs. (26) and (27) alsotend to Eq. (28), but when RLM�1 (not shown). In thiscase, the required RLM value is four orders of magni-tude lower than the reaction scheme of Eq. (5). Thisfact points to a strong levelling off effect due to theligand adsorption equilibrium, which guarantees a con-stant ligand concentration close to the electrode andvalidates the employment of the quasi-reversible maxi-mum for the evaluation of ks [11,12,20].

The dependences of Ep on log(ks) show three well-dif-ferentiated zones (Fig. 2B). When log(ks)� −2, allsystems are controlled basically by the kinetic contribu-tion, with a slope of −2.3RT/nF V dec−1, regardlessof the effects of RLM and u values. A completelydifferent behaviour is observed for reversible redoxreactions for which log(ks)�2. In this case, Ep isstrongly dependent on RLM and on the complex stoi-chiometry, although it is not affected by ks. Finally,there is an intermediate zone corresponding to thequasi-reversible redox reactions, in which the differentvariables analysed, ks, RLM and u, affect the Ep valuesjointly. As for Fig. 2A, all curves approach that of Eq.(28) as RLM is raised (dashed line).

The effect of RLM on the theoretical SW voltammet-ric profiles calculated according to Eqs. (24), (25) and(27) is examined in Fig. 3A, B and C, respectively.These responses were obtained for ks=0.3 s−1, f=100Hz, changing RLM over four orders of magnitude. If thereduction process does not affect the ligand gradient atthe electrode surface, all the reaction schemes consid-ered will give the same voltammetric response. The

Fig. 1. Theoretical �–E profiles from Eq. (5) for u=1 (A) and u=2 (B) and from Eq. (6) for u=2 (C), f=100 Hz, Esw=50, dE=5 mV,Kad,ML=0.1 cm, RLM=10, n=1, �=0.5 and ks=0.01 (a); 0.1 (b); 0.3 (c); 0.6 (d); 1 (e) and 103 s−1 (f).


Fig. 2. Dependence of (A) ��p and (B) Ep as functions of log(ks),obtained from Eq. (5) for u=1 (open symbols) and u=2 (fullsymbols). RLM=104 (× ); 103 (�); 102 (�) and 101 (�). Otherparameters as in Fig. 1.

high enough to ensure that the ligand gradient will notbe affected by the electrochemical redox reaction, Fig.3C. However, completely different current–potentialprofiles for this scheme are observed, when RLM isbelow the lower limit. As RLM is diminished, the back-ward current (in which the complex re-oxidation takesplace) is also diminished until it becomes undetectable.Provided that cL* is very low, the ligand desorption isalmost complete during the reduction pulse, followedby its diffusion towards the bulk. Consequently, duringthe following oxidation pulse, �L will be very low,hindering the oxidation of the amalgamated metal atthis potential.

Fig. 4 shows how ��p (Fig. 4A) and Ep (Fig. 4B),arising from the scheme of Eq. (5) vary as functions oflog(RLM) for different values of ks. Open and fullsymbols stand for complex stoichiometries 1:1, Eq. (24),and 1:2, Eq. (25), respectively. Three distinct types ofbehaviour are observed. First, for log(RLM)�3, bothparameters are controlled basically by ks, and conse-quently, neither RLM nor the complex stoichiometry hasa noticeable effect on the voltammetric responses. Sec-ond, there is a transitional region for 0� log(RLM)�3,where complex behaviour is observed. The strongchanges produced by ks and RLM on ��p predict thateither the enhancement or the diminution of ��p canbe commanded by changing the ligand–complex ratio.These tendencies depend directly on the value of ks withregard to the quasi-reversible maximum. The enhance-ment of the peak current when RLM is diminisheddepends on the driving force of the cL gradient, whichpromotes the occurrence of the redox reaction. There-fore, experiments performed with low values of RLM areuseful to differentiate the complex stoichiometry,whereas the higher values help to determine the ks

value. Finally, when log(RLM)�0, the responses aredetermined by diffusion and ��p becomes independentof ks. In this region, ��p has a constant value, charac-teristic of the complex stoichiometry.

In the case of Ep (Fig. 4B), provided log(RLM)�3,the kinetic contribution essentially controls the voltam-metric responses and hence the Ep values remain con-stant for increasing log(RLM) values. For log(RLM)�0,Ep varies linearly with log(RLM) presenting slopes of0.060 and 0.120 V dec−1 and pointing to a diffusionalcontrol for both stoichiometries, Eqs. (24) and (25),respectively. This dependence of Ep on both RLM and ks

is similar, since both parameters govern the reversibilityof the global reaction, the first controlling the chemicalkinetics of the complex formation and the second com-manding the electrochemical rate. For the reactionscheme of Eq. (6), which considers the ligand adsorp-tion step, Ep values are constants provided log(RLM)�−2, since they are determined mainly by ks (notshown).

latter can be achieved when RLM�1, Eq. (28), or whenthe ligand adsorption produces the levelling off effectpreviously described, Eqs. (26) and (27). Differencesbetween each mechanism start to be evident at thevoltammograms when RLM is lower than a given limit-ing value, which is strongly dependent on the ligandstate at the electrode surface (Fig. 3A (curve e), B(curve c) and C (curve i), respectively). However, whenthe ligand–complex ratio is diminished by three ordersof magnitude in relation to the limiting value, thecurrent–potential profiles maintain their shapes, but ashift in the Ep value towards more negative potentials isobserved (Fig. 3A (curves g and h), B (curves e–g) andC (curves k and l)). The diffusion contribution to thecurrent is evident in Fig. 3A (curves e–g) and B (curvesc–e) in which the diffusional current tails at morepositive potentials than Ep are increased as RLM isdecreased. This effect is more important for u=2, Fig.3B. On the other hand, if the ligand remains adsorbedafter the complex reduction, an RLM value of just 0.1 is


Fig. 3. Theoretical �–E profiles obtained from Fig. 5 for u=1 (A) and u=2 (B) and from Fig. 6 for u=1 (C). ks=0.3 s−1 and RLM=104 (a);103 (b); 102 (c); 30 (d); 10 (e); 3 (f); 1 (g); 0.1 (h); 10−3 (i); 3×10−4 (j); 10−4 (k) and 10−5 (l). Other parameters as in Fig. 1.

Fig. 5 compares, according to the scheme of Eq. (6),the dependence of ��p with log(Kad,L) obtained fordifferent values of RLM (symbols), for u=1 (dashedlines) and for u=2 (full lines). Considering ks=1 s−1,below the quasi-reversible maximum (ks,max), the curvesshow two kinds of well-defined behaviour. At lowlog(Kad,L), ��p increases steadily with log(Kad,L) until itreaches a constant value. High RLM and 1:1 stoi-chiometries expand the range of Kad,L values for which��p is constant, which is achieved for RLM×Kad,L�0.1 (for u=1) and �0.3 (for u=2), respectively. Con-sequently, Eqs. (26) and (27) will present similarvoltammetric responses if RLM×Kad,L exceeds theselimiting values.

Fig. 6 shows the theoretical dependence of ��p

versus log(Kad,L) for ks values below and above theks,max value, considering u=1 and RLM=10. The curvefor ks=1 s−1, shown in Fig. 5, is included here forcomparison. The tendency of increasing ��p withlog(Kad,L) can be reversed when ks�ks,max or remainindependent of Kad,L for highly irreversible reactions.When ks�ks,max, ��p will be controlled completely byks, regardless of the ligand desorption or its surfaceconcentration. The condition of RLM×Kad,L�0.1holds for all ks values analysed, since ��p does notchange with Kad,L for log(Kad,L)� −2. For slow elec-tron transfer reactions (ks�ks,max), the adsorption equi-librium can be restored before the current sampling atthe end of the pulse; therefore, the diminution of RLM

and Kad,L will decrease the value of �L, diminishing inthis way the value of ��p. On the other hand, for morereversible redox reactions (ks�ks,max), ��p is enhancedby decreasing Kad,L. This effect depends directly on the

increase of the ligand concentration close to the elec-trode surface. After the ligand desorption takes place,�L will not be high enough to drain the whole reduced

Fig. 4. Dependence of (A) ��p and (B) Ep as functions of log(ks),obtained from Eq. (5) for u=1 (open symbols) and u=2 (fullsymbols). ks=103 (�); 1 (�); 0.3 (�) and 0.1 s−1 (�). Otherparameters as in Fig. 1.


Fig. 5. Dependence of ��p versus log(Kad,L) obtained from Eq. (6)for u=1 (open symbols) and u=2 (full symbols), ks=1 s−1, RLM=10 (�); 1 (�) and 0.1 (�). Other parameters as in Fig. 1.

the net peak height. Likewise, the Ep shifts slightlytowards more positive values, Fig. 7B. By nEsw�100mV, the skew in the �f and �b has defined a shoulderat the �� (curves d and e).

For irreversible redox reactions, the peak width re-mains constant, with �Ep/2= (63.50.5)/�n mV, forevery SW amplitude and the increment of Esw will justenhance the ��p value [11]. Nevertheless, for reversibleand quasi-reversible redox reactions, larger values ofEsw increase the peak width linearly (not shown).

The calculated dependences of the ��p �Ep/2−1 ratio

versus Esw are shown in Fig. 8A and B, consideringreversible and quasi-reversible charge transfer reactions,respectively. Both figures show the changes in��p �Ep/2

−1 when RLM is varied from 1 to 103 (symbols)according to the scheme of Fig. 5 and for u=1. Thebehaviour for both charge transfer kinetics indicatesthat ��p �Ep/2

−1 is enlarged by the increment of RLM

and Esw. Particularly for Esw�40 mV, the analyticalsignal increases more rapidly. For the case of a quasi-reversible system (ks=1), Fig. 8B, the response qualityincreases for Esw up to 100 mV, except for the case ofRLM=103, which has a maximum at 80 mV. For thereversible case (ks=104), Fig. 8A, the maximum atEsw=40 mV is only for RLM=103, whereas for RLM=102 the maximum is shifted to 50 mV. For RLM=10and 1 there is no maximum in the Esw range exploited.

Fig. 6. Dependence of ��p versus log(Kad,L) obtained from Eq. (6)for u=1, ks=104 (�); 10 (�); 1 (�) and 10−2 s−1 (�). Otherparameters as in Fig. 1.

Fig. 7. Theoretical �–E profiles from Fig. 5 for u=1, ks=1 s−1,Esw=5 (a); 10 (b); 20 (c); 40 (d) and 100 mV (e). Other parametersas in Fig. 1.

species at the beginning of the oxidation pulse. Lowvalues of Kad,L will favour the ligand desorption duringthe reduction pulse, increasing the cL near the electrodesurface and producing an enhancement of ��p.

Fig. 7 shows how the morphology of forward andbackward currents (Fig. 7A) as well as of �� (Fig. 7B)varies with Esw, considering a 1:1 complex according toEq. (5). Reduction currents are observed for direct andreverse pulses for E=5 mV, Fig. 7A. As the SWamplitude increases, an oxidation current is observed atthe reverse pulse making a substantial contribution to


Fig. 8. Dependence of ��p �Ep/2−1 on Esw obtained from Eq. (6) with

u=1. For (A) ks=104 and (B) ks=1 s−1, RLM=1 (�); 10 (�); 102

(�) and 103 (�). Other parameters as in Fig. 1.

From the analytical point of view, the theoreticalcurves provide a comprehensive way to maximise the��p �Ep/2

−1 ratio. Also, the elucidation of the mecha-nism points out how the ligand concentration and theother experimental parameters should be tuned to max-imise the current response and to produce one well-shaped peak at a favourable potential.

Acknowledgements

Financial support from the Consejo Nacional deInvestigaciones Cientıficas y Tecnologicas (CONICET),Consejo de Investigaciones de la Provincia de Cordoba(CONICOR) and Secretarıa de Ciencia y Tecnologıa dela Universidad Nacional de Cordoba is gratefully ac-knowledged. The author also wishes to thank DrMilivoj Lovric and Dr Velia Solis for very helpfuldiscussions.

Appendix A

A electrode surface area (cm2)=DML

1/2 Kad,ML−1 (s−1/2)a1

a2 =DL1/2Kad,L

−1 (s−1/2)cL* ; cMLu

* bulk ligand and complex concentra-tions (mol cm−3)

cM(Hg); cMLu; concentrations of M°, MLu and L−

cL near the electrode surface (mol cm−3)D diffusion coefficient (cm2 s−1)dE SW step amplitude (V)�Ep/2 half-peak width (V)E(t) SW potential program (V)E° standard potential for a simple redox

reaction of free soluble species (V)Ep peak potential (for the net current)

(V)Esw half peak-to-peak SW potential ampli-

tude (V)f SW frequency (Hz)

Faraday constant (C)Fi =m−j+1

current (A)I(t)

Kad,L; Kad,ML ligand and complex adsorption con-stants (cm)

ks standard reaction rate constant (s−1)the value of ks at the quasi-reversibleks,max

maximum (s−1)Kst,o stability constant of oxidised species

(cm3 mol−1)stability constant of reduced speciesKst,r

(cm4 mol−1) or (cm7 mol−2)n number of electronsP(i ) ={�S(i )+Y2(i )}a2

−1 (s)=40; number of time increments inqeach SW period

Consequently, a SW amplitude of 100 mV is preferredfor reversible reactions in which RLM101.

In the case of irreversible charge transfer reactions, asuitable ��p �Ep/2

−1 value was obtained for Esw=100mV, regardless of the RLM value (not shown).

4. Conclusions

A description of the SW voltammetric behaviourconsidering the ligand concentration effect over non-labile metallic complexes has been formulated. Thetheoretical voltammograms obtained according to theproposed schemes permit the morphology of the re-sponse for a wide range of experimental parameters tobe examined. These schemes can be found for broadcategories of reactions, especially for the case of SWstripping analysis of adsorbed metallic complexes.

The schemes in this study provide a means to charac-terise the electrochemical properties of adsorbed metal-lic complexes without employing a great ligand excess.Low ligand concentrations can be adjusted to discernthe complex stoichiometry as well as to determine theligand chemical state. In contrast, high ligand–complexratios emphasise features concerning the rate process.Thus, the quasi-reversible maximum is increased andthe effect of ks is stronger.


={�S(i )+Y1(i )}a1−1 (s)Q(i )

gas constant (J mol−1 K−1)RRLM ligand–complex concentration ratio

=1 cmrs

= (i )1/2−(i−1)1/2S(i )

T temperature (K)=DM(Hg)

1/2 {�rs exp[�(m)]}−1T(m)

complex stoichiometryu�(m) =exp[��(m)]k s

−1+Q(1) (s)={exp[ay

2(�i )] erfc[ay(�i )1/2]Yy(i )

−exp[ay2(�(i−1))] erfc[ay(�(i−1))1/2]}

ay−1 (s1/2)

Za =RLMKad,MLDL1/2[ f� ]−1

Zb =RLMKad,ML[Kad,LfP(1)]−1

=�m−1j=1 �( j )S(i )�a(m)

=�m−1j=1 �( j )Q(i ) (s)�b(m)

=P (1)−1�m−1

j=1 �( j )P(i )�c(m)

charge transfer coefficient�

time increments in each SW period (s)�

dimensionless SW potential program�(t)initial ligand and complex surface con-�L

ini; �MLu

ini

centrations (mol cm−2)ligand and complex surface concentra-�L; �MLu

tions (mol cm−2)� =2(�/�)1/2 (s1/2)

=I(t)/(nFAf�MLu

ini ) dimensionless�(t)current function

forward and backward normalised cur-�f, �b

rents=�f−�b; normalised net current��

normalised net peak current��p

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