Comparison Between Derivative and Differential Pulse Voltammetric Curves of EC, CE and Catalytic Processes at Spherical Electrodes and Microelectrodes

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1 Int. J. lectochem. Sci., (7) Compaison Between eivative and iffeential Pulse Voltammetic Cuves of, and Catalytic Pocesses at Spheical lectodes and Micoelectodes. Molina and I. Moales epatamento de Química Física, Univesidad de Mucia, spinado 3, Mucia, Spain -mail: Received: 5 Febuay 7 / ccepted: 8 pil 7 / Published: May 7 nalytical expessions fo deivative voltammety (V) and diffeential pulse voltammety () coesponding to, and alytic mechanisms at spheical electodes ae deduced. Though compaison of these esponses, citeia to disciminate between these thee mechanisms ae established. Methods fo detemining the kinetic constants of the chemical eactions based on the measuement of coodinates ae also poposed. Keywods: Homogeneous kinetic; Steady state; Spheical electodes; Micoelectodes; iffeential pulse voltammety.. INTROUCTION In pevious papes we have tackled alytic and mechanisms in voltammety with constant potential at spheical electodes unde kinetic steady state conditions and also supposing a puely diffusive behaviou fo pseudo-species ζ ( c B + c C ) and fo species c [-3]. This last appoximation has been intoduced by us [, 3], leading to simple equations which have been shown to be valid fo any electode adius (if the kinetic is fast enough) o fo any value of the kinetic constants of the chemical eaction (when the electode adius is less than a cetain value) [, 3]. In all the mechanisms we have consideed that all the species paticipating in the pocesses ae initially pesent in the solution. In this pape we have deduced fist the esponse obtained fo an mechanism in potential constant voltammety in ode to compae the expessions of the I t cuves of the thee mechanisms. ftewads, we obtain the coesponding esponses to the, and alytic mechanisms when deivative voltammety (V) and/o diffeential pulse voltammety () ae used,

2 Int. J. lectochem. Sci., Vol., since, as is demonstated in this pape, unde the ight conditions, the expessions of the deivative voltammetic esponse can be used in diffeential pulse voltammety (as occus fo a evesible pocess) if a sufficiently small potential step is used [4-8]. The use of the above techniques with spheical electodes o micoelectodes allows us to distinguish clealy between, and alytic pocesses and also to popose methods to detemine the kinetic paametes of the chemical steps, fom the measuement of the height and position of the obseved s. We also demonstate that unde the above conditions [, 3] the I cuves coesponding to, and alytic mechanisms can be witten in a fomally identical way to those fo an mechanism whee the half wave potential / fo each pocess is dependent on the ' homogeneous kinetic ate constants in the case of and mechanisms, but coincides with in the alytic mechanism. This I behaviou gives ise to the paametes of V and coesponding to the thee above mechanisms having an analogous fom to that coesponding to the mechanism, efeing to the suitable /.. KINTIC STY STT PPROXIMTION SUPPOSING PURLY IFFUSIV BHVIOUR FOR ζ N c OR c We will simultaneously analyse the following thee eaction schemes [4, 9]: + ne mechanism (I) B k C k mechanism (II) B k C+ ne k Catalytic mechanism (III) k B C+ ne B k k C+ ne B C k Schemes (I) and (II) coespond to fist o pseudo-fist ode and mechanisms in which only one of the species paticipating in the electode pocess (the educed one, B, in the pocess, o the oxidized one, C, in the pocess) is chemically coupled to diffeent species. Moeove, a alytic pocess is epesented by Scheme (III). Fo these thee mechanisms we define the equilibium constant K: and k c K () k c B C κ k + k () c i (ib o C) being the bulk concentations of species B and C and k and k the homogeneous eaction ate constants.

3 Int. J. lectochem. Sci., Vol., If we conside the kinetic steady state appoximation supposing a puely diffusive behaviou fo pseudo-species ζ and species c ( mechanism) o c ( mechanism) [, 3], the following expessions fo the I t cuves ae obtained fo these thee mechanisms, when consideing identical diffusion coefficients:.. mechanism Fo an pocess this appoximation [, 3] supposes an intemediate situation between the kinetic steady state (whose equations can be obtained following the pocedue indied in Ref. [3]) and the tue steady state []. Theefoe, unde these conditions (see Refs. [, 3]) and supposing that all species paticipating in Scheme I (, B and C) ae initially pesent in the solution, we obtain the following expession fo the I t cuve in the case of pocess (see ppendix): being I d I ( c ) d ( ) + K Ke ζ c (3) K e K ( ) ( ) I c nfc (4) ζ c + c (5) B C nf ( ' ) (6) RT and ae the eaction and diffusion laye thicknesses fo a spheical electode [], espectively, which have been shown to be independent of the applied potential [3], given by: κ + + κ (7) π t + π t + π t (8).. mechanism s in the case of the pocess, this appoximation [, 3] leads to a handy, time dependent expession fo the I t cuve, supposing that all species paticipating in Scheme II (B, C and ) ae initially pesent in the solution: I ( + K ) e c ζ (9) I ζ + ( + K ) e + K d

4 Int. J. lectochem. Sci., Vol., whee Id ζ nfζ () Note that q. (9) can be obtained fom q. (3) coesponding to an pocess eplacing K, and n by / K, and -n, espectively, and taking into account that c c..3. Catalytic mechanism In the case of the alytic mechanism (fo which ζ c + c ζ c + c ), and due to B C B C ctn the singulaities that this mechanism shows in its oute to the steady state [], the supposition of a puely diffusive behaviou does not have an additional meaning egading the kinetic steady state since both assumptions lead to the same time independent esponse, in contast to the cases of and mechanisms: I Ke () nfζ + K + e.4. Compaison between,, alytic and pocesses It is vey inteesting to highlight that the simplified equations fo the cuent coesponding to and mechanisms (qs. (3) and (9), espectively) pesent two inteesting limit physical situations. Indeed, if we suppose that the diffusion laye thickness tuns into the eaction laye thickness, the kinetic plays its maximum ole and the esponses obtained fo o [, 3] mechanisms (qs. (3) and (9), espectively), by making c c in q. (3) o C convet into that coesponding to a alytic mechanism [] (q. ()), i.e., c c in q. (9), B I I I if () c cc c cb On the othe hand, if we suppose that the eaction laye thickness eaches the value of the diffusion laye thickness, qs. (3), (9) and () obtained fo, and alytic mechanisms, espectively, become that coesponding to the simple chage tansfe pocess + ne B (fo mechanism) o C + ne (fo and alytic mechanisms), i.e., ( B) ( ) ( C ) + C + if + I I ne I I ne I I ne (3) The main achievement of these expessions fo, and alytic pocesses, which ae applicable fo κt 9.7 and/o t /.63 [3], is that the esponses can be lineaized in a fomally identical way to that coesponding to an evesible pocess, in such a way that, even if all the species ae initially pesent in the solution, they fulfil:

5 Int. J. lectochem. Sci., Vol., 7 39 RT I I / nf I + I lim, c ln fo, and alytic pocesses lim, a (4) whee: RT + K + ln nf K + ' / RT + nf ' / ln ' / + K + K (5) (6) (7) I lim,c and I lim,a ae the hodic and anodic limiting cuent of the coesponding mechanism: ( ) I I c (8) lim, c d I I ζ (9) lim,c d lim, c I nfζ + K () K I I c ζ c lim, a d ( K + ) ( ζ ) lim, a d () I I c ζ () lim, a I KnFζ + K (3).4.. Paticula cases a) Tue steady state If the electode adius is esticted to the inteval / κ / π t /, the spheical eaction laye thickness (q. (7)) emains unalteed, wheeas the spheical diffusion laye thickness mico, given by q. (8), simplifies to. In this paticula case, the half wave potentials coesponding to and mechanism become independent of time (see qs. (5)-(6)). b) Plana electode In plana diffusion and theefoe, the eaction and the diffusion laye thicknesses,, plan plan and, given by qs. (7) and (8), simplify to / κ and π t, espectively. Note that in this case the half wave potentials of and pocesses behave as that deduced by Koutecký applying the eaction laye model []. Moeove, note that in plana electodes an independent of time esponse can not be eached.

6 Int. J. lectochem. Sci., Vol., IFFRNTIL PULS VOLTMMTRY We can establish a diect elationship between the deivative esponse di / d and the diffeential pulse voltammety (). is a double pulse potential technique in which two potential pulses of amplitude and and length t and t, espectively, with t >> t, being, ae applied to an electode with the potential scanned in the negative diection ( < ) o in the positive diection ( > ) in such a way that a delay between each pai of pulses is intoduced in ode fo the equilibium to be e-established [6]. In this potentiostatic technique the diffeence I I ( t + t ) I ( t ) is plotted vs.. s is well known, the esponse ( I / ) vs. can be consideed pactically identical to the deivative of the I cuve (deivative voltammety) fo a evesible pocess unde the ight conditions [4-8]. This citeion can be genealised to the thee mechanisms we ae consideing in this pape,, and alytic, once we have demonstated (see above) that thei I- esponses can be lineaized as that coesponding to an evesible pocess. Then, if we conside a sufficiently small diffeence between the applied potentials (which has been established in the bibliogaphy as << RT nf [4, 5, 7], we can take an appoximation of the esponse I to the deivative esponse di / d : di RT I if << (4) d nf nalysing the heights, we find that ( I ) coincides with ( I ) V, with an eo of less than 5%, if <.6 RT nf. On the othe hand, on compaing the potentials we find that the diffeence between both techniques is less than 5% if < mv, accoding to the above efeences [, 3]. Howeve, on plotting I vs. ( + ) /, this last estiction disappeas, since in this case the potential coincides with that obtained fo V. Theefoe, the di / d esponses applicable to the technique can be easily deduced fo, and alytic pocesses unde these conditions, fom the deivative of qs. (3), (9) and (), espectively. 3.. mechanism Fom q. (3) and (4) we obtain: ( ζ + ( + ))( + ) + + ( + ) nf K c K K e I Id ( c ) (5) RT K K e The paametes ae as follow: ( ) (6) /

7 Int. J. lectochem. Sci., Vol., 7 39 nf + 4RT K ( I ) Id ( c ) Kζ c + (7) whee p / is the half width. Note that if we define: p / 9 mv (8) nf / ( / ) (9) RT q. (5) can be ewitten as: nf K ζ c I Id ( c ) + RT K + 4cosh (3) ( / ) Fom qs. (6) and (7) it can be seen that both the potential (q. (6)) and the height ( I ) ( I, q. (7)) ae in elation to the kinetic paametes (κ ( k + k ) ) and to the equilibium constant ( K ) of the homogeneous chemical eaction. Howeve, note that in the most usual case in which ζ, the height coincides with that obtained fo a simple chage tansfe pocess ( ( I ) ( ζ ) ( nf / 4 RT ) I ( c ) ) and it does not give any infomation about the homogeneous kinetic. Taking into account q. (7), we can ewite q. (3) as: d I ( I ) cosh ( / ) (3) 3.. mechanism Fom q. (9) and (4) we obtain: ( + ( + ) ζ )( + ) + + ( + ) nf K c K e I Id ( ζ ) (3) RT K K e The paametes ae as follows: ( ) (33) / nf c + 4RT ζ + K ( I ) Id ( ζ ) (34) p / 9 mv (35) Intoducing q. (3) can be ewitten as: nf / ( / ) (36) RT

8 Int. J. lectochem. Sci., Vol., nf c I Id ( ζ ) + (37) RT ζ + K 4cosh ( / ) Fom qs. (33) and (34) it can be seen that both the potential (q. (33)) and the height ( I ) ( I, q. (34)) ae in elation to the kinetic paametes (κ ( k + k ) ) and to the equilibium constant ( K ) of the homogeneous eaction. Taking into account q. (34), we can ewite q. (37) as: I ( I ) cosh ( / ) (38) 3.3. Catalytic mechanism Fom q. () and (4) we obtain: I nf nfζ 4cosh RT The paametes ae as follows: ( ) (39) (4) / nf nfζ I 4 RT ( ) (4) p / 9 mv (4) and, as can be seen, the half width value is identical fo these thee mechanisms (see also qs. (8) and (35)), indiing that χ ( κt) is sufficiently high o that ξ ( t / ) takes the convenient value to satisfy the kinetic steady state. Fom these equations it can be seen that only the height (q. (4)) is in diect elation to the kinetic paametes (κ ( k + k) ) of the homogeneous eaction. Taking into account q. (4), we can ewite q. (39) as: I ( I ) cosh ( ) (43) 4. RSULTS N ISCUSSION In Fig. we have studied the diffeent behaviou of the thee mechanisms (Fig. a), (Fig. b) and alytic (Fig. c) with the vaiation of χ ( κt ) in ode to emphasize that this technique allows us to disciminate them. The figue shows the nomalized diffeential pulse

9 Int. J. lectochem. Sci., Vol., voltammogams fo diffeent values of χ ( κt ) obtained fo these thee mechanisms. It can be seen that the potentials of the I cuves ae shifted towads moe positive and moe negative values with an inceasing of χ fo and pocesses, espectively, wheeas in the case of the alytic mechanism, the potentials emain at the fomal potential value. On the othe hand, the height is independent of the homogeneous kinetic in the case of the mechanism (in the moe usual case fo which ζ ), wheeas it inceases with χ ( κt ) fo the and alytic pocesses, with this incease being moe noticeable in the last case. Taking all of these esults into account, the citeia to disciminate between, and alytic pocesses analyzing the influence of the kinetic constants χ ( κt ) in I cuves, which can be easily deduced fom qs. (6),(7), (33), (34), (4) and (4), ae summed up in Table. Table. ffects on potential and height of, and alytic pocesses when the dimensionless chemical ate constant χ ( κt ) inceases. Mechanisms Peak potential Peak height moves towads moe positive potentials does not vay ( nf / 4RT ) moves towads moe negative potentials inceases Catalytic ' does not vay ( ) notoiously inceases Hence, it is possible to chaacteize, and alytic mechanisms by changing the dimensionless ate constants, which can be done by modifying the expeimental time pulse t o by modifying the expeimental conditions (such as ph in the case of the eduction of an acid in a buffeed solution o the concentation of ligand in the case of the eduction of a metal complex) if the chemical eaction is of pseudo-fist ode []. In Fig. the esponses of and mechanisms in have been compaed with those ' obtained fo alytic and mechanisms. In this figue the I / / 4π nfc vs. cuves (whee c c fo mechanism and c ζ fo and alytic mechanisms), coesponding to, (solid lines in Figs. a and b espectively) and alytic (dashed lines) mechanisms have been plotted fo diffeent values of ξ ( t / ). Thus, qs. (3), (37) and (39) can be ewitten, espectively, as: I nf Kζ c + (44) 4π nfc RT K + 4cosh / I nf c + (45) 4π nfζ RT ζ + K 4cosh / RT I nf (46) 4π nfζ 4cosh

10 Int. J. lectochem. Sci., Vol., a mechanism II / I/Id (c ) / V χ b χ 5 mechanism 5 II / I/Id (ζ )/ V c χ 5 Catalytic mechanism II / I /4π nfζ / V ' / V Figue. I / / I ( c ) (a, q. (3)), d I / / 4π nfζ (c, q. (39)) vs. mechanism (a): K, ζ ; mechanism (b): 3 K. ξ ( 3. cm, consideing t s and cuves. I / / Id ζ (b, q. (37)) and κ ). ' cuves fo seveal values of χ ( t K, 5 c cm ; alytic mechanism (c): - s ). Values of χ ae on the

11 Int. J. lectochem. Sci., Vol., s can be seen in this figue, a decease of the electode adius (i.e., an incease of ξ ( t / )) gives a decease of the heights in all the cases. Howeve, wheeas the potential of the alytic cuves emains independent of the spheicity of the electode, and is equal to ' the fomal potential (see q. (7)), the coesponding to o mechanisms moves away fom ' towads moe negative and moe positive potentials, espectively, as deceases, i.e., ξ inceases (qs. (5) and (6)), in such a way that fo high values of the spheicity electode (see cuves fo ξ ), the cuves obtained fo these mechanisms ae almost supeimposable with that obtained fo the alytic pocess (dashed line) and also pactically coincide (with an eo of less than %) with the one coesponding to a simple chage tansfe pocess ( mechanism, see the dotted lines) unde stationay conditions, i.e. when deceases. This fact indies that, in this case ( χ ), the homogeneous kinetic of the chemical eaction cannot be studied with spheicity values of ξ ( χ [4, 5]) since it has been masked o immobilized. The movement of the potential fo and mechanisms with the vaiation of ξ can also be used as a citeion in ode to identify and disciminate these thee mechanisms. Thus, taking all of these esults into account, the citeia to disciminate between, and alytic pocesses analyzing the influence of the spheicity of the electode ξ ( t / ) in I cuves, which can be easily deduced fom qs. (6),(7), (33), (34), (4) and (4), ae summed up in Table. Table. ffects on potential and height of, and alytic pocesses when the spheicity of the electode ξ ( t / ) inceases. Mechanisms Peak potential Peak height moves towads moe negative potentials slightly deceases moves towads moe positive potentials deceases Catalytic ' does not vay ( ) notoously deceases It is inteesting to note that qs. (3), (37) and (39), fo, and alytic mechanisms, espectively, ae explicit expessions and such that it is possible to analytically deduce the paametes (qs. (6)-(7), (33)-(34) and (4)-(4)), fom which the homogeneous kinetic ate constants can be obtained. Thus, in the case of an pocess we can deduce κ ( k + k ) fom the potential (q. (6)): with: κ θ θ ( + K Kθ ) nf exp RT ( ) ' (47) (48)

12 Int. J. lectochem. Sci., Vol., a ξ Compaison between and alytic (- - -) mechanisms I (I / ) I / 4π nfc / V - (cc (), ζ (alytic)) 5 5 b mechanism ξ Compaison between and alytic (- - -) mechanisms I (I / ) I / 4π nfζ / V mechanism ' / V Figue. I nfc / / 4π vs. ' cuves ( c c fo mechanism and c ζ fo and alytic mechanisms) fo seveal values of ξ coesponding to (solid lines in a, q. (44)), (solid lines in b, q. (45)) and alytic (dashed lines, q. (46)) mechanisms. The dotted line is obtained fo an mechanism unde stationay conditions. χ. mechanism: K, ζ. mechanism: K, Consideing t s and values ae: c. Catalytic mechanism: K. Values of ξ ae on the cuves. 5 cm 3. 4,.6 4 and s the coesponding values (in cm) fom mino to majo ξ The kinetic constants, κ ( k + k ), fo a pocess can be obtained fom the potential (q.(33)),

13 Int. J. lectochem. Sci., Vol., with: κ o fom the height (q. (34)): κ θ K (( + K ) θ ) nf exp RT ( ) ' c ζ K 4 c ζ + ( I ) 4RT nf Id ( ζ ) ( I ) RT nf Id ( ζ ) (49) (5) (5) In the case of a alytic pocess we can deduce κ fom the height (q. (4)), obtaining: κ 4 I RT nf nf ζ (5) Moeove, as is shown in the following figues, we can popose woking cuves to obtain the homogeneous kinetic ate constants fom the potentials (Fig. 3) and the heights (Figs. 4 and 5), which, as has been shown, depend on the value of the ate constants and the equilibium constant of the chemical eaction. In compaison with dc pocedues fo the detemination of the ate constants, eveals a much moe accuate due to that the is scacely affected by no desied influences as ae double laye and ohmic dop effects. In Fig. 3 we have plotted the vaiation of the potential with χ ( κt ) fo (Fig. 3a, q. (6)) and (Fig. 3b, q. (33)) mechanisms, fo diffeent values of the spheicity of the electode, ξ ( t / ). Fom this figue it can be seen that the potential shifts towads moe positive (Fig. 3a) and moe negative (Fig. 3b) values when χ inceases o ξ deceases. The figue shows that the potentials become independent of the homogeneous kinetic paametes when the electode spheicity ξ inceases and χ deceases. Thus, the cuve fo ξ diffes by less than % fom that coesponding to an mechanism (dotted line) if χ <. Fig. 4 shows the vaiation of the nomalized height with χ fo a mechanism (q. (34)) fo diffeent values of the equilibium constant K. Fom this figue it can be seen that, fo this mechanism, the height always inceases with χ and deceases with K. Moeove, the detemination of χ is clealy moe accuate with lowe values of the equilibium constant. Thus, once the height has been calculated, and the equilibium constant K is known, we can obtain χ fom the x-axis of the coesponding cuve.

14 Int. J. lectochem. Sci., Vol., a mechanism ξ.5.6 / V 5 p - ' b Catalytic and mechanisms Catalytic and mechanisms / V p - ' mechanism ξ χ Figue 3. Vaiation of the potential with χ fo (3a, q. (6)) and (3b, q. (33)) mechanisms fo diffeent values of the spheicity of the electode, ξ. K, 3a); K, 5 ζ ( mechanism, c ( mechanism, 3b). Values of ξ ae on the cuves. Consideing t s and cm s, the coesponding values (in cm), fom mino to majo ξ values, ae: 6.3 4,.6 4, , In Fig. 5 we can see the vaiation of the nomalized height of a alytic mechanism with χ (q. (4)) fo diffeent values of the spheicity of the electode, ξ. Fom this figue it can be seen that the height always inceases with χ and deceases with ξ. Moeove, the detemination of χ is clealy moe accuate with lowe values of the electode spheicity. Thus, in the case of a alytic pocess it is possible to obtain appopiate ξ value. χ fom the x-axis of the cuve coesponding to the

15 Int. J. lectochem. Sci., Vol., 7 4 mechanism 6 K 5 I(I,n ) / I V χ Figue 4. Vaiation of the nomalized height of a mechanism ( I ) / / I ( ζ ) with d χ (q. (34)) fo diffeent values of the equilibium constant K (see q. (34)). ξ ( 3. cm consideing t s and 5 cm s ), c. Values of K ae on the cuves. 3 b ξ.5 I(I ) / I /4π nfζ mechanism 5 χ 5 Figue 5. Vaiation of the nomalized height of a alytic mechanism ( ( I ) / / 4π nfζ ) with χ (q. (4)) fo diffeent values of the spheicity of the electode. Values of ξ ae on the cuves. Consideing t s and values (in cm), fom mino to majo ξ values, ae: 3. 4, , 5 cm s, the coesponding 3. 3,.6 3, 6.3 4,

16 Int. J. lectochem. Sci., Vol., 7 4 Note that the dependence of the nomalized height, ( I ) / / 4π nfζ, with χ is linea, as can be deduced fom q. (4). In this case, it is inteesting to note that the dependence of ( I ) / with ξ and χ is fictitious, in such a way that the tue dependence is with and κ, since, as has been pointed out, this esponse is independent of time unde kinetic steady state conditions. 5. CONCLUSIONS We have deduced the voltammetic I esponse coesponding to an mechanism, and also the deivative voltammetic esponses, di / d, fo, and alytic mechanisms which ae also applicable to diffeential pulse voltammety ( I ) when << RT / nf. This study has been caied out analytically, fom the explicit equations obtained unde the kinetic steady state appoximation with the additional condition of consideing a puely diffusive behaviou fo ζ, c and c. The I cuves of, and alytic mechanisms can be lineaized in a fomally identical way to the coesponding to an evesible pocess, being the half wave potential / dependent on the chemical ate constant κ of each pocess and on the electode adius in the case of and pocesses. We have found citeia to disciminate between, and alytic pocesses based on the influence of the dimensionless chemical ate constants ( χ κt ) and the electode adius (ξ t / ) in I cuves. The esults ae summaized in Tables and. We have given easy analytical expessions to detemine the kinetic constants of the chemical eaction fom the paametes ( potentials and/o heights) of the I cuves. 6. ppendix Moeove, we have also plotted woking cuves in ode to calculate these kinetic constants of the chemical eactions. The mass tanspot to/fom the electode suface when a constant potential,, is applied to a spheical electode of adius is descibed by the following diffeential equation system and bounday value poblem: ( t) ˆ c, (a) ˆ BcB, t kcb, t + kcc, t (b) ˆ CcC, t kcb, t kcc, t (c) () whee: ˆi i + t ()

17 Int. J. lectochem. Sci., Vol., 7 4 t, t >, (, ), (, ), (, ) c t c c t c c t c B B C C (, ) (, ) c t cb t t >, } B whee: c ( t) C, (, ) (, ) B (3) (4) (5) c t e c t (6) and is the applied potential. The cuent obtained is given by: nf ( ' ) (7) RT (, ) (, ) c t cb t I nf nfb F and having thei usual electochemical meanings. With the intoduction of vaiables ζ and φ, defined as []: whee (, t) (, t ) c (, t) c (, t ) B C (8) ζ + (9) (, t) c (, t ) Kc (, t) φ () B C φ measues the petubation of the chemical equilibium (see q. ()), and consideing equidiffusivity conditions, which set that B C, the equation system () and the bounday value poblem (3)-(6) simplify to: t, t >, t ( t) ( t) ( t) κφ ( t ) ˆ c, (a) ˆ ζ, (b) ˆ φ,, (c) ζ ζ φ c, t c,, t c + c,, t B C } (, ) ζ (, ) c t t >, (, t) φ (, t) ζ () () (3) (4)

18 Int. J. lectochem. Sci., Vol., 7 43 with the cuent (q. (8)) given by: ( + K ) c (, t ) e K ζ (, t) + φ (, t) (5) ζ c, t, t I nf nf (6) The kinetic steady state appoximation only affects the vaiable φ (q. ()) and supposes that the petubation of the chemical equilibium is independent of time, i. e., φ ( t) Unde this condition, the solution fo the vaiable φ in q. ()b is immediately obtained,, / t [, 3]. φ κ φ ( ) exp ( ) in such a way that taking into account qs. (4) and (6) also, we can wite: (, t) ( ) I ζ φ φ nf (7) (8) with κ + + κ (9) We now intoduce an additional appoximation which has been shown in a ecent pape to be fundamental fo the geneal definition and undestanding of the spheical eaction and diffusion layes [6]. In this appoximation, the vaiable φ (q. ()) fulfils the kinetic steady state condition ( φ t ). In elation to vaiables c and ζ, we conside also that c t and ζ t, and theefoe both vaiables must veify qs. ()a and ()b. t this point, we simplify this poblem by assuming that the solutions of qs. ()a and ()b have the same fom as that fo species that would only suffe spheical diffusion and would keep independent of time values at the electode suface []. Thus, fo any value of the applied potential,, the solutions fo c and ζ have the fom: c (, t) c ( c c ( )) efc () t ζ (, t ) ζ ( ζ ζ ( )) efc t () whee c ( ) and ζ ( ) ae the value of c and ζ at the electode suface at any constant value. Fom qs. (7) and () the concentation pofiles of species B and C can be immediately obtained:

19 Int. J. lectochem. Sci., Vol., 7 44 κ c ( ) B, t Kζ K ζ ζ efc φ ( ) exp ( ) + K t κ c ( ) C, t ζ ζ ζ efc + φ ( ) exp ( ) + K t qs. () and () fulfil, espectively: (, ) c t c c (, t) ( ) ζ ζ ζ () (3) (4) (5) whee is the diffusion laye thickness: + (6) π t Fom qs. (3), (5), (8), (4) and (5) we obtain: c K + ζ c + e c + K + K + e ( ) ( ) ( ) ( ) ( + )( + ) + ( ) ζ K ζ c e ζ c c + K + K + e ( ) φ + K Ke ζ c c K K e + + ( + ) (7) (8) (9) and taking into account qs. (9) and () we deduce the suface concentations of species B and C: cb ( ) + K ( + ζ c ) (3) c + K + K + e ( ) + ( + e ) c ζ c c + K + K + e C ( ) (3) The concentation pofiles of species, B and C ae obtained by intoducing q. (7) in q. () and qs. (8)-(9) in qs. ()-(3), espectively: c (, t) + K Ke ζ c efc c K ( K ) e (3) t c, t ζ + K Ke ζ c B K + Kefc exp ( + ) c + K c + K + ( K + ) e t κ (33)

20 Int. J. lectochem. Sci., Vol., 7 45 c, t ζ + K Ke ζ c C + efc exp ( ) c + K c + K + ( K + ) e t κ (34) Finally, fom q. (8), and taking into account qs. (4)-(9), the following expession fo the cuent is obtained: I + K Ke ζ c (35) nfc + K + e K + CKNOWLGMNTS The authos geatly appeciate the financial suppot povided by the iección Geneal de Investigación (M) (Poject Numbe CTQ6-55/BQU) and by the Fundación SN (xpedient numbe 379/PI/5). lso, I. M. thanks the Ministeio de ducación y Ciencia fo the gant eceived. Refeences []. Molina, I. Moales, J. lectoanal. Chem. 583 (5) 93. []. Molina, I. Moales, M. López-Tenés, lectochem. Commun. 8 (6) 6. [3] I. Moales,. Molina, lectochem. Commun. 8 (6) 453 [4] J. Heyovský, J. Kuta, Pinciples of Polaogaphy, cademic Pess, New Yok, 966, Ch. XVII [5].P. Pay, R.. Osteyoung, nal. Chem. 37 (965) 634. [6].J. Bad, L.R. Faulkne, lectochemical Methods. Fundamental and ppliions, second ed., Wiley, New Yok, [7]. Molina, J. González, lectochem. Commun. (999) 477. [8] J. González,. Molina, M. López-Tenés, C. Sena, J. lectochem. Soc. 47 () 349. [9] Z. Galus, Fundamentals of lectochemical nalysis, second ed., llis Howood, Chicheste, 994. [] K.B. Oldham, J. lectoanal. Chem. 33 (99) 3. [] J. Koutecký, R. Bdicka, Coll. Czech. Chem. Commun. (947) 337. [] M. Vukovic,. Cukman, V. Pavdic, J. lectoanal. Chem. 54 (974) 9. [3] M. Vukovic,. Cukman, V. Pavdic, J. lectoanal. Chem. 6 (975) 387. [4] J. Koutecký, Chem. Listy 47 (953) 33. [5] J. Koutecký, Collection Czech. Chem. Commun. 8 (953) 597. [6] C. Sena,. Molina, J. lectoanal. Chem. 466 (999) 8. 7 by SG (