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New methods for the application of an alternating current
97
J. Electraanal. Chem., 308 (1991) 97-112 Elsevier Sequoia S.A., Lausanne
New methods
for the application
of an alternating
current
Part 1. Planar electrodes F. Martinez-Ortiz, Departamento (Received
A. Molina
and C. Serna
de Quimica Fisrca, Uniuersidad de Marcia, Espinardo, 30100 Marcia (Spain)
24 July 1990; in revised form 22 November
1990)
Abstract
The theoretical analysis of the application of a pure alternating current to stationary and non-stationary planar electrodes has been carried out. The response obtained is essentially different from any other previously appearing in the literature. The conditions under which the response is never accompanied by the reaction of the supporting electrolyte have been found. The degree of reversibility of the process can be characterized from the shape of the E/t curves. The applicability of the equations deduced is shown for several experimental systems.
INTRODUCTION
As has been frequently demonstrated, chronopotentiometric techniques in which the applied current changes in sign are of great use for the study of electrode processes [l-3]. They can be classified into two groups: (1) techniques in current reversal, which have been developed extensively [l, 3-51, with many applications in several fields; and (2) techniques with alternating current. These have been applied only to stationary electrodes, and their use has been restricted to the two following cases: (a) Alternating current with small amplitude superimposed on a dc current of a high level [3,6,7]. The transition time is not affected by the alternating current and, for a plane electrode, it can be obtained directly from Sand’s equation. (b) A pure alternating current with large amplitude. In this case, the electrode process is always accompanied by the reaction of the supporting electrolyte, which makes the theoretical analysis difficult [2]. In this paper, we study the application of a pure alternating current to expanding and stationary plane electrodes from a new point of view. We find the conditions 0022-0728/91/$03.50
0 1991 - Elsevier Sequoia
S.A.
98
under which reaction of the supporting electrolyte never occurs, and analyse three situations, depending on the amplitude of the alternating current: (1) There is depletion of the electroactive species in the electrode surface. At this moment, the transition time of the oxidized species is reached and the experiment must be stopped. In these conditions, the use of a pure alternating current is not substantially different from other current-time functions, and we will not consider this case exhaustively in this paper. (2) The surface concentration of the oxidized species remains different from zero, but after the change in sign of the current the depletion of the reduced species, and therefore its transition time, takes place. In this case, we obtain a typical E/t response that allows the characterization of the degree of reversibility of the electrode process by means of simple visual inspection. (3) Neither oxidized nor reduced species are depleted in the electrode surface. As a consequence, an oscillating E/t response governed by the kinetics of the charge transfer is obtained. For any of the above situations, the analysis of the E/t response allows the parameters implied in the heterogeneous process to be obtained easily. Finally, we apply an alternating current as described above to some well-known experimental systems in order to verify the theoretical predictions. EXPERIMENTAL
Waveforms and data collection were performed with an ADDA converter interfacing a Personal Microcomputer (IBM XT) and an AMEL 553 PotentiostatGalvanostat. Experimental E/t curves were sampled at a frequency 2000 times higher than that of the alternating current and then digitally filtered in order to reduce the noise. The working electrodes were a Metrohm Carbon Paste Electrode and a platinum foil. The electrochemical area of these electrodes was determined by using Sand’s equation, taking literature values for the diffusion coefficient of Fe(CN)i[8]. The values obtained were 0.55 f 0.02 cm* for carbon paste and 0.20 f 0.02 for platinum. The reference was a SCE and the counter-electrode was a platinum one. The amplitude of the alternating current I,, was selected in the range 25 PA-5 mA, which yields values of NA (see eqn. 8) between 0.1 and 3 s-l/*. The alternating current frequency w was in the range 0.1-5 s-i. These ranges are appropriate for achieving the conditions described in the Introduction and for preventing distortion in the experimental E/t curves by a non-faradaic process. The chemicals were of Merck reagent grade and they were dissolved in twice-distilled water. THEORY
Let us consider A+ne-$B k,
the charge-transfer
reaction
(1)
99
When a pure alternating
current
Z,(t) (i = s or c) of the form
I,( 1) = Z, sin( wt)
(1)
Z,(t)
(2)
= z, cos( wt)
is applied to an expanding plane electrode, the surface concentrations of species A and B can be calculated by following a procedure similar to that described in ref. 9, by using the power expansion for the sine and cosine functions. In this way, we find that
where p = cB*/cA*
(5)
Y = ( wkl)1’2
(6)
p = ( r/r,)1’3
(7)
NA = 2Z,,/nFA,(
DA)1’2c;
(8)
r, = r, + r
(9)
r, is a blanking time prior to the application of the current. This time must be used with a non-stationary electrode when a net current is applied at the beginning of the experiment, e.g. for a current-time function given by eqn. (2) (see ref. 10 for other advantages derived from the use of rl). Gi (i = s or c) is a functional series which has the form
(10) s,(p)
=
1+
P3
7
P6
3(4j + 5) + 18 (4j + 5)(4j
20 P’ +27 (4j+5)(4j+7)(4j+9)
+ 7)
+ “.
(11) (12)
C,(P)
=
1+
P6 P3 7 3(4j + 3) + i% (4j + 3)(4j
P9 20 + 27 (4j + 3)(4j + 5)(4j
+ 5)
+ 7) + ”
(13)
100
Moreover,
Setting
c,(O, t) = 0 (eqn. 3) we deduce
where &, is the value of the variable j3 for t = rA. When the transition time rA of species A is reached, the experiment must be stopped, since electrolysis of the supporting electrolyte cannot be avoided. In this situation, the use of an alternating current is not advantageous compared with other types of current-time functions [9]. However, there exist values of NA or Z,, (lower than a predetermined one) for which species A remains undepleted in the electrode surface and, therefore, there is no value of rA which fulfils eqn. (15) (see Results and Discussion). In this case, species B can or cannot be depleted in the electrode surface after the first change in sign of the alternating current, depending on the value of CL.If p< pli,,, (the value of pli,,, is given in the Results and Discussion), ra always exists and is given by l/2
'8
-
-
-(tl
+
7B)2’3
(16)
N&&pd
Na = YN./P
(17)
where &, is the value of fi for t = rB. Although eqns. (15) and (16) do not provide explicit expressions for T, its calculation can be carried out easily by means of any standard mathematical procedure for this purpose. Otherwise, if ZJ> plirn, ~a is not reached and the E/t response shows an oscillating behaviour around E o controlled by the kinetics of process (I). The general E/f response, which can be obtained from eqns. (3), (4) and (15), is NA~0;;‘210”9”‘Z, (t )/I,, = t,2’3 - NAt1’2G, ( fi, ot) - 10q”‘( p~t,2’~+ TN~~“~G, (,8, tot)) e0.
A
n(t)
=
(19)
4k,2/DA
= (nF/RT
(18)
In lO)[E(t)
-E”]
(20)
Equation (18) is notably simplified for a reversible process (k, + co). For an irreversible process (k, +C 1 cm s-i), owing to the change in sign of the alternating current, the following two limiting equations are deduced: (a) Z,(t) > 0. Equation (18) becomes E(t)-EO=-!$ln
nFA kc* i
sA
0
+~hg,?h
(21)
101
(b) Z,(t) < 0. Equation E(t)
-E”
= (1 _RGnF
(18) becomes In
10 nFA,k,c,*
+ (1 _Rz) nF In g,““”
(22)
where g,4’ =
I,{ ty3 - NAt”*Gi( /I, tit)} (23)
I,(t) g,=d
-‘iCt)
=
I,,{ ptf’3 + yNAtl’*Gi( P, tit>} Stationary
(24)
plane electrode
The E/t response electrode, with fixed AotF’3 = A in all the (eqns. 11 and 13) and
and the transition time corresponding to a stationary plane area A, can be deduced by setting t, x=-t (j3 + 0) and previous equations. In these conditions, S,(O) = C,(O) = 1 the Gj series takes the simple form
(25)
Moreover, NA = 2Z,,/nFA(
D*)“*c,*
(26)
RESULTS AND DISCUSSION
In order to determine the conditions for the existence of the transition time for species A (eqn. 15) or B (eqn. 16), we must rewrite eqns. (3) and (4), taking into account eqns. (5) (8) and (17), in the form
CA(O, t> ?-Hi
(27)
CB(O, t> =$+H,
(28)
CA*%
GNB
A
B
where H, = sGi(/3, s
wt)
(29)
-02C
12 t Is
Fig. 1. Deccndence of H,(B, wt) on t for an expanding plane electrode (cqn. 36). w =1 s-‘. of r1 (in s) are shown on the curves.
The values
for an expanding plane electrode and H,,r = t”*G;,J
wt)
(30)
for a stationary plane electrode. The function H, (i = s or c), given by eqns. (29) and (30), is very useful for the analysis of the E/t response because it is directly related to the surface concentrations of the electroactive couple (eqns. 27 and 28) [9]. The dependence of Hi with time is shown in Figs. 1 and 2 for several values of w and t,. Depending on the values of NA - or I, - and p, the following three situations can occur: (1) When Hi = l/N,, there exists a transition time 7A for species A, as can be deduced from eqn. (27). Moreover, for any values of w and t,, H, reaches its absolute maximum H,,max (see Figs. 1 and 2), verifying for t = rA the relationship H r,max =
l/NA,min
(4
Therefore, it is obvious that this transition time exists only if NA ’
NA,min
(ii)
When condition (ii) is verified, the E/t response obtained with an alternating current is not essentially different from that obtained with other programmed currents. Table 1 lists the values of NA,minfor several values of w and t, when the alternating current given by eqn. (1) is used. (2) If NA < NA,min,there is no transition time for the reduction process. In this case, the transition time of species B can be reached during the reoxidation process.
103
-0.21 0
2
4
6
8
10
t Is Fig. 2. Dependence of H,(B, wt) on t for an expanding w (in s-‘) are shown on the curves.
plane electrode
(eqn. 36). t, =1 s. The values of
We deduce from eqn. (28) that TV exists when Hi = -l/N,. Taking into account the fact that Hi takes its absolute minimum H,,,i, (see Figs. 1 and 2), the condition for the existence of ~a is
NB’ TABLE
(iii)
N*,min
1
Values of NA.min (eqn. i) and P,~,,, (eqn. vi) deduced current given by eqn. (1) is applied
for different
values of t, and o when an alternating
O/SC'
NA,mdS‘/6
Plim
0
0.5
0 1 1 2 2 3 3
2.0 0.5 2.0 0.5 2.0 0.5 2.0
2.41 1.96 2.90 3.90 3.40 4.30 3.80 5.30
0.09~ 0.11~ 0.09~ 0.10~ 0.09~ 0.09~ 0.08~ 0.08~
Cd/SC’
NA,min/S-“2
Pltm
0.5 2.0
1 .os 2.11
0.55~ NA 0.27~ NA
h/S EPE a
NA N, NA NA NA NA NA NA
SPE =
a EPE: expanding
plane electrode;
SPE stationary
plane electrode.
0.4
0
0.8 t Is
1.2
1.6
Fig. 3. Potential-time curves for a reversible process on an expanding plane electrode. The alternating current applied is given by eqn. (2). tl =1 s, y =l, n =l, /I = 0, T= 298 K, w =1.5 SC’. The values of N, (in s’j6) are shown on the curves. The transition time ~a ( = 1.581 s) does not depend on NA (eqn. 18).
where N B,min
=
‘/I
Condition P<
Hi,rnin
1
(iii) is equivalent
Plim
(iv> to (see eqn. 17) (v>
with plim
=
yNA/NB.min
(vi)
The values of pLli,,,for an alternating current like (1) and several values of o and t, are given in Table 1. Obviously, rB always exists if species B is not initially present in the solution (p = 0), verifying (see eqns. 5, 17 and 28) that Hi (&#‘%)
= o
(vii)
In this case, 7B is given by the first intercept of the function Hi with the t-axis (see Figs. 1 and 2). As a consequence, for fixed values of t, and w, 7B does not depend on NA (ie. I,,, CA*and the electrode area). This behaviour is shown in Fig. 3 for a reversible process on an expanding plane electrode. (3) If neither condition (ii) nor (v) is verified, an oscillating E/t response (around E “) controlled by the heterogeneous kinetics is obtained. This response is shown in Fig. 4 for a stationary plane electrode. For an expanding plane electrode the E/t curves are similar, but the amplitude of the oscillations decreases with time, owing to the growth of the electrode area.
105
t/.5
Fig. 4. Potential-time curves for a reversible process on a stationary plane electrode when there is no transition time for either species A or species B. The alternating current applied is given by eqn. (1). p = 0.5, NA = 0.8 SC”*, y = 1, n =l, T= 298 K. The values of w (in s-‘) are shown on the curves.
From the above, we can conclude that the existence of the transition time of species A or B is determined mainly by the alternating current amplitude I, and p. The angular frequency of the alternating current, w, is responsible for the value of 7, but does not determine its existence substantially.
-300J 0
0.8
1.6
2.4
t/s
Fig. 5. Influence of applied is given by Reversible process; conditions as in Fig.
k, on the E/r curves on an expanding plane electrode. The alternating current eqn. (2). NA =l s”~, o =1 SC’, DA =10-s cm2 S-I, a= 0.5, ~a= 2.385 S. (a) (b) k, = 5x 10m4 cm s-l; (c) k, =10-4 cm S-‘; (d) k, =10-s cm S-I. Other 3.
106
600
-6007
a3 0
0.8
1.6
2.4
3.2
tls Fig. 6. Influence of n on the E/r curves on an expanding plane electrode. The alternating current applied is given by eon. (1). NA = 3 s’/~, w =1.5 SC’, p = 0.2, k, =10e4 cm SC’, 7B = 3.219 s. The values of 01are shown on the curves. Other conditions as in Fig. 5.
The influence of cy and k, on the E/t curves under several conditions is shown in Figs. 5-7. In all these plots, we can observe one (Figs. 5 and 6) or several (Fig. 7) typical cross-potential points corresponding to the values of wt for which Z,(t) = 0
60
8
16
24
t Is Fig. 7. Influence of k, on the E/t curves on an expanding plane electrode when there is no transition time for either species A or species B. The alternating current applied is given by eqn. (1). NA =l s’j6, w = 0.5 s-i, ).t = 1. The values of k, are shown on the curves. Other conditions as in Fig. 5.
107
6OC-
500.
> 400. E a :
300.
200.
100, 0
1 log
2 (k, /cm
3 s-’ )
4
Fig. 8. Variation of AE, with log k, on a stationary plane electrode. The alternating current applied is given by eqn. (2). NA = 2 SC”‘, w =1.5 SC’, DA =lO-’ cm2 SC’, y =l, n =l, T= 298 K. The values of (Y are shown on the curves.
[8]. When 78 does exist, the E/t curves become more and more distorted as k, diminishes, leading to the appearance around the cross-potential of a shoulder that transforms into a jump for a totally irreversible process (see Fig. 5). Figure 6 shows the influence of cx (with k, = 1 X lop4 cm SC’) on the E/t curves, establishing clearly that a decrease in (Yshifts the curves to more negative potentials. In Fig. 7 the influence of KS on the E/t curves when there is no transition time for either species A or species B is shown. In these curves, the potential difference between consecutive peaks increases as k, diminishes. This last phenomenon can be used to obtain an estimate of k,. In Fig. 8 we have represented AEr vs. log k, for two extreme values of (Y (0.3 and 0.7) to show how these plots can be used as working curves to obtain k,. When a current of the type given by eqn. (2) is applied, it is also possible to obtain an estimate of the (Yand k, values from the E/t curves, since from eqn. (21) we obtain for t --j 0
E-E”=-&$. Finally, irreversible
In
nFA,,c,*tfj3k
S
4
it is possible to obtain process by plotting E(t)
(31) accurate values of cr, k, and E” for an vs. In g,ca*h (eqns. 21 and 23) and E(t) vs. In
108
(eqns. glanod
22 and 24) [4,5]. From
RT aI-=l-nFP, ln
E”
RT nFP,
k,=!?-(“A-uB) s
=
these plots we obtain
nF
‘APB
OApB ‘A
+
‘,‘A
+
‘B
(32) +ln
‘0
where 0, and P, are the intercepts (i = B) processes, respectively. EXPERIMENTAL
(33)
nFA,c,*
(34) and slopes for the cathodic
(i = A) and anodic
RESULTS
To carry out the recording of experimental E/t curves on plane electrodes a platinum foil and a commercial carbon paste electrode filled with commercial paste were selected. Although the heterogeneous kinetic response on this type of electrode is seriously affected by the paste composition [ll], no special characteristics were required for the selection of this paste since our aim was just to show that the theoretical equations obtained describe the experimental E/t curves properly, rather than to obtain accurate kinetic measurements. The alternating current given by eqn. (1) was applied to two well-known experimental systems: Fe(CN)i-/Fe(CN)%on the carbon paste and platinum electrodes and Fe3+/ Fe’+ on the carbon paste electrode. The first system behaves more irreversibly on carbon paste than on platinum [12], as shown in Figs. 9 and 10.
Fig. 9. Experimental E/t curves obtained for 10 mM Fe(CN)iin 0.4 M KNO, on a Pt electrode. The alternating current applied is given by eqn. (1). I, = 280 pA. The values of w (in s-‘) are shown on the curves.
109
1
1.3
Fig. 10. Experimental E/t curves obtained for 10 mM Fe(CN)iin 0.4 M KNO, on a carbon paste electrode. The alternating current applied is given by eqn. (1). I, = 500 pA. The values of w (in s-l) are shown on the curves.
The second one behaves as totally irreversible (Fig. 11). In these plots, the gradual appearance of the shoulder characteristic of an irreversible process can be observed. In Fig. 12 we represent the experimental E/t curves for the system Fe3+/Fe2+ with lt = 0.27, Z, = 875 PIA (note that under these conditions the transition time of species B does not exist) and Z, = 2400 PA (now the transition time of species B does exist). The theoretical curves obtained using OL= 0.38 and log(k,/cm SC’) =
2
4
6
1lS Fig. 11. Experimental E/r curves obtained for 10 mM Fe3+ in 1 M H2S04 on a carbon paste electrode. The alternating current applied is given by eqn. (1). I, = 500 PA. The values of w (in s-‘) are shown on the curves.
110
600/
1.2
0.6
1.8
2'.4
tlS Fig. 12. Comparison between the experimental and theoretical E/t curves for the system 1 M H,SO, on a carbon paste electrode. The alternating current applied is given by eqn. Fe3+, 6 mM Fe’+, o = 2 s-l, I,, = 2400 pA. (b) I, = 875 PA. Other conditions as in values used in the theoretical calculations were (a) NA = 1.95 s-l/*, y = 0.95, log(k,/cm a = 0.38, /.I = 0.27; (b) NA = 0.7 s- ‘I2 Other conditions as in curve (a). The solid experimental curves and the symbols correspond to the theoretical ones.
Fe3+/FeZ* in (1). (a) 22 mM curve (a). The s-l) = - 5.94, lines are the
-5.94 are also plotted in this figure. These values of (Y and k, were obtained following the method described above. The application of this method to the experimental curves in Fig. 12 is shown in Fig. 13 We obtained the following results: (Y= 0.36 + 0.02, 1 - (Y= 0.60 + 0.02 and log(k,/cm s-‘) = - 5.90 + 0.05.
60
! 4
-1
b
1
Fig. 13. Dependence of E on (I) In gFth (eqn. 23) and (II) In gpd curves in Fig. 12. (in) I, = 2400 PA; ( + + + ) 1, = 875 PA.
(eqn. 24) for the experimental
E/t
111
The correlation coefficients were error in the determination of the the values of NA and N, were transition time corresponding to As can be seen, the agreement predictions is quite good.
in all cases larger than 0.995. In order to make the electrode area or the diffusion coefficients smaller, obtained from Sand’s equation by measuring the a current step of magnitude I,. between the experimental results and theoretical
CONCLUSIONS
The new procedure described in this work for the application of alternating currents to different electrodes can be considered as an easy, versatile and effective technique for the analysis of the behaviour of electrode processes. Double-layer charge effects usually have a complex nature and have not been considered here. However, we used values of w, CA*and I, for which we did not observe experimentally any appreciable distortion of the curves by non-faradaic processes. The amplitude Z, of the alternating current and the value of /J are the main factors determining the existence of the transition time in the reduction or the reoxidation processes, while the angular frequency w is the main factor responsible for the value of the transition time. This technique is particularly interesting when T* does not exist due to the following reasons: (a) The independence of 7B (when cg = 0) with NA (I,,, c,*, DA and electrode area) is found to be very advantageous for the detection of any kinetic complications relating to the behaviour of reduced species B. (b) The heterogeneous kinetic parameters can be obtained independently from the anodic and cathodic responses. This fact can also be used to detect complications in process (I). (c) The degree of reversibility of the charge-transfer reaction can be estimated from the shape of the E/t curves. ACKNOWLEDGEMENTS
The authors greatly appreciate the financial support of the Direction General de Investigation Cientifica y TCcnica (Project No. PB87-0700) and also of the Direction Regional de Education y Universidad de la Comunidad Autonoma de la Region de Murcia (Project No. PCT89/19). We also acknowledge the critical reading of this manuscript by Dr. Rafael Chicon (University of Murcia). NOTATION AND DEFINITIONS
z,(t) Z0 w CA*,cB*
applied alternating current (Z, sin( wt) if i = s or I0 cos( ot) if i = c) amplitude of the alternating current angular frequency (27r times the conventional frequency in hertz) bulk concentrations of species of A and B
112
ks 01
E0 E(t) AE AEP 7A 7B t1
t t, A(h) A0 A P.Y r
heterogeneous rate constants of the forward and the reverse chargetransfer reaction apparent heterogeneous rate constant of charge transfer at E o transfer coefficient formal standard potential of the electroactive couple time-dependent electrode potential = E(t) - E’= difference between the first two peaks (anodic and cathodic) in an oscillating E/t response transition time for reduction process transition time for reoxidation process blank period used only with non-stationary electrodes time elapsed between the application of the alternating current and the measurement of the potential =t,+t time-dependent area for an expanding plane electrode (= Aoti13) electrode area when t, = 1 s fixed area of a stationary plane electrode = 2l31 + x/2)/T(1/2 +x/2) Euler gamma function
Other definitions
are conventional.
REFERENCES 1 A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, Ch. 7. 2 M. Heyrovsk$ and K. Micka in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 2, Marcel Dekker, New York, 1967, p. 193. 3 R.K. Jain, H.C. Gaur and B.J. Welch, J. Electroanal. Chem., 79 (1977) 211. 4 A. Molina and J. Albaladejo, J. Electroanal. Chem., 256 (1988) 33. 5 A. Molina, C. Serna and M. Lbpez-Tents, J. Electroanal. Chem., 284 (1990) 21. 6 Y. Takemori, T. Kambara, M. Senda and I. Tachi, J. Phys. Chem., 61 (1957) 968. I N.P. Bansal and J. Plambek, J. Electroanal. Chem., 78 (1977) 205; 81 (1977) 239. 8 J. Heyrovskq and J. Ktita, Principles of Polarography, Academic Press, New York, 1966. 9 A. Molina, M. L6pez-Tents and C. Sema, J. Electroanal. Chem., 278 (1990) 35. 10 A. Molina, F. Martinez-Ortiz, J. Zapata and J. Albaladejo, J. Electroanal. Chem., 227 (1987) 1. 11 C. Urbaniczky and K. Lundstriim, J. Electroanal. Chem., 176 (1984) 169. 12 N. Tanaka and R. Tamamushi. Electrochim. Acta, 9 (1964) 963.
J. Electraanal. Chem., 308 (1991) 97-112 Elsevier Sequoia S.A., Lausanne
New methods
for the application
of an alternating
current
Part 1. Planar electrodes F. Martinez-Ortiz, Departamento (Received
A. Molina
and C. Serna
de Quimica Fisrca, Uniuersidad de Marcia, Espinardo, 30100 Marcia (Spain)
24 July 1990; in revised form 22 November
1990)
Abstract
The theoretical analysis of the application of a pure alternating current to stationary and non-stationary planar electrodes has been carried out. The response obtained is essentially different from any other previously appearing in the literature. The conditions under which the response is never accompanied by the reaction of the supporting electrolyte have been found. The degree of reversibility of the process can be characterized from the shape of the E/t curves. The applicability of the equations deduced is shown for several experimental systems.
INTRODUCTION
As has been frequently demonstrated, chronopotentiometric techniques in which the applied current changes in sign are of great use for the study of electrode processes [l-3]. They can be classified into two groups: (1) techniques in current reversal, which have been developed extensively [l, 3-51, with many applications in several fields; and (2) techniques with alternating current. These have been applied only to stationary electrodes, and their use has been restricted to the two following cases: (a) Alternating current with small amplitude superimposed on a dc current of a high level [3,6,7]. The transition time is not affected by the alternating current and, for a plane electrode, it can be obtained directly from Sand’s equation. (b) A pure alternating current with large amplitude. In this case, the electrode process is always accompanied by the reaction of the supporting electrolyte, which makes the theoretical analysis difficult [2]. In this paper, we study the application of a pure alternating current to expanding and stationary plane electrodes from a new point of view. We find the conditions 0022-0728/91/$03.50
0 1991 - Elsevier Sequoia
S.A.
98
under which reaction of the supporting electrolyte never occurs, and analyse three situations, depending on the amplitude of the alternating current: (1) There is depletion of the electroactive species in the electrode surface. At this moment, the transition time of the oxidized species is reached and the experiment must be stopped. In these conditions, the use of a pure alternating current is not substantially different from other current-time functions, and we will not consider this case exhaustively in this paper. (2) The surface concentration of the oxidized species remains different from zero, but after the change in sign of the current the depletion of the reduced species, and therefore its transition time, takes place. In this case, we obtain a typical E/t response that allows the characterization of the degree of reversibility of the electrode process by means of simple visual inspection. (3) Neither oxidized nor reduced species are depleted in the electrode surface. As a consequence, an oscillating E/t response governed by the kinetics of the charge transfer is obtained. For any of the above situations, the analysis of the E/t response allows the parameters implied in the heterogeneous process to be obtained easily. Finally, we apply an alternating current as described above to some well-known experimental systems in order to verify the theoretical predictions. EXPERIMENTAL
Waveforms and data collection were performed with an ADDA converter interfacing a Personal Microcomputer (IBM XT) and an AMEL 553 PotentiostatGalvanostat. Experimental E/t curves were sampled at a frequency 2000 times higher than that of the alternating current and then digitally filtered in order to reduce the noise. The working electrodes were a Metrohm Carbon Paste Electrode and a platinum foil. The electrochemical area of these electrodes was determined by using Sand’s equation, taking literature values for the diffusion coefficient of Fe(CN)i[8]. The values obtained were 0.55 f 0.02 cm* for carbon paste and 0.20 f 0.02 for platinum. The reference was a SCE and the counter-electrode was a platinum one. The amplitude of the alternating current I,, was selected in the range 25 PA-5 mA, which yields values of NA (see eqn. 8) between 0.1 and 3 s-l/*. The alternating current frequency w was in the range 0.1-5 s-i. These ranges are appropriate for achieving the conditions described in the Introduction and for preventing distortion in the experimental E/t curves by a non-faradaic process. The chemicals were of Merck reagent grade and they were dissolved in twice-distilled water. THEORY
Let us consider A+ne-$B k,
the charge-transfer
reaction
(1)
99
When a pure alternating
current
Z,(t) (i = s or c) of the form
I,( 1) = Z, sin( wt)
(1)
Z,(t)
(2)
= z, cos( wt)
is applied to an expanding plane electrode, the surface concentrations of species A and B can be calculated by following a procedure similar to that described in ref. 9, by using the power expansion for the sine and cosine functions. In this way, we find that
where p = cB*/cA*
(5)
Y = ( wkl)1’2
(6)
p = ( r/r,)1’3
(7)
NA = 2Z,,/nFA,(
DA)1’2c;
(8)
r, = r, + r
(9)
r, is a blanking time prior to the application of the current. This time must be used with a non-stationary electrode when a net current is applied at the beginning of the experiment, e.g. for a current-time function given by eqn. (2) (see ref. 10 for other advantages derived from the use of rl). Gi (i = s or c) is a functional series which has the form
(10) s,(p)
=
1+
P3
7
P6
3(4j + 5) + 18 (4j + 5)(4j
20 P’ +27 (4j+5)(4j+7)(4j+9)
+ 7)
+ “.
(11) (12)
C,(P)
=
1+
P6 P3 7 3(4j + 3) + i% (4j + 3)(4j
P9 20 + 27 (4j + 3)(4j + 5)(4j
+ 5)
+ 7) + ”
(13)
100
Moreover,
Setting
c,(O, t) = 0 (eqn. 3) we deduce
where &, is the value of the variable j3 for t = rA. When the transition time rA of species A is reached, the experiment must be stopped, since electrolysis of the supporting electrolyte cannot be avoided. In this situation, the use of an alternating current is not advantageous compared with other types of current-time functions [9]. However, there exist values of NA or Z,, (lower than a predetermined one) for which species A remains undepleted in the electrode surface and, therefore, there is no value of rA which fulfils eqn. (15) (see Results and Discussion). In this case, species B can or cannot be depleted in the electrode surface after the first change in sign of the alternating current, depending on the value of CL.If p< pli,,, (the value of pli,,, is given in the Results and Discussion), ra always exists and is given by l/2
'8
-
-
-(tl
+
7B)2’3
(16)
N&&pd
Na = YN./P
(17)
where &, is the value of fi for t = rB. Although eqns. (15) and (16) do not provide explicit expressions for T, its calculation can be carried out easily by means of any standard mathematical procedure for this purpose. Otherwise, if ZJ> plirn, ~a is not reached and the E/t response shows an oscillating behaviour around E o controlled by the kinetics of process (I). The general E/f response, which can be obtained from eqns. (3), (4) and (15), is NA~0;;‘210”9”‘Z, (t )/I,, = t,2’3 - NAt1’2G, ( fi, ot) - 10q”‘( p~t,2’~+ TN~~“~G, (,8, tot)) e0.
A
n(t)
=
(19)
4k,2/DA
= (nF/RT
(18)
In lO)[E(t)
-E”]
(20)
Equation (18) is notably simplified for a reversible process (k, + co). For an irreversible process (k, +C 1 cm s-i), owing to the change in sign of the alternating current, the following two limiting equations are deduced: (a) Z,(t) > 0. Equation (18) becomes E(t)-EO=-!$ln
nFA kc* i
sA
0
+~hg,?h
(21)
101
(b) Z,(t) < 0. Equation E(t)
-E”
= (1 _RGnF
(18) becomes In
10 nFA,k,c,*
+ (1 _Rz) nF In g,““”
(22)
where g,4’ =
I,{ ty3 - NAt”*Gi( /I, tit)} (23)
I,(t) g,=d
-‘iCt)
=
I,,{ ptf’3 + yNAtl’*Gi( P, tit>} Stationary
(24)
plane electrode
The E/t response electrode, with fixed AotF’3 = A in all the (eqns. 11 and 13) and
and the transition time corresponding to a stationary plane area A, can be deduced by setting t, x=-t (j3 + 0) and previous equations. In these conditions, S,(O) = C,(O) = 1 the Gj series takes the simple form
(25)
Moreover, NA = 2Z,,/nFA(
D*)“*c,*
(26)
RESULTS AND DISCUSSION
In order to determine the conditions for the existence of the transition time for species A (eqn. 15) or B (eqn. 16), we must rewrite eqns. (3) and (4), taking into account eqns. (5) (8) and (17), in the form
CA(O, t> ?-Hi
(27)
CB(O, t> =$+H,
(28)
CA*%
GNB
A
B
where H, = sGi(/3, s
wt)
(29)
-02C
12 t Is
Fig. 1. Deccndence of H,(B, wt) on t for an expanding plane electrode (cqn. 36). w =1 s-‘. of r1 (in s) are shown on the curves.
The values
for an expanding plane electrode and H,,r = t”*G;,J
wt)
(30)
for a stationary plane electrode. The function H, (i = s or c), given by eqns. (29) and (30), is very useful for the analysis of the E/t response because it is directly related to the surface concentrations of the electroactive couple (eqns. 27 and 28) [9]. The dependence of Hi with time is shown in Figs. 1 and 2 for several values of w and t,. Depending on the values of NA - or I, - and p, the following three situations can occur: (1) When Hi = l/N,, there exists a transition time 7A for species A, as can be deduced from eqn. (27). Moreover, for any values of w and t,, H, reaches its absolute maximum H,,max (see Figs. 1 and 2), verifying for t = rA the relationship H r,max =
l/NA,min
(4
Therefore, it is obvious that this transition time exists only if NA ’
NA,min
(ii)
When condition (ii) is verified, the E/t response obtained with an alternating current is not essentially different from that obtained with other programmed currents. Table 1 lists the values of NA,minfor several values of w and t, when the alternating current given by eqn. (1) is used. (2) If NA < NA,min,there is no transition time for the reduction process. In this case, the transition time of species B can be reached during the reoxidation process.
103
-0.21 0
2
4
6
8
10
t Is Fig. 2. Dependence of H,(B, wt) on t for an expanding w (in s-‘) are shown on the curves.
plane electrode
(eqn. 36). t, =1 s. The values of
We deduce from eqn. (28) that TV exists when Hi = -l/N,. Taking into account the fact that Hi takes its absolute minimum H,,,i, (see Figs. 1 and 2), the condition for the existence of ~a is
NB’ TABLE
(iii)
N*,min
1
Values of NA.min (eqn. i) and P,~,,, (eqn. vi) deduced current given by eqn. (1) is applied
for different
values of t, and o when an alternating
O/SC'
NA,mdS‘/6
Plim
0
0.5
0 1 1 2 2 3 3
2.0 0.5 2.0 0.5 2.0 0.5 2.0
2.41 1.96 2.90 3.90 3.40 4.30 3.80 5.30
0.09~ 0.11~ 0.09~ 0.10~ 0.09~ 0.09~ 0.08~ 0.08~
Cd/SC’
NA,min/S-“2
Pltm
0.5 2.0
1 .os 2.11
0.55~ NA 0.27~ NA
h/S EPE a
NA N, NA NA NA NA NA NA
SPE =
a EPE: expanding
plane electrode;
SPE stationary
plane electrode.
0.4
0
0.8 t Is
1.2
1.6
Fig. 3. Potential-time curves for a reversible process on an expanding plane electrode. The alternating current applied is given by eqn. (2). tl =1 s, y =l, n =l, /I = 0, T= 298 K, w =1.5 SC’. The values of N, (in s’j6) are shown on the curves. The transition time ~a ( = 1.581 s) does not depend on NA (eqn. 18).
where N B,min
=
‘/I
Condition P<
Hi,rnin
1
(iii) is equivalent
Plim
(iv> to (see eqn. 17) (v>
with plim
=
yNA/NB.min
(vi)
The values of pLli,,,for an alternating current like (1) and several values of o and t, are given in Table 1. Obviously, rB always exists if species B is not initially present in the solution (p = 0), verifying (see eqns. 5, 17 and 28) that Hi (&#‘%)
= o
(vii)
In this case, 7B is given by the first intercept of the function Hi with the t-axis (see Figs. 1 and 2). As a consequence, for fixed values of t, and w, 7B does not depend on NA (ie. I,,, CA*and the electrode area). This behaviour is shown in Fig. 3 for a reversible process on an expanding plane electrode. (3) If neither condition (ii) nor (v) is verified, an oscillating E/t response (around E “) controlled by the heterogeneous kinetics is obtained. This response is shown in Fig. 4 for a stationary plane electrode. For an expanding plane electrode the E/t curves are similar, but the amplitude of the oscillations decreases with time, owing to the growth of the electrode area.
105
t/.5
Fig. 4. Potential-time curves for a reversible process on a stationary plane electrode when there is no transition time for either species A or species B. The alternating current applied is given by eqn. (1). p = 0.5, NA = 0.8 SC”*, y = 1, n =l, T= 298 K. The values of w (in s-‘) are shown on the curves.
From the above, we can conclude that the existence of the transition time of species A or B is determined mainly by the alternating current amplitude I, and p. The angular frequency of the alternating current, w, is responsible for the value of 7, but does not determine its existence substantially.
-300J 0
0.8
1.6
2.4
t/s
Fig. 5. Influence of applied is given by Reversible process; conditions as in Fig.
k, on the E/r curves on an expanding plane electrode. The alternating current eqn. (2). NA =l s”~, o =1 SC’, DA =10-s cm2 S-I, a= 0.5, ~a= 2.385 S. (a) (b) k, = 5x 10m4 cm s-l; (c) k, =10-4 cm S-‘; (d) k, =10-s cm S-I. Other 3.
106
600
-6007
a3 0
0.8
1.6
2.4
3.2
tls Fig. 6. Influence of n on the E/r curves on an expanding plane electrode. The alternating current applied is given by eon. (1). NA = 3 s’/~, w =1.5 SC’, p = 0.2, k, =10e4 cm SC’, 7B = 3.219 s. The values of 01are shown on the curves. Other conditions as in Fig. 5.
The influence of cy and k, on the E/t curves under several conditions is shown in Figs. 5-7. In all these plots, we can observe one (Figs. 5 and 6) or several (Fig. 7) typical cross-potential points corresponding to the values of wt for which Z,(t) = 0
60
8
16
24
t Is Fig. 7. Influence of k, on the E/t curves on an expanding plane electrode when there is no transition time for either species A or species B. The alternating current applied is given by eqn. (1). NA =l s’j6, w = 0.5 s-i, ).t = 1. The values of k, are shown on the curves. Other conditions as in Fig. 5.
107
6OC-
500.
> 400. E a :
300.
200.
100, 0
1 log
2 (k, /cm
3 s-’ )
4
Fig. 8. Variation of AE, with log k, on a stationary plane electrode. The alternating current applied is given by eqn. (2). NA = 2 SC”‘, w =1.5 SC’, DA =lO-’ cm2 SC’, y =l, n =l, T= 298 K. The values of (Y are shown on the curves.
[8]. When 78 does exist, the E/t curves become more and more distorted as k, diminishes, leading to the appearance around the cross-potential of a shoulder that transforms into a jump for a totally irreversible process (see Fig. 5). Figure 6 shows the influence of cx (with k, = 1 X lop4 cm SC’) on the E/t curves, establishing clearly that a decrease in (Yshifts the curves to more negative potentials. In Fig. 7 the influence of KS on the E/t curves when there is no transition time for either species A or species B is shown. In these curves, the potential difference between consecutive peaks increases as k, diminishes. This last phenomenon can be used to obtain an estimate of k,. In Fig. 8 we have represented AEr vs. log k, for two extreme values of (Y (0.3 and 0.7) to show how these plots can be used as working curves to obtain k,. When a current of the type given by eqn. (2) is applied, it is also possible to obtain an estimate of the (Yand k, values from the E/t curves, since from eqn. (21) we obtain for t --j 0
E-E”=-&$. Finally, irreversible
In
nFA,,c,*tfj3k
S
4
it is possible to obtain process by plotting E(t)
(31) accurate values of cr, k, and E” for an vs. In g,ca*h (eqns. 21 and 23) and E(t) vs. In
108
(eqns. glanod
22 and 24) [4,5]. From
RT aI-=l-nFP, ln
E”
RT nFP,
k,=!?-(“A-uB) s
=
these plots we obtain
nF
‘APB
OApB ‘A
+
‘,‘A
+
‘B
(32) +ln
‘0
where 0, and P, are the intercepts (i = B) processes, respectively. EXPERIMENTAL
(33)
nFA,c,*
(34) and slopes for the cathodic
(i = A) and anodic
RESULTS
To carry out the recording of experimental E/t curves on plane electrodes a platinum foil and a commercial carbon paste electrode filled with commercial paste were selected. Although the heterogeneous kinetic response on this type of electrode is seriously affected by the paste composition [ll], no special characteristics were required for the selection of this paste since our aim was just to show that the theoretical equations obtained describe the experimental E/t curves properly, rather than to obtain accurate kinetic measurements. The alternating current given by eqn. (1) was applied to two well-known experimental systems: Fe(CN)i-/Fe(CN)%on the carbon paste and platinum electrodes and Fe3+/ Fe’+ on the carbon paste electrode. The first system behaves more irreversibly on carbon paste than on platinum [12], as shown in Figs. 9 and 10.
Fig. 9. Experimental E/t curves obtained for 10 mM Fe(CN)iin 0.4 M KNO, on a Pt electrode. The alternating current applied is given by eqn. (1). I, = 280 pA. The values of w (in s-‘) are shown on the curves.
109
1
1.3
Fig. 10. Experimental E/t curves obtained for 10 mM Fe(CN)iin 0.4 M KNO, on a carbon paste electrode. The alternating current applied is given by eqn. (1). I, = 500 pA. The values of w (in s-l) are shown on the curves.
The second one behaves as totally irreversible (Fig. 11). In these plots, the gradual appearance of the shoulder characteristic of an irreversible process can be observed. In Fig. 12 we represent the experimental E/t curves for the system Fe3+/Fe2+ with lt = 0.27, Z, = 875 PIA (note that under these conditions the transition time of species B does not exist) and Z, = 2400 PA (now the transition time of species B does exist). The theoretical curves obtained using OL= 0.38 and log(k,/cm SC’) =
2
4
6
1lS Fig. 11. Experimental E/r curves obtained for 10 mM Fe3+ in 1 M H2S04 on a carbon paste electrode. The alternating current applied is given by eqn. (1). I, = 500 PA. The values of w (in s-‘) are shown on the curves.
110
600/
1.2
0.6
1.8
2'.4
tlS Fig. 12. Comparison between the experimental and theoretical E/t curves for the system 1 M H,SO, on a carbon paste electrode. The alternating current applied is given by eqn. Fe3+, 6 mM Fe’+, o = 2 s-l, I,, = 2400 pA. (b) I, = 875 PA. Other conditions as in values used in the theoretical calculations were (a) NA = 1.95 s-l/*, y = 0.95, log(k,/cm a = 0.38, /.I = 0.27; (b) NA = 0.7 s- ‘I2 Other conditions as in curve (a). The solid experimental curves and the symbols correspond to the theoretical ones.
Fe3+/FeZ* in (1). (a) 22 mM curve (a). The s-l) = - 5.94, lines are the
-5.94 are also plotted in this figure. These values of (Y and k, were obtained following the method described above. The application of this method to the experimental curves in Fig. 12 is shown in Fig. 13 We obtained the following results: (Y= 0.36 + 0.02, 1 - (Y= 0.60 + 0.02 and log(k,/cm s-‘) = - 5.90 + 0.05.
60
! 4
-1
b
1
Fig. 13. Dependence of E on (I) In gFth (eqn. 23) and (II) In gpd curves in Fig. 12. (in) I, = 2400 PA; ( + + + ) 1, = 875 PA.
(eqn. 24) for the experimental
E/t
111
The correlation coefficients were error in the determination of the the values of NA and N, were transition time corresponding to As can be seen, the agreement predictions is quite good.
in all cases larger than 0.995. In order to make the electrode area or the diffusion coefficients smaller, obtained from Sand’s equation by measuring the a current step of magnitude I,. between the experimental results and theoretical
CONCLUSIONS
The new procedure described in this work for the application of alternating currents to different electrodes can be considered as an easy, versatile and effective technique for the analysis of the behaviour of electrode processes. Double-layer charge effects usually have a complex nature and have not been considered here. However, we used values of w, CA*and I, for which we did not observe experimentally any appreciable distortion of the curves by non-faradaic processes. The amplitude Z, of the alternating current and the value of /J are the main factors determining the existence of the transition time in the reduction or the reoxidation processes, while the angular frequency w is the main factor responsible for the value of the transition time. This technique is particularly interesting when T* does not exist due to the following reasons: (a) The independence of 7B (when cg = 0) with NA (I,,, c,*, DA and electrode area) is found to be very advantageous for the detection of any kinetic complications relating to the behaviour of reduced species B. (b) The heterogeneous kinetic parameters can be obtained independently from the anodic and cathodic responses. This fact can also be used to detect complications in process (I). (c) The degree of reversibility of the charge-transfer reaction can be estimated from the shape of the E/t curves. ACKNOWLEDGEMENTS
The authors greatly appreciate the financial support of the Direction General de Investigation Cientifica y TCcnica (Project No. PB87-0700) and also of the Direction Regional de Education y Universidad de la Comunidad Autonoma de la Region de Murcia (Project No. PCT89/19). We also acknowledge the critical reading of this manuscript by Dr. Rafael Chicon (University of Murcia). NOTATION AND DEFINITIONS
z,(t) Z0 w CA*,cB*
applied alternating current (Z, sin( wt) if i = s or I0 cos( ot) if i = c) amplitude of the alternating current angular frequency (27r times the conventional frequency in hertz) bulk concentrations of species of A and B
112
ks 01
E0 E(t) AE AEP 7A 7B t1
t t, A(h) A0 A P.Y r
heterogeneous rate constants of the forward and the reverse chargetransfer reaction apparent heterogeneous rate constant of charge transfer at E o transfer coefficient formal standard potential of the electroactive couple time-dependent electrode potential = E(t) - E’= difference between the first two peaks (anodic and cathodic) in an oscillating E/t response transition time for reduction process transition time for reoxidation process blank period used only with non-stationary electrodes time elapsed between the application of the alternating current and the measurement of the potential =t,+t time-dependent area for an expanding plane electrode (= Aoti13) electrode area when t, = 1 s fixed area of a stationary plane electrode = 2l31 + x/2)/T(1/2 +x/2) Euler gamma function
Other definitions
are conventional.
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