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Electrode kinetics by analysis of polarographic current-time curves
Electroanalytical Chemistry and Interracial Electrochemistry. 55 (1974) 187 199 @ Elsevier Sequoia S.A., Lausanne
187
Printed in The Netherlands
ELECTRODE KINETICS BY ANALYSIS OF POLAROGRAPHIC CURRENT TIME CURVES
YUZO AYABE
Tokyo Kogyo Shikensho (National Chemical Laboratory for Industry) I-chome, Honmachi, Shibuyaku, Tokyo (Japan) (Received 19th February 1974; in revised form 30th May 1974)
INTRODUCTION
A method of analysis for polarographic current~otential curves of metal-ion complexes has been proposed in previous papers 1 3. In order to evaluate the kinetic parameters of electrode reactions (transfer coefficient and rate constant) by using this method, it is desirable to increase the accuracy of the measurements. If the recent improvements in electronic instrumentation are taken into account, it may be better to record the instantaneous current during the drop life by a rapid-response recorder. For the theoretical expression of the instantaneous d.c. potarographic current, a good approximate equation has been derived by Smith et al. 4. Employing this approximate function, the analysis of the instantaneous current-time curve (~t curve) of the simple metal ion and of metal-ion complexes can easily be performed. The reversible half-wave potential and forward formal rate constant can be obtained by calculating the ratio of instantaneous limiting current to instantaneous current as a function of the time during drop growth from the measured i-t curve at a given electrode potential and at a given composition of electrolyte. The transfer coefficient and the formal rate constant of the electrode reaction can be evaluated from the dependence of the forward formal rate constant on the electrode potential and the concentration of the complexing agent. In the present paper, the method described above is applied to the zinc ion in various concentrations of sodium perchlorate solution and to the zinc-acetate complexes in sodium perchlorate solution. THEORETICAL
For a simple electrode reaction: Ox + n e ~ Red,
(1)
a general expression for the instantaneous polarographic current during the drop life has been derived in the previous paper 5, as follows: i=
id /\3re~,
188
Y. AYABE
where = 2t +
(3)
= (nF/RT)(E-E
(4)
r)
(5)
= - Dox+ DL
The definitions of symbols used in this paper are summarized at the end of the discussion. The approximate expression for the function (37r/7)~0(~) given by Smith et al. 4 is (3~/7)~0(~) -~ ( 1.349 ~)1. 091f{ [ + (1.349~) 1°91 }
(6)
Thus eqn. (2) is reduced to id (5.3492t~) 1"°91 i = 5+e¢ l+(1.3492t~)a.o9a
(7)
By using the relation; (8)
e~ = k D L ~ / f D o 2
we obtain, after some rearrangements,
5
idi = (l+e:-) + --1.386(5+ e~)°'°91(kDo~) 1 " ° 9 f t - 1 ° 9 1 / 2
(9)
It can easily be seen from eqn. (9), that if the quantity (ia/i) is plotted against t -1°91/2, we obtain a straight line, from whose slope and intercept with the ordinate we can obtain the values of exp ( and kDo~~, respectively. Therefore, from these values we can calculate the values of the forward and backward formal rate constants fc and ~'. The above argument can easily be transposed to the system of metal-ion complexes with the following reduction mechanism: (1) A series of different species of complex ions coexists in t h e solution and successive dissociation and association reactions between different species are assumed to be in practical equilibrium, even if the current is flowing. MX(p. - pb, + ~ MX~"_-~p - " b ' + + X -b (p=0, 5,2 .... N)
(5 O)
(2) All species in the solution can participate in the discharge reactions with their own kinetic parameters. MX~"-pb)+ +ne.~M(Hg)+pX -b
(15)
(p=0, 5,2 .... N) As shown in the previous papers 3"6, ~Do2 and ~DL~ in eqn. (8) should be replaced by the sum of the individual functions for each species of complex. Thus we have N
kD-~=e-~" Z
Mv(E)'cPx
(52)
p=0 ~--
N
t
k D ; " = e~ fcD - ~ = =~-"om v ( E)cPx P
(13)
189
ANALYSIS OF P O L A R O G R A P H I C C U R R E N T TIME CURVES
where -- ~p
Do ( 1 -ap)/2Da
~:P/2
exp [{(1 - %)nF/R T} (E -
D= Z-v
(E~) o)]
(14)
(15)
In eqn. (14), (k°h is the formal rate constant at the formal potential E ° of the overall reaction, whose standard state is defined as ip = 0 N
Z (Cp)s=(c,)~=Cx = 1 moll -1 p=0
(k°)B and E ° are related to k° and Ep°, respectively, as follows"
E° = E°p+(RT/'W)ln fiP/~-o fl~
(16)
( k°), = k°( flp/ ~=o fl~) l -~P
(17)
where E ° is the formal potential of the charge-transfer reaction (11) of the complex m X._p ' ( n - p b ) + and kp .0 is the corresponding formal rate constant. Ep° can be related to the formal potential of simple metal ion E °, as
E°v = E ° - (RT/nF) In tip
(18)
and the reversible half-wave potential of complexes, E~, is related to the reversible half-wave potential of the simple ion (E~)o, as follows:
Er~=(E~)°+(RT/;W)ln(D°/D)~-(RT/nF)ln[,,;o~ fl~c:l
(19)
The procedure used to evaluate (k°)B from eqn] (13) is just the same as described in the previous papers 6-s. EXPERIMENTAL
Guaranteed reagents and twice distilled water were used throughout. A standard solution of Zn(II) perchlorate was prepared by dissolving a known amount of pure zinc metal (99.99%) in perchloric acid and diluting to 10 2 m. For zinc-acetate complex, the ionic strength of all sample solutions (sodium acetate + sodium perchlorate) was held constant (4.0 M), and the pH value of the sample solutions was regulated to 6.0_+011 by Beckman pH meter "Century SS-I". The concentration of acetate ion at pH=6.0 was corrected by using the acid dissociation constant of acetic acid of pK=4.61 (ref. 9). Instantaneous polarographic currents were measured using instruments shown schematically in Fig. 1. The potentiostat (Yanagimoto Co. Ltd., pS-PT)
190
Y. AYABE
I I 4. m°
Fig. 1. Block diagram of apparatus: (A) potentiometer, (B) potentiostat, (C) recorder (input impedance 1 M~)), (D) working electrode, (E) reference electrode, (F) counter electrode, (G) resistance 10 kfL
used has a potential stability within 1 mV, under conditions of i ~ 3 - 4 pA (corresponding to the current at the end of the drop life). The recorder used (San-ei Sokki Co. "Rectigraph') is a pen-writing electromagnetic oscillographic recorder, which has a response of 70 Hz and a maximttm chart speed of 250 mm s-1. The temperatures of the mercury eotumn and the sample solution were controlled at 25_+0.2°C by "Cootnies" (Komatsu glectronies, CTR-1A, CTE-I~), i.e., by continuously circulating water of 25°C through the jackets of the cell and of the connecting tube between the mercury pool and capillary. As a working electrode, Smoler's 10 dropping mercury electrode (DME) was used. The characteristics of the D M E as measured in a working solution were z (drop time)=4.50 s and m (flow rate of mercury)=0.560 mg s 1 when h (height of the mercury h e a d ) = 4 0 cm. A saturated calomel electrode (SCE) with surface area of 11 cm 2 was used as both the counter and the reference electrodes. Currents during the drop life were evaluated after correcting by comparing with the calibration curve which was previously prepared using the standard resistance instead of the electrolytic cell. The faradaic current was measured as the difference between the total current and the residual current as measured separately in pure supporting electrolyte at the same potential. RESULTS AND DISCUSSION
Simple zinc ion
The method presented above is applied to the evaluation of the kinetic parameters of the zinc ion in sodium perchlorate solution. The dependence of id/i o n ~-1.091/2 obtained at various electrode potentials in 1.5 M NaC104 solution is shown in Fig.2. F r o m the slope and the intercept with the ordinate the values of E'~, k, and k can easily be calculated by the use of eqns. (8) and (9), where D(1.5M)= 6.93 x 10- 6 c m 2 S- 1 was calculated from the measured values of id by the use of the revised Ilkovic equation 11 for the instantaneous diffusion current and Da = 1.72 x 10 -5 cm 2 s 1 was taken from the literature 12. The values of E~ determined by this method a r e plotted against the concentration of sodium perchlorate in Fig. 3, in which we can see that E~ has the constant value of -0.9956 V vs. SCE within the concentration range from 0.1 M to 0.6 M, and beyond 0.6 M it shifts gradually to the positive values with increasing concentrations. If it is assumed that the positive shift of E~ with the increase of concentration of sodium perchlorate is due to the increase of the
ANALYSIS OF POLAROGRAPHIC
CURRENT-TIME
191
CURVES
A 2.~
0.5 \
B
\x
o o.o
2.C
ff
-- 0.5
12
1£
O.5 t - 1.091 / 2
-1.0
1.0
m
I
-0.95
/
I
I
-0.97
I
-0.99
I
-1.01
E~I2/V. vs. SCE Fig. 2. Relations between id/i and t 1.o91/2 obtained from the measured i-t curves of zinc ion in 1.5 M s o d i u m perchlorate solution at various electrode potentials. E = ( A ) 4).986, (B) - 0 . 9 9 Z (C) - 0 . 9 9 7 , ( D ) - 1.003, (E) - 1.019, (F) - 1.036 V
vs.
SCE.
Fig. 3. Reversible half-wave potential E~ obtained from i t curves for zinc ion as a function of the
logarithm of the concentration of sodium perchlorate log Cc~o4, the dashed curve corrected for the liquid-junction potentials calculated from the Henderson e q u a t i o n 16 by using the values of mobility at infinite dilution.
activity coefficient of the zinc ion, the activity coefficient of the zinc ion in x M perchlorate solution, fxM~, can be evaluated as a value relative to the activity coefficient of the zinc ion in 0.1 M NaC104 solution, f 0 . 1 ~t), as follows;
J;x~,l, =exp{~TT(E+)(x,,_(E½)(o.eM)]+ln(nD_~)"}"} nF
r
r
/ D
f{ 0.1 M )
\a-'(0.1M)/
(20)
J
The results obtained are listed in Table 1, from which we can see that the activity coefficient of the zinc ion increases tremendously with the increase of perchlorate concentration, just as for the mean activity coefficients in perchlorate solutions of polyvalent cations 13. According to the usual kinetic theory of electrode reactions, the dependence of the forward and backward formal rate constant on the electrode potential can be expressed as = kf e x p
I
-
~nF . gr(
k'= kf exp [(1-~)nF
.7
(21)
~ ,]
(22)
-Ef)]
Hence,
= kf exp
[-
~nF ( RT\
+ ~
In nF
t l ll \
(~M)
.J
(23)
192
Y. AYABE
-2 .C -2.C
-2.5,
~-2 ~
\,
'~ -3.C
\ '\
-4.0
\, \
.
-3.5 --
ZX - 0.98
-1.00
-1.02
E/V. vs.SCE
I
-1.03 -1.0
- 0.5
log CC104/
I
mOl1-1)
I + O. 5
Fig. 4. Dependence of log ~ and log *k on the electrode potential E for the sample solution (1.0 m M Zn(II) + 1.5 M NaC104). Fig. 5. Dependence of log kf and log [k~(f 0.1 ~o) 1 ~] on log Cclo4. (A)log k,-, ( B ) l o g [ k ~ 0.1 m)) l -~].
k=kfexp[ (1-c0nF(E-E~+ ~: \ -
R T l n ( Da ~ l nF \D~xM)/ /J
(24)
Therefore, the transfer coefficient and the formal rate constant can be calculated from the slope of the log k vs. E plot and the value ~: at the potential E = E; - (R T/nF) In (D.)'O~-~M), respectively. In Fig. 4, the values determined for log k and]og } are plotted against the potential E. The transfer coefficient thus obtained has the constant value of 0.24_+0.01 throughout all the measurements for various concentrations of sodium perchlorate. The values of the formal rate constant calculated are given in Fig. 5 as a function of the concentration of sodium perchlorate. If the activity coefficients of all the species concerned are known, we can calculate the standard rate constant,/% according to the following equation:
l~(fx~,)) (f~) (25) The amalgams formed at the DME are so very dilute that we are justified in assuming that they behave ideally and that f~ is practically equal to 1. Unfortunately, it is generally impossible to obtain the single activity coefficient of the metal ion, which is present in a large excess of the supporting electrolyte. Therefore, using the relative values of the activity coefficient of the zinc ion given in Table 1, the variations of the standard rate constant, multiplied by (fo.x M~)1-~, with the perchlorate concentration were calculated. The results obtained are given in Fig. 5, from which we can see that the rate constants corrected for the variation of activity coefficient decrease linearly with increasing perchlorate concentrations. On the basis of the idea that this decrease in / ~ 0 . ZM))a-~ may be attributed to the double-layer effect, we attempted to make the Frumkin correction for the ~ ( f 0 . ~v)) a-~ values. Thus, the standard rate constant can be corrected to =
ANALYSIS OF POLAROGRAPHIC CURRENT-TIME CURVES
/~s(f( O. 1 M))I-~ = [(ks)....
(f~o. 1M))l-a] exp [
193
(1-c0nFq52i RT
(26)
where (/)2 is the potential at the outer Helmholtz plane. According to the G o u y C h a p m a n theory of the diffuse "double layer, if the electrode potential is sufficiently m o r e negative than the point of zero charge, the potential ~bz can be fairly well expressed as 0 2 = 40 + ( R T / F ) i n
Cc,o,
(27)
Since the values of E~ obtained are ranged only from - 0 . 9 5 V to - 1 . 0 0 V vs. SCE, the value of ~bo Can be assumed to be constant, and it has the value of - 4 7 + 1 mV, as calculated by using the Russell 14 table for 1 M solution and for the potential range from - 0 . 9 5 V to - 1 . 0 0 V vs. SCE. Substituting eqn. (27) into (26) yields log [(/~)~o~(f(0.1M))I-~] = log [k~(f o., M))t-a] + (1 - c0n log Cclo~- 1.22
(28)
The results obtained by the use of eqn. (28) are given in Table 1. "ABLE 1 ELATIVE VALUES OF THE ACTIVITY COEFFICIENT OF THE ZINC ION IN VARYING CONCENTRA'IONS OF PERCHLORATE SOLUTION AND CORRECTED STANDARD RATE CONSTANT 7oneentration of fxM)/fO. IM) lectrolyte/mol l- ~
).1 ).3 ).6 ).9 _5 !.0 !.5 1.0 k5 ~.0
1.0 1.0 1.18 2.15 3.59 5.42 8.48 14.1 38.5
{f,~)/fo.1M))~or,
0.9 0.9 1.0 1.65 2.63 3~91 5.79 9.43 24.9
Corrected standard rate constant/cm s- ~ 104x(ks)co,,(Jlo. lM)) ' ~
lO'~x[(k~)co,r(fol~f,)l-~]cor
0.2 0.7 1.5 1.6 1.7 1.6 1.6 1.7 1.5 1.0
0.2 0.76 1.7 2.0 2.1 2.1 2.2 2.3 2.2 1.5
l?hese quantities are corrected for the liquid-junction potentials calculated from the Henderson equation 16 by using te values of mobility at infinite dilution. As can be seen from Table 1, the corrected rate constant (/g)corr has a constant value over the concentration range 0 . 6 4 . 0 M, indicating that the F r u m k i n correction can be applied to such high concentrations, when the activity coefficient of the zinc ion is corrected. This fact seems to be surprising although the reason is not clear. Zinc-acetate complexes
As an example of the determination of electrode kinetics by the m e t h o d discussed here, the zinc-acetate complexes in perchlorate solution were investigated. Since the activity coefficient of zinc-acetate complexes and of the complexing agent
194
Y. AYABE
in the concentrated mixed solution generally cannot be obtained, the ionic strength of the electrolyte solution was maintained at a constant value of 4 M throughout all the measurements for the variation of concentration of the acetate ion. According to the previous paper 3, the limiting current of zinc-acetate complexes does not involve any kinetic complication and is diffusion-controlled. The relations between ia/i and f-l.o91/2 obtained at the free acetate concentration of 1.34 M and at several electrode potentials are shown in Fig. 6. The reversible half-wave potential E~ was calculated from the intercepts of the straight lines with the ordinate. Figure 7 shows the dependence of the reversible half-wave potential on free acetate concentration. By using D e F o r d - H u m e ' s method 1s, the overall conditional stability constants//p and the conditional successive stability constants Kp w e r e evaluated, as fit=7, fl2=92, fl3=70,//4=140,//5=152,
K 2 = 13.1, K 3 = 0 . 8 ,
K4=2.0,
Ks=
1.1.
2.0 1.9 1.8 1.7
*0.5
J..d 1.6
,L 1.5
E "~ 0.0
m
U
1.4 1.~ 1.2
-0.5
1.1
I
0.5
I
(,F~-~ 1"091
1.0
- ~E&~/ I -0.95
-1.00
I d/2/V
-1.05 ~.SCE)
I -1.10
Fig. 6. Relation between ia/i and f-1.091/2 obtained from i-t curves of zinc-acetate complex ions in the solution (1.34 M sodium acetate+2.6 M sodium perchlorate). E = ( A ) -1.043, (B) -1.054, (C) - 1.065, (D) - 1.076, (E) - 1.087 V vs SCE. Fig. 7. Reversible half-wave potential E~ obtained from i ~ curves for zinc acetate complex ions as a function of the logarithm of the acetate concentration in 4 M (sodium acetate +sodium perchlorate) solution.
The quantities exp ~ and ~D -~ on the left-hand side of eqn. (13) were calculated from (id/i)-t 1.o91/2 plots at various potentials and at various acetate concentrations. Then the quantity exp ~ x ~D -~ was obtained as a function of the acetate concentration at a given value of electrode potential, and it is denoted as the L 0 function. Next, L p function ( p = 1, 2 . . . . ) is defined as Z p = [Zp - 1 - M p _ I(E)]/CAc
Then,
Mp(E) can
be evaluated by the following extrapolation
(29)
ANALYSIS O F P O L A R O G R A P H I C C U R R E N T - T I M E CURVES
c~m -* 0 Lp
=
Mp(E)
195 (30)
Figures 8 and 9 show the Lp (p= 1, 2)-Cat plots at various potentials and Mp (p=0, 2)-E plots, respectively. From Fig. 8, we can deduce the following conclusions: The value of M1 at every potential is negligibly small and thus the species Zn(Ac) ÷ does not directly participate in the electrode process. The discharging species are Zn e+ and Zn(Ac)e. Therefore, zinc-acetate complexes are reduced to zinc amalgam according to the following two simultaneous charge-transfer reactions: Zn 2 + + 2e~-~-Zn(Hg) Zn(Ac)2 + 2e~- Zn(Hg) + 2Ac-
(31) (32)
The transfer coefficient can be expressed as 2.30 e z . [ A log Mp/ nF L AE J
0(p= 1
(33)
Thus, we obtain, after introducing the values of the slopes from Fig. 9 into
eqn. (33), c~0 = 0.24,
~2 = 0.31 A
0.5 0.4 0.3
--
B
--
C
0.2 D 0.1
-
E
1.0
3.0
2.0
4.0
CAC/mol F1)
(b) o
O
0
r)~
o
B
o
u
~
u
o
o
o
u
o
n
o
0.1 0 L2 C
0.05
I
~
1.0
2.0 CAc/mOl 14)
o
o I
3.0
-o- D
O-
E
O------O-
F
o
o-(3 1
4.0
Fig. 8. Variations of the functions i p (p = 1, 2) with acetate concentration at various values of the electrode potential. E = (A) - 1.065, (B) - 1.076, (C) - 1.087, (D) - 1.098, (E) - 1.109, (F) - 1.120, (G) -1.131 V v s . SCE. (a) p = l , (b) p=2.
196
Y. AYABE
0.0
log Mp
-1.C
-2.(
I
- 0.95
I
I
- 1.00
E/V
I
-1.05 (vs. S C E )
- 1.10
Fig. 9. Dependence of the function log Mp (p=0, 2) on the electrode potential E. (A) log Mo, (B) log M2."
The rate constants (k°)B and (k°), were evaluated by introducing into eqn. (14) the values of/3~ (v= 1-5), ~p ( p = 0 , 2) and Mp (p=0, 2) obtained above and the following data: Da 1.72x10 -5 cm 2 s 1 (ref. 12), D o = 0 . 5 2 x 1 0 -5 cm 2 s -1 (calculated from id in solution of 4.0 M NaC104 by the use of the revised Ilkovic equation11), and (E~)o = -0.9464 V vs. SCE. We have (k°)B =3.4 x 10 .5 cm s -1 (kO)B = 5 . 1 × 10 -3 c m s -1 M - 2 The formal potential E ° is evaluated from (Er)o as E ° = - 1.033 V vs. SCE Moreover, the formal rate constant k° can be calculated by eqn. (17) and also Ev° by eqn. (16), as follows; k° = 3 . 5 x l 0 - 3 c m s kO=l.5xl0-2cms
1, -1,
E °=-0.954Vvs.SCE
E°=-I.O12Vvs.
SCE
')(( n If ip denotes the current resulting from the reduction of the species M~._._p and i the total current, then the ratio can be given as
p b ~+
,
N
ip/i = Mp(E).cxP/p~=o Mp(E)..CPx
(34)
Thus, it can be seen that when CAc >0.5 M, i2/i~l, i.e., the faradaic Current results only from the discharging of the species Zn(Ac)2. This conclusion differs from the results of the previous paper 3, in which the discharging species were found to be Zn 2+, Zn(Ac) + and Zn(Ac)3 in 4 M (sodium nitrate+sodium acetate) solution. The discrepancy between these results may be attributable to the different supporting electrolytes employed.
ANALYSIS OF POLAROGRAPHICCURRENT TIME CURVES
197
APPENDIX The theoretical equation for the polarographic current-time curves used in the text is that derived on the basis of the "so-called" expanding plane electrode model. In this Appendix, the influence of the sphericity of the DME on the behaviour of the instantaneous current will be discussed. If the electrode reaction given by eqn. (1) is reversible and the product of the electrode reaction diffuses into the DME, the rigorous theoretical equation of the instantaneous current at any point along the wave for the expanding sphere electrode is given, according to Koutecky a7 and Weber 18, by the relationship: 1 '~ph. = id(1 +AO) 1 + e ~
(35)
A = 39.7 m-~D~x t ~
(36)
0 = 1--(DR~d/DOx)~e~
(37)
with
1 +e ~
where ia is given by the llkovic equation, which is derived for the expanding plane electrode. From eqns. (35) and (37), the instantaneous limiting current (exp ff~0, 0 = 1) can be given as (id)sph. = id(l + A)
(38)
For the expanding plane electrode, when the electrode reaction is reversible, v~e have, from eqn. (2), i = id/(1 + e ~)
(39)
Thus, comparing eqn. (35) with eqn. (39), we have ( id/i)~ph. = ( id/i) e
(40)
e = (1 + A)/(1 + AO)
(41)
with Therefore, e represents a correction factor for sphericity of the DME, and depends on the electrode potential, as can easily be seen from eqns. (37) and (41), In Table 2, the dependence of e on the electrode potential for t = 1 and 3.5 s is given. Unfortunately, no theoretical equation of the instantaneous current for the expanding sphere electrode has ever been developed for the case of slow electrode reaction, in which the current is controlled partly by the rate of diffusion and partly by the rate of electrode reaction. However, since the sphericity of the D M E affects only the diffusion process, it may be safely assumed that the effect of the sphericity for the slow electrode reaction is smaller than that for the reversible electrode reaction. The (id/i) vs. t - 1 " 0 9 1 / 2 plots given in this paper were carried out in the
198
Y. AYABE
TABLE 2 THE VALUES O F e AT F O U R D I F F E R E N T POTENTIALS CALCULATED BY THE USE O F EQN. (41) m = 1.0 mg s-1, Dox = 5 x 10 .6 cm 2 s-1, DRed/Dox= 2.
(E-E~)/mV
-
e ( t = l s)
~(t=3.5 s)
10 0
1.16 1.11
10
1.07
1.19 1.I4 1.08 1.04 1.01
20 -40
1.04 1.01
-
potential range from about E~ to E r-21 50 mV. Thus, the errors caused by the sphericity of the D M E may be smaller than about 10~o, as can be seen from Table 2. ACKNOWLEDGEMENTS
The author is indebted to Professor H. Matsuda, Tokyo Institute of Technology, for his interest and discussion of this work. Thanks are also due to Mr. H. Tanaka of Tokai University for his assistance in carrying out the experiments. SUMMARY
The method, which enables the evaluation of the kinetic parameters of electrode reactions by analysing the polarographi c current-time curve, is presented. As examples of the application of this method, the electrode reactions of the simple zinc ions and of the zinc acetate complex ions were experimentally examined. For the former case the current-time curves were measured as a function of the electrode potential and the perchlorate concentration. It was found that the standard rate constant, corrected both for the increase of the activity coefficient of the zinc ion and for the Frumkin double-layer effect, has a constant value over the concentration range of perchlorate 0.6-4.0 M. For the latter case the current-time curves in the solution with constant ionic strength (4 M) are measured as a function of the electrode potential and of the acetate concentration. It was concluded that the potential-determining reaction of zinc-acetate complexes is referred only to the species Zn(Ac)2 over a range of acetate concentrations larger than 0.5 M. D E F I N I T I O N O F SYMBOLS
i ia Dj CC104
Ef
= instantaneous d.c. polarographic current = instantaneous limiting diffusion current = diffusion coefficient of species j = concentration of the sodium perchlorate solution = cathodic transfer coefficient of the charge-transfer reaction of eqn. (1) = formal potential of charge-transfer reaction of eqn. (1) measured against a reference electrode
ANALYSIS OF POLAROGRAPHIC CURRENT TIME CURVES
kf Cx CAc (Cv)~
(cA 0¥
Eo ko t
199
= forward a n d b a c k w a r d formal rate c o n s t a n t s of the charge transfer reaction of eqn. (1) or eqn. (11) = s t a n d a r d rate c o n s t a n t of the charge-transfer reaction of eqn. (1) at the s t a n d a r d potential ---formal rate c o n s t a n t of the charge-transfer reaction of eqn. (1) at the potential E = E ~ - ( R T / n F ) l n ( D ~ e d D o x ~) -- c o n c e n t r a t i o n of the free form of c o m p l e x i n g agent X -b =- c o n c e n t r a t i o n of the acetate ion --- c o n c e n t r a t i o n of the metal complex M~(n _._._p pb) + at the electrode solution interface c o n c e n t r a t i o n of the a m a l g a m a t e d metal at the e l e c t r o d e - s o l u t i o n interface - - c o n d i t i o n a l overall stability c o n s t a n t of the metal complex MX(p"-pb~+ = cathodic transfer coefficient of the charge-transfer reaction of eqn. (11) - - f o r m a l potential of the charge-transfer reaction of eqn. (11) m e a s u r e d against a reference electrode - - f o r m a l rate c o n s t a n t of the charge-transfer reaction of eqn. (11) at the formal potential E ° = time in seconds for d r o p growth
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
H. Matsuda and Y. Ayabe, Z. Elektrochem., 63 (1959) 1164. H. Matsuda and Y. Ayabe, Z. Elektrochem., 66 (1962) 469. H. Matsuda, Y. Ayabe and K. Adachi, Ber. Bunsenges. Phys. Chem., 67 (1963) 593. D. E. Smith, T. G. McCord and H. L. Hung, Anal. Chem., 39 (1967) 1149. H. Matsuda and Y. Ayabe, Bull. Chem. Soc. Japan, 28 (1955) 422. H. Matsuda and Y. Ayabe, Bull. Chem. Soc. Japan, 29 (1956) 134. Y. Ayabe and H. Matsuda, Denki Kagaku , 39 (1971) 635. Y. Ayabe, Denki Kagaku , 42 (1974) 93. L. G. Sillen and A. E. Martelk Stability Constants of Metal-Ion Complexes~ The Chemical Society, London, 1964, p. 364. I. Smoler, J. Electroanal. Chem., 6 (1963) 465. H. Matsuda, Bull. Chem. Soc. Japan. 26 (1953) 342. Landolt-Bornstein, Physikalisch-Chemische Tahellen, Springer, Berlin, 3rd. edn., 1935, p. 239. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edn., 1968, p. 497. C. D. Russell, J. Electroanal. Chem., 6 (1963) 486. D. D. DeFord and D. N. Hume, J. Amer. Chem. Soc., 73 (1951) 5321. P. Henderson, Z. Phys. Chem., 59 (1907) 118:63 (1908) 325. J. Koutecky, Czech. J. Phys., 2 (1953) 51. J. Weber, Collect. Czech. Chem. Commun., 24 (1959) 1424.
187
Printed in The Netherlands
ELECTRODE KINETICS BY ANALYSIS OF POLAROGRAPHIC CURRENT TIME CURVES
YUZO AYABE
Tokyo Kogyo Shikensho (National Chemical Laboratory for Industry) I-chome, Honmachi, Shibuyaku, Tokyo (Japan) (Received 19th February 1974; in revised form 30th May 1974)
INTRODUCTION
A method of analysis for polarographic current~otential curves of metal-ion complexes has been proposed in previous papers 1 3. In order to evaluate the kinetic parameters of electrode reactions (transfer coefficient and rate constant) by using this method, it is desirable to increase the accuracy of the measurements. If the recent improvements in electronic instrumentation are taken into account, it may be better to record the instantaneous current during the drop life by a rapid-response recorder. For the theoretical expression of the instantaneous d.c. potarographic current, a good approximate equation has been derived by Smith et al. 4. Employing this approximate function, the analysis of the instantaneous current-time curve (~t curve) of the simple metal ion and of metal-ion complexes can easily be performed. The reversible half-wave potential and forward formal rate constant can be obtained by calculating the ratio of instantaneous limiting current to instantaneous current as a function of the time during drop growth from the measured i-t curve at a given electrode potential and at a given composition of electrolyte. The transfer coefficient and the formal rate constant of the electrode reaction can be evaluated from the dependence of the forward formal rate constant on the electrode potential and the concentration of the complexing agent. In the present paper, the method described above is applied to the zinc ion in various concentrations of sodium perchlorate solution and to the zinc-acetate complexes in sodium perchlorate solution. THEORETICAL
For a simple electrode reaction: Ox + n e ~ Red,
(1)
a general expression for the instantaneous polarographic current during the drop life has been derived in the previous paper 5, as follows: i=
id /\3re~,
188
Y. AYABE
where = 2t +
(3)
= (nF/RT)(E-E
(4)
r)
(5)
= - Dox+ DL
The definitions of symbols used in this paper are summarized at the end of the discussion. The approximate expression for the function (37r/7)~0(~) given by Smith et al. 4 is (3~/7)~0(~) -~ ( 1.349 ~)1. 091f{ [ + (1.349~) 1°91 }
(6)
Thus eqn. (2) is reduced to id (5.3492t~) 1"°91 i = 5+e¢ l+(1.3492t~)a.o9a
(7)
By using the relation; (8)
e~ = k D L ~ / f D o 2
we obtain, after some rearrangements,
5
idi = (l+e:-) + --1.386(5+ e~)°'°91(kDo~) 1 " ° 9 f t - 1 ° 9 1 / 2
(9)
It can easily be seen from eqn. (9), that if the quantity (ia/i) is plotted against t -1°91/2, we obtain a straight line, from whose slope and intercept with the ordinate we can obtain the values of exp ( and kDo~~, respectively. Therefore, from these values we can calculate the values of the forward and backward formal rate constants fc and ~'. The above argument can easily be transposed to the system of metal-ion complexes with the following reduction mechanism: (1) A series of different species of complex ions coexists in t h e solution and successive dissociation and association reactions between different species are assumed to be in practical equilibrium, even if the current is flowing. MX(p. - pb, + ~ MX~"_-~p - " b ' + + X -b (p=0, 5,2 .... N)
(5 O)
(2) All species in the solution can participate in the discharge reactions with their own kinetic parameters. MX~"-pb)+ +ne.~M(Hg)+pX -b
(15)
(p=0, 5,2 .... N) As shown in the previous papers 3"6, ~Do2 and ~DL~ in eqn. (8) should be replaced by the sum of the individual functions for each species of complex. Thus we have N
kD-~=e-~" Z
Mv(E)'cPx
(52)
p=0 ~--
N
t
k D ; " = e~ fcD - ~ = =~-"om v ( E)cPx P
(13)
189
ANALYSIS OF P O L A R O G R A P H I C C U R R E N T TIME CURVES
where -- ~p
Do ( 1 -ap)/2Da
~:P/2
exp [{(1 - %)nF/R T} (E -
D= Z-v
(E~) o)]
(14)
(15)
In eqn. (14), (k°h is the formal rate constant at the formal potential E ° of the overall reaction, whose standard state is defined as ip = 0 N
Z (Cp)s=(c,)~=Cx = 1 moll -1 p=0
(k°)B and E ° are related to k° and Ep°, respectively, as follows"
E° = E°p+(RT/'W)ln fiP/~-o fl~
(16)
( k°), = k°( flp/ ~=o fl~) l -~P
(17)
where E ° is the formal potential of the charge-transfer reaction (11) of the complex m X._p ' ( n - p b ) + and kp .0 is the corresponding formal rate constant. Ep° can be related to the formal potential of simple metal ion E °, as
E°v = E ° - (RT/nF) In tip
(18)
and the reversible half-wave potential of complexes, E~, is related to the reversible half-wave potential of the simple ion (E~)o, as follows:
Er~=(E~)°+(RT/;W)ln(D°/D)~-(RT/nF)ln[,,;o~ fl~c:l
(19)
The procedure used to evaluate (k°)B from eqn] (13) is just the same as described in the previous papers 6-s. EXPERIMENTAL
Guaranteed reagents and twice distilled water were used throughout. A standard solution of Zn(II) perchlorate was prepared by dissolving a known amount of pure zinc metal (99.99%) in perchloric acid and diluting to 10 2 m. For zinc-acetate complex, the ionic strength of all sample solutions (sodium acetate + sodium perchlorate) was held constant (4.0 M), and the pH value of the sample solutions was regulated to 6.0_+011 by Beckman pH meter "Century SS-I". The concentration of acetate ion at pH=6.0 was corrected by using the acid dissociation constant of acetic acid of pK=4.61 (ref. 9). Instantaneous polarographic currents were measured using instruments shown schematically in Fig. 1. The potentiostat (Yanagimoto Co. Ltd., pS-PT)
190
Y. AYABE
I I 4. m°
Fig. 1. Block diagram of apparatus: (A) potentiometer, (B) potentiostat, (C) recorder (input impedance 1 M~)), (D) working electrode, (E) reference electrode, (F) counter electrode, (G) resistance 10 kfL
used has a potential stability within 1 mV, under conditions of i ~ 3 - 4 pA (corresponding to the current at the end of the drop life). The recorder used (San-ei Sokki Co. "Rectigraph') is a pen-writing electromagnetic oscillographic recorder, which has a response of 70 Hz and a maximttm chart speed of 250 mm s-1. The temperatures of the mercury eotumn and the sample solution were controlled at 25_+0.2°C by "Cootnies" (Komatsu glectronies, CTR-1A, CTE-I~), i.e., by continuously circulating water of 25°C through the jackets of the cell and of the connecting tube between the mercury pool and capillary. As a working electrode, Smoler's 10 dropping mercury electrode (DME) was used. The characteristics of the D M E as measured in a working solution were z (drop time)=4.50 s and m (flow rate of mercury)=0.560 mg s 1 when h (height of the mercury h e a d ) = 4 0 cm. A saturated calomel electrode (SCE) with surface area of 11 cm 2 was used as both the counter and the reference electrodes. Currents during the drop life were evaluated after correcting by comparing with the calibration curve which was previously prepared using the standard resistance instead of the electrolytic cell. The faradaic current was measured as the difference between the total current and the residual current as measured separately in pure supporting electrolyte at the same potential. RESULTS AND DISCUSSION
Simple zinc ion
The method presented above is applied to the evaluation of the kinetic parameters of the zinc ion in sodium perchlorate solution. The dependence of id/i o n ~-1.091/2 obtained at various electrode potentials in 1.5 M NaC104 solution is shown in Fig.2. F r o m the slope and the intercept with the ordinate the values of E'~, k, and k can easily be calculated by the use of eqns. (8) and (9), where D(1.5M)= 6.93 x 10- 6 c m 2 S- 1 was calculated from the measured values of id by the use of the revised Ilkovic equation 11 for the instantaneous diffusion current and Da = 1.72 x 10 -5 cm 2 s 1 was taken from the literature 12. The values of E~ determined by this method a r e plotted against the concentration of sodium perchlorate in Fig. 3, in which we can see that E~ has the constant value of -0.9956 V vs. SCE within the concentration range from 0.1 M to 0.6 M, and beyond 0.6 M it shifts gradually to the positive values with increasing concentrations. If it is assumed that the positive shift of E~ with the increase of concentration of sodium perchlorate is due to the increase of the
ANALYSIS OF POLAROGRAPHIC
CURRENT-TIME
191
CURVES
A 2.~
0.5 \
B
\x
o o.o
2.C
ff
-- 0.5
12
1£
O.5 t - 1.091 / 2
-1.0
1.0
m
I
-0.95
/
I
I
-0.97
I
-0.99
I
-1.01
E~I2/V. vs. SCE Fig. 2. Relations between id/i and t 1.o91/2 obtained from the measured i-t curves of zinc ion in 1.5 M s o d i u m perchlorate solution at various electrode potentials. E = ( A ) 4).986, (B) - 0 . 9 9 Z (C) - 0 . 9 9 7 , ( D ) - 1.003, (E) - 1.019, (F) - 1.036 V
vs.
SCE.
Fig. 3. Reversible half-wave potential E~ obtained from i t curves for zinc ion as a function of the
logarithm of the concentration of sodium perchlorate log Cc~o4, the dashed curve corrected for the liquid-junction potentials calculated from the Henderson e q u a t i o n 16 by using the values of mobility at infinite dilution.
activity coefficient of the zinc ion, the activity coefficient of the zinc ion in x M perchlorate solution, fxM~, can be evaluated as a value relative to the activity coefficient of the zinc ion in 0.1 M NaC104 solution, f 0 . 1 ~t), as follows;
J;x~,l, =exp{~TT(E+)(x,,_(E½)(o.eM)]+ln(nD_~)"}"} nF
r
r
/ D
f{ 0.1 M )
\a-'(0.1M)/
(20)
J
The results obtained are listed in Table 1, from which we can see that the activity coefficient of the zinc ion increases tremendously with the increase of perchlorate concentration, just as for the mean activity coefficients in perchlorate solutions of polyvalent cations 13. According to the usual kinetic theory of electrode reactions, the dependence of the forward and backward formal rate constant on the electrode potential can be expressed as = kf e x p
I
-
~nF . gr(
k'= kf exp [(1-~)nF
.7
(21)
~ ,]
(22)
-Ef)]
Hence,
= kf exp
[-
~nF ( RT\
+ ~
In nF
t l ll \
(~M)
.J
(23)
192
Y. AYABE
-2 .C -2.C
-2.5,
~-2 ~
\,
'~ -3.C
\ '\
-4.0
\, \
.
-3.5 --
ZX - 0.98
-1.00
-1.02
E/V. vs.SCE
I
-1.03 -1.0
- 0.5
log CC104/
I
mOl1-1)
I + O. 5
Fig. 4. Dependence of log ~ and log *k on the electrode potential E for the sample solution (1.0 m M Zn(II) + 1.5 M NaC104). Fig. 5. Dependence of log kf and log [k~(f 0.1 ~o) 1 ~] on log Cclo4. (A)log k,-, ( B ) l o g [ k ~ 0.1 m)) l -~].
k=kfexp[ (1-c0nF(E-E~+ ~: \ -
R T l n ( Da ~ l nF \D~xM)/ /J
(24)
Therefore, the transfer coefficient and the formal rate constant can be calculated from the slope of the log k vs. E plot and the value ~: at the potential E = E; - (R T/nF) In (D.)'O~-~M), respectively. In Fig. 4, the values determined for log k and]og } are plotted against the potential E. The transfer coefficient thus obtained has the constant value of 0.24_+0.01 throughout all the measurements for various concentrations of sodium perchlorate. The values of the formal rate constant calculated are given in Fig. 5 as a function of the concentration of sodium perchlorate. If the activity coefficients of all the species concerned are known, we can calculate the standard rate constant,/% according to the following equation:
l~(fx~,)) (f~) (25) The amalgams formed at the DME are so very dilute that we are justified in assuming that they behave ideally and that f~ is practically equal to 1. Unfortunately, it is generally impossible to obtain the single activity coefficient of the metal ion, which is present in a large excess of the supporting electrolyte. Therefore, using the relative values of the activity coefficient of the zinc ion given in Table 1, the variations of the standard rate constant, multiplied by (fo.x M~)1-~, with the perchlorate concentration were calculated. The results obtained are given in Fig. 5, from which we can see that the rate constants corrected for the variation of activity coefficient decrease linearly with increasing perchlorate concentrations. On the basis of the idea that this decrease in / ~ 0 . ZM))a-~ may be attributed to the double-layer effect, we attempted to make the Frumkin correction for the ~ ( f 0 . ~v)) a-~ values. Thus, the standard rate constant can be corrected to =
ANALYSIS OF POLAROGRAPHIC CURRENT-TIME CURVES
/~s(f( O. 1 M))I-~ = [(ks)....
(f~o. 1M))l-a] exp [
193
(1-c0nFq52i RT
(26)
where (/)2 is the potential at the outer Helmholtz plane. According to the G o u y C h a p m a n theory of the diffuse "double layer, if the electrode potential is sufficiently m o r e negative than the point of zero charge, the potential ~bz can be fairly well expressed as 0 2 = 40 + ( R T / F ) i n
Cc,o,
(27)
Since the values of E~ obtained are ranged only from - 0 . 9 5 V to - 1 . 0 0 V vs. SCE, the value of ~bo Can be assumed to be constant, and it has the value of - 4 7 + 1 mV, as calculated by using the Russell 14 table for 1 M solution and for the potential range from - 0 . 9 5 V to - 1 . 0 0 V vs. SCE. Substituting eqn. (27) into (26) yields log [(/~)~o~(f(0.1M))I-~] = log [k~(f o., M))t-a] + (1 - c0n log Cclo~- 1.22
(28)
The results obtained by the use of eqn. (28) are given in Table 1. "ABLE 1 ELATIVE VALUES OF THE ACTIVITY COEFFICIENT OF THE ZINC ION IN VARYING CONCENTRA'IONS OF PERCHLORATE SOLUTION AND CORRECTED STANDARD RATE CONSTANT 7oneentration of fxM)/fO. IM) lectrolyte/mol l- ~
).1 ).3 ).6 ).9 _5 !.0 !.5 1.0 k5 ~.0
1.0 1.0 1.18 2.15 3.59 5.42 8.48 14.1 38.5
{f,~)/fo.1M))~or,
0.9 0.9 1.0 1.65 2.63 3~91 5.79 9.43 24.9
Corrected standard rate constant/cm s- ~ 104x(ks)co,,(Jlo. lM)) ' ~
lO'~x[(k~)co,r(fol~f,)l-~]cor
0.2 0.7 1.5 1.6 1.7 1.6 1.6 1.7 1.5 1.0
0.2 0.76 1.7 2.0 2.1 2.1 2.2 2.3 2.2 1.5
l?hese quantities are corrected for the liquid-junction potentials calculated from the Henderson equation 16 by using te values of mobility at infinite dilution. As can be seen from Table 1, the corrected rate constant (/g)corr has a constant value over the concentration range 0 . 6 4 . 0 M, indicating that the F r u m k i n correction can be applied to such high concentrations, when the activity coefficient of the zinc ion is corrected. This fact seems to be surprising although the reason is not clear. Zinc-acetate complexes
As an example of the determination of electrode kinetics by the m e t h o d discussed here, the zinc-acetate complexes in perchlorate solution were investigated. Since the activity coefficient of zinc-acetate complexes and of the complexing agent
194
Y. AYABE
in the concentrated mixed solution generally cannot be obtained, the ionic strength of the electrolyte solution was maintained at a constant value of 4 M throughout all the measurements for the variation of concentration of the acetate ion. According to the previous paper 3, the limiting current of zinc-acetate complexes does not involve any kinetic complication and is diffusion-controlled. The relations between ia/i and f-l.o91/2 obtained at the free acetate concentration of 1.34 M and at several electrode potentials are shown in Fig. 6. The reversible half-wave potential E~ was calculated from the intercepts of the straight lines with the ordinate. Figure 7 shows the dependence of the reversible half-wave potential on free acetate concentration. By using D e F o r d - H u m e ' s method 1s, the overall conditional stability constants//p and the conditional successive stability constants Kp w e r e evaluated, as fit=7, fl2=92, fl3=70,//4=140,//5=152,
K 2 = 13.1, K 3 = 0 . 8 ,
K4=2.0,
Ks=
1.1.
2.0 1.9 1.8 1.7
*0.5
J..d 1.6
,L 1.5
E "~ 0.0
m
U
1.4 1.~ 1.2
-0.5
1.1
I
0.5
I
(,F~-~ 1"091
1.0
- ~E&~/ I -0.95
-1.00
I d/2/V
-1.05 ~.SCE)
I -1.10
Fig. 6. Relation between ia/i and f-1.091/2 obtained from i-t curves of zinc-acetate complex ions in the solution (1.34 M sodium acetate+2.6 M sodium perchlorate). E = ( A ) -1.043, (B) -1.054, (C) - 1.065, (D) - 1.076, (E) - 1.087 V vs SCE. Fig. 7. Reversible half-wave potential E~ obtained from i ~ curves for zinc acetate complex ions as a function of the logarithm of the acetate concentration in 4 M (sodium acetate +sodium perchlorate) solution.
The quantities exp ~ and ~D -~ on the left-hand side of eqn. (13) were calculated from (id/i)-t 1.o91/2 plots at various potentials and at various acetate concentrations. Then the quantity exp ~ x ~D -~ was obtained as a function of the acetate concentration at a given value of electrode potential, and it is denoted as the L 0 function. Next, L p function ( p = 1, 2 . . . . ) is defined as Z p = [Zp - 1 - M p _ I(E)]/CAc
Then,
Mp(E) can
be evaluated by the following extrapolation
(29)
ANALYSIS O F P O L A R O G R A P H I C C U R R E N T - T I M E CURVES
c~m -* 0 Lp
=
Mp(E)
195 (30)
Figures 8 and 9 show the Lp (p= 1, 2)-Cat plots at various potentials and Mp (p=0, 2)-E plots, respectively. From Fig. 8, we can deduce the following conclusions: The value of M1 at every potential is negligibly small and thus the species Zn(Ac) ÷ does not directly participate in the electrode process. The discharging species are Zn e+ and Zn(Ac)e. Therefore, zinc-acetate complexes are reduced to zinc amalgam according to the following two simultaneous charge-transfer reactions: Zn 2 + + 2e~-~-Zn(Hg) Zn(Ac)2 + 2e~- Zn(Hg) + 2Ac-
(31) (32)
The transfer coefficient can be expressed as 2.30 e z . [ A log Mp/ nF L AE J
0(p= 1
(33)
Thus, we obtain, after introducing the values of the slopes from Fig. 9 into
eqn. (33), c~0 = 0.24,
~2 = 0.31 A
0.5 0.4 0.3
--
B
--
C
0.2 D 0.1
-
E
1.0
3.0
2.0
4.0
CAC/mol F1)
(b) o
O
0
r)~
o
B
o
u
~
u
o
o
o
u
o
n
o
0.1 0 L2 C
0.05
I
~
1.0
2.0 CAc/mOl 14)
o
o I
3.0
-o- D
O-
E
O------O-
F
o
o-(3 1
4.0
Fig. 8. Variations of the functions i p (p = 1, 2) with acetate concentration at various values of the electrode potential. E = (A) - 1.065, (B) - 1.076, (C) - 1.087, (D) - 1.098, (E) - 1.109, (F) - 1.120, (G) -1.131 V v s . SCE. (a) p = l , (b) p=2.
196
Y. AYABE
0.0
log Mp
-1.C
-2.(
I
- 0.95
I
I
- 1.00
E/V
I
-1.05 (vs. S C E )
- 1.10
Fig. 9. Dependence of the function log Mp (p=0, 2) on the electrode potential E. (A) log Mo, (B) log M2."
The rate constants (k°)B and (k°), were evaluated by introducing into eqn. (14) the values of/3~ (v= 1-5), ~p ( p = 0 , 2) and Mp (p=0, 2) obtained above and the following data: Da 1.72x10 -5 cm 2 s 1 (ref. 12), D o = 0 . 5 2 x 1 0 -5 cm 2 s -1 (calculated from id in solution of 4.0 M NaC104 by the use of the revised Ilkovic equation11), and (E~)o = -0.9464 V vs. SCE. We have (k°)B =3.4 x 10 .5 cm s -1 (kO)B = 5 . 1 × 10 -3 c m s -1 M - 2 The formal potential E ° is evaluated from (Er)o as E ° = - 1.033 V vs. SCE Moreover, the formal rate constant k° can be calculated by eqn. (17) and also Ev° by eqn. (16), as follows; k° = 3 . 5 x l 0 - 3 c m s kO=l.5xl0-2cms
1, -1,
E °=-0.954Vvs.SCE
E°=-I.O12Vvs.
SCE
')(( n If ip denotes the current resulting from the reduction of the species M~._._p and i the total current, then the ratio can be given as
p b ~+
,
N
ip/i = Mp(E).cxP/p~=o Mp(E)..CPx
(34)
Thus, it can be seen that when CAc >0.5 M, i2/i~l, i.e., the faradaic Current results only from the discharging of the species Zn(Ac)2. This conclusion differs from the results of the previous paper 3, in which the discharging species were found to be Zn 2+, Zn(Ac) + and Zn(Ac)3 in 4 M (sodium nitrate+sodium acetate) solution. The discrepancy between these results may be attributable to the different supporting electrolytes employed.
ANALYSIS OF POLAROGRAPHICCURRENT TIME CURVES
197
APPENDIX The theoretical equation for the polarographic current-time curves used in the text is that derived on the basis of the "so-called" expanding plane electrode model. In this Appendix, the influence of the sphericity of the DME on the behaviour of the instantaneous current will be discussed. If the electrode reaction given by eqn. (1) is reversible and the product of the electrode reaction diffuses into the DME, the rigorous theoretical equation of the instantaneous current at any point along the wave for the expanding sphere electrode is given, according to Koutecky a7 and Weber 18, by the relationship: 1 '~ph. = id(1 +AO) 1 + e ~
(35)
A = 39.7 m-~D~x t ~
(36)
0 = 1--(DR~d/DOx)~e~
(37)
with
1 +e ~
where ia is given by the llkovic equation, which is derived for the expanding plane electrode. From eqns. (35) and (37), the instantaneous limiting current (exp ff~0, 0 = 1) can be given as (id)sph. = id(l + A)
(38)
For the expanding plane electrode, when the electrode reaction is reversible, v~e have, from eqn. (2), i = id/(1 + e ~)
(39)
Thus, comparing eqn. (35) with eqn. (39), we have ( id/i)~ph. = ( id/i) e
(40)
e = (1 + A)/(1 + AO)
(41)
with Therefore, e represents a correction factor for sphericity of the DME, and depends on the electrode potential, as can easily be seen from eqns. (37) and (41), In Table 2, the dependence of e on the electrode potential for t = 1 and 3.5 s is given. Unfortunately, no theoretical equation of the instantaneous current for the expanding sphere electrode has ever been developed for the case of slow electrode reaction, in which the current is controlled partly by the rate of diffusion and partly by the rate of electrode reaction. However, since the sphericity of the D M E affects only the diffusion process, it may be safely assumed that the effect of the sphericity for the slow electrode reaction is smaller than that for the reversible electrode reaction. The (id/i) vs. t - 1 " 0 9 1 / 2 plots given in this paper were carried out in the
198
Y. AYABE
TABLE 2 THE VALUES O F e AT F O U R D I F F E R E N T POTENTIALS CALCULATED BY THE USE O F EQN. (41) m = 1.0 mg s-1, Dox = 5 x 10 .6 cm 2 s-1, DRed/Dox= 2.
(E-E~)/mV
-
e ( t = l s)
~(t=3.5 s)
10 0
1.16 1.11
10
1.07
1.19 1.I4 1.08 1.04 1.01
20 -40
1.04 1.01
-
potential range from about E~ to E r-21 50 mV. Thus, the errors caused by the sphericity of the D M E may be smaller than about 10~o, as can be seen from Table 2. ACKNOWLEDGEMENTS
The author is indebted to Professor H. Matsuda, Tokyo Institute of Technology, for his interest and discussion of this work. Thanks are also due to Mr. H. Tanaka of Tokai University for his assistance in carrying out the experiments. SUMMARY
The method, which enables the evaluation of the kinetic parameters of electrode reactions by analysing the polarographi c current-time curve, is presented. As examples of the application of this method, the electrode reactions of the simple zinc ions and of the zinc acetate complex ions were experimentally examined. For the former case the current-time curves were measured as a function of the electrode potential and the perchlorate concentration. It was found that the standard rate constant, corrected both for the increase of the activity coefficient of the zinc ion and for the Frumkin double-layer effect, has a constant value over the concentration range of perchlorate 0.6-4.0 M. For the latter case the current-time curves in the solution with constant ionic strength (4 M) are measured as a function of the electrode potential and of the acetate concentration. It was concluded that the potential-determining reaction of zinc-acetate complexes is referred only to the species Zn(Ac)2 over a range of acetate concentrations larger than 0.5 M. D E F I N I T I O N O F SYMBOLS
i ia Dj CC104
Ef
= instantaneous d.c. polarographic current = instantaneous limiting diffusion current = diffusion coefficient of species j = concentration of the sodium perchlorate solution = cathodic transfer coefficient of the charge-transfer reaction of eqn. (1) = formal potential of charge-transfer reaction of eqn. (1) measured against a reference electrode
ANALYSIS OF POLAROGRAPHIC CURRENT TIME CURVES
kf Cx CAc (Cv)~
(cA 0¥
Eo ko t
199
= forward a n d b a c k w a r d formal rate c o n s t a n t s of the charge transfer reaction of eqn. (1) or eqn. (11) = s t a n d a r d rate c o n s t a n t of the charge-transfer reaction of eqn. (1) at the s t a n d a r d potential ---formal rate c o n s t a n t of the charge-transfer reaction of eqn. (1) at the potential E = E ~ - ( R T / n F ) l n ( D ~ e d D o x ~) -- c o n c e n t r a t i o n of the free form of c o m p l e x i n g agent X -b =- c o n c e n t r a t i o n of the acetate ion --- c o n c e n t r a t i o n of the metal complex M~(n _._._p pb) + at the electrode solution interface c o n c e n t r a t i o n of the a m a l g a m a t e d metal at the e l e c t r o d e - s o l u t i o n interface - - c o n d i t i o n a l overall stability c o n s t a n t of the metal complex MX(p"-pb~+ = cathodic transfer coefficient of the charge-transfer reaction of eqn. (11) - - f o r m a l potential of the charge-transfer reaction of eqn. (11) m e a s u r e d against a reference electrode - - f o r m a l rate c o n s t a n t of the charge-transfer reaction of eqn. (11) at the formal potential E ° = time in seconds for d r o p growth
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