Faradaic Impedance with Polarographic Generation of Reactants THOMAS BIEGLER and H. A. LAITINEN Noyes Chemical laboratory, University o f Illinois, Urbana, 111.
Application of the theory of faradaic impedance to cases where reactants are generated polarographically is discussed and an equation is derived which allows evaluation of the transfer coefficient, ct, from impedance measurements at several points along a reversible polarographic wave. Kinetic parameters obtained in this way generally agree well with available data determined under true equilibrium conditions. Departures from theory in the slopes of plots OF faradaic resistance and reactance against w-1/2 are attributed to the influence of spherical diffusion on the surface concentrations of reactants.
of faradaic impedance as a means of obtaining kinetic parameters of rapid electrode reactions is generally carried out with an electrode a t its equilibrium potential. However, the theory is also applicable, with certain limitations, to the case where the electrode passes a net direct current, as for example in a x . polarography. Accordingly, polarographic generation of reactants, because of its experimental simplicity, becomes an attractive method, especially when the impedance of an electrode reaction involving a metal amalgam or an unstable product is to be examined ( 5 , 29, 25). Several workers have obtained kinetic data by measuring the faradaic impedance in the presence of a direct current flow. Tamamushi and Tanaka (25) pointed out the advantages of this method and used an equation derived by hlatsuda (16) to calculate the rate constant k," and transfer coefficient a from the potential dependence of cot where + is the phase angle of the faradiac ax. Randles (19) showed that the classical theory of the faradaic impedance, derived for an electrode a t equilibrium, remains valid in the presence of a direct current provided that the electrode reaction is sufficiently rapid and obtained kinetic parameters under these conditions. Study of the faradaic impedance with a x . polarography also involves in situ generation of reactants and several authors have used this approach in examining electrode kinetics ( 2 , 21, 2 2 ) . EASUREMENT
+
572
ANALYTICAL CHEMISTRY
The present work employs the classical approach developed by Randles (17) in which the resistive and reactive components of the faradaic impedance are plotted against u-l'z where w is the angular frequency of the alternating voltage. The exchange current is obtained from these measurements through the following relations R , = Rj
-
l/wCj
(1) (2)
A R , = RT/nFi,,
and +
i,,
=
ai
+ (1 - a)i
c
(3)
where R , and CI are the series faradaic resistance and capacitance, A is the -
+
c
electrode area, and i and i are the forward (cathodic) and backward current densities, respectively (9). For an electrode at equilibrium, i,, is identical to the apparent exchange current denWhen the electrode passes a sity I,'. small net current it can be shown (19) that for a polarographically reversible process-Le., one for which the apparent rate constant exceeds about 2 x cm. sec.-l-i,, over most of the wave will differ from the corresponding exchange current a t equilibrium by no more than 2 to 3y0. For faster reactions (k,' > lo-' cm. sec.-l) and/or CY close to 0.5, the difference is well within the experimental error for such measurements. Diffusion control of the d.c. process is also desirable for simplifying the calculation of reactant concentration. The apparent rate constant is obtained from the exchange current ( = i,, under the aforementioned conditions) through the relation
I,"
=
nFK,o/Coxl-aCRa
(4)
where the concentrations of oxidant ( C o x ) and reductant (C,) are measured at the surface. We consider now the case of a reversible cathodic wave for a bulk concentration C B of oxidized species. Inserting the surface concentrations and
into Equation 4, taking logarithms, and rearranging, we obtain
a
log
i/(id
- i) (7)
where (Zao)c is the exchange current measured when current i is flowing and id is the diffusion current. Determination of exchange currents at various points along a reversible wave can therefore readily be used to evaluate a. This method is quite analogous to the classical determination of a by measuring the variation of exchange current with concentration of one of the reactants. It seems experimentally simpler than the approach used by Tamaniushi and Tanaka (65), which involves determining cot + as a function of potential and applying the relation
where E,,, is the potential for maximum is the reversible halfcot + and ETlI2 wave potential. The results are obviously sensitive to small experimental errors in the potentials especially when 0.4 < a < 0.6. Recording of cot + us. potential (25) should be important in simplifying experimental determination of E,,, to the required accuracy. The method holds the advantage that Equation 8 is applicable to quasireversible as well as reversible systems but in the former case, it may be difficult to determine ET1/2. At the half-wave potential
( I a o ) ~=l ,~, F ~ , " ( D o ~ / D R ) ~(9)W B / ~ I n principle, a single measurement of I,' a t is insufficient for calculation of k,'. In this work, rate constants for some simple metal ion reductions have been calculated from Equation 9 using a values determined or taken from the literature. Errors introduced by uncertainties in such values depend on how far the ratio Dox/Dx differs from unity; this ratio itself may be uncertain in cases of amalgam formation because the literature contains seriously discrepant values for diffusion coefficients of metals in mercury (4, 7 , 2 2 ) .
Figure 1 . o.
far 6.0 X lO-'M cadmium in 1.OM KNOa E l / , = -0.591 volt VS. S.C.E.
EXPERIMENTAL
Impedance measurements were carried out with a Wien bridge of conventional design ( 8 ) , some details of which have been described in an earlier communication ( 3 ) . The cell impedance was balanced a t six frequencies (100, 200, 400, 600, 1000, and 2000 c.p.s.) against decade resistance and capacitance boxes in series. The other t'wo arms of the bridge were precision resistors whose values could be chosen to give ratios of I : 1 to 10: 1. Capacities of up to 20 pf. could be balanced in this way. Measurement of the time of bridge balance (3) was preferred to the more tedious procedure of balancing a t a predetermined instant. Other advant,ages of this arrangement were that any variations with time of the double layer capacity per unit area were easily detected and time dependence of adsorption of surface active material (not reported here) could be followed. The time dependence of solution resistance between a growing mercury drop and a large counter electrode has been given (12) in the form
R
=
u
+ bt-1'3
(10)
and this relation was verified repeatedly during the course of this work. Accordingly, about' five measurements of R s , the ohmic cell resistance (solution capillary), over a wide time range during drop life were sufficient to allow Rs a t any given time t,o be read off a plot of Rs us. t - l ' 3 . Resistance values ranged from 3.5 Q in molar acid to around 14 Q in a molar salt solution and accuracy of measurement was k O . 1 Q . Capillaries of resistance in the range 0.7 to 1.0 Q were constructed of an upper section of 0.045-mm. i.d. marine barometer tubing sealed to a lengt,h of 6-mm. soft glass tubing drawn to a fine tillj 0.d. about 0.3 mm. .S, plat'inum contact was sealed in about 3 cm. below the joint. Such capillaries supported
+
Components of the faradaic impedance at €I/, b.
for El/p
2.02 X 10-4M l e a d in 0.1M N a O H = -0.690 volt VI. S.C.E.
mercury drops of area about 0.06 sq. em. and were used with drop times and flourates of 10 to 14 seconds and 1.6 to 2.1 mg. per see., respectively. Frequency dispersion of the double layer capacity for a supporting electrolyte was less than 1% over the range 100 to 2000 c.p.s., indicating absence of significant shielding of the electrode. A Leeds and Northrup student potentiometer was used to polarize the D M E against a saturated XaC1 calomel reference electrode (reference potential = -5 mv. us. SCE). A 200 Henry choke (UTC type h'I&L-3), chosen for its low d.c. resistance of 820 Q , was used to block a s . from the polarizing circuit. Deionized water was shaken with activated charcoal, filtered, and distilled from alkaline permanganate in an all-glass apparatus. Reagent grade chemicals were used without further purification. The all-glass cell was immersed in a water bath whose temperature was maintained a t 25' % 0.5' C. A salt bridge containing supporting electrolyte connected the cell to a beaker containing saturated XaC1 solution into which dipped a bridge from the reference electrode. I n a typical experiment 100 ml. of supporting electrolyte were placed in the cell, and after deaeration, capacity measurements at 1000 c.p.s. were made a t small intervals in the potential range of interest. The ohmic resistance, Rs, was measured a t several times during drop life. A stock solution of the depolarizer in supporting electrolyte was then added, the polarogram recorded, and the impedance measured at E l i zor, in the case of CY determinations, a t several points along the rising portion of the wave. The total cell resistance, generally in the range 700 to 2500 Q , was noted so that the necessary iR drop corrections could be made to the electrode potential. A set of measurements a t six frequencies could be carried out in about 30 minutes and
+ 0.9M N a C 1 0 4
impedance data needed for CY determinations (at five value5 of potential) were typically obtained in less than 3 hours. The faradaic components of the overall impedance were calculated from the experimental quantities with the usual network conversions (15). In more recent work the tedium of these calculations was avoided with the use of an 11311 7094 computer. RESULTS A N D DISCUSSION
Kinetic Data. I n t h e systems described below in detail, the faradaic impedance measured with polarographic generation of reactants showed the theoretically expected type of behavior, as has previously been indicated in most of these cases for impedance data a t equilibrium (19, 20). Some representative plots of the components of the faradaic impedance are shown in Figure 1 for a very fast and a moderately fast reaction. Cnder favorable conditions the upper limit of k,' measurable with the present apparatus is in the range of 5 to 10 cm. sec.-' (19). .Inomalous behavior was found, as expected, in the reduction of cobalt (111) trisethylenediamine (15) and organic compounds such as quinone and riboflavin. These systems were not studied further. The reduction of Cd+* was examined in several supporting electrolytes. I n 1 . O M HC1O4 k,' was 0.35 i 0.03 em. see.-' with results of three separate runs (CB = 0.269, 0.280, and 0.537mM) agreeing within the experimental error. Randles (19) has also examined this system under both polarographic and equilibrium conditions, presenting his results as plots of log kl (forward rate constant) us. electrode potential. If it is assumed that the standard VOL. 37, NO. 4, APRIL 1965
573
potential lies near the middle of the potential range, one finds k,' = 0.35 cm. sec.-l for the dropping mercury electrode and k,' = 0.7 cm. set.-' for the dropping amalgam electrode. Randles tentatively attributed the latter value to the use of a more rigorously purified solution. On this basis, agreement between his lower value and the present result would lead to the unlikely conclusion that the two solutions were contaminated to the same extent. Further, a t the concentration levels expected for traces of surface active contaminants one predicts diffusion control of adsorption and hence time dependence of the double layer capacity (8). N o such effect was observed in the present experiments. Two determinations of (Y for this system were carried out. Plots of log [ (Zao), .&/(id- i) ] us. log i/'(id-i) (Equation 7) are shown in Figure 2. Agreement between the slopes is good, with CY = 0.14 i .02, the value used to calculate the above mentioned rate constant. We can find no evidence for a decrease in CY corresponding to ~tlimiting value of cathodic rate constant at the cathodic end of the potential scale as reported by Randles (19). I n 0.lM HClOa 0.9M NaC1O4, exchange currents were slightly higher than in 1.OM HC10,. If it is assumed that (Y is also 0.14 in this electrolyte, we find ',k = 0.46 f 0.03 cm. sec.-l (mean of three determinations). Behavior of the faradaic impedance for Cd+2 in 0.5M HC1 was of interest because of anomalous results reported by Bauer and Elving (2). With a solution of similar composition ( C B = 0.557mM) , the impedance was diffusion controlled-Le. within experimental error, Rj = I / d J and cot $J = 1, where + is the phase angle of the faradaic ax. The reduction of 0.60mM C d f 2 in 1.OM KN03, another system for which Bauer
+
Table 1.
lO-'M
Cd + 2
3 72
Cd + 2
4 96
Cd _ _ +2
Cd + 2 Pb +x
HPb02-
Cg,
3.63
2 80 5 37 5.57 6 00 3 05 2 02
HPbOz-
4 06
T1+
2 09
574
'
'
Supporting electrolyte 0.lMHClOa 0 . 9 M NaC10. 0 1M HClOn 0 9M NaC104 0 1M HC104 0 9M NaC104 1 OM HCIO4 1 OM HClOi 0.5MHCl 1 OMKNOs 1 OM HClOi 0 1M NaOH 0 9M NaC104 0 1MNaOH 0 9M NaC104 0 1M HClO4 0 9M NaC104
ANALYTICAL CHEMISTRY
+
+' +
+ + +
DOX, Ilkovic equation 0.93
HCIOd Exchange currents expressed in ma. p e r square centimeter
h
od C!. 1.30 d 0
c e = 280 x 1.2-
d '0
-J,Q
-05
1 0 - 4 ~
14
05
0
IO
LOG i h d - i )
and Elving (2) found anomalies (cot
