CHEMICAL REACTIONS INITIATED BY A CHARGE TRANSFER
a45
A Potential Step-Linear Scan Method for Investigating Chemical
Reactions Initiated by a Charge Transfer
by W. M. Schwarz and Irving Shain Department of Chemistry, University of Wisconsin, Madison, Wisconsin
(Received September 27, 1966)
A simple method has been developed for the investigation of electrode processes in which the product of the electron-transfer step is involved in a further chemical reaction to k ne e R -+ Z. The technique involves two steps. I n produce an inactive species: 0 the first, R is generated a t a stationary electrode under diffusion-controlled conditions by applying a constant potential for a timed interval. During this interval, substance R diffuses into the solution and simultaneously reacts. Then in a second step, the potential is scanned rapidly in the anodic direction to reoxidize R back to 0. The resulting anodic peak current is a measure of the unreacted R and can be related to the rate constant k. An approximate solution to the boundary value problem for this combined diffusionelectron transfer-kinetic system was obtained for the case of a plane electrode. The method was applied to the reduction of azobenzene to hydrasobenzene which, in turn, undergoes the benzidine rearrangement. The pseudo-first-order rate constants in 50 w t % ethanol-water were found to range from 0.10 sec-' in 0.4 M HClOI to 2.4 sec-' in 1.0 M HC104, in agreement with previously reported data.
+
Recently, several two-step electrochemical metho d ~ ' - ~have been described for the study of the general reduction-kinetic process4 ne
O Z R k
R+Z where the product of the charge-transfer step, R, is unstable and subsequently undergoes an irreversible homogeneous first-order chemical reaction to give an electroinactive species, Z. I n these methods, a stationary electrode is used, and the electrolysis conditions are varied in a step functional manner: the first step corresponds to the reduction of 0-the generation step; the second step corresponds to the reoxidation of the unreacted R-the measuring step. Experimentally, step functional electrolysis conditions can be obtained easilv and from the theoretical standDoint the necessary diffusion-kinetic equations can be solved rigorously. On the other hand, the dependence of the measured k l is quantities On the rate complex, and working curves usually are required to estimate k.
In this work, an alternate two-step, controlled-potential method was investigated which combines the step functional method with linear scan voltammetry (stationary electrode polarography), I n the first step, the potential is jumped to a value where the rate of generation of R (the cathodic current) is determined solely by the diffusion of substance 0 to the electrode surface. Then after a timed interval, the potential is scanned linearly in the anodic direction. The resulting anodic peak current gives a measure of the unreacted R which remains in the vicinity of the electrode. By using a linear voltage scan for the measuring step, a reasonably accurate value of the rate constant can be determined rapidly without the use of theoretical working curves. In addition, this method provides a rapid (1) (a) A. c. Testa and w. H. Reinmuth, Anal. Chem., 32, 1512 (1960); (b) 0. Dracka, Collection Czech. Chem. Commun., 2 5 , 338 (1960); (c) W.Jaenicke and H. Hoffmann, 2. Elektrochem., 66, 803 814 (1962)* (2) H. B. Herman and A. J. Bard, A n d . Chem., 36, 510 (1964). (3) W.M. Sohwara and I. Shain, J. Phys. Chem., 69, 30 (1965). (4) Although the discussion emphasizes cases in which the initial charge transfer is a reduction, extension to oxidations is obvious.
Volume 70. Number 9 March 1966
W. M. SCHWARZ AND IRVING SHAIN
846
semiquantitative way of examining the products which are formed in the reduction and kinetic steps. To test the applicability of this potential steplinear scan method for kinetic measurements, the azobenzenehydrazobenzene system was investigated. This system follows reaction scheme I. Azobenzene is reduced to hydrazobenzene which, in acid solution, undergoes the benzidine rearrangement. Numerous studies have been made of both the electrochemical reduction step5 and the benzidine rearrangement reaction.6 The azobenzene system has been studied recently using the step functional, controlled-potential method, and the results of that study were used for direct comparison with the rate constants determined here.
Theory The usefulness of the potential steplinear scan method depends on obtaining a simple but accurate relation among the electrolysis time, r , the anodic peak current, i,, and the rate constant, k. This can be done if the experimental conditions are arranged so that the rate of voltage scan during the second step is rapid compared to the rate of the kinetic process. Except for the approximations which can be introduced as a result of this restriction, the boundary value problem and its solution are very similar to the step functional, controlled-potential method. For a system following reaction I under conditions of semiinfinite linear diffusion, the dependence of the current on the surface concentration of R is3
t
> 0 , [co + (CR d- c z ) d D ~ / D o ] ~ ==o co*
(4) After the appropriate transformations, eq 2, 3, and 4 are used to eliminate c R ( z = ~ from ) eq 1 t
> 0, Z
= nFAdDR(s
S,
+ k) X
( v ‘ D ~ / D ~ c ~ *- Cz),=oe -stdt] 1 m R e e a ( t - ’ )
+
(5)
This expression still contains CZ(,=o)-a quantity which is not known explicitly for times greater than T . As a first approximation, however, it can be assumed ) to its value a t time T , and that it that C Z ( ~ =is~equal does not change greatly during the scanning cycle
t
> 7 , 1 / D o l D R C O * - cZ(z=O) CO*d D o / D ~ -e kT’210( k ~ / 2 ) (6)
~) Actually, a t the start of the scan, C Z ( ~ =increases slightly due to the continued predominance of the kinetic process over diffusion. Then, as the surface concentration of R becomes small (roughly at a time, t,, corresponding to the appearance of the anodic peak current) the kinetic process diminishes and Cz(z=o)begins to decrease. A quantitative idea of the accuracy of the approximation in eq 6 can he obtained by comparing dDo/DRCo* - C Z ( = =at ~ )time T with the value which would be expected ai. time t, in the absence of the anodic scan. The ratio of these two terms is approxit >; 0, i = nFAdDR(s k)(cR)z=o (1) mately equal to e - k ( T - t p ) / z indicating that the error in l/o,/o,CO* - C Z ( ~ = Ocalculated ) from eq 6 is no where the bar signifies the Laplace transform of the greater than 3% as long as k ( r t,) is less than about variable, s is the transform parameter, and the other 0.06. Even with this approximation, however, the symbols have their usual meaning. integral in the second term of eq 5 cannot be evaluated During the initial potential step, CR(z=O) varies with explicitly, but as will be shown below the exact time time dependence of this term is not needed to obtain a useO 7,
~ k ( f - ' i ~ ~ * h l n e ' ' d. .hdXn l. -
[ U t - 711" nFACo * %'?%e-
[
G
-
a
T
{l/dr(t -
7'210( k 7/2) d:- X e-8(t-r)
m e
n FA CO* *e-
J
o
1
d(t -
+ v'&TGP-r)
1
where i, and i, refer to the cathodic and anodic currents, respectively. Equation 7 describes the cathodic current-time curve for the first potential jump, and as expected, the cathodic current does not depend on the homogeneous kinetic processes. Equation 8 represents the anodic current-voltage curve which is obtained during the linear potential scan. The first term on the right side of this expression is the result expected for an anodic potential jump.3 The second term modifies this result and gives the dependence of the current on the anodic scan rate. As the scan rate becomes very large, a + m l and the second term approaches zero. Another limiting case of importance is that in which the rate of the kinetic step is negligible, k: + 0. Then the bracketed part of the first term becomes unity, and eq 8 reduces to
t > 7, i, nFACo*=
+
=
nFACo*G X
( ~ / d r-( t7 ) - 6-1
The right side of eq 9 is one of several equivalent expressions for the current for a reversible electron transfer with a linear potential scan at a stationary electrode,' except that the time scale is measured from the point t - 7 . Thus, for this limiting case, the anodic peak current would be proportional to the original bulk concentration of substance 0, to the square root of the rate of potential scan, and to the other experimental parameters of stationary electrode polarography. Equation 8 can be simplified further by restricting k ( t - T ) to values less than 0.05, as indicated above. Then the quant,ities under the summation sign in the first term of eq 8 can be neglected since they always will
7)
kr/21 0(k7/2) X
- d:-1[e-%Tif(o,a,s)lJ
(10)
Here, i, is the anodic current measured to the extension of the cathodic i-t curve as a base line, and f(O,a,s) is the integral function of scan rate shown in eq 9. The right side of eq 10 consists of the product of two distinct terms. One depends only on k and (t T)-not on 7-and is represented by the quantity in the braces. This "scan" term is the sole factor determining the shape of the anodic current-voltage curve. In general, it is not necessary to know the explicit form of this term, but only that it reniains the same for all values of r , for a constant rate of potential scan. The other term depends only on k and r and is given by the quantities preceding the braces. In effect, this term acts as an amplitude term and determines the magnitude of i, for different values of r . The r dependence itself is contained in the expression for the surface concentration of R at time 7. That is, i, is directly proportional to the surface concentration of R at the switching time, T , as long as the rate of the voltage scan is fast compared to the rate of the kinetic process. The same considerations also apply to cases in which the charge transfer is irreversible. The steps in the derivation are analogous, eq 10 is still valid, and the only change which results is in the form of the function f(e,a,s). The features which are important with respect to the chemical kinetics, however, remain unchanged-the anodic peak current is still directly proportional to the surface concentration of the reactive species at the switching time, and the same procedure can be used to determine the rate constant. In the usual method of analysis using eq 10, the value of i, is measured at a fixed value of (t - r ) (or the equivalent voltage) for a series of different switching times. Since the peak current i, can be measured most accurately (Figure l), (t - T ) invariably is chosen as the peak time (t, - 7). Xormally, a provisional value of k is obtained from the limiting slope of a In i, (7) A. Sevcik, Coll. Czech. Chem. C o m m u n . , 13, 349 (1948); R. S. Nicholson and I. Shain, Anal. Chem., 36, 706 (1964) and references therein.
V o l u m e 70, Number 3 March 1966
848
W. M. SCHWARZ AND IRVING SHAIN
- - - ---____
W
r r o
\
t-
3
o
ilv Ii
T’
I
-0.zc
B
Figure 1. A Typical curves for the potential step-linear M azobenzene-50 wt % scan method, with 2.0 X ethanol-water, scan rate = 4.15 v/sec. Solid line: 0.797 M HClO,, half-life of the benzidine rearrangement 7. Dashed line: 0.10 M HClO,, half-life of the benzidine rearrangement >> T . B. Time dependence of the applied cell voltage. T plot. Then, a second plot including Io(kr/2), which normally is close to unity, gives an accurate value of k.
us.
Experimental Section Instrumentation. The potentiostatic setup used in this work has been described previously as “potentiostat B” by Schwarz and S h a h 3 For the present application, the booster amplifier was omitted. The wave form shown in Figure 1 was obtained by combining a, symmetrical triangular pulse with a square pulse of opposite polarity in a simple passive adding circuit. The square pulse was obtained from the gate output of a Tektronix Model 162 wave form generator. This unit, along with a Tektronix Model 161 pulse generator, provided a delayed trigger signal which was used to activate the triangular pulse generator. The length of the delay corresponded to the time r and could be varied continuously. The triangular wave itself was obtained from an operational amplifier multivibrator siniilar to that described by Underkofler and Shains The accessory electronic equipment, as well as the cells, the electrodes, the chemicals, and the solution preparation have been described previ~usly.~ Procedures. For all measurements the initial dc level of the working electrode was set at 0.3 v. us. sce. The cathodic polarization was carried out a t -0.1 v us. sce. The Journal of Physical Chemistry
I n the theoretical discussion, the switching time, T , and the generation time were considered as the time prior to the start of the linear scan. Actually, generation of the reactive species R continues during the scan cycle until the voltage reaches the value of E” for the system. I n practice, therefore, an effective switching time, T I , was defined as the time required for the potential to reach Eo-i.e., the current to reach roughly 85% of the peak current on the reverse scan.7 This measure of switching time was used in place of T for all of the rate constant calculations, I n experiments involving long generation times, an anomalous stirring effect appeared. This was detected by a sharp increase in the cathodic current followed by rather large periodic fluctuations in the current-time curve. In addition, a direct observation of stirring was made by carrying out a reduction-oxidation cycle in the presence of 1,l1-diethyl-4,4’-dipyridinium ion, the reduced form of which is intensely colored. The onset of stirring was characterized by a very rapid tangential transport of material from the bottom of the mercury electrode to the top. In general, the stirring effect was independent of inert electrolyte concentration, added surface-active agents like gelatin or Triton X-100, cell geometry, electrode shielding, temperature, auxiliary stirring, added products of the reduction or kinetic steps-i.e., benzidine, hydrazobenzene, or previously rearranged hydrazobenzene, and the observation technique-ie., chronopotentiometry, polarography, etc. On the other hand, stirring was less for more dilute azobenzene solutions and more cathodic reduction potentials and always seemed to be roughly proportional to the square of the acid concentration-ie., to the rate of the homogeneous kinetic process. One possible explanation assumes the formation of a surface-active side product intermediate by the kinetic process. As soon as this species reaches significant concentration levels, it becomes adsorbed on the electrode and disturbs the properties of the interface to the extent that stirring results. That such an effect can cause violent convection has been reported previously. By completing the kinetic measurements within the first half-life of the rearrangement reaction, however, the electrochemical results were totally unaffected by stirring effects. For the actual kinetic measurements, 15 to 30 anodic current-voltage curves were obtained with each azobenzene solution. Each curve corresporided to the (8) W. L. Underkofler and I. Shain, Anal. Chem., 35, 1778 (1963).
(9) J. T. Davies and E. K. Rideal, “Interfacial Phenomena,’ Academic Press Inc., New York, N. Y., 1961, p 309.
CHEMICAL REACTIONS INITIATED BY A CHARGE TRANSFER
same scan rate but to a different value of r'. I n addition, each curve was measured on a fresh hanging mercury-drop electrode. Reproducibility of any single curve was of the order of 2% in the most dilute solutions. The principal source of error was 60-cps pickup. I n all cases a blank correction for the charging current was made based on current-voltage curves measured on solutions containing no azobenzene. These corrections were generally small and amounted to no more than 5% of the peak current in the most dilute s o htions .
849
0.20-
i, /c: 0.15-
0.10
-
Results and Discussion I n this work, as in the previous in~estigation,~ the dependence of the benzidine rearrangement rate on acidity was used to separate the kinetic and nonkinetic effects. Thus, one series of experimental curves was obtained in dilute acid solution where the rate of the kinetic process was negligible. These measurements were used to evaluate the general procedures of the potential steplinear scan method and to test the purely electrochemical behavior of the azobenzenehydrazobenzene system. Then, a second series of measurements was obtained under identical experimental conditions but in more acidic solutions where the kinetic step was important. It was found that the electrochemical behavior was influenced by adsorption and other surface phenomena, and empirical corrections had to be applied in order to use the potential step-linear scan method to evaluate the rate constants for the rearrangement reaction. Nonreacting Systems. The first set of experiments was carried out on azobenzene solutions (1, 2, 3, and 4 mM) containing only 0.1 M perchloric acid. Under these conditions, the half-life of the kinetic step was about 500 see--a time very much longer than the switching times that were used. For each solution, eight scan rates were tested ranging from 0.417 to 8.34 v/sec, and for each scan rate 10 to 15 values of were selected in such a way that the product V T covered the range of 0.4 to 4.0 v. The anodic peak currents, i,, were measured as shown in Figure 1 and analyzed according to eq 9. The experimental values of i, were found to be independent of r' for all scan rates and concentrations. These results indicate, as expected, that the reduction of azobenzene during the cathodic step is a simple diff usion-controlled process. On the other hand, the dependence of the measured peak currents on both the scan rate and the surface concentration of R ( i e . , CO*)showed marked deviations from theory. Plots of ~,/CO*as a function of l / C O * , which should be horizontal straight lines are shown in Figure 2. The values of n F A 6 0 used in the theo-
I 0
0.5
I/C.,,
1.0
(tnJ)-'
Figure 2. Comparison of experiment with theory when the half-life of the benzidine rearrangement >> T': dashed lines, theoretical; points, experimental for azobenzene in 0.10 M HC104-50 wt % ethanol-water solutions a t 25'. The scan rates (v/sec) are: A, 7.73; B, 3.84; C, 1.93; D, 0.764; E, 0.378; F, 0.194.
retical calculations were obtained experimentally from the diffusion-controlled current-time curves using eq 7. Depending on the scan rate, the values of i,/Co* tended to be up to 15% too high for the solutions containing 1 mM azobenzene, and 11-15% too low for those with 3 and 4 mM azobenzene. The deviations were greatest with the fastest scan rates. Normally, some deviations of i,/Co* from eq 9 are expected as a result of stirring in the diffusion layerloan effect caused by movement of the mercury drop as the potential is varied. In general, such deviations are independent of concentration and amount to only 3 4 % . Thus, these factors cannot explain the large discrepancies observed with the azobenzene system. The anomalous behavior of the azobenzene-hydrazobenzene system probably can be attributed to surface phenomena. Recently, Holleck and co-workersSh-j have studied the effect of traces of surface-active materials on the polarographic behavior of both azobenzene and hydrazobenzene in 30% methanol-water solutions of pH 1 to 12. From shifts in the half-wave potential, they concluded that such materials became adsorbed on the electrode and inhibited the electron-transfer step. I n the oxidation of hydrazobenzene, not only were known adsorbers like gelatin and triphenylphosphine oxide effective inhibitors but so was azobenzene itself. (10) I. Shain and K. J. Martin, J. P h y s . Chem., 6 5 , 254 (1961).
Volume 70,Number 3 March 1966
850
W. M. SCHWARZ AND IRVING SHAIN
Further evidence has been presented by Nygard,5f who studied the azobenzene system with several methods, including cyclic-scan experiments with stationary electrodes. Results in agreement with those of Holleck were found in 0. I M HC104--50 wt % ethanol-water solutions in the presence of 0.01% gelatin. The anodic currentvoltage curves, obtained with single-scan stationaryelectrode polarography were broader, and the peak current was about 25% lower than for comparable curves without gelatin. If adsorption of azobenzene has an analogous effect in this solution, then the low values of ip/Co* could be explained on the basis of inhibition of the normally reversible electron-transfer step." The concentration dependence arises because of the fact that the aaobenaene is produced during the rising portion of the anodic current-voltage curve, and the greater the concentration, the more rapidly the adsorbed surface layer is formed. One method of detecting the inhibition effect directly was attempted-the measurement of the separation of the cathodic and anodic peak voltages in cyclic-scan experiments. The results, however, were inconclusive, and, although the peak potential showed a slight variation with scan rate, the magnitude of the anodiccathodic peak separation was always within 20 mv of the reversible value. If the inhibition effect acts to decrease the peak current, a second effect must also be present to explain the high peak currents observed with low azobenzene concentrations. Again assuming an adsorption step, an additional current may result not from a faradaic process but from a change of the double-layer capacity during the adsorption process. For low azobenzene concentration, a capacitive current would be more noticeable and could cause an apparent net increase in the normal peak current. Whether the inhibition or capacitive effects would predominate for any given solution would depend on the depolarizer concentration. Although the factors which influence the experimental data of Figure 2 are not known with certainty, one result is of particular importance in connection with the subsequent kinetic measurements. That is, the experimental line for each scan rate is nearly straight over a rather wide range of concentrations. Thus, Co*-Le., CR(r=O)-can be expressed as a linear function of i, 1
> 7 ) (i,
- m) = (i,/CO*),C,*
(11)
where wz is the slope of the experimental line and (ip/ Co*)O is the extrapolated y axis intercept. This equation can be considered as the experimentally derived analog of eq 9 and must be used in place of eq 9 in estiThe Journal o,f Physical Chemistry
mating C R ( 2 = ~from ) measured peak currents for the case where the kinetic step is negligible. Reacting Systems. I n the second set of experiments, measurements were obtained on azobenzene solutions (1.4, 2, and 3 mM) containing from 0.4 to 1.0 M perchloric acid. For these acid concentrations the halflife of the rearrangement reaction ranged from about 6 to 0.3 sec. For each solution, from 15 to 30 curves were obtained each corresponding to a different value of r' and the same scan rate. The values of 7' were selected in the range of 0.1 to 1.0 times the half-life of the reaction. In general, the scan rates were chosen such that the product vr' remained between 0.4 and 4.0 v, as with the nonreacting systems. Normally, the experimental measurements would be analyzed in terms of eq 10. However, in view of the results of Figure 2 for dilute acid solutions, the direct proportionality between i, and surface concentration of R predicted by eq 10 seems unlikely for the more acidic azobenzene solutions. As a result, a semiempirical equation analogous to eq 11 was used to estimate the rate constants t
> 7,
(ip
- m)
= (ip/co*)ocR(z~o)= (i*/CO
*)oCo *e - k"'2Ll(kr '/2)
(12)
If it is assumed that the factors which cause the deviations from eq 9 in the dilute acid case cause similar deviations from eq 10 in the presence of the kinetic step, for a given scan rate the values of m and (~,/CO*)O should be identical with the slope and intercept obtained experimentally from Figure 2. Since m and ( i , / ~ o * )as ~ well as vl" are constant for a given series of measurements, the rate constant, IC, can be evaluated directly from the slope of a plot of ln[(i, - m)/Io ( k r f / 2 ) ]os. r'. Several such plots are shown in Figures 3 and 4. I n spite of the relatively good precision noted previously for the peak current measurements, there is considerable scatter in the points. This is primarily a result of the insensitivity of the peak current measurements to the kinetic process. The problem is one of detecting small changes in rather large currents and is common to all of the two-step electrochemical methods with systems of this type. Nevertheless, in all eases the experimental points describe a straight line. If eq 10 had been used directly without the correction term, the lines would be slightly curved and the apparent (11) The low results were not caused by excessive uncompensated IR drop since solutions containing an additional 1.0 M NaClOd and the minimum possible Luggin capillary-HMDE separation gave results identical with Figure 2.
CHEMICAL REACTIONS INITIATED BY A CHARGE TRANSFER
851
1
I
Table I: Kinetic Data for the Perchloric Acid-Catalyzed Rearrangement of Hydrazobenzene in 50 Wt % Ethanol-Water a t 25"
0.7
Ic
y
Q! o
0
O
2
I
k
1
r I sec.
0
3
4
Figure 3. Rearrangement of hydrazobenzene in 50 wt % ethanol-water a t 25" and 2.00 X M azobenzene with: A. 0.395 M HC104, v = 0.417 v/sec; B. 0.497 M HC104, v = 0.834 v/sec; C. 0.641 M HC104, v = 2.07 v/sec. (Curves have been normalized for presentation.)
1.0
0.9
T
e Y
Y
7OO.B 1
E I
M
Range of 7 ' ) sec
.4nodic wan rate, v/sec
sec -1
0.395 0.395
1.40 2.00
0.60-4.90 0.60-3.90
0.417 0.417
0.111 0.106
0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497 0.497
1.40 2.00 3.00 1.40 2.00 3.00 1.40 2.00 3.00
0.70-3.70 0.60-2.90 0.50-1.90 0.60-3.80 0.45-3.15 0.30-1.90 0.50-3.60 0.35-3.00 0.25-1.70
0.534 0.534 0.534 0.834 0.834 0.834 1.66 1.66 1.66
0.203 0,191 0.203 0.202 0.204 0.211 0.203 0.202 0.200
0.641 0.641 0.641
1.40 2.00 3.00
0.24-1.56 0.24-1.70 0.24-1.39
2.07 2.07 2.07
0.503 0.485 0.502
0.797 0.797 0.797 0.797 0.797 0,797 0.797 0.797 0.797
1.40 2.00 3.00 1.40 2.00 3.00 1.40 2.00 3.00
0.15-0.86 0.16-0.88 0.15-0.87 0.12-0.86 0.12-0.86 0.12-0.70 0.10-0.88 0.10-0.83 0.10-0.83
2.77 2.77 2.77 4.15 4.15 4.15 8.34 8.34 8.34
1.26 1.07 1.01 1.15 1.06 1.01 1.18 1.11 0.95
0.988 0.988 0.988
1.40 2.00 3.00
0.06-0.47 0.07-0.42 0.06-0.37
8.34 8.34 8.34
2.64 2.33 2.25
Perchloric acid concn, M
Asobenzene concn
x IO',
k,
.-a Lt;
-C 0 . 7 O
0
0.2
F
o
0.4
O\
0.6
rlI sec.
0.8
Figure 4. Rearrangement of hydrazobenzene in 50 wt % M azobenzene with: ethanol-water a t 25" and 2.00 X A. 0.797 M HC104, u = 4.15 v/sec; B. 0.988 M HC104, v = 8.34 v/sec. (Curves have been normalized for presentation.)
rate constants much lower-of the order of 30y0 for 0.988 M acid to about 12% for 0.395 M acid. Thus, it is apparent that the "adsorption" corrections are by no means small in the azobenzene-hydrazobenzene system. The observed rate constants are summarized in Table I, along with the range of 7' values and the scan rates used in the measurements. For two of the acid concentrations (0.497 and 0.797 M ) the scan rate was
varied over a threefold range and was shown to have no effect on the calculated rate constants. This result is in accord with eq 10 and indicates that a complete decoupling of r' and (t - r') terms does occur with the scan rate to reaction rate ratios that were tested here. For each of the acidities less than 0.65 M , the rate constants calculated from the three azobenzene concentrations are in good agreement. In general, they fall within f3% of an average value. For the more acidic solutions, however, the apparent rate constants tend to increase as the azobenzene concentration decreases. I n magnitude the increase is of the order of 10-15% for a concentration change from 3.0 to 1.4 mM. Ordinarily, a slight trend-but in the opposite directionwould be expected as a result of second-order side reactions involving hydrazobenzene. 3,6 In the present case, such small kinetic deviations are completely masked by other effects, namely surface phenomena. The semiempirical method of treating the data corrects for most of these effects; however, uncertainties enter Volume YO, Number 3
March 1.966
W. M. SCHWARZ AND IRVING SHAIN
852
rate constants for the benzidine rearrangement with the potential step-linear scan method.
Conclusion
Figure 5. Dependence of the rate of the hydrazobenzene rearrangement on the square of the hydronium ion concentration: solid points, data from the step functional, controlled-potential method; open points, data obtained here with the potential step-linear scan method. E is the average rate constant a t each acidity.
into the calculations if the adsorption behavior of the azobenzene-hydrazobenzene system changes with acid concentration. When considering solutions whose acidities are as different as 0.1 and 1.0 M , such changes are probably significant and are no doubt the cause of the apparent dependence of k on concentration. An idea of the accuracy of the rate constants in Table I can be obtained from a direct comparison with rate constants determined by the step functional, controlledpotential m e t h ~ d . ~ The adsorption effects encountered here do not affect the rate constant measurements with the latter method. Figure 5 shows the dependence of li on acid concentration for these two methods. The points for both series of rate constants fall on the same smooth curve well within the expected experimental error. These results indicate that the approximations, especially the correction for the dependence of i,/C,* on concentration, are valid and that such corrections are necessary to obtain accurate
T h e Journal of Physical Chemistry
I n most aspects, the potential step-linear scan method is very similar to the closely related step functional, controlled-potential method. For both methods, the generation step is the same and so are their general applicability and the range of rate constants that can be measured with an ideal system. I n favorable cases, the accuracy of both methods is comparable, but, in general, because of the linear scan reoxidation step, rate constants measured with the potential step-linear scan method are subject to greater errors. Perhaps the greatest source of difficulty with the potential steplinear scan method arises from interfacial phenomena, such as adsorption, which interfere with accurate peak current measurements. Also, for rapid scan rates, the charging current may be an appreciable fraction of the faradaic current. On the other hand, the method is very flexible. By taking the view that the anodic peak current is a direct measure of the surface concentration of the reactive species, it is very easy to modify the theoretical working equation in a semiempirical way to correct for interfering effects. I n any case, the rate constants can still be determined rapidly from the measurements without resorting to previously calculated working curves. The application of the potential step-linear scan method to the azobenzene system illustrates the type of problem that can arise and the semiempirical method of analysis. In general, however, it is more difficult to use the semiempirical approach; seldom is it possible to determine the behavior of the system in the absence of the kinetic step. I n such cases, the potential steplinear scan method is still useful in obtaining a rapid semiquantitative estimate of the rate constant.
Acknowledgment. This work was supported by funds received from the National Science Foundation under Grant No. G 15741.
