30
W. M. SCHWARZ AND IRVING SHAIN
-
Investigation of First-Order Chemical Reactions Following Charge Transfer by a Step-Functional Controlled Potential Method. The Benzidine Rearrangement'
by W. M. Schwarz and Irving Shain Department of Cht"8tTy,
University of Wisconsin, Madison; Wisconsin
(Received March IS, 1904)
A 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 produce k an inactive species: 0 ne $ R + Z. The technique involves producing substance R a t a stationary electrode under diff usion-controlled conditions by applying a constant potential for a timed interval. During this interval, substance R diffuses into the solution and simultaneously reacts. Then the potential is suddenly switched to a value where R is reoxidized to 0. The anodic current is an indication of the amount of R which has not reacted and can be related to the rate constant k . The boundary value problem for this combined diffusion-electron transfer-kinetic system has been solved for the case of a plane electrode. The method was applied to the reduction of azobenzene to hydrazobenzene which, in turn, undergoes the benzidine rearrangement. The pseudo-first-order rate constants for the rearrangement in 50 wt. % ethanol-water were found to range from 0.58 sec. -l in 0.64 M perchloric acid to 87 set.-' in 2.5 M acid. These values were shown to be compatible with conventional kinetic data.
+
Although there have been many studies of electrode processes which involve chemical reactions coupled to a charge-transfer step, several reaction sequences which are of particular interest in organic electrochemistry have received relatively little attention. One of the more important of these is the reaction scheme2 OfneI-R
the vicinity of the electrode surface by the chemical reaction. Displacement of this equilibrium as indicated by the measured electrode potential can be related to the rate constant, k . Examples of such methods include the analysis of the shape and the half-wave potential shifts of polarographic waves3 and the potential-time behavior of chronopotentiometric curves. Unfortunately, the one-step methods are subject to several serious limitations: the charge-transfer step
k
R-Z
in which the product of the charge-transfer step, substance R , is subsequently involved in an irreversible first-order chemical reaction to produce the electroinactive species, Z. Two basic electrochemical approaches have been used for generating R and measuring its rate of disappearance. In the one-step methods, R is generated under conditions where the electrochemical equilibrium can be shifted significantly by the removal of R from The Journal of Physical Chemistry
(1) Presented in part before the Division of Physical Chemistry, 142nd National .Meeting of the American Chemical Society, Atlantic City, N. J., Sept. 1962. (2) Although the discussion emphasizes cmes in which the initial charge transfer is a reduction, extension to oxidations is obybus. (3) (a) D. M. H. Kern, J . Am. Chem. SOC.,7 5 , 2473 (1953); 7 6 , 1011 (1954); (b) J. Koutecky, Collection Czech. Chem. Commun., 2 0 , 116 (1955). (4) (a) T.R. Rosebrugh and W. L. Miller, J . Phys. Chem., 14, 816 (1910); (b) P.Delahay, C. C. Mattax, and T. Berzins, J. A m . Chem. Soc., 7 6 , 5319 (1954); (c) W.K. Snead and A. E. Remick, ibid., 7 9 , 6121 (1957); (d) A. C.Testa and W. H. Reinmuth, Anal. Chem., 3 2 , 1518 (1960).
INVESTIGATION OF ELECTRODE PROCESSES BY CONTROLLED POTENTIAL
must be reversible, the potential shifts are small and relatively insensitive to changes in the rate constant, and in most cases E o for the charge-transfer reaction in the absence of kinetic complications must be known. The two-step methods, on the other hand, are more generally applicable. During the first step, R is elec-' trochemically generated a t a controlled rate for a short period of time. Then, in a second step, the electrolysis conditions are changed so that the unreacted R can be measured-usually by converting it back into 0. Here the electrochemical reaction need be reversible only in the gross sense, i.e., some conditions must be available where R can be converted back to 0. Thus, these methods are not restricted to cases in which the electron-transfer step is reversible, nor does E ohave to be known. Chronopotentiometry with current reversalb is the only previous example of a two-step method which has been applied to this reaction scheme (eq. I). This method, however, has serious limitations when applied to systems involving rapid chemical reactions since an appreciable fraction of the current in both the generating step and the measuring step is required for charging the electrical double layer. In an attempt to minimize this source of error, Reinmuth6&has suggested the use of abnormally high depolarizer concentrations. Unfortunately, this introduces other problems, such as instrumental difficulties in rapidly switching high currents, the need for more concentrated buffer solutions of suitable speed and capacity, and the increased chance of streaming phenomena. In this work, an alternate two-step method was developed in which the potential is controlled rather than the current. In the first step of the electrolysis, 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 at some switching time, T , the potential is suddenly returned to the initial value. The resulting anodic current, which is determined by the diffusioncontrolled reoxidation of R , gives a measure of the unreacted R. This controlled potential method has the advantage that the faradaic current can be separated easily froni the charging current and thus relatively fast chemical reactions following a charge-transfer step can be studied by a simple analysis of the cathodic and anodic current-time curves.
The theoretical relationship between the current, the time of electrolysis, and the rate constant, k , for a system following eq. I can be derived by solving Fick's
31
laws of diffusion to a plane (modified by the appropriate kinetic terms) (dCo/dt )
=
D O ( ~ ~ C O / ~ X ~ ) (1)
( b C ~ / b t= ) D R ( ~ ~ C R / ~-XkCp, ')
(2)
where Co and C R are concentrations of 0 and R, t is the time, z is the distance from the electrode, and D O and D R are the diffusion coefficients of 0 and R. The electrolysis conditions resulting from the application of a single large amplitude square voltage pulse to the working electrode are expressed in the initial and boundary conditions t =
t 0
0,
> 0, X
< t < r, x
> 0:
+
:
=
CO
CO*,
CR =
(3)
0
co + Co*, C R +0
(4)
= 0 : Co = 0 , D o ( b C o / b ~ = )
- D R ( ~ C R / ~(~5 )) t
> T, 2
=
0: CR = 0, Do(bCo/bX) = - D R ( ~ C R / ~(6) ~)
i = nFADo(bCo/bz), = o
=
-
- ~ F A D R ( ~ C R / ~ Xo )(7) ,
Here, CO* is the bulk concentratism of substance 0, T is the switching time, n is the number of electrons involved in the charge transfer, F is the Faraday, and A is the area of the electrode. The solution to this problem can be obtained by a relatively straightforward application of the Laplace transform. Transforming eq. 1 and 2, solving the resulting equations in the conventional manner, and applying boundary conditions ( 3 ) and (4),one obtains two general relationships which are valid for all values of time
(bC'o/bz),= o = C o * / G -
(~CR/~X),= = O- d ( s
a
(Co),-o
+ ~ ) / D (CR),=O R
(8) (9)
where s is the transform variable and the bar signifies the Laplace transform of the concentration variable. The problelll is now to evaluating CR(Z o ) for all times and untransforniing eq. 9. The time dependence of CR(, o) for times less than r can be obtained easily by combining eq. 8 and 9 and the boundary condition ( 5 ) . The result is
-
( 5 ) (a) A . C. Testa and W. H. Reinmuth, A n a l . Chem., 32,1512 (1960) ; (b) 0 . Dracka, Collection Czech. Chem. C o m m u n . , 2 5 , 338 (1960) ; (0) W. Jaenicke and H. IIoffmann, Z . Elektrochem., 66, 803, 814 (1962); (d) H. B. Herman and A . J. Bard, Anal. Chem., 36, 510
(1964).
V o l u m e 6 9 , .\lumber
1
J a n u a r u 1966
W. M. SCHWARZ AND IRVING SHAIN
32
where $ ' 1 ( I / 2 , 1 , k t ) is a confluent hypergeometric series6with the properties
arid lFl(., 7 , 0 ) = 1
(12)
The evaluation of C R ( z 0) can be extended to all. times by combining eq. 10 and boundary condition ( 6 ) , and noting the fundamental definition
CR(*-O)=
1
+
C R (= ~o ) e-stdt
CR(==O) e-"dt
(14) Substituting this into eq. 9
+
(15)
Integrating the right-hand side of eq. 15 by parts repeatedly
(dCR/dz)z = O - C ~ * ( ~ D O / Dx R) -(8
]/Z,
a/~,
...
n!(s
+
k)T
+
. (n - ' / ~ ) k "
+ k)("
+
'*)
,--
i,/nFA
=
DR(bCR/dx),,o
=
C O * ~ D XO
(13)
The result is
( ~ C R / ~ Z ) ( ~==-OC )o * ( d D o ( ~ k ) / D R ) X + k)f& lFl(l/z, 1,
where i, and i. refer to the cathodic and anodio currents, respectively. Equation 17 describes the cathodic current-time curve for the first potential jump. As expected, the cathodic current does not depend on the kinetics of the subsequent reaction. Equation 18 represents the anodic current-time curve obtained when the potential suddenly reverts to its original value. The functional form of this expression can be seen more clearly by letting 4 represent the terms in the brackets.
X
For the case where k = 0, 4 reduces to unity-a result expected in the absence of kinetic complication^.^ In the limit as ( t - 7) approaches zero-a time infinitesimally greater than the switching time-the summation terms in become zero and the instantaneous current is directly proportional to the surface concentration of species R a t the switching time T (see eq. 10). In considering the current-time curves for finite values of t - r and k, the anodic current is still largely determined by the "surface concentration term." However, with the usual experimental values of (t - r ) and 7, it is necessary to retain several terms in the infinite series. For example, assuming that kr = 1.2 and ( t - T ) / T = 0.5, approximately 0.4% accuracy in the current is obtained using three terms of the series; 3% using two terms. More rapid convergence for the kr and ( t - r ) / r values of greatest interest, i.e., kr < 2 and (t - r ) / r < 1 , can be obtained by algebraically manipulating 4 into the form
.c
4 = e-k'/210(kT/2) + 2e-kr/2e-k(t
5 In(kr/2)
n=l
E:quation 16 can be untransformed easily, and after recognizing the expression for the series expansion of e - k t , one obtains for t < T
~ ~ ( b ~ ~ =/ -bi , / ~ n F)A , == -~C o * d D o / T t (17)
lk"-".
..
-
7)
X
h n l & j h l .. . . dh,
[k(t -
(20) T)l"
where In(kt/2)represents the modified Bessel functions.* Using this expression with the above conditions, convergence to the 0.2% error level is obtained using- only one term of the infinite series.
and fort > r D R ( ~ c R / ~ X ) , = O=
i,/nFA
=
+c
[e-x'lFl(l/z, 1 , ~ c r )
n=l
-
c ~ * Z / D ~ / T ( ~7)
e-kf[(t - r)k]" 1Fl(n n!
x
+
- C o * d D o / T t (18)
The Journal of Physical Chemistry
'(6) (a) A. Erdelyi, "Higher Transcendental Functions," Vol. I, MoGraw-Hill Book Co., Lnc., New York, N. Y.,1953, Chapter VI;
(b) L. J. Slater, "Confluent Hypergeometric Functions," Cambridge University Press, London, 1960; (c) I. N. Sneddon, "Special Functions of Mathematical Physics and Chemistry," Interscience Publishers, Inc., New York, N. Y., 1956, Chapter 11. (7) (a) T. Kambara, Bull. C h a . SOC.Japan, 27, 527 (1954); (b) J. Weber, Collection Czech. Chem. Commun., 24, 1770 (1959). (8) See ref. 6c. p. 113.
33
INVESTIGAWON OF ELECTRODE PROCESSES BY CONTROLLED POTENTIAL
Although eq. 18 could be used to calculate the rate constant directly from the measured anodic current, in practice it is more convenient to eliminate the dependence on the initial depolarizer concentration, the electrode area, and the diffusion coefficient by simultaneously ineasuring the cathodic current, i,, and working with the dimensionless ratio aa/ic.A further simplification results if the anodic current measured at a time tl is always coupled with the cathodic current measured at the time tl - 7 as shown in Figure 1. The final working equation for ia/icis obtained by dividing eq. 18 by eq. 17 and introducing the restrictions on the time of measurement
In this case, the rate constants can be obtained directly from working curves constructed from eq. 21 in which the current ratios ia/icare plotted against the dimensionless parameter ICT for various values of the time ratio, ( t - T ) / T . Several of these working curves are shown in Figure 2.
Figure 1. Typical cathodic-anodic current-time curves for the azobenzene-hydrazobenzene system, 2.0 X 10-8 M azobenzene-50 wt. % ethanol-water; T = 0.060 sec. : solid line 1.588 M HClOd; half-life of the benzidine rearrangement = T ; dashed line 0.40 M HCIOa; half-life of the benzidine rearrangement >> T .
"
Kinetic and Electrochemical Characteristics of the Azobenzene System
+
2H'
+
2e-
n
0.8
0.7
In order to test the theoretical calculations, the reduction of azobenzene (eq. 11) was investigated. QN=NQ
O
0.)
0.6
.-c
0.5
-'. A
I-\
ic
0.4
0.3
0.2
Here the initial compound, azobenzene, is reduced to hydrazobenzene which, in strong acid, undergoes the benzidine rearrangement. The reduction of azobenzene has been studied polarographicallyg under conditions where the rate of the benzidine rearrangement is negligible. Although there are some inconsistencies among the various investigations, the results all indicate that in the pH range 1-13 both cis- and trans-azobenzene undergo a pHdependent , two-electron reduction as shown in eq. , ~reverse ~ ~ ~process, * ~ the IIa. In several c ~ s ~ sthe oxidation of hydrazobenzene to azobenzene, was also
kT
Figure 2. Theoretical working curves for the step-functional controlled potential method; charge transfer a t a plane electrode followed by an irreversible first-order rate process.
reported and shown to occur a t potentials close to those corresponding to the reductions. The rearrangement of hydrazobenzene (eq. IIb) (9) (a) A. Foffani and M.Fragiacomo, Ric. Sca.. 22, 139 (1952);(b) P. J. Hillson and P . P. Birnbaum, Trans. Faraday Soe., 48, 478 (1952); (c) C. R. Castor and J. H . Saylor, J. A m . Chem. Soe.. 75, 1427 (1953); (d) S.Wawsonek and J. D.Fredrickson, ibid., 77,3985, 3988 (1955); (e) B. Nygard, Arkic Kemi, 20, 163 (1963).
Volume 69,Xirmber 1
January 1966
34
has been the subject of numerous kinetic investigations’o in which spectrophotometric,” potentiometric,l* or titrationI3 techniques were used to follow the rate of the reaction. These studies show that in dilute acid (> 7 : solid lines, theoretical; points, experimental for 2.0 X M . azobenzene in 50 wt. yo ethanol-water; 1, 0.10 M HClO,, T = 1.2sec.; 2,0.40MHClO4, T = 0.30sec.; 3, 0.40 M HCI04, T = 0.060 sec.; 4, 1.0 M HC104, 7 = 0.012 sec.; 5, 1.0 M HCIO,, T = 0.0040 sec.
the theoretical behavior for diff usion-controlled currents. The theoretical curves were calculated from eq. 21 assuming 6 to be unity. These results showed that for all values of T , the anodic currents were essentially controlled by the diffusion of hydrazobenzene to the electrode surface. For each value of T , the three azobenzene concentrations that were tested gave practically identical results. More important, the agreement between the experimental points and theory was reasonably good for all values of T . In most cases, slight deviations of in/& from theory could be ascribed to deviations in i, rather than i,. The tendency for the experiinental points to fall uniformly below the theoretical curves for the longer values of 7 was probably the result of stirring in the hydrazobenzene diffusion layer initiated by momentary movement of the mercury drop at the switching time.2* The tendency for the experimental values to be higher than theory at the shortest switching tiines probably is related to the adsorption phenomena involving hydrazobenzerie discussed above. Reacting Systems. In the second set of experiments, cathodic--anodic current-time curves were obtained in a series of 14 acid concentrations ranging from 0.65 to 2.6 M. For a number of these acid con-
0.60 1.oo 2.00
Potentio-
atat
k, eec. -1
circuit
11.7 11.4 11.2
0.570 0.555 0.616
Bb
0.694
1.05
1.OO
11.4
0.775
Bb
0.793 0.793 0.793
0.95
0.60 1.oo 2.00
11.2 10.6 10.4
1.02 1.04 1.08
A
0.886
0.86
1.oo
11.4
1.61
Bb
0,989 0.989 0.989
0.79
0.60 1.oo 2.00
10.6 10.7 10.5
2.16 2.32 2.40
A
1.090
0.72
1.oo
10.3
2.96
Bb
1.185 1 185 1.185
0.66
0.60 1.OO 2.00
11.0 10.9 10.4
4.57 4.45 4.20
A
1.385 1.390
0.56
1.oo 2.00
11.2 11.o
6.60 7.60
Bb B
1.588 1.588 1.588 1.582
0.46
0.60 1.00 2.00 2 .oo
11.2 10.8 10.1 11.1
11.3 11.2 10.9 12.6
1.775
0.37
2.00
11
.o
19.4
1.979 1.979 1.979 1.955 1.98 1.98
0.28
0.60 1.OO 2.00 2.00 1.oo 2.00
11.6 11.2 10.5 10.9 11.5 10.8
29.2 28.6 27.8 29 1 29.3 29.9
A
2.150
0.20
2.00
10.8
45.0
B
2.36 2.36 2.345
0.12
1.00 2.00 2.00
11.8 11.8 10.7
63.0 65.5 61.2
B
2.00
10.0
86.5
B
T.0.004
1
Asobensene Slope/ conon. concn., i, us. 1/ x lo', moles/l. curve8
2,545
A
B B
B
The values of HOin this solvent were obtained by applying a small solvent correction, as indicated in the work of E. A. Braude and E. S. Stern, J. ChenL. SOC.,1976 (1948), to the H O values given by 11. P. N. Satchell, ibid., 2878 (1957), for perchloric acid in 67 wt. % ethanol-water. I n this case, the potentiostatic circuit contained no booster amplifier.
centrations (see Table I), three different azobenzene concentrations (0.6, 1.0, and 2.0 m M ) were investi(24) I. Shain and K. J. Martin, J . Phys. Chem., 6 5 , 254 (1961). Note t h a t in the present case the bulk concentration of hydrasobenzene is zero, so that stirring causes transport of material away from the electrode and results in low anodic currents.
V o l u m e 69, N u m b e r 1
J a n u a r y 1B65
38
-
gated; in others, only the most concentrated solution was used. As with the nonreacting systems, T-values were in the range from 1.6 to 0.005 sec., depending on the acid concentration. Cathodic Current-Time Curves. In addition to their use in the analysis of the rate data, the cathodic current-time curves also served as a inethod for detecting such side reactions as the disproportionation of hydrazobenzene nientioned previously. 1 3 f ~ 1l 4 If such side reactions were present , the regeneration of depolarizer by the chemical step would result in an enhancement of the cathodic current. As before, the cathodic current-time curves were analyzed according to eq. 17, which is still applicable. 111 general, the results were very similar to those obtained under conditions where the rate of the chemical reaction was negligible. The plots of i , l C ~ *us. l/’z/t for each azobenzerie concentration yielded straight lines for all 14 acid concentrations. The slope for each individual plot, as shown in Table I, deviated less than .5yo froin the average-a greater degree of scatter, however, than obtained previously. The iiiagnitudes of the slopes tended to be 2 to 4% greater thLan those observed at the same t h e under comparable conditions in the less acidic norireacting solutions. This difference in slope would result if 4 to 8% of the initially generated hydrazobenzene were involved in a disproportionation side reaction. On the other hand, a trend toward slightly greater deviations a t lower azobenzene concentrations indicates that other factors are also present, and the estimate of the amount of side reaction based on a siinple second-order disproportionation is probably too high. 111 order to exaiiiine these side reactions further, several large-scale controlled potential coulometric electrolyses were carried out on 1 niM azobenzene solutions a t acid coiiceiitratioris ranging froni 0.5 to 4.0 Jl . The observed current efficiencies ranged from 100yo (0.5 -11) to 95% (4.0 AI), i.e., in the more acid solutions, the quantity of electricity required for coinplete reduction was about 5% higher than expected for the stoichionietric reduction of the azobenzene present. Although these results cannot be compared directly with the poteritiostatic current-time curves because of the differences in the mass-transfer processes, the iiiagiiitude of the side reaction is similar in the two cases arid probably does not exceed the 3 to 5% previously reported. Anodac Current-Tzme Curves. The shape of the curves in Figure 2 indicates that the maximum accuracy in the kinetic iiieasurenieiits is achieved for switching tiines, T, which are the same order of magnitude as the half-life of the chemical reaction. Thus, the switching T h e Journal of Phgsical Chemistrv
W. M. SCHWARZ AND IRVING SHAIN
times used here were selected to be approximately 2/3, 1, and times the half-life of the rearrangement reaction for each azobenzene solution at each acidity. This permitted a significant amount of reaction to take place, but at the same time enough hydrazobenzene reiiiained so that the anodic currents could be iiieasured with reasonable accuracy. Froin any individual cathodic-anodic current-time curve, several estimates of the kinetic parameter were obtained. I n principle, i, and i, could be measured a t any arbitrarily selected value of t and (t - T) as is shown in Figure 1. However, the analysis was greatly siniplified by making nieasurements along the curves at values o f t such that the ratio ( t - T ) ’ r corresponded to values used in constructing the working curves, ie., (t - T ) / T equal to 0.1, 0.2, 0.3, 0.4, and 0.5. The measured current ratios, i,’ zc, mere then used with the working curves (Figure 2 ) to obtain the kinetic parameter, k r . Thus, for a given cathodic-anodic current-time curve (at a particular value of 7 ) five estimates of k~ were obtained. Siinilarly, five more nieasuremeiits were made on the current-time curves obtained for each of the other two values of r used. For each azobenzene solution, all of these data were combined to obtain the best possible estimate of the rate constant. The values of k r were converted to k(t - 7) so that data obtained with the three values of T could be combined in a form which gives equal weight to each value. Then plots were made of k ( t - r ) us. ( t - T ) , and the rate constant was calculated directly from the slope.25 I n each case, the experiiiierital data determined a straight line which passed through the origin, as required by theory. Several of these plots are shown in Figures 4 and 5. A summary of the kinetic data is given in Table I. These results were obtained (in no particular order) over a period of several months, using two different potentiostats. For any given acidity, the rate constants calculated for the three different azobenzene concentrations are in good agreement. Generally, they fall within +5y0 of an average value. The observed values of k show no systematic trend with azoberizene concentration, i e . , for soiiie acidities (0.8, 1.0 M),k increased slightly with concentration; for others (1.2, 1.6 M),IC decreased. Thus, it was riot possible to detect any effect due to the presence of second-order side reactions. This is riot surprising, however, in view of the magnitude of the scatter in the experimental points (Figures 4 and 5 ) , particularly for 0.6 i i d l solutions. IJrider these circumstances, ( 2 5 ) Working curves calculated directly in terms of k ( t - r ) could not be used conveniently because the plots for various values of the ratio (t - r ) / r intersect
39
INVESTIGATION OF ELECTRODE PROCESSES BY CONTROLLED POTENTIAL
(t - 1 ) ( m a s . )
Figure 4. Rearrangement of hydrazobenzene in 50 wt. % ethanol-water; 1.0 X 10-8 M azobenene with: 1, 1.588 M HClO,; 2, 1.185 M HClOI; 3, 0.989 M HC10,; 0 , T sz ' / d l / , ; d , 7 t 1 / * ; 0, % 2/stl/,.
( 1 - 7 ) (msec)
Figure 5. Rearrangement of hydrazobenzene in 50 wt. % ethanol-water; 2.0 X M azobenzene with: 1, 2.545 M HC10,; 2, 2.150 M HClOr; 3, 1.979 M HClO,; 0 , T 3 / ' 2 t ~ / , ;8 , T t l / * ; 0, T = 2 / d 1 / 2 .
-
the observed values of IC accurately reflect the precision with which rate constants can be measured by this met hod. An estimate of the accuracy of these kinetic measurements can be made easily. If it is assumed that the major errors are caused by phenomena which interfere with the niass-transfer processes, the ratio of the observed to the true value of &./&should be approximately the sanie for both the nonreactirig and the reacting systems. Corrected values of iR/& for the reacting system then can be calculated by dividing the observed values of i R / &by the above ratio obtained from the
Figure 6. Dependence of the rate of hydrazobenzene rearrangement on the Hammett acidity function. h is the average value of k at each acidity.
analysis of the corresponding nonreacting system (Figure 3). The rate constants estimated in this way show that the measured values of k given in Table I could not be more than 25% too high for the slow reactions, nor niore than 15% too low for the fast reactions. Another source of error, the presence of 3-5% second-order side reaction, would lead to sinall positive errors in the observed rate constants. The dependence of the rate constants on acidity can be seen in Figure 6 where the log of the average rate constant, &, is plotted against the Hammett acidity function, Ho. The dashed line indicates the magnitude and direction of the error limits discussed above. (The two low acidity points show a positive deviation which is due in part to the spherical nature of the hanging mercury drop electrode.) The experimental points define a reasonable straight line with a slope of 2.2, a slightly higher slope than the theoretical value of 2.0. These results are similar to those found by Bunton, Ingold, and MhalaI3g for the perchloric acid catalyzed rearrangement of hydrazobenzene in 60 vol. % dioxane-water. These workers observed a slope of 2.6 for acidities in the range of 0.05 to 1.0 M . The deviation of these slopes from the theoretical value has been discussed a t great length by Banthorpe and ~ o - w o r l i e r s 'and ~ ~ has been shown to be caused primarily by a positive kinetic salt effect. The better agreement with theory observed in the present case is probably because of a smaller salt effect in the more polar ethanol-water solvent. With the exception of the work in dioxane-water, classical kinetic iiieasurenients on the hydrazobenzene Volume 69,Number I
January 1966
40
-
system have been limited to acid concentrations less than 0.1 M . As a result, a direct comparison cannot be made between the electrochemical and the classical rate measurements. On the other hand, some checks are possible since the electrochemical measurements can be extrapolated to much lower acidities by using the linear relationship between log K and Ho. For instance, a rate constant of 1.4 X lop3 sec.-' was estimated from the data of Croce and G e t t l e ~ ?for ~~ the rearrangement reaction in 50 wt. yo ethanolwater a t 2.5' catalyzed by 0.093 JI acid (Ho = 2.16). Extrapolation of the data of Figure 6 to this acidity yields a rate constant of 2.4 X set.-'. A second extrapolation following the dashed line (points of maximum error limit) gives a value of 1.0 X sea.-'. The rate constant determined by classical methods falls in between the two extrapolated values indicating general agreement between the step-functional controlled potential measurements and the classical results.
Conclusion 'The rate constants listed in Table I indicate the range of applicability of the step-functional controlled potential method to the azobenzene system. Adsorption effects involving both azobenzene and hydrazobenzene limited measurements to acidities in which the half-life was greater than about 5 msec. Severtheless, this represents a significant extension of the kinetic data for this system to a range in which the more classical
The Jownal of Physical Chemistry
w.hl. SCHWARZ .4SD I R V I N G SHhIN
methods cannot be used. The method was not extended to acidities where the half-life was more than about 1.5 sec., because the assumption of electrode planarity breaks down. The general applicability of the method is much greater than illustrated here. Cnder favorable conditions, reactions with half-lives of the order of 20 bsec. could probably be measured although more elaborate potentiostats than those used here would be required. In addition, a number of problems concerning cell design, I R drop, and the effect of the electrical double layer on the kinetic process would have to be considered. Unfortunately, in most cases, measurement of these very fast rates is already prohibited by properties of the system such as adsorption. At the other limit, slow reactions can be investigated as long as diffusion is the only means of mass transfer. Thus, a half-life of the order of 30 sec. probably represents the slowest reaction which could be studied conveniently, provided, of course, that a plane electrode is used, or that deviations from planarity could be accounted for. Because of this wide range of applicability, it is expected that the step-functional controlled potential method will be extremely useful in the investigation of other systems in which a chemical reaction follows the initial charge-transfer step. Acknowledgment. This work was supported by funds received from the Sational Science Foundation under Grant S o . G 15741.
