Adsorption Effects in Stationary Electrode Polarography with a Chemical Reaction Following Charge Transfer Robert H. Wopschall and Irving Shain Chemistry Department, Unirrersity of Wisconsin, Madison, Wis. The effect of adsorption of reactant i n stationary electrode polarography (including cyclic scan experiments) was evaluated in a kinetic case involving a chemical reaction after the charge transfer. The system which was investigated was the reduction of azobenzene to hydrarobenzene and the subsequent benzidine rearrangement. The absorption of the azobenzene prevented the direct application of the theory previously presented [Anal. Chem., 36, 706(1964)], and therefore the boundary value problem including adsorption was solved numerically in this work. Direct correlation of the theory with the experimental results was not completely successful, but a semi-empirical method was developed which appears to be generally applicable. Rate constants determined for the benzidine rearrangement were in good agreement with previous results.
FORCASES in which a homogeneous chemical reaction is coupled with a charge transfer reaction, stationary electrode polarography provides a powerful method for investigating the kinetic parameters (1-5). However, mechanisms involving a chemical reaction coupled with the charge transfer reaction where either product or reactant is adsorbed have not been discussed previously. When adsorption of an electroactive species and a chemical reaction are both coupled with charge transfer, numerous effects can be observed, depending on the type of chemical reaction (preceding, succeeding, etc.), which species is adsorbed, and the strength and type of adsorption. This work has therefore emphasized an example of experimental importance: a reversible charge transfer followed by an irreversible chemical reaction with weak adsorption of the reactant (Langmuii isotherm): Osoln
0
*
+ ne-
Oads
(1)
R
(11)
k
R+Z
(111)
Qualitatively, the adsorption of reactant 0 enhances the cathodic peak and causes more product R to be produced than in the absence of adsorption. As R is lost because of Reaction 111 and also by diffusion, the increase in the anodic peak is relatively less than the cathodic peak. This causes the ratio of the anodic peak current to the cathodic peak current to be smaller than predicted from the work of Nicholson and Shain (3). Thus, direct application of the theory previously presented without consideration of the effect of adsorption leads to a high value for the rate constant of the irreversible chemical reaction. (1) J. M. Saveant and E. Vianello, Electrochim. Acta, 8, 905 (1963).
(2) J. M. Saveant and E. Vianello, in "Advances in Polarography," I. S. Longmuir, Ed.; Vol. I, Pergamon Press, New York, 1960, p.
The azobenzene-hydrazobenzene system, which undergoes the benzidine rearrangement in acid solution, is an example of a succeeding chemical reaction where the reactant is known to be adsorbed. This system was selected for study to determine the applicability of adsorption models to systems with coupled chemical reactions. Reliable kinetic data has been obtained previously using electroanalytical methods less affected by adsorption of reactant (6), thus providing a means of evaluating this work. BOUNDARY VALUE PROBLEM
For an irreversible succeeding chemical reaction with adsorption of the reactant (Reactions 1-111) the boundary value problem is identical to that presented previously (7), except that the Fick's Law diffusion equation for substance R must be modified to include the kinetic term, and in addition, it is assumed that substance R is not adsorbed: x o i a t = Do(a2co,iax2)
(1)
bCR/dt = DR(b2CR/dX2)- kCR
(2)
t = 0, x
2 0 : Co
= Co", C R = CR* ( a O )
(34
ro = ro*,rR= rR*( - 0 ) t
t
> 0,
2 0 ) x -+ x = 0:
co:
CojCR
Do(aco/ax) -
CO
=
(3b)
* C O " , CR * 0
exp[(nF/RT) ( E
aro/at =
(4)
- E")]
(5)
-DDR(acR/ax)
(6)
ro = roSco/(Ko +co)
(7)
Here, k is the formal first-order (or pseudo first-order) rate constant for the succeeding chemical reaction, and all other terms are the same as defined previously (7). Substance Z is not electroactive in the potential range of interest, and therefore the concentration term for this species does not appear in the boundary value problem. The potential function was defined previously in Equation 11 of Reference 7. Using the same procedures as before, Equations 1 through 4 can be converted to integral equations relating the surface concentrations and the fluxes: Co(O, t )
=
CO* - (I/&)
J'' 0
[fo(~)/dt
- r]dr
(8)
Changing variables as before, converting to dimensionless form, and combining Equations 5 through 9 gives:
367.
(3) R. S . Nicholson and I. Shain, ANAL.CHEM., 36, 706 (1964). (4) R. S. Nicholson and I. Shain, Ibid., 37, 178 (1963). (5) D. S.Polcyn and I. Shain, Ibid.,38, 376 (1966).
(6) W. M. Schwarz and I. Shain, J. Phys. Chem., 69, 30 (1965);
70, 845 (1966). (7) R. H. Wopschall and I. Shain, ANAL.CHEht., 39, 1514 (1967) VOL. 39, NO. 13, NOVEMBER 1967
0
1535
cause of this increased complexity, it is beyond the scope of this work to include all of the results obtained. Therefore, the discussion was restricted to a qualitative description of the influence of the two experimental variables, scan rate and concentration, along with the effect of varying the rate constant for the succeeding chemical reaction. Because the adsorption is weak, the effects of the free energy of adsorption and Tos have only a minor influence and therefore are only mentioned briefly. Effect of Rate of Chemical Reaction. For a succeeding chemical reaction in the absence of adsorption, it has been shown (3) that the cathodic peak increases about 10% and changes slightly in shape as the value of k / a increases. For the case where the reactant is adsorbed, one observes an analogous increase of the cathodic current function as the kinetic parameter k/a increases. However, the amount of this increase is also dependent on the values of the adsorption parameters, and is less than the 10% observed for cases without adsorption. Typical results are shown in Figure 1, illustrating the change in the stationary electrode polarograms with the rate constant k for representative values of the adsorption constants. While the cathodic current function is somewhat dependent on the scan rate, it is primarily dependent on the adsorption of reactant as shown by the increased cathodic peak current function and cathodic peak symmetry. On the other hand, even in the presence of adsorption of the reactant, the anodic current function is still very much dependent on the rate constant k. For low values of the kinetic parameter k/a, the anodic scan is the same as for the case without the kinetic complication (Figure 1, curve A ) ,
0.8
0.6
z
s
G z
0.4
3
LL
5w
0.2
a: LT
3
0.0
-0.2
0.2
0
- 0.1 (E-E,,*)n,
-0.2
v
Figure 1. Stationary electrode polarograms for a reversible charge transfer coupled to an irreversible succeeding chemical reaction, with the reactant weakly adsorbed-variation with chemical rate constant Adsorption parameters PO and vo are held constant, and thus, u is constant. PO = 58.0; 90 = 0.23. k / u equals: A , 0.001; B,O.01; C, 0.1; D,1.0; E,10.0
0.
z
0 Io
0.
z 3
LL
where
I-
(12)
z w a. E LT
and =
co*z/rrDo~4ro~d~
3 0
(13)
The theoretical polarograms are determined by these two adsorption parameters PO and PO, and, just as in the case without adsorption, by the kinetic parameter kla. Using the numerical methods (8), Equations 10 and 11 were solved t o obtain theoretical stationary electrode polarograms for selected values of these parameters. For correlation of the theoretical stationary electrode polarograms, the current function was defined as $(at) = i / n F A C o * d G , as described previously (7). THEORETICAL CORRELATIONS
While the adsorption of reactant and the irreversible chemical reaction following charge transfer are each reasonably easy to describe when they occur alone, when combined as in eactions 1-111 the behavior becomes very complex. BeB
ANALYTICAL CHEMISTRY
C
-0.
I
100
I
0
I
-100
I
-200
(E-E,12)n, mV Figure 2. Stationary electrode polarograms for a reversible charge transfer coupled to an irreversible succeeding chemical reaction, with the reactant adsorbed-variation of scan rate
k = 1.0 see-', P o / d i
= 2.0 sec1'2, and o .& = 2.0 see:1112.a (=nFc/RT) = A , 1 sec-1; B, 10 sec-'; C, 100
see -1
1.0 -
0.6 -
0.2 -2
-I
0
I
log(v”2)
Figure 3. Variation of ratio of anodic to cathodic peak current, ( i P ) & J C with , scan rate for stationary electrode polarogram of a succeeding chemical reaction with reactant adsorbed Pop0 = 13.0, Ex = 240 mV. A, chemical reaction negligible, k = 0;
B, k = 0.13 sec-’; C, k = 1.3 sec-1; D,k = 13 sec-1; E, k = 130 sec-l
As the rate constant increases, the concentration of R is diminished and the anodic current decreases correspondingly (curves B, C, and D), and finally disappears entirely (curve E). If the rate constant is increased still further, the cathodic peak reaches its maximum and shifts to less cathodic potentials with no further change in shape, just as in the case without adsorption. Effect of Scan Rate. With a change in the scan rate, all the parameters (PO, pO, and kia) are different, causing a marked change in both anodic and cathodic peaks, and in the influence of the rate constant k . Thus, the overall behavior is significantly different from either the succeeding chemical reaction case or the adsorption case alone. At low scan rates the adsorption effects become negligible (7), and the effects of the succeeding chemical reaction become dominant. Under these conditions, the behavior is the same as the succeeding chemical reaction without adsorption (3), where the anodic peak is absent (Figure 2, curve A ) . As the scan rate is increased, two changes occur. First, because of the adsorption of reactant, there is an increase in the cathodic current function. This can also cause an increase, although smaller, in the anodic current function because of the increased amount of oxidizable material present. Second, the anodic current function must increase with scan rate because of the decreased influence of the succeeding chemical reaction. Therefore, as the scan rate is increased, both anodic and cathodic current functions must increase, as shown in Figure 2. Even though an increased scan rate causes both anodic and cathodic current functions to increase, the anodic current function is primarily dependent on the rate of the succeeding chemical reaction, while the cathodic current function is almost entirely dependent on the adsorption isotherm. Thus, because the adsorption process and chemical reactions are independent, if the effects become important in different ranges of scan rate, it is possible for the ratio of the peak currents, (iu)u,i(in)c, to be dependent on kinetics at one scan rate and adsorption at another. This is shown in is given as a function of Figure 3 where the ratio (ip)u/(ip)c scan rate for several relative values of the rate constant k , with all else constant. If the rate constant is zero (Le,, no
-0.4
I100
0
(E-E,,Jn,
-100
-200
mV
Figure 4. Effect of variation of bulk comcentration on stationary electrode polarogram for the succeeding chemical reaction with reactant adsorbed k/a = 0.05, PO = 5.0. (00 (which is proportional to CO*) = A , 0.1; B, O S ; C, 5.0 and greater
succeeding chemical reaction), the effects are entirely due to adsorption. At slow scan rates, the case reduces to the uncomplicated reversible charge transfer and the anodiccathodic peak current ratio is unity. As the scan rate is increased, the cathodic peak current function increases faster than the anodic, and the ratio (ip)a:(ip)c decreases. Ultimately the peak current ratio levels off at a constant value which is dependent on the switching potential Ex. This is because only adsorbed material contributes significantly to the current at high scan rates, and as EA becomes more cathodic, more of the reduced material diffuses away from the electrode and is not oxidized on the subsequent anodic scan. The other limiting case occurs when k is very Jarge (curve E ) , where an anodic peak is not observed until the scan rate is so high only the material initially adsorbed contributes to the current, Therefore, the peak current ratio approaches the high scan rate value of curve A as a limit. For larger values of the rate constant, curve E is shifted to higher scan rates. Thus, the peak current ratio can never be higher than curve A . For intermediate values of the rate constant (curves B, C, and 0 ) the peak current ratio increases in the range of scan rates where the kinetics of the succeeding reaction dominate the characteristics, and decreases in the range of scar, rates where the adsorption becomes important. The overall behavior depends, of course, on the relative values of the kinetic and adsorption parameters, and the curves are shifted along the scan rate axis accordingly. Effect of Concentration. In general, the theoretical results indicate that an increase in the bulk concentration causes the adsorption of reactant to have relatively less influence on (8) R. H. Wopschall, Ph.D. thesis, University of Wisconsin, Madison, Wis., 1967. VOL. 39, NO. 13, NOVEMBER 1967
0
1537
expected, an increase in the scan rate, or a, causes the maximum value of (ip),/(ip)cto decrease significantly. For application of the above theory to an experimental case, prior knowledge would be required of the adsorption parameters, the switching potential, and the scan rate used. With this information, it would be possible to obtain theoretical polarograms and determine the ratio of (ip)a/(ip)cfor various values of the kinetic parameter k/a at particular values of a. Construction of a plot similar to Figure 5 would then permit the calculation of the rate constant from experimental values of (iP),/(i&. -4
-2
-3
-I
log (k/a)
Figure 5. Variation of ratio of anodic to cathodic peak current, (ip)a/(ip)c, with rate constant (Le., k/u) for the succeeding chemical reaction with reactant adsorbed Poqo = 13.0. A, limiting behavior at low scan rates where adsorption is negligible; B, a = 125 sec-I; C,a = 635 sec-1; D , a = 6460 8ec-1
the stationary electrode polarograms. In addition, the anodic current function is less influenced by a concentration change than the cathodic current function. These results are summarized in Figure 4 for a typical value of the kinetic parameter kia, where curve C (high concentrations) corresponds very closely to the case in the absence of adsorption. As the bulk concentration is decreased (Curve B), the cathodic current function increases because of adsorbed reactant. Simultaneously the anodic peak broadens, the amount being dependent on the free energy of adsorption AGO. In general, this case implies that AGO is large enough so that significant adsorption occurs but not SO large as to cause a separate postpeak. Thus, the broadening of the anodic peak, also shown in Figure 1, is due to the “adsorption” part of the current appearing slightly cathodic of the “diffusion” part of the current. A further decrease in the concentration (curve A ) causes a significant increase in the cathodic peak current function. The anodic current function is also increased slightIy and the peak shifts further cathodically as the reduction of adsorbed material becomes more important in the overall process. Still further decrease in the concentration causes both the anodic and cathodic current functions to increase markedly. Determination of the Rate Constant. To determine k from experimental results it is necessary to have a plot analogous to that of Figure 12 in Reference 3, where (ip)a/(ip)c is plotted as a function of k . For the case including adsorpis dependent on, rate constant, adtion, the ratio (ip)a/(ip)c sorption parameters, and switching potential. The interdependence of these parameters makes it impractical to present solutions for all values of the parameters in this work, and therefore only a single example is presented. For the kinetic case without adsorption, the ratio (ip),/(ip)c decreases from unity for low values of k/a to a lower limit which becomes undefined as the anodic peak disappears. With the adsorption of reactant present, the cathodic peak is enhanced more than the anodic peak and the upper limit of (ip)a/(ip)c is generally less than unity, the actual value being dependent on the adsorption parameters and the switching potential. Illustrative plots of (ip)a/(ip)c as a function of k/u are presented in Figure 5 for several scan rates and representative values of the isotherm parameters. As 1538
e
ANALYTICAL CHEMISTRY
EXPERIMER-TAL
All voltammetric experiments were carried out in a threeelectrode cell, using a controlled potential circuit based on operational amplifiers. The instrumentation and procedures were essentially the same as described previously (9). transAzobenzene (Eastman White Label) was recrystallized three times from hot 95% ethanol and dried over phosphorus pentoxide in a vacuum. The melting point was 68.0”68.5” C. Stock solutions of azobenzene (4mM) and perchloric acid (4M) were made up in 50 wt. ethanol-water, and were diluted as necessary with the same solvent. EXPERIMENTAL VERIFICATION The Reduction Mechanism for Azobenzene. The extensive previous work on the kinetic and electrochemical characteristics of the azobenzene system has been summarized by Schwarz and Shain (6). Polarographic studies, conducted under conditions where the rate of the benzidine rearrangement is low, indicate that both cis- and trans-azobenzene undergo a pH dependent, two-electron reduction. However, a recent polarographic and coulometric study in dimethylformamide (10) reported two one-electron reductions separated by 480 mV. Both the one- and two-electron products were easily air-oxidized to the original azobenzene. The intermediate was reported to be a highly colored dark red species exhibiting ESR activity. Using stationary electrode polarography, the reduction of azobenzene was studied in ethanol-water to detect any evidence of two one-electron charge transfers rather than the previously reported two-electron charge transfer. Experiments were carried out over a range of scan rates from 80 mV/second to 100 V/second using the cyclic scan technique under conditions where the adsorption of the reactant was relatively negligible (high concentration) and where the succeeding chemical reaction could be ignored (weakly acid solutions). Within the accuracy of the measurements, which were made from photographs of oscilloscope displays, the azobenzene-hydrazobenzene system appeared to be reversible. This has also been confirmed by more accurate measurements made by Nicholson and coworkers (11) who performed cyclic. experiments using a potentiometric recorder. For example, they obtained a value of 29.0 mV for Ep - Epizfor the reduction of l0-3M azobenzene in 0.1M HCIOl and 50 wt ethanol at a scan rate of 140 mV/second, compared with the theoretical value of 29.2 mV for a twoelectron reversible reaction. Thus, it was not necessary to consider the possibility of a stepwise mechanism in this work. (9) R. H. Wopschall and I. Shain, ASAL.CHEM., 39, 1527 (1967). (10) G. H. Aylward, J. L. Garnett, and J. H. Sharp, Chen?. Corn???., 137 (1966). (11) R. S. Nicholson, Michigan State Univ., East Lansing, Mich., unpublished work, 1967.
Figure 6. Stationary electrode polarograms of azobenzene at several concentrations Scan rate
=
19.2 V/sec, HC104 = 1M. Azobenzene B, 1.OmM; C,0.4mM
= A , 2.0mM;
---
Theoretical for a reversible charge transfer
Adsorption of Azobenzene. Previous work has noted the presence of adsorbed azobenzene on mercury electrodes (6,12, 13), but quantitative measurements were not reported. To verify these observations and to determine the adsorption parameters, stationary electrode polarograms were obtained for the cathodic scan over a range of azobenzene concentrations and scan rates in 1M HC10,. Since the cathodic scan is relatively unaffected by the succeeding chemical reaction, the effects which were observed were primarily due to adsorbed azobenzene. Even though azobenzene is adsorbed readily, there is no evidence for significant adsorption of hydrazobenzene. It is reasonable that azobenzene, which can be made completely planar and therefore can be adsorbed flat on the electrode surface, may be adsorbed to a significantly higher extent than hydrazobenzene, which cannot assume a planar configuration. Because of this, it should be reasonable to ignore any small amount of adsorbed hydrazobenzene which might exist. Using the diagnostic criteria suggested previously (7), the polarograms were first checked for appreciable symmetry around the peak potential. Typical polarograms are shown in Figure 6 for several concentrations. Curve A , for a concentration of 2mM shows some symmetry, while in curves B and C (1.0 and 0.4mM azobenzene) the symmetry is quite apparent. This increased symmetry at low concentrations is indicative of adsorption (see Figure 2 in Reference 7). Next, a plot of ( i , ) c : C ~ * ~asaa function of scan rate for several concentrations was constructed (Figure 7). Comparison with Figure 3 of Reference 7 confirmed that the behavior of azobenzene is what would be expected for the weak adsorption of reactant. The largest deviations from a constant value for (i,),/CO*di are observed at lower concentrations, as predicted for this case. (12) P. Delahay, S . Oka, and H. Matsuda, J . Am. Chem. SOC., 82, 329 (1960). (13) C. A. Stredi and W . D. Cooke, AKAL.CHEW,26, 963(1954).
To calculate the theoretical polarograms, from which the relations necessary for the determination of the rate constant can be obtained, the values of DO,KO,and roSmustbe known. Then, for each experimental value of CO”, u, and EA,the theoretical polarograms for several values of k can be calculated. To determine the adsorption parameter PO,from which KO can be obtained, the normal approach would involve construction of a plot of the peak current function as a function of concentration, similar to Figure 5 of Reference 7, and from this the ratio of the limiting low concentration and limiting high concentration peak current functions could be obtained. Then, from this ratio, Figure 21 of Reference 7 can be used to determine PO. In this work, however, the ratio was determined by an alternative procedure. The high concentration value was estimated from Figure 7, although it would have been desirable to obtain data at higher concentrations if it had not been for the limited solubility of azobenzene. The low concentration value was difficult to obtain accurately because of charging current, and the general uncertainty of oscillographic measurements at relatively high scan rates. Therefore the low concentration value of ( i p ) c , / C ~ * dwas D (for several scan rates) as estimated from a plot of (i&/d a function of Co*. At low concentrations (Henry’s law behavior) the current is proportional to CO”,and ( i , ) J C o * d ; was obtained from the limiting value of the slope at low concentrations. From the ratio of ( i , ) c / C ~ * dati low and high concentrations an average value for pO/dG = ~ T O ~ , I / K O of 1 . 1 secondl‘2was obtained. This value is considered to be in error by less than 25 %.
0.0
0
I
log(v”2) Figure 7. (i,),/Co*d/z;for stationary electrode polarograms of azobenzene as a function of scan rate Units are: i, amps; CO* moles,Oiter; c, volts/sec. HCI04 = 1M. Azobenzene = A , 0.08mM; B, 0.2mM; C, 0.4mM; D,l.OmM; E, 2.0mM. - - Estimated low scan rate, high concentration, limiting value
-
VOL. 39, NO. 13, NOVEMBER 1967
e
153
~ ~ D ~
Using the method of Qsteryoung, Lauer, and Anson (14, the surface concentrations were calculated at several solution concentrations. From this, the value of POs for azobenzene mole/cm2. Combining this was found to be 1.0 X with the value for Po/%/;, and the diffusion coefficient for azobenzene, 0.34 X IOT5 cm2/second (6), a value for KO of 1.1 X 10-7 mole/cm3 was calculated. Another method for calculating KO is from a plot of surface concentration os. solution concentration, since KO = Co when ro = rOs/2. Calculation of KO in this manner resulted in a value of 0.5 X 10-7 mole/cm3. Both values of KO are considered to have an uncertainty of at. least 20 %, with the lower value probably being more reliable. Nevertheless, if experiments are conducted at approximately millimolar concentrations or greater, the surface is nearly covered and the value of KO should have only minor influence on the theoretical polarograms. This was checked, and it was found that a IO-fold change in KO will cause at most a two-fold change in the calculated rate constant k. On this basis a value of KO equal to 0.75 X 10-7 mole/cm3 (which was considered correct within 50%) was used in the calculations. This possible error in ICO will result in less than a 20z error in k. Determination of Rate Constant. The experimental study of the azobenzene-hydrazobenzene system was carried out using perchloric acid concentrations ranging from about 0.6M to 2.8M (which in turn determined the rate constant for the benzidine rearrangement). Three different azobenzene concentrations were used (0.6, 1.0, and 2.0mM), and the scan rates ranged from 0.08 to 169 Visecond. For a general evaluation of the applicability of stationary electrode polarography in investigating a system such as the azobenzene reduction, rate constants were first calculated ignoring the adsorption, using the theory presented previously (3). These results, referred to as kl values, are listed in Table I, where they can be compared with the results (listed under k 4 ) obtained using the potentiostatic method (6). Using kinetic theory (without considering adsorption) gives abnormally high values, which are in some cases more than an order of magnitude high. Because of the large difference between the values using the potentiostatic method and those using the uncomplicated theory, as well as the apparent dependence of the results on the scan rate: it is obvious that, when present, adsorption cannot be ignored in any kinetic study. To correlate experimental results with the theory developed above, it was necessary, for each scan rate, bulk concentration of azobenzene, and switching potential, to calculate theoretical stationary electrode polarograms for several values of the kinetic parameter kia. From these polarograms a plot of (ip)a/(ip)c as a function of k,'a was constructed, similar to Figure 5. Then, from the ratio (ip)a/(ip)c obtained experimentally, the rate constant k was calculated. For many individual polarograms it was not possible to determine a value for k because for the experimental conditions used, the experimental value of (&'(j& which was obtained exceeded the theoretical maximum value for this ratio. Those values which could be obtained are listed in Table I under ks. In general, rate constants could not be calculated for the faster scan rates at each acid concentration. Where the calcuiation was possible, results were betrer than results obtained neglecting the adsorption of azobenzene. But because there were so many polarograms for which a rate constant could not be calculated, it is evident that the proposed theory does (14) R. A. Osteryoung, G. Eauer, and F. C . Anson, 1. Electrochern. Soc., 110, 926 (1963).
15-40
e
ANALYTICAL CHEMISTRK
not adequately describe the azobenzene-hydrazobenzene system. It is possible that the poor results obtained can be accounted for to some extent by inaccurate values for the adsorption parameters. First, these parameters were calculated for azobenzene in 1M HC1Q4 and may be different at other acid concentrations. However, even at this acid concentration the rate constant could not be evaluated under all conditons. Thus, even if the adsorption parameter had been determined at other acid concentrations, it is unlikely that the results would have been greatly improved. Also, the inaccuracy is far greater than that estimated from the uncertainty in KO. Other reasonable explanations for the difficulties are the possibility of undetected but important changes in the charging current compared with the blank, possible adsorption of hydrazobenzene, a rate-controlled adsorption-desorption, or the inapplicability of the kangmuir isotherm to the azobenzene. Because the largest deviations are observed at high scan rates and low concentrations and the ratio (ip)a/(ip)c is high, the results possibly imply the adsorption of hydrazobenzene. However, before the reasons for the observed deviations could be determined, extensive additional studies of the adsorption of both azobenzene and hydrazobenzene would be required. The above discussion points to the general difficulty encountered in developing a theoretical model to include both coupled chemical reactions and an adsorption process. Inclusion of all known phenomena may lead to a mathematical problem which is too complex to be satisfactorily solved, and in many systems all the important factors may not be known. For example, in the azobemene system, consideration of the adsorption of hydrazobenzene would introduce at least one additional parameter. But each additional parameter, unless known accurately in advance, markedly increases the computing complexity, Thus, even if the theoretical solutions are obtainable, it may be exceedingly difficult to separate the effects of each parameter to obtain useful correlations. EMPIRICAL METHOD FOR DETERhiINING KINETIC PARAMETERS
Because of the difficulty in determining the rate constant using the theoretical model involving both coupled chemical reaction and adsorption, a more general method was sought which could be easily applied. The application of the theory failed because the maximum theoretical value of the ratio (ip)al(ip)c was often less than the measured value for the azobenzene system. On the other hand, it was always possible to calculate a value of the rate constant (even though incorrect) from the theory for the succeeding chemical reaction without adsorption. Thus, although this method gave high values, it appeared to be a reasonable starting point for an empirical method. The basis for handling the data is that in the presence of adsorbed reactant, the characteristics of the stationary electrode polarograms approach the behavior which they would exhibit in the absence of adsorption as the bulk concentration is increased andlor as the scan rate is decreased. Thus, under some conditions, it may be possible to use sufficiently high bulk concentrations to eliminate interference due to adsorption. But if this is not possible, extrapolation of data to zero scan rate should provide the same results. In selecting the quantity to extrapolate to E = 0, the choice is limited to the rate constant calculated at a given scan rate, k,. Another quantity which might be used is (ip)a,'(ip)c. However, at v = 0, the anodic peak must vanish
Table I. Rate Constants for Benzidine Rearrangement Calculated by Different Methods
(Rate constants are given in second-') [H-1, M
[azobenzene], mM
0.62
I .oo
0.82
0.60
0.82
1.0
0.82
2.0
1.03
0.60
1.03
1 .o
1.03
2.0
1.44
0.60
1.44
1 .o
1.44
2.0
2.05
1.o
2.46
2.87
1 .o
1.0
D,
V/second
0.082 0.162 0,423 0,823 1.61 3.93 0.423 0.823 1.61 3.93 8.15 0,423 0,823 1.61 3.93 8.15 0.423 0,823 1.61 3.93 8.15 0.823 1.61 3.93 8.15 15.8 0.469 0.823 1.61 8.15 15.8 0,823 1.61 3.93 8.15 1.61 3.93 8.15 15.8 40.0 1.61 3.93 8.15 16.2 40.0 3.93 8.15 15.8 40.0 8.15 15.8 40.0 87.0 169 15.8 40.0 82.9 169 40.0
82.9 169
kia 0.651 0,773 0.799 1.03 1.95 4.68 1.93 2.04 2.55 6.98 12.9 1.55 1.89 2.29 5.57 11.7 2.06 2.40 2.45 4.19 8.76 4.08 5.30 8.91 15.2 24.6 3.43 3.97 5.40 14.8 30.0 3.33 3.33 6.49 10.8 9.42 14.4 21.8 28.5 67.8 10.7 14.8 18.3 44.0 72.1 13.3 16.2 29.6 75.2 49.2 54.4 86.9 152 255 102 115 159 270 192 231 339
kzl'
0.44 0.57 0.51 0.47 0.88
0.55
0.46
1.05 1.06
0.86
1.20
1.1
1.20
1.3
1.20
2.8
2.75
1.53 2.40 2.26
2.6
2.75
2.56 2.63 3.98 3.43 3.26 1.84
2.4
2.75
7.8
8.6
5.01 5.20
9.1
8.6
10.0
8.6
... 1.02 1.03 1.06
... 1.65 1.86 1.88 2.08 1.78 1.79 1.44
.*.
...
...
...
... ..
... 8.26 7.62 6.40
...
12.7
38
35
75
74
142
148
... 20.9
... ... 12.5
...
Calculated using theory which assumes no adsorption. Calculated using theory which includes adsorption of reactant. 0 Determined by extrapolating kl to zero scan rate. Determined from best line through potentiostatic results (6). a
b
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Figure 8. Ratio of observed rate constants for the benzidine rearrangement (using the theory for a succeeding chemical reaction without adsorption), to the value of the rate constant determined by extrapolation to zero scan rate, as a function of scan rate A. 1.44M HCIQ4,0.6mM azobenzene L: = abscissa X 40.0 V/sec; k,=o = 7.8 sec-1 B. 2.05M HC104, l.OmM azobenzene L: = abscissa X 169 V/sec; k,=$ = 38 sec-1 C. 1.03MHCIQ4,0.6mM azobenzene LI = abscissa x 8.15 V/sec; k,=o = 2.8 sec-1
Because of the wide range of values for k , the plots have been normalized to show the ratio k/k,=o. However, determining the of kvPO initially, k was plotted against L: and extrapolated to I: = 0.
so (ip)a/(ip)e is undefined. But even if a value can be obtained for [(ip)a/(ip)c]t,-O, and one uses this with Figure 12 of Reference 3, to obtain kr, the value of r is undefined and k cannot be calculated. The only remaining possibility is some function of k,, plotted against a function of u and extrapolated to L; = 0 to obtain k,=o. This method has several advantages. First, as long as the ratio (ip)Q,'(ip)c is less than about 0.9, a value for k , can be easily obtained from the theoretical results for a succeeding chemical reaction without adsorption (3). This eliminates the difficulty encountered above in calculating rate constants when (ip)Q/(ip)e was greater than the maximum theoretical value. Experimentally, (ip>Q/(ip)c was never greater
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than unity in the case of the azobenzene system. 'In addition, the extrapolation method does not require knowledge of what materials are adsorbed, the interdependence of adsorbed materials, adsorption-desorption kinetics, or the isotherms, whereas the attempt to obtain an accurate theoretical model might be markedly affected by a slight inaccuracy in either the model itself or the values of the parameters used. The extrapolation technique was applied to each azobenzene solution at each perchloric acid concentration. To determine what functions of k , and u to plot, several functions of each parameter were tried to determine which gave the best line for extrapolation to o = 0. Of those tried, a direct plot of k , against c appeared most suitable. Extrapolation of the best line through the points (typical plots are shown in Figure 8 where the coordinates are normalized for ease in presentation) was used to calculate kt=o,and the results are presented in Table I under kS. These results are considered to be reasonably accurate, with the largest deviation (assuming again the potentiostatic results are correct) being only 28%. Considering the general accuracy of rate constant determinations and the problems encountered in the azobenzene system, the agreement is very good, and indicates that the extrapolation method can obtain accurate results even when it is difficult to describe the system completely. Although the extrapolation method works well when applied to the azobenzene system and appears to be a general method, there may be other kinetic cases where the technique will not be directly applicable. For example, in the case of a succeeding chemical reaction with the product of the charge transfer, R, adsorbed, neither the adsorption of R nor the succeeding chemical reaction causes much change in the cathodic scan. Therefore, while the cathodic peak remains relatively unaffected, the anodic peak may be increased greatly because of adsorption, or vanish because of the succeeding can range chemical reaction. The result is that (i&/(iJC from a minimum value where the anodic peak vanishes to a maximum value much greater than unity at high scan rate, making it impossible to determine k , in a direct application of the theory which does not include adsorption for some of the stationary electrode polarograms. But, as v becomes smaller, kia increases and the anodic peak must decrease, and a range of scan rates will be observed where values of (ip)Q/(ip)c less than unity will be obtained. Therefore, even though every polarogram may not yield a value of It,, it would still be possible to determine k,,o by extrapolation of the data which are in this usable range. Thus, it would appear that the extrapolation to u = 0 will work for many other cases where adsorption and a chemical reaction are both coupled to a charge transfer reaction. RECEIVED for review March 24, 1967. Accepted August 18, 1967. During the 1965-66 academic year, R. H. Wopschall held a Public Health Service Fellowship. Work was supported also by the National Science Foundation under Grant No. GP 3907.
