Kinetics of Comproportionation: A Spectroelectrochemical Approach

This work was performed under the auspices of the Office of Basic Energy Sciences, Division of. Chemical Sciences, US-DOE, under contract no. W-31-109...
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4741

J. Phys. Chem. 1991, 95,4741-4748 not been seen previously. Because of the formation of a new species in solution, new chemical pathways are possible using radiolysis which have not been available photochemically or thermally. Acknowledgment. We thank Drs.J. R. Miller, P. Piotrowiak, and N. Liang for experimental assistance. These experiments

could not have been run without the assistance of G. L. COXand D. T. Ficht in tuning and maintaining the linac. Helpful discussions with Drs. M.c. Sauer, Jr., and A. D. Trifunac are gratefully acknowledged. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, US-DOE, under contract no. W-31-109ENG-38.

Kinetics of Comproportionation: A Spectroelectrochemical Approach Richard

G.Compton,* Barry A. Coles, and R. Anthony Spackman

Physical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ,U.K. (Received: October 29, 1990; In Final Form: January 30, 1991)

-.

A general spectroelectrochemical approach to the kinetics of the comproportionation reaction Y + Y2- 2Y- is presented which combines both rotating-disc double-step chronoamperometry and quantitative in situ electrochemical ESR spectrosc~py. It is shown that when the electroreduction of Y to Y2- proceeds via two separate one-electron waves and Y'-is ESR active, the combination of the two techniques allows the unambiguous determinationof the kinetics together with the diffusion coeffdents of the three species involved. The method is applied to the case where Y = p-chloranil (tetrachloro-p-benzoquinone),and good agreement between theory and experiment is found.

Introduction This paper is concerned with investigating and quantifying the effect of a homogeneous comproportionation reaction on an electrode reaction that proceeds stepwise via two single-electron transfers: A+e-+B

(i)

B+e-+C

(ii)

A+C,-2B

(iii)

where E0Ap is more positive than E O B p Hitherto, electrochemical measurements, largely cyclic voltammetry, have yielded thermodynamic information (notably comproportionation constants, K(mmp)= exp[-F/RT(EoBlc - EoAlB)])via the determination of the standard electrode potentials E O A p and EoB/C and numerous applications of this type have been Theoretical considerations of the kinetics of comproportionation (reaction iii) have been carried out by Smith, Feldberg, and Ruzic6.' in respect of ac voltammetry but otherwise are limited!*9 It will be shown below that when (i) and (ii) are electrochemically reversible and A, B,and C have equal diffusion coefficients, purely electrochemical experiments are blind to (iii), a conclusion reached by Guidclli' and also by Magno and Bontempelligin respect of potential step chronoamperometry, although the latter technique was shown to be viable when the first cathodic step (A to B) in the above mechanism is sufficiently hindered by overvoltage that

a single irreversible tweelectron process is observed. In this case, the homogeneous comproportionation kinetics are accessible because of the enhanced currents due to (iii). In other cases the Occurrence of irreversible chemical reactions coupled to the electrode reaction makes visible the effects of comproportionation in voltammetric experiments. Some other techniques have also been employed in studying comproportionation kinetics, including stopped flow,1° various optical techniques at transparent electrodes (OTE!S),~'-'~and ESR studies."J* In this paper, we present a general electrochemical approach to the study of comproportionation kinetics involving a combination of double potential step chronoamperometry at the rotating disc electrode (RDE) and quantitative in situ electrochemical ESR spectroscopy. Both these experiments are highly sensitive to the diffusion coefficients, D, of A, B, and C and the rate constant, khi,for (iii). Typically DB, Dc, and kiiiare unknown, and it is for this reason that the combination of techniques is essential so that all of these parameters may be unambiguously determined. Theory is developed for the chronoamperometric response to a potential step at a RDE for the particular case when a stepwise two-electron transfer is coupled to a comproportionation reaction as in (i)-(iii). The precise form of the potential step sequence to be examined can be understood by reference to Figure 1 and recognizing that the bulk solution contains only one electroactive species A. The initial potential El is one at which no current flows. (IO) Bennion, B. C.; Auborn, J. J.; Eyring, E. M. J . Phys. Chcm. 1972,

76, 701. (1) Polcyn. D. S.;Shain, 1. Anal. Chcm. 1966, 38, 370. (2) Heize, J. Angcw. Chcm., fnr. Ed. Engl. 1984, 23, 831. (3) Kaiffer, A. E.; Bard, A. J. J . Phys. Chcm. 1985,89, 4876. (4) Mohammed, M. Elecrrochlm. Acra 1988, 33, 417. (5) Richardson, D. E.; Taube, H.fnorg. Chcm. 1981, 20, 1278. (6) Ruzic, 1.; Smith, D. E.; Feldberg, S.W. J . Elecrroanal. Chem. 1974, 52, 157. (7) Ruzic, 1.; Smith, D. E. J . Elecrroanal. Chcm. 1975, 58, 145. (8) Guidelli, R. Anal. Chcm. 1971.13. 1715. (9) Magno, F.; Bontempelli. G. Anal. Chcm. 1981, 53, 599.

(1 1) Kuwana. T.; Winograd, N. Anal. Chcm. 1966,38, 1810. (12) Winograd, N.; Kuwana, T. J . Am. Chcm. Soc. 1970, 92, 224. (13) Winograd, N.; Kuwana, T. J . Am. Chcm. Soc. 1971, 93, 1353. (14) Strojek, J.; Kuwana, T.; Feldberg, S. J. Am. Chcm. SOC.1968. 90, 4343. (15) Kuwana, T.; Strojek, J. Discuss. Furaday Soc. 1968. No. 15. 134. (16) Armstrong, N. R.; Vanderborgh, N. E. J . Phys. Chcm. 1976. 80, 2140. (17) Compton, R. G.; Monk, P. M. S.; Rosseinsky, D. R.; Waller, A. M. J . Elecrroanal. Chem. 1989, 267, 309. (1 8) Male, R.;Samotowka, M. A.; Allendoerfer, R. D. Elccrrocrnalysb 1989, I, 333.

0022-365419112095-4741S02.50/0 0 1991 American Chemical Society

Compton et al.

4142 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991

t

ET

1"

dependent mass transport equations relating to this case are as follows:

TimeFigure 1. Schematic representation of the potential step sequence used in the rotating-disc chronoamperometry experiments.

Then a potential E2 is applied of sufficient magnitude and for enough time that a steady-state transport-limited current for the reduction of A to B is established. The potential is then further stepped to E3, which is such that a steady-state transport-limited current for the reduction of B (to C) is established. Last, the reverse processes are investigated by stepping the potential back first to E2 and subsequently to E , . This corresponds to C being oxidized back to B and then to A. Our models of these chronoamperometric measurementsemploy an explicit finite-difference computational strategy that makes use of a spatial transformation first suggested by Hale,Ie2' and the current transients are found to be highly sensitive to the values of DB, Dc, and kiii. Additionally we utilize in situ electrochemical ESR measuremenh2' It has been previously demonstrated that a channel electrode made of synthetic silica could be located at the center of the cavity of an ESR spectrometer with negligible perturbation of its sensitivity, thus allowing the identification of radical species formed during electrochemical reactions.22Moreover, the known pattern of flow in the channel enables the calculation of the concentration distribution of electrogenerated radicals if the homogeneous chemistry of the radical is known.23 In this paper, we apply quantitative in situ electrochemical ESR studies to the mechanistic scheme outlined above. Experimentally, the ESR signal from the radical anion, B, is measured at the two potentials E2 and E3 (cf. Figure 1) and the ratio of the two signals is recorded as a function of electrolyte flow rate. A backward implicit finite difference methodN (vide infra) is used to simulate the experimental results, which are again sensitive to the choice of kiii,Dgrand Dc. In combination with RDE measurements, they permit the kinetics of (iii) to be established. To illustrate these experimental techniques, we present results for the reduction of p-chloranil (tetrachloro-p-benzoquinone)in acetonitrile solution. This system is especially challenging to experimental and theoretical investigation because, first, the markedly different diffusion coefficients exhibited by the species involved (vide infra) and, second, the common planar geometry of the parent and its dianion suggest that the comproportionation kinetics may be very fast. This is found to be the case.

Theory In this section, we describe the theoretical models for RDE chronoamperometry and in situ channel electrode ESR experiments outlined above. In both cases, theory is developed for the mechanistic scheme contained in (i)-(iii). We first, however, establish a general result relating to the examination of the reaction scheme under scrutiny by means of electrochemical measurements alone in the circumstance that A, B, and C all have an identical diffusion coefficient. The time (19) Compton, R. 0.;Laing, M. E.; Mason, D.; Northing, R. J.; Unwin, 113. (20) Hale, J. M. J. Elcctroanal. Chcm. 1963, 6, 187. (21) Hale, J. M. J . Elecrroanal. Chcm. 1964. 8, 332. (22) Coles, B. A.; Compton, R. G. J . Elccrroanal. Chcm. 1983, 144, 87. (23) Compton, R. G.; Page, D. J.; Scaly,G. R.J . E/ecrmna/. Chem. 1984, 161, 129; 1984, 163,65. (24) Compton, R. G.; Pillrington, M. B. G.; Stearn, G. M. J . Chem. Soc., Faraday Trans. I 198(1,84, 2155.

P.R.Proc. R.Soc. 1988, ,4418,

-a [-c I - Trc[CI - kiii[Al [CI

-El

El-

at

(c)

where Tr is a (linear) operator describing the diffusive and convective transport to the electrode of interest. In the event that DA = DB = Dc this operator is constant, i.e. TrA = TrB = Trc = Tr. Equations a, b, and c are subject to the following boundary conditions where E is an unspecified function describing the variation of the electrode potential with time. At the electrode surface, p = 0: [Al/[Bl

~xP(~A/B)

[B]/[c1 = exP(@B/C)

eLIL'= F / R T ( E - EoL/Lt) a[Bl ac1 -a[AI =-=-= aP

In the bulk solution, p

-

aP

aP -:

[AI

--

[AI"

[BI

0

[CI

0

where p is a coordinate normal to the electrode surface and [A]" is the bulk concentration of A. If the redox couples A/B and B/C are electrochemically reversible the electrode current is solely governed by diffusive fluxes and is given by

or

or

where integration over the entire electrode surface (7)is implied. It is readily seen that, first, by taking appropriate linear combinations of (a), (b), and (c) and, second, by using the Nemstian boundary conditions above to show that the surface concentration of A, B, and C depend only on [A]" and E, the functions f,= 2[C] + [A]& = [C] - [A], andf, = 2[A] + [B] are independent of kiii. It follows from (d), (e), and (f) that the observed current is also independent of this parameter. We thus arrive at the general conclusion that the kinetics of the comproportionation scheme defined by (i)-(iii) cannot be investigated by electrochemical experiments alone, whatever potentialltime sequence (function E ) is used to interrogate the system, subject to A, B, and C all having identical diffusion coefficients and with (i) and (ii) being electrochemically reversible. This conclusion is irrespective of the electrode geometry employed. We now consider the RDE experiments. The time-dependent convective-diffusion equations describing mass transport to the RDE coupled with the homogeneous com roportionation kinetic scheme shown in the scheme above are2 P

The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4143

Kinetics of Comproportionation

0-

Xe

flar

where z is the distance normal to the disc surface. The term Cz2 describes the convective flow normal to and close to the electrode (C = 8.032W%-'/*; W is the rotation speed in hertz; Y is the kinematic viscosity in centimeters per second). These equations may be cast into dimensionless form26by writing WL

= (C/DL)'/~Z

t ~ * (C/DL)'/3t kL = kiii[A]"C2/3DL-I/3 where L = A, B, or C. We now introduce the x coordinate in place of w, according to the Hale transformation:20,2'

-

iw exp(-l /3w2/7 dw

x"

Figure 2. Schematic diagram of the channel electrode showing the coordinate system used under Theory.

our A-Hale space so that W, = w. We may .then calculate the corresponding values of wB and wc from the relation

Since these values of wB and wc are unlikely to correspond to an exact grid point in A-Hale space, concentrationsat the two nearest grid points are interpolated by averaging their values weighted by the distance between their coordinates and the required point. The numerical solution of the convective-diffusion equations is then carried out by using the procedure adopted in refs 19, 29, and 30 (with a value of N = loo), subject to the following boundary conditions: For z = 0 a/b = ~XP(~A/B)

exp(-1 / 3 d / 3 ) dw

This uses the description of the rotating disc diffusion layer, originally formulated by Levich2' for steady-state conditions, so that the convective diffusion equations are transformed into

atA*

-aa-

-atc*- -

exp(-2/3wA3)

[

x"

-aza -

. kAaC

exp(-1 /3w3) dw]2

- - kcac

[ &" exp(-1 /3w3) dwI2 ax2

where a = [A]/[A]", b = [B]/[A]", and c = [C]/[A]". Notice that the act of Hale transformation reduces the two terms, corresponding to diffusion and convection, into just one expression. Moreover, because the z coordinate changes from zero (at the disc surface) to infinity (in bulk solution), the transformed coordinate, x, changes from 0 to 1. Thus the simulation of the transient response to a perturbation of some existing concentration distribution can be found by splitting the interval of unity (Hale space) into n boxes defined by N + 1 grid points where the coordinates of each grid point depend on the diffusion coefficient of the species under consideration. Consequently, our calculations use three different Hale spaces: one each for A, B, and C. The above equations each involve A, B, and C so that if we choose to evolve the concentration profile of A in A-Hale space, B in B-Hale space, and C in C-Hale space it is essential to interchange concentrations between the three coordinate systems. The procedure we adopt is as follows: For each value of x, 0, (N + l)-', 2(N + I)-', ..., 1, we calculate a value of w using the third-order RungtKutta method2*and we arbitrarily use this as (25) Albery, W. J. Electrode Kinetics; Clarendon Press: Oxford, U.K., 1975: D 51. (26) Albery, W. J.; Hitchman, M. L. Ring-disc Electrodes; Clarendon Press: Oxford.. U.K.. 1971. - .~ (27) Levich, V. G. Physicochemical Hydrodynamics, Rentice-Hall: Englewood Cliffs, NJ, 1962. (28) Abramowitz, C.; Stegun, I. A. Handbook of Mathematical Functions: Dover: New York, 1965.

For z

-

a-1 b-0 c-0 where E is defined by Figure 1. This produces the time-dependent concentration distributions for A, B, and C (a = a(t,z), b = b(t,z), c = c(t,z)) from which the current flowing at time t* may be readily deduced as beinglg Z(t*) = (1 .65894)-'/2FA,C1/3[exp(-!/3W'/3)] ( D A ~ / ~- JDc2/3jc) A where A, is the area of the electrode and jLis the flux of species L at the electrode surface. We next present the theoretical model for the in situ electrochemical ESR experiments. In particular, we predict both the shape of the current-voltage curves and the magnitude of the ESR signal and its flow rate dependence for species B at the channel electrode for the case of the mathematical scheme defined by (i)-(iii). This necessitates the solution of the following steady-state mass-transport equations:

where x and z are distances parallel to and normal to the electrode surface (Figure 2) and kiiiis the rate constant for (iii). The term u, describes the convective flow velocity along the x axis: V, = ~ o ( l- ( z ' / h ) 2 ) where vo is the velocity at the center of the channel, I' = h - z, and h is the half-height of the channel (see Figure 2). (29) Harland, R. G.; Compton, R. G. J . Colloid Interface Sci. 1989, 131, 83.

(30) Compton, R. G.:Harland, R. G.J . Chem. Soc., Faraday Trans. I

1989, 85, 761.

4144 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991

Compton et al.

Theoretical Results and Discussion Theoretical chronoampcrometric transients for the RDE were generated for the potential step sequence defined in Figure 1 by using the parameters DA = 1.75 x 10-5 cm2 s-I, DB/DA= 0.7, Dc/DA 0.5 and (a) k~ 0,(b) kA 10, and (C) kA = 100. The computed current transients for the four potential steps El E2, Et El, E2 E3,and E, E2are shown in Figures 8-1 1, respectively. First consider the effect of stepping the potential from E, E2. For this case, the time-dependent current is only proportional to the flux of A at the electrode and hence analysis of the current transient yields directly a value for the diffusion coefficient of A as confirmed experimentally for various systems in our earlier However, the steady-state concentration distribution of B which is established by the reduction of A depends on the diffusion coefficient of B so that for DA > DB the surface concentration of B is larger than [A]" whereas for DA < DBthe reverse is true. Thus, when we reverse the potential step (E2 E , ) so that B is oxidized back to A, we observe (Figure 9) larger or smaller oxidative currents relative to the forward transient for the respective cases of DA > DB and DA < DB when compared to DA = Db Consider next the effect of stepping the potential from E2 E,. Notice from Figure 10 that, at any value of t*A, as kA is increased from zero to effective diffusion control, the current demases. The effect of comproportionationis to consume A and produce B so that provided DB < DA (as is the case for

Figure 10) reduced mass transport (diffusion) ensues compared to the forward transient. On the other hand, for DB > DA comproportionation increases mass transport (diffusion) of B to the electrode and enhanced reductive currents are observed. We now turn to the E, E2potential step for which the time dependent current has a contribution from the flux of C. A consequence of the homogeneous comproportionation kinetics is to alter the distribution of C so that its concentration falls to zero (bulk concentration)closer to the electrode. When C diffuses away from the electrode it reacts with A to give B, which may then diffuse back to the electrode to give an increased surface concentration of C. Consequently when the potential is stepped from E3 E2 (see Figure 11) the magnitude of the oxidative currents increases as kA is increased from zero to diffusion control. We turn to the calculations for the in situ electrochemical ESR experiments. As an initial exercise, the validity of our numerical approach was assessed by calculating the current(I)-voltage(E) behavior at the channel electrode flow cell for the first reduction B) where (iii) has no influence. Mass-transportwave (A corrected Tafel analysis32(plots of E vs log [ZIim(l/I- 1/11im)I, where Ilimis the transport limited current) of the computed voltammograms gave a slope of 59 mV/decade and a half-wave potential equal to E O A l B . This was considered to vindicate our general computational strategy and its implementation. We next consider the result of our calculations for the second reduction, which unlike the first wave can be influenced by the coupled comproportionation kinetics. For the case of no kinetics and DA = DB = Dc, a one-electron reversible wave centered on OeIc = 0 was found, as expected. Moreover, in the case of equal diffusion coefficients the simulated voltammograms were found to be independent of kiii. This confirms the predictions of the Theory section in respect of purely electrochemical experiments being "blind" to the presence of homogeneous kinetics for the equal diffusion coefficients case. Finally, we examine the case of unequal diffusion coefficients. Computed voltammograms are shown in Figure 3 for the extreme cases of kiii= 0 and 1 X 108 mol-' cm3s-' for (a) DA = &,DA/Dc = 4 and (b) DA/DB = DA/Dc = 2, where DA = 1.75 X Cm2 s-I. These illustrate the sensitivity of the mechanistic scheme proposed in (i)-(iii) to the diffusion coefficients of B and C. Notice that one effect of decreasing Dc relative to DB is to shift the reduction wave cathodically so that its half-wave potential occurs at a negative value of dB/C (see Figure 3a). The origin of this shift is an increased concentration of C in the vicinity of the electrode owing to its diminished diffusion coefficient. A second effect of decreasing Dc (Figure 3a) is a relatively increased mass-transport-limited current as kii increases: the enhanced concentration of C in the vicinity of the electrode leads to more reaction with A, giving more B than for the case of equal diffusion coefficients. An overall increased electroreduction of B ensues. On the other hand, a decrease in DBrelative to DA produces a reduced masstransport-limited current as kih:increases (see Figure 3b), analogous to the discussion above in the context of RDE chronoamperometry for the potential step E2 E,. It may be expected from the above that for these cases of mixed diffusion coefficients there is an accompanying effect of comproportionation kinetics on the wave shape of the computed voltammograms for the B C wave. Indeed mass-transportcorrected Tafel analysis (seeabove) of these voltammograms shows small but distinct deviations from simple one-electron behavior. For the case of no kinetics and equal diffusion coefficients, we observe a Tafel slope of 59 mV/decade and a half-wave potential equal to E o Bc. However, h e n we introduce the complications of mixed d i f t b o n coefficients and comproportionation kinetics, the plots may be shifted anodically or cathodically and/or may become curved. Figure 4 shows the Tafel plots for the voltammograms in Figure 3a. The apparent curvature can be explained: where the current is high, at the top of the wave (I N IIim), comproportionation is more effective than at the foot if DB > Dc.

(31) Poole, C. P.EIecrron Spin Resononce; Wiley: 210.

1975; p 79.

The convective-diffusion equations are subject to the following boundary conditions, where x, is the length of the electrode: For z = Oand 0 < x < x,

a/b

~XP(~A/B)

b / c = exp(OB/C)

eLIL,= F/RT(E - EoL/~,) aa DAaz

For z = 0,x

For 0 < z

ac + DB-ab az + Dc-az = 0

-

> xe, and z = 2h, for all x

< 2h and x

I0

a=l b=O c-0 The backward implicit method is used to solve the sets of coupled partial differential equations and boundary conditions presented above, using the protocol found in ref 24. The dimensions of the hypothetical cell modeled to give the theoretical results discussed below are as follows: width of channel = 0.591 cm, height of channel = 0.036 cm, width of electrode = 0.402 cm, and length of electrode = 0.425 cm. A grid size of 4000 X 2000 (K X J, see ref 24) over the zone of the electrode and a correspondingly spaced grid downstream is employed so as to give accurate and converged results. We use as our measure for the ESR signals the integral of the concentration of B over the whole volume of the ESR cavity weighted by a cos2 sensitivity profile" along the length of the cavity. The following equation shows the individual contributions to the ESR signal (S) from the regions adjacent to and downstream from the electrode: S

0:

x2h( -42

+ XM- cos2

(E)+

sx' xJZ

+ X- b cos2

(E))

dx d r

where 2xLis the length of the ESR cavity and xdw is the distance (offset) from the center of the electrode to the center of the cavity. Full details of the computational procedure appear in ref 24.

- - -

-

-

-

-

New York,

1967; p

-

-

(32) Albery, W. J. Electrode Kinetics; Clarendon Press: Oxford, U.K.,

The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4145

Kinetics of Comproportionation rkiii=ld

I

-M

00

+

40

eBK Figure 5. Tafel plots for channel electrode voltammograms calculated cm3s-' by using the geometry specified in the text for a flow rate of for the case of kiii= lo* mol-' cm3s-', Dc/DA = 0.5 with Dn/DA = 1.0, Dn/DA= 0.7,and Dn/DA = 0.5.

1.2. 1.0-

00

-4.0

40

~ B K Figure 3. Computed channel electrode voltammograms for the second reduction wave (B C) for rate constants kUi= 0 and 10' mol-' cm3 c', from the cell geometry specified in the text (calculated at a flow rate of 1W2cm3 s-' and a concentration of A of 0.243 mM): (a) D A = Dn = 1.75 X lW5 cm2s-', DA/Dc = 4; (b) D A = 1.75 X lW5om2 s-l, Dn = Dc 0 ~ / 2 Also . depicted in Figure 3a is the curve for D A = Dn = Dc = 1.75 X IO-' cm2s-'. Values of 10A/n - BB ,-I 30 were used to ensure that the wave was far removed from the Arstteduction wave.

-

-

-

50

60

I0

80

90

~og,(kiii/mol1cm3s11

Figure 6. 'Working curve" showing the relationship between S2/Sland log (kiii)for the channel electrode flow cell geometry specified in the text and D A = Dn = Dc = 1.75 X lW5cm2s-' and a volume flow rate of lW2 cm3 s-l.

-

> DBand DA< DB. Thus, Figure 5 shows the Tafel plots for the

/ Figure 4. Tafel analysis of the three voltammograms shown in Figure 3a.

This occurs since the rate of formation of B is proportional to the concentration of C, and when Dc < DB a relative buildup of C to B occurs in the vicinity of the electrode, as compared to the case of DB = Dc. This increases the rate of turnover of C into B so that more current flows near the top of the wave. Note that these changes in waveshape and half-wave potentials are only observed when DBand Dc have differing diffusion coefficients: when DB = Dc the waveshape is that of a one-electron kinetically uncomplicated reversible process centered on Eonlc, and the only evidence of comproportionation then is a reduced or increased absolute transport-limited current for the respective cases of DA

B C wave for ki = 1 X 108 mol-' cm3C1,Dc/DA = 0.5, D D = 1.0,0.7,and 0.5, respectively, where DA = 1.75 X lW5cm4 PIA. For the case where DB/DA = 0.5 (Le. DB = Dc), it can be seen that the Tafel plot exactly coincides with that predicted for a simple oneelectron process with equal diffusion coefficients. The other two cases give rise to cathodically shifted Tafel plots. It is evident from the foregoing that channel electrode steady-state voltammetry measurementswill not be precise enough that they alone define the mechanism under consideration. This provides the impetus for our ESR computations to which we now turn. We consider first the behavior of the steady-state ESR signals from B and as an initial exercise, assess the validity of our numerical approach by calculating the ESR signal (SI) of B at a potential corresponding to the steady-state limiting current for the reduction of A to B. We have shown previously22that for a stable radical this should be related to the current, I, generating the radical at a channel electrode by the equation SI = I/V,Z/3

where V,is the mean solution velocity (an2s-'). Our computations gave signal/current/flow rate data in exact agreement with this equation over the flow rate range 5 X lo-' > V,> 1 X lW3 cm3 s-1.

The potential was then stepped so that the steady-state limiting current for the reduction of B to C had been established and the new value for the ESR signal, S2,was calculated. Figures 6 and 7 show the dependence of the dimensionless parameter, S2/SI, on log (kiii)for the respective cases of equal and mixed diffusion

Compton et al.

4746 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991

50

60

7.0

8.0

90

loqdkill /mol"cm3s" 1 Figure 7. "Working curves" showing the relationship between Sz/Sland log (kiii) for the channel electrode flow cell geometry specified in the text and DA = 1.75 X cm3 s-I, Dc = 0.875 X IO-' cm3s-', flow rate = cm3s-I with DB = 0.875 X IO", DB = 1.20 X and DE = 1.75 x 10-5 cm* s-l.

coefficients calculated for the cell dimensions mentioned above and for a bulk concentration of A of 0.243 mM. When kiii is initially increased from zero, S2/SIincreases rapidly, but when the kinetics approach mass transport control, S2/S, tends towards a limit. It is important to note, first, that S2/SIdoes not reach a limiting value of 2, which is what we would expect if all of the C produced by electroreduction of A is converted into B by comproportionation. Rather, for infinitely fast comproportionation kinetics two separated spatial zones are established, one close to the electrode containing C (but not A) and the other extending into the center of the channel containing A (but not C), with a sharp interface existing between the two zones. Second, we note the effect of unequal diffusion coefficients on the limiting value of S2/S1.If DBis decreased relative to DA,assuming an initial condition DA = DB,both SIand S2are increased but SIis increased relatively more than S2so that the quotient S2/SIis decreased. We explain the increased magnitude of both SIand S2by noting that decreasing D Bnarrows the diffusion layer of B and consequently reduces the flow rate of B out of the channel (the parabolic dependence of the solution velocity on the coordinate normal to the electrode is found in the Theory section). Decreasing DB thus leads to an accumulation of radicals in the channel flow cell.

Experimental Section Rotating-disc measurements were carried out by using an Oxford Electrodes (Oxford, U.K.) rotating-disc assembly and motor controller. This permitted rotation speeds in the range 1-50 Hz to be utilized. A 1286 electrochemical interface (Schlumberger Electronics (U.K.) Ltd., Farnborough, Hampshire) was used for voltammetric measurements and current-voltage curves were recorded on a Lloyd Instruments PL3 chart recorder. The former enabled positive feedback ohmic drop compensation to be employed during the chronoamperometric measurements. The current transients were recorded by using a purpose built transient recorder controlled by an Opus AT personal computer.29 The working electrode was a platinum rotating-disc electrode with teflon insulation of diameter 0.675 cm, which had been polished with a succession of finer diamond lapping compounds (Engis Ltd., Maidstone, Kent) down to 0.25 pm in size. A graphite rod served as the counter electrode. All measurements were made under thermostatted conditions at 25 O C (f0.2OC) and potentials were recorded against a saturated calomel electrode (SCE) as a reference electrode. The basic apparatus and techniques used for in situ electrochemical ESR work have been described previously.z33 The silica channel-electrode unit was 30 mm long and had approximate cross-sectional dimensions of 0.4 mm X 6.0 mm. Precise values for all the cell dimensions were found by using a traveling mi-

croscope and the cell depth was obtained from the gradient of the Levich plot (of current against V$j3)for an electroactive species of known diffusion coefficient.22 Platinum foils (4.0 mm X 4.0 mm), used as electrodes in these units, were cemented onto the silica cover plates and were carefully polished flat with a succession of finer diamond lapping compounds as described above. The ESR spectrometers used were as reported before.2xM Electrochemical measurements associated with ESR studies were carried out by using an Oxford Electrodes potentiostat modified to boost the counter electrode voltage.34 A silver-wire pseudoreference electrode was located in the flow system upstream of the ESR cavity. A platinum-gauze counter electrode was placed downstream of the working electrode. Flow rates in the range 104-10-1 cm3 s-I were employed. Dropping mercury electrode measurements were made on a home-built assembly of conventional design. The pool of mercury produced by these measurements was then further used as a working electrode for the exhaustive electrolysis of p-chloranil to its radical anion. The solvent used throughout was acetonitrile (Fisons, dried, distilled) and tetrabutylammonium perchlorate (TBAP) (Kcdak, puriss) served as the background electrolyte. p-Chloranil (Aldrich, 99%) was used as received. Solutions were thoroughly purged of oxygen by bubbling through the solution argon that had been dried with calcium chloride and then presaturated with acetonitrile. Supporting theory for the chronoamperometric and ESR measurements was generated from programs written in FORTRAN 77 on the Oxford University VAX cluster.

Experimental Results and Discussion Preliminary steady-state experiments were carried out using a platinum rotating-disc electrode on solutions approximately 1.O mM in p-chloranil in acetonitrile/O.l M TBAP. Two waves were observed with half-wave potentials a t -0.02 f 0.002 V and -0.70 f 0.02 V, respectively (vs saturated calomel electrode). These values are in reasonable agreement with the reported literature values.3s The shape of the current-voltage curves measured for the first wave was consistent with a reversible one-electron transfer: gave a slope of Tafel analysis (a plot of E/Vvs log (r'- Ilim-l)) approximately 59 mV/decade. A similar slope was obtained for the second reduction wave: the effects of comproportionation on Tafel slopes discussed in the Theoretical Results and Discussion section are too small to be accurately measured. The relative heights of the two waves were found to be in the ratio 1:0.7 but showed little rotation speed dependence. This result is consistent with the theoretical discussion of Figure 3b, assuming D, > DB and a value for kiiisuch that the kinetics are approaching effective mass transport control. Under these conditions there is only a slight rotation speed dependence of comproportionation kinetics on the second reduction wave. The transport-limiting current for the first wave was found to be proportional to the square root of the rotation speed and a value for the diffusion cm2 s-l (literature value coefficient ofp-chloranil of 1.75 X cm2 s-I 3s) was derived from a Levich analysis: 1.79 X The diffusion coefficient of the radical anion was determined as follows: The radical was prepared in bulk by exhaustive

-

(34)

(33) Waller, A. M.;Compton, R.G. Compr. Cham. Kinet. 1989, 29, 297.

-

electrolysis of the parent (approximately 2 h) over a pool of mercury and voltammograms for both its oxidation (B A) and reduction (B C) at a platinum RDE were recorded. Two waves of equal height were observed and Levich analysis of the transport-limited current in both cases yielded a diffusion coefficient of 1.2 x 10-5 cm2 s-I. The difference in heights between the two current-voltage curves for the reduction of p-chloranil can be understood from our theoretical discussions above on the effect of mixed diffusion coefficients on fast comproportionation kinetics. For the case of no kinetics, the steady-state transport-limited current is only Compton, R,G.;Waller, A. M. J. ElactrGQMI. Chem. 1985,195,289. E.J . Cham. Soc. 1962, 4540.

( 3 5 ) Peover, M .

The Journal of Physical Chemistry, Vol. 95, No. 12, 1991 4141

Kinetics of Comproportionation

450-

400-

350-

3002 50200-

0'2

04

06

0.2

10

08

04

t:

0.6 :t

-

Figure 8. Plot showing analysis of chronoamperometrictransients corrwponding to a potential step from El E2for a 1.05 mM solution of pchloranil in acetonitrile/O.l M TBAP for four different rotation speeds: 4 Hz ( 0 ) ,9 Hz (A), 16 Hz (+), and 25 Hz (X). The solid line shows the transient computed for DA = 1.75 X lo-' cm2S-I and DB/DA = 0.7.

0%

x)

-

Figure 10. Plot showing analysis of chronoamperometric transients corresponding to a potential step from E2 E3 for a 1.OS mM solution of p-chloranil in acetonitrile/O.l M TBAP for four different rotation speeds: 4 Hz ( 0 ) ,9 Hz (A), 16 Hz (+), and 25 Hz (X). The solid line shows the transient computed for DA = 1.75 X 10" cm2s-I, DB/DA = 0.7, Dc/DA = 0.5, and kA = 0, 10, and 100.

t

0.2

1

04

0;6

1.0

0;8 n-

-150

i

z- -250 -300:

ill;

2 -100 \

1

I1L

i-2501"

-%: -lSO: - -200

f'p

-300' Figure 11. Plot showing analysis of chronoamperometric transients corresponding to a potential step from E3 E2for a 1.05 mM solution of p-chloranil in acetonitrile/O.l M TBAP for four different rotation speeds: 4 Hz (O),9 Hz (A),16 Hz (+), and 25 Hz (X). The solid line shows the transient computed for DA = 1.75 X lo-' cm*s-I, DB/DA = 0.7, Dc/DA = 0.5, and kA = 0, lo, and 100.

- - -

-

-

steps El E,, E2 El, E2 E,, and E, E2 for a 1.05 mM solution of p-chloranil are shown in Figures 8-1 1, respectively. For each case measurements were made at four different rotation speeds, 4, 9, 16, and 25 Hz. All the different rotation speed data for a particular step were displayed on the same normalized plot of current Ili,,,/Wl2against normalized time r(dLD,)'/3 where C is equal to 8.032W/2u-'/2, as suggested by the theory presented above. Theoretical data for the case of DA = 1.75 X lW5 cm2 s-', DBf DA = 0.7, Dc/DA = 0.5 and for values of kA = 0,10, and 100 are plotted together with the experimental points. The potential step measurements for the first wave ( E , E2,E2 E,) confirm the ratio of diffusion coefficients for A and B as determined above from steady-state measurements. Analysis of the transients for the second reduction wave required the inclusion of the comproportionation kinetics proposed above and a value for Dc. We observe a marked dependence on kA for the range 0 < kA < loo, but for higher k A the calculated transients merge and the reaction is effectively diffusion controlled. It is apparent from Figures IO and 11 that the diffusion coefficients cited above together with a value for kA lying between 10 and 100 (1 X lo7 < kiii< 1 X lo9mol-' cm3SI) gives a good fit between experiment and theory. ESR studies were next conducted using the flow cell described above with the electrode at potential E2and positioned at the center of the ESR cavity. A single line was observed (Figure 12) with

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4748 The Journal of Physical Chemistry, Vol. 95, No. 12, 1991

-1.25 -1.20 VI

N

-1.15z,

-1.10 -1.05 Figure 12. ESR spectrum obtained by electroreduction (at -0.3 V vs SCE)of a 0.25 1 mM solution of pchloranil in acetonitrile/O.l M TBAP and attributed to the corresponding radical anion.

p20

VI h)

,3i

\

krsSxr0’ -0.95

-3.0

-2:6

-I$

-2:2

log,, @f/cm3 s,’)

Figure 13. Theoretical and experimental plots showing the variation of &/SIagainst electrolyte flow rate for the cell of geometry as specified in the text and with the 4 mm long electrode located with its leading edge aligned with the center of the ESR cavity (of total active length 24 mm) ( x ~ =~2 mm). ~ , The three lines shown are calculated for DA = 1.75 X IO- cm2 s-l, DB/DA= 0.6 and Dc/DA = 0.5and for values of kiiiof 5 X IO7, 7 X IO7, and 9 X IO7 mol-’ om3 s-l.

an approximate peak-to-peak width of 300 mG. The signal can be attributed to the pchloranil radical anion and analysis of the peak-to-peak height of the signal against volume flow rate data confirmed that the radical was stable over the range of flow rates 1x < Vf < 5 x IO-’ cm3 s-l. Specifically a plot of log (Sl/Ili,,,)vs log ( V f )gave a straight line of slope -2/3. The potential was then stepped to E3 and the changed peak-to-peak height for the ESR signal (S,) was recorded again as a function of flow rate. The electrode was then repositioned 90 that its center was 2 mm downstream of the center of the ESR cavity and a new set of SIand S2data as a function of flow rate were recorded. Figures 13 and 14 show both experimental and theoretical plots of S2/SIvs log (V,)for a concentration of p-chloranil of 0.251 mM. Good agreement between experiment and theory was achieved for DA = 1.75 X IO-$ cm2 s-I, DB DA = 0.6,Dc-DA = 0.5,and kiii= (7 i 2) X IO7 mol-’ cm3 s- by utilizing the two locations of the electrode stated above (relative to the ESR cavity) and for the following measured cell dimensions: width of channel

I

-26

-24

-22

-20

-18

kqxltVf /cm3s4)

Figure 14. Theoretical and experimental plots showing the variation of &/SIagainst electrolyte flow rate for the cell of geometry as specifided in the text and with the 4 mm long electrode located with its leading edge at a distance 2 mm upstream of the center of the ESR cavity (of total active length 24 mm) (xdW = 0). The three lines shown are calculated for DA = 1.75 X IO-’ cm2s-I, DB/DA = 0.6, and DC/DA = 0.5 and for values of k,,,of (bottom) 5 X lo7,(middle) 7 X lo7, and (top) 9 X IO7 moP cm3 s-I.

= 0.600 cm, height of channel = 0.034 cm, width of electrode = 0.293 cm, and length of electrode = 0.397 cm. Notice that S2/S1varies not only with flow rate but is also highly sensitive to the position of the electrode within the ESR cavity. This may be explained by the following: When the electrode is at potential E,, B is generated by electroreduction of A and then flows unperturbed out of the channel so that a concentration profile of B within the channel is established such that [B] decreases in both the x and z directions. However, when the electrode potential is stepped to E3, the concentration of B no longer varies monotonically with the x or z coordinates (see Figure 2) but the precise distribution is determined by two competing processes: the electroreduction of B to C and the comproportionation reaction between A and C. In particular, at the downstream edge of the electrode the concentration of B rises rapidly with x and z,since B is no longer being destroyed electrochemically but then passes through a maximum when C has been consumed. Consequently, B’s concentration profile develops a characteristic “plumen and the position of this plume within the ESR cavity determines the magnitude of S2relative to SI. Conclusions

The successive electroreduction of p-chloranil to the corresponding radical anion and then the dianion proceeds via two separate reversible one-electron waves the second of which is coupled to a comproportionation reaction between the dianion and the parent molecule. A combination of rotating-disc double-potential-step chronoamperometry and quantitative in situ electrochemical ESR measurement allows the unambiguous determination of the comproportionation rate constant kiii = (7 f 2) X lo7 mol-’ cm3 s-’ together with the diffusion coefficients, 1.75 X lo-$cm2 s-’, (1.12 f 0.08) X cm2s-’, (0.9 f 0.1) X lO-$cm2 s-l for the parent, the radical anion, and the dianion, respectively. Acknowledgment. We thank SERC for a studentship for R.A.S.