Spectrochemical evaluation of first order and pseudo-first order reactions following charge transfer for chronoamperometry at transparent electrodes

ELECTROANALYTICAL CHEMISTRY AND INTERFACIAL ELECTROCHEMISTRY Elsevier Sequoia S.A., Lausanne Printed in The Netherlands

11

SPECTROCHEMICAL EVALUATION OF FIRST ORDER AND PSEUDOFIRST ORDER REACTIONS F O L L O W I N G CHARGE TRANSFER FOR C H R O N O A M P E R O M E T R Y AT TRANSPARENT ELECTRODES

GEORGE C. GRANT AND THEODORE KUWANA Department o f Chemistry, Case Western Reserve University, Cleveland, Ohio 44106 (U.S.A.) (Received November 1 lth, 1968; in revised form May 27th, 1969)

INTRODUCTION

In the study of mechanism and kinetics of homogeneous chemical reactions which accompany charge transfer processes, there are many electrochemical methods available. The choice is often made according to the experimental convenience or the ease of numerically analyzing the results in accord with reliably established theoretical treatments. More than one method may be applied but the additional information is still basically limited by the same variables of charge, potential and time. In recent times there has been considerable interest in nonelectrochemical methods, particularly aimed toward in situ measurements. Of these, optical methods using optically transparent electrodes (OTE) show a great deal of promise and have been applied to elucidation of mechanism l'a and to the evaluation of kinetic parameters 2'3 by transmission spectroscopy. Since changes in the optical absorbance, A, are primarily a consequence only of the faradaic process (i.e., post-faradaic homogeneous processes), such methods are of particular advantage for kinetic evaluations in short time experiments where purely electrochemical techniques are limited by correction for the non-faradaic component of the current. Small changes in A can occur due to non-faradaic processes4, but usually these are of negligible magnitude in transmission spectroscopy. It also appears that since products generated in parallel electrode processes have, in general, different spectra, resolution of such complicated processes seems possible by following optical absorbance changes simultaneously at several wavelengths. In this paper, application of OTE using transmission spectroscopy with the light beam normal to the face of a planar electrode for evaluation of the first and pseudo-first order rate constant, k, of irreversible chemical reactions following charge transfer (EC mechanism) will be discussed. The rate of the electrochemical reaction (eqn. 1) is assumed to be controlled solely by diffusion. A

+_he, B

k

, c

(1)

THEORY

If the potential is stepped to a point where current flows at a rate controlled J. Electroanal. Chem., 24 (1970) 11-21

12

G . C . GRANT, T. K U W A N A

by semi-infinite linear diffusion of species A, the current-time behavior is described by the Cottrell equation 5 and the net rate of production of species B for a first order reaction is : dNB(t) _ i(t) dt nF

kN.(t)

(2)

where NB(t) is the total number of moles of species B, n is the number of electrons, F is the faraday, i is the current and k is the first order homogeneous rate constant. Now, the number of moles of B and the optical absorbance are related by:

NB(t) = f ; CB(x, t)Adx

(3)

and

A,(t) = eB

f

~

Ca(x, t)dx

(4)

0

where Ca (x, t) is the concentration, A is the electrode area, x is the distance from the electrode, AB (t) is the absorbance of species B and ea is the molar extinction coefficient of B. Combining eqns. (3) and (4) and substituting the Cottrell equation for i(t) yields:

kAB(t)

d A , (t) _ e, C°~/DA dt lr~ t ~

(5)

where DA is the diffusion coefficient of species A. This equation is readily solved with the aid of Laplace transforms 6 and gives

Aa(t) -- 2eBC°x/DA ~ [exp(--kt) Jo f k(°~

(6)

The function in brackets (eqn. 6) cannot be evaluated exactly, but has been evaluated numerically 7 and is known as Dawson's Integral. Now the amount of C at any time is : Uc(t) =

f o*kNB(t)dt

(7)

Converting to absorbance as before, substituting from eqn. (6) and integrating by parts leads to the result: Ac(t) = 2ec(DA)~ Co [-

(~tk)~

L(kt)6-exp(-kt)

f(k,)~ Jo exp (22)d2-1

(8)

where At(t) is the absorbance of species C and ~c is the molar extinction coefficient. The total absorbance, assuming no absorption by species A, is : AT (t) = A. (t) + Ac (t)

(9)

RESULTS OF NUMERICAL CALCULATIONS

General case Since AT is not an explicit function, theoretical curves for J. Electroanal. Chem., 24 (1970) 11-21

AB(t), Ac(t)

and

CHRONOAMPEROMETRY AT TRANSPARENT ELECTRODES

13

AT(t) were calculated by numerical analysis with the aid of a Univac 1108 computer. The shapes of the AT vs. time curves are markedly dependent upon the relative values

of EB and ec, as shown in Figs. 1 and 2. In Fig. 1 is shown the effect of variable ec on the AT vs. t curves for the case where eB/ec > 1 ; i.e., where most of the absorbance in the time range of interest is due to the electrochemical product, B. It can be seen

14

e

c(10)

,'

£1

~8 / ,

..

o

o.2

0.4

o:e

o.8

0.-

1.o

112

q.4

q.e

4Y/-

Fig. 1. Variation of time-dependent absorbance with the molar absorptivity ratio (1 ~< e~/ec ~< oo). ( ) AT at 2 (eB/ec in parentheses). Curves (b) (e) were obtained by addition of the corresponding curve for Ac to the curve a (oo), calculated for eB = 10,000. ( - - - - - ) Ao Curves (b)-(e): ec = 500, 1000, 2000, and 5000, respectively. T h e constants used were 2C°~/DA/x/rr=2.5 x 10 6. 36 32 28 24

1.(

"~ 20 X3

.< 8

-

-0.5 0.0

4 o o

0,2

o.4

0.6

o.8

1:o

1:2

1'.,*

1:6

1.8

Fig. 2. Variation of time dependent absorbance with the molar absorptivity ratio (0< eB/ec ~ 1). While 0 < e. ~< 10,000, ec = 10,000, 2C°x/DA/x/lr=2.5 x 10 6.

J. Electroanal. Chem., 24 (1970) 11 21

14

G.C.

GRANT,

T. KUWANA

that for cases where ec is less than 2 0 ~ of eB (eB/eC ~>5), the absorbance passes through a maximum and this provides a means of calculating rate constants directly from the experimental maximum without curve fitting which will be illustrated below. Figure 2 shows the effect of variable eB on the A T vs. t curves for the case where eB/e c < 1 ; i.e., where most of the absorbance is due to the final product, C. It can be seen that the shapes of the total absorbance vs. time curves are relatively insensitive to a change in/~a and in no case is there any observed maximum in AT. In Fig. 3 is shown the effect of varying rate constants on the curves for conditions similar to Fig. 2, where AT is primarily the result of species C, the chemical reaction product. It is apparent that the AT vs. t curve is considerably more sensitive to a change in k than to a corresponding change in eB. In other words, an error in the determination of e a (e.g., for an unstable intermediate) does not affect the value of the rate constant appreciably. For a reaction such as the coupling of oxidized phenylenediamines with q8 16 5,0

20

14

"

1.0

i2

0.5

8

02

'~ 4

0

O.

0.4

0.6

0.8 1.0 Time/sec

1.2

1.4

1.6

1.8

2.0

Fig. 3. Variation of absorbance-timecurveswith rate constant (0.1~
AB(t)

•

2eB C A (DAt)~ -

J. Electroanal. Chem., 24 (1970)11~1

(10)

CHRONOAMPEROMETRY AT TRANSPARENT ELECTRODES

15

whence a plot ofAB(t) vs. x/t yields eB from the slope. The molar extinction coefficient of the stable products may be simply measured by coulometric generation of B followed by spectroph0tometric examination of the solution after sufficient time has elapsed to assure complete formation of species C. Absorbance determined primarily by eB As indicated in Fig. 1, in cases where ec is less than 2 0 ~ of eB the total absorbance vs. time curve passes through a characteristic maximum due to the maximum amplitude in Dawson's integral of 0.541 where kt=0.85. It is also apparent that, as ec increases relative to eB, the maximum amplitude is larger and occurs at longer times. This information is summarized for any ratio of eB/ec in Fig. 4, where each of the points was taken from the maximum in the computer calculated curves. For any chemical system, where at a given wavelength the condition es/ec > 5 exists, the rate constant may be calculated from the A T maximum by two methods which do not require a tedious curve-fitting procedure. Method 1. The amplitude factor [AF] is taken from the known eB/ec ratio (see Fig. 4) and, AT(max) =

2g.

C° (DA)~ [AF] (~k)¢

(11)

or

[2e, C°(DA)* [AF] l 2

k=L

(12)

J

Method 2. The time (Fig. 4) of the maximum [TF] occurs when kt = [ T F ]

(13)

k-

(14)

or

rTF] t(max) 1.8 1.6

0.66 0.64

1.4

\\

\\

o.62 "<'0.60

1.2

1.o ,t-,

\

\

0.58

0.8

0.56

5

i l l 10

~B/6 c

50

100

Fig. 4. Variation of the amplitude factor, [AF], and time factor, [TF], with eB/eC. J. Electroanal. Chem., 24 (1970) 11 21

16

G. C. GRANT, T. KUWANA

Evaluation of the rate constant is considerably more precise using Method 1, because experimental A vs. t curves exhibit broad maxima and AT(max) can be determined with greater precision than t(max). Method 2, however, offers distinct advantages when eB is either unknown or uncertain, since [TF] does not depend on DA, CA, or eB (but on the ratio ea/ec). One important class of reactions which is particularly suited for study by the latter method is the solvolytic reactions of unstable aromatic free radicals, many of which are too short-lived for accurate determination of molar absorptivities by conventional spectrophotometry. To achieve maximum sensitivity by monitoring at the wavelength of absorbance maximum for such species, spectra are obtained using a rapid scanning spectrophotometer 1°. Thus, set wavelength studies using Method 2 are then feasible. It is to be emphasized that the shape of the AT vs. t curves is governed solely by the ratio eB/ec and not by the absolute magnitudes. It can be seen in Fig. 1 (for large es/ec) that for small values of kt, the absorbance follows eqn. (10) and deviates at progressively shorter times for increasing rate constants. The special case where eB=ec also gives the same absorbance curve as eqn. (10) (where k=0). For first-order reactions, it is also convenient to evaluate the rate constant by disconnecting the working electrode during the chronoamperometric potential pulse and following A T during the relaxation. A typical AT vs. t curve is given in Fig. 5 where, for the limiting case of very large es/ec, it can be seen that the time of the halflife for first-order relaxation is only slightly less than the time at which absorbance maximizes during the potential pulse. For a more general case, where ec is not negligible, the time dependence of total absorbance for a first-order relaxation is AT(t') = ec(C'B + C'c) + (e s - ec)C'~ e x p ( - kt)

(15)

where t' is the time elapsed after disconnection and where C~ and C~ are defined by C~ =

f

oO

CB(x, t' = 0)dx

(16)

Cc(x, t ' = O)dx

(17)

0

• co

C'c = t .! 0

If In [Av(t')- A T ( t ' ~ oo)] vs. t' is plotted, the slope is - k . The relative error in AT depends on the relative values of ea(2) and ec(2), and the time of disconnection, A v ( t ' ~ o o ) being larger the longer the pulse duration. Surprisingly, large errors can result from neglect of the absorbance due to species C under conditions where the simpler plot of In [Av(t')] vs. t' would seem appropriate. If eqn. (15) is differentiated with respect to time and evaluated at t'= t'~, assuming disconnection occurs at the absorbance maximum, the error in k resulting from the incorrect slope of the simpler plot is 2, 4, 7, 16, and 2 9 ~ for ratios of eB/ec of 200, 100, 50, 20 and 10, respectively. Furthermore, a plot of In [AT(t')] vs. t' for eqn. (15) over two half-lives shows the curvature is very small for the above cases and would probably be obscured by random noise in the absorbance. Comparison of the pulse and relaxation methods from an experimental viewpoint shows on the one hand that the relaxation method has the advantages of (a) measurement of smaller rate constants, because the breakdown of semi-infinite linear diffusion conditions prevents long pulse experiments, and (b) experimental simplicity, J. Electroanal. Chem., 24 (1970) 1121

17

CHRONOAMPEROMETRY AT TRANSPARENT ELECTRODES

Disconnection

14 12 10 8 ×

~6

f

J3

o 4

<

2

0

0:2 or4 0'.6 0'.8 110 1.'2 114 116 1.'8 2'.0 Time (sec)

Fig. 5. Comparison of pulse and pulse-relaxation methods. Constants used were ~B=10,000, ~c=0, k = l , 2C°A~/DA/x/n=2.5×10 6.

since knowledge of D, e and C is not required. On the other hand, the pulse method, either alone or in conjunction with relaxation, has the advantages of (a) less relative uncertainty in [A T(t') - AT ( t ' ~ ~ ) ] , since absorbance is monitored during competitive generation and kinetic reaction, and (b) greater insight into the overall e.c. mechanism. Since both techniques in this paper lead to inhomogeneous solution concentrations, it is necessary to show that the simple equations for relaxation in homogeneous solutions are equally valid in this work. The Laplace transform of the absorbance following disconnection of the working electrode is .4B(S') = eB f °

CB(X,s')dx

(18)

If the transform of Ca(x, t) is substituted according to the first-order rate law, the result is

AB(s') = eB

fo

CB(X, O) dx - eB s'+k s'+k

o

CB(X, O)dx

(19)

where CB(X, 0) is given by the concentration profile at the time of disconnection (t' = 0). According to eqn. (19), the absorbance at any time does not depend on the distribution of species B in solution and the inverse transformation leads to a prediction of absorbance-time behavior which is identical to the homogeneous case. The relaxation method for second or higher order reactions in inhomogeneous solutions cannot be applied without knowledge of the concentration profiles. REACTION KINETICS OF THE 9,10-DIPHENYLANTHRACENECATION RADICAL WITH WATER IN ACETONITRILE

The solvolytic reaction of DPA + with water in acetonitrile has been recently discussed by Sioda 10. This radical cation, substituted in the reactive 9 and 10 positions, is long-lived compared to other aromatic radical cations.

J. Electroanal. Chem.i24 (1970) 11 21

18

G.C. GRANT, T. KUWANA

The rate data 1° apparently show the reaction of DPA + to be first order with respect to DPA + and H20, and Sioda proposes the overall reaction: 2 DPA + + 2 H 2 0 ~ 2 H + + D P A + D P A ( O H ) 2

(20)

where DPA(OH)2 is trans-9,10-dihydroxy-9,10-diphenylanthracene. The radical has absorption bands with high molar absorptivities in the visible region of the spectrum. The known solvolytic reaction and spectral properties of DPA + make it suitable for illustrating the present technique. EXPERIMENTAL Cyclic voltammetry and potential step experiments were performed with conventional circuitry similar to that previously described 1'2, except that I R compensation with positive feedback was employed in all experiments. A Moseley model 7035A X-Y recorder, Tetronix model 564 storage oscilloscope and a Midwestern Instruments model 800 R light beam oscillographic recorder with a maximum chart speed of 100 in. s- 1 and multichannel capability were employed for signal monitoring. A sandwhich-type cell, similar to that previously described 1'2, was used with the electrode surface masked to a constant area. Thermostatting of the sample solution was accomplished by wrapping the body of the cell as well as the auxiliary and reference entrance ports with copper foil and tightly wrapping the jacketed cell with in. copper tubing*. In addition, the aluminum restraining plate previously used in contact with the OTE was replaced by a ~ in. brass plate in which ~ in. copper tubing was inset and thermostatting fluid was circulated through both plate and cell jacket. Antimony-doped tin oxide glasses, obtained from the Corning Glass Co., were used as OTE with surface resistances of 3-6 ~) sq 1 ; platinum OTE had resistances of approximately 20 f~ sq-1. All potentials were measured with respect to an aqueous saturated calomel electrode. Fixed wavelength studies were performed with optics previously described 1'2 and u.v.-visible spectra were recorded on a Cary 15 spectrophotometer. Lithium perchlorate (anhydrous) was obtained from G. F. Smith Chemical Co. and used without further purification; 9,10-diphenylanthracene (Aldrich Chemical Co.) was recrystallized from ethanol. Water was doubly distilled, the second distillation being from alkaline permanganate through a one-meter glass-packed column. Spectroquality acetonitrile (Matheson, Coleman and Bell) was further purified by passage through an alumina column freshly activated at 350°C for at least 12 h; the water content was evaluated chromatographically using a Porapak Q (Waters Associates, Inc., Framingham, Mass.) column 5 ft. in length and 0.125 in. in diameter**. Column temperature was 140°C and the helium flow rate was 35 ml min-1. RESULTS AND DISCUSSION Cyclic voltammetry at a bright platinum electrode or platinum OTE of millimolar solutions of DPA in acetonitrile (0.1 M LiC104) gave a nearly reversible * 1 in.=2.54× 10-2 m. ** 1 ft.=0.3048 m. J. Electroanal. Chem., 24 (1970) 11-21

CHRONOAMPEROMETRY AT TRANSPARENT ELECTRODES

19

first oxidation wave with peak potential, Ep, of + 1.22 V and an irreversible second wave a t E p = 1.62 V. At tin oxide electrodes, the first wave occurred at Ep= + 1.18 V, showing quasireversible behavior with Ep(anodic)-Ep(cathodic)=90 mV. At all electrodes the waves showed characteristics of increased irreversibility as the water content in the acetonitrile was increased. The spectrum for DPA in the 300400 nm region was identical to that previously reported 1° , but the absorption peaks for DPA+ were consistently 7 nm lower in the present work. The diffusion coefficient for DPA was evaluated from the Cottrell equation and a plot of ip vs. v ~ in experiments with no added water resulting in the values (1.12 _+0.08) x 10- 5 and (1.19 _+0.02) x 10- 5, respectively. The concentration of D PA + was monitored at a 2max of 653 nm using the value e(DPA+)=(8.75+0.2)x 10 3, evaluated in pulse experiments using eqn. (10) and the average value: ~/D = 3.4 x 10- 3 cm s-% The experimental variation of absorbance during a pulse experiment with time is compared with the computer generated curve in Fig. 6, where the maximum is

30

~" 2o

o

8 1:)

,<

Time (sec)

Fig. 6. Comparison of exptl, time dependent absorbance ( [ H 2 0 ] = 1.8 M) with the computer generated pulse response. Constants used were eB = 8.75 x 103, ec = 0, x/DA = 3.4 x 10 3, k = 0.096 s - 1, CA = 0.55 mM.

slightly broader than that predicted by theory.At 22(+ 0.5)°C the rate constant values 0.26 ( [ H 2 0 ] =3.7 M) and 0.15 ([H20] =2.7 M) for relaxation agree well with the corresponding values, 0.25 and 0.17, calculated from the data of Sioda 1°. However, to avoid any ambiguity in semi-infinite linear diffusion conditions, due to convection at the long times required at his temperature, the majority of data were obtained at 27.4_ 0.2° C and are presented in Fig. 7. The same pattern as that shown in Fig. 7 was observed also at the lower temperature. Computer fitted least-squares lines are shown in Fig. 7 for each set of points and fit the equations: k / s - ~ =0.062 [ H 2 0 ] - 0 . 0 2 5 (pulse, eqn. (14)), k=0.081 [ H 2 0 ] -0.053 (pulse, eqn. (12)) and k=0.107 [ H 2 0 ] 0.077 (relaxation, eqn. (15)). Additional points are shown for the relaxation method only at water concentrations where the time of the experiment would have been too long for diffusion conditions to hold. The plot of log AB(t') vs. time for relaxation was J. Electroanal. Chem., 24 (1970) 11-21

20

G . C . GRANT, T. KUWANA

linear and independent of [DPA +] and [ H 2 0 ] in agreement with Sioda 1°. The discrepancies between values of kobsdetermined by the three methods may arise from several sources: (a) greater complexity in the mechanism than a simple bimolecular reaction of D PA + and H2 O ; (b) irreversibility in the electrode reaction; (c) concurrent oxidation of water. First, although excellent agreement is seen between pseudo-first order relaxation for inhomogeneous solutions in the present work and previous work for homogeneous solutions, the variation of kobs with [H20] clearly shows the mechanism to be complex. Electroactivity in any of the species in the mechanism other than DPA + could result in disagreement between pulse and relaxation methods. Second, the slight irreversibility in the electrooxidation of DPA increases with increasing water concentration. Further examination of the effects of a slow electron transfer is in progress and preliminary results indicate that kobs would necessarily be lower in a given experiment if determined from tmax rather

0,3C

D 0.20

Cg 0.10

o o o

EH2o~/M

b

~

4

Fig. 7. Observed pseudo-first order rate constant vs. water concn, for three methods. (O) Relaxation, (D) pulse (eqn. 12), (A) pulse (eqn. 14).

than Amax when the equations for a reversible electron transfer are used for an experimentally irreversible electron transfer. The results in Fig. 7 are in agreement with this prediction. Third, cyclic voltammetry of solutions containing DPA and water or water alone reveals that oxidation of water consumes a significant fraction of the current (ca: molar water concentrations) at potentials beyond Ep for the first oxidation of DPA. The interaction of these water oxidation products with DPA + is not known. With present instrumental capabilities of detecting 10- 2 (.jr_ 1,°~oo) or 10- 3 (_+ 10~) absorbance change with time resolution of 10 -4 s, the evaluation of rate constants in the order of 103 - 104 s- 1 appears feasible. In this regard, many organic intermediates, especially aromatic free radicals which possess allowed doublet state electronic transitions, have high molar absorptivities in the easily accessible wavelength region of the spectrum. Therefore, optical techniques are eminently suitable for characterization of these radical systems. J. Electroanal. Chem., 24 (1970) 11-21

CHRONOAMPEROMETRYAT TRANSPARENT ELECTRODES

21

ACKNOWLEDGEMENT Financial support by G r a n t No. G M 14036 from the Research G r a n t Branch of National Institutes of Medical Sciences, N a t i o n a l Institutes of Health and by the N a t i o n a l Science F o u n d a t i o n , G r a n t No. G P 6479, is gratefully acknowledged. SUMMARY Relationships for spectrochemical evaluation of first-order and pseudo-firstorder reactions following reversible electron transfer have been developed for chronoa m p e r o m e t r y at optically transparent electrodes. C o m p u t e r generated curves for the time dependent absorbance are presented for the general case where b o t h electrochemical and chemical products a b s o r b at 2, permitting determination of rate constants as fast as 103-104 s - 1. F o r the special case where the m o l a r absorptivity of the chemical reaction p r o d u c t is less than 2 0 ~ of that for the electrochemical product, the rate constant m a y be calculated without recourse to curve-matching and m a y be estimated without knowledge of the m o l a r absorptivity of either product. C o m p a r i s o n has been made between pulse and pulse-relaxation methods for determination of the rate constant. The latter case has been applied to the solvolytic reaction of the 9,10diphenylanthracene cation radical in aqueous acetonitrile. REFERENCES 1 J. W. STROJEKANDT. KUWANA,J. Electroanal. Chem., 16 (1968) 471. 2 T. KUWANAANDJ. W. STROJEK,Trans. Faraday Soc., 45 (1968) 134. 3 J. W. STROJEK,T. KUWANAANDS. W. FELDBERG,J. Am. Chem. Soc., 90 (1968) 1353. 4 N. WINOGRADANDZ. KUWANA,J. Electroanal. Chem., in press. 5 P. DELAHAY,New Instrumental Methods in Electrochemistry, Interscience, New York, 1954. 6 R. V. CHURCHILL,Operational Mathematics, McGraw-Hill, New York, 1958. 7 M. ABRAMOWITZ ANDI. A. STEGUN(Eds.), Handbook of Mathematical Functions, Nat. Bur. of Std. U.S., Appl. Math. Ser., 55, 1964. 8 L. K. J. TONGAND M. C. GLESMANN,Jr. Am. Chem. Soc., 79 (1957) 383. 9 J. W. STROJEK,G. A. GRUVERANDZ. KUWANA,Anal. Chem., 41 (1969) 481. 10 R. E. SIODA,J. Phys. Chem., 72 (1968) 2322. J. Electroanal. Chem., 24 (1970) 11-21