Effects of homogeneous kinetics on chronocoulometric relaxations in quasi-reversible adsorbing systems

Electroanalytical Chemistry and Interracial Electrochemistry

41

Elsevier Sequoia S.A.,Lausanne- Printed in The Netherlands

EFFECTS OF HOMOGENEOUS KINETICS ON CHRONOCOULOMETRIC RELAXATIONS IN QUASI-REVERSIBLE ADSORBING SYSTEMS

w. H. REINMUTH Department of Chemistry, Columbia University, New York, N.Y. 10027(U.S.A.) (Received24th January 1972)

In recent work we examined the results to be expected in chronocoulometric experiments conducted in the potential region corresponding to the rising portion of the polarographic wave for diffusion limited systems subject to adsorption coupled with reversible 1'2 and quasi-reversible 3'4 charge transfer. In the present work we extend our examination to include systems in which homogeneous chemical reactions couple the electroactive species with electroinactive forms. THEORY At a stationary planar electrode: i/nFA =fo - dFo/dt = - J R + dFR/dt = koRCo - -

kgoCR

(1)

where i is faradaic current (cathodic current taken as positive); F, the faraday; A, the electrode area ;fo andfR the fluxes of the oxidized and reduced forms respectively at the electrode; Co and CR their homogeneous concentrations at the electrode; Fo and FR the surface excesses of their adsorbed forms, koR and kRo the rate constants for their charge transfer reactions; and t, time. As we have remarked previously4, the charge-transfer rate law need not be of the form assumed in eqn. (1), and the possibility of its not being of this form should not be lightly dismissed in systems subject to non-linear adsorption. For systems in which mass transport to the electrode is by semi-infinite linear diffusion coupled with first or pseudo-first order homogeneous reactions of the type: kyo

Y.

kZR

"O koY

Z...-'R

(2)

kRZ

equations relating the concentrations and fluxes of O and R at the electrode can be derived from the diffusion equations. In Laplace space these equations take the form: (C*/s - Co) = f o To,

(C~/s - CR) =fR TR

(3)

where C~ and C~ are the initial homogeneous solution concentrations of O and R; fo, fR, CO and CR are the Laplace transforms of their fluxes and concentration at the electrode; To and TR are what may be termed their transfer functions5 and s is the transform variable. For simple semi-infinite diffusion uncomplicated by homogeneous kinetics : T =

J. Electroanal. Chem., 39 (1972)

(4)

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w.H. REINMUTH

For cases of the type described by eqn. (2), T, as given by Matsuda 6, is of the form: TA = {1 + [D~A(s + k,A)/ Oi s~ (s + k,A + kAl)~] }/ DA (r+ + r~ )

(5)

where subscript A denotes the electroactive form (O or R), and subscript I the corresponding electroinactive form (Y or Z); D denotes diffusion coefficient and r+ are defined by" r+ = (1/2DADI)[DA(S q- klA) q- DI(S + kAi) + {[DA(s+k,A)+D,(s+kA,)]2--4SDATZ,(s+k,A +kAI)} ~ }

(6)

Equations (1) and (3) can be combined to give an explicit expression for faradaic charge (time integral of faradaic current) in Laplace space of the form: Q/nFA = [(koR C8 - kRo C~)/s 2 (1 + koR To + k,o TR)] + { [ ( r ~ -- Sffo)(1 + koR To) - (ff~ - SIVR)kRoTR]/S [ 1 + koR To -1-kRo TR] } (7)

The first term on the right-hand side of eqn. (7) is identical to the expression for Q/nFA in the absence of any complication due to adsorption (F~, F], fro and fiR equal to zero), while the second term reflects the contribution of adsorption to the relaxation process. Thus Q can be thought of as composed of the sum QK + QA where QK corresponds to the first and QA to the second term on the right-hand side of eqn. (7). We discuss these contributions in turn. RESULTS IN THE ABSENCE OF ADSORPTION

Because the first term on the right-hand side of eqn. (7) is given explicitly in terms of the transform variable, in principle it can be inverse-transformed to give an explicit expression for QK as a function of t. In practice the expression is so cumbersome that inverse transformation can be accomplished readily only by series expansion in Laplace space and term by term inverse transformation. In the usual method of analysis of chronocoulometric data, Q is plotted vs. t ~, and the long-time limiting slope and the extrapolated value of Q,=o) are obtained from this plot. The expansion of QK corresponding to this data analysis is in powers of s~, resulting on inverse transformation in an asymptotic series 7 representation of QK in descending powers of t ~ of the form : QK/nF A = at ½+ b + ct- ~ +...

(8)

Where a is the slope and b the intercept of the Q vs. t ~ plot. The explicit value of a is * ~a -~- 2(OCToDMo

* ~ 21(1 -]- O) CTRDMR)/7~

(9)

where : C~o= C$ + C * ; C*R = C~ +C*

(10)

DMO = (kvo Do + koy Dr)~ (kvo + koy); DMR = (kzR DR + KRZ Dz)/(kzR + kRz) (11) + kov)D~o 0 = koR kyo(kzR + kRz) DMR/kRokzR(kvo ½

(12)

Equation (9) is of the same form as the result in the case of no kinetic complications, but without complications, DMo=Do, DMR=DR, C~o=C~, CTR--CR, ** and 0= ½ ½ kORDR/ kRoDo. J. Electroanal. Chem., 39 (1972)

KINETIC EFFECTS ON CHRONOCOULOMETRY

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The result conforms to the intuitive expectation that at sufficiently long times the effect of finite rates of homogeneous and charge-transfer kinetic complications are negligible and the system behaves as it would if the equilibria were achieved instantaneously. The value of b in eqn. (8) is given by : 1 -- "/U1 a |r ~, k R Z + kZR) D~m

b - 2(1+ (~) L

kzakRo

ODykov + D~oDMokvo(kvo + kov)~

Dzkaz 1 d- DRDMRkzR(kzR+ kRz) ~

(13)

This equation has a strikingly simple significance. Each of the three terms in brackets represents the contribution of one of the slow kinetic processes to the (negative) extrapolated value of Qt=o. The first term represents the effect of the charge-transfer process, and has the same form as it would have if the O-Y and R-Z interconversions were infinitely rapid. The second term reflects the effect of the 0-5( process, and has the same form it would have if its O-R and R - Z interconversions had infinite rates. The last term represents the effect of the R - Z process and has the same form it would have if the O-R and O-Y interconversions were infinitely rapid. Thus, the effect of the three slow kinetic processes is the linear combination of the effects to be expected if each were present alone. The effects of slow charge-transfer have been described in detail by Christie and co-workers 8'9. What the present result shows is that a negative extrapolated value of Qt=o can arise not only as the result of slow charge-transfer kinetics, but as the result of slow homogeneous chemical kinetics as well. From a single chronocoulometric relaxation there is no a priori method of deciding which possibility (if indeed only one of them) is responsible for an observed negative extrapolation. Fortunately, there is some reason to hope that examining the variation of the extrapolated value of Q as a function of the potential of relaxation will enable the experimenter to distinguish between the three possibilities here considered. The usual experimental technique is one in which the initial potential is more anodic thanthe foot of the polarographic wave (where C~ is essentially zero), and Qt=o is plotted as a function of the potential at which relaxation is observed. The limiting slope of the Q vs. t ~ curve has the form of a conventional polarographic wave. That is : a = constant. 0/(1 + 0)

(14)

where : 0 = exp [(nF/RT)(E~ - E)]

(15)

For charge-transfer kinetics in which the rate constant is given by absolute rate theory: b = constant. 02 -~t/(1 -t- 0) 2

(16)

where ct is the charge-transfer coefficient for the electrode reaction. In the usual circumstance in which 0 < ct< 1, the form of the b vs. potential curve is an asymmetric peak with maximum at 0 = (2 - c~)/~, corresponding to a point cathodic of E~ (where E~ occurs at 0 = 1). For a following chemical reaction:

J. Electroanal. Chem.,39 (1972)

44

W. H. REINMUTH b = constant. 0/(1 + 0) 2

(17)

which is of the form of a simple derivative of the normal polarographic wave, i.e. a symmetric peak with maximum at 0 = 1; while for a preceding chemical reaction: b = constant. OZ/(1 + O)2

(18)

which has the form of the square of the usual polarographic wave (i.e. a plot of l n [ ( b m a x - b ~ ) / b ½] vs. E is linear and of slope ( R T / n F ) ) . In this case the curve of b vs. E shows a plateau rather than a peak, and this provides convenient qualitative distinction from the other two cases. It should be noted that distinctions based on the form of b vs. E curves are not entirely unambiguous, because if e approaches the extremes zero or unity, the form of the charge-transfer b vs. E curve approaches the forms of b vs. E curves for preceding or following chemical kinetics, respectively. If such a complication arises, only detailed analysis of Q vs. t curves or resort to other relaxation techniques can resolve the ambiguity. Still another practical difficulty may attend the application of this analysis to experimental data. Implicit in the analysis is the assumption that at times corresponding to the measured Q's from which extrapolation is conducted, the third and subsequent terms in the expansion given in eqn. (8) are negligible. Such times may not be conveniently accessible experimentally, and, if the requirement is overlooked, nominal values of b obtained by extrapolation may be in error. In general, the relative error in b introduced by truncating the series in eqn. (8) after the second term is approximately equal to the ratio of the third to the second term, and thus, in order to estimate the error, it is necessary to determine the magnitude of the third term. Because the expansion is cumbersome algebraically, we confine our attention to the case in which D o = Dy and DR----Dz. This simplification materially reduces the algebraic complexity without substantially changing the qualitative nature of the results. With this simplification: c = 2bZ/na

(19)

b/(an~/2) = crt~/b

(20)

or:

Equation (20) implies that in the expansion of eqn. (8) the third term is to the second as the second term is to the first. This provides a convenient experimental estimate of the systematic error in the nominal value of b induced by truncation after the second term. The relative error is approximately equal to the ratio of the extrapolated value of Qt=0 divided by the actual value of Q at the time from which the extrapolation is performed. Examining the explicit form of c/b indicates that the time required for extrapolation to be useful is a function of potential. In the case of preceding kinetics the longest times are required in the potential region corresponding to the plateau of the polarographic wave, for following chemical reactions the critical region is at the foot of the wave, and for charge-transfer kinetics the critical region is near the half-wave potential. In each case, if the condition is satisfied in the critical region it will be satisfied at all other potentials as well. J. Electroanal. Chem., 39 (1972)

KINETIC EFFECTS ON CHRONOCOULOMETRY

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CONTRIBUTION OF ADSORPTION The detailed time dependence of the second, adsorption, term on the right-hand side of eqn. (7) cannot be analyzed without knowledge of the specific adsorption isotherms for O and R. However, at sufficiently long times, the values of Fo and FRasymptotically approach limiting values (here denoted as F~ and F~, respectively) for reasons we have previously discussed t - 4. Using this fact and invoking the Laplace transform theorem 7 that: lim Jio = lim sf(s) t~oo

(21)

s~O

the limiting value of the adsorption term (denoted as Q~) can be evaluated as" Q ; / n F A = [ O ( F ~ - F~ ) - (F* - F~°)]/(1 + 0)

(22)

Equation (22) is identical in form to the corresponding equation we have previously derived for the case in which O and R do not undergo homogeneous chemical reactions 4. There are differences between the two cases, however. In the present case, the effect of adsorption is likely to be small, because, depending on the equilibrium constants for the O-Y and R-Z interconversions, the concentrations of O and R, and hence the Fo and FR with which they are in equilibrium, are likely to be much smaller than if conversion to electroinactive forms did not occur. This, however, does not imply generally that the effect of adsorption is less when kinetic complications occur; rather the result stems from the postulate of our model, that Y and Z are not adsorbed. It may be noted that the model could be generalized to include adsorption of Y and Z. One reason for not doing so in the present work was to minimize algebraic complexity, but a second reason was the thought that if both O and Y (or R and Z) were adsorbed in an actual system, a model which ignored the possibility of their heterogeneous (as well as homogeneous) interconversion might well prove inappropriate. CONCLUSIONS Within the limitations imposed by possible oversimplifications of the model here chosen, a relatively simple and straightforward rule of thumb emerges for the prediction of results to be expected on application of the chronocoulometric technique. Extrapolated values of Qt=o in the presence of adsorption, charge-transfer and homogeneous kinetic complications equal simply the sums of the Q values which would be observed if each complication were present alone. For kinetic complications the contributions are negative (i.e. opposite in sign to the observed total charge) while for adsorption contributions may be either positive or negative 1 depending on the extent of initial adsorption and the variation of adsorption with potential. SUMMARY The theory is given of chronocoulometric relaxations in systems in which the reactive couple undergo quasi-reversible charge-transfer, first order homogeneous reaction to electroactive forms and adsorption. As in simpler cases extrapolation of J. Electroanal. Chem., 39 (1972)

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w . H . REINMUTH

Q vs. t ~ curves from long times to give Qt=o provides a convenient method of data analysis. The extrapolated Q is simply the sum of the Q's which would be observed if each complication were present alone. Kinetic complications inevitably produce negative extrapolated Q's. Variation of Q with potential is diagnostic for different complications. REFERENCES 1 W. H. Reinmuth and K. Balasubramanian, J. Electroanal. Chem., 38 (1972) 271.

W. H. Reinmuth and K. Balasubramanian, J. Electroanal. Chem., submitted. W. H. Reinmuth and K. Balasubramanian, J. Electroanal. Chem., submitted. W. H. Reinmuth, J. Electroanal. Chem., submitted. H. Keller and W. H. Reinmuth, Anal. Chem., in press. H. Matsuda, J. Amer. Chem. Soc., 82 (1960) 332. G. A. Korn and T. M. Korn, Mathematical Handbook Jot Scientists and Engineers, McGraw-Hill Book Co. Inc., New York, 1961, chap. 8. 8 J. H, Christie, G. Lauer, R. A. Osteryoung and F. C. Anson, Anal. Chem., 35 (1963) 1979. 9 J. H. Christie, G. Lauer and R. A. Osteryoung, J. Electroanal. Chem., 7 (1964) 60. 2 3 4 5 6 7

J. Electroanal. Chem., 39 (1972)