Effects of langmuirian adsorption on potentiostatic relaxation of diffusion-limited reversible electrode reactions

Electroanalytical Chemistry and Inter[acial Electrochemistry

79

Elsevier Sequoia S.A., Lausanne - Printed in T h e Netherlands

EFFECTS OF LANGMUIRIAN ADSORPTION ON POTENTIOSTATIC RELAXATION OF DIFFUSION-LIMITED REVERSIBLE ELECTRODE REACTIONS

W. H. REINMUTH and K. BALASUBRAMANIAN*

Department of Chemistry, Columbia University, New York, N.Y. 10027 (U.S.A.) (Received 10th December 1971)

The effect of adsorption on diffusion-limited charge-transfer processes has received considerable attention in the thirty years since Brdi~ka's initial investigations. Much of the work has concerned itself with polarographic situations, where the time base of observation renders a stationary state approximation very useful in the theoretical description of results. Thus, exact theoretical treatments of the time-dependent behavior of adsorptive reacting systems at expanding electrodes have been limited to Holub's elegant treatments of linear adsorption 2'3 and Koryta's treatment of the limiting case of strong adsorption 4. Recently, chronocoulometric studies at stationary electrodes have begun to yield a wealth of information on adsorption in reactive systems 5. For the most part experimental studies have been confined to relaxations occurring under conditions such that the concentration of the adsorbing species at the electrode is zero, and the time-dependency of the results is independent of the form of the adsorption isotherms. This condition is unfortunately not applicable in the potential regions in which both forms of the reactive couple are present to a significant extent at equilibrium, which region is of particular interest with respect to elucidation of the mechanism of their interconversion. Rigorous theoretical treatments applicable to such situations have been confined to systems obeying linear adsorption isotherms 6 - 8. While these systems form an experimentally important class, their theory leaves some doubt as to the results to be expected in systems in which surface saturation becomes prominent because such saturation is associated specifically with non-linearity of adsorption isotherms. Levich and coworkers 9 have shown that the Nernst diffusion layer approximation leads to reasonably accurate results in cases of simple diffusion-limited adsorption without reaction. However, this approximation is less useful in cases in which adsorption is coupled with reaction, because, particularly in cases of non-linear isotherms, it usually does not materially reduce the mathematical complexity of the problem. Nor is the stationary state approximation generally helpful, because with stationary electrodes, in contrast to expanding ones, it describes a situation in which the most prominent effects of adsorption on the electrode process are negligible. Thus Guidelli 1° in a treatment of Langmuirian adsorption based on the stationary state approximation was led to conclude that equivalents of polarographic pre- and * Present address: Ciba Geigy Corporation, Saw Mill River Road, Ardsley, N.Y., U.S.A.

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postwaves should not be observable in stationary electrode voltammetric studies, a conclusion at striking variance with the results of the present investigation. For these reasons it seemed of interest to examine the rigorous theory of relaxation at stationary electrodes in reacting systems obeying non-linear isotherms. For simplicity, the present work confines its attention to reactions obeying the Nernst equation, and in quantitative detail to adsorption obeying a Langmuir isotherm. Some of the conclusions, however, will be shown to be independent of the detailed form of the isotherm. The work is divided into four sections dealing respectively with the statement of the boundary value problem, limiting behavior at long times, time-dependency, and extension to non-Langmuirian systems. THE BOUNDARY VALUE PROBLEM

Mathematically, the problem can be stated in the form: (~CR/~t) = DR(~32 CR/SX2), (SCp/~t) = Dp(~ 2 ¢p/SX 2)

(1)

t = 0 ; CR=C °, Cp=C °

(2)

CRx = 0

t >0;

c °,

¢o

D R (OCR/OX) + O v (c~cv/~x) = ( d F R / d t ) + (dFp/dt)

(3) (4)

Cp = Oc R

(5)

i / n F = DR (OCR/C3X) -- ( d F R / d t )

(6)

where CR and Cp are the concentrations of reactant and product respectively; c ° and c ° the initial concentrations of these species; FR and Fp their surface excesses; D R and Dp their diffusion coefficients ; t is time ; x distance from the electrode : i the current density due to conversion of reactant to product according to a sign convention which considers positive conversion to correspon& to positive current; n the number of electrons per molecule involved in the process; F Faraday's constant and 0 the factor relating the ratio of concentrations of R and P according to the Nernst equation, i.e.

0 = exp ( _ n F / R T ) (E - E °')

(7)

where R is the gas constant; T absolute temperature; E °' the formal standard potential of the R - P couple, E the constant potential applied after t = 0 and the + or - is to be taken accordingly as the reactant is the oxidized or reduced form of the reactive couple. We assume the isotherms relating the c's and F's are of the form F R = KRCRFm/(1 + KRC R + Kpep)

(8)

rp = I,:pcprm/(1 + KReR + K~C.)

(9)

where KR and Kp are the adsorption coefficients of reactant and product respectively and Fm is the surface excess corresponding to surface saturation. These isotherm forms constitute Langmuirian adsorption in which a finite number of adsorption sites exist that can be occupied by either reactant or product. The same model was used by Guidellil o. The model thus neglects self-interactions and cross-interactions between adsorbed molecules and such complications as might arise due to different adsorption J. Electroanal. Chem., 38 (1972)

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sites for R and P on inhomogeneous surfaces or differences in the effective areas of adsorbed molecules on homogeneous surfaces. As will become apparent later many of our most interesting conclusions are qualitatively independent of possible oversimplifications of the model, and the only eventuality likely to produce pronounced deviations would be specific chemical interaction between adsorbates i.e. surface compound formation. The problem as stated above is incomplete without further specification of the values of F R and Fp at time zero. While any values consistent with Nernstian equilibrium between the homogeneous concentrations of R and P at the electrode can be assumed, we shall confine our attention to two specific cases of particular interest. One of these is the case in which FR and Fp before the application of the potential step are given by eqns. (8) and (9) with c o = COoand CR= C°. The other case is that in which FR and Fp before application of the potential step are identically zero. The latter case might be expected to arise when R and P were neutral species and the initial potential corresponded to a value more extreme than their tensammetric desorption peaks while the potential after step application did not. For the case in which there is adsorption before potential-step application, some care must be exercised in the specification of initial condition, that is t = - 0 corresponds to the situation before application of the potential step and t = + 0 corresponds to the situation immediately after step application. At t-- + 0 the application of the potential step may have resulted in some net conversion of adsorbed reactant (if present initially) to adsorbed product, but in no finite amount of interconversion between material in adsorbed and unadsorbed states. Such interconversion would require mass transfer over finite distances, which, of course, would require finite times. It might appear that our exclusion of instantaneous interconversion between adsorbed and unadsorbed states at time zero conflicts with the mathematical model proposed, for example, by Christie et al. 11 for an adsorbing system subjected to a double potentiostatic pulse. There, impulsive conversion of adsorbed reactant to diffusing product is assumed. It will subsequently become evident, however, that the conflict is more apparent than real because the finite times required for conversion of adsorbed to unadsorbed material according to the present model become sufficiently short to be considered instantaneous in any actual physical system under the conditions assumed in the other work. According to the present model, the surface excesses of reactant and product and the homogeneous concentrations of reactant and product at the electrode surface with which they are in equilibrium (F~ o, F~O, c + o, c~-O respectively) can be calculated readily by combining eqn. (5) (for t = + 0) with the equation expressing the fact that ira + Fp is the same for t = + 0 and t = - 0 , and eqns. (8) and (9) expressing the relations between the F's and c's at t = + 0 and t = - 0 . The results are: c +° = (KRcO)/(KR + OKp)

(10)

c~ ° = Oc+°

(11)

F[ o = KRFm (KR c° + Kpc°)/(KR + OK.)(1 + KR c° + K.C °)

(12)

Fp+°= (OKp/KR)F~ o

(13)

The charge per unit area required for the instantaneous conversion of reactant to product at time zero (Q~) is then given by: J. Electroanal. Chem., 38 (1972)

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W . H . REINMUTH, K. BALASUBRAMANIAN

QI/nF = FR o _ F~ o = KR KpFm(OCORcO)/ (KR + OKp) (1 + KR c° + Kvc °) For the case in which there is no initial adsorption of R or P, F + o, Fp+o and Ql are, of course, zero. LIMITING BEHAVIOR AT LONG TIMES

Equations (1)-(7) can be manipulated (see Appendix I) to show that the total faradaic charge passed during relaxation is: QT/nF =

(i/nF)dt' = [2(DRDpi)~(Oc ° - Ca)r/c~o ~(D{~•+ ODf,)]'-0

+(Go_r o) + ~[DR(Fp-F; o)_ ODg(FR-Ff°)]/(DR+D~)} ,

(15)

where the first term on the right-hand side is identical to the charge that would be observed in the absence of any adsorption of reactant or product, the second term corresponds to the charge resulting from instantaneous conversion of adsorbed reactant to adsorbed product at t = 0 (discussed in the preceding section) and the third term constitutes the (time-dependent) contribution of adsorption to the observed faradaic charge after time zero. Deferring consideration of the nature of the timedependency of the third term, it suffices to note for the present that it ultimately approaches a limiting value as t ~ oe. Physically, the reason is as follows. In the absence of adsorption in a system of planar geometry it can be shown that at all times at the electrode surface : o D~'- T± ~p ,~on~ CRDf~, + cpDp = ¢R ~V

(16)

In the reversible potentiostatic case the ratio CR/Cpis fixed by the Nernst equation, and this, coupled with eqn. (16) implies that the individual values of cR and Cp at the electrode are time-independent as well. The effect of the adsorption process is to perturb the concentration so that eqn. (16) is no longer generally true. However, in terms of amount of material, the perturbation produced by adsorption is a finite one, and eventually the finite excess (or deficiency) of material corresponding to the adsorption is dissipated by diffusion. Thus at sufficiently long times eqn. (16) becomes applicable even in the presence of adsorptive complications. The time-independent values of CR and Cv ultimately achieved in the stationary state situation can be calculated by combination of eqn. (16) with the Nernst equation (eqn. (5)) and the values of FR and Fp can then be calculated from eqns. (8) and (9) or their equivalents if adsorption obeys other isotherms. We emphasize that eqn. (15) and the ultimate achievement of a stationary state do not depend on the particular adsorption isotherm or isotherms obeyed by the reactive couple, and the foregoing discussion describes results to be expected in reversibly reacting adsorbing systems generally. Specializing to the case of Langmuirian adsorption, the limiting value of the time-dependent component of adsorptive faradaic charge (the third term of eqn. (15)) in the absence of initial adsorption is : (Q~DA/nF) = { OFm(DR Kv - D~ KR) (C 0 DR "]-C° Op) / (DR -1--OD~) x [DR + OD~ + (KR+ / ~ ) (c° DR + co D~) ] } and in the presence of initial adsorption (as given by eqns. (12) and (13)) is: J. Electroanal. Chem., 38 (1972)

(17)

L A N G M U I R A D S O R P T I O N IN P O T E N T I O S T A T I C R E L A X A T I O N

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(Q~Dp/nF) = (Q2DA/nF) -- [OFm(D~RI~ - D~KR)(KR c° + I ~ c ° ) / ( D ~ + OD~) ×

(KR + 0Kp)(1 + KRc° + Kec°) ]

(18)

Figures 1-3 show the contributions of components of charge due to impulsive conversion of R to P at time zero, time-distributed adsorptive charge (calculated from eqn. (17) or (18)) and simple diffusion to the total charge under various conditions. The abscissae may be identified with potential (0 = 1 corresponds to E = E °' and a tenfold change in 0 corresponds to 60/n mV at 30°C). The figures are not entirely accurate representations of experimental results to be expected at the stated times because QAD has not reached Q2D in all cases. The curves will nonetheless serve as a useful basis for qualitative discussion. Figure 1 concerns itself with an experimental situation in which the reactant is initially adsorbed to a negligible extent (ca. 1 ~ of a monolayer) and is converted to a product much more strongly adsorbed (up to essentially Fro). Note that the significance or negligibility of components of charge due to adsorption is strongly timedependent, since the component of charge due to diffusion increases with t~ while the maximum values of the adsorptive components are fixed. Thus, the impulsive contribution, if not identically zero, must become the major component of faradaic charge at sufficiently short times, and the term "negligible" as here applied is within the context of a specified time. In the situation of Fig. 1 the negligible initial adsorption of reactant implies negligible impulsive contribution to charge. The total charge (curve A) differs from the appearance it would have in the absence of any adsorption (curve B) by the addition of a prewave. The prewave occurs because the free energy of reaction of bulk reactant to adsorbed product is more negative than that of reaction of bulk reactant to bulk product. I f / ~ increased without other modification of the parametric values of Fig. 1, the prewave would shift to the left and curve C would

I0

A

I

10-6

10-4

I 0 -2

I

0

10 2

io 4

Fig. 1. Contributions to chronocoulometric charge under "prewave" conditions. (A) Total faradaic charge, sum of B + C ; (B) diffusion component from eqn. (15); (C) distributed adsorptive component from eqn. (17) or (18). All curves calcd, for K a = 104 cm 3 mol-1, K p = 109 cm 3 mol-1, CO= 10-6 mol cm -3, cO=0, Fro=5× 10 - l ° mol cm -2, D p = D R = 1 0 - s cm 2 s -1, t = 7 . 8 6 × 10 -2 s.

J. Electroanal. Chem., 38 (1972)

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W . H . R E I N M U T H , K. BALASUBRAMANIAN

assume the form of a flat-topped plateau of limiting height 5 x 10-lO mol cm -2 (the limiting value governed by the magnitude of Fm). I f / ~ were decreased, curve C would narrow, decrease in height and shift to the right ultimately disappearing completely when Ks = Kp. Figure 2 concerns itself with the experimental situation in which reactant is not initially adsorbed, b u t its adsorption coefficient is large enough to render it adsorbable to the extent of essentially full monolayer coverage, while product would be adsorbed to a negligible extent (ca. 1 ~ of monolayer coverage) at equilibrium. The lack of initial adsorption of reactant again implies the lack of an impulsive component of adsorptive charge. In this case the time-distributed adsorptive contribution is negative. Physically, in the potential region in which this component is significant, reactant diffusing to the electrode that would be converted to product in the absence of adsorption, and thus contribute to faradaic charge, is instead adsorbed. However, at potentials sufficiently to the right of the standard potential, reaction occurs in preference to adsorption. The result is the appearance of a postwave with an E~ having a significance roughly akin to a standard potential for conversion of adsorbed reactant to bulk product. If KR increased without other modification of the parametric values of Fig. 2, the postwave would shift to the right and curve C would assume the form of a flat-topped plateau of height approaching the limit 5 x 10-lo tool cm-2 (again determined by Fro). If K R decreased the postwave would diminish in height

iO

8

A

5 oJ

8"o

t

-5 IO" 4

I 0 -2

I

I0 2

I0 4

i0 6

Fig. 2 Contributions to chronocoulometric charge under "postwave" conditions without initial adsorption. (A) Total faradaic charge, sum of B + C; (B) diffusion component from eqn. (15); (C) distributed adsorptive component from eqn. (18). All curves calcd, for same values of parameters as in Fig. 1. J. Electroanal. Chem., 38 (1972)

L A N G M U I R A D S O R P T I O N IN P O T E N T I O S T A T I C RELAXATION

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and merge with the diffusion wave, corresponding to a shift to the left, narrowing and decrease in height of curve C resulting in its ultimate disappearance when KR----/~. Figures 1 and 2 depict behavior in all ways analogous to the polarographic behavior of the same systems, the only major difference being the relative magnitudes of the adsorptive and diffusive contributions because of the longer times associated with polarographic observation. Note that the charge curves here depicted are equivalent to average current curves often determined polarographically since average current is simply Q/t. Figure 3, however, shows a number of features which may be novel to those used to thinking in polarographic terms. The system is the same as that in Fig. 2, the resultant charge-potential curve shows a postwave due to the adsorptive complication. However, the postwave adds on to the top of the diffusion wave so that in its presence the charge exceeds that due to diffusion alone. Such situations do not arise polarographically (except in the presence of convective or migrational contributions to mass transfer), and the difference is due to the possibility with stationary electrodes of the accumulation of reactant at the electrode surface by adsorption prior to electrolysis. Another interesting feature of Fig. 3 is that although the postwave has the same form as in Fig. 2 it is in fact made up of two distinct contributions: the timedistributed component of curve C and the impulsive component of curve D. In principle, because curve D corresponds to a component that occurs at time zero, while

15

lO ¢q i

E E

~5 o P

0

I 0 "2

I

I0 2

I0 4

I0 6

I0

a

0 Fig. 3 Contributions to chronocoulometric charge under "postwave" conditions with initial adsorption. (A) Total faradaic charge, sum of B + C + D; (B) diffusion component from eqn. (15); (C) distributed adsorptive component from eqn. (17); (D) impulsive adsorptive component from eqn. (13). All curves calcd. for K R: 109 cm 3 mol-1, cO= 10-6 mol cm-3, C° : 0 , Fm= 5 × 10-10 mol cm -2, DR= D e = 10-5 cm 2 s-1, t = 7 . 8 6 x 10 -2 s.

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w.H.

R E I N M U T H , K. B A L A S U B R A M A N I A N

curve C corresponds to a component distributed in time, proper selection of the time base of observation of instantaneous current (rather than charge) should allow separation of these components. With a relatively long time base (thus excluding the contribution of curve D) the result would be similar in form to a conventional polarographic wave with a "maximum" of form of curve C superimposed upon it. In fact Barker and Bolzan 6 using pulse polarographic techniques have performed" essentially this experiment in lead bromide systems, and observed such maxima which they attribute to basically this cause. The region in which curve C makes major contribution to the adsorptive component of charge corresponds to reaction of R to produce bulk P while curve D becomes prominent when the potential becomes sufficiently reactive to cause direct conversion of adsorbed reactant to product, the latter process being the thermodynamically less favorable one under the conditions assumed. TIME-DEPENDENT

C H A R A C T E R I S T I C S OF A D S O R P T I V E R E L A X A T I O N

Elaboration of the time-dependent characteristics of QAD requires detailed solution of the problem stated in eqns. (1)-(14). The problem can be transformed to an integro-differential equation of the form: (v-lp)/(1-0) =

(dtp/du)~-~(z-u)-~du

(19)

0

where the details of the transformation and the definitions of the dimensionless parameters v, ~k, and ~ are given in Appendix I. The same equation applies whether or not initial adsorption is assumed, but the range of values assumed by the dimensionless parameters differs in the two cases. Equation (19) is a non-linear integro-differential equation which admits of no general closed-form solution. Its major features, however, can be elucidated readily. The quantity ~, proportional in magnitude to QAD, is a function of two parameters : z, proportional to time; and v, equal to the limiting value of ~ as z ~ oo. The values of v of interest in the present case lie in the range - oe < v < 1. Positive values of v correspond to the physical situation in which there is net increase in adsorption during the relaxation process. Such situations arise when there is no initial adsorption of reactant, or when there is such adsorption, but the adsorption coefficient of the product is larger than that of the reactant. In either of these cases the amount of adsorbed material is greater after relaxation than before. Negative values of v correspond to net desorption during relaxation, which situations arise when the reactant is adsorbed initially, but the product is less strongly adsorbed (Ka > Kp). As v ~ + 1, the variation of q~ with z-~ approaches more and more closely a limiting form consisting of two straight line portions, the earlier of slope 2v/x/~ and the later of slope zero corresponding to the limiting value 0 = v (note that when v = l the fraction on the left-hand side of eqn. (19) reduces to unity corresponding mathematically to a singular point in the solution of eqn. (19) and physically to the limiting case of infinitely strong adsorption). This form of curve with its abrupt transition at a time corresponding to saturation of the surface with adsorbate is a common feature of the electrochemical behavior of strongly adsorbing systems 4. As v decreases from unity toward zero, the transition of the ~O vs. z ~ curve J. Electroanal. Chem., 38 (1972)

LANGMUIR ADSORPTION IN POTENTIOSTATIC RELAXATION

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between its initial slope of 2v/x/rc and its final slope of zero becomes more and more gradual, and the relaxation becomes relatively featureless. Equation (19) with the parametric range 1 > v > 0 is of precisely the same mathematical form as the equation describing diffusion-limited Langmuirian adsorption of a single species at a plane electrode which we have reported previously 12. The value of ~9 can be represented in the form of a power series in ascending powers of ¢ (see ref. 12 for mathematical details). The series converges relatively rapidly for values of z corresponding to values of ~, up to those differing insignificantly from the limiting value. Rampazzo 13 in her numerical treatment of diffusion-limited Frumkin isotherm adsorption and Holub and Nemec ~4 in their applications of analogue computing methods to diffusionlimited adsorption have also given results equivalent to solutions of eqn. (19) for selected positive values of v. For v in the range + 0.01 >~v >~ -0.01 the value of ~ can be approximated in closed form by noting that, because Ivl ~>1~'1/>0, the aforementioned range of v corresponds to the condition 1 >>I~'1. Thus, the right-hand side of eqn. (12) is approximately equal to v - ft. With the substitution of this approximation the equation becomes linear, and can be solved by conventional Laplace transformation to yield -~ v [ l - e x p ( z ) erfcx/z ] for Ivl~ 1

(20)

The range [vl ~< 0.01 is perhaps somewhat more conservative than it need be. Thus for v=0.1, eqn. (20) is accurate ~2 to 1 ~ for ~ < 0.05. Physically, the approximation corresponds to the situation in which the extents of adsorption of reactant and product are sufficiently small that the Langmuir isotherm reduces to a linear isotherm, and eqn. (20) is equivalent to results previously derived for this c a s e 6'7. Note that the approximation of eqn. (20) is a special form of the more general approximation valid for [ffJ~ 1 in which the right-hand side of eqn. (19) becomes approximately equal to v + v~9- ¢. ~' ~ I-vI(1 - v)] {1 - exp [ (1 -

Y)2T]

erfc [ (1 - v)x/z] } I~1 ~ 1

(21)

The condition v = 0 results in ~k-=0 and zero time-distributed adsorption current associated with the relaxation process. Physically the situation arises when reactant is adsorbed initially, but is converted to product with an equal adsorption coefficient (or more precisely when D~ KR = D~Kp). The result is no adsorption or desorption of adsorbate during the relaxation process and no interconversion between adsorbed reactant and product except during the initial transient. Hence, aside from the transient, the adsorption has no effect on the relaxation. The condition v = 0 also corresponds to the trivially uninteresting situation in which the adsorption coefficients are so small that essentially no adsorption occurs. An equivalent physical situation arises when the reactant is initially adsorbed strongly enough to produce surface saturation, and the product is also very strongly adsorbed. Again, notwithstanding the fact that the adsorption coefficients may differ, the surface remains saturated throughout relaxation, and interconversion of adsorbates occurs only at time zero. Still a third case resulting in no distributed adsorption current, is that in which the reactant is not initially adsorbed and the adsorption coefficients of reactant and product are equal. In contrast to the above-mentioned cases, the concentration profiles of reactant and product in solution are affected by the adsorption process. The increase J. Electroanal. Chem., 38 (1972)

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W . H . REINMUTH, K. BALASUBRAMANIAN

in amount of adsorbate with time corresponds to an enhanced flux of reactant to the surface over the value in the absence of adsorption. When product is more strongly adsorbed than reactant, the enhanced flux is reflected in a positive distributed adsorption current. When reactant is more strongly adsorbed, enhancement of flux does not compensate for the removal of reactant by adsorption, and the difference, the fraction of the flux available for reaction, is less than it would be in the absence of adsorption, corresponding to a decrease in diffusion current or a negative distributed adsorption current. In the situation in which the adsorption coefficients of reactant and product are equal, however, the enhancement of flux is just equal to the amount of reactant adsorbed, and the current assumes the same value it would have in the absence of adsorption. In this unique circumstance, the presence of adsorption has absolutely no effect on the relaxation process, because, in contrast to the corresponding case with initial adsorption of reactant, not only is there no time-distributed adsorption current, but there is no initial transient conversion of adsorbed reactant to product either. In the range - 1 ~< v ~< 0 the series representation for ~, in powers of z~ can again be used for calculation. However for v < - 1, the series rapidly becomes inaccurate (for roughly the same reason that the power series expansion of e x becomes inaccurate for calculation when x = - 10). Brief examination suggested that conversion of the series to a continued fraction according to Viskatov's algorithm ~5 would alleviate, if not remove, the difficulty, but we have conducted no detailed calculations in this region. The most useful method of obtaining data from systems of the type here considered appears to be the chronocoulometric method in which the readout is in the form of the current-time integral rather than current as function of time. The inherent virtue of the integral readout derives from the fact that it includes information concerning the initial transient conversion of adsorbed reactant to product as well as the time-distributed component. Analysis of the initial transient by direct observation of the current during the transient is likely to prove highly inaccurate, and in the mathematical extreme of the present model, namely a delta-functional form, would become meaningless. The d e l t a - f u n c t i o n representation should not be taken literally because it corresponds to an experimental system containing a potentiostat of infinite rise time, but the point is that the current-time form of the initial transient is a function primarily of the limitations of the experimental arrangement, while its integral is the physically significant quantity. Chronocoulometric data are commonly analyzed by extrapolation of Q vs. t~ plots to their zero-time intercepts. In systems of the type considered here, such plots would have two linear regions. At long times the time-distributed adsorption charge reaches its limiting value and the slope of the Q vs. t ~ plot assumes the value it would have in the absence of adsorption. The zero-time intercept corresponding to extrapolation from this region includes the sum of the initial transient charge and the timedistributed charge (plus also a double-layer contribution we have not here explicitly considered). At short times the variation of (2 with t ~ is again linear (corresponding to times at which only the first term in the expansion of ~ in powers of z~, i.e. 2vr~/rc I, is important). Extrapolation from this region yields only the initial transient charge (plus double-layer contribution). For the purpose of estimating the time at which the distributed adsorption charge has essentially reached its limiting value (i.e. the time at which the Q vs. t ~ plot J. Electroanal. Chem., 38 (1972)

LANGMUIR ADSORPTION IN POTENTIOSTATIC RELAXATION

89

has reached its long-time limiting linear slope) the following inequality is useful (the derivation is given in Appendix II). v {1 - e x p Jr/(1 - v)2] erfc [z~/(1 - v)]} .>/~9 ~>v {1 - exp (r) erfc (z~)}

(22)

The value of exp Z 2 erfc Z decreases monotonically from unity at Z = 0 to zero as Z ~ oe and reaches the value 0.1 when Z = 5.6. Thus 0 (and the time-distributed adsorption charge to which 0 is proportional) reaches 90 ~ of its limiting value when x/v is between 5.6 and 5.6(1 - v) (which is the upper limit and which the lower depends on whether v is positive or negative). The short-time linear-slope region of the Q vs. t ~-plot corresponds (except when v is very close to unity) to the region in which the second term of the expansion of in powers of z~ is negligible compared with the first. This corresponds to the inequality 1 ~> (1 -v)x/g~. Whether the linear regions of the Q vs. t ~ relation are accessible to experimental observation will depend primarily on experimental limitations having to do at short times usually with finite electronic rise times, and at long times often with advent of convective mass transfer. It is noteworthy that the range of times corresponding to linear Q vs. t ~ plots is not unlikely to vary drastically in systems with unexceptional values of adsorption parameters as the applied potential varies over the region of experimental interest. One of the implications of eqn. (22), though the proof is too lengthy to justify presentation here, is that the time required for 90 % relaxation cannot exceed the value t = 31.8 F~/CR 2 02 D though it often becomes much smaller. In the system corresponding to Fig. 3, the time corresponding to the long-time limiting slope ranges from the order of 10 -1 s at 0 = 10 a to 10-'* S at 0 = 10 7. For Fig. 1 the corresponding times range from 3 x 10 . 4 s at 0 = 10 -5 to ~ 1 s at 0 > 10 -2. The value for Fig. 2 range from 3 x 10- 4 s at 0 = 105 to ~ 1 s at 0 < 10 2. The short-time linear range in Fig. 3 is infinitesimal. For Fig. 1, it ranges from 10 -6 s at 0 = 10 - 4 to essentially the entire relaxation time at 0 = 10-1. For Fig. 2 it ranges from i 0 - 6 s at 0 = 103 to essentially the entire relaxation time for 0 ~< 1. Generally short-time linear regions tend to be substantially smaller for processes involving net desorption than for those involving net adsorption. A case of particular interest is that of Fig. 3, but with the modification Kp = 0 instead of 104 cm 3 m o l - 1, i.e. the case in which the reactant is strongly adsorbed, but the product not at all. For this case there is no impulsive contribution because of the absence of initial adsorption. The limiting form of the time-distributed adsorption charge in this case is the same as the form of the total adsorption charge curve depicted in Fig. 3. This suggests a material difference between cases in which there is small but finite adsorption of product at equilibrium and these in which there is none at all, in the region of large 0 where the former situation yields exclusively time-distributed current, and the latter exclusively impulsive current. The distinction evaporates however, on examination of the time required for relaxation of the time-distTibuted adsorption charge to its limiting value. At 0 = 108, for this case that time is 10-lo s. Even if the mathematical model could be taken literally, the experimentally observable difference between an impulse of zero duration and a relaxation of finite duration 10-10 s would be insignificant. However, the model should not be taken literally, because quite apart from limitations imposed by the assumption of reversible charge-transfer, two other sources of difficulty could be expected to arise in application 'to very short-time behavior. One of these is that at

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W. H. REINMUTH, K. BALASUBRAMANIAN

sufficiently short times the diffusion layer thickness would no longer be large in comparison with the thickness of double-layer inhomogeneity. The second problem is peculiar to situations corresponding to desorption, and can be illustrated concretely by reference to Fig. 3.At 0 = 10s, the impulsive current is sufficient to convert essentially all adsorbed reactant into adsorbed product. Before the start of the experiment the surface was saturated with adsorbed reactant in equilibrium with millimolar homogeneous concentration of reactant. After the impulsive conversion of adsorbed reactant to product, the adsorbed product, at least according to the mathematical model, must be in Langmuirian equilibrium with the homogeneous concentration of product in the immediate vicinity. However, because the adsorption coefficient of product is a factor 105 less than that of reactant, the homogeneous concentration of product in equilibrium with the adsorbed product must be a factor of 105 greater than the concentration of reactant required to maintain equilibrium before time zero. This implies a homogeneous concentration of product of the order of 100 mol 1-1 ( a difference of a factor of 109 in adsorption coefficients would correspond to 106 mol 1-1 !). Clearly such predictions cannot be taken at face value. ADSORPTION OBEYING OTHER ISOTHERMS

While we have considered thus far only Langmuirian adsorption, it is of interest to inquire to what extent our conclusions are applicable to systems described by other isotherms. As indicated previously, the expressions for the impulsive charge and the limiting value of time-distributed charge can be given in forms independent of the particular isotherm (eqn. (15)). This implies that contributions of qualitatively the same form as those depicted in Figs. 1-3 are to be expected whatever the form of the isotherm. The major difference is that in the general case the adsorption coefficients would not be potential-independent, and the curves of Figs. 1-3 would be distorted to a greater or lesser degree depending on the magnitude of this effect. The detailed form of the relaxation vs. time is, of course, a direct function of the adsorption isotherm. However the two most prominent features of the relaxation are not. The existence of a long-time linear portion of the Q v s . t ~ relation relies merely on the fact that the adsorption process eventually reaches a stationary state, while a short-time linear portion is to be expected for any isotherm in which deviations from the initial state of the system can be expressed in the form of a Taylor expansion around the initial state (see Appendix III for details). QAD =

2t~[c°KpDRO--c°KRDR+C

+°

K R D ~• O - c P+o K p D ~1 ] / ~ (1K R q ' - O K p )

(23)

The last significant difference between Langmuirian systems of the kind here considered and adsorptive systems in general, has to do with the effect of doublelayer charging on the relaxation process. Our assumption of lack of variation of adsorption coefficient with potential implies thermodynamically that the charge of the electric01 double layer is independent of the extent of adsorptions of reactant and product, since variation of surface charge with chemical potential of a species is proportional in magnitude to the variation of its adsorption with potential x5. Adsorption coefficients are, of course, not generally potential-independent, and when they are not, variation of adsorption with time is accompanied by a flow of non-faradaic current as well as the faradaic current we have here considered. The net result is that in the J. Electroanal. Chem., 38 (1972)

L A N G M U I R A D S O R P T I O N IN P O T E N T I O S T A T I C R E L A X A T I O N

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general case the distributed adsorption charge as well as the initial impulsive charge contains a non-faradaic component in addition to the faradaic component. It is the presence of this added component that produces the complication in the separation of faradaic and non-faradaic currents pointed out by Delahay iv and discussed in terms more directly applicable to the present case by Holub 3. While we have explicitly considered here only single-step potentiostatic perturbations, there appears to be substantial experimental virtue in employing double step 11, the second step returning the system to its initial potential. Brief remarks concerning the implications of the results of the present investigation for such a technique may be useful. The major difference between the systems described here and those usually examined chronocoulometrically is that in the latter case experimental conditions are so arranged that there is no significant time-distributed adsorption component of current. This allows data analysis by extrapolation to time-zero of relatively simple functions i.e., Q. vs. t ~ for the first step, and Q vs. t ~ - ( t - tR)~ (where tR is the time of the start of the second step) for the second step. As we have'indicated, the same extrapolation is useful for the initial step in the present case, and has its usual chronocoulometric significance provided that the extrapolation is conducted from times sufficiently long that the time-distributed adsorptive component of charge has reached its limiting value. Provided also that the second step is initiated at times corresponding to this limiting value, the usual chronocoulometric plot can be used for analysis of the second step, if either no time-distributed adsorptive component accompanies this step or the extrapolation is again conducted from times sufficiently long to correspond to the limiting value of the time-distributed component. We forego the lengthy proof of this assertion, but note that intuitively it derives from the fact that when the time-distributed adsorption component has relaxed to its limiting value, the concentration profiles in solution have become the same as they would have been in the absence of the complication. APPENDIX I

In order to avoid confusion over impulsive contributions arising at time zero we begin with the boundary value problem of eqns. (1)-(14) with the modification that the potential step occurs not at t = 0 but at t = b > 0, and defer the process of taking the limit of the result as b ~ + 0. From eqns. (1)-(3) it can be shown readily by conventional Laplace transform techniques that in Laplace space at x = 0: O.(OeR/~X) = [(cO~s) -- e.] (SOR) ~

(A1)

Op(~ep/~X) = [(cO~s)-- ep] (SOp)~

(A2)

and

where s is the transform variable and barred functions represent the Laplace transforms of their real-time analogues. The transform of eqn. (4) is : D R(~??R/t3X)+ Dp (03p/OX) = sFR- r ° + sFp - Fp°

(A3)

where FRo and Fp° are the values of FR and Fp at t = 0. Substituting the values of DR (dCR/ J. Electroanal. Chem.. 38 (1972)

92

w.H. REINMUTH, K. BALASUBRAMANIAN

0x) and Dp(~Cp/~X) from eqns. (A1) and (A2) into (A3), dividing both sides by s~ and inverse transforming gives : DR(COR--CR)+Dp(CO--CP) = f'o [ ( d f i R / d t ) + ( d f i p / d t ) ] ~ ½ ( t - t ) - ½ d t '

(A4)

For t > b, the integral on the right-hand side of eqn. (A4) can be split into three segments, the first covering the time interval 0 < t < b, the second covering the discontinuity at b and the third the time interval t > b. During the first of these intervals fir and fip do not vary, their time derivatives are zero and the integral is zero. At t = b both fir and fip are subject to step-functional discontinuities. The derivatives of these discontinuities are Dirac delta-functions of the form (fiR(t= b ) - fi°)6(t: b) and (fiP(t = b ) F°)6(t=b). However, (fiR(t=b)--fiO)=--(fia(t=b)--fiO), that is, the change in fir at t = b is equal and opposite to the change in fip so that the delta-functional contributions cancel. Thus eqn. (A4) is equivalent to the corresponding equation in which the lower limit of the integral on the right-hand side is replaced by b. Taking the inverse Laplace transform of eqn. (A1) yields : DR(OCR/OX) = --

i

t (OCR/Ot')D~z~ . . .~.(t-. . . . t ) ~dt 0

(A5)

Again the interval of integration can be divided into three segments; t < b for which the result is zero, t > b for which the integral is that on the right-hand side of eqn. (A5) with the lower limit of zero replaced by b, and the point of discontinuity t = b. Because CR undergoes step functional change from c ° to c b at t = b , its derivative is again a delta-function (CbR- C°)6t= b" The integral of the product of a delta-function. Thus eqn. (A5) becomes, for t > b : DR(OCR/OX) = (cO , c ~ ) D R l Z _ ~ ( t _ b ) _ ~ _

j

.t (~CR/~t ,)D~Tz~(t--t') -~dt' • 1 b

(A6)

Inverse transformation of eqn. (A6) yields an analogous expression for Dp(t~Cp/C~x). Adding eqn. (A6) to its analog for Cp and noting that, lot t > b, Cp= OCR gives : DR(C~CR/C~X) + DP(~cv/Bx) =

~ 0

0 -- OCR)] b 7r-~"(t-- b) + D~~ (Cp

Equation (A7) can be combined with eqn. (4) to eliminate the partial derivative with respect to x. The result can be manipulated algebraically to give an explicit expression for the integral of eqn. (A7). This expression can be substituted into eqn. (A6) to eliminate the integral from that equation. The result of these manipulations is an expression for DR(OCR/OX) in which cb as well as the convolution integral involving it have been eliminated. Substituting this result into eqn. (6), integrating from b to t, taking the limit as b ~ + 0, and adding the impulsive contribution of faradaic charge at t = 0 given by nF (fiRO_fi+o) yields eqn. (15) of the text. In eqn. (A4) (with the lower limit of integration set equal to b), Cp, CR and fip can be eliminated by the successive substitutions Cp= OCR, CR----fiR/KR(fim-- fiR--'fiP) and fip=OKRfiR/Kp. This results in an integro-differential equation expressing FR as a function of t and various constants. The value of QAD/nF (the last, term on the right-hand side of eqn. (15) of the text) can also be expressed as a function of fiR and constants by eliminating fip through the substitution fiR = OKpfiR/KR. Combining the J. Electroanal. Chem.. 38 (1972)

LANGMUIR ADSORPTION IN POTENTIOSTATIC RELAXATION

93

integro-differential equation with the expression for QAD/nF to eliminate F R results in an integro-differential equation for QAD/nF as a function of t and various constants. This reduces to eqn. (19) of the text on the substitution of the following dimensionless parameters : = QAD(KR + OKp)(DR + OD~)/nFO(Fm - Fd o _ F f o)(KRDR _/~D~) v =

(AS)

(c° D R + co U~)(KR+ 0F,e)(Fm - F~ o _ F~- o) _ (DR + OD~)(F~ o + F~ o) (A9) (Fm - F~ o __Fp+o) [ (KR + Oil)(C ° D~ + co D~) -- (D~ + OD~)]

z = [(DR+OD~,) ~ + (CRD~+c°D~)(KR+OI~)]2t/(Fm--Ff°--~F~-°)2(KR+OI~)2 o ±

(AI0)

APPENDIX II Since ~ of eqn. (19) lies in the range 0~< 10[~< Ivl, we have from eqn. (19): f~ (d~/du)~-}(z, - u)-}du

(v-~)/(1 -~) >/(v-0)

(All)

0

Now ¢ can be written in the form: ~9 = v [ 1 - exp (z) erfc (z~)] + e,

(A 12)

where e, is a function of unknown form. Substituting (A12) into (A11) gives :

f i (de,/du)lr-i ( z - u ) - ~ d u >~ - e ,

(113)

but applying the mean value theorem to the integral of eqn. (A 13), this integral can be replaced by e,Tr-~(Z-Zo) -~ where 0~< Zo ~< z. Or ~ , ~ - ~ ( r - Z o ) -~ ~> - e ,

(A14)

Since (z-~o) -~ is positive the inequality of eqn. (A14) implies that e, >~0 or from eqn. (112) >~v[i - exp (z) erfc (z~)]

(A15)

which is the right half of the multiple inequality of eqn. (24) of the text. Similarly from (v - ~)/(1 - ~) ~< (v - ~9)/(1 - v)

(A16)

~9= v {1 - exp [z/(1 - v)2] erfc [z~/(1 - v)] } + e

(A17)

and the left-hand side of the inequality of eqn. (22) of the text derives. APPENDIX III Take a system of the type described by eqns. (1)-(7) with equilibrium between adsorbing and diffusing species so that under potentiostatic conditions: t > / + 0 ; FR=fR(~.),

rp=fptcp)

(A18)

wherefR andfp represent arbitrary adsorption isotherms. Note that it is sufficient to specifyfR as a function of CRrather than as a function of both Cp and CR because CR and Cpare related through eqn. (5). IfJR andfp are continuous functions in the region around c~-o and c{ o then they can be Taylor expanded e.g."

J. Electroanal. Chem., 38 (1972)

94

W. H. REINMUTH, K. BALASUBRAMANIAN

FR 7--Fr~ ° = (~FR/OCR)c~ =c~O(CR--C+ 0) +...

(A19)

where for sufficiently small variations from the initial condition, corresponding to CR close to c~ o and t close to + 0, higher order terms in the Taylor expansion can be neglected. However eqn. (A19) and its analog for P represent linear isotherms. Thus coupling these equations with eqns. (1)-(7) and solvingv gives a result of the form of eqn. (20) in which: KR I'm = (OFR/OCR)c, =c~+o;KeFm= (OFp/~Cp)~p - ~ o

(A20)

But, since eqn. (A19) and its analog for P are applicable only for short times in any case, the exp erfc function of eqn. (20) can be expanded in a power series of its argument : exp (z2) erf(z)- 1 - (2z/re~) +... Retaining only the first two terms in the expansion gives eqn. (23) of the text. SUMMARY

A theoretical treatment is given of the effect of Langmuirian adsorption on Nernstian, diffusion-limited electrode reactions at a plane electrode under potentiostatic conditions. Relaxation currents generally consist of three components, an instantaneous initial conversion of adsorbed reactant to product, the same diffusional component that would be observed in the absence of adsorption, and a time-distributed adsorptive component due to interconversion between adsorbed and diffusing species. The magnitude, sign, and time-constant associated with the time-distributed adsorptive component vary widely with experimental conditions. In favorable situations the magnitudes of both adsorptive components can be determined conveniently by extrapolation of two separate linear segments of a chronocoulometric plot of charge vs. square root of time to time zero. This feature and other prominent effects of adsorption are shown to be qualitatively independent of the detailed form of the adsorption isotherms. REFERENCES 1 2 3 4 5

R. Brdi~ka, Z. Electrochem., 48 (1942) 278. K. Holub, Collect. Czech. Chem. Commun., 31 (1966) 1461. K. Holub, J. Electroanal. Chem., 16 (1968) 433. J. Koryta, Collect. Czech. Chem. Commun., 18 (1953) 206. (a) F. C. Anson, Anal. Chem., 38 (1966) 54; (b) F. C. Anson and D. J. Barclay, Anal. Chem., 40 (1968) 1791; (c) F. C. Anson and J. Caja, J. Electroehem. Soc., 117 (1970) 306. 6 G. C. Barker and J. A. Bolzan, Z. Anal. Chem., 216 (1966) 215. 7 R. Guidelli, J. Electroanal. Chem., 18 (1968) 5. 8 K. Holub, J. Koryta, Collect. Czech. Chem. Commun., 30 (1965) 3785. 9 V. G. Levich, B. I. Khaikin and E. D. Belokolos, Electrokhimiya, 1 (1965) 1273. 10 R. Guidelli, J. Phys. Chem., 74 (1970) 95. 11 J. H. Christie, R. A. Osteryoung and F. C. Anson, J. Eleetroanal. Chem., 13 (1967) 236. 12 W. H. Reinmuth, J. Phys. Chem., 65 (1961) 473. 13 L. Rampazzo, E~eetrochim. Acta, 14 (1969) 733. 14 K. Holub and L Nemec, J. Electroanal. Chem., 11 (1966) 1 ; 18 (1968) 209. 15 A. N. Khovanski, Applications of Continued Fractions and Their Generalizations to Problems in Approximation Theory, P. Noordhoff, Groningen, 1963. 16 D. M. Mohilner in A. J. Bard (Ed.), Electroanalytical Chemistry, Vol. I, Marcel Dekker, New York, 1966, p. 288. 17 P. Delahay, J. Phys. Chem., 70 (1966) 2373. J. Electroanal. Chem., 38 (1972)