On the influence of electrode curvature and growth in d.c. and a.c. polarography: The e.e. mechanism with amalgam formation

Electroanalytical Chemistry and lnterfacial Electrochemistry, 57 (1974) 129-139 © Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

129

ON THE INFLUENCE OF ELECTRODE CURVATURE AND GROWTH IN D.C. AND A.C. POLAROGRAPHY: THE E.E. MECHANISM WITH AMALGAM FORMATION

IVICA RU~I0* and DONALD E. SMITH**

Department of Chemistry, Northwestern University, Evanston, lllinois 60201 (U.S.A.)

(Received 12th July 1974; in revised form 6th September 1974)

INTRODUCTION

Influences of expansion and curvature attending the dropping mercury electrode (DME) are widely acknowledged to be significant second-order effects in d.c. a-24 and a . c . 2 5 - 3 3 polarography. It has been shown that contributions of these geometric properties of the DME can be quantitatively, or even qualitatively, rather important in many situations. Because of the complexity of the expanding sphere boundary value problem 4-22, rigorous d.c. or a.c. polarographic rate law derivations based on this model are rare s' 6,12,13,17-22, 32, 33 and usually are confined to simple single-step electron transfer processes. Most attempts to account for these effects are approximate. For example, the so-called expanding plane model 5' 6 is invoked, thus ignoring curvature effects, and even this approximate model often is subject to further approximation via steady-state treatments of various types 5' 23. 34-36

Recently, a detailed study of d.c. polarographic current-potential curves for single-step irreversible and quasi-reversible electrode processes has revealed that certain approaches which have been used in the interpretation of results obtained at a DME are in error due simply to imprecise treatment of the expanding plane boundary value problem 23. Similarly, when comparisons have been made 5-22' 26-33, it is indicated that neglect or inexact treatment of electrode curvature can have moderate-to-serious consequences in applying rate laws to d.c. and a.c. data obtained at a DME, particularly when amalgam formation occurs I 2,13,19, 27-31. Con- sequently, despite the somewhat sporadic interest to date, we feel that use of d.c. and a.c. polarographic rate laws based on rigorous solutions of the expanding sphere boundary problem should be encouraged for the analysis of data obtained at the DME. Derivational difficulties are no longer severe because the digital simulation technique 37 has been applied successfully to predict the d.c. response with an expanding sphere in a manner which is applicable to any desired mechanistic scheme with and without amalgam formation 38. The scheme is readily adapted to a.c. poiarography 32'33. In theoretical studies to date which have used this development, the importance of rigorously accounting for drop growth and curvature has been

* On leave from the Center for Marine Research, Rudjer Bogkovi~ Institute, Zagreb, Yugoslavia, 1972-1975.

** To whom correspondence should be addressed.

1 3 0 I. RUZIC , D. E. S M I T H

demonstrated for a.c. polarography with several mechanistic schemes 27-33' 39 and, in several instances, the realization of satisfactory theory~experiment agreement has depended on proper introduction of the spherical correction 32, 33, 39. The importance of the coincidence of amalgam formation with the geometric properties of the DME, as suggested in earlier, more approximate work 27 31 is reaffirmed in these latest studies 39.

The latter conclusions primarily were deduced from consideration of mech- anisms involving simple single-step heterogeneous charge transfer. However, an intuitive extrapolation to cases involving multi-step heterogeneous processes suggests that rigor in the treatment of the mass-transfer rate procCess should be equally, if not more important, in these cases. This conjecture is supported by a recent careful study of drop growth effects and the influence of various levels of rigor in rate law derivations (excluding curvature effects) with the so-called "e.e.-mechanism "4°,

ks, l, ~1 A + n l e ~ B ("first" step, E °, E~,I) (I)

B-t-nEe ~ C ("second" step, E °, E~,2) ks, 2, ~2

These results strongly emphasize the need for rigorously-derived rate laws for mechanism I. A similar study of the sensitivity of the a.c. polarographic rate law to derivational rigor has not been provided, and a.c. rate laws which have been published for this mechanistic scheme 4~,42 are all quite approximate for the DME.

The latter observations, together with recognition that many stepwise reduc- tive electron-transfer processes lead to amalgam formation, has led us to examine and compare with more approximate models the predictions of expanding sphere d.c. and a.c. polarographic rate laws for mechanism I, with emphasis on cases involving amalgam formation. As expected, substantial drop growth and curvature effects are found. Typical results of this study are reported here.

THEORY

A description of the mathematical procedure for evaluation of the boundary value problem of mechanism I will be omitted here because it is only a special case of a separately-published derivation 43 of greater generality which includes influences of the homogeneous redox reaction ("nuance"), A + C~2B. Also, two other publications 4 o, 44 are recommended for detailed descriptions of the derivational methods used in solving the d.c. part of the boundary value problem.

These derivations do not consider depletion effects ¢5~9, or consequences of non-uniform current densities arising from capillary shielding 12 and non-concentric drop growth 24. However, the simple expedient of using a sharp-tipped capillary reduces perturbations due to depletion effects and capillary shielding to ~< 1~o (approximately) ~2'5°, which is more than an order-of-magnitude smaller than the amalgam-formation-induced curvature effects on which this report focuses. Further, depletion effects are completely eliminated with so-called "first drop" experiments 45, which can be automated conveniently with modern electronic tools51,52. Only the non-concentric drop growth problem 24 is immune to simple experimental strategies, but calculations have shown convincingly z4 that this

C U R V A T U R E E F F E C T S I N E . E . - M E C H A N I S M W I T H A M A L G A M S 131

perturbation amounts to 1-2%, which is, again, considerably sm~iller than the curvature effects addressed here.

FORTRAN programs used for calculations presented in this paper are available from the authors on request. The programs allow calculations to be performed on the basis of stationary plane, expanding plane, and expanding sphere (with or without amalgam formation) models.

R E S U L T S A N D D I S C U S S I O N

I. Predictions for the case of a very unstable intermediate (E~ ~ E °) In this situation, a single polarographic reduction wave is observed, unless

~, 2 ~ ~, 1'and/g, 2 also is extremely small in the absolute sense (the latter possibility is not considered here). This unstable intermediate case is presumably very common in inorganic systems, although it is often difficult to definitively establish that a

+1

4.0

-2

A ~ / /

~/~:" /t.~Xlrreverslble /i~I / port

~il!! // IJ//.!IY ///

//.2 .- l

......

~0 d

o 20 =

1.0

I 0.20

"9-

i

4o I

0.10 0 -0.10 -0.20 -030

EDC / volts

20

-o.'~o -o~.lo ~ 0'.,0 o'.~o EDC/ volts

e mode ls

i 0.20 Ok.lO 0 -0110 -0.20

E D O / v o l t s

Fig. 1. Exam ple s of d.c. and a.c. polarographic responses predicted by rate laws based on various electrode models for the e.e.-mechanism with unstable intermediate. Parameter values: T = 2 9 8 K, DA=DB=Dc=4.0×IO -6 cm 2 S 1, t = l . 0 0 s, ~J=250 s 1, EO= _ E O = - 0 . 5 0 V, / g , 1 = / ~ , 2 = 2 0 c m s a, ~1 =~2 =0.50, m = 1.0 m g s -1. ( - - ) Expanding sphere result with amalgam formation; ( • - - ) expanding sphere result without amalgam formation; ( - - ) expanding plane result; ( . . . . . . ) stationary plane result; ( . . . . . . ) irreversible and reversible segments for approximate method. Alternating current is given in units of 102 RTl(o~t)/FZA(2o~O)~c * (same notation used as in literature 32' 33).

132 I. RUZIC, D. E. SMITH

two-step mechanism is operative, rather than a single multi-electron step. Some earlier papers dealing with this case 4°'41 have pointed out that apparent "overall" heterogeneous rate parameters (see ref. 40 for discussion of the meaning of overall rate parameter) deduced by assuming a single-step process will be quite different from the actual individual rate parameters, k. 1 and k,. 2, associated with the stepwise process. For example, if k, 1 and/g, 2 are of the order of 1 cm s- 1, and the stability constant of the intermediate, Species B, is about 10-15 (E 0 _ E o_~ 0.4 V for T = 25°C, n = 1), the apparent overall heterogeneous rate constant will be about 10-a cm s-l . Figure 1 shows typical results of d.c. and a.c. polarographic calculations for this kind of situation. It is evident in this illustration that, despite very large lg, 1 and /g,2 values, the predicted system behavior is decidedly quasi-reversible, regardless of the electrode geometry model invoked. In the d.c. polarographic log plot (Fig. 1A), it can be seen that the influence of charge-transfer kinetics follows the order: stationary plane < expanding plane < expanding sphere without amalgam formation (Species C soluble in solution phase)~ expanding sphere with amalgam formation. This ordering is as expected, because the same ordering applies to the mass-transfer rates which are driving the heterogeneous process. The implication is that application of an approximate model (e.g., stationary or expanding plane) to DME data would lead to the conclusion that the heterogeneous process is slower than is actually the case. The d.c. polarographic diagram in Fig. IA also includes the so- called irreversible and reversible parts of the log profile deduced from an approximate steady-state method described elsewhere 4°. The fundamental harmonic a.c. polarographic current amplitudes are shown for the same case in Figure lB. For this observable, the status Of the d.c. process, including attendant electrode geometry factors, is quite important 26-33, which leads to significantly different predictions from the various electrode models regarding the magnitude and shape of the a.c. polarogram. Within the framework of the expanding sphere model, the influence of amalgam formation is seen to be non-trivial. The phase angle, which is less sensitive to the status of the d.c. process under the conditions in question, shows negligible dependence on the electrode model employed (Figure 1C). The latter is not a general result, as will be shown subsequently.

II. Predictions Jor the case of a moderately stable intermediate (E ° ~ E °) Here we consider the situation where the E°-values for the two heterogeneous

steps are comparable--i.e., Species B has moderate thermodynamic stability (significant equilibrium concentration). Under these circumstances incompletely resolved polarographic waves are observed, although wave shape characteristics usually provide clear evidence that a stepwise process is operative, in contrast to the result with highly unstable intermediates. The fundamental harmonic a.c. polarographic response will be emphasized in this discussion because the effects under consideration are more significant, as well as more often ignored, than with d.c. polarography.

Figure 2 illustrates results of some calculations for conditions where ks. 1 and ks.2 are sufficiently large that, given the small E°-value separations, the d.c. process is diffusion-controlled (nernstian). In these circumstances electrode curvature effects are quite small in the absence of amalgam formation 27'29. Thus, polarograms in Figure 2 which are based on the expanding plane model are almost identical to those predicted by the expanding sphere model when Species C is soluble in the

133

4.0

5.0 5.Ù

4.0 40

3.0 ~ 3.0 ; °

~ 2o E 2.o

i o t .o

+010 0 -0.10

EOC / volts

E 3,0

~ 2.0

1.0

+0.10 0 -010

EDc/VOlfS

CURVATURE EFFECTS IN E.E.-MECHANISM WITH AMALGAMS

I r

+0.10 0 -OlO -0.20

EDC / volts

D 5.0

4.C

) 3.0

"9-

2.0

1.0

i i i i

+020 +010 0 -0.10

EDC / volts

Fig. 2. Examples of a.c. polarographic responses predicted by rate laws based on various electrode models for the e.e.-mechanism with moderately stable intermediate. Parameter values: same as Fig. 1, exceptDa=DB=Dc= 1.0 x 10 5 cm 2 s 1, and(A) k~.l = ks,2=0.10 cm s l, el =a2 =0.50, E ° = E ° = 0 . 0 0 V; (B) ks,a=k~.2=0.10 cm s -1, ~,=0.20, ~2=0.80, E ° = - E ° = 0 . 0 5 0 V; (C) ks j = 0 . 0 1 0 cm s l, ks,2= 1.00 cm s -1, ~1=0.80, ~2=0.20, E ° = -E2°=0.050 V; (D) (1) conditions of Fig. 2A. (2) conditions of Fig. 2B, (3) conditions of Fig. 2C. ( - - ) Expanding sphere result with amalgam formation; ( ) expanding sphere result without amalgam formation and expanding plane result; ( . . . . . . ) stationary plane result. Alternating current given in units of IORTI(vJt)/F2A(2v3D)~c *.

solution phase (except where diffusion coefficients of the various species differ inordinately). However, when Species C is assumed to exist in the amalgam state, a.c. polarograms based on the expanding sphere calculation are strikingly different from those predicted by the other electrode models considered (Figs. 2A-2C). In Fig. 2B, current magnitudes in the vicinity of the second (more negative) peak cal- culated from the expanding sphere model with amalgam formation are nearly double

134 [. RUZlC, D. E. SMITH

those yielded by the planar diffusion models. As with the conditions of Fig. 1, one observes small to totally negligible effects of electrode geometry in the predicted behavior of the phase angle cotangent (Fig. 2D).

Figure 3 shows some calculational results for conditions where/~. 1 and ks, 2

t A 40 4.0

g = 30 3.0

V

+O.I

,el 3.0

"0-

2.0

i 0 -0.10 -020

E o c / v o l t s B

LO

.!!, -e-

1.0

*o~o 6 -o'~o -d2o

+0.10 0 -OllO 1020 E DC/ volts

4.0 / ' : I O

3.0

20

1.0

+0.10 0 -0. I0 -020

EDC / volts Eoc / volts

Fig. 3. Examples of a.c. polarographic responses predicted by rate laws based on various electrode models for the e.e.-mechanism with moderately stable intermediate. Parameter values: same as Fig. 2, except: (A), (B) /~,1=0.030 cm s -1, k. 2=3.0 cm s -1, e1=~2=0.50, E °= -E°=-0.050 V; (C), (D) /~.1 =0.0030 cm s 1, /~.2=0.3 cm s 1, cq=~2=0.50 ' EO= _EO=0.010 V. Notation same as Fig. 2.

differ appreciably. Further, k~, 1 is sufficiently small that the first heterogeneous step is nearly irreversible, even in the d.c. sense. Under these circumstances, it is known that a number of interesting effects can arise 4l. For example, the existence of a stepwise process may be more clearly manifested in the phase angle observable than in the current amplitude polarograms (e.g., compare Figs. 3C and 3D). Small, but noticeable electrode geometry contributions can be ascertained in absence of amalgam formation under these conditions (not shown), but here again they are relatively trivial compared to the rather large amalgamation-induced curvature effects illustrated in Fig. 3. Here one also finds that the phase angle cotangent is geometry dependent (Fig. 3B, D).

CURVATURE EFFECTS IN E.E.-MECHANISM WITH AMALGAMS 135

v E ®

c)

"~ 03

1.5

lot) 92 ro-J 2 0

i O.,2S "~ 2 02e2 Z 0 ~e4 ~ 1.5

c

0 5

z: i r I

+0.20 +O.iO 0 -0 . I0 -0 .20 - 0 3 0

E o c / v o l t s

B

4 0

"B- 3.0

2 0

I.O i - L _

0 2 0 0110 0 -0110 ~ 0,120 ~0.30

EOC / v o l t s

5.0

4.0

"~" 30

2.0

I 0

C

___ J +020 +0~0 0 -010 -020 -0.30

EOC / volts

O

+020 +0.I0 0 ~010 - C j ~

EDC / volts

Fig. 4. Examples ofa.c, potar0graphic responses predicted by expanding sphere model for e.e.-mechanism: effect of sphericity. Parameter values: same as Fig. 3C, D, except sphericity values shown on Figure. (A), (B) Results without amalgam formation; (C), (D) results with amalgam formation.

Figures 4 and 5 show effects of varying the "sphericity parameter", (D@/r o (D = diffusion coefficient = DA = DB ---- Dc, t = time and r o = electrode radius) on the a.c. polarographic response. Figure 4 considers conditions similar to those depicted in Figs. 3C, D, whereas Fig. 5 involves a situation where the second heterogeneous step is slowest, which leads to two distinguishable waves. The same effects on d.c. polarographic log plots are illustrated in Fig. 6. Figures 4 and 6 provide comparisons of these effects in the presence and absence of amalgam formation.

III. Predictions for the case o f a very stable intermediate (E ° ~ E °) We will not present here explicit examples of predictions for this case where,

except for very small ks, l, one obtains totally resolved waves corresponding to the two heterogeneous steps. Electrode geometry effects, including those attending amalgam formation, are very similar to those found with single-step processes which have been discussed elsewhere 26-31'39. Under these circumstances, the ubiquitous curvature effects attending amalgam formation 27-31'39 will be manifested on the second wave, whereas the first wave will show negligible electrode geometry effects unless it is quasi-reversible or irreversible in the d.c. sense 26' 27.


0.40 I

l_

~0.20 oJ

A (D t ) ~/2 r(~ I

I 0.077 2 O, lf16 3 0.283

4 0,449

0.10

+0.20 +0.10 0 -0.10 -0.20 -0,30 -0.40

EDC / vo l ts

12 B

I0

8.0 "e-

6.0

4.0

2.0

+o.'2o -.o.'lo ; -o'.,o -&o -o!3o E o c / v o l t s

Fig. 5. Examples of a.c. polarographic responses predicted by expanding sphere model for e.e.-mechanism: effect of sphericity. Parameter values: same as Fig. 4, except/g. 1 = 0.030 cm s 1,/g. 2 = 3.0 x 10- 5 cm s and all results with amalgam formation.

CONCLUSIONS

From the few sample predictions given above, it is evident that the detailed geometric characteristics of an electrode can influence substantially the properties of d.c. and a.c. polarographic responses with the e.e.-mechanism. While this conclusion is widely recognized for d.c. polarography, it has been largely ignored for a.c. polarography, where the effects in question actually are more significant, some- times yielding current enhancements approaching a factor of two. When amalgam formation is operative, geometry effects on the a.c. response magnitudes are both ever-present and substantial. An inescapable conclusion of the results illustrated is that one must invoke a rate law based on the expanding sphere model when dealing quantitatively with D M E current amplitude data from an amalgam-forming e.e.-mechanism.


o ~

I

+1

0

- I

- 2

( D t ) Vz ro- I

I 0 J 2 5

2 0.381

3 0 .564

4 0 .0769

5 0 .283

6 0 , 4 4 9

I I I i I +0.20 +0.10 0 -0.10 - 0 2 0 -0.50

E DC/ vo l t s

Fig. 6. Examples of d.c. polarographic responses predicted by expanding sphere model for e.e.-mechanism: effect of sphericity. Parameter values: sphericity values shown on Figure. ( - - ) Conditions of Fig. 5; ( - - ) conditions of Fig. 4A, B; ( . . . . . . ) conditions of Fig. 4C, D.

Negligible geometry effects can attend certain combinations of a.c. polarographic observables and experimental conditions, such as: (a) current amplitudes in absence of amalgam formation when the d.c. process is nernstian; (b) phase angle cotangents with nernstian d.c. processes. In these latter situations use of the usual approximate DME models, like the expanding plane, is acceptable, although this option should be applied with extreme care based on sufficient information to establish that geometry effects are unlikely. Fortunately, in a.c. polarography the existence or absence of geometry effects is readily established by examining the mercury column height dependence (time-dependence at constant area) of the observable of interest, because significant geometry effects always are accompanied by a column height dependence, and vice versa 26-a°. In DME work, we recommend as a necessary condition for use of rate laws based on approximate electrode models the empirical observation of negligible column height dependence. Otherwise, use of the expanding sphere model is essential for quantitative data analysis.

ACKNOWLEDGMENTS

This work was supported by NSF Grant GP-28748X and by financial support from the Northwestern University Computing Center.


SUMMARY

Predictions of d.c. and a.c. polarographic responses with the e.e.-mechanism obtained from various electrode models, ranging from the stationary plane to the expanding sphere, are compared. Electrode curvature effects are found to be substantial, not only in d.c. polarography where this effect is long-acknowledged, but also in a.c. polarography where the problem is usually ignored. Large curvature effects attend electrode reactions which lead to amalgam formation, the a.c. polarographic current amplitude being most susceptible. The results presented suggest that proper quantitative analysis of a.c. data obtained with a DME often will require use of rate laws based on rigorous treatment of the expanding sphere boundary value problem.

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and Instrumentation Vol. 2, Marcel Dekker, New York, 1972. 45 J. Kfita and I. Smoler in P. Zuman, L. Meites and 1. M. Kolthoff (Eds.), Progress in Polarography,

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Documents

On the influence of electrode curvature and growth in d.c. and a.c. polarography: The e.e. mechanism with amalgam formation