Theory of polarographic maxima of the first kind

Elwtrochimrro km,

1973. Vol. 1X, pp. 427-431.

THEORY

Pergamon Pres.*.Printed tn Great Britain

OF POLAROGRAPHIC FIRST KIND H. H.

Department

MAXIMA

OF THE

BALJER

of Chemistry, The University, (Received 4 December

Southampton,

England*

1972)

Abstract-Maxima of the first kind involve streaming that arises from inequalities of the surface tension at different parts of the mercury drop. These inequalities arise from differences in the local interfacial potential and the latter arise from a non-uniform current-distribution. It is pointed out that the requisite differences in interfacial potential can arise only in the region of the rising part of the polarographic step, and only if the concentration of depolarizer exceeds some particular lower value. Thus, one expects quite narrow streaming maxima: contrary to previous ideas, it is negative maxima that are “normal” and positive ones that are anomalous. Positive maxima cover a wider range of potentials because the flux of depolarizer, induced by the streaming, prevents concentration polarization at the neck of the drop. Experiments with fine-tipped capillaries, designed to decrease the non-uniform current-distribqtion. have been inconclusive because the measured currents were at least partly controlled by the resistance and not solely by the extent of shielding.

INTRODUCTION

It is widely agreed that polarographic maxima of the first kind reflect streaming of the solution in the vicinity of the electrode as a result of movement of the mercurysolution interface due to inequalities of surface tension at different places on the surface. The inequalities of tension are due to differences in electrode potential which arise from a non-uniform distribution of current across the surface[ l-91. However, there has been no agreement concerning other details of the processes, chiefly the following: (1) Why do maxima cease at some potential? (2) Why are negative maxima (reduction potential cathodic to the potential of zero charge, pzc) different in character from positive maxima (observed at positively charged surface)? (3) How do the non-uniform current-distribution and interfacial potential arise? (4) Why do the maxima only appear at sufficiently high concentrations of the depolarizer? Answers are suggested in the present article.

nearly linear, variation of current with applied voltage, ie Ohm’s law behaviour. The current here is limited by the resistance of the circuit, and the actual electrode potential differs increasingly from the applied potential as the latter is made more cathodic, until the current drops at cessation of the maximum[lO]. The major discussions of the surface-tension theory of these maxima are due to von Stackelberg[7, 91 and to Frumkin and Levich, summarized by Levich[S]. There is agreement that a non-uniform distribution of current arises from the geometry of the electrode, with the top or neck of the drop being shielded by the presence of the capillary from which the mercury Rows, or the mount on which a hanging drop is suspended. De Levier1 l] has also shown that the

E

1. REVIEW

2

OF DATA AND THEORIES

5 0

Polarograms held to be typical of first-kind maxima are shown in Fig. 1. Negative maxima are comparatively narrow and small, while positive maxima may span a wide region of potential and attain heights that exceed the diffusion current by factors greater than ten. The rising portion of the curve shows a linear, or

-0.25

-0-50

Voltage. v

I

- 0.75

-I 0

-1.5

Voltage.

-2.0

v

Fig. 1. Typical first-kind maxima, positive (a-d) and negative (e-g); after Fig. X1X-11, ref. [I]. 10m3N depolarizer in 5 x 1Om3N KCIO,--a--S,O;; b-0,; c-Cu*+; d-Pbz+; in O.lhJ KCI, e-Zn’+;f-Mn’+: g-Ba’+.

* Permanent address: Department of Chemistry, University of Kentucky. Lexington, Ky. 40506, U.S.A. 427

42X

H. H.

BAUER

current density at the bottom of a growing drop will exceed that at the neck by a factor of two as a result of the unequal mass-transport generated by the eccentric growth of the drop. (However, Barker and Bolzan[lZ] have commented that de Levie’s calculation is likely to overestimate the degree of nonuniformity of the mass transport.) De Levie further pointed out that thin films of solution are known to penetrate between mercury and glass, offering a very high resistance and thus producing a non-uniform primary current-distribution (similar effects have been found at mercury drops suspended from platinum wires sealed into glass[ 131). Experiments with capillaries having finely pointed rather than blunt tips produced no clear evidence that maximum formation was diminished by this presumed decrease in the shielding of the drop: there have been reports that the maxima decreased[l4, 151, but also that they showed no appreciable change[M-181 even when less disturbance of the solution was indicated by interferometric observations[l9]. According to von Stackelberg[7, 91, streaming is autocatalytic at positively charged surface and selfinhibiting at negative charge. Positive maxima cease when the electrode potential approaches the pzc, since the variation of surface tension with potential becomes zero there. Negative maxima are explained in a manner that requires the charge density to play a decisive role, and also requires an increased flux of depolarizer to the neck of the drop even though depleted solution is transported there; this appears an untenable assumption, and the theory has been criticized by Frumkin[ZO]. The most widely applied attempt at a quantitative theory is due to Frumkin and Levich[21]. The treatment begins by considering a mercury drop placed between two electrodes which produce a potential drop horizontally across the drop. The expected velocity of streaming, v, results as

Potential

Fig. 3. First-kind maximum in absence purified with charcoal, m = 1 mg . set-‘, a--O.lM;

vs

see.

d

d

c

e f

Potential.

Fig. 2. Variaton the surface (2); of concentrations

with

potential

V

of the specific

mobility

of

after Fig. XIX-14 in ref. [l]; KC1 solutions a4GOl;

b4.01;

c---0.1;

dAl.5;

e-1;

f-3N. where AE is the effective difference of potential, q the surfacecharge density, p and p’ the dynamic viscosities of the solution and of the mercury respectively, and IC is the conductivity of the solution. The “speciiic mobility” of the interface, z, ie the extent of motion arising from application of unit difference of electrical potential, then is z=

4

2/l + 3/I’ + q2/lc

The variation of z with electrode potential is shown in Fig. 2. This equation has been substantiated experimentally for the case where a mercury drop is subject to an external field between two electrodes[22]. It has also been said that the shapes of maxima under polarographic conditions are similar to the curves in Fig. 2; however, the data used[6] were obtained under conditions where the polarographic currents were limited by circuit resistance. Polarograms obtained in the absence of resistance control have a quite different appearance, and change in a different way with changing concentration of supporting electrolyte (Fig. 3).

V

of resistance control[l3]; t = 6.8 s; three-electrode ba.25M; c--1M Na,SO,.

8 x lO_“M Cu ‘* in Na2S0,, circuit with positive feedback:

Theory

of polarographic

Frumkin and Levich[u), 231 recognised that (2) could not be applied directly to the case of polarographic maxima, where the variations in interfacial potential arise in a different way. They derived the approximate expression, for the difference in local interfacial potential, AE = wl,o Ii, lim - w2,0 b, lim i (3) i,,,(l + w&) where w is the interfacial resistance (change of potential for unit change of current), w,,~ and w~,~ are the resistances to the different parts of the drop through the solution, i,, I,mand i,, ,,,,,the limiting current densities to the two parts of the surface (shielded US unshielded), i is the (average) current density and &, and w. are also average values. Consideration of (3) serves to show that positive maxima should be higher and more pronounced than negative maxima: the flux of depolarizer caused by the initiated streaming accentuates the non-uniformity of potential at positively charged mercury and inhibits it at negatively charged mercury (see discussionIS]). 2. CRITIQUE

Resistance-controlled

OF PREVIOUS

THEORIES

currents

Virtually all experiments dealing with first-kind maxima have provided data in which the magnitude of the current in the region of the maximum was limited by the resistance of the circuit. It follows that the magnitudeof the current cannot be taken to indicate directly the magnitude of the force, due to differences in interfacial tension, that is generated by a nonuniform current distribution. The reported experiments[ 14-191 with fine-tipped capillaries are therefore inconclusive. Origin of non-uniformity

The demonstration that non-uniform mass-transport can arise at a dme in virtue of the eccentric dropgrowth[ll] cannot explain why maxima are also observed at a hanging stationary drop, and why the streaming apparently was not appreciably different at growing, stationary,and contracting drops[16]. Moreover, the treatment[ 1l] shows that non-uniform mass-transport arises from geometrical considerations at a given electrode potential. This non-uniformity cannot, therefore, be invoked to explain differences of interfacial potential across the drop, which are required to induce streaming. Neither this theory, nor the discussions given by von Stackelberg[7, 91 and by Levich[8], provide grounds for understanding why maxima appear only above a given concentration of depolarizer. Theory

of Frumkin

and Levi&

Equation (3) provides a basis for qualitative understanding of differences between positive and negative maxima. However, the question of the cessation of

maxima of the

first kind

429

streaming is not resolved; the equation predicts a finite AE in the region of the limiting current, so that in general one might expect maxima to persist beyond the rising portion of the step, to an extent that depends on the actual magnitudes of the resistances and limiting currents. 3. SUGGESTED

RESOLUTION

Origin of non-uniform

OF THE PROBLEM

current-distribution

Shielding of the drop leads to a non-uniform primary current-distribution. The primary current-distribution is determined solely by the resistance of the solution, and depends on the geometry of the electrodes, of the cell, and of any other structures forming boundaries for the solution. The actual current-distribution is the same as the primary one when there is no polarization at the electrodes; ie, when the resistance of the interface, defined below, is negligible compared to that of the solution. It remains to be discussed, under precisely what conditions a non-uniform primary currentdistribution will lead to non-uniformity of the interfacial potential. It has already been pointed-out that non-uniformity of the current across the interface resulting from eccentric drop-growth will not lead to a non-uniform interfacial potential. Non-unijTorm interfacial

potential

The actual distribution of current is determined by the combined effects of the resistance in the solution and the resistance of the interface, the latter defined by the variation of interfacial potential for unit change of current. Thus, the resistance of the interface varies with potential, and is extremely high both before and after the polarographic step. Before the step, only charging current flows. There is nothing in the nature of the charging process to give rise to non-uniformity, so that the latter will arise only if the resistance of the solution is comparable to that of the interface. Even with the most dilute electrolytes used, this will not be the case. Newman[24] has shown how the current distribution varies according to the relative impedances of solution and interface for the case of alternating currents, and at zero frequency-orresponding to the present case-the primary current-distribution does not induce a nonuniformity in the charging current flowing across the interface. On the plateau of the polarographic step, the resistance of the interface is again very high, and the current distribution is determined by conditions in the interface and not by the primary current-distribution. The current distribution will be non-uniform to some extent due to the unsymmetrical geometry of mass transport, including the fact that drops grow eccentrically. However, this does not lead to non-uniformities in the interfacial potential. At all places on the electrode, diffusion limits the current and the diffusion

H. H. BAUER

430

occurs between the bulk solution and the interface, where the concentration of depolarizer is zero. Nonuniformity of current-density arises as a result of geometrical factors and does not imply a nonuniformity of the interfacial potential. In the region of the rising portion of the polarographic step, the resistance of the interface is much lower than before or after the step. When this resistance is comparable to that of the solution, the nonuniformity of the primary current-distribution will force some non-uniformity of the current across the interface. Now, the interfacial potential is a function of the current density in this region of potential, so that non-uniform current-density occasioned by the variation surface

of solution

resistance

to d&rent

parts of the

now leads to differences in the interfacial potential at various parts of the surface. Then, differences of interfacial tension arise, and lead to streaming in the way previously outlined[l-91. The lacuna in the earlier treatments was the failure to realise that it is only in the region of the rising portion of the polarographic step that the requisite non-uniformity of interfacial potential will arise. Since the interfacial resistance must not be much higher than that of the solution for this to occur, the non-uniformities will arise only when the concentration of depolarizer exceeds a certain value, so that the interfacial resistance is sufficiently low. This explains why maxima are observed only above a given concentration of depolarizer; why streaming commences at some point on the rising part of the step rather than at the moment that faraclaic current starts to flow[25]; and why the height of the maximum increases with concentration taken to a power greater than unity[6]-the extent of non-uniformity increases as the concentration increases. It is also evident that once streaming begins, the interfacial resistance decreases, and the non-uniformity of the interfacial potential increases since the nonuniformity of the actual current-distribution becomes more pronounced. Thus, streaming is autocatalytic, and the rapidity with which streaming sets in and ceases is not surprising. Positive

and negative

the neck, where the current density is also lower than at the bottom. This flux of depolarizer prevents the neck of the drop from reaching the state of total concentration polarization until the mean electrode potential is considerably cathodic to the reduction potential. Thus, positive maxima persist because the initial non-uniformity is accentuated by the increased flux of depolarizer, causing the neck of the drop to remain partly unpolarized even though the average interfacial potential would imply complete concentration polarization. Positive maxima then cease when the neck of the drop is in a state of complete concentration polarization, but in any case as soon as the mean potential approaches the pzc so that even a nonuniform interfacial potential does not produce nonuniform interfacial tension. The streaming does not begin again beyond the pzc and there is no reason why it should since the potential is no longer in the region of the half-step potential. This explanation is basically the same as that of Frumkin and Levich (discussion in ref. [S]), and has similarities to that of von Stackelberg[7, 91. However, it is perhaps more straightforward in a qualitative way: positive maxima persist because the streaming causes the neck of the drop to remain, so to speak, “on the rising part of the curve,” which is the only region of potential in which a non-uniformity of interfacial potential can persist. Acknowledgements:-Theauthor wishes to record his thanks to Profess& Graham Hills and his colleagues at the University of Southampton for the splendid hospitality afforded him during the-sabbatical lea&in which this work was carried out. He is also grateful to the University of Kentucky for the granting of a sabbatical year. The work forms part of a study of electrochemical processes under Project Themis, Contract DAABO7-69-C-0366, administered by the U.S. Army Electronics Command. REFERENCES

maxima

Since it was implicitly assumed in earlier discussions that non-uniformity of interfacial potential would arise at all potentials cathodic to the reduction potential, the problem was seen as one of explaining the small and narrow nature of negative maxima-positive maxima were regarded as the “normal” case. From the present point of view, we see the problem in reverse: maxima are expected to persist only in the region of the step, and to have a width of the order of 100 mV or so. This is the case with negative maxima. The question is, why do positive maxima persist over a much wider region of potentials? With positive maxima, the solution streams towards the neck of the drop and then downwards along the surface of the drop. Fresh depolarizer is brought to

5. 6. 7. 8. 9. 10. 11. 12. 13.

J. Heyrovsky and J. Kuta, Principles of Polarography. Czechoslovak Academy of Sciences (1965). L. Meites. Polaroaraohic Techniaues. 2nd edn. Interscience, New YorkY(lb65). A 1.M. Kolthoff and J. J. Lingane, Polarography, 2nd edn. Interscience, New York (1952). G. W. C. Milner, Principles and Applications of Polarography and other Electroanalytical Processes. Longmans, Green and Co., London (1957). J. Heyrovsky and P. Zuman, Practical Polarography. Academic Press, New York (1968). T. A. Kryukova, Z. phys. Chem. 212, 247 (1959). M. von Stackebexg, Fortschr. them. Forsch. 2,229 (195 1). V. G. Levich, Physicochemical Hydrodynamics. PrenticeHall, New York (1962). M. von Stackelberg and R. Doppelfeld, Advances in Polarography 1, 68 11960). R. Brdicka. Colln Czech. them. Commun. 8. 419 (19361. R. de Levi;, J. electroanal. Chem. 9, 311 (1965). ’ ’ G. C. Barker and J. A. Bolzan, Z. analyt. Chem. 216, 215 (1966). S. La1 and H. H. Bauer, unpublished work.

Theory

of polarographic

14. A. N. Frumkin, Conf, electrode processes, Moscow, 1956; reported in H. Berg, Chem. Technik. 9,245 (1957). 15. F. Strafeldaand M. Stastny, Colln Czech. them. Commun. 32, 1836 (1967). 16. H. J. Antweiler, Z. Elektrochem. 44, 831 (1938). 17. J. Flemming and H. Berg, J. electroan& Chem. 8, 291 (1964). 18. F. Willeboordse, J. electroanal. Chem. 2, 408 (1961). 19. H. Triebel and H. Berg, J. elecfroanal. Chem. 2, 467 (1961). 20. A. N. Frumkin, Zh.$z. Khim. 29, 1318 (1955); Chem. Abstr., SO, 1491i. 21. A. N. Frumkin and V. G. Levich, Zh. fiz. Khim 19,

maxima

of the first kind

431

573 (1945); Acta Phvsicochim. U.R.S.S. 20, 769 (1945); V. G. Levich, Zh.ji. Khim. 21,689 (1947); summarized in English in ReE [8]. 22. 1. Bagotskaya and A. N. Frumkin, Zh. jz. Khim. 2L, 1033 (1947); Compr. Rend. Acad. Sci. U.R.S.S. 55, 131 (1947); I. Bagotskaya, Zh. jiz. Khim. 23, 1231 (1949); summarized in English in Ref. [El. 23. A. N. Frumkin and V. G. Levich, Zh. jz. Khim. 21, 1335 (1947); T. A. Popova and T. A. Kryukova. Zh.

fiz. Khim. 25, 283 (1951); see Ref. [8]. 24. J. Newman, J. electrochem. Sot. 117, 198 (1970). 25. F. M. Hawkridge, T. W. Holt and H. H. Bauer. Chim. Acta 58, 203 (1972).

Anal.