Use of Polarography and Pulse-Polarography in the Determination of the Kinetic Parameters of Totally lrreversible Electroreductions K. B. Oldham and E. P. Parry Science Center of the North American Rockwell Corp., Thousand Oaks, Calif. 91360
An analysis of the current-voltage relationships applicable to polarography (instantaneous and time-averaged) and to normal pulse-polarography, shows that the conventional “log-plot” yields truly straight lines only in the cases of reversible waves. Alternatives to the usual log plots a r e suggested which a r e linear for irreversible waves and from which kinetic parameters a r e determinable. A method based upon time-dependence is also proposed.
IN POLAROGRAPHY, the faradaic current resulting from a n electrochemical reaction a t a potentiostated dropping mercury electrode is either measured a t a fixed time in the life of a drop, usually a t the instant of drop fall (instantaneous polarography), or averaged over the drop life (averaged polarography). Pulse-polarography ( I ) (in this paper we are concerned solely with normal pulse polarography) differs from polarography in that the potential is applied only for a brief period toward the end of drop life. Throughout this article we restrict attention to a single electroreduction uncomplicated by chemical reactions, and for such a process, the variation of current with potential is qualitatively similar for both types of polarography and for pulse-polarography, being zero a t a sufficiently positive potential but rising with increasing negative potential to approach a constant diffusion-limited value. In polarography, the reversibility of the electrode reaction may be detected from the linearity and slope of the so-called “polarographic log plot” ( 2 ) ,and the same diagnosis applies in pulse-polarography (see below). The kinetic parameters play no role in determining the shapes of current-potential curves for reversible waves and so are not determinable therefrom. Reversible waves are characterized by large standard rate constants, but as the latter becomes progressively smaller, the wave passes through a domain of “quasireversibility” (3) t o become “totally irreversible.” The classification of waves into “reversible,” “quasireversible,” and “totally irreversible” applies equally t o polarographic and pulse-polarographic waves. However, the status of a given electrode reaction may be different according to the two techniques in that a reaction which is reversible polarographically may exhibit quasireversibility when examined by pulse-polarography and one which shows quasireversibility by ordinary polarography will usually become totally irreversible on the time scale appropriate to the pulse-polarograph (4). Ever since Koutecky (5)and Barker (6),respectively, derived (1) E. P. Parry and R. A. Osteryoung, ANAL. CHEM.,37, 1634 (1965). (2) I . M. Kolthoff and J. J. Lingane, “Polarography” 2nd. ed., Interscience, New York, 1952, Chap. XI. (3) P. Delahay, J . Am. Cliem. Soc., 75, 1430 (1953). (4) J. H. Christie, E. P. Parry, and R. A. Osteryoung, Electrochim. Acta, 11, 1525 (1966). (5) J. Koutecky. Chem. Listy, 47, 323 (1953); Collection Czech. Cliem. Commun., 18,597 (1953). (6) G. C . Barker and A. W. Gardner, C/R 2297, A.E.R.E., Harwell, England (1958).
equations for polarographic and pulse-polarographic waves in the absence of electrochemical reversibility, the determination of kinetic parameters (rate constant and transfer coefficient) from irreversible polarographic waves has been feasible. However, such determinations have rarely been carried out by polarographers. This omission arises, we believe, largely from the difficulty of making the necessary analysis of the polarographic data and from the unfamiliarity of the mathematical functions involved. Irreversibility causes nonlinearity in the polarographic log plots, mildly in the case of polarography, but pronouncedly for pulse-polarography. Provided current data are collected from a restricted potential range, the log plot for totally irreversible polarographic waves is sufficiently linear that a straight line may be drawn with little uncertainty and empirical methods based on such a procedure have been advocated as a means of determining kinetic parameters (7, 8). No such method can be used for irreversible pulse-polarographic waves and even for the polarographic case, the method is somewhat unsatisfactory in that the derived value of the transfer coefficient depends slightly on the potential range spanned by the data. In this article we show how simple alternatives to the log plots produce lines which are straight for totally irreversible polarographic waves and for totally irreversible pulse-polarographic waves. The linearity of these plots establishes the total irreversibility of the electrode process and the slope and position of the straight line yield the kinetic data without recourse t o empirical parameters. In principle, the dependences on (drop or measuring) time of the polarographic or pulse-polarographic half-wave potentials provide methods of determining the kinetic parameters for irreversible electroreductions. While the expected dependence, in the case of polarography, has been demonstrated ( 9 ) , the methods lack the necessary precision for the accurate determination of the parameters. However, as we will demonstrate below, a combination of polarographic and pulse-polarographic data enables the kinetic parameters to be measured accurately from time dependence. In a subsequent article we will present experimental data exemplifying the methods discussed in the two foregoing paragraphs. GENERAL RIATHEMATICAL DISCUSSION When a n n-electron reduction is occurring at potential E a t a dropping mercury electrode, the polarographic current (instantaneous or averaged) is given by (5,I O ) (7) L. Meites and Y . Israel, J . Am. Cliem. SOC.,83,4903 (1961). (8) D. M. H . Kern, Ibid., 76,4234 (1954). (9) Pekka Kivalo, K. B. Oldham, and H. A. Laitinen, J . Am. Cliem. Soc., 75,4148 (1953). (10) H. Matsuda and Y. Ayabe, Birll. Cliem. Snc. Japan, 28, 422 (1955). VOL 40, NO. 1, JANUARY 1968
65
il
=
nAFC
dz[+ @ {g 1
D’
3 nt
-
( E - E,)}]-’@l(A)
exp
(1)
where Cand D are the bulk concentration and diffusion coefficient of the reactant and D‘ is the diffusion coefficient of the initially absent soluble reduction product. F, R, and T are Faraday’s constant, the gas constant, and the absolute temperature. The corresponding equation for pulse-polarograPhY is ( 6 ) iz = nAFC
43nt [l + PI exp {$( E - E,)}]-’@z(X) D
In both these equations, t is the time at or during which the current is measured (after the birth of the drop in instantaneous and averaged polarography or after the imposition of the potential in pulse-polarography) and A is the drop area at the conclusion of measurement. In both equations X = k , & e x p { - a E ( ERT -
E,)
where k, is the heterogeneous rate constant for either direction of the electrode reaction at the standard potential E, of the system, and a is the cathodic transfer coefficient. and aZ are different functions and has a form (ali)for instantaneous polarography different from that (+lo) appropriate to averaged polarography. We shall discuss these three functions below in some detail but they are similar in that all approach zero for small argument, have values close to a moiety for unity argument, and approach a unity limit as the argument approaches infinity. Equations 1 and 2 neglect any influence that double-layer structure may have upon electrode kinetics. They also ignore the curvature effect resulting from electrode sphericity. Though there are instances in which one or other of these effects is important, the deviations resulting from their neglect is generally of comparable magnitude to those arising from other artifacts (convection, capillary shielding, nonsphericity, polarographic maxima, etc.). Suitable criteria of reversibility and total irreversibility are that, at all potentials embraced by the polarographic or pulsepolarographic wave: Reversible Totally irreversible
X
>> 1
exp { $ ( E
[
1+
d’
@(A) = @ (k,
-exp
D‘
ANALYTICAL CHEMISTRY
4 1 exp {-a D
2 ( E - E,)}) RT
(7)
in the irreversible, though different 9 functions must be used for the instantaneous polarographic, averaged polarographic, and pulse-polarographic cases. Equation 6 may be readily recast (assuming a 25 “Ctemperature and measuring potentials in volts) as :
where
This relation is very well known in the case of polarography (2),being the basis of the “log plot” method for evaluating reversibility. The remainder of this paper seeks ways of utilizing Equation 7 to determine k, and CY from measured values of x, t , and E and known values of D , n, and E,. There are cases, however, in which the standard potential is unknown and in these cases it may be of value (e.g., in comparing rate constants for reactions involving a series of related compounds) to determine the rate constant k, for reduction at some convenient reference potential E,. As the two rate constants are in the ratio k,/k8 given by exp( -anF[E, - E,]/RT],Equation 9 below may be used to cover totally irreversible cases where E, is unknown. x =
CP (k,
dg
exp {-a
RT
( E - E,)
INSTANTANEOUS POLAROGRAPHIC CASE
Koutecky (5) and, independently, Matsuda and Ayabe (IO) showed that G I i is expressible as either of two infinite summations :
where y1 = 6o = 1 and other coefficientsare most concisely defined by the recurrence formulae
- E,)}
