John G. Mason
Virginia Polytechnic Institute Blacksburg, Virginia
Applying a Steady State Approach to Polarography
The teaching of polarography a t the undergraduate level is generally limited to one section of a comprehensive course in instrumental analysis. This restriction forces emphasis upon the purely analytical aspects of the polarographic limiting current. The current voltage curves for completely reversible redox systems are treated but irreversible processes and kinetic effects are in general ignored. The time element and the lack of necessaly background on the part of the students preclude a rigorous treatment such as that developed by Delahay (1). It would be desirable to have a general treatment of the polarographic process in which the mathematics was rather simple but which was capable of treating reversible, irreversible, and reaction-affected reductions within the same framework. The application of a simple steady-state rate treatment requires very little mathematics and offers much insight into polarographic processes in particular and electrochemical processes in general. Rate Processes at Electrodes'
The reduction or oxidation of any material a t an electrode surface involves several steps, any one of which may be rate determining: (1) transport of the electroactive material to the electrode, (2) transfer of electrons from (or to) the material by the electrode, and (3) removal of the product of the electrode reaction. Steps (1) and (3) can he accomplished by diffusion, migration, or convection. I n practice, migration is effectively eliminated by the presence of indifferent electrolyte and t,he mass transfer accomplished by either diffusion or convection (i.e., stirring the solution or the electrode). Step (2), t,he electron transfer step, involves the introduction of several concepts. Let us consider the electrochemical reaction, .iZ ne = Bz-* (1)
+
where A and B are both soluble and the electron transfer process is a first order reaction. The net rate of the cathodic process (1) will he the difference between the rate of reduction of A and B and the rate of oxidation of B to A. The rates of these reactions may be expressed as Forward rate = k,,&~.
(2)
Reverse rate = ~,.CB.
(3)
Presented in part before the Division of Chemical Education, 140th Meeting of the American Chemical Society, Chicago, Illinois, September, 1961. 1 The treatment which follows is derived from reference (lb), pp. 32-6.
where CA, and CB. represent the concentrations of species A and B a t the electrode surface; k..d and k, are heterogeneous first order rate constants referred to unit electrode area and having the units cm sec-'. From the application of absolute rate theory k..a
= ks,b
exp [ - a n S ( E - Eco)IRTI
(4)
= k.,s exp [(I - o)n5 ( E - E.')/RT] (5) is the speci6c heterogeneous rate constant at where the formal equilibrium potential, EE. where by definition the rate of oxidation and the rate of reduction are equal and the net rate (current) is equal to zero. The symbol a is a transfer coefficientrelated to the fraction of the potential operating in the direction of reduction. The current is then the difference between these two rates
k..
i
=
(6)
n5 A(k..dC*. - k 0 S h )
where n is the number of electrons in equation (I), 5 is Faraday's constant, and A the area of the electrode in cm2. Application of the Steady-State Kinetics to Polarographic Waves
Steady state treatments of polarographic waves have been developed by Evans and Hush ( 2 ), Laitinen and co-workers (S),Kivalo (4), and Tanaka, et al. (5) with a great deal of success in the interpretation of complex polarographic waves. Jordan (6) has used a similar treatment for steaming solutions. All treatments of polarographic phenomena are in essence steady state treatments since the average or maximum current is presumed to be independent of time and experimentally reproducible. It is assumed for the following discussion that no measurable change in the concentration of the speciesin the hulk occurs due to the electrolysis. Consider the scheme outlined by the following k*
Ab F==2 A.
kmd
Bs
LB
Bb
(7)
ko.
Ab and Bbrepresent species in the bulk of the solution; A. and B, refer to species a t the electrode surface; kn and .k designate diffusion rate constants with units cm sec-I. Assuming that steady state conditions are maintained a t the electrode then the rate of supply of A to the electrode equals the rate of loss of A from the electrode. A is supplied a t the electrode by two processes, diiusion of A from the bulk and oxidation of B a t the electrode surface. Therefore, the rate of supply of A to the electrode is given by rate of supply =
+ k,Ca
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has been confirmed bv Okinaka and Kolthoff (9) ~,for the reduction of normally reversible systems a t the rotating mercury electrode. Application of the Steady State Treatment to Reaction Affected Currents
For the historical development of the treatment of kinetic waves, reference is made to Kolthoff and Lingane ( 1 O),Wiesner ( 1 I ) , and Delahay (1 b). A rigorous theoretical treatment of reaction affected polarographic waves has been recently presented by Brdickaet al. (13) and Koutecky and Koryta (13). Consider the following reactions
kr and kb are formal heterogeneous rate constants in cm sec-' for the forward and reverse reactions of the equilibrium represented by equation (27). If A is the only electroactive species the following steady state equations may be written k*C*,
+ CbkOx+ C-kb
+ kred + KdClr, kDCm + Chkr = (kb + ~DJCD. =
~ A C A .= (ks
(kr
+ k.,)Ca
(29) (30) (31)
The solution of these equations is simplified if the following reasonable assumptions are made. First, that the concentration of D a t the electrode is equal to that in the bulk and secondly, k,* is very much larger than kka,knl kb, k,, k,. Since the concentration of D is always equal to C-, equation (30) does not have to be utilized. The expression for the current, equation (6)is reduced to i
n5A (k..dClr,)
=
(32)
Equation (29) is reduced to
C*.
=
k*C*, 4- Cmka kred
Substitution of (34) into (32) yields
+
i = n5A ( k r C ~ ~C D ~ ~ ~ )
(35)
which demonstrates that under the conditions described i is independent of potential and is therefore a limiting current consisting of two components, a diffusion component and a kinetic component. Rewriting (35) yields it =
id
+ w5ACnskb
(36)
Equation (36) demonstrates that the formation of A from D has increased the current over that expected from the concentration of A in the bulk of the solution. If the equilibrium in equation (27) greatly favors D, so that the bulk concentration of A is very low, equation
(36) reduces to
with the result that the limiting current shows complete kinetic control. The rate constant, kb,has the dimensions cm sec-1 rather than the conventional units of sec-I. In favorable cases, the homogeneous rate constant is related to the heterogeneous rate constant by where DA is the diffusion coefficient of species A in cm2 sec-1. The homogeneous rate constants for the reaction represented in equation (27) are kbf and kt'. This treatment has been discussed by Delahay (Ib) and Kern (1.4). Pedagogically, the reaction layer treatment has great value only if sufficient time is available for a rather complete presentation. Qualitative uudentanding of the nature of the process is pron~otedif the complexity of the quantitative determination of rate constants by polarography is only indicated rather than dwelt on. I t should be indicated to the student that these results are only approximations of the truth and that rigorous solutions to these problems have been developed. In the opinion of the author, the particular advantages of this simpljfied kinetic approach to current voltage curves are (1) increased student awareness of what is meant by "reversibility" and "irreversibility", ( 2 ) possible extension to other electroanalytical techniques, and (3) profitable digression into the diagnosis of irreversible organic reductions and the relation between polarographic reversibility and structural changes in complex ions. literature Cited ( 1 ) ( a ) DELAHAY, P., "Instrumental Analysis,'' The MacMillan Company, Inc., New York, 1957, Chap. 4, (h) DELAHAY, P., "New Instrumental Methods in Electrochemistry," Interscience Publishers, Inc., 1954, Chaps. 2-5. ( 2 ) EVANS,M. G., AND HUSH,N. 8., J. Chim. Phvs., 49, (2159 (1952). H . A., OLDHAM, K. B., AND ZIEQLER,W. A,, ( 3 ) LAITINEN, J . Am. Chem. Soc., 75, 3048 (1953). ( 4 ) KIVALOP., Ada Chem. Scand., 9 , 221 (1955). N., , TAMAMUSHI, R., AND KODAMA, M., Z. fur. (5) TANA~A Physilc. Chenz. AT. F. 14, 141 (1958). ( 6 ) JORDAN, J., A n d . Chem., 27, 1708 (1955); JORDAN, J. .&no JAVICK, R. A., Electrochim. A&, 6 , 23 (1962). (7) VETTER, K., Z. fur Physik. Chem., 194,199 (1950). R., LORCH,A. E., AND HAMMETT,L. P., J . ( 8 ) ROSENTAAL, Am. Chem. Sac. 59,1795 (1937). Y., AND KOLTHOFF, I. M., J . Am. C h m . Sac., ( 9 ) OKINAKA, 82, 324 (1960). I. hl. AND LINGANE,J. J., L'P~lar~graphy," (10) KOLTHOFF, 2nd ed., Vol. I, Intencience Publishers, Inc., New York, 1952, pp. 268-294. (11) WIESNER,K., Anal. Chen~.,27, 1712 (1955). R., HANUS,V., AND KOUTECKY, J., in "Progress ( 1 2 ) BRDICKA, in Polamgraphy," Vol. I, edited by ZUMIN, P. AND KOLTHOFF, I. IM., Interscience Publishers, Inc., New York, 1962, pp. 14.5-200. J. A N D KORYTA, J . , Electrochim. Aela, 3 , 318 (13) KOUTECKY, 1 19fili~ ~.--.,.
(14) K E ~ ND. , M. H., J . Am. Chem. Soc., 75, 2473 (1953)
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