~ V O L FVIELSTICI-IAND PAUL DELAHAY
1S74
[ C O N T R I B U T I O N FROX THE COATES CHEMICAL
Vol. 79
LABORATORY, LOUISIANA STATE UNIVERSITY]
Voltage-step Method for the Kinetic Study of Fast Electrode Reactions BY WOLFVIELSTICH’AND PAUL DELAHAY RECEIVED JULY 30, 1956 A method is described for the kinetic study of fast electrode reactions in which a voltage impulse represented by a step function is applied to a cell composed of the working a n d the unpolarized reference electrodes. The resulting current-time curve is recorded by means of a cathode-ray oscilloscope. The exchange current is calculated from a plot of current against squ,ire root of time. This plot is linear for sufficiently short times and a voltage step not exceeding 2-5 mv. The transfer coefficient is obtained from a log-log plot of exchange current against concentration of one reactant, the concentrations of the other reactants being kept constant. The selection of conditions in which the charging or discharging of the double layer c a n be neglected is discussed Results for the cadmium amalgam electrode are comparable with d a t a obtained by other authors by a c electrolysis arid the curreiit-;tep method Coniparison with other relaxation methods for the kinetic study of fast electrodc reactions is made.
Introduction Three relaxation methods have been developed for the study of fast electrode reactions, namely, the potential-step method2 (potentiostatic method), electrolysis with superimposed alternating voltage3 (Iiandles, Ershler, Gerischer) and the current-step meth~d.~ A modified and simplified form of the potential-step method is described here. The principle is as follows. The working electrode whose kinetics is studied is coupled with an unpolarized electrode, and the voltage corresponding to equilibrium a t the working electrode is applied t o the cell. The voltage applied to the cell is then varied by a small increment not exceeding 2 to 5 mv. and the resulting current-time curve is recorded with a cathode-ray oscilloscope. The kinetic parameters for the electrode reaction are deduced from the current-time variations for different concentrations of one reactant, the concentration of the other reactants remaining constant. In this method, the voltage across the cell during electrolysis is constant but the potential of the working electrode changes because of variations of ohmic drop. The method differs in this respect from the potential-step method of Gerischer and Trielstich2 in which the potential of the working electrode is kept constant. Current-Time Relationship The current-time relationship is derived from the currcnt-potential characteristic of the electrode reaction, due allowance being made for the variations of the potential and the concentrations of reactants during electrolysis. The currentpotential equation for an electrode process symne = R can be written bolized by the reaction 0 in the form5
+
wliere io = n F k , Coo(’- 00 CRO~
(2 )
(1) Postdoctoral fellow, 1955-1956. ( 2 ) (a) €1. Gerischer and W. Vielstich, Z . p h y s i k . Chem., AT. F., 3, 1 G (1955): (b) W. Vielstich and H. Gerischer, i b i d . , 4, 12 (1955). ( 3 ) F o r a survey see, for instance, P. Delahay, “ N e w Instrumental Xleihods in Electrochemistry,” Interscience Publishers, Inc., S e w Vork, h-.Y . , 1954, p p . 146-175. ( 4 ) 1’. Bcrzins and P . Delahay, l’ms J O I ‘ R N A I . , 77, 6448 (1955). ( 5 ) H . Gerischer, L. Elekfvochern., 5 5 , 93 (19.51); see also ref 4.
is the exchange curreizt per unit area and the notations are as follows: i the current (positive for a net cathodic reaction) for an electrode of area A ; CO and CR the concentrations of 0 and R a t the electrode surface; the CO’s the bulk concentrations; a the transfer coefficient; E the potential (European convention) : E, the equilibrium potential for the concentrations Coo and CR”; k, the rate constant characterizing the electrode process a t the standard potential for the couple 0 ne = R ; F the faraday; R the gas constant and T the absolute temperature. The exponentials in (1) can be expanded and, if (E - Ee) 6 RTIanF, Le., in general ( E - Ee) 6 2-5 mv. a t room temperature, only the first two terms in the expansion need be retained. Thus
+
Furthermore E
- E,
- iR, = V
(4)
where Vis the total voltage variation, and Rt is the total resistance of the cell and the calibrated resistance (connected to oscilloscope) in series with the cell. The quantity Vis positive when E > E , and vice versa. The combination of (3) and (4) in which CO and CR are functions of time is the current-time relationship for a given voltage variation. The explicit form of this relationship depends on the functions CO and CR and, consequently, on the mass transfer process by which 0 and R are brought to and removed from the electrode. The concentrations will be calculated by assuming that a large excess of supporting electrolyte (in comparison with Cooand CRO)is present and that conditions of semiinfinite linear diffusion prevail. The latter assumption is valid even for spherical or cylindric electrodes provided that the radius of curvature of the electrode is not too small (perhaps not less thaii 0.01 to 0.1 cm.) and the electrolysis is of short duration. These conditions can easily be fulfilled. T h e concentrations Co(x,t) and C ~ ( x , tare ) then solutions of Fick’s equation, as written for substances 0 and R , f o r the following conditions: Co(x,1) = Coo and C ~ ( x , t )= CEO at x > 0 and t = 0, x being the distance from the electrode, and t the time elapsed after the voltage variation; C o ( x , t ) -+ Coo and C n ( r , t ) -+ Coo for t >c 0 and x -+ m; and i, as calculated from ( 3 ) and (4), is such t h a t i = n FA Do[a C ~ ( x , , t ) / d ~ ] , - o= - n F A D E[aC p . ( ~ , t ) / b ~ ] z - ot ,h e D’s being the diffusion cocfiicients of 0 and R.
April 20, 1957
KINETICS T U D Y
OF F4ST
The solution of this boundary value problem, as obtained by the Laplace transformation, is
ELECTRODE REACTIONS
IS73
--
80 70 60 -
*
50 -
40-
N
d
I
4
3020-
d and where the notation "erfc" represents the complement of the error function. i Since the function 4 = exp(X2t)erfc(Xt1/z)decreases with increasing values of t , the current de- .- 108creases continuously during electrolysis. At t = 76 0, 4 = 1, and one verifies that the current has the 5 I 1 / 1 1 value obtained by combining (3) and (4) and by 0.I 0.2 03 0.4 .5 I 2 3 noting that CO = Coo and CR = CRO for t = 0; the current is then entirely controlled by the kiccd*+ IYILLIMOLES LIT-']. netics of the electrochemical reaction. Conversely, Fig. 1.-Plot for the determination of a: 1, current-step for sufficiently long times the current is entirely method; 2, a.c. electrolysis; 3, voltage-step method. diffusion controlled. This can be shown readily by expanding the error function in (5) for argu- methods to the same solution for identical condiments larger than unity. An order of magnitude tions of exposure of the electrode (contamination). of time T a t which diffusion becomes prevalent can I n the preparation of the plot i vs. t l / z current should not be determined by setting for instance A d 2 = 1, be read a t very short times when the charging or discharging I
v
0
I
i.e.,
T
= 1/X2.
Finally, eqs. 5 and 6-as written for Rt = O-reduce to the result for the potential-step method derived by Gerischer and Vielstich.2 The latter result itself is a modified form of the currentpotential-time for irreversible processes in polarography and related methods6 (Smutek, Evans and Hush, Kambara and Tachi, Delahay). Determination of io, ks and a! If X t 1 / 2
