6448
TALIVALDIS BERZINSAND [CONTRIBUTION FROX THE COATES CHEMICAL
PAUL
DELAHAY
Vol. 77
LABORATORY, LOUISIANA STATE UNIVERSITY]
Kinetics of Fast Electrode Reactions BY TALIVALDIS BERZINS'AND PAULDELAHAY RECEIVED JULY 25, 1955 A method is described for the study of the kinetics of fast electrode reactions. The working electrode, whose kinetics is being studied, is coupled with an unpolarized electrode, and the potential of the former electrode is adjusted a t its equilibrium value. A current impulse represented by a step function is passed through the electrolytic cell, and the potential of the working electrode is recorded. The parameters characterizing the kinetics of electron transfer-the rate constant k. a t the standard potential and the transfer coefficient a-are determined from potential-time curves. A rigorous mathematical analysis, which takes into account the double layer capacity, is developed for potential-time curves. Values of k, up to 102 cm. sec.-l can be determined, and application to the study of the kinetics of so-called "reversible" electrode reactions in aqueous solutions and in fused salts is thus possible. Experimental methods are described in detail: cell with hanging drop electrode, generator of current impulses having t h e shape of a step function (transients shorter than one microsec.), compensation of ohmic drop, etc. The application to the reduction of cadmium ions on cadmium amalgam yielded experimental results which were in excellent agreement with theory.
Introduction I n a recent note and a more detailed paper2 a iiiethod was described for the kinetic study of fast electrode reactions. In this method, the potential of the electrode whose kinetics is being studied is adjusted at its equilibrium value, and a current impulse represented by a step function flows through the electrolytic cell. This cell may be composed of two identical polarized electrodes or one polarized and one unpolarized electrode. The solution is not stirred, and a large excess of supporting electrolyte is present: diffusion is thus the sole mode of mass transfer. The polarized electrode is a plane, and conditions of semi-infinite linear diffusion are assumed to prevail. The voltage across the cell is recorded during electrolysis, and parameters characterizing the kinetics of electron transfer are calculnted from voltage-time curves. A theoretical analysis of these curves was presented in our previous publications for the simplified case in which the double layer capacity is neglected. A rigorous analysis in which this simplification is not made is now presented, and experimental methods are discussed in detail. The rate of an electrode reaction represented by the symbolic equation, 0 ne- = R , and involving only a single rate-determining step is
thermore, it is assumed that substance 0 is soluble in solution, and that the reduction product is soluble either in solution or in the electrode as in the deposition of an amalgam-forming metal on a mercury electrode. The modification of equation 1 and the subsequent treatment is trivial when R is an insoluble species (metal) .3 At the equilibrium potential Ee the current is equal to zero, and one has
where Ck and Ck are the concentrations at equilibrium. The quantities equated in (2) can also be written in the form
ctC1- a) C:Uka where k, is defined by k,
=
k! exp
[-* a RT '1
=
k: exp
[" -RT
I n view of equations 2 and 3 the current-potential relationship (1) can now be written in a form containing only the rate constant k,, i.e.
+
an FE
(4)
k0being the standard potential for the system 0 ne- = R. The product where the notations are as follows : if,the current density, which is positive for a cathodic reaction; n, the number of electrons involved in the electrode reaction; F,the faraday; R, the gas constant; T , the absolute temperature; COand CR, the concentrations of substances 0 and R, respectively, a t the electrode-solution interface; E , the electrode potential referred to the normal hydrogen electrode ; a, the transfer coefficient; ko and k:, the formal rate constants a t E = 0 for the forward and backward reactions, respectively. Equation 1 is written on the assumption that the number of electrons involved in the rate-determining step is equal to the number of electrons in the over-all reaction. Fur(1) Predoctoral fellow, 1983-1055. (2) (a) T. Berzins and P. Delahay, J . C h e w Phys., 23, 972 (1953); (b) T. Berzins and P. Delahay, Z. Elckl?'ochcm., in press (proceedings of the 103.3 annual meet:ng of the Bunsen Gesellschaft).
nFk.C:('-*)C:"
=
io
+
(5)
is the exchange current density, and can be used to characterize the kinetics of electrode reactions. The main drawback in the use of the exchange current density results from the dependence of io on the equilibrium concentrations C& and C i , ;.e., on the potential. This is not the case for the rate constant k . which is therefore more characteristic of the kinetics of electron t r a n ~ f e r . ~ The kinetic parameters a and K s or io cau be determined by conventional polarization experiments for values of k , up to about 10-3-10-2 cm. set.-'. Faster reactions with values of k. up to about 1 cm. set.-' can be studied by electrolysis with supcriiii( 3 ) For a detailed discussion of equation 1 see an olectrochemistry text. (4) J. E. B. Randles. Trans. Pavodoy S o c . , 48, 828 (1952)
KINETICS OF FASTELECTRODE REACTIONS
Dec. 20, 1955
6449
posed alternating voltage.6 Such processes and possibly faster reactions can be studied by the method a t constant potential recently developed by Vielstich and Gerischer.s
are introduced. After linearization the boundary conditions are
Boundary Value Problem The electrolysis current density i t is the sum of a faradaic component it corresponding to electron transfer and the capacity current density i, used to charge, or discharge, the double layer. The faradaic component is given by equation 4 in which the exchange current density, as written in equation 5, can be introduced for the sake of simplicity. The capacity current density is
with
io =
- CI d ( E - Ee)/dt
( 6)
= Xo[($o)z;=o
(d$o/dx)z-o
- ( $ R ) ~ - o - nF(E - E,)/RT] (14)
XO = io/nFDoC;
and a similar condition in terms of more one has $0 $0 --f
for x Z 0 and t = 0 ----f m and t > 0
= $R = 0
0 and
$R
(15)
qk. Further-
+0 for x
(16) (17)
The solution, which is obtained by means of the Laplace transformation as shown in appendix, is E-E,=
where GI is the differential capacity of the double layer. Since the sum of the faradaic and capacity currents is constant, one has if
+ i, =
it
(7)
The potential-time relationship is derived by solving Fick's equation for linear diffusion for the following initial and boundary conditions CO = CL and CR = Ck for z 2 Oand t = 0 ( 8 ) CO + CL and CR ---t & for x --f and t = 0 it
=
- cld(E - Ee)/dt + nFDo(dCo/dx)z,o
-o
t~FDo( d C o / d ~ ) z
- D~(dC~/dx),,o
1
(9) (10)
= io
Do(dCo/dx),-o =
(18)
where the notation "erfc" represents the complement of the error function; 0 is defined by the following relationship
(12)
and y is defined by the same equation as (19) but with a minus sign in front of the quantity between brackets. Potential-Time Relationship The potential-time relationship (18) will be first discussed for the case in which the conditions CL = Cg = Coand DO = D R = D are fulfilled. The interpretation is then greatly simplified. However, the general equation will be utilized in the calculation of k, and a. I n the simplified case the constants 0 and y, which are now
The new notations in the above conditions are as follows: the D's, the diffusion coefficients; x , the distance from the electrode; and t, the time elapsed since the beginning of electrolysis. Condition (10), in which the second term on the right-hand side is the faradaic current density written in terms of the flux of substance 0 a t the surface of the electrode, expresses the condition that the total current density is constant (see equation 7). Condition (11) expresses the equality of the flux of substance 0 a t the electrode surface to the faradaic current density can be real or complex quantities according to calculated from (4) and (5). Finally, condition (12) whether the quantity nsPCOD expresses the equality of the fluxes of substances 0 RTcik. and R a t the electrode surface. The solution of this boundary value problem will is smaller or larger than unity. By taking the folbe given for the particular case in which the expo- lowing representative values, n = 1, T = 300"K., nentials in (11) can be replaced without any appre- D = 10-6 sec.-I, q = 2 X farad. cm.-2, ciable error by the first two terms of their expansion Co = lov6mole ~ m . - ~one , deduces that ,8 and y are in series. This is the case when the condition E - real when k, is of the order of unity or larger. The E,
