K. HOLUB,G. TESSARI, AND P. DELAHAY
2612
Electrode Impedance without a Priori Separation of Double-Layer Charging and Faradaic Process
by Karel Holub, Gino Tessari, and Paul Delahay Chemistry Department, New York Universay, New York, New York 10009 (Received January 89,1067)
The impedance of an electrode with charge-transfer reaction is derived on the basis of the three previously proposed general equations and a new explicit form of the time derivatives of surface excesses. Perturbation of the double layer in the absence of equilibrium between potential and the concentrations of reactant and product is discussed. It is shown that the time derivatives of the surface excesses must be expressed in terms of three variables-potential and the concentrations of reactant and product. Only two variables are needed when there is equilibrium between potential and the concentrations of reactant and product. A posteriori separation of the electrode impedance into faradaic and double layer components is analyzed. Whenever a priori separation is not justified for all practical purposes, previous analyses of fast electrode processes actually yield composite quantities depending on kinetic and double-layer parameters.
that current analysis It was recently pointed of nonsteady-state and periodic electrode processes can be misleading because of the a priori separation of the double-layer charging and the faradaic process. Actually, this separation has always been introduced in the numerous papers on this subject over the past 20 years. This classical approach is much simpler than the one involving no a priori separation and it probably suffices in many cases. However, this approach glosses over the inherent difficulty of interpretation and seems inadequate for certain processesspecific adsorption of reactant and/or product, not sufficiently concentrated supporting electrolyte, and possibly also strong chemisorption, although we have no definite opinion on this point. Three general equations were proposed’ which, upon solution for any particular experimental conditions, should allow analysis without a priori separation. These equations contain the time derivatives (dI’/dt) of the surface excesses of reactant and product, which must be stated in explicit forms for their application. An inadequate expression for the drldt’s based on equilibrium considerations was proposed and applied in a recent analysis of the electrode impedance with charge-transfer r e a ~ t i o n . ~Further consideration led us to the idea of a double-layer perturbation without p2
The Journal of Physical Chemistry
equilibrium between potential and the concentrations of reactant and product involved in the charge-transfer r e a ~ t i o n . ~A new explicit form of the dI’/dt’s was proposed which is far more general than the relationship previously applied. An analysis of the electrode impedance with charge-transfer reaction, based on this development, is presented here and is preceded by a detailed discussion of the new form of the dr/dt’s. It will be seen that the general conclusions previously reached3hold, but final formulas are, of course, different from those previously derived. Analysis is given for amalgam electrodes of the type M+z ze = M(Hg), but ideas carry over to other types of processes. It is assumed that the over-all electrode reaction involves no coupled chemical reaction. A large excess of supporting electrolyte is present.
+
Double-Layer Perturbation without Equilibrium between Potential and the Concentration of M+zand M InJinite Exchange Current. When the exchange(1) P.Delahay, J . Phys. Chcm., 70, 2067 (1966). (2) P.Delahay, ibid., 70, 2373 (1966). (3) P.Delahay and G. G. Susbielles, ibid., 70, 3150 (1966). (4) P. Delahay, K. Holub, G. G. Susbielles, and G. Tessari, ibid., 71, 779 (1967).
ELECTRODE IMPEDANCE
26 13
current density io is infinite, the surface excess of M+z is a function of two independent variables among the following three quantities-the equilibrium potential E, and the bulk concentrations c'+ and C'M of M+z and M.6 The potential E, is, of course, related to c'+ and cSM by the Nernst equation. Selecting the concentrations as independent variables, we write l+ ' = I'+(c'+, c'M). The change of r+ resulting from variations dc'+ and dCM ' of the bulk concentrations is
+
(dr+)iO+- = (~r+/~C8+)c~MdC'+ (br+/bCsM)c.+dCsM (1) Perturbation from the equilibrium potential corresponding to c'+ and C'M, by some external action (variation of potential or flow of current) causes the concentrations of M f 2 and M just outside the double layers in the solution and the amalgam to vary from cs+ and C'M to (c+),& and ( C M ) ~ The ~ . subscript II: is the distance (in solution or amalgam) from a plane just outside the double layers according to the previously discussed model2 The potential E is still related to (c+)~*and ( c M ) ~by - ~the Nernst equation since we assume io + a . If E - E, is small enough, the partial derivatives in eq 1 can be assumed to be constant in the interval E - E,. Hence
+
(dr+/dt)ieco = (br+/bc'+,>C*,(bc+/bt>z - - ~ (br+/bc8M)c8+(acM/at)zd
(2)
Finite Exchange Current. Perturbation for a finite io, different from zero, causes E to be different from the Nernst value calculated for (c+),d and ( c M ) , ~ . The charge-transfer overvoltage qct is then required to drive the electrode reaction. Because of nonequilibrium between E and (c+),& and ( C M ) ~ =r+ O ,is now a function of the three variables (c+)~&, ( c M ) , ~ ,and E . Hence one has for a small perturbation (dr+/dt)o
