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A S D P A U L DELAHAY
ELECTRODE PROCESSES WITH SPECIFIC OR NON-SPECIFIC .IDSORPTION : FA4RSDAICIMPEDA4NCEAND RECTIFICATION BY
~ ~ I T S U GSEh-DAl I AZJD PAEL
LIELAHAY
C o d e s Chemical Lahwatory, Louisiana Stute Unizlersity, Ruton Rouge, Louisiana Received April 2 1 , 1961
Faradaic impedance and faradaic rectification are discussed for processes with specific or non-specific adsorption of reactants and/or reaction products. S o explicit form of the rate equation for adsorption kinetics is postulated. Analysis of the faradaic impedance for processes with specific adsorption includes: derivation of a general current-potential characteristic for small amplitudes without the postulating of an explicit form for the rate equation for charge transfer kinetics; an equivalent circuit for electrode processes in which adsorption processes are independent of one another (e.g., as for metal deposition with amalgam formation); variations of the phase angle with frequency and analysis of some existing data in the !iterature; correction for the double layer capacity. Equivalent circuik for processes with non-specific adsorption of reactants, i.e., with repulsion or attraction of the discharged species, are also derived for the faradaic impedance; and comments are made on a related problem as solved by previous authors (Levich, Matsuda and Delahay). Faradaic rectification is treated for the above cases, and the frequency dependence of the rectification voltage is analyzed in detail. Implications in t,he determination of kinetic parameters are pointed out.
The characteristics of electrode processes with specific adsorption of reactants and/or products of reaction in relaxat’ion methods have been analyzed by a few authors. Laitinen and Randles2 and Llopis and co-morkers3discussed the corresponding faradaic impedance and showed that, under simplifying assumptions, adsorpt’ion is accounted for by an additional frequency-independent capacity and resistance in the equivalent circuit. Barker4 reached a, similar conclusion for absolutely equivalent circuits of the faradaic impedance. Matsuda and Delahay6 analyzed the effect of reactant adsorption in t’he potent’iostatic and galvanostatic methods. The adsorption isotherm (Langmuir) is postulated at the onset and the adsorption and desorption rate constants are assumed to be potent’ial-independentin these investigations.6 These assumptions render t’he model for the influence of adsorption quite simple but they are not substantiated by experiment. Specific adsorption of ions does not follow a Langmuir isotherm as was shown by Parsons’ and Grahame8; and nothing is known about the influence of potential on rate constants for ionic specific adsorption. A more general analysis in which these assumptions are not made a priori is discussed for the faradaic impedance and rectification methods. Some comments are also made on electrode processes with non-specific adsorption which had been treated previously in this Laboratory for the conditions prevailing in rc1axat)ion method^.^ (1) Postdoctoral research associate, January 1960-March 1961; on leave f r o m the Department of .4gricultural Chemistry, Kyoto University, Kyoto, Japan. ( 2 ) H. 8.Laitinen and J. E. B. Eandles, Trans. Faraday Soc., 61, 54 (1955). (3) (a) J. L!opis, J. Fernandez-Biarge a n d M. Perez Fernander, EZectrochirn. Acta. 1, 130 (1969); (b) “Transactions of the Symposium on Electrode Processes, Phi!adelphia,” E. Yeager editor, John Wiley and Sons, Kerv York, P;. Y., 1959, in course of publication. (4) G. C. Barker, “Transactions of the Symposium on Electrode Processrs, Philadelphia,” E. Yeager editor, John Wiley a n d Sons, New York. h-. Y.,1959, in course of pubiication. (5) 13. Matsuda and P. Delahay, Coll. Czeehoslou. Chem. Communs., 26, 2977 (1960). (6) With the possible exception of Barker’s work for which not enough details are available thus far. (7) R. Parsons, Trans. Faradau floc., 61, 1518 (1955). ( 8 ) D. C. Grahame, J . Am. Chem. SOC.,80, 4201 (1958). (9) (a) H. Matsuda a n d P. Delahay, J . Phgs. Chem., 64, 332 (1960); (b) H. Matsuda, ibid., 64, 339 (1960).
Faradaic Impedance for Processes with Specific Adsorption Curren t-Potential Characteristic for Small Amplitudes.-We consider the over-all reaction 0 ne = R for which 0 and R are soluble in solution and distinguish the sequence of steps in Fig. 1, where arrows correspond to a positive flux of matterlo: mass transfer of 0 and R, adsorption of 0 and R as heterogeneous processes, and charge transfer between adsorbed species (to the exclusion of charge transfer involving species 0 and R in solution). Since the current density I is a function of the potential E and the surface concentrations I’o and r R , one has for the first harmonic and for small variations of potential, as prevail in the faradaic impedance and rectification methods1‘
+
6 1 = (aI/bE)6E
+ (aI/mo)6ro + (bT/dh)8h
(1)
where the derivatives are taken a t equilibrium. Our problem is to evaluate 6I’o and 6 r R and derive, if possible, the elements of an equivalent circuit. ro and r R in eq. 1are such that dPi/dt = =t( I / n F ) i +ia
(2)
There i represents 0 and R; the upper and lower sign correspond t’o 0 and R, respectively; the ~ 8 ’ sare the fluxes for the adsorption processes; n is the number of electrons in the charge transfer reaction; and F is the faraday. The flux &a in eq. 2 for species i depends on I’o, r R , E , and the concentration Ci* of species i a t the electrode surface. Thus 6+ia
= (a+ia/aro)6ro
+
(b+in/3rR)6rR $(d+is/dCi*)GCi* (a+ia/aE)6E (3)
+
One further has $i* = i +ia where &*is t’heflux for the mass transfer process a t the electrode surface for species i. For sinusoidal variations, 6Ci* in eq. 3 is related to $ia by 6Ci* = =k (h,i
- jhxi)6$i”
(4)
where j is the operator (-l)*’s and h r i and jhxi are the real and complex parts of a function h i (IO) This convention is the opposite of the one generally adopted for electrode processes, except in the German literature. “Awkward” signs are avoided with this convention. (11) (a) T h e use of eq. 1 as written in terms of volume concentrations is due t o D. C. Grahame, J . Electrochem. Soc., 99, C370 (1952): (b) see summary in P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, Inc., New York, N. Y., 1954, pp. 146-178.
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E L E C T R O D E P R O C E S S E S WITH S P E C I F I C OR X O K - S P E C I F I C ,4DSORPTION
c"0
whose explicit form, which only depends on mass transfer, will be determined below. One further has d(6ri)/dt = jw6ri
(5)
where w = 2nf, f being the frequency. There results by combination of the foregoing equations 6ri = f (Hri
- jHx,)6I =k (GrL- jGx,)6E
(6)
where the function H and G can be written (see Appendix) in terms of the function h of eq. 4 and the partial derivatives of eq. 3. By introducing the 6ri's from eq. 6 in eq. 1, one obtains the I-E characteristic for sinusoidal variations of small amplitude 61
[(gr
.igx)/(r
- j~)laE
(7)
where the functions g, r and x can be written (see Appendix) in terms of the functions H and G in eq. 6 and the partial derivatives in eq. 1. Equivalent Circuit.-The analysis of the general eq. 7 in terms of an equivalent circuit would be very involved, and it is useful to consider the case in which the adsorption processes for 0 and R are independent, i.e., the case in which d$oa/bI'a = 0 and dcj&a/dI'o = 0. Thus, we assume that the adsorption kinetics for 0 is independent of the surface concentration of R and vice versa. This is the case, for instance, in the deposition of metal with amalgam formation, and the foregoing simplification is not unrealistic. Equations for g, r and x in eq. 7 are then greatly simplified and are given in the Appendix. One further simplification can be made when the parameter j-i defined by ti
= (ar,/3E)a/(arddE)c
(8)
6I = [l/(?- jX)]SE
(9)
is such that Ij-iI 0 (repulsion) and x,Acp < 0 (attraction), respec-
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J I I T S U G I S E N D A A N D P A K L IIELAHAY
REPULSION
AT TR A C TIO N *I
Fig. 4.-l:quivalent circuit of faradaic impedance for the reaction 0 n e = R, only 0 being non-specifically adsorbed.
+
tively. Equation 32 for G i applies only when the conditions are fulfilled (see Note after the Appendix). (l/K)(U/Di)l/z
