HIROAXI MATSUDA AND PAULDELAHAY
332
Vol. 64
DOURLE LAYER STRUCTURE AKD RELAXATION METHODS FOR FAST EIXCTItODE PltOCESSES-THE DOUBLE PULSE GALVAXOSTATIC METHOD BY HIROAKI MATSUDA~ AND PAUL DELAIIAY Contes Chemicd Laboratory, Louisiana Slab University, Bnbn Rouge, Louisiana Receired September S I , 1969
An equation is derived for the overvoltage-time characteristic in the double pulse galvanostatic method, the structure of the double layer being considered in the boundary value problem. It is shown that a previous treatmentwithout consideration of the double layer structure can be applied with simple modifications in the expression of the measured exchange current density. The measured transfer coefficient ams,is seriously in error in case of strong repulsion and, especially, ab traction of the discharged species in the diffuse double layer. Abnormally high transfer coefficients recently reported by other investigators are interpreted. A detailed mathematical analysis is given.
The structure of the double layer a t electrodeelectrolyte interfaces is neglected in the solutions of boundary value problems which have been given for relaxation methods2 in electrochemical kinetics. Such a simplification is entirely justified for not too rapid electrode processes for which the thickness of the diffusion layer is so much larger than the diffuse double layer thickness. Under such conditions kinetic parameters can be corrected3 very simply for the double layer structure without change in the boundary value problem, i.e., the original ideas of Frumkin4 on the effect of the double layer structure on electrochemical kinetics then can be applied without further mathematical analysis. However, as will be shown below, boundary value problems must be reconsidered for rapid prmesses, that is, for processes having a standard rate constant larger than perhaps 0.010.1 cm. sec.-’. Such fast processes can be studied quite easily thanks to the recent development of the faradaic rectification6 and the double pulse galvanostatic6 methods. The latter method, which is the simpler to apply, is analyzed here in detail for the double layer effect. I t will be seen that correction of kinetic parameters can be very significant and that the abnormally high transfer coefficients reported by Barker,6 and confirmed by R a n d l e ~ for , ~ certain processes can be interpreted. Such double layer effects cannot be dismissed even in rather concentrated electrolytes ( 2 I Jl). Boundary Value Problem Consider the electrode reaction 0.0
+ ne = R”R
(1)
~
(1) Research Aeoociate, 1958-1959; on leave from the Government Chemical Industrid Research Institute. Tokyo. (2) For reviews see for instance, P. Delahsy, “New Instrumental Methods in Electrochemratry.” Interscience Publ., Inc., New York, N. Y., 1954; P. Delahay, Ann, Rev. Phus. Chem., 8, 229 (1957). (3) M. Breiter, M. Kleinerman and P. Delahay. J . Am. Chem. SOC.. 80, 5111 (1958). (4) For a recent review, see A. N. Frumkin, 2. Elektrochem., 69, 807 (1955). See alao ref. 3. (5) G. C. Barker, R. L. Faircloth and J. A. W. Gardner. Ndure, 181. 247 (1958); G. C. Barker, Anal. Chin. A&, 18, 118 (19583; G. C. Barker, “Proceeding8 of the Symposium on Electrode Processes,” Philadelphia. May, 1959, E. Yeager ed.. John Wilry and Sons, lnc., New York, N. Y.. in course of publication. (6) (a) H. Geriacher and M. Krause. 2. phyeik. Chcm. (Frankfurt), 10, 264 (1957); 14, 184 (1958). (b) H. Matsuda. 8. Oka and P. Delahay, J Am. C,bm. Soc.. 81, 5077 (1959). (7) J. E. B. Randles. “Proceedings of the Symposium on Electrode ProceMes,” Philadrlphis, May, 1959, E. Yeager ed.. John Wiley and Sons, Inc., New York. N. Y., in coume of publication.
where 0 and R are soluble species having the ionic valencies a and Z R ( ~= ZR n). R is soluble either in solution or in the electrode (amalgam; ZR = 0). The solution contains a large excess of z-z indifferent electrolytes which are assumed to determine entirely the double layer structure. In the absence of specific adsorption the gradient of potential in the diffuse double layer is, according to the Gouy-Chapman theory
+
dp/dz = -(2RT/lzlF)~ sinh (IzIFpplZET)
(2)
with I/. = ( R T ~ / ~ ~ z * F * C ~ ) ” ~
(3)
where p is the electrical potential referred to the potential in the bulk of solution, e is the dielectric constant, and Ct is the sum of t,he concentrations of z-z indifferent electrolytes. The concentrat,ion of species i (0 or R) involved in the electrode reaction a t a plane electrode is 11, solution of8
+
dCi/bt = D i h / d ~ [ & C i / b ~ (~iF/RT)Cibp/d~l (4)
where zi and Di are the ionic valence and diffusion coefficient of species i, respectively. The origin of the coordinate x is taken as the plane of closest approach (Helmholtz plane). At equilibrium, ie., at 1 = 0, one has the Boltzmann distribution Ci = CiO exp (--ziFv/RT)
(5)
where C i o is the bulk concentration of species i outside the double layer region. Furthermore, one has the boundary conditions z
z = 0: Di[dCl/bz
+
OD:
Ci --?t CO i
+ (ziF/RT)Cdp/dz] = i Z J n F
(6) (7)
where Ifis the faradaic current density. The plus sign in front of It holds for species 0 , the minus sign for species R. General Solution in the Form of Laplace Transform We shall first obtain a general solution of eq. 4 in the form of a Laplace transform and then derive the inverse transform for the articular case of the double pulse galvanostatic method. set z l i = CCi as the Laplace trnnsformsof the function Ci. Equations 4, 6 and 7 become after transformation sui CiQexp ( -ziFv/RT) = Did/dz[dui/dx (ziF/RT) uid~/dxl ( 8 )
d
-
+
( 8 ) See for instance. B. Levich. Dokl. Acod. Nauk. 67, 309 (1949).
(9)See for instance, R. V. Churchill, “Modern Operational Mathematics in Engineering,” McCraw-Hill Book Co., Ne# York. N. Y., 1944.
llurch, 1900
h l J l 3 L I i PULSE GALVANOSTATIC h l E T H O D FOR FAST ELECTlLODE 1'RWESSES 2
+ -:
x = 0: Di[dui/dz
ui
+CiO/S
333 (9)
+ (~iF/RS')uidv/d~]= C( fIr/nF)
(10)
s being the par:tmctcr for transformation The general solution, derived in the Appendix, is a t z = 0, i . e , in the Helmholtz plane (ui = ui*a t z = 0) ui* = exp ( -ziFpo/ET)((CiO/s) - [d:(&If/nF)](l/Dis)'/iG i ) with
(11)
+
KP+'/s [(2/~)(S/Di)"iEO-'] (ZP+'/,[(~/K)(S/D~)'/~II P + ' / ~ [ ( ~ / K ) ( S / D ~I ) ' / ' ~
(13) P = f(lz,l/lzl) - 1 = exp (lzlFIv0~/2RT) (14) Further notations: w is thc potential in the Helmholtz plan?; and thc 1,'s and tlir K,'s are modified Bcssrl functions of thc v-th order and of the first and second kind, respectively. The upper and lower signs in eq. 13 correspond to the cases z , p > 0 and z l p < 0 , respectively.
EO
G.enera1 Current-Overvoltage Relationship in the Form of Laplace Transform The faradaic current density I f for processes corresponding to eq. 1 is for a rate-determining step involving n electrons2 cup (zoFqo//i?') c'xp [( -anF/R?')(E - E, )] / exp ( z n Fpp,/fiy') exp [ { ( I - o l ) n ~ / ~( E~ ' ] E,)] VIo = - c , sV(E - E,)
(15)
I
with
(22) and that the ccll current density I t = Z, To,.one obtains (l,,),,,,,, = 1, cxI, - z o ) ~ a o / ~ (16) ~ ~ ~ ~ the general currenkovervoltaye characteristic in the form I O = nFk,Coo( 1 - a)CROa (17) of Laplace transform
where IO is the exchange current density, (I0)app the apparent exchange current density, IC. the standard rate constant, a the transfer coefficient, E the electrode potential, E e the equilibrium electrode potential corresponding to the bulk concentrations, (00 the potential in the Helmholtz plane, the C*'s the concentrations in the Helmholtz plane, and the (70's the bulk concentrations. When (&'-&'e) tl, and this equation negative when the discharged ion IS sufficiently with only the first term on the right-haud side a t t r a c t d in the d8use double layer. This implies that (10)- is negative or that one has for holds for 0 < t < tl. By transposition of a previous result," one the extrapolated overvoltage at t l = 0, E - E, > 0 deduces that the overvoltage, E h - E,, at time tl in a reduction or E - Ee < 0 in an oxidation. for the ovemltage-time curve with a horimntal Under these conditions, the overvoltage deduced from the plot of Eh - E. against tl'/: according to tangent at tl is, for d c i e n t l y dd u e s of tl ~
+ + +
1
t-
l
*
%-Ee= -$Iz)
[1/(zO)sppl
+ (l/np)[(m/dOoDO) + (aR/flRoDR)l + t + + .. .
(4/3r1/:)(l/nP)[(1/C00D0'/:) l/CRo&'/l)]tl'/l
(31)
- E, against t>l (spo the potential in the Helmholta plane, i.e., the largeat possible value of (p)
-
W b = -(d2)€Yd/dl)
(44)
Equation 8 now hecomes (d/dE)I€-WWdl)J
- (2/K)qS/Di)€-2p-4ui
0
(45)
p being defined by e ~ 13. . Equation 45 is equivalent to a modified Besael equataon of (p l/t)-th order of the formla
+
+
d%i/d~* (l/v)(dWi/dv)
- [1 + (P + l / ~ ) t / ~ * l =~ i 0
(46) as one can &ow by eetting r) (2/1c)(U/Di)'/*t-' and (ti = p+'/r(~i(47) in eq. 45. The general solution of eq. 45 is ui = { [e(.tZr/nP)]/(Dp)'/a)~+'/~+'/a x { A J P + ' / ~ [ ( ~ / ~ ) ( S / ~ ~ ) ' ~At~p+'/a[(2/~X~/Di)"5-'1 ~~-~I 1 (48) (13) T. V. g.rmM and M. A. Biot. "Mathenutid Method. in =I
+
Enginea&&" MoGnr-Hill Book CO.. New York, N. Y.. 1940, p. W.
Fig. l.-Variatiom of (Z*xpC/(Zl),., with the potential spo in the Helmbolts plane for different exchange current densiti- Zo (in amp. ~m.-'; w eq. 17). Solid and dad cumcorrespond to I/K = 3 X 10- cm. and l / ~ = '01 cm., respectively, i e . , to approximately 1 and 0.1 M solution of a 1-1 electrolyte, respectively. Note that I c ~ , ( P / R T= 1 corresponds approximately to 1 ~ =1 0.025 volt at 2 5'. Data used in ddatiorm: Q = 0.5, zo = n = 2, ZE = 0, CO = 10- mole em--: DO = 10'~m.~ gec.-l.
I~IROAKI MATSUDA
336
F’ol. 6.1
DOUBLE LAYER STRUCTURE AND ELECTRODE PHOCESSES WITH A PRECEDING CHEMICAL REACTION BY HIROAKI MATSUDA’ Coales Chemical Laboratory, Louisiana Slate University, Baton Rouge, Louisiana Received September 81, 1969
r-
An analysis is made of the influence of the double layer structure on electrode processes wit,h a pseudq-first order cedin chemical reaction in which a non-reducible (non-oxidizable) substance ( Y ) is transformed into a reducible (oxldlza le) in presence of a large excess of another reactant (X). The concentration of 0 in the Helmholtz plane is denved, one and several particular forms of the resulting equation are reported. It is shown that the magnitude of thc double layer effect depends, among several factor, on the ratio of the diffuse double layer thickness (l/u) to the reaction layer thickness (p). When ( 1 / ~ )< < p there is hardly any double layer effect; when ( 1 / ~ >>p ) the double layer effect can be analyzed in terms of a simple Boltzmann distribution. The general e uation is applied to the derivation of the transition time in chronopotentionietry and the limiting current in polarograpiy.
(8)
The influence of the double layer structure on electrode processes with a preceding chemical reaction (so-called kinetic processes) was recently discussed in a paper from this Laboratory* and by Gierst.* It was shown in these investigations that the polarographic and galvanostatic characteristics of kinetic processes are accounted for by a simple Boltzmann correction for the concentrations of reactants in the plane of closest approach (Helmholtz plane). A more rigorous approach‘ in which the double layer structure is considered in the solution of the boundary value problem was announced b y Gierst.3b The same problem is considered here but for conditions somewhat more general than those mentioned by G i e r ~ t .Equa~~ tions are dcrived by the limiting current in polarogmphy and the transition time in chronopotentiometry (galvanostatic method) for the reactions Y(W)
z? O ( W ) + v?i(.x)
Y b Y ) 4, t X ( z x )
20
+
YZX)
Y2X
=
20)
(zy =
O ( d (zy
+
(I) (11)
where Y aiid X are not reduced or oxidized at the potential a t wliich substancc 0 is reduced. General Assumptions For the sake of simplicity, we introduce the asxumptions: (i) The solution contains a largc cxcess of indifferelit z-z electrolytes, which esscntially dctermine the double layer structure. According to the Gouy-Chapinan theory of the diffuse double layer one has dq/tlz == - 2 ( R T / j z i F ) ~sinh (lzlFq/2RZ’)
(1)
with ( l / ~= ) ( R T I / ~ ~ Z * F ~ C , ) ’ / ~ (2)
where cp is the electrical potential, B the dielectric constant and Ct the sum of the bulk concentrations of indifferent electrolytes. (ii) The thickness of the diffuse double layer 6 and the reaction layer p are very small in comparison with the diffusion layer thickness a’, and consequently the time-variation of the concentrations of substances 0 and Y within the former two layers can be neglected. (iii) The solution contains a large excess of substances X in comparison with the concentrations of 0 and Y. The conccntration distribution of X thus is independent of time during electrolysis and is not influenced by current flowing. One has CX = CXO exp ( - z x F q / R l ’ ) (3) where CXO is the bulk concentration of X. (iv) The equilibrium between the substances 0 and Y is greatly in favor of Y, ie. (COO/CY~)
