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A comparison between some recent theories on the effect of ionic specific adsorption upon electrode kinetics
Electroanalytical Chemistry and Interfacial Electrochemistry, 53 (1974)205..218
205
<1i.Elsevier Sequoia S.A., l..ausannc -. Printed in The Netherlands
A COMPARISON BETWEEN SOME RECENT T H E O R I E S ON THE EFFECT
O F IONIC SPECIFIC A D S O R P T I O N U P O N E L E C T R O D E KINETICS
ROLANDO GUIDELI.I
Institute o]" Analytical ('hemistry, Unirersity of Florence. Florence (Italy) (Received 2nd July 1973; in revised form 5th February 1974)
INTRODUCTION
The present author and Foresti recently considered the effect of ionic specific adsorption on electrode processes I starting from Hush's "charge-centric" theory of electrode kinetics 2. According to our approach the current density for a simple electrode reaction of the type OxZ+e __, Red z- 1
(1)
is expressed by the equation
i = Fkf C~x = const., c*x exp
~
I~)M exp [
(z-~)FRT49~t
(2)
where c8, is the concentration of Ox just outside the diffuse layer, q~M is the potential drop across the whole double layer, q~2c is the local potential relative to the bulk solution at the position x = x , occupied by the reacting particle in the transition state, and c~ is the charge-transfer coefficient. In practice the preceding equation differs from the well-known Frumkin relation 3 only in that the ( z - 7 ) factor in eqn. (2) is explicitly regarded as identical to the charge z, of the activated complex. Upon identifying qS~'~ with its average value q~,, as derived on the basis of a simple compact double-layer model, we showed that the ratio kf/kf.q,=o of the rate constant for the electrode reaction in the presence of ionic specific adsorption to that in the absence of ionic specific adsorption, at the constant applied potential q~M,depends on the charge density q~ of the adsorbed ions i at the inner Helmholtz plane (i.h.p.) according to the equation In
k~ _ kr,ql= 0
(z-~)F 4r~lqi RT e
(3)
with
=B(l-x,/d) {ll=Tx,/d
for x , > ~ B for x , ~ B
Here B and d denote the distances of the i.h.p, and of the outer Helmholtz plane (o.h.p.) from the electrode surface plane x=O, y equals (d-B), and ~, is the dielectric constant in the compact double layer ( O < x < d ) .
206
R. GUIDELLI
Somewhat different approaches to the problem at hand were proposed by Parsons 4 and, more recently, by Fawcett and Levine 5. As distinct from these two approaches, which regard the activated complex as occupying the i.h.p., our approach considers the possibility of the activated complex occupying any position within the compact layer ( 0 < x ,
4R/3 3
(4)
into eqn. (4), ref. 1. By so doing, after simple rearrangement we obtain
i = fkrc;x • exp I
= const.-c~ exp
~-~ 4~
4rt(z-c0F ('/r")(/3/~')qi] RT fl/% + V/e,,
exp
-
RT
4~
(5)
where ~, = ~ +
°/Icy
/3/~ + ~/~
(z-~)
(5a)
Incidentally, ~a designates the average potential at x=fl relative to the bulk solution; moreover e,p and e~.denote the dielectric constants between the electrode surface plane and the i.h.p, and between the i.h.p, and the o.h.p, respectively. If no specifically adsorbed ions are present at the given applied potential ~M (i.e., if qi=0), the potential at the o.h.p, will assume a value q~d.qi=0 which is generally different from qSd. Hence eqn. (5) will take the form
THEORY OF EFFECT OF ADSORPTION IN KINETICS
i = Fkf.q, =0 Cgx = const." c~x exp
~
~bM exp
207 RT
~d.q, = 0
(6)
where kf.q,= 0 is the rate constant for the present situation. From eqns. (5)and (6) it follows that
k, In kf.q~o__ 0 -
(ct'-z)F 4 r t ( a - z ) F (fl/ea)(~//ey) q, R T - - (~d--~d.q,=O) + RT fl/e.a+~//e~
(7)
It should be noted that according to eqns. (6) and (5a) the experimental value of the charge-transfer coefficient as determined in the absence of ionic specific adsorption is equal to :t' and is therefore different from the "true" value a ( a t least when x~ =fl). In particular eqn. (5a) shows that ~' is greater or less than ct according to whether ( z - ct) is positive or negative. Such a difference between the "apparent" value ct' and the "true" value ~t for the charge-transfer coefficient was already pointed out by Frumkin 3 (see also ref. 6). In ref. i we recognized the approximation implied in identifying the local potential (often referred to as the micropotential) at the position occupied by the activated complex, ~bl2~, with its average value ~ , but at the same time we stressed the difference between 4 , and the micropotential ~m~¢~o, as commonly employed in double-layer studies. In fact this latter micropotential is defined as the potential at the centre of an adsorbed ion with that ion absent but with all other charges, including image charges, as they would be in the presence of the ion 7's. In other words, in estimating ~m~,o it is assumed that an adsorbed ion, besides preventing the charge centre of any other adsorbed ion from approaching its own charge centre closer than a minimum distance, which is exclusively determined by the "hard cores" of the contacting ions, also perturbs the charge density of the surrounding adsorbed ions by way of long-range coulombic forces 9'~°. In the following we shall designate by ~b~~¢'° the micropotential at the position occupied by a given adsorbed species j. Moreover, we shall designate by ~q~T~c~°the contribution to ~,i¢ro due to the charge-free "exclusion disc" created on a perfectly uniform charge distribution at the i.h.p, around one adsorbed particle ofj by hard-core interactions; the further contribution to qS~'~'° due to the perturbation in the charge density of all adsorbed ions beyond this exclusion disc will be designated by 2tkT~¢'°. Incidentally, Levine and coworkers 9-t2 found it convenient to write ~bJ~'ic~° in the form:
~micro --(/)d = ((lift-- (~d) "~-(j
(8)
with ~t~_(;bo - 4rc2.'(qM+q~) and
~j-
4rtfl)'qigJ
(8a)
Here (qSp--~bd) is the average potential difference between the i.h.p, and the o.h.p., and (j(self-atmosphere potential) expresses the deviation of the micropotential ~ c , o from the average potential qSa. Obviously (j, as well as the parameter gj through which (j is expressed in eqn. (8), depend on the particular double-layer model chosen. In estimating the micropotential at a metal-aqueous electrolyte interphase it is generally held that the role of the diffuse layer is accounted for rather satisfactorily by regarding the o.h.p, as a perfectly conducting plane 8'~°. With this assumption both the hexagonal array model and the cut-offdisc model lead to the conclusion that,
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R. G U I D E L L I
provided ~:~= ~:?,the micropotential gradient within the compact layer is close to linear and tends to perfect linearity for sufficiently low qi values 8. The parameter qj was, indeed, chosen so as to be close to unity for any value of qi and to become practically unity for qi~0. In fact it is readily seen that, for gj= 1 and ~:p=~:~., eqn. (8) reduces to the equation 4,7'i¢" - ~ba= (7/d)(q~ - qSd),expressing a linear micropotential change in the compact layer. If the reactant particle occupies the o.h.p, in the equilibrium position just prior to the charge transfer proper, it is likely that its passage to the transition state occurs fast enough so as not to perturb appreciably the charge density of the surrounding adsorbed ions beyond the exclusion disc, which the hard-core interactions create around the adsorbed activated complex. The same considerations also apply to a neutral reacting particle which occupies the i.h.p, in the equilibrium position just prior to charge transfer; in fact, the charge z¢ = - ~ assumed by this particle in the transition state is likely to persist for a short time with respect to the relaxation time of the two-dimensional ionic atmosphere, which is expected to form at the i.h.p, around any stably adsorbed charged particle. In this sense the preceding considerations are analogous to those which account for the absence of the ionic atmosphere around an ion moving in the bulk solution under the action of a very high field 13 (Wien effect). On the basis of the previous considerations, the local potential qS~~ can be identified with lck~i~°, which accounts only for the presence of a charge-free exclusion disc around the adsorbed activated complex. The radius of this disc equals the distance of closest approach between the activated complex and the electroinactive specifically adsorbed ions i, and therefore should be close to 4-5/~.. More precisely, the local potential 4)IU in which we are interested will be identified with the electric potential at the centre of a charge-free exclusion disc of radius a on a planar perfectly uniform charge distribution of charge density q~, the electrode surface plane and the o.h.p, behaving as perfectly conducting planes. The local potential for the present situation, under the simplifying assumptions that the two fictitious dielectric constants e,t~and e~. are equal (ea=e;.=_Q, is given by the relation ~° 2': - as,~ = 4~-t
(q~ + ¢)lr.- 4~flv q~~alr.d
(9)
with
=
1-
-~fl'/. = ,
K1 n
(71
(9a)
In the preceding equation K1 is the Bessel function of the second kind with imaginary argument and of order 1. In practice the rapidly converging series of eqn. (9a)can be arrested at the third term to a very good approximation 1°,12, yielding d2
10 = I - ~7; [¼f(r)+s°212(~)+So~&(~)]
with - a/d,
So - ( f l - 7 ) / 2 d
f(z) = __8z ~ I rc ,, = t 2 m - 1 K1 [rc (2m - I) T]
THEORY OF EFFECT OF ADSORPTION IN KINETICS
209
1
I z (z) = 2z 2 ~_,
.,=~ [z2+(2m_ 1)2] ~ - 1
m=l
[z2+( 2m- l)Z] ~-
Tabulated values of the ( l - a 9 ) function for several values of r and for /3=3 ,h and ],= 1 or 2 A are reported in ref. 12. In practice, for reasonable values of /3, 7, and a(say, for 2 ~ < / 3 < 3 ]~; 1 ] k < 7 < 2 ~; 4 ,h,
)'
4'.) +
4~z/37qi (1 - Iq)
A comparison of the preceding equation with eqn. (4), in which co and e>. are set equal to c, shows that (4'~¢-4'd) differs from the corresponding average potential difference (4'p-~ba) only by the ( 1 - 19) factor. Hence the local character of the potential at the position occupied by the adsorbed activated complex is accounted for quite simply by introducing ( 1 - 1 9 ) as the multiplying factor of qi in eqns. (5) and (7). In particular this latter equation will become
In
kf _ (ot'-z)F 4rt(ct-z)F fly (1-Xg)qi kr.qi: o RT (4'0- 4'd.q,=O) + RT d
(10)
once we set e0 =c~.=~; as usual. According to the preceding equation, which accounts for discreteness-of-charge effects, plots of In kr.qi= °
RT
(4'd--4'd.q,=0
VS. qi
should yield straight lines of slope 4~z(~-z)F fly 1 - 19 RT
d
e
This theoretical slope differs from that predicted by eqn. (7), neglecting discreteness of charge, by the factor ( 1 - 1 g ) and hence is at least one order of magnitude smaller. In this connection it must be recalled that eqn. (7) accounts rather satisfactorily for the experimental slopes relative to the electroreductions of Zn 2 + and H + in the presence of halide ions, once e is given the value ~ 4.5. It follows that the slopes of the kf
In kf.qi= °
)]
(og- z ) F RT
(4'd--4'd.q,=O
VS. qi plots
210
R. GUIDELLI
for H + and Zn z+ discharge as predicted by eqn. (10) (which accounts for discreteness of charge) on giving ~: the value 4.5 or, even worse, the value 15 adopted in ref. 5, are decisively much smaller than the corresponding experimental slopes. Incidentally, the value r,= 13 was chosen 5 on the basis of the experimental dependence of the charge-dependent part of the average potential difference (4)M--q~d) upon qM and qi for halide adsorption 14'15. The experimental plots of
kr In kr.q,=0
)]
(~'-z)F
RT
(~d--~bd'q'=0
VS. qi
relative to BrOg reduction in the presence of halide and pseudohalide ions are perfectly linear for - 1 0 pC cm - 2 < q i < - 3 0 pC cm -2 and exhibit slopes of 0.12 to 0.14 cm 2 p C - t (ref. 16). The analogous plots for $4062- electroreduction both in the presence of specifically adsorbed halide and pseudohalide ions and in the presence of TIF (in which latter case the specifically adsorbed species is the TI ÷ ion and q~ ranges from 0 to 5 pC cm-2) are also straight lines of slope 0.17 to 0.20 cm 2 pC- t (ref. 17). These further experimental slopes are also too high to be accounted for by eqn. (10) if in this equation ~ is given some value in the range from 5 to 15. Thus setting a = 5 A, f l = 3 ,~, 7=1 A, e.= 10, and ~=0.5, the theoretical slope predicted by eqn. (10) is 0.0146 cm 2 p C - t for BrOw7 electroreduction and 0.0244 cm 2 #C- t for S,O~- electroreduction. It must be noted, however, that once discreteness of charge is accounted for in estimating the potential at the position occupied by the activated complex, thedielectric constant employed must also account for the microscopic structure ofthe compact layer: Now, c is commonly derived from the experimental dependence of the average potential difference (q~M-q~n) in the presence of surface-active anions upon ~ and qi. Incidentally ( ~ - - ~ d ) is obtained by subtracting the G o u y Chapman value for q~d from the applied potential relative to the point of zero charge in the absence ofsurfactants 14'' 5. The resulting dielectric constant is hardly sensitive to the microscopic structure of the compact layer; moreover it accounts for polarization effects in the direction normal to the electrode surface. On the other hand the dielectric parameter with which we are herein concerned must express to what extent the interaction energy of two neighbouring adsorbed ions is modified by the presence of polarizable matter in the compact layer s . In this latter case polarization effects are produced by a very limited number of adsorbed solvent molecules in the neighbourhood of the two ions, because of the strong screening exerted by the o.h.p.; moreover they are directed parallel to the electrode surface, along which direction the electric field component is quite small, certainly much less than the normal component, it has been shown that the dielectric constant which satisfies the previous requirements and is sensitive to discreteness effects, is of the order of and possibly less than unity 8. It is evident that if we set e,= 1 rather than 10 in eqn. (10), the slope of the In kr.q,= °
RT
(q~d--q~d'q'=O) VS. qi plot
as predicted by this equation becomes one order of magnitude greater and therefore agrees satisfactorily with experiment.
T H E O R Y Ok" E F F E C T O F A D S O R P T I O N
IN K I N E T I C S
211
It is interesting to observe that whether we use eqn. (7) with 5<~,< 15, thus disregarding discreteness effects in estimating ~b~ and e, or else eqn. (10) with ~:= 1, thus accounting for these effects, the resulting theoretical slope for the Ii
(~'-z)F
kf
RT
n kf,qi= 0
)] (q~d--t~d'qt=O VS. qi plot
is approximately the same. Apart from the similarity in the final conclusions, we now feel that the approach accounting for discreteness effects is more appropriate.
Comparison with Parsons" approach Let us now examine the analogies and the main distinguishing features between the foregoing approach and Parsons' approach 4. Parsons accounts for ionic specific adsorption by considering its effect on the activity coefficient ~, of the activated complex of the electrode reaction. To determine ~,~, he assumes that the activated complex is adsorbed according to an isotherm proposed by Temkin 18, which is analogous to a Frumkin isotherm in which the apparent standard free energy of adsorption is regarded as a linear function of the surface concentrations of all adsorbed species. In practice only the adsorption free energy term 2 RTB~ .iF~ which is proportional to the surface concentration F~ of the specifically adsorbed electroinactive ions i, of charge z~, is retained, and the second virial coefficient B,.i relative to the interaction between the activated complex and these ions is set equal to z~ B~.~/z~, where B~.~ is the virial coefficient expressing the first-order interaction between the specifically adsorbed ions themselves 4. This amounts to assuming that the energy of the electrostatic interaction of the activated complex with the surrounding specifically adsorbed ions is perfectly analogous to that between the specifically adsorbed ions themselves, once the difference between z , and zi is taken into account; in other words the activated complex is assumed to be located at the same distance x=fl from the electrode surface and to be subject to the same micropotential q~m~croas the electroinactive adsorbed ions. This implies that the adsorbed activated complex, besides creating a charge-free exclusion disc in the uniform charge distribution qi at the i.h.p, by its very presence, also perturbs the charge density of the surrounding adsorbed ions i in exactly the same way as an adsorbed electroinactive ion does. It is evident that, even if we ascribe to the activated complex a perturbing action upon the distribution of the surrounding adsorbed ions, thus disregarding the transitory nature of the complex, the magnitude of such an action should depend on the value of z~ rather than on that of z~, as tacitly assumed in Parsons' approach. Following the preceding line of reasoning, Parsons obtains an expression for the/q/kf q~= o ratio4, In
kf kf.q, = o
_
2z,Bi,iq i z2 F
(11)
which he then applies to the electroreduction of H ÷ and Zn 2 ÷ in the presence of adsorbed halide ions. In his paper 4, Parsons does not give details on the relation between the quantity Bi.i which he introduces into eqn. (11) and double-layer parameters. We deem it useful to examine such a relation. To this end we note that
212
R. GUIDELLI
according to Parsons the adsorption of halides, as well as of azide ~9 and aromatic sulphonates 2°, satisfies the isotherm qi _ a, exp (Cb - z._,F qb'~;cr° ) q~-q~ \ RT
(12)
as a first approximation. In the preceding equation q.~ is the saturation value of q~, ai is the bulk activity of the specifically adsorbed ions, and 4) is Stern's "specific adsorption potential", which should not depend upon the electric field in the double layer. The micropotential ~b~'~'r° at the position occupied by the adsorbed ions i is expressed by Parsons through the relation
~m~. . . . ,~
7 = ~ q,.(q~M--~ba)+2,,(q~M--q~.)
(13)
where the contribution qM(thm--thd) to the average potential drop (4)M--thd) across the compact layer due to the charge density qM on the metal is approximately given by 14 qM(q~U--4)d) -- 4nqM~d
(14)
whereas the analogous contribution q,(~bu-4)d) due to the charge density q~ at the i.h.p, is qt((~bM--~d)
-
-
(15)
4nqi7
Parsons derives ;t (which he designates by "//(fl+y)) from experimental data 19. A comparison between eqns. (13)-(15) on the one hand and eqn. (8) with e~=e.y-=e on the other reveals, as already pointed out by Parsons himself 19, that ), and the theoretical parameter gi are related by the equation // ).=l-~g~ (16) Hence, recalling that gi is close to unity, tending to this value when q~ --* 0, 2 is expected to be close to 7/d. Comparing eqn. (12), in which (~b~'i . . . . ~bd) is expressed by eqns. (13)-(15) and the potential difference ~bd across the diffuse layer is regarded as negligible, with the Frumkin isotherm in the form 2t qiq~-ql
fliaiexp (
2Bi.iqi) ziF
(17)
where - R T In fl~ is the standard free energy of adsorption on a bare surface, we immediately realize that, for a constant charge density qM on the metal, the virial coefficient B~.~ is given by B i i = 2nz2i F2? • r,R T
~ 2nzZ F272 ~.RTd
This is actually the parameter that Parsons introduces into his eqn. (11), thus obtaining a theoretical slope equal to - z¢ F/(R T). 4n°12/(ed) for the In (/q/kr.q, = o) vs. q~ plot. This slope differs from that expressed in our eqn. (3), which does not
THEORY OF EFFECT OF ADSORPTION IN KINETICS
213
account for discreteness-of-charge effects, only by the factor y/f;, once we set x~ =[3 and z , ~_z-c~. This explains why the experimental data relative to Zn 2+ and H + electroreductions are approximately in accord with Parsons' approach 4 and with our eqn. (3) t, once in this latter equation x=x~ is set close to x=[3 and ~: is given a value consistent with the experimental integral capacitances at constant q.u and at constant q~. It must be noted, however, that the virial coefficient Bi.~ to be introduced into Parson's eqn. (11) should be referred to a constant potential drop (qSM--qSd) across the compact layer rather than to a constant qM, since the comparison between the rate constants kr and kf.q,= 0 is actually made at constant ~ . For this purpose, noting that the overall potential difference across the compact layer is given by
('/'M--'C'd) = ,,,,('#,,,-- '#,,) +,,,('¢'M--'#,,) and taking eqn. (15) into account, eqn. (13) can be written
7
(~'~cr°--tkd) = 3 (~M--~bd) +
( d2 ) --
4nqi~/ g
(18)
A comparison of eqn. (12), in which (~'icr°--q~d) is expressed by eqn. (18) and q~d is regarded as negligible, with the Frumkin eqn. (17), shows that the virial coefficient Bi. i at constant (tkM--~bd) is given by
Bi i - 2nz'2' e.RT F27 (2 -
(19)
Replacing B~.~ from the preceding equation into eqn. (I 1) and taking eqn. (16) into account, we obtain the proper expression for the kf/kf,q~= o ratio as derivable on the basis of Parsons' starting hypotheses In
kf
kf.qi=o
_
_
4nz~ F [37 (1--,qi) RT d ~ . qi
(20)
Upon considering z~ equal to (z-~), the preceding equation, henceforth referred to as "Parsons-type" equation, differs from our eqn. (10) accounting for discreteness of charge in two points" (1) it contains the factor ( 1 - g i ) in place of the factor ( 1 - l q); (2)it neglects the term (o:'-z)F/(RT)(dpd-dPd.qi=o). In view of the dependence of gi upon qi, eqn. (20) does not actually predict a rigorous linear dependence of ln(kf/kf.qi=o) upon qi. Thus, according to Levine and Robinson 1°, (1-g~) varies by about 14% in passing from qi = - 5 /uC c m - 2 to qi------I0 /rE cm -z and by a further 20% in passing from q~= - 1 0 / ~ C cm -2 to q ~ = - 2 0 /.tC cm -2 for typical values of a, fl, and 7 (of. ref. 10, Table 3). Incidentally, the above values of ( 1 - g i ) refer to the adsorption of a single ionic species at a mercury-aqueous electrolyte interphase and therefore must be regarded as only indicative for the present situation. Quite probably the major difference between our eqn. (10) and the "Parsons-type" eqn. (20) consists in the neglect of the tkd-containing term in the latter equation. In fact, especially for qi values which are not too high, the experimental value of the first term on the right-hand side of eqn. (10) is often found to be comparable with the value of the second term, as determined
214
R. G U I D E L L I
from the expcrimental slope of the
I In kf,qi= kf °
(o~'-z)F RT (~bd--~bd'q'=°
VS. qi P IOtl6"a7.
Comparison with Fawcett and Levine's approach In their approach to the problem under study, Fawcett and Levine 5 start from Parsons'"potential-centric" theory of electrode kinetics z2 thus assuming that the potential-dependent part AGone of the standard free energy of activation for the electrode reaction (1) is a constant fraction ~ of the potential-dependent parts of the free cnergies of the species Ox, Red, and the electrons at their equilibrium positions just prior to charge-transfer. Incidentally, ~ is simply regarded as an experimental parameter. The authors accept Krishtalik's point of view z3 according to which, in order that the electrode reaction be energetically feasible, both the reactant and the product must be adsorbed at the i.h.p., x=/3, at least to a small extent, before charge-transfer. Discreteness of charge is accounted for by expressing the potential-dependent parts of the electrochemical potentials of Ox and Red at x =/3 in the equation for AG~),c under the form micro ZqSox =z(~b#+~Ox) micro __
(Z--1)(PRcd --(Z--1)((~bp+~Rcd)
for
Ox
for
Red
where the micropotentials ~bS~c~° and • micro at the positions occupied by the oxidized (;bRed and reduced species as well as the corresponding self-atmosphere potentials and the self-atmosphere potential ~o~ and ~,Rcd ~ are defined as in eqn. (8). ~ox, - ~,~d, for the activated complex, ~, , are related to the corresponding activity coefficients 7o~, ~'R~d,and 7', by the equation 5
RT In 7j= zjF (j with j = O x , Red, #
(21)
which is analogous to that relating the bulk activity coefficient of an ion to the potential produced by its own ionic atmosphere. In order to express (o~ and ~Rcd in terms of ( , , Fawcett and Levine apply the equation ~'~ = 7~)x~7~cd derived by Krishtalik 24 on the basis of the Bronsted relation. In view of eqn. (21) the preceding equation may be written in the form z,.~, = (1-ot)Z~o=+Ot(z- 1)~Red
(22)
The adsorption of Ox is assumed to satisfy an adsorption isotherm proposed by Levine et al. 2s, which, apart from minor differences, is analogous to eqn. (12). For ease of comparison with the two previous approaches, in reporting Fawcett and Levine's expression of ln(kf/kf q,=o) as a function of qi, the surface coverage term therein contained 5 will be regarded as negligible. This is surely a perfectly legitimate assumption for the majority of inorganic surface-active ions, whose fractional surface coverages rarely exceed 0.3. With such an assumption and using the present notations Fawcett and Levine's equation s reads
215
T H E O R Y O F E F F E C T O F A D S O R P T I O N IN K I N E T I C S
in
kf (u'-z)F 4rt(~-z)F kf.q,= o RT (~bd--q~d,~,=0)+ RT
(1 - z~ 0 (fl/e#)('ffe, y) z--a g -fl/etj+ ~,/e~ qi
(23)
where ~' is still defined by eqn. (5a). The charge z , of the activated complex is regarded as generally different from ( z - ~ ) and the parameter g , is related to the self-atmosphere potential ( , of the activated complex via eqn. (8a). Equation (23) differs from our eqn. (7) not accounting for discreteness effects by the factor 1
z¢
,
z--~
) go'
in the q~-containing term. The comparison between eqn. (23) and our eqn. (10) accounting for discreteness effects appears more interesting. Thus if we set e,o= e~-e and z , = z - ~ in eqn. (23), it differs from our eqn. (10) only by the presence of the parameter g~ in place of 19. We already pointed out that t 9 accounts only for the charge-free exclusion disc around the activated complex and is independent of qi; on the other hand, the .q, parameter also accounts for the (in our opinion nonexisting) charge-density perturbation beyond the exclusion disc and varies somewhat with qi. In spite of this conceptual difference, the values for 19 and g , are likely not to differ too much over a reasonably wide range of qi values, at least when the activated complex and the specifically adsorbed ion bear charges of the same sign (of., for instance, ref. 10, Table 3). A more fundamental difference between eqns. (10) and (23) stems from the different way in which they are applied. Thus, in our equation, ( z - ~ ) is regarded as identical to the charge z , of the activated complex in view of Hush's theory 2. Moreover, t o is regarded as a well-defined qi-independent parameter once reasonable values are given to a, fl, and ~,. In addition, e is considered to assume a value close to unity so as to account for the doublelayer microstructure. On the other hand, in applying their eqn. (23), Fawcett and Levine 5 regard the product z, g , as a parameter to be determined from experiment and attribute to ,: a value ~ consistent with the experimental dependence of the charge-dependent part of the average potential difference (~b~--4~d) upon qi and qr~. Thus in examining the effect of halide adsorption upon H ÷ electroreduction, they 5 set g/? equal to 7.55 ,~-1, which amounts to setting g=15 when 7 is given the reasonable value of 2/~. For ease of comparison, let us write eqn. (10) and eqn. (23) with a # = ~ - ? ~ in the unified form
RT
c~
4n (a - z) F
IIn k f ,kfq l : 0 (or'z)F )1 R ~ (q~d--q~d'q'=O - Oqi
-
fl'},' --
d
A
(24)
with '
in our eqn. (10)
(24a)
in Fawcett and Levine's eqn. (23)
(24b)
216
R. GUIDELLI
In the electroreduction of BrOg 16 8402- 17, and CC1426 in the presence of halides and pseudohalides, the experimental parameter on the left-hand side of the preceding equation, henceforth referred to as A, assumes values ranging from 0.02 to 0.05 ~,. A somewhat higher value of about 0.2 is observed in H + electroreduction in the presence of I- ion 4. It is readily seen that these experimental A values can be satisfactorily accounted for by our eqn. (10) once we set e,~ 1 and we recall that the (1 - 19) parameter ranges from 0.02 to 0.09. In fact, with the above values of ~, and ( 1 - lg) and ascribing reasonable values to fl and ), (say, fl~-7~2 ,~) the flTA/d term assumes values ranging from 0.02 to 0.09 ]~. The relatively high experimental value of A for H + electroreduction in the presence of halide ions 4 may be explained by the particularly low value which is likely to be assumed by the exclusiondisc radius a around an adsorbed unsolvated hydrogen ion, and by the consequently higher value of the (1 - ~g) parameter. If this assumption is correct, the percentage decrease of a in passing from l- to CI- should be greater for H + ion than for the other previously cited reacting particles, which may explain the particularly high increase'* in the A value for H + electroreduction in passing from I- to CI-. If the above experimental A values are compared with Fawcett and Levine's expression (eqns. 24 and 24b) in which g is set about equal to 15, one concludes that ( z , g , ) / ( z - c t ) m u s t be appreciably less than unity. Hence, recalling that 9* is very close to unity, one is also forced to conclude that Iz, l is appreciably less than [z-ctl. In the particular case of H ÷ electroreduction in the presence of iodide, (z, g , ) / ( z - ct)is ~ven negative, which would require z , and (z-ct) to be of opposite sign. In fact Fawcett and Levine 5, on the basis of their approach, conclude that the activated complex for H + electroreduction in the presence of I- has a charge z~ = -0.6. To explain the above unexpected result, the above authors assume that in the transition state the discharging H + particle interacts (probably electrostatically) with a neighbouring adsorbed iodide ion giving rise to a single unit with charge ( z - c~- 1) - 0 . 5 ; the micropotential tk'~~c~°is assumed to act on this unit rather than on the single H + particle. It is evident that this explanation cannot hold when the reacting particle and the specifically adsorbed ion bear charges of the same sign. Hence the values from 0.25 to 0.7 which are obtained for the z,/(z - :t) ratio on applying Fawcett and Levine's equation to the kinetics ofBrO2, $4062 -, and CC14 elect roreduction in the presence of halides and pseudohalides remain unexplained. In our opinion Fawcett and Levine's explanation, according to which the presence of specifically adsorbed ions alters not only the fluctuation potential ( , but also the charge z , of the activated complex, is in no case consistent with the starting hypotheses on which their approach is based. In fact the distance between the centres ofcharge of the two particles composing the single unit postulated by Fawcett and Levine, i.e. the reacting particle proper and some oppositely charged adsorbed ion, can be no less than the exclusion disc radius a, unless we assume that the interaction between these two particles is chemical in nature. (The possibility of a chemical interaction between the reactant and one or more adsorbed anions, which is the basis of the "anion-bridging mechanism "27, was ruled out in all the treatments herein examined t'4"5.) In other words any electrostatic interaction between the reacting particle and the neighbouring adsorbed ions, strong as it may be, should be adequately accounted for by the fluctuation
T H E O R Y O F E F F E C T O F A D S O R P T I O N IN KINETICS
217
potential (-,, as actually stated by Fawcett and Levine themselves in ref. 5 (c:J'. for instance, ref. 5, eqn. (3.9) in which z~ is considered to be the same both in the absence and in the presence of adsorbed electroinactive ions). In practice, thanks to the screening by the diffuse layer, the electrostatic interaction between the reacting particle and an oppositely charged ion, even if their "hard cores" are in direct contact, should be relatively weak. Thus, in the case of a single adsorbed ion i and for typical values of fl, ~', and a (fl=2.46 A, 7=2.019 ]k, a = 5 ]k) it was shown that the 19 parameter, which accounts only for the charge-free exclusion disc, equals 0.941 whereas the ,qi parameter, which also accounts for the perturbation in the charge density beyond the exclusion disc, equals 0.921 when q~ is equal to - 2 0 / a C cm -2 (ref. 10). Incidentally at this charge density and for an hexagonal array distribution of monovalent anions, the nearest neighbour separation of the array is equal to 9.6 ~, and therefore not much greater than the exclusion disc radius a. In conclusion, the gj parameter in the expression (eqn. 8) for the micropotential ~m~¢~o of a given adsorbed species j is expected to be close to unity even if it J accounts for the electrostatic interactions between the j particle in question and any other charged particle, close as it may be to j. Once it is realized that the 9j parameters for the species Ox, Red, and for the activated complex are all close to unity, then from eqn. (8a) it follows that the corresponding fluctuation potentials (ox, ~,R~d, and ~ are almost equal. Hence in light of eqn. (22) z~ cannot be appreciably different from (z-ct). Finally it should be noted that the activity coefficients 7¢ .qt=0 and ",,~ for the activated complex both in the absence and in the presence of adsorbed electroinactive ions can hardly be brought in relation to one another if in the ideal Guntelberg charging process which conceptually relates these coefficients to the corresponding fluctuation potentials via eqn. (21) 5, the charge of the activated complex is regarded as different in the two cases. ACKNOWLEDGEMENTS
It is a pleasure to acknowledge the helpful discussion of the topics considered in this note with Dr. Ronald Fawcett. This work was supported by the C.N.R. under contract 72.01135.64. SUMMARY
A theoretical expression for the rate constant of an electrode process as a function of the charge density q~ of specifically adsorbed electroinactive ions, derived by Guidelli and Foresti I, is modified taking discreteness-of-charge effects into account, for the particular case in which the activated complex is specifically adsorbed. The modified equation is compared with analogous theoretical expressions derived by Parsons 4 and by Fawcett and Levine 5, so as to point out both the analogies and the main distinguishing features. REFERENCES l R. Guidclli and M. L. Foresti, Electrochim. Acta, 18 (1973) 301.
218 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
R. GUIDELLI
N. S. Hush, J. Chem. Phys., 28 (1958) 962. A. N. Frumkin, Aduan. Electrochem. Electrochem. Eng., 1 (1961) 65. R. Parsons, J. Electroanal. (_'hem., 21 (1969) 35. W. R. Fawcett and S. Levine, J. Electroanal. Chem., 43 (1973) 175. W. R. Fawcctt. J. Electroanal. (7hem., 22 (1969) 19. J. R. Macdonald and C. A. Barlow, Jr. in J. A. Friend and F. Gutmann (Eds.), Proc. of the First Australian ('onference on Electrochemistry, Pergamon Press, Oxford, 1965, pp. 199 247. J. R. Macdonald and C. A. Barlow, Jr., Adt,an. Electrochem. Electrochem. Eng., 6 (1967) 1. S. Levine, K. Robinson, G. M. Bell and J. Mingins, J. Electroanal. Chem., 38 (1972) 253. S. Levine and K. Robinson, J. Electroanal. Chem., 41 (1973) 159. S. Levine, G. M. Bell and D. Calvert, Can..I. Chem., 40(1962) 518. S. Lcvinc. J. Mingins and G. M. Bell, Can. J. Chem., 43 (1965) 2834. G. Kortiim, Treatise on Electrochemistry, Elsevier Publishing Company, Amsterdam, 1965, p. 202. D. C. Grahame and R. Parsons, J. Amer. (..'hem. Soc., 83 (1961) 1291. J. Lawrence, R. Parsons and R. Payne, J. Electroanal. Chem., 16 (1968) 193. M. L. Foresti, D. Cozzi and R. Guidelli, submitted to the J. Electroanal. Chem., M. L. Foresti and R. Guidclli, submitted to the J. Electroanal. Chem., M. I. Temkin, Zh. Fiz. Khim., 15 (1941) 296. C. V. D'Alkaine, E. R. Gonzales and R. Parsons, J. Electroanal. Chem., 32 (1971) 57. J. M. Parry and R. Parsons. Trans. Faraday Soc., 59 (1963) 241. J. M. Parry and R. Parsons, J. Electrochem. Soc., 113 (1966) 992. R. Parsons. Trans. Faraday Soc., 47 (1951) 133. L. I. Krishtalik, Elektrokhialiya, 6 (1970) 1165; J. Electroanal. Chem., 35 (1972) 157. L. !. Krishtalik, Zh. Fiz. Khim., 41 (1967) 2883; Electrochim. Acta, 13 (1968) 715. S. Levine, G. M. Bell and D. Calvert, Can. J. Chem., 40(1962) 518. G. Pezzatini and R. Guidelli, unpublished data. R. De Lcvie, J. Electrochem. Soe., 118 (1971) 185 C.
205
<1i.Elsevier Sequoia S.A., l..ausannc -. Printed in The Netherlands
A COMPARISON BETWEEN SOME RECENT T H E O R I E S ON THE EFFECT
O F IONIC SPECIFIC A D S O R P T I O N U P O N E L E C T R O D E KINETICS
ROLANDO GUIDELI.I
Institute o]" Analytical ('hemistry, Unirersity of Florence. Florence (Italy) (Received 2nd July 1973; in revised form 5th February 1974)
INTRODUCTION
The present author and Foresti recently considered the effect of ionic specific adsorption on electrode processes I starting from Hush's "charge-centric" theory of electrode kinetics 2. According to our approach the current density for a simple electrode reaction of the type OxZ+e __, Red z- 1
(1)
is expressed by the equation
i = Fkf C~x = const., c*x exp
~
I~)M exp [
(z-~)FRT49~t
(2)
where c8, is the concentration of Ox just outside the diffuse layer, q~M is the potential drop across the whole double layer, q~2c is the local potential relative to the bulk solution at the position x = x , occupied by the reacting particle in the transition state, and c~ is the charge-transfer coefficient. In practice the preceding equation differs from the well-known Frumkin relation 3 only in that the ( z - 7 ) factor in eqn. (2) is explicitly regarded as identical to the charge z, of the activated complex. Upon identifying qS~'~ with its average value q~,, as derived on the basis of a simple compact double-layer model, we showed that the ratio kf/kf.q,=o of the rate constant for the electrode reaction in the presence of ionic specific adsorption to that in the absence of ionic specific adsorption, at the constant applied potential q~M,depends on the charge density q~ of the adsorbed ions i at the inner Helmholtz plane (i.h.p.) according to the equation In
k~ _ kr,ql= 0
(z-~)F 4r~lqi RT e
(3)
with
=B(l-x,/d) {ll=Tx,/d
for x , > ~ B for x , ~ B
Here B and d denote the distances of the i.h.p, and of the outer Helmholtz plane (o.h.p.) from the electrode surface plane x=O, y equals (d-B), and ~, is the dielectric constant in the compact double layer ( O < x < d ) .
206
R. GUIDELLI
Somewhat different approaches to the problem at hand were proposed by Parsons 4 and, more recently, by Fawcett and Levine 5. As distinct from these two approaches, which regard the activated complex as occupying the i.h.p., our approach considers the possibility of the activated complex occupying any position within the compact layer ( 0 < x ,
4R/3 3
(4)
into eqn. (4), ref. 1. By so doing, after simple rearrangement we obtain
i = fkrc;x • exp I
= const.-c~ exp
~-~ 4~
4rt(z-c0F ('/r")(/3/~')qi] RT fl/% + V/e,,
exp
-
RT
4~
(5)
where ~, = ~ +
°/Icy
/3/~ + ~/~
(z-~)
(5a)
Incidentally, ~a designates the average potential at x=fl relative to the bulk solution; moreover e,p and e~.denote the dielectric constants between the electrode surface plane and the i.h.p, and between the i.h.p, and the o.h.p, respectively. If no specifically adsorbed ions are present at the given applied potential ~M (i.e., if qi=0), the potential at the o.h.p, will assume a value q~d.qi=0 which is generally different from qSd. Hence eqn. (5) will take the form
THEORY OF EFFECT OF ADSORPTION IN KINETICS
i = Fkf.q, =0 Cgx = const." c~x exp
~
~bM exp
207 RT
~d.q, = 0
(6)
where kf.q,= 0 is the rate constant for the present situation. From eqns. (5)and (6) it follows that
k, In kf.q~o__ 0 -
(ct'-z)F 4 r t ( a - z ) F (fl/ea)(~//ey) q, R T - - (~d--~d.q,=O) + RT fl/e.a+~//e~
(7)
It should be noted that according to eqns. (6) and (5a) the experimental value of the charge-transfer coefficient as determined in the absence of ionic specific adsorption is equal to :t' and is therefore different from the "true" value a ( a t least when x~ =fl). In particular eqn. (5a) shows that ~' is greater or less than ct according to whether ( z - ct) is positive or negative. Such a difference between the "apparent" value ct' and the "true" value ~t for the charge-transfer coefficient was already pointed out by Frumkin 3 (see also ref. 6). In ref. i we recognized the approximation implied in identifying the local potential (often referred to as the micropotential) at the position occupied by the activated complex, ~bl2~, with its average value ~ , but at the same time we stressed the difference between 4 , and the micropotential ~m~¢~o, as commonly employed in double-layer studies. In fact this latter micropotential is defined as the potential at the centre of an adsorbed ion with that ion absent but with all other charges, including image charges, as they would be in the presence of the ion 7's. In other words, in estimating ~m~,o it is assumed that an adsorbed ion, besides preventing the charge centre of any other adsorbed ion from approaching its own charge centre closer than a minimum distance, which is exclusively determined by the "hard cores" of the contacting ions, also perturbs the charge density of the surrounding adsorbed ions by way of long-range coulombic forces 9'~°. In the following we shall designate by ~b~~¢'° the micropotential at the position occupied by a given adsorbed species j. Moreover, we shall designate by ~q~T~c~°the contribution to ~,i¢ro due to the charge-free "exclusion disc" created on a perfectly uniform charge distribution at the i.h.p, around one adsorbed particle ofj by hard-core interactions; the further contribution to qS~'~'° due to the perturbation in the charge density of all adsorbed ions beyond this exclusion disc will be designated by 2tkT~¢'°. Incidentally, Levine and coworkers 9-t2 found it convenient to write ~bJ~'ic~° in the form:
~micro --(/)d = ((lift-- (~d) "~-(j
(8)
with ~t~_(;bo - 4rc2.'(qM+q~) and
~j-
4rtfl)'qigJ
(8a)
Here (qSp--~bd) is the average potential difference between the i.h.p, and the o.h.p., and (j(self-atmosphere potential) expresses the deviation of the micropotential ~ c , o from the average potential qSa. Obviously (j, as well as the parameter gj through which (j is expressed in eqn. (8), depend on the particular double-layer model chosen. In estimating the micropotential at a metal-aqueous electrolyte interphase it is generally held that the role of the diffuse layer is accounted for rather satisfactorily by regarding the o.h.p, as a perfectly conducting plane 8'~°. With this assumption both the hexagonal array model and the cut-offdisc model lead to the conclusion that,
208
R. G U I D E L L I
provided ~:~= ~:?,the micropotential gradient within the compact layer is close to linear and tends to perfect linearity for sufficiently low qi values 8. The parameter qj was, indeed, chosen so as to be close to unity for any value of qi and to become practically unity for qi~0. In fact it is readily seen that, for gj= 1 and ~:p=~:~., eqn. (8) reduces to the equation 4,7'i¢" - ~ba= (7/d)(q~ - qSd),expressing a linear micropotential change in the compact layer. If the reactant particle occupies the o.h.p, in the equilibrium position just prior to the charge transfer proper, it is likely that its passage to the transition state occurs fast enough so as not to perturb appreciably the charge density of the surrounding adsorbed ions beyond the exclusion disc, which the hard-core interactions create around the adsorbed activated complex. The same considerations also apply to a neutral reacting particle which occupies the i.h.p, in the equilibrium position just prior to charge transfer; in fact, the charge z¢ = - ~ assumed by this particle in the transition state is likely to persist for a short time with respect to the relaxation time of the two-dimensional ionic atmosphere, which is expected to form at the i.h.p, around any stably adsorbed charged particle. In this sense the preceding considerations are analogous to those which account for the absence of the ionic atmosphere around an ion moving in the bulk solution under the action of a very high field 13 (Wien effect). On the basis of the previous considerations, the local potential qS~~ can be identified with lck~i~°, which accounts only for the presence of a charge-free exclusion disc around the adsorbed activated complex. The radius of this disc equals the distance of closest approach between the activated complex and the electroinactive specifically adsorbed ions i, and therefore should be close to 4-5/~.. More precisely, the local potential 4)IU in which we are interested will be identified with the electric potential at the centre of a charge-free exclusion disc of radius a on a planar perfectly uniform charge distribution of charge density q~, the electrode surface plane and the o.h.p, behaving as perfectly conducting planes. The local potential for the present situation, under the simplifying assumptions that the two fictitious dielectric constants e,t~and e~. are equal (ea=e;.=_Q, is given by the relation ~° 2': - as,~ = 4~-t
(q~ + ¢)lr.- 4~flv q~~alr.d
(9)
with
=
1-
-~fl'/. = ,
K1 n
(71
(9a)
In the preceding equation K1 is the Bessel function of the second kind with imaginary argument and of order 1. In practice the rapidly converging series of eqn. (9a)can be arrested at the third term to a very good approximation 1°,12, yielding d2
10 = I - ~7; [¼f(r)+s°212(~)+So~&(~)]
with - a/d,
So - ( f l - 7 ) / 2 d
f(z) = __8z ~ I rc ,, = t 2 m - 1 K1 [rc (2m - I) T]
THEORY OF EFFECT OF ADSORPTION IN KINETICS
209
1
I z (z) = 2z 2 ~_,
.,=~ [z2+(2m_ 1)2] ~ - 1
m=l
[z2+( 2m- l)Z] ~-
Tabulated values of the ( l - a 9 ) function for several values of r and for /3=3 ,h and ],= 1 or 2 A are reported in ref. 12. In practice, for reasonable values of /3, 7, and a(say, for 2 ~ < / 3 < 3 ]~; 1 ] k < 7 < 2 ~; 4 ,h,
)'
4'.) +
4~z/37qi (1 - Iq)
A comparison of the preceding equation with eqn. (4), in which co and e>. are set equal to c, shows that (4'~¢-4'd) differs from the corresponding average potential difference (4'p-~ba) only by the ( 1 - 19) factor. Hence the local character of the potential at the position occupied by the adsorbed activated complex is accounted for quite simply by introducing ( 1 - 1 9 ) as the multiplying factor of qi in eqns. (5) and (7). In particular this latter equation will become
In
kf _ (ot'-z)F 4rt(ct-z)F fly (1-Xg)qi kr.qi: o RT (4'0- 4'd.q,=O) + RT d
(10)
once we set e0 =c~.=~; as usual. According to the preceding equation, which accounts for discreteness-of-charge effects, plots of In kr.qi= °
RT
(4'd--4'd.q,=0
VS. qi
should yield straight lines of slope 4~z(~-z)F fly 1 - 19 RT
d
e
This theoretical slope differs from that predicted by eqn. (7), neglecting discreteness of charge, by the factor ( 1 - 1 g ) and hence is at least one order of magnitude smaller. In this connection it must be recalled that eqn. (7) accounts rather satisfactorily for the experimental slopes relative to the electroreductions of Zn 2 + and H + in the presence of halide ions, once e is given the value ~ 4.5. It follows that the slopes of the kf
In kf.qi= °
)]
(og- z ) F RT
(4'd--4'd.q,=O
VS. qi plots
210
R. GUIDELLI
for H + and Zn z+ discharge as predicted by eqn. (10) (which accounts for discreteness of charge) on giving ~: the value 4.5 or, even worse, the value 15 adopted in ref. 5, are decisively much smaller than the corresponding experimental slopes. Incidentally, the value r,= 13 was chosen 5 on the basis of the experimental dependence of the charge-dependent part of the average potential difference (4)M--q~d) upon qM and qi for halide adsorption 14'15. The experimental plots of
kr In kr.q,=0
)]
(~'-z)F
RT
(~d--~bd'q'=0
VS. qi
relative to BrOg reduction in the presence of halide and pseudohalide ions are perfectly linear for - 1 0 pC cm - 2 < q i < - 3 0 pC cm -2 and exhibit slopes of 0.12 to 0.14 cm 2 p C - t (ref. 16). The analogous plots for $4062- electroreduction both in the presence of specifically adsorbed halide and pseudohalide ions and in the presence of TIF (in which latter case the specifically adsorbed species is the TI ÷ ion and q~ ranges from 0 to 5 pC cm-2) are also straight lines of slope 0.17 to 0.20 cm 2 pC- t (ref. 17). These further experimental slopes are also too high to be accounted for by eqn. (10) if in this equation ~ is given some value in the range from 5 to 15. Thus setting a = 5 A, f l = 3 ,~, 7=1 A, e.= 10, and ~=0.5, the theoretical slope predicted by eqn. (10) is 0.0146 cm 2 p C - t for BrOw7 electroreduction and 0.0244 cm 2 #C- t for S,O~- electroreduction. It must be noted, however, that once discreteness of charge is accounted for in estimating the potential at the position occupied by the activated complex, thedielectric constant employed must also account for the microscopic structure ofthe compact layer: Now, c is commonly derived from the experimental dependence of the average potential difference (q~M-q~n) in the presence of surface-active anions upon ~ and qi. Incidentally ( ~ - - ~ d ) is obtained by subtracting the G o u y Chapman value for q~d from the applied potential relative to the point of zero charge in the absence ofsurfactants 14'' 5. The resulting dielectric constant is hardly sensitive to the microscopic structure of the compact layer; moreover it accounts for polarization effects in the direction normal to the electrode surface. On the other hand the dielectric parameter with which we are herein concerned must express to what extent the interaction energy of two neighbouring adsorbed ions is modified by the presence of polarizable matter in the compact layer s . In this latter case polarization effects are produced by a very limited number of adsorbed solvent molecules in the neighbourhood of the two ions, because of the strong screening exerted by the o.h.p.; moreover they are directed parallel to the electrode surface, along which direction the electric field component is quite small, certainly much less than the normal component, it has been shown that the dielectric constant which satisfies the previous requirements and is sensitive to discreteness effects, is of the order of and possibly less than unity 8. It is evident that if we set e,= 1 rather than 10 in eqn. (10), the slope of the In kr.q,= °
RT
(q~d--q~d'q'=O) VS. qi plot
as predicted by this equation becomes one order of magnitude greater and therefore agrees satisfactorily with experiment.
T H E O R Y Ok" E F F E C T O F A D S O R P T I O N
IN K I N E T I C S
211
It is interesting to observe that whether we use eqn. (7) with 5<~,< 15, thus disregarding discreteness effects in estimating ~b~ and e, or else eqn. (10) with ~:= 1, thus accounting for these effects, the resulting theoretical slope for the Ii
(~'-z)F
kf
RT
n kf,qi= 0
)] (q~d--t~d'qt=O VS. qi plot
is approximately the same. Apart from the similarity in the final conclusions, we now feel that the approach accounting for discreteness effects is more appropriate.
Comparison with Parsons" approach Let us now examine the analogies and the main distinguishing features between the foregoing approach and Parsons' approach 4. Parsons accounts for ionic specific adsorption by considering its effect on the activity coefficient ~, of the activated complex of the electrode reaction. To determine ~,~, he assumes that the activated complex is adsorbed according to an isotherm proposed by Temkin 18, which is analogous to a Frumkin isotherm in which the apparent standard free energy of adsorption is regarded as a linear function of the surface concentrations of all adsorbed species. In practice only the adsorption free energy term 2 RTB~ .iF~ which is proportional to the surface concentration F~ of the specifically adsorbed electroinactive ions i, of charge z~, is retained, and the second virial coefficient B,.i relative to the interaction between the activated complex and these ions is set equal to z~ B~.~/z~, where B~.~ is the virial coefficient expressing the first-order interaction between the specifically adsorbed ions themselves 4. This amounts to assuming that the energy of the electrostatic interaction of the activated complex with the surrounding specifically adsorbed ions is perfectly analogous to that between the specifically adsorbed ions themselves, once the difference between z , and zi is taken into account; in other words the activated complex is assumed to be located at the same distance x=fl from the electrode surface and to be subject to the same micropotential q~m~croas the electroinactive adsorbed ions. This implies that the adsorbed activated complex, besides creating a charge-free exclusion disc in the uniform charge distribution qi at the i.h.p, by its very presence, also perturbs the charge density of the surrounding adsorbed ions i in exactly the same way as an adsorbed electroinactive ion does. It is evident that, even if we ascribe to the activated complex a perturbing action upon the distribution of the surrounding adsorbed ions, thus disregarding the transitory nature of the complex, the magnitude of such an action should depend on the value of z~ rather than on that of z~, as tacitly assumed in Parsons' approach. Following the preceding line of reasoning, Parsons obtains an expression for the/q/kf q~= o ratio4, In
kf kf.q, = o
_
2z,Bi,iq i z2 F
(11)
which he then applies to the electroreduction of H ÷ and Zn 2 ÷ in the presence of adsorbed halide ions. In his paper 4, Parsons does not give details on the relation between the quantity Bi.i which he introduces into eqn. (11) and double-layer parameters. We deem it useful to examine such a relation. To this end we note that
212
R. GUIDELLI
according to Parsons the adsorption of halides, as well as of azide ~9 and aromatic sulphonates 2°, satisfies the isotherm qi _ a, exp (Cb - z._,F qb'~;cr° ) q~-q~ \ RT
(12)
as a first approximation. In the preceding equation q.~ is the saturation value of q~, ai is the bulk activity of the specifically adsorbed ions, and 4) is Stern's "specific adsorption potential", which should not depend upon the electric field in the double layer. The micropotential ~b~'~'r° at the position occupied by the adsorbed ions i is expressed by Parsons through the relation
~m~. . . . ,~
7 = ~ q,.(q~M--~ba)+2,,(q~M--q~.)
(13)
where the contribution qM(thm--thd) to the average potential drop (4)M--thd) across the compact layer due to the charge density qM on the metal is approximately given by 14 qM(q~U--4)d) -- 4nqM~d
(14)
whereas the analogous contribution q,(~bu-4)d) due to the charge density q~ at the i.h.p, is qt((~bM--~d)
-
-
(15)
4nqi7
Parsons derives ;t (which he designates by "//(fl+y)) from experimental data 19. A comparison between eqns. (13)-(15) on the one hand and eqn. (8) with e~=e.y-=e on the other reveals, as already pointed out by Parsons himself 19, that ), and the theoretical parameter gi are related by the equation // ).=l-~g~ (16) Hence, recalling that gi is close to unity, tending to this value when q~ --* 0, 2 is expected to be close to 7/d. Comparing eqn. (12), in which (~b~'i . . . . ~bd) is expressed by eqns. (13)-(15) and the potential difference ~bd across the diffuse layer is regarded as negligible, with the Frumkin isotherm in the form 2t qiq~-ql
fliaiexp (
2Bi.iqi) ziF
(17)
where - R T In fl~ is the standard free energy of adsorption on a bare surface, we immediately realize that, for a constant charge density qM on the metal, the virial coefficient B~.~ is given by B i i = 2nz2i F2? • r,R T
~ 2nzZ F272 ~.RTd
This is actually the parameter that Parsons introduces into his eqn. (11), thus obtaining a theoretical slope equal to - z¢ F/(R T). 4n°12/(ed) for the In (/q/kr.q, = o) vs. q~ plot. This slope differs from that expressed in our eqn. (3), which does not
THEORY OF EFFECT OF ADSORPTION IN KINETICS
213
account for discreteness-of-charge effects, only by the factor y/f;, once we set x~ =[3 and z , ~_z-c~. This explains why the experimental data relative to Zn 2+ and H + electroreductions are approximately in accord with Parsons' approach 4 and with our eqn. (3) t, once in this latter equation x=x~ is set close to x=[3 and ~: is given a value consistent with the experimental integral capacitances at constant q.u and at constant q~. It must be noted, however, that the virial coefficient Bi.~ to be introduced into Parson's eqn. (11) should be referred to a constant potential drop (qSM--qSd) across the compact layer rather than to a constant qM, since the comparison between the rate constants kr and kf.q,= 0 is actually made at constant ~ . For this purpose, noting that the overall potential difference across the compact layer is given by
('/'M--'C'd) = ,,,,('#,,,-- '#,,) +,,,('¢'M--'#,,) and taking eqn. (15) into account, eqn. (13) can be written
7
(~'~cr°--tkd) = 3 (~M--~bd) +
( d2 ) --
4nqi~/ g
(18)
A comparison of eqn. (12), in which (~'icr°--q~d) is expressed by eqn. (18) and q~d is regarded as negligible, with the Frumkin eqn. (17), shows that the virial coefficient Bi. i at constant (tkM--~bd) is given by
Bi i - 2nz'2' e.RT F27 (2 -
(19)
Replacing B~.~ from the preceding equation into eqn. (I 1) and taking eqn. (16) into account, we obtain the proper expression for the kf/kf,q~= o ratio as derivable on the basis of Parsons' starting hypotheses In
kf
kf.qi=o
_
_
4nz~ F [37 (1--,qi) RT d ~ . qi
(20)
Upon considering z~ equal to (z-~), the preceding equation, henceforth referred to as "Parsons-type" equation, differs from our eqn. (10) accounting for discreteness of charge in two points" (1) it contains the factor ( 1 - g i ) in place of the factor ( 1 - l q); (2)it neglects the term (o:'-z)F/(RT)(dpd-dPd.qi=o). In view of the dependence of gi upon qi, eqn. (20) does not actually predict a rigorous linear dependence of ln(kf/kf.qi=o) upon qi. Thus, according to Levine and Robinson 1°, (1-g~) varies by about 14% in passing from qi = - 5 /uC c m - 2 to qi------I0 /rE cm -z and by a further 20% in passing from q~= - 1 0 / ~ C cm -2 to q ~ = - 2 0 /.tC cm -2 for typical values of a, fl, and 7 (of. ref. 10, Table 3). Incidentally, the above values of ( 1 - g i ) refer to the adsorption of a single ionic species at a mercury-aqueous electrolyte interphase and therefore must be regarded as only indicative for the present situation. Quite probably the major difference between our eqn. (10) and the "Parsons-type" eqn. (20) consists in the neglect of the tkd-containing term in the latter equation. In fact, especially for qi values which are not too high, the experimental value of the first term on the right-hand side of eqn. (10) is often found to be comparable with the value of the second term, as determined
214
R. G U I D E L L I
from the expcrimental slope of the
I In kf,qi= kf °
(o~'-z)F RT (~bd--~bd'q'=°
VS. qi P IOtl6"a7.
Comparison with Fawcett and Levine's approach In their approach to the problem under study, Fawcett and Levine 5 start from Parsons'"potential-centric" theory of electrode kinetics z2 thus assuming that the potential-dependent part AGone of the standard free energy of activation for the electrode reaction (1) is a constant fraction ~ of the potential-dependent parts of the free cnergies of the species Ox, Red, and the electrons at their equilibrium positions just prior to charge-transfer. Incidentally, ~ is simply regarded as an experimental parameter. The authors accept Krishtalik's point of view z3 according to which, in order that the electrode reaction be energetically feasible, both the reactant and the product must be adsorbed at the i.h.p., x=/3, at least to a small extent, before charge-transfer. Discreteness of charge is accounted for by expressing the potential-dependent parts of the electrochemical potentials of Ox and Red at x =/3 in the equation for AG~),c under the form micro ZqSox =z(~b#+~Ox) micro __
(Z--1)(PRcd --(Z--1)((~bp+~Rcd)
for
Ox
for
Red
where the micropotentials ~bS~c~° and • micro at the positions occupied by the oxidized (;bRed and reduced species as well as the corresponding self-atmosphere potentials and the self-atmosphere potential ~o~ and ~,Rcd ~ are defined as in eqn. (8). ~ox, - ~,~d, for the activated complex, ~, , are related to the corresponding activity coefficients 7o~, ~'R~d,and 7', by the equation 5
RT In 7j= zjF (j with j = O x , Red, #
(21)
which is analogous to that relating the bulk activity coefficient of an ion to the potential produced by its own ionic atmosphere. In order to express (o~ and ~Rcd in terms of ( , , Fawcett and Levine apply the equation ~'~ = 7~)x~7~cd derived by Krishtalik 24 on the basis of the Bronsted relation. In view of eqn. (21) the preceding equation may be written in the form z,.~, = (1-ot)Z~o=+Ot(z- 1)~Red
(22)
The adsorption of Ox is assumed to satisfy an adsorption isotherm proposed by Levine et al. 2s, which, apart from minor differences, is analogous to eqn. (12). For ease of comparison with the two previous approaches, in reporting Fawcett and Levine's expression of ln(kf/kf q,=o) as a function of qi, the surface coverage term therein contained 5 will be regarded as negligible. This is surely a perfectly legitimate assumption for the majority of inorganic surface-active ions, whose fractional surface coverages rarely exceed 0.3. With such an assumption and using the present notations Fawcett and Levine's equation s reads
215
T H E O R Y O F E F F E C T O F A D S O R P T I O N IN K I N E T I C S
in
kf (u'-z)F 4rt(~-z)F kf.q,= o RT (~bd--q~d,~,=0)+ RT
(1 - z~ 0 (fl/e#)('ffe, y) z--a g -fl/etj+ ~,/e~ qi
(23)
where ~' is still defined by eqn. (5a). The charge z , of the activated complex is regarded as generally different from ( z - ~ ) and the parameter g , is related to the self-atmosphere potential ( , of the activated complex via eqn. (8a). Equation (23) differs from our eqn. (7) not accounting for discreteness effects by the factor 1
z¢
,
z--~
) go'
in the q~-containing term. The comparison between eqn. (23) and our eqn. (10) accounting for discreteness effects appears more interesting. Thus if we set e,o= e~-e and z , = z - ~ in eqn. (23), it differs from our eqn. (10) only by the presence of the parameter g~ in place of 19. We already pointed out that t 9 accounts only for the charge-free exclusion disc around the activated complex and is independent of qi; on the other hand, the .q, parameter also accounts for the (in our opinion nonexisting) charge-density perturbation beyond the exclusion disc and varies somewhat with qi. In spite of this conceptual difference, the values for 19 and g , are likely not to differ too much over a reasonably wide range of qi values, at least when the activated complex and the specifically adsorbed ion bear charges of the same sign (of., for instance, ref. 10, Table 3). A more fundamental difference between eqns. (10) and (23) stems from the different way in which they are applied. Thus, in our equation, ( z - ~ ) is regarded as identical to the charge z , of the activated complex in view of Hush's theory 2. Moreover, t o is regarded as a well-defined qi-independent parameter once reasonable values are given to a, fl, and ~,. In addition, e is considered to assume a value close to unity so as to account for the doublelayer microstructure. On the other hand, in applying their eqn. (23), Fawcett and Levine 5 regard the product z, g , as a parameter to be determined from experiment and attribute to ,: a value ~ consistent with the experimental dependence of the charge-dependent part of the average potential difference (~b~--4~d) upon qi and qr~. Thus in examining the effect of halide adsorption upon H ÷ electroreduction, they 5 set g/? equal to 7.55 ,~-1, which amounts to setting g=15 when 7 is given the reasonable value of 2/~. For ease of comparison, let us write eqn. (10) and eqn. (23) with a # = ~ - ? ~ in the unified form
RT
c~
4n (a - z) F
IIn k f ,kfq l : 0 (or'z)F )1 R ~ (q~d--q~d'q'=O - Oqi
-
fl'},' --
d
A
(24)
with '
in our eqn. (10)
(24a)
in Fawcett and Levine's eqn. (23)
(24b)
216
R. GUIDELLI
In the electroreduction of BrOg 16 8402- 17, and CC1426 in the presence of halides and pseudohalides, the experimental parameter on the left-hand side of the preceding equation, henceforth referred to as A, assumes values ranging from 0.02 to 0.05 ~,. A somewhat higher value of about 0.2 is observed in H + electroreduction in the presence of I- ion 4. It is readily seen that these experimental A values can be satisfactorily accounted for by our eqn. (10) once we set e,~ 1 and we recall that the (1 - 19) parameter ranges from 0.02 to 0.09. In fact, with the above values of ~, and ( 1 - lg) and ascribing reasonable values to fl and ), (say, fl~-7~2 ,~) the flTA/d term assumes values ranging from 0.02 to 0.09 ]~. The relatively high experimental value of A for H + electroreduction in the presence of halide ions 4 may be explained by the particularly low value which is likely to be assumed by the exclusiondisc radius a around an adsorbed unsolvated hydrogen ion, and by the consequently higher value of the (1 - ~g) parameter. If this assumption is correct, the percentage decrease of a in passing from l- to CI- should be greater for H + ion than for the other previously cited reacting particles, which may explain the particularly high increase'* in the A value for H + electroreduction in passing from I- to CI-. If the above experimental A values are compared with Fawcett and Levine's expression (eqns. 24 and 24b) in which g is set about equal to 15, one concludes that ( z , g , ) / ( z - c t ) m u s t be appreciably less than unity. Hence, recalling that 9* is very close to unity, one is also forced to conclude that Iz, l is appreciably less than [z-ctl. In the particular case of H ÷ electroreduction in the presence of iodide, (z, g , ) / ( z - ct)is ~ven negative, which would require z , and (z-ct) to be of opposite sign. In fact Fawcett and Levine 5, on the basis of their approach, conclude that the activated complex for H + electroreduction in the presence of I- has a charge z~ = -0.6. To explain the above unexpected result, the above authors assume that in the transition state the discharging H + particle interacts (probably electrostatically) with a neighbouring adsorbed iodide ion giving rise to a single unit with charge ( z - c~- 1) - 0 . 5 ; the micropotential tk'~~c~°is assumed to act on this unit rather than on the single H + particle. It is evident that this explanation cannot hold when the reacting particle and the specifically adsorbed ion bear charges of the same sign. Hence the values from 0.25 to 0.7 which are obtained for the z,/(z - :t) ratio on applying Fawcett and Levine's equation to the kinetics ofBrO2, $4062 -, and CC14 elect roreduction in the presence of halides and pseudohalides remain unexplained. In our opinion Fawcett and Levine's explanation, according to which the presence of specifically adsorbed ions alters not only the fluctuation potential ( , but also the charge z , of the activated complex, is in no case consistent with the starting hypotheses on which their approach is based. In fact the distance between the centres ofcharge of the two particles composing the single unit postulated by Fawcett and Levine, i.e. the reacting particle proper and some oppositely charged adsorbed ion, can be no less than the exclusion disc radius a, unless we assume that the interaction between these two particles is chemical in nature. (The possibility of a chemical interaction between the reactant and one or more adsorbed anions, which is the basis of the "anion-bridging mechanism "27, was ruled out in all the treatments herein examined t'4"5.) In other words any electrostatic interaction between the reacting particle and the neighbouring adsorbed ions, strong as it may be, should be adequately accounted for by the fluctuation
T H E O R Y O F E F F E C T O F A D S O R P T I O N IN KINETICS
217
potential (-,, as actually stated by Fawcett and Levine themselves in ref. 5 (c:J'. for instance, ref. 5, eqn. (3.9) in which z~ is considered to be the same both in the absence and in the presence of adsorbed electroinactive ions). In practice, thanks to the screening by the diffuse layer, the electrostatic interaction between the reacting particle and an oppositely charged ion, even if their "hard cores" are in direct contact, should be relatively weak. Thus, in the case of a single adsorbed ion i and for typical values of fl, ~', and a (fl=2.46 A, 7=2.019 ]k, a = 5 ]k) it was shown that the 19 parameter, which accounts only for the charge-free exclusion disc, equals 0.941 whereas the ,qi parameter, which also accounts for the perturbation in the charge density beyond the exclusion disc, equals 0.921 when q~ is equal to - 2 0 / a C cm -2 (ref. 10). Incidentally at this charge density and for an hexagonal array distribution of monovalent anions, the nearest neighbour separation of the array is equal to 9.6 ~, and therefore not much greater than the exclusion disc radius a. In conclusion, the gj parameter in the expression (eqn. 8) for the micropotential ~m~¢~o of a given adsorbed species j is expected to be close to unity even if it J accounts for the electrostatic interactions between the j particle in question and any other charged particle, close as it may be to j. Once it is realized that the 9j parameters for the species Ox, Red, and for the activated complex are all close to unity, then from eqn. (8a) it follows that the corresponding fluctuation potentials (ox, ~,R~d, and ~ are almost equal. Hence in light of eqn. (22) z~ cannot be appreciably different from (z-ct). Finally it should be noted that the activity coefficients 7¢ .qt=0 and ",,~ for the activated complex both in the absence and in the presence of adsorbed electroinactive ions can hardly be brought in relation to one another if in the ideal Guntelberg charging process which conceptually relates these coefficients to the corresponding fluctuation potentials via eqn. (21) 5, the charge of the activated complex is regarded as different in the two cases. ACKNOWLEDGEMENTS
It is a pleasure to acknowledge the helpful discussion of the topics considered in this note with Dr. Ronald Fawcett. This work was supported by the C.N.R. under contract 72.01135.64. SUMMARY
A theoretical expression for the rate constant of an electrode process as a function of the charge density q~ of specifically adsorbed electroinactive ions, derived by Guidelli and Foresti I, is modified taking discreteness-of-charge effects into account, for the particular case in which the activated complex is specifically adsorbed. The modified equation is compared with analogous theoretical expressions derived by Parsons 4 and by Fawcett and Levine 5, so as to point out both the analogies and the main distinguishing features. REFERENCES l R. Guidclli and M. L. Foresti, Electrochim. Acta, 18 (1973) 301.
218 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
R. GUIDELLI
N. S. Hush, J. Chem. Phys., 28 (1958) 962. A. N. Frumkin, Aduan. Electrochem. Electrochem. Eng., 1 (1961) 65. R. Parsons, J. Electroanal. (_'hem., 21 (1969) 35. W. R. Fawcett and S. Levine, J. Electroanal. Chem., 43 (1973) 175. W. R. Fawcctt. J. Electroanal. (7hem., 22 (1969) 19. J. R. Macdonald and C. A. Barlow, Jr. in J. A. Friend and F. Gutmann (Eds.), Proc. of the First Australian ('onference on Electrochemistry, Pergamon Press, Oxford, 1965, pp. 199 247. J. R. Macdonald and C. A. Barlow, Jr., Adt,an. Electrochem. Electrochem. Eng., 6 (1967) 1. S. Levine, K. Robinson, G. M. Bell and J. Mingins, J. Electroanal. Chem., 38 (1972) 253. S. Levine and K. Robinson, J. Electroanal. Chem., 41 (1973) 159. S. Levine, G. M. Bell and D. Calvert, Can..I. Chem., 40(1962) 518. S. Lcvinc. J. Mingins and G. M. Bell, Can. J. Chem., 43 (1965) 2834. G. Kortiim, Treatise on Electrochemistry, Elsevier Publishing Company, Amsterdam, 1965, p. 202. D. C. Grahame and R. Parsons, J. Amer. (..'hem. Soc., 83 (1961) 1291. J. Lawrence, R. Parsons and R. Payne, J. Electroanal. Chem., 16 (1968) 193. M. L. Foresti, D. Cozzi and R. Guidelli, submitted to the J. Electroanal. Chem., M. L. Foresti and R. Guidclli, submitted to the J. Electroanal. Chem., M. I. Temkin, Zh. Fiz. Khim., 15 (1941) 296. C. V. D'Alkaine, E. R. Gonzales and R. Parsons, J. Electroanal. Chem., 32 (1971) 57. J. M. Parry and R. Parsons. Trans. Faraday Soc., 59 (1963) 241. J. M. Parry and R. Parsons, J. Electrochem. Soc., 113 (1966) 992. R. Parsons. Trans. Faraday Soc., 47 (1951) 133. L. I. Krishtalik, Elektrokhialiya, 6 (1970) 1165; J. Electroanal. Chem., 35 (1972) 157. L. !. Krishtalik, Zh. Fiz. Khim., 41 (1967) 2883; Electrochim. Acta, 13 (1968) 715. S. Levine, G. M. Bell and D. Calvert, Can. J. Chem., 40(1962) 518. G. Pezzatini and R. Guidelli, unpublished data. R. De Lcvie, J. Electrochem. Soe., 118 (1971) 185 C.