A new approach to the analysis of electrode kinetics with specifically adsorbed reactants

Electroanalytical Chemistry and Interfacial Electrochemistry, 58 (1975) 81-93

81

© Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

A NEW APPROACH TO THE ANALYSIS OF ELECTRODE KINETICS WITH SPECIFICALLY ADSORBED REACTANTS*

MICHAEL J. WEAVER and FRED C. ANSON

A. A. Noyes Laboratory of ChemicalPhysics**, California Institute of Technology, Pasadena, Calif. 91125

(V.S.A.) (Received 8th August 1974)

Although approaches to the analysis of electrode kinetics for specifically adsorbed reactants have been presented previously by several authors 1-3, experimental tests of the analyses have remained rare. A number of factors have contributed to this circumstance, including the experimental difficulties often encountered in ascertaining the extent to which observed rates correspond to the reaction of adsorbed, rather than unadsorbed reactants present at the outer Helmholtz plane. We have obtained new kinetic data for several adsorbed chromium isothiocyanate complexes under conditions where they are reduced entirely in the adsorbed state. These data are presented and analyzed in the succeeding paper 4. In this paper we wish to present and discuss the theoretical relations that are used to analyze the kinetic data. The approach employed has much in common with the previous treatments of Mohilner and Delahay 1"2 along with some significant differences. In order best to expose the differences (and what we believe to be their advantages) the essential core of both present and previous analyses will be outlined at the outset. CURRENT-POTENTIAL RELATIONS WITH ADSORBED REACTANTS

The current density for the simple, one-step reduction of an adsorbed reactant under conditions where the back reaction is negligible can be expressed as follows 5 :

~ k T at i = n~-ff~y~ exp ( -

AG°~ RT/

(1)

n, F, k, T, R, and h have their usual significance, at is the activity of the reactant in the adsorbed state, V, is the activity coefficient of the activated complex and AG ° is the difference in standard free energies between the activated complex and the ground state of the adsorbed reactant. This definition of AG° differs from that used by most previous authors1'Z'6'v who define their activation energy with respect to the ground state energy of the unadsorbed reactant in the bulk of the solution because this is the parameter that is customarily held constant in rate-potential studies. We have chosen * Dedicated to Dr. J. E. B. Randles on the occasion of his retirement from the Chemistry Department, University of Birmingham. ** Contribution No. 4905.

82

M . J . WEAVER, F. C. A N S O N

an alternative definition for the standard free energy of activation in eqn. (1) in anticipation of deriving kinetic relationships for experiments carried out at constant surface concentration rather than at constant hulk activity of the reactants*. There are two further differences between eqn. (1) and the previous general approach of Mohilner and Delahay 1'2. These authors did not include the activity coefficient of the activated complex in their analysis and chose to express what we have called the activity of the adsorbed reactant as the ratio of two terms which they described as the activities of the adsorbate and of the adsorption sites. We prefer to use a single variable, ar, to express the same parameter which may also be expressed as the product of the surface concentration of adsorbate, Fr, and its activity coefficient, ~r, a r ~ /'rTr

(2)

In order to use eqn. (1) to calculate current-potential curves, it is necessary to have values for the potential dependences of A G ° , ~ , , and at. We will take up the evaluation of each of these three parameters in turn. The potential dependence of AG ° is usually evaluated (cf. ref. 1), by considering four states and their standard electrochemical free energies: State I: The reactant in its standard state in the bulk of the solution and n electrons in the electrode at potential tkm, G ° = ~o + nttOa,e_ n F ~ m

(3)

State II: The reactant adsorbed on the electrode in its standard state and n electrons in the electrode at potential q~,,, G ° = fi° r--~ ]~],~°.e - nrc~m

(4)

State III: The reduced product adsorbed on the electrode in its standard state

GO =

(5)

State IV: The product in the bulk of solution in its standard state GO = tip0

(6)

where/~o and/~o denote the standard electrochemical potentials of the reactant and product in the bulk of the solution, flop and PAr-0denote the corresponding electrochemical potentials for the species in the adsorbed state, and #m,e o is the standard chemical potential for electrons in the metal electrode. The electron-transfer step, from states II to III is assumed to be rate-determining. It is assumed that AG° can be written as the sum of a potential-independent and a potential-dependent part 1, (AG°)E, and that the latter can be expressed in terms of the potential-dependent parts of the standard free energies of the adsorbed reactant, (G°)E, and product, (G°I)e, by introducing the transfer coefficient, 0~1"10,11:

(AG °)E = [(G°,)E-- (G°)E]

(7)

* It is convenient (and conventional 8) to take the standard state for the adsorbate on the surface at a very low surface concentration (e.g., 1 molecule c m - 2), where it would be expected to exhibit ideal (i.e., Henry's law) behavior so long as the ionic strength is high enough to represss diffuse layer contributions to the free energy of adsorption 9. This is the standard state we shall adopt.

83

KINETICS WITH SPECIFICALLY ADSORBED REACTANTS

The arbitrariness of this separation of the total free energy of activation into two hypothetical components" is not a source of concern for the present purpose because only the potential dependences of the free energies will be calculated. The standard electrochemical free energies of adsorption of the reactant, ~G A,' and product, ~G~p, can be conveniently introduced at this point: ~G~,

= GR-G?

(8)

~G~p=GRI-G?V

(9)

The potential-dependent parts of these free energies are then: (~Gth = (GRh+ nF4>m

(10)

because

(11 ) and (~G~p)E = (GRI)E

(12)

because G?v contains no potential-dependence. Thus (GRI)E-(GRh = (~G~p)E-(~G~JE+nF4>m

(13)

So eqn. (7) can be rewritten as: (~G~)E

= ct[(~G~p)E-(~G~,h+nF4>m]

(14)

and eqn. (1) can be expressed as: In i = In K -In I'r+ In rr-In I';t - RctT [(~G~p)E-(~G~JE+nF4>m]

(15)

where K is a combination of constants that includes the potential-independent part of the standard free energy of activation. From the form of eqn. (15), it is clear that the evaluation of ct from currentpotential curves will be simplified if measurements are made (or the data analyzed) under conditions that maintain r n the surface concentration of the reactant, constant. The dependences of I'r and I';t on T, are then also held constant and only their dependence on the electrode potential, if any, has to be known or measured. Kinetic measurements made at constant F, have fundamental appeal because they serve to hold constant the mutual interactions among adsorbed reactant molecules as well as interactions among reactant molecules and activated complexes. (This should be true whether or not the adsorption isotherms exhibit congruency12.13; note that failure to observe congruency is normally attributed to an incorrect choice of electrical variable which will necessarily lead to values of I'r which appear to depend on the electrical variable as well as upon T; The surface concentration itself is always taken as the appropriate chemical variable which controls the extent of particle-particle interactions.) The resulting Tafel slopes then reflect primarily adsorbate-electrode interactions. Thus,

i)

= (8 In I'r) _ (8 In I';t) _ ~[8(~G~P)E _ 8(~G~JE 8 In ( 8E r; 8E t, 8E T, RT et: oE

+

nFl

(16)

84

M. J. WEAVER, F. C. ANSON

If the two derivatives of the activity coefficients in eqn. (16) are not zero (i.e., the adsorption isotherms are not congruent with respect to potentiaI 1 2 . 1 3 ) the implication is that the use of the electrode potential as the electrical variable does not achieve complete separability of electrode-adsorbate, and adsorbate-adsorbate interactions. However, the use of electrode potential in eqn. (16) is obligatory since there is no choice about the way to express the free energy of the reacting electron. If, following Krishtalik 14, we make the assumption that the logarithm of the activity coefficient of the activated complex, "I"" can be related to those of the adsorbed reactant and product, ff and YP' in the same way that the standard free energies of the three species are related, that is, via fractional coefficient, o', it follows that

(17)

(1n ff-In "I",) = iX'(1n "If-In Yp ) Equation (16) can then be rewritten:

-iX'

[(0 l~ fP) uE

_

(a cE l~ "If) J

(18)

t,

1.

Inasmuch as

(19) eqn. (18) can also be written in terms of the experimentally accessible total standard free energies:

!n i) (aoE r, _ iX'

[(a In yp) _ (0 ~n "If) J oE

r,

uE

(20)

1.

To proceed further it is necessary to have a relation between iX and c', If we assume that these two coefficients can be taken as equal, eqn. (20) can be rewritten in the following form: In i) \(0 aE. r, = -

iX [0

a

RT oE (!1G~p + RT In Yp)r, - aE (!1G~, +RT In Yf)r, +nF] (21)

The first two terms inside the brackets on the r.h.s. of eqn. (21) can be evaluated from the adsorption isotherms for the reactant and product. For adsorption equilibrium of the reactant:

fi? +RT In Q~ = fi2, +RT In zz, = fit +RT In(Tf"If) where Q~ is the activity of the reactant in the bulk of the solution and the other symbols were defined earlier (cf. eqns. 3-6).

85

KINETICS WITH SPECIFICALLY ADSORBED REACTANTS

The standard electrochemical free energy of adsorption, i1G~" is given by

sa:A, --

(22)

-0 _ -0 /lA,. u;

so, RT In a~ = RT In Yr+RT In rr+i1G~, and

o

b

3 In a r ) (- DE

Iny )

=D - (i1G A,

--+

3E

I,

(23)

RT

(24) I,

r

Similarly, for the reaction product: 0 In a~) ( -oE- I,

(i1G~P + ln j

=.0- - - -

st:

RT

) P

(25)

t,

Thus, eqn. (21) becomes

(

3 In

oE

i)

(.

I,

2.3

)

== 'Tafel slope" =

I,

_l(O l~ a~) _(0 l~ a~) L

cE

I,

cE

+ t,

nFlJ

RT

(26)

Note that eqn. (26) should be valid no matter which theoretical isotherm (if any) the adsorption data appear to obey*. The two differential coefficients on the r.h.s. of eqn. (26) can be evaluated in various direct ways. For example, from sets of chronocoulometric data giving surface concentrations as a function of potential for a series of solutions containing different bulk activities of the reactant, plots can be prepared in which the bulk activity necessary at each potential to maintain the surface concentration constant is plotted vs. the potential". The slopes of the resulting curves give the needed differential coefficients. Knowing the values of these differential coefficients allows the slopes of (constant coverage) Tafel plots to be calculated without fitting the data to any theoretical isotherm. This is an important advantage that results when the current-potential measurements are made at constant coverage rather than at constant bulk activity of the adsorbate as in previous analyses v". If eqn. (20) is not simplified by assuming that !Y.=!Y.', it is still possible to proceed with the analysis for cases in which the adsorption data exhibit congruence 12.13 with respect to potential, i.e., plots of surface concentration versus the chemical potential of the adsorbate in solution are parallel to one another at each (constant) potential where the adsorption is measured. For such congruent isotherms (

0 l~n Yr) = cE I,

(3 l~cEYl')

=0

(27)

I,

* It is relevant here to express disagreement with the arguments given by Delahay!" that with very strongly adsorbed (i.e., "chemisorbed") systems it is often unnecessary to include a potentiaI- or chargedependent parameter in writing the adsorption isotherms. The derivatives with respect to potential of the free energy quantities appearing in eqn. (20) which give rise to the differential coefficients in eqn. (26) can continue to be significant even when the overall free energies of adsorption become very large and negative.

86

M . J . WEAVER, F. C. A N S O N

In this case, eqn. (20) reduces to eqn. (26) directly. Even when the adsorption data do not exhibit congruency at all coverages and ~ is allowed to differ from ~', Tafel slopes evaluated at coverages low enough for the adsorption to obey Henry's law will obey eqn. (26)• This is true because 7~= 7p = l in the Henry's law region of an isotherm where all adsorbates must exhibit congruency. For the analysis of the kinetics of adsorbed reactants at moderate and high coverages under conditions where non-congruent adsorption isotherms are obeyed, the form of eqn. (26) may still be used if the transfer coefficient appearing in it is redefined in a less conventional way: in place of the definitions of ~ and ~' embodied in eqns. (7) and (17), a single, composite coefficient, ~", may be defined as follows:

[(G~+RT In ?~)~-(G°+RT In )'~)E]rr

=~"[(G°OE-(G°)e+RT(G°+RTln7r)E]rr (28) where the (ln 7)e terms denote the potential-dependent parts of the corresponding activity coefficients evaluated at a constant value of F~. Equation (28) presents a relationship among particular, non-standard-statefree energies of the reactant, activated complex and product at each constant surface concentration, F~, of the reactant. If these non-standard-state free energies are defined by*: Gut = Gh0 + R T In 7~; G~t = G°+RTln ~ , • and Gi11 ' = Gnl o -k R T In ~p, eqn. (28) can be rewritten in the form [(G~)~ - (G~0E] = ~" [(G~,,)e - (Gi,)e]

(29)

Equation (29) is the direct analog of eqn. (7). The (G°)E and (RT In 7)~ terms which appear together in eqn. (28) were previously treated separately in eqns. (7) and (17) where ~ and ~' were defined. These two terms represent conceptually distinguishable contributions to the three non-standard-state free energies defined above: The (G°)~ terms represent the contribution to (G')~ arising from interactions between adsorbed solvent and each of the adsorbates (reactant, activated complex and product) because the standard state chosen for each corresponds to an extremely dilute surface concentration. The (RT In 7)E terms represent the contributions to (G')E from interactions between each of the three adsorbed species and the adsorbed reactant present at concentration F~. (For congruent isotherms these contributions will, of course (eqn. 27), be zero.) No attempt is made in eqns. (28) and (29) to separate these factors which contribute to the overall symmetry factor, ~t. However, an advantage of eqn. (28) is that it will always lead to an equ~/tion identical to eqn. (26) but with ~" (defined by eqn. 28) replacing ~. Thus, eqn. (26) provides a general basis for analyzing the kinetics of adsorbed reactants measured at a constant surface concentration of the reactant. Since the electrode reaction whose rate is to be measured consumes the adsorbed reactant as it occurs, the constraint of constant surface concentration implies that only initial rates will be measured. This requirement poses no problem in most circumstances. However, the faradaic consumption of the reactant does introduce a difficulty in evaluating the coefficient (~In a~/t3E)rrwhich appears in * These non-standard-state free energies (G O+ RT In 7) are akin to the "apparent standard free energies of adsorption" frequently employed by Conway x6. /

87

KINETICS WITH SPECIFICALLY ADSORBED REACTANTS

eqn. (26). This coefficient needs to be measured at potentials where the electrode reaction proceeds at an appreciable rate• It cannot, therefore, be measured by equilibrating the electrode with a solution of the reactant at each potential of interest• One route around this difficulty which is followed in the succeeding paper, takes advantage of the following thermodynamic relationship 17 ~F~/~+_

r ~_

where qm is the electronic charge density on the electrode, Pr is the electrochemical

potential of the adsorbate in solution, and the other symbols have their customary significance• For dilute solutions of adsorbates in media of high ionic strength (or in solutions of constant ionic strength) lz eqn. (30) can be replaced by eqn. (31) 3qm'] =

-RT(Dlnab~

(31)

Thus, a measurement of the change in the electronic charge on the electrode produced by the adsorption of the reactant provides another route to the differential coefficient required in eqn. (26)• At potentials where the electrode reaction proceeds, the coefficient (Oqn~/~F) may often be evaluated by rapid pulse methods, (e g, coulostatic charge lnjectxo~ in which the charge-potential characteristxcs at a series of surface concentrations are constructed by abruptly altering the charge on the electrode and measuring the resulting potential so rapidly that neither the faradaic reaction nor the mass transfer-limited adsorption~lesorption equilibrium has time to change the surface concentration ofadsorbate significantly• The value of the coefficient (Oqm/dF~)E at the required potentials can then be determined by combining these measurements with values of the same coefficient obtained by differential capacitance or chronocoulometric4 methods at initial potentials where no faradaic reaction occurs. For many of the chromium(Ill) isothiocyanate complexes examined in the succeeding paper 4 the adsorption~tesorption rates were well below the diffusion-limited values t s so that pulse perturbation measurements of charge-potential relations were facilitated. However, even when the adsorption equilibrium is diffusion-controlled, "snap-shots" of the charge-potential characteristic at constant coverage by the adsorbate should be obtainable provided suitably rapid perturbation techniques can be employed. The differential coefficient in eqn. (26) pertaining to the adsorption of the product of the reaction can be easily evaluated for stable reaction products by equilibrating solutions of them with the electrode at each potential of interest in the Tafel region because no faradaic reaction which consumes them will occur. However, the coefficient required in eqn. (26) is associated with the adsorption of product on an electrode containing a constant surface concentration of adsorbed reactant, i.e., (~ In a b / O E ) r r . This precise coefficient would be difficult if not impossible to evaluate by equilibrium measurements. Nevertheless, for reactant and product species which both exhibit adsorption congruent with respect to potential, it is to be expected that •

.

r I

(Olnabp~ = 1 ~E Jrr \ ¢3E Jrp--RT

(~3 l_n ab~ =

•

' d~AGA~ ~

dE

•

•

(32)

88

M.J. WEAVER, F. C. ANSON

and the necessary coefficient may be evaluated in reactant-free solutions of stable products. For cases in which the reaction product is chemically unstable or is adsorbed in quantities too small to be measured this differential coefficient must be estimated in some other way. Equation (26) contains three terms that contribute to the observed Tafel slope. Parsons 11 has suggested that it may be useful to distinguish between two classes of effects that control the apparent transfer coefficients (i.e., the observed slopes of Tafel plots) which he terms "intrinsic" and "environmental" effects. The former is the "direct effect" of the potential upon the charge-transfer rate, and the latter a secondary effect arising from the potential dependence of the interfacial structure surrounding the reactant. This separation was recognized by Parsons 11 to be ultimately arbitrary but it does serve the useful function of identifying the disparate origins of the factors which control Tafel slopes. Two kinds of environmental effects can normally be distinguished: (i) the concentration of reactant in the preelectrode state may be different from its value in the bulk of the solution; (ii) the potential dependences of the standard free energies of the reactant and product states are usually neither zero nor equal. For unadsorbed reactants whose preelectrode state is the outer Helmholtz plane these effects can be described in purely electrostatic terms and the well-known Frumkin correction results 2' 11. In the case of specifically adsorbed reactants, environmental effect (i) can be eliminated from the analysis by working at a constant concentration of adsorbed reactant rather than at a constant bulk concentration. Effect (ii) is precisely accounted for by the two differential coefficients on the r.h.s, of eqn. (26). Therefore the transfer coefficient ~ in eqn. (26) may be termed an "intrinsic transfer coefficient", ~, which, by virtue of its definition via eqn. (7), expresses the symmetry properties of the elementary barrier to electron transfer (e.g., for a symmetrical barrier, ~ = 0.5). The effects of interparticle interactions on the rate-potential dependence are embodied in the ~' term in eqn. (17) and should not affect the rate law at constant coverage, at least for systems which adhere to isotherms that are congruent with respect to the electrode potential. In the companion paper 4 the use of eqn. (26) is shown to rationalize a rather divergent set of Tafel slopes for the reduction of otherwise similar, adsorbed chromium(III) isothiocyanate complexes. In a recent study of the kinetics of reduction offac-Cr(OH2)3(NCS)3 in the adsorbed state 19 a rather low value of the apparent transfer coeffÉcient (i.e., a high Tafel slope) was observed. This was rationalized in terms of the loss in driving potential that the reactant experiences when it reacts at a site on the electrode side of the outer Helmholtz plane. In the light of eqn. (26) it is evident that this rationalization amounts to an assumption that only coulombic interactions are involved in determining the effect of electrode potential on the free energies of adsorption of the reactant and product. Thus, if ~b1 is the potential at the site where the adsorbed reactant and product reside, and their charges are z and z', respectively, eqn. (14) becomes (AG °)E = ~ [~blF (z' -

z) + nF~b~,]

(33)

Or, since n = ( z - z') (AG ° )e =

~nF(4., - 491)

which, with eqn. (1), leads to

(34)

KINETICS WITH SPECIFICALLY ADSORBED REACTANTS

- (J. = RT

din i ---c--------,-nF d(¢m-¢d

89 (35)

~

Equation (35) is the same as the one used in ref. 19 to account for the low values of

(J. obtained. Note that eqn. (33) can be substituted for eqn. (14) only when the sole

source of the potential dependence of the free energies of adsorption of the reactant and product are given by zF ¢ 1 and z' F¢ 1, respectively. In general, the effect of potential on L1G2, and L1G1 p will include additional factors so that eqn. (14), and eqn. (26) to which it leads, must be used to evaluate intrinsic transfer coefficients from measured Tafel slopes with adsorbed reactants. RATE-COVERAGE DEPENDENCES

To determine the effect of varying surface concentrations (i.e., varying interparticle interactions) on the reaction rates of specifically adsorbed reactants, it is necessary to evaluate both a r and Y", in eqn. (1) as a function of T; In order to hold the free energy of the reacting electron constant, this evaluation must be made at constant electrode potential. From eqn. (15) it follows that . 3lni) ( 3 In T E = 1 r

+

Y"')

[(3In yr) (3 In 3 In Tr E 3 In Tr

E

l

(36)

And, if eqn. (17) is used to eliminate Y", :

-1+(J.'[(~) - (~) J* (~) 3 In F, 3 In F, 3 In T, E -

E

(37)

E

Thus, the coverage-dependence of the activity coefficients for the reactant and its initial reduction product will be reflected in the current--coverage response at constant potential. The coefficient (31n Yr/3ln Trh must be obtained from the adsorption isotherm for the reactant evaluated at each potential where the rate--coverage dependence is sought. The required isotherms can be constructed, as described above, by combining the results of rapid charge-injection measurements with eqn. (31) and an isotherm determined at any convenient potential outside the faradaically active range. The evaluation of the coefficient (3 In Yp/3 In Tr)E presents greater difficulties since it represents the dependence of the activity coefficient of the adsorbed reaction product on changes in the quantity of adsorbed reactant: in some instances it may prove just as satisfactory (i.e., no more unsatisfactory) to use eqn. (36) and to estimate (3 In 'Y '"/ 3 In T)E on the basis of coulombic models for the interparticle interactions 2 o . 2 1 • But, regardless of the method used to estimate one or the other of the required differential coefficients,it must be recognized that the inevitable uncertainties in their values limit the precision with which molecularities may be assigned to complex reactions on the basis of experimentally observed reaction orders. * Allhough the transfer coefficient ()( in eqn. (37) appears in ref. 2 in analogous algebraic expressions involving a "coverage parameter", the two coefficients may not be identified with each other as they arise from lines of reasoning that are conceptually distinct.

90

M.J. WEAVER, F. C. ANSON

Procedures for the evaluation of 7r and ~, from adsorption isotherms have been discussed 22. When the Frumkin isotherm is used, one obtains 19"22 In y r - l n 7, -- ( A r - A , ) O

(38)

where 0 = Fr/Fm is the coverage, Fm is the concentration of adsorbed reactant on a saturated surface, and A~ and A ~ are the Frumkin interaction parameters for the reactant and transition state. The rate-coverage relation at constant potential can then be obtained from eqn. (15) 2.3 (log i - log 0) = (A r - A , ) 0 + const.

(39)

This relationship is used in the succeeding paper 4. SURFACE ACTIVITY AND REACTION ORDERS

Several recent publications have dealt with the problems of inferring the molecularities of reactions from reaction orders observed with adsorbed reactants 22-24. In general, the surface activity of an adsorbed reactant (as conventionally defined 22'24) depends on both its surface concentration and the electrode potential (or charge) z3'24. The latter dependence was utilized by Pospisil and de Levie za to vary the surface activity of adsorbed anionic reactants in solutions of fixed bulk activity of the anions in order to determine the order of the reactions studied with respect to adsorbed anions. (The need to include the activity coefficient of the activated complex in such analyses has already been noted22.) This is a convenient method for changing the surface activity of adsorbates but it has a serious drawback that has not been previously emphasized: Changes in potential affect not only the pre-exponential term in eqn. (1) but also the exponential term. This will be true even for reactions, such as those studied by Pospisil and de Levie 23, in which a chemical rather than the electron-transfer step is rate determining. In such cases the two differential coefficients on the r.h.s, of eqn. (26) will produce changes in the measured rates as the potential changes even if the surface activity of the adsorbed reactant, at, is kept constant (by appropriate adjustment of the bulk activity and, therefore, the coverage of the reactant). Thus, unless the values of and the differential coefficients in eqn. (26) are known, it is not possible to obtain moleculariti'es from the dependence of reaction rates on the surface activities of adsorbed reactants unless these surface activities are varied while keeping the electrode potential constant. PARTIAL CHARGE TRANSFER

The "partial charge-transfer coefficients" introduced by Lorenz and Salie z5 were conceived as measures of the extent of the partial oxidation (or reduction) of an adsorbate produced by its adsorption. Parsons 26 and Damaskin 27 have shown that the thermodynamic methods that have been used to evaluate partial charge-transfer coefficients25 cannot be used to argue that a change in the oxidation state of an adsorbate accompanies its adsorption although neither do they demonstrate the contrary. It has been suggested 28 that more definitive evidence of the partial reduction (or oxidation) ofadsorbates upon their adsorption might result from investigations of

KINETICS WITH SPECIFICALLY ADSORBED REACTANTS

91

their kinetic behavior. For this reason it seems worthwhile to examine how the analysis given here would be affected if partial charge transfer is considered. Partial charge-transfer coefficients (as well as the "electrosorption valencies" discussed by Schultze and Vetter 29) refer to the value of the coefficient (1/F)(0qm/~F)~, which was shown in eqns. (30) and (31) to be equal to - ( R T / F ) ( d In ab/dE)r . The latter coefficient appears in eqn. (26) where it serves to account for the effects on the observed kinetics of the potential dependence of the free energy states for the reactant and product (or for the reactant and its transition state). Thus, if partial charge transfer is explicitly considered in writing eqns. (4) and (5) and the identities given in eqns. (24) and (25) are used to express the partial charge-transfer coefficients, the resulting modifications in the electron-free energy terms will lead to a kinetic relationship identical to eqn. (26). Hence, the presence of partial reduction (or oxidation) of adsorbed reactants will not be apparent from this phenomenological treatment of their electrode kinetics. Plieth and Vetter 28 have argued that when both the oxidized and reduced forms of a reactant are adsorbed the exchange of electrons between the two oxidation states will occur too rapidly for them to retain their identities. Instead, a mean redox state involving a tautomeric equilibrium is thought to result. This proposal seems reasonable for adsorbed redox couples comprised of structurally very similar oxidation states, but readily distinguishable oxidation states can result in cases where irreversible cleavage of chemical bonds accompanies the redox process, as in the Cr(III)/Cr(II) examples described in the succeeding paper 4. DIFFUSE LAYER EFFECTS

Inasmuch as the concentrations and activities of adsorbed reactants can be measured directly without the need to employ a calculational model (such as is used in the usual Gouy-Chapman-Stern theory of the double layer), corrections of observed reaction rates for the presence of the diffuse layer should be simpler for adsorbed than for non-adsorbed reactants. The situation is similar to the case of uncharged, non-adsorbed reactants whose concentrations at the O.H.P. are unaffected by the presence of the diffuse layer 6. To take account of the diffuse layer, eqn. (15) can be written in the modified form shown in eqn. (40). In i = In K + l n 7r+ln F r - l n 7, R T { [(AGAp)E -- (AGAr)E -- ]~Fq~2] "~ tlFq~m}

(40)

The two primed terms inside brackets in eqn. (40) represent the same quantities as in eqn. (15) with the exception that the contributions to the two free energies of adsorption resulting from the electrostatic work of crossing the diffuse layer by reactant and product (whose charges will differ by n) have been extracted and are expressed by the nF(o 2 term. This division of the overall free energies, of adsorption should be valid providing that the adsorbed reactant and product lie on the electrode side of the O.H.P. as is usually assumed. It follows that shifts in measured potentials accompanying changes in ionic strength at constant F and i can be directly related to changes in q~2:

92

M.J. WEAVER, F. C. ANSON

([~E)F,i =

A(~m = A(~2

(41)

A test of this simple relation is described in the experimental paper which follows. ACKNOWLEDGMENTS

This work was supported by the National Science Foundation and the U.S. Army Research Office (Durham). Helpful comments from D. M. Mohilner are a pleasure to acknowledge. SUMMARY

A procedure for analyzing the kinetics of electrode processes involving adsorbed reactants is described and the necessary equations derived. Methods for determining or estimating the various contributions to the potential dependences of the standard free energy of activation, the activity coefficient of the activated complex, and the surface activity of the adsorbed reactant are discussed. The procedures for analysis of kinetic data suggested here are contrasted with previous treatments and the differences and relative advantages of the present treatment are pointed out. REFERENCES 1 2 3 4 5

D. M. Mohilner and P. Delahay, J. Phys. Chem., 67 (1963) 588. D. M. Mohilner, J. Phys. Chem., 73 (1969) 2652. A. B. Ershler, D. A. Tedoradze and S. G. Mairanovskii, Dokl. Akad. Nauk SS_S_R, 145 (1962) 1324. M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 58 (1975) 95. S. Glasstone, K. J. Laidler and H. Eyring, The Theory o f Rate Processes, McGraw-Hill, New York, 1941. 6 P. Delahay, Double Layer and Electrode Kinetics, Interscience, 1965, ch. 7. 7 R. Parsons, Proc. 3rd Europ. Symp. on Corrosion Inhibitors, Universita degli Studi di Ferrara, 1971, p. 3. 8 For example, see R. Parsons, Proc. Roy. Soc. (London), A261 (1961) 79. 9 R. de Levie, J. Electrochem. Soc., 118 (1971) 185c; J. E. B. Randles, J. Electroanal. Chem., 41 (1973) 272. 10 R. Parsons, Trans. Faraday Soc., 47 (1951) 1332; also, see ref. 6, p. 154 et seq. 11 R. Parsons, Croat. Chem. Acta, 42 (1970) 281. 12 E. Dutkiewicz and R. Parsons, J. Electroanal. Chem., 11 (1966) 100. 13 For a review, see R. Payne, J. Electroanal. Chem., 41 (1973) 277. 14 L. I. Krishtalik, Zh. Fiz. Khim., 41 (1967) 2883. 15 Ref. 6, p. 240. 16 B. E. Conway, Electrode Processes, Ronald Press, New York, 1965, p. 82. 17 R. Parsons in J. H. Schulman (Ed.), Proc. 2nd Int. Congress on Surface Activity, Vol. 3, Butterworths, London, 1957, p. 38 18 S. N. Frank, M. J. Weaver and F. C. Anson, J. Electroanal. Chem., 54 (1974) 387. 19 F. C. Anson and R. S. Rodgers, J. Electroanal. Chem., 47 (1973) 287. 20 R. Parsons, J. Electroanal. Chem., 21 (1969) 35. 21 W. R. Fawcett and S. Levine, J. Electroanal. Chem., 43 (1973) 175. 22 F. C. Anson, J. Electroanal. Chem., 47 (1973) 279. 23 L. Pospisil and R. de Levie, J. Electroanal. Chem., 25 (1970) 245. 24 R. de Levie, J. Electrochem. Soc., 118 (1971) 185c. 25 W. Lorenz and G. Salie, Z. Phys. Chem. N.F., 29 (1961) 390, 408; Z. Ph~vs. Chem., 218 (1962) 259. Additional relevant references are given in 26 and 27.

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26 R. Parsons in P. Delahay (Ed.), Adv. in Electrochem. and Electrochem. Engineering, Vol. VII, 1970, p. 177. 27 B. B. Damaskin, Soy. Electrochem., 5 (1969) 717. 28 W. J. Plieth and K. J. Vetter, Collect. Czech. Chem. Commun., 36 (1971) 816. 29 J. W. Schultze and K. J. Vetter, J. Electroanal. Chem., 44 (1973) 63.