II. ORIGIN OF POTENTIAL DEPENDENCE OF ELECTROCHEMICAL REACTION RATES
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108
Brian E. Conway
significance of the a parameter in b [Eq. (5)]; in fact, a is added
somewhat as an "afterthought" (as a parameter y = I/a) in this
otherwise important and seminal paper.
Almost simultaneously, Erdey-Gruz and Volmer17 and Butler18
wrote a generalized kinetic equation for a one-electron transfer
process corresponding to the form of Eq. (2), namely
i = io{exp(aVF/RT) - exp[-(l - a)r,F/RT]}
(6)
which took into account: (1) both the forward and backward
reaction directions of the process corresponding to net i; and (2)
the exponential potential dependence of the forward and backward
current components through the transfer coefficients a and 1 - a,
respectively. [The representation in Eq. (6), with the a and 1 - a
factors, only applies to the one-electron process considered here
and in the original papers. More complex relations must be written
for multielectron, multistep processes; see Ref. 13. For the oneelectron case in Eq. (6), a = /3, the symmetry factor (see Section
H.2).]
From Eq. (6), it is obvious that for appreciable polarizations
where i » i0, the Tafel equation in the form of Eq. (4) is obtained
so that b is defined as RT/aF, as in Eq. (5). This is the essential
origin of the conventional representation of b in terms of a transfer
coefficient a.t
While the derivative drj/d In i is used commonly to characterize the dependence of current density on potential and is referred
to as the Tafel slope, b = RT/aF, we suggest that there is some
advantage to using its reciprocal, d In i/ dr] = aF j RT, as this corresponds directly to the exponential term in the electrochemical free
energy of activation [Eq. (9)]. Then reciprocal Tafel slopes^ can
conveniently be referred directly to factors that affect the activation
process in charge transfer reactions [Eqs. (4) and (9)].
A more explicit physical significance of the a factor and the
potential-dependence functions at]F/RT or (1 - a)r]F/RT in Eq.
(6) is given when the energy course of the reaction is represented
t A useful general review on the topic of transfer coefficients has been given by
Bauer19 but does not cover the important question of temperature dependence of
b or a discussed in the present paper.
t If a name is required for this reciprocal quantity, it might be suggested that it be
called the "Lefat" slope.
Electrochemical Reaction Rates and the Tafel Equation
a) GURNEY
H+/H2O
109
b) BUTLER
H + /H 2 O
H/H-,0
Figure 1. Potential energy profile diagrams for a charge transfer
process as in H+ ion discharge with coupled atom transfer (based
on representations by Gurney15 and Butler21. In (b), curve / represents the H + /H 2 O proton interaction potential and m that for
discharged H with the metal M. R is the repulsive interaction of H
with H2O and A the resultant interaction curve for H with M.
in terms of a potential (or free) energy profile or surface diagram
(Fig. 1). Such an approach was given qualitatively in the papers of
Gurney15 and Horiuti and Polanyi,20 and more quantitatively in
Butler's 1936 paper21 where the importance of metal-to-H adsorption energy as well as solvational interactions in the kinetics of the
h.e.r. was also deduced. This paper, it may be noted, provided the
foundation for future work on the dependence of "electrochemical
catalysis" on electrode properties, e.g., the role of H adsorption
energy and work function. Historically, it should be noted that the
role of solvational activation in the kinetics of electrode reactions
was already recognized in papers by Gurney,15 Gurney and
Fowler,16 and Butler21 (Butler also introduced quantitatively the
effect of chemisorption energy, as mentioned above, e.g., of H in
the h.e.r.), and was given more detailed treatment in various papers
of Bockris with Parsons and with Conway in the 1950s. The solvational activation factor is what became known later as the "reorganization energy" in Marcus's representations14 of the activation
process.
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Brian E. Conway
A formally more explicit representation of the effect of electrode potential and associated role of temperature on
electrochemical reaction rates follows from the transition-state
theory applied by Glasstone et al.22 soon after the appearance of
Butler's paper.21 The rate constant A of a chemical process is written
as
kT
A = K— exp(-&Got/RT)
(7)
h
where k is Boltzmann's constant and K a transmission coefficient
for passage of activated complexes, $, over the energy barrier; K
is normally taken as near 1.
For an electrochemical rate process, the rate constant A is
determined by an electrochemical free energy of activation, AG°*,
related to AG°*, the "chemical" free energy of activation, by
kT
A= K— exp(-AG°7tfr)
n
(8)
where
AG°* = AG°*-0VF
(9)
and j3 is a barrier symmetry factor, analogous to Br0nsted's a factor
in linear free-energy relationships and V is the metal/solution
potential difference across part of which electron transfer normally
takes place. The significance of /3 is illustrated in the well-known
way by reference to the working diagram, Fig. 2, which shows how
an electrical energy VF applied to the initial state of an electron
transfer reaction, e.g., H3O+ + e + M -> MHads, changes the relative
energy of the transition state by (1 - fi)VF and hence the activation
energy for the forward direction of the process by ($VF as in Eq.
(6).t For a symmetrical barrier, based on two approximately anharmonic potential energy curves (cf. Refs. 10 and 21), as shown
schematically in Fig. 2, it is obvious that p — 0.5 over an appreciable
range of change of V. For the H2 evolution reaction at Hg, the
range of constancy of (3 is surprisingly large (cf. the work of
Nurnberg). For a barrier based on harmonic curves (see p. 150), /3
is not constant with changing V.
t Equation (6) is written in a general way with a transfer coefficient a. Here a = (S
(see later discussion in Section II.2).
Electrochemical Reaction Rates and the Tafel Equation
111
ADSORBED
STATE
INITIAL STATE
WITH e IN METAL
Figure 2. Working diagram showing how the "linear
free-energy relationship," common in electrode process kinetics, arises from changes in electrode potential. j8 is a symmetry factor. An extreme case of an
anharmonic oscillator energy profile is shown in schematic form (cf. Ref. 25). This representation assumes
changes in V affect only the energy of electrons in the
initial state at the Fermi level.
The principles involved in expressing /3 in this way are
evidently quite analogous to those that define Br0nsted's a factor23
(cf. Ref. 2). However, one matter requires special mention: in the
case of electrode reactions, the free-energy change of the initial
state of a cathodic electrochemical reaction or of the final state of
an anodic one associated with a change of potential has, in previous
work, been almost entirely attributed to the change the Fermi level
of electrons:
eV
(10)
Thus the form of the energy hypersurface itself is not normally
considered to vary with the potential or corresponding field across
the double layer. Hence the effect of potential in electrode kinetics
has usually been represented simply by "vertical movement" on
the energy axis of potential energy curves without much or any
change of their shape.f However, we shall see that the conclusions
t In a recent paper by Kuznetsov, discussed later on p. 153, some change of shape
is attributed to the interaction of medium fluctuations on the partially charged,
transferred H atom in the case of the h.e.r.
112
Brian E. Conwav
of the present paper and of recent new experimental work in this
Laboratory (see p. 179) indicate that the entropy of activation of
some electrode processes is materially dependent on electrode
potential in addition to the usual effect of potential on the energy
of activation through changes in the electron Fermi level energy
(Eq. 10). Thus electrode potential evidently can have a substantial
effect on the structure and/or solvational environment of the reacting molecular species in the interphase as reflected in this entropy
function, in addition to changing the energy of electrons.
In Br0nsted linear free-energy relations for homogeneous
acid/base proton transfers, modification of p ^ ' s of a series of acids
reacting with a given base normally changes not only the depth of
the proton's energy but also the shape of the potential energy curve
for its transfer since force constants for BH+ bonds are approximately proportional to bond strength, hence pJTs, reflecting
Badger's rule. Because the main effect of electric energy change is
on O v (Eq. 10), the "linear free-energy relations" for electrode
processes (Tafel behavior) are often linear over wider ranges of
energy change (but below, of course, those corresponding to
approach to the diffusion-controlled limit24) than are Br0nsted
relations for acid/base processes over comparable ranges of pK;
see e.g., Refs. 23 and 24.
Introduction of Eq. (9) into~A[(Eq. (8)] gives
t=K ^exp[-(AG°* - f3VF)/RT]
(11)
n
Equation (11) is the transition-state equation for electrochemical
rates (i is, of course, proportional to~kand concentration of reactants
in the double layer at the electrode interface in the usual way25)
and is obviously equivalent to the Tafel equation in exponential
form [Eq. (4)]. From Eq. (11) it is seen that the Tafel slope for a
simple electron transfer process is RT//3F, i.e., b is linear in temperature. We shall return later to a more critical examination of
Eq. (11) insofar as energy and entropy components of the free
energy of activation are concerned.
It was mentioned that manipulation of energy profile diagrams
(Fig. 2) gives a simple significance to /3. However, this is a grossly
oversimplified representation since the course of activation in a
charge transfer process involves intermolecular and orientational
Electrochemical Reaction Rates and the Tafel Equation
113
fluctuations in the solvation shell of the reacting ion or molecule
at the electrode, as well as some physical transfer of an atom down
a single reaction coordinate in the case of atom transfer reactions,
as with anodic Cl2 evolution, metal deposition, or the h.e.r. (in the
latter case, the atom transfer component may be nonclassical26'28).
Other representations19'25 of the significance of/3 must accordingly be included here.
In the case of multicoordinate redox processes, Hush29 has
suggested that /3 can be represented as the fractional charge transferred when the transition-state configuration is attained: for symmetrical redox reactions such as Fe(CN)6~/Fe(CN)e~ or
Fe(aq)3+/Fe(aq)2+, p is then obviously 0.5 except for any asymmetry that may be introduced by a different degree of specific
adsorption of the ox (oxidized) form from the red (reduced) form
of the conjugate redox pair.30 This representation, now no longer
believed to be correct (cf. Marcus914), also gives an indication of
how (3 is related to the way in which the transition-state configuration may be more similar to the initial state than to the final state,
or vice versa, in the case of unsymmetrical redox pairs. The potential
energy diagram representation also gives similar indications, as
may be seen for limiting cases of highly exoenergetic or highly
endoenergetic electron transfers.
Alternatively, in a related qualitative sense, /3 can be interpreted as measuring25'35 fractionally "how far along" the reaction
coordinate is the transition state attained and hence over what
fraction of the potential difference, A V, across the Helmholtz layer
does the electron charge have to be transferred to this transition
state. This interpretation is related to the "half-jump distance"
characterizing activated charge transfer across, e.g., oxide films.31
For symmetrical processes, each of these ways of representing
the significance of fi gives the latter quantity a value of 0.5. However,
the unsymmetrical processes, the different representations do not
lead to identical values of /3 for a given reaction; note that atom
transfer processes are normally highly unsymmetrical, so how /3 is
considered from the theoretical point of view is especially important
for such types of electrode reaction.
The significance of ]8 as a measure of the fraction of the
interface p.d. involved in the formation of activated complexes is
formally explained as follows. In terms of a supposed quasiequili-
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Brian E. Conway
brium between initial-state ions i and transition-state complexes $
in a discharge reaction,
where tyx is the potential at the outer Helmholtz plane and <£* is
the local potential in the compact double layer at which activated
complexes begin to pass over to product configurations. AGot is
defined as
/4-M,
and nf-tf
+
RTln^/c^^-zFi^-^)
Then * -
i, where
<£M is the electrode metal Galvani potential. The same result, but
not having the same physical significance, is obtained if a fraction
(3 of a unit e charge is envisaged as being transfered to $ as it is
formed from the initial state.29 This representation is not, however,
consistent with that shown above in terms of local potential <£*;
nor can both types of representation be introduced simultaneously
since then evidently (3 would be 0.25!
It is clear from the foregoing proposition that p could be |,
that a representation of the approach to the transition state in
terms of traversal of a charged reactant particle across an electrical
potential profile to * (approximately half the p.d. A across the
Helmholtz layer) cannot be combined with an adiabatic readjustment of electron distribution giving (cf. Hush29) a charge of approximately e/2 on the transition state for a one-electron transfer process.
Thus, the representation of the process of attainment of the transition state in which activation of the reacting particle takes place
through solvation-shell reorganization and/or ligand coordinationshell distortions in the case of ionic redox reactions or, for atom
transfer processes, through a coupled bond stretching and solvationshell reorganization, implies that the transfer of the electron takes
place by a tunneling process, as proposed by Gurney,15 to (or from)
the reactant particle when the latter has attained, multidimensionally, the required transition-state configuration so that a radiationless electron transition arises in the usually supposed manner.
The discussion given above serves to indicate that the question
of "how" the electron transfer event actually takes place, if such
a microscopic and normally nonclassical process can be considered
in those terms, is not yet fully answered.
Electrochemical Reaction Rates and the Tafel Equation
115
2. Relation of p to a
In Section II. 1 the transfer coefficient a was introduced in the usual
general way in relation to the definition of 6, the Tafel slope. In
electrode reaction mechanisms involving two or more consecutive
steps, the dependence of a time-invariant current density on potential must be evaluated by one of the following:
1. The steady-state method in which differential equations
are set up defining the time dependence, d[x]/dt, of concentrations
of intermediates x involved in the process. In the steady state,
d[x~\/dt = 0 giving a rate equation containing potential-dependent
terms.
2. Using the quasiequilibrium hypothesis, for steps prior to
the rate-determining one, to enable an explicit relation between
(surface) concentration of x species and potential to be substituted
in a rate equation for the velocity of the rate-determining step which
itself may or may not be directly dependent on potential.
Normally,25'32 method 1 does not give a rate equation that
leads to a simple explicit expression for potential dependence of
the current density and hence no simple-valued Tafel slope arises.
Numerical evaluations of the dependence of In i on rj can, however,
usually be made, e.g., for a range of assumed or evaluated rate
constants. Bockris32 has given the most sophisticated application
of method 1, using Christiansen's approach;33 the multistep O2
evolution reaction was taken as an example. It must be stated that
experimentally meaningful results from this method only arise after
limiting assumptions are made about relative values of the rate
constants for the component steps (in forward and backward directions) and associated surface coverages by adsorbed intermediates.
Usually, the assumptions required to derive the electrode-kinetic
behavior for the limiting cases are equivalent to those that are
adopted in the quasiequilibrium method 2; then, the Tafel slopes
for most kinetically significant cases can be evaluated "by inspection" after a little experience with writing the equations.
For a reaction scheme involving n charge transfer events prior
to the rate-determining step (for most electrode reactions n > 3),
the Tafel slopes work out to be
b = RT/(n + P)F
(i.e., a = n + 0)
(12)
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Brian E. Conway
or
b = RT/nF
(i.e.,a = n)
(13)
depending on whether [Eq. (12)] or not [Eq. (13)] charge transfer
is involved in the final rate-controlling step. "Chemically" controlled rate-determining steps thus have Tafel slopes determined by
even fractions of RT/F, while those that involve a charge transfer
process in the rate-controlling step have slopes that are odd fractions
of RT/ F, depending on the value n and of /3. It will be important
for the subsequent discussion to note that processes of the former
type [slopes characterized by Eq. (13)] are independent of any
barrier symmetry factor or of its possible temperature dependence.
Possible values of b for steps in the h.e.r. have been discussed
by various workers and the limiting cases have been considered in
detail by Bockris,13 Parsons,3 and Conway;34 similar treatments
have been given for the O2 evolution reaction,32 the Kolbe reaction,36 and some metal-deposition processes.35
Finally, here it should be noted formally that a is, of course,
identical with /3 in cases where the initial step in a reaction sequence
is a one-electron charge transfer process and is itself the ratecontrolling process. Also b can be related to the total number of
electrons passed in the overall reaction or in the rate-controlling
step through a and the stoichiometric number v. This matter has,
however, been treated in various earlier works by, e.g., Horiuti and
Ikusima,37 Bockris,13 and Gileadi,38 and so need not be examined
again here since no involvement of the temperature variable arises
apart from that in b itself.
III. THE EXPERIMENTAL SITUATION REGARDING THE
TEMPERATURE DEPENDENCE OF TAFEL SLOPES
1. General Remarks
Contrary to common belief, the Tafel equation as represented by
Eqs. (1) or (4), with b given by Eq. (5), virtually never represents
the electrode-kinetic behavior of electrochemical processes (except
probably simple ionic redox reactions that have minimal chemical
coupling of one kind or another, p. 125) in particular with reference
Electrochemical Reaction Rates and the Tafel Equation
117
to: (1) the supposed linear proportionality of b to T (i.e., a is a
constant, temperature-independent parameter); and/or (2) the form
of b with respect to T. These experimental facts are central to the
topic of the present paper.
In this section the bases of the above remarks are documented
from previous literature and from new results recently reported.
2. Documentation and Examples of the Experimentally Observed
Dependence of b on T and the Behavior of a with T
(i) The H2 Evolution Reaction at Hg
Historically, the h.e.r. has often been taken as a prototype
process for discussions of the principles of electrode
kinetics;I317'18'2O'21U however, the behavior of this process at
various metals is far from that represented by Eqs. (1) or (4) with
(5). For the h.e.r. at Hg, four sets of relevant data that are to be
considered reliable from the point of view of system purity and
experimental technique,39'40 and cover a wide range of temperature,
are available for discussion: the work of Post and Hiskey41 in
aqueous HC1; the work of Conway et al2 in methanolic HC1 from
173 to 353 K; the work of Bockris et al43 in methanolic HC1 above
and below the mp of Hg; and the work of Conway and Salomon42
in methanolic HC1 and MeOD/DCl, also down to low temperatures.
Following the first indication in the work of Stout48 that b can
be independent of temperature, Bockris and Parsons,11 and Bockris
et al43 showed that a similar effect arose in the h.e.r. at Hg in
methanolic HC1 between 276 and 303 K; below these temperatures,
b apparently varied in the conventional way with T. However, the
derived a values showed a considerable spread. Variations of the
temperature effect in b were discussed in terms of the possible
influence of impurities but an overall assessment of all other, more
recent, observations of the dependence of b on T for various types
of reactions leads to the conclusion that the "unconventional"
dependence is not due to some incidental effect of impurities. In
fact, in another paper, Bockris and Parsons11 suggested that the
temperature dependence of fi for the h.e.r. at Hg arose because of
expansion of the inner region of the double layer with temperature.
They also noted that, formally, for b to be independent of T, the
entropy of activation should be a function of electrode potential,