Ch 2. The Temperature and Potential Dependence of Electrochemical Reaction Rates,
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Brian E. Conway
of qualitative exceptions to this assumed behavior that a major
reassessment of the form of the Tafel equation is required with
regard to the way T enters into the behavior of the potential
dependence of rates.
Conway et al.2 in 1970 made the first detailed study of this
situation experimentally and also discussed comparatively several
pertinent theoretical aspects of this problem regarding the commonly observed actual behavior where a is itself a function of T
or b is represented as a f(T) in some way other than by the
conventional definition given above. Notwithstanding the treatment
of this problem in Ref. 2, already some 14 years ago, very little
attention has been given to it since, except in discussion contributions of Yeager at meetings (see p. 129) and some private communications between Gileadi and the present author and others during
the past year. This situation is surprising since the problem of the
proper form of the temperature dependence of Tafel's b factor is
central in the understanding of effects of potential on
electrochemical reaction rates and the molecular mechanism of
activation in charge-transfer processes.
This paper examines this whole problem in a comprehensive
way and reports some new examples of nonconventional behavior
of b with respect to temperature.
First, we establish some preliminary definitions and summarize
the historical background.
2. Preliminary Definitions
Conventionally, the Tafel equation represents the potential dependence of electrochemical reaction rates, expressed as currents (/)
or current densities (/), according to the following empirical
relation:1'2
r] = a + b\ni
(1)
where rj is an overpotential defined as the difference of electrode/solution potential difference, V, when a current of / A c m " 2
is passing from that, Vn when the electrode process is at equilibrium:
i = 0, rj = 0. The net current density / can be defined generally as
the difference of forward (i) and backward (/) components:
i=i-i
(2)
Electrochemical Reaction Rates and the Tafel Equation
105
and, at equilibrium, this leads to the definition of the exchange
current density, i0:
*o = 1(77=0) = r (77=O )
when i = 0
(3)
This definition, using Eq. (1), leads to an alternative form of
the Tafel equation, written exponentially as
i = ioexp(iy/fc)
(4)
from which it is evident that a = -bin i0. Conventionally, 17 is
taken positive for an anodic reaction with b positive.
Initially, the parameter b which defines the slope of the 17 vs.
In 1 relation (the 'Tafel slope" for this relation) was taken as an
empirical quantity, usually found to be a multiple or simple fraction
of RT/F.
Tafel's original work in 19051 was concerned with organic
reactions and H2 evolution at electrodes, and Eq. (1) was written
as an empirical representation of the behavior he first observed. A
particular value of b = RT/2F has come to be associated specifically with Tafel's name for the behavior of the cathodic H2 evolution
reaction (h.e.r.) when under kinetic control by the recombination
of two (adsorbed) H atoms following their discharge from H3O+
or H2O in a prior step.3 Such kinetic behavior of the h.e.r. is observed
under certain conditions at active Pt electrodes4 and in anodic Cl2
evolution at Pt.5 (We note here, in parentheses, that an alternative
origin for a Tafel slope of RT/2F for the h.e.r. at Pt has been
discussed by Breiter4 and by Schuldiner6 in terms of a quasiequilibrium diffusion potential for H2 diffusing away from a very active
Pt electrode at which H2 supersaturation arises).
The empirical representation of electrode process rates according to a relation such as Eq. (1) or its exponential form, Eq. (4),
takes into account that, for many electrode processes, In i is linear
in 77 over an appreciable range (> ~0.2 V say) of potentials. More
will be said about this later with regard to specific examples;
however, it must be stated here that for some processes such as
rapid redox reactions7 (high i0 values) and some organic electrode
reactions,8 a quadratic term in rj may also arguably appear in Eq.
(4), giving a curved 17 vs. In i relation, the observation of which
has been claimed experimentally. Such behavior follows theoretically (see Section IV.7) from a harmonic potential energy rep-
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Brian E. Conway
reservation, e.g., of Marcus,9 of the course of the energy change of
the reaction along its reaction coordinates rather than from the
anharmonic representation used in earlier works.10"12
3. The Tafel-Slope Parameter b
While the form of the Tafel equation with regard to the potential
dependence of i is of major general interest and has been discussed
previously both in terms of the role of linear and quadratic terms
in 7]7"9 and the dependence of the form of the Tafel equation on
reaction mechanisms,13 the temperature dependence of Tafel slopes
for various processes is of equal, if not greater general, significance,
as this is a critical matter for the whole basis of ideas of "activation"
and "reorganization" processes14 in the kinetics of electrode reactions.
The conventional form of fe, which follows from the recent
IUPAC recommendation for the so-called transfer coefficientt a,
is
b = RT/anF
(5)
where n is the electron charge number for the reaction. In terms
of the derivative of the current / with potential V, a/v =
(RT/nF)(d\n I/dV)T.p.c where v is the stoichiometric number.^
For simplicity, in some of the material that follows, we shall write
the conventional form of b (with n and v = 1) simply as b = RT/ aF,
i.e., with the commonly assumed linear dependence on T. This form
for b, with the transfer coefficient a to be discussed in Section II
with respect to its possible linear dependence on T, is, however,
rarely found to be followed2 experimentally, so that very basic
ideas, extant and accepted for many years, about the mechanism
of activation and potential dependence of electrochemical rates are
t In this article, the symbol ft is used for the barrier symmetry factor and a for the
charge transfer coefficient. For a l e process, ft = a in the usual way (see Section
II.2). In various places in the text, where generality is implied and the process
concerned is not necessarily a l e , single-step reaction, the transfer coefficient a is
written. In the latter case, e.g., when charge transfer is involved in a rate-controlling
desorption step, a ft factor is included in the relevant value of a (see p. 115).
$ Introduction of the stoichiometric number v, with a and n, in b is sometimes
confusing. Some discussion of this matter is to be found in Ref. 13 and in Gileadi's
contribution in Chapter 8 of the present author's monograph, Theory and Principles
of Electrode Processes, Ref. 25.
Electrochemical Reaction Rates and the Tafel Equation
107
put at issue. Attention was first directed to this rather remarkable
situation in the paper by Conway et al. in 1970.2
The main purpose of this paper is to examine the real temperature dependence of the Tafel-slope parameter b for various
processes and to discuss the mechanisms of activation in electrode
processes in the light of the observed behavior of b as a function
of temperature.
Before proceeding to direct attention to the real temperature
dependence of Tafel slopes as found experimentally for a number
of systems, it will be necessary to review the conventional behavior
usually assumed and describe its theoretical and historical origins.
The remarkable contrast of the behavior actually observed, to be
described in Section III, to that conventionally assumed will then
be apparent and thus the present major gap in our understanding
of the fundamental aspect of potential dependence of electrode
reaction rates will be better perceived.
II. ORIGIN OF POTENTIAL DEPENDENCE OF
ELECTROCHEMICAL REACTION RATES AND THE
CONVENTIONAL ROLE OF TEMPERATURE
1. Theoretical Representations of Rates of Electrode Processes
Here, for obvious reasons, we confine the discussion to activationcontrolled processes thus avoiding complications due to concentration and nucleation overpotentials which are not relevant to the
present discussion.
The first attempts at deriving a theoretical rate equation for a
heterogeneous electrochemical process involving only electron
transfer were given by Gurney15 and Gurney and Fowler.16 Gurney's
treatment recognized the essential quantum-mechanical aspect of
electron transfer at electrodes, a feature rather neglected subsequently for some 30 following years. The electron transfer rate
was calculated in terms of the integral of the product of probabilities
of electron states, and acceptor or donor states as functions of their
energy distributions. A quantum-mechanical tunneling of electrons
was involved. This paper provided the basis of later treatments of
electron transfer but did not explicitly give an account of the
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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.