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Ch 2. The Temperature and Potential Dependence of Electrochemical Reaction Rates,

Ch 2. The Temperature and Potential Dependence of Electrochemical Reaction Rates,

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104



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-



106



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



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.



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