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Electrochemical Reaction Rates and the Tafel Equation


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,


Brian E. Conway

0 13




0 10


(CONWAY et ol)




0 07













b and a as f(T) at Hg

0 62


Solid Hg

0 54





0 09

0 08

0 07

0 06


. Solid Hg







Electrochemical Reaction Rates and the Tafel Equation


but the matter was not pursued further until the treatment of

Conway et al2 referred to in more detail elsewhere in this article.

In Fig. 3a, the Tafel slopes from these authors' works are

plotted comparatively as a function of temperature T. It is evident

that the Eq. (5) for b a s / ( T ) , commonly assumed, is not followed,

and the T dependence of b is fitted2 by an equation of the form



rather than that of Eq. (5) with a s p. (Note, that for Hg, almost

all authors, with the exception of Horiuti,44 agree that the ratecontrolling step in H2 evolution at Hg from acidic solutions is the

discharge of the hydronium ion, H 3 O + , or the related species H9C>4,

so that a = fi; cf. Section II.) The data of Bockris et al43 are

somewhat different (Fig. 3b) and (3 appears to be dependent on T

over part of the T range investigated.

Thus from Refs. 2,41, and 42, b has a temperature-independent

component, X, which for the h.e.r. at Hg has the value 40 mV. Here,

in Eq. (14), /3' is the symmetry factor which must be evaluated

from the derivative of b with respect to T; thus

db/dT= R/p'F


P' = (R/F)[l/(db/dT)]


so that

The data of Fig. 3a show that p' is constant with T but has a value

appreciably different from 0.5 ± 0.05 usually obtained using Eq. (5)

at T =* 298 K. It is evident, and important to note, that for the h.e.r.

results at Hg it is incorrect to evaluate j3 from Eq. (5) at some

particular values of T since, in fact, that equation erroneously

represents the form of b with T; it is Eq. (14) that must be employed,

using Eq. 16.

It is certainly a remarkable coincidence that (3 evaluated from

Eq. (5) at the "laboratory temperature" —298 K comes out very

Figure 3. (a) Tafel slopes b for the h.e.r. at Hg in aqueous and methanolic acid

solutions as a function of temperature (from data of Post and Hiskey, Conway et al.,2

and Conway and Salomon42). Comparison with behavior for Br2 at graphite also shown

(see Fig. 8). (b) Tafel slopes b for the h.e.r. at liquid and solid Hg from the results of

Parsons and Bockris11 in methanolic HC1 (from Ref. 43).


Brian E. Conway

near the conventionally "expected" value of 0.5; a similar point

has been emphasized by Yeager et al.45b (see below) with regard

to his results for O2 reduction at Pt in H3PO4 over a wide temperature

range. However, it evidently just is a coincidence since it is quite

evident that the true "/3", written as j3' in Eq. (14), must be evaluated

from Eq. (16) rather than from Eq. (5). If /3 is evaluated from Eq.

(5) it has an apparent temperature dependence, the origin of which

is clear if Eq. (5) is coupled with Eq. (14), which truly represents

the results of Fig. 3a, namely


Then j8 is evidently given by

1//3 = l/p' + KF/RT




so that p is apparently T dependent; thus it is seen that fi apparently

increases with T as has been noted empirically in some earlier

works using Eq. (5) with experimental b values. We emphasize

again that the symmetry factor for the h.e.r. at Hg must be evaluated

from Eq. (14); then the true /3, written as /3', is found to be

independent of T and the Tafel slope has the form represented by

Eq. (14) which is fundamentally different from that normally

assumed [Eq. (5)]. We shall comment on possible nontrivial explanations for the form of Eq. (14), including the significance of K,

in Section IV.

(«) The H2 Evolution Reaction at Ni

Conway et al.2 also studied the h.e.r. at Ni under high-purity

conditions in methanolic HC1 over a wider range of temperature.

Under these conditions, the "Tafel lines" appear as two linear

segments,t the slopes of each of which depend on T as shown in

Fig. 4. Comparative data for the D2 evolution reaction (d.e.r.) from

MeOD/DCl are shown in Fig. 5.

t In aqueous HC1, lines with a small degree of curvature are usually observed with

b = 0.10 ±0.005 V a t 298 K.

Electrochemical Reaction Rates and the Tafel Equation







Figure 4. Tafel relations for the h.e.r. at Ni in methanolic HC1 over a wide range

of temperature (from Ref. 2). Note that upper-region slopes increase with increasing

T, while lower-region slopes decrease for the same electrode process.

The two linear Tafel regions at Ni in methanolic HC1 vary

with T in a continuous and complementary way: one has a slope

that increases with T while the slope of the other simultaneously

decreases with T (Fig. 4), so there is a singular temperature at

which the Tafel relation is one line over the whole c.d. range.2 The

directions of change of the slopes of the two Tafel lines at each

temperature, other than at the singular temperature, correspond

apparently to reaction mechanisms that are consecutive ( ^ ^ 0

or parallel (/""""). However, we believe, based on new data

obtained from potential-decay experiments,77 that the two regions

correspond to desorption mechanisms in the h.e.r. taking place

through different populations of adsorbed H which have different

dependencies of their coverage on T and rj.

It is evident that the 7-dependence of b again does not follow

that conventionally assumed [Eq. (5)] and, moreover, for the same

reaction at the same metal, Ni, the two Tafel-line segments have

slopes that vary in opposite directions with temperature; Fig. 4

shows that this is a systematic variation and not the result of some


Brian E. Conway



0 26 -

^ D C ! MeOD




022 -






^^^^^S^Mupper ' O \



0.14 -

H (lower i)

" ^

0.10 -






C/) 0.09 150






Figure 5. Tafel slopes for the d.e.r. at Ni in d-methanolic DC1 over a wide

range of temperature in comparison with those for the h.e.r. in ethanolic HC1

(from Ref. 2).

arbitrary comparison of the behavior at say just two different

temperatures. It is interesting that in methanolic HC104, made up

from the 70% aqueous acid, so that the methanolic solution contains

an unavoidable small mole fraction of H2O, two Tafel regions are

not observed. Hence the behavior in anhydrous methanolic HC1

may be connected with anion adsorption effects in that solvent [see

Section IV.9]. As will be discussed in that section, the influence of

temperature on the effects of adsorbed anions on the kinetics of

Electrochemical Reaction Rates and the Tafel Equation


an electrode process must always be recognized as a possible reason

for unconventional temperature variation of Tafel slopes, unless

conditions otherwise preclude the possibility of such effects, e.g.,

as at Hg at appreciable overpotentials in the h.e.r. where anion

adsorption is known to be insignificant.93'94

(Hi) The H2 Evolution Reaction at Cd

Data for the h.e.r. at Cd were also found2 to behave unconventionally.

(it?) The H2 Evolution Reaction at Electrocatalytic Ni-Mo Alloys

Recently, much interest has centered on the electrocatalytic

behavior of Ni-Mo alloyst (prepared electrolytically or by thermal

reduction of oxide or molybdate mixtures) for cathodic H2 evolution

in alkaline-water electrolyzers or as cathodes in electrolytic Cl2

cells. The h.e.r. at these electrode materials exhibits46 remarkably

low Tafel slopes, in the range 22-26 mV, at elevated temperatures,

~363 K. This is one of the reasons for their excellent behavior as

electrocatalysts for the h.e.r. However, the Tafel slopes exhibited

by these materials depend on temperature in a most unusual way,

decreasing with increasing temperature, as do some of the results

at Ni2 (Fig. 4).

The rj vs. In [current-density] relations (Fig. 6) exhibit two

linear regions (as with Ni under certain conditions—see Fig. 4),

each of which has a slope that decreases with T as shown in Fig.

7. This is not seemingly due to some change of mechanism as

temperature is increased since the decrease of the slopes with T is

evidently (see Fig. 7) a continuous one for both regions of the In i

vs. rj relations as is seen from Fig. 7.

This behavior must correspond to variation of a with a power

of T > 1 in the relation b = RT/a(T)F in order for b to decrease

with increasing T, or b is of the form b = K' - cT for the particular

case where b decreases linearly with T as in some experiments.

$ There is some question whether these are true equilibrium metallic alloys (it is

unlikely) and also what is the state of Mo—is it metallic or in the form of MoO

in a Ni-metal host structure.

Brian E. Conway








log i (mA cm

Figure 6. Tafel relations for the h.e.r. at Ni-Mo electroplated electrocatalysts at

several temperature curves (1-4), 341, 319, 298, and 278 K, compared with behavior

of metallic nickel (curve 5) (Real apparent area factor for Ni-Mo coated electrode

450 x ); electrolyte is 1.0 M aqueous KOH (from Ref. 46, see also Fig. 19).

(v) Redox Reactions at Electrodes Involving Werner Complex Ions

The behavior of b as /(T) for ionic redox reactions at electrodes, especially those processes that involve only outer-sphere

changes of state, and both red and ox species which are not

specifically adsorbed, is of great interest. From some of Weaver's

work107108 information on the dependence of b on T for such

reactions is available and some attempts have been made, e.g., by

Parsons and Passeron,7 to establish if the potential-dependent factor

in the electrochemical rate equation in fact includes a quadratic

term in rj as well as the usual linear one; however, this is a different

question (cf. Ref. 8) related to the harmonicity or, otherwise, of

the fluctuations involved in the activation process.14

Later in this article, we refer to Weaver's work107108 on the

evaluation of real entropies of activation of some one-electron redox

reactions of complex metal ions at Hg, and corresponding data on

Electrochemical Reaction Rates and the Tafel Equation



NJ:Mo:Cd I

7 9 2 0 : I I TAFEL SLOPES






u- 50










Figure 7. Decrease of slope b for h.e.r. at Ni-Mo electrocoated cathodes with

temperature: (O) upper c.d. region, and (x) lower c.d. region (from Ref. 46).

temperature variation of the transfer coefficient. These results are

of great interest.

The problem with redox reactions of this type is that their rate

constants are usually too large for regular steady-state techniques

to be reliably applied, a or (3 then have to be determined through

the reaction order or by some method such as Faradaic rectification.

Usually, such methods require evaluation of the double-layer

behavior in order to make "double-layer corrections." This is often

an unsatisfactory business, especially when corrections would be

required over a range of temperatures. We conclude that for this

important class of electrochemical reactions more data for

examination of b(T) or a(T) are required. However, for certain

ionic redox reactions that are sufficiently slow, Weaver107 has been

able to evaluate a as /(T) from Tafel plots over a range of 0.3 ~

0.4 V: a is in fact found to be independent of T for these cases (see

p. 179), an important result.

The Tafel-slope behavior as /(T) for solvated electron formation47 in, e.g., liquid NH3 or HMPA solutions would be another

interesting case for investigation.


Brian E. Con way

(vi) Anodic N2 Evolution from N3-ion at Pt

Probably the earliest, well-documented case of unconventional

behavior of the Tafel slope with temperature was reported by Stout48

for the anodic evolution of N2 from discharge of azide ion at Pt in

aqueous solutions. The overall reaction is

2N3 -> 3N2 + 2e


but detailed steps in the reaction have not been characterized;

presumably the N3 radical is an intermediate but whether its

decomposition to N2 is homogeneous or heterogeneous is as yet

unknown and the question of recombination to give an unstable

(and probably chemically unlikely) N6 species is unresolved.

For the azide discharge reaction at Pt, b in Eq. (5) is found

to be independent of T so that a itself is apparently linear in TI

This case can be regarded qualitatively as different from that for

the h.e.r. at Hg or limitingly as a special case of Eq. (14) where

K » RT/ p'F. Theoretical aspects of this type of case will be treated

in Section IV.

Although very little is known about the surface electrochemistry of the azide discharge reaction, analogies with the

situation in Cl~ or R.COO"36 discharge from aqueous solutions

would suggest that N^ is discharged at Pt on a surface covered or

partially covered by discharged OH or O species (cf. Ref. 49) with

possible coadsorption of N^ ion (regarded as a pseudohalogen).

Then the state of surface oxidation of the Pt and the temperature

dependence of the state of surface oxidation will determine, in part,

the temperature dependence of the Tafel slope of the N^ discharge

process. That a process occurring on a surface oxide is involved is

indicated by the unusually high Tafel slopes of —ART/F (as in

the Kolbe reaction36) characteristic (cf. Refs. 50 and 51) of charge

transfer across a barrier-layer film. Competition between specific

adsorption of N^ and electrodeposition of OH and O species49 may

be expected in this reaction. In this case, it is difficult to assign any

single reason for the independence of b on temperature. Qualitatively, a tends to become smaller with increasing film thickness but

greater with increasing film conductivity. The surface electrochemistry involved (cf. Ref. 49) is obviously complex and little

is known about its temperature dependence except that surface

Electrochemical Reaction Rates and the Tafel Equation


oxidation of Pt is facilitated, as may be expected, by elevation of

temperature ;52 the charge transfer characteristics of the oxide film

are not, however, known as a function of temperature, but conductance measurements were made by Shibata and Sumino53 at room

temperature for thin and thick films in the absence of Cl" or

pseudohalide ions.

(vii) Anodic O2 Evolution at Pt

In studies of the O2 evolution reaction at Pt from aqueous

H2SO4, it is found that the Tafel slopes determined for the steadystate kinetics of this reaction at several temperatures do not correspond to the conventionally expected behavior according to Eq.

(5), so that a is somewhat temperature dependent as in the case

of Section III.2, (ii). However, more work on this case is required,

taking into account the temperature dependence of oxide-film

growth, etc.

(viii) Anodic Br2 Evolution from Br~ in CH3CN Solution

In an experiment designed to determine if unconventional

temperature dependence of Tafel slopes (where a seems to be itself

linear in T) originates in some way on account of the temperature

dependence of the structure54'55 of associated solvents such as H2O

and MeOH (cf. Ref. 2) in the electrode/solution interphase or of

orientation of solvent dipoles of the structured solvent, Conway et

al2 determined b for anodic Br2 evolution at graphite from Br~ ion

in CH3CN over an 80 K temperature range: b was found to be

70 ± 7 mV and independent of T over the whole temperature range

covered so that a is again, for this case, independent of T (Fig.

8a). Since CH3CN, although a strongly polar solvent, is generally

considered to be "unstructured" in the sense that H2O or MeOH

are, it can be concluded that it is unlikely that the temperature

dependence of a can be directly attributed to changes of structure

of the solvent in the double layer or of the structural factor55 in

the solvation of the reactant ion. Recent results from this laboratory

on anodic Cl2 evolution at Pt also show that b (=0.04 V in this

case, prior to a limiting current) is almost independent of T. This

result is surprising since two complicating aspects of the electrodekinetic behavior must be involved: temperature dependence of Cl~

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