DOI 10.1515/pac-2014-5026
Pure Appl. Chem. 2014; 86(2): 245–258
IUPAC Technical Report
Rolando Guidelli*, Richard G. Compton, Juan M. Feliu, Eliezer Gileadi,
Jacek Lipkowski, Wolfgang Schmickler and Sergio Trasatti
Defining the transfer coefficient in electrochemistry:
An assessment (IUPAC Technical Report)
1
Abstract: The transfer coefficient α is a quantity that is commonly employed in the kinetic investigation of
electrode processes. In the 3
rd
edition of the IUPAC Green Book, the cathodic transfer coefficient α
c
is defined
as –(RT/nF)(dlnk
c
/dE), where k
c
is the electroreduction rate constant, E is the applied potential, and R, T, and
F have their usual significance. This definition is equivalent to the other, -(RT/nF)(dln|j
c
|/dE), where j
c
is the
cathodic current density corrected for any changes in the reactant concentration at the electrode surface with
respect to its bulk value. The anodic transfer coefficient α
a
is defined similarly, by simply replacing j
c
with
the anodic current density j
a
and the minus sign with the plus sign. It is shown that this definition applies
only to an electrode reaction that consists of a single elementary step involving the simultaneous uptake of
n electrons from the electrode in the case of α
c
, or their release to the electrode in the case of α
a
. However,
an elementary step involving the simultaneous release or uptake of more than one electron is regarded as
highly improbable in view of the absolute rate theory of electron transfer of Marcus; the hardly satisfiable
requirements for the occurrence of such an event are examined. Moreover, the majority of electrode reactions
do not consist of a single elementary step; rather, they are multistep, multi-electron processes. The uncriti-
cal application of the above definitions of α
c
and α
a
has led researchers to provide unwarranted mechanistic
interpretations of electrode reactions. In fact, the only directly measurable experimental quantity is dln|j|/
dE, which can be made dimensionless upon multiplication by RT/F, yielding (RT/F)(dln|j|/dE). One common
source of misinterpretation consists in setting this experimental quantity equal to αn, according to the above
definition of the transfer coefficient, and in trying to estimate n from αn, upon ascribing an arbitrary value to
α, often close to 0.5. The resulting n value is then identified with the number of electrons involved in a hypo-
thetical rate-determining step or with that involved in the overall electrode reaction. A few examples of these
unwarranted mechanistic interpretations are reported. In view of the above considerations, it is proposed to
define the cathodic and anodic transfer coefficients by the quantities α
c
= –(RT/F)(dln|j
c
|/dE) and α
a
= (RT/F)
(dlnj
a
/dE), which are independent of any mechanistic consideration.
Keywords: electrode kinetics; IUPAC Physical and Biophysical Chemistry Division; reorganization energy;
symmetry factor; Tafel slope; transfer coefficient.
1
Sponsoring body: IUPAC Physical and Biophysical Chemistry Division: see more details on p. 257.
*Corresponding author: Rolando Guidelli, Department of Chemistry “Ugo Schiff”, University of Florence, Via della Lastruccia 3,
50014 Sesto Fiorentino (Firenze), Italy, e-mail: rolando.guidelli@libero.it
Richard G. Compton: Department of Chemistry, University of Oxford, Physical and Theoretical Chemistry Laboratory, South
Parks Road, Oxford OX1 3QZ, UK
Juan M. Feliu: Department of Physical Chemistry, University of Alicante, Ap. De Correos, 99, 03080, Alicante, Spain
Eliezer Gileadi: School of Chemistry, Faculty of Exact Sciences, University of Tel-Aviv, Tel-Aviv, Israel
Jacek Lipkowski: Department of Chemistry and Biochemistry, University of Guelph, 50 Stone Road East, Guelph, Ontario,
N1G2W1, Canada
Wolfgang Schmickler: Institute of Theoretical Chemistry, University of Ulm,
Albert Einstein Allee 11, D-89069 Ulm, Germany
Sergio Trasatti: Department of Chemistry, University of Milan, Via Camillo Golgi 19, 20133 Milan, Italy
© 2014 IUPAC & De Gruyter
246
R. Guidelli etal.: Transfer coefficient: An assessment
1 Historical overview
The transfer coefficient, α, was originally introduced in electrochemistry by Butler [1] and by Erdey-Gruz and
Volmer [2]; it was defined as the fraction of the electrostatic potential energy affecting the reduction rate in
an electrode reaction, with the remaining fraction (1 – α) affecting the corresponding oxidation rate. For a
generic cathodic reaction
OeRn
+→
(1)
the electrostatic potential energy is given by nFΔφ, where Δφ is the potential difference across the electrified
interface, which differs from the applied electric potential E by a constant depending on the choice of the
reference electrode. In this connection, it should be noted that electrons in a metal differ from the molecular
species that are dealt with in chemical kinetics because they satisfy the Fermi statistics instead of the Boltz-
mann statistics. While the latter statistics lead to an expression of the electrochemical potential of the species
that includes a term proportional to the logarithm of its concentration, this is not the case for the Fermi sta-
tistics. Differently stated, there is no sense in referring to an “electron concentration” in a metal. This makes
a major difference between electrochemical kinetics and chemical kinetics. Inserting this expression into the
equation of the absolute reaction rate theory yields the following relationship between the cathodic current
density j
c
and the applied potential E:
cc
exp( /)
j
nFERT
α∝−
(2)
where F, R, and T have their usual significance. This expression holds in the absence of depletion and diffuse-
layer effects that may alter the reactant concentration at the electrode surface with respect to its bulk value.
The dimensionless “cathodic” transfer coefficient α
c
is immediately obtained from the slope of the plot of ln
|j
c
| against E:
cc
(/)(dln| |/
d)
RT nF
jE
α =−
(3)
The derivative dE/dln|j
c
| is referred to as the cathodic Tafel slope. Incidentally, here and in the following the
symbol ln|j
c
| implies that the argument of the logarithm is of dimension one, obtained by division with the
corresponding unit, e.g., ln|j
c
| meaning ln(|j
c
|/A m
–2
), and similarly for the other quantities j
a
, k
a
, and k
c
. By
analogous considerations, for an anodic reaction
ROne
→+
(4)
we have
aa
exp(
/)
j
nFERTα∝
(5)
and
aa
(/)(dln/d)RT nF jE
α =
(6)
where dE/dlnj
a
is the anodic Tafel slope. It should be borne in mind that eqs. 3 and 6 hold strictly only if the
cathodic reaction of eq. 1 and the anodic reaction of eq. 4 consist of a single elementary step involving the
simultaneous release or uptake of n electrons by the electrode, respectively. The notations of eqs. 2, 3, 5, and
6 have been used in a number of textbooks.
In the proximity of the equilibrium potential, E
eq
, of a redox couple O/R, both the cathodic and the anodic
current densities contribute to the net current density j, which is given by the sum of the two:
ac aa
cc
[R]exp(/)[O]exp(
/)
j
jjnFknFE RT nF
kn
FE RT
αα
=+=−−
(7)
R. Guidelli etal.: Transfer coefficient: An assessment
247
Here, k
a
and k
c
are the anodic and cathodic rate constants; the concentrations between square brack-
ets are “volume concentrations on the electrode surface”, which can be expressed in mol·m
–3
units; conse-
quently, the rate constants k
a
and k
c
can be expressed in m
3
·mol
–1
·s
–1
units. Under equilibrium conditions, j
equals zero, yielding
0
aaeq cceq
[R]exp(/)[O]exp(
/)
j
nFknFE RT nF
kn
FE RT
αα
==−
(8)
or
ac ca eq
[O]/[R]( /)exp[()
/]
kk nFERTαα=+
(9)
In eq. 8, E
eq
is the equilibrium potential and j
0
, called the exchange current density, is the common absolute
value of the anodic and cathodic current densities, when they match at E
eq
. Since at equilibrium the Nernst
equation applies, from eq. 9 it follows that (α
c
+ α
a
) = 1 and k
a
/k
c
= exp(-nFE°/RT), where E° is the formal
potential of the O/R couple. Combining eqs. 7 and 8, we obtain the so-called Butler–Volmer equation
0
cc eq
[exp(1 )/ exp( /)]with:
j
jnFRTnFR
TE
Eαη αη η=− −− ≡−
(10)
where η is called overpotential (see Fig. 1).
From Fig. 2 it is apparent that the linear dependence of the anodic or the cathodic current density upon
the applied potential E is only attained when E is sufficiently apart from the equilibrium potential E
eq
, i.e.,
when the absolute value of η is > > RT/(nF). When this condition is fulfilled, the extrapolation of the resulting
Tafel plot to the η = 0 axis yields the natural logarithm of the exchange current density.
Fig. 2Plot of the overpotential η against the natural logarithm of the anodic and cathodic current densities, according to the
Butler–Volmer equation with α
c
= 0.5. The upper and lower dashed lines are the anodic and cathodic Tafel plot, respectively.
Fig. 1Plots of the anodic current density j
a
(upper dashed curve), the cathodic current density j
c
(lower dashed curve), and the
net current density j (solid curve) against the overpotential η, according to the Butler–Volmer equation with α
c
= 0.5.
248
R. Guidelli etal.: Transfer coefficient: An assessment
If n equals unity, α
c
is frequently denoted by β, termed the symmetry factor (see below), yielding
0
[exp(1 )/ exp( /)] with:(/)(ln| |/ )
c
j
jFRT FRTRTF djdEβη βη β=− −− =−
(11)
Delahay, in his well-known monograph on instrumental methods in electrochemistry [3], used eq. 2 for the
kinetics of a “single electrode process” involving a generic number n of electrons in a single rate-determining
step (rds). In considering a number of “consecutive electrochemical reactions”, with the k-th reaction charac-
terized by a rds involving a number n
k
of electrons, he used the expression
c,
kk
exp(
/)
j
nFERTα∝−
(12)
to differentiate n
k
from the total number of electrons, n, involved in the overall electrochemical reaction. In
his book Double Layer and Electrode Kinetics [4], published in 1965, Delahay also considered an electrode
reaction consisting of two consecutive one-electron transfer steps.
In his 1965 book on electrode processes [5], Conway treated some examples of consecutive elementary
one-electron transfer steps by both the steady-state and the quasi-equilibrium method, consequently setting
n = 1 in eq. 2 and using the notation β in place of α. In treating these processes, he did not use the notation
of the transfer coefficient, but simply that of the “Tafel slope”, dE/dln|j|. The effect of reactant adsorption on
Tafel slopes was also considered (see also ref. [6]).
In his comprehensive textbook on electrochemical kinetics [7], Vetter adopted a similar approach, which
he applied principally to metal/(metal ion) electrode reactions using the valence z of a polyvalent ion in place
of n. He also examined a sequence of two elementary one-electron transfer steps, which he treated on the
basis of the steady-state method. A practically identical approach, apart from the use of n in place of z, was
adopted by Brenet and Troare in their monograph on transfer coefficients in electrode kinetics [8]. An analo-
gous approach was adopted by Erdley-Gruz in his textbook [9]. He used again the valence z in place of n, but
he also treated a sequence of elementary one-electron transfer steps on the basis of both the steady-state and
the quasi-equilibrium method. Moreover, he also reported the following expression of the “overall transfer
coefficient” for a sequence of consecutive elementary electron transfer steps and chemical steps character-
ized by a single rds, as derived by Bockris and Reddy [10] two years before:
cf r
/nvnα β
=+
(13)
Here n
f
is the number of electrons released by the electrode before the rds, ν is the number of occurrences
of the rds in the electrode reaction as written, n
r
is the number of electrons involved in the rds, and β is the
symmetry factor, which is usually assumed to take values close to 0.5.
A simple example of an electrode mechanism involving the occurrence of the rds twice is a possible
mechanism for hydrogen evolution:
rdsrds
ad
sa
ds ad
s2
He H;He H;2H H
++
+→+
→⇔
(14)
If n
r
equals zero, eq. 14 denotes a rate-determining chemical step preceded by the reversible release of
n
f
electrons by the electrode. Let us consider, for the sake of simplicity, the common situation in which
ν equals unity and let us assume that the backward electrode reaction, namely, that proceeding in the
anodic direction, is characterized by the same rds. In this case, the anodic transfer coefficient α
a
is clearly
given by
ab
(1 )
r
nnα β=+ −
(15)
where n
b
is the number of electrons taken up by the electrode before the rds. Since the sum n
f
+n
r
+n
b
is neces-
sarily equal to the number, n, of electrons involved in the overall electrode reaction, summing α
a
in eq. 15 to
α
c
in eq. 13, with ν = 1, yields
R. Guidelli etal.: Transfer coefficient: An assessment
249
ca
nα α
+=
(16)
In the more general case in which the rds occurs ν times in the electrode reaction, α
c
+ α
a
is given by
ca
/
nv
α α+=
(17)
The presence of ν in the denominator of eq. 17 can be easily understood by considering the electrode reaction
of eq. 14 as an example. Here, the overall electrode reaction 2H
+
+ 2e → H
2
involves two electrons, but a single
rds involves only one. It must be stressed that eqs. 16 and 17 hold only if the forward and backward electrode
reactions are characterized by the same rds, a situation that is not necessarily encountered when the negative
overpotential for the cathodic process and the positive overpotential for the corresponding anodic process
are relatively high [11].
The approach leading to eq. 13 is similar to that previously adopted by Mauser [12] in 1958. A treatment
leading to an expression for α practically identical to eq. 13 was also developed by Gileadi, Kirowa-Eisner,
and Penciner in their textbook on interfacial electrochemistry [13]. The expression of eq. 13 was subsequently
reported again by Bockris, Reddy, and Gamboa-Aldeco [14] in an updated textbook on fundamentals of elect-
rodics. A more general treatment of a sequence of elementary one-electron transfer steps and chemical steps
was subsequently reported in an exhaustive monograph by Lefebvre [15].
Parsons’ recommendations in the framework of IUPAC’s Physical Chemistry Division [16, 17] deserve a
special mention. They were based on his previous treatment of electrode kinetics published in 1961 [18]. To
introduce the concept of electrode reaction orders, Parsons adopted a generic electrode reaction in which the
cathodic partial current has the form
i,c
cc
i
i
(/)I nvFA
kc
ν
=−
∏
(18)
Here, A is the electrode area, n is the charge number of the electrode reaction “as written”, ν
i,c
is the order
of reaction for the i-th reactant of concentration c
i
, and ν is the “stoichiometric number giving the number
of identical activated complexes formed and destroyed in the completion of the overall reaction as formu-
lated with the transfer of n electrons”. With this expression of the cathodic current, its dependence upon the
applied potential takes the form
cc
exp(
/)
I
nFEvRTα∝−
(19)
It follows that the transfer coefficient is given by
cc
(/)( dln| |/
d)
RT nF
IE
α ν=−
(20)
The electrode reaction adopted by Parsons is not entirely general. In fact, Parsons stated that eq. 19 and the
corresponding equation for the anodic current “are not general definitions since not all reaction rates can be
expressed in this form. For example, the rate of a multistep reaction or a reaction involving adsorbed species
may not be expressible in this form” [16]. In the majority of electrode reactions, n and ν are equal and cancel
each other out in eq. 20, yielding
cc
(/)( dln| |/
d)
RT FIEα =−
(21)
However, Parsons did not exclude the possibility for n and ν to be different, when he wrote [17]: “In general,
the quantity n/ν is not arbitrary and is a true characteristic of the reaction kinetics. This ratio is the charge
number that always appears in the kinetic equations. In principle, the use of ν could be avoided by choosing
n so that ν is always unity but this requires more knowledge about the detailed kinetics of a complex reaction
than is often available”.
