8
Phenomenological treatment of
electron-transfer reactions
8.1 Outer-sphere electron-transfer
Electron-transfer reactions are the simplest class of electrochemical reactions.
They play a special role in that every electrochemical reaction involves at
least one electron-transfer step. This is even true if the current across the
electrochemical interface is carried by ions since, depending on the direction
of the current, the ions must either be generated or discharged by an exchange
of electrons with the surroundings.
In general electron-transfer reactions can be quite complicated, involving
breaking or forming of chemical bonds, adsorption of at least one of the redox
partners, or the presence of certain catalysts. So far our understanding is
limited to the simplest possible case, so-called outer-sphere electron-transfer
reactions, in which from a chemist’s point of view nothing happens but the
exchange of one electron – as we shall see later, the simultaneous transfer of
two or more electrons is highly unlikely. In the course of such a reaction, no
bonds are broken or formed, the reactants are not specifically adsorbed, and
catalysts play no role. If one of these conditions is not fulfilled, the reaction
is said to proceed via an inner-sphere pathway. Unfortunately, there are not
many examples for outer sphere reactions; here are two:
[Ru(NH3 )6 ]2+ ! [Ru(NH3 )6 ]3+ + e−
2+
[Fe(H2 O)6 ]
3+
! [Fe(H2 O)6 ]
+ e−
(8.1)
In aqueous solutions these reactions seem to proceed via an outer-sphere mechanism on most metals. Typically such reactions involve metal ions surrounded
by inert ligands, which prevent adsorption. Note that the last example reacts
via an outer-sphere pathway only if trace impurities of halide ions are carefully
removed from the solution; otherwise it is catalyzed by these ions.
72
8 Phenomenological treatment of electron-transfer reactions
8.2 The Butler-Volmer equation
In this chapter we treat electron-transfer reactions from a macroscopic point
of view using concepts familiar from chemical kinetics. The overall rate v of
an electrochemical reaction is the difference between the rates of oxidation
(the anodic reaction) and reduction (the cathodic reaction); it is customary to
denote the anodic reaction, and the current associated with it, as positive:
v = kox csred − kred csox
(8.2)
where csred , csox denote the surface concentrations of the reduced and oxidized
species, and kox and kred are the rate constants. Using absolute rate theory,
the latter can be written in the form:
!
"
∆G†ox (φ)
kox = A exp −
RT
$
#
∆G†red (φ)
(8.3)
kred = A exp −
RT
The phenomenological treatment assumes that the Gibbs energies of activation Gox and Gred depend on the electrode potential φ, but that the preexponential factor A does not. We expand the energy of activation about the
standard equilibrium potential φ00 of the redox reaction; keeping terms up to
first order, we obtain for the anodic reaction:
∆G†ox (φ) = ∆G†ox (φ00 ) − αF (φ − φ00 ),
%
1 ∂∆G†ox %%
with α = −
F
∂φ %φ00
(8.4)
∆G†red (φ) = ∆G†red (φ00 ) + βF (φ − φ00 ),
%
1 ∂∆G†red %%
with β =
%
F
∂φ %
(8.5)
The quantity α is the anodic transfer coefficient; the factor 1/F was introduced, because F φ is the electrostatic contribution to the molar Gibbs energy,
and the sign was chosen such that α is positive – obviously an increase in the
electrode potential makes the anodic reaction go faster, and decreases the corresponding energy of activation. Note that α is dimensionless. For the cathodic
reaction:
φ00
where the cathodic transfer coefficient β is also positive. One would expect
that higher terms in the expansion of the Gibbs energy of activation will
become important at potentials far from the standard equilibrium potential
φ00 ; we will return to this point in the next chapter. The Gibbs energies of
activation are related by:
8.2 The Butler-Volmer equation
73
potential
energy
αF(φ−φ 00)
F(φ−φ 00)
reaction
coordinate
Fig. 8.1. Potential energy curves for an outer-sphere reaction; the upper curve is for
the standard equilibrium potential φ00 ; the lower curve for φ > φ00 .Potential energy
curves for an outer-sphere reaction; the upper curve is for the standard equilibrium
potential φ00 ; the lower curve for φ > φ00 .
∆G†ox (φ) − ∆G†red (φ) = Gox − Gred
(8.6)
to the molar Gibbs energies Gox and Gred of the oxidized and reduced state;
in particular:
∆G†ox (φ00 ) = ∆G†red (φ00 ) = ∆G†00
(8.7)
When the electrode potential is changed from φ00 to a value φ, the Gibbs
energy of the electrons on the electrode is lowered by an amount −F (φ −
φ00 ), and so is the energy of the oxidized state. If the reactants are so far
from the metal surface that their electrostatic potentials are unchanged when
the electrode potential is varied, then the Gibbs energy of the reaction is
also changed by −F (φ − φ00 ). This condition is generally fulfilled for outersphere reactions in the presence of a high concentration of an inert electrolyte
which screens the electrode potential; it is not fulfilled when the reactants are
adsorbed as in inner-sphere reactions. When it is fulfilled we have:
∆G†ox (φ) − ∆G†red (φ) = −F (φ − φ00 )
(8.8)
By differentiation we obtain for the sume of the two transfer coefficients the
relation:
α+β =1
(8.9)
Since both coefficients are positive, they lie between zero and one; we can generally expect a value near 1/2 unless the reaction is strongly unsymmetrical.
The transfer coefficients have a simple geometrical interpretation. In a
one-dimensional picture we can plot the potential energy of the system as
74
8 Phenomenological treatment of electron-transfer reactions
a function of a generalized reaction coordinate (see Fig. 5.1). The reduced
and the oxidized states are separated by an energy barrier. Changing the
electrode potential by an amount (φ − φ00 ) changes the molar Gibbs energy
of the oxidized state by −F (φ − φ00 ); the Gibbs energy of the transition state
located at the maximum will generally change by a fraction −αF (φ − φ00 ),
where 0 < α < 1. The relation α + β = 1 is easily derived from this picture.
The current density j associated with the reaction is simply j = F v.
Combining Eqs. (5.2)-(5.4) and (5.9) gives the Butler-Volmer equation [25, 26]
in the form:
αF (φ − φ00 )
"
! RT
(1 − α)F (φ − φ00 )
exp −
RT
j = F k0 csred exp
−F k0 csox
where
!
"
∆G† (φ00 )
k0 = A exp −
RT
(8.10)
(8.11)
Using the Nernst equation:
φ0 = φ00 +
RT
cs
ln sox
F
cred
(8.12)
for the equilibrium potential φ0 , and introducing the overpotential η = φ − φ0 ,
which is the deviation from the equilibrium potential, we rewrite the ButlerVolmer equation in the form:
!
"'
&
αF η
(1 − α)F η
j = j0 exp
− exp −
(8.13)
RT
RT
where
j0 = F k0 (csred )(1−α) (csox )α
(8.14)
is the exchange current density. At the equilibrium potential the anodic and
cathodic current both have the magnitude j0 but opposite sign, thus cancelling
each other. The exchange current density for unit surface concentration of the
reactants is the standard exchange current density j00 = F k0 , which is a
measure of the reaction rate at the standard equilibrium potential.
According to the Butler-Volmer law, the rates of simple electron-transfer
reactions follow a particularly simple law. Both the anodic and the cathodic
current densities depend exponentially on the overpotential η (see Fig. 5.2).
For large absolute values of η, one of the two partial currents dominates, and
a plot of ln |j| – or of log10 |j| – versus η, a so-called Tafel plot [27] (see Fig.
5.3), yields a straight line in this region. From its slope and intercept the
transfer coefficient and the exchange current density can be obtained. These
two quantities completely determine the current-potential curve.
8.2 The Butler-Volmer equation
75
10
α=0.6
j / j0
6
2
α=0.5
α=0.4
-2
-6
-10
-0.1
-0.06
-0.02
0.02
0.06
0.1
η/V
Fig. 8.2. Current-potential curves according to the Butler-Volmer equation.
For small overpotentials, in the range |F η| " RT , the Butler-Volmer equation can be linearized by expanding the exponentials:
j = j0
Fη
RT
(8.15)
The quantity η/j = RT /j0 F is called the charge-transfer resistance. Note
that the transfer coefficient does not appear in the current-voltage relation for
small overpotentials, and hence cannot be determined from measurements at
small deviations from equilibrium, they give the exchange current density only.
However, α can be obtained by varying the surface concentrations, measuring
the exchange current density, and using Eq. (5.14). We will discuss a few
examples of outer-sphere electron-transfer reactions in Chapter 11.
We conclude these phenomenological considerations with a few remarks:
1. The transfer coefficient is equivalent to the Broenstedt coefficient well
known from ordinary chemical kinetics. Both describe the change in the
energy of activation with the Gibbs energy of the reaction.
2. The transfer coefficient α has a dual role: (1) It determines the dependence of the current on the electrode potential. (2) It gives the variation
of the Gibbs energy of activation with potential, and hence affects the
temperature dependence of the current. If an experimental value for α is
obtained from current-potential curves, its value should be independent of
temperature. A small temperature dependence may arise from quantum
effects (not treated here), but a strong dependence is not compatible with
an outer-sphere mechanism.
3. For small overpotentials the linear approximations of Eqs. (8.5) and (8.6)
should be sufficient, but at high overpotentials higher-order terms are
expected to contribute.
76
8 Phenomenological treatment of electron-transfer reactions
α=0.6
2
ln(j / j 0 )
1
α=0.5
α=0.4
0
-1
-2
0
0.02
0.04
0.06
0.08
0.1
η/V
Fig. 8.3. Tafel plot for the anodic current density of an outer-sphere reaction.
4. The transfer coefficient determines the symmetry – or lack thereof – of the
current-potential curves; they are symmetric for α = 1/2. For this reason
the transfer coefficient is also known as the symmetry factor.
5. The surface concentrations are generally not known, and may vary with
time as the reaction proceeds. One way to circumvent this problem is
to work under conditions of controlled convection, so that the surface
concentrations can be calculated from the bulk concentrations. Another
technique consists in the use of potential or current pulses, which allows
an extrapolation back to the time of the onset of the pulse when surface
and bulk concentrations are equal. These techniques will be discussed in
detail in Chapters 18 and 19.
6. Inner-sphere electron-transfer reactions are not expected to obey the
Butler-Volmer equation. In these reactions the breaking or formation of
a bond, or an adsorption step, may be rate determining. When the reactant is adsorbed on the metal surface, the electrostatic potential that
it experiences must change appreciably when the electrode potential is
varied.
8.3 Double-layer corrections
When the concentration of the inert electrolyte is low, the electrostatic potential at the reaction site differs from that in the bulk and changes with the
applied potential. This results in two effects [28]:
1. The surface concentrations csox and csred differ from those in the bulk even
if the surface region and the bulk are in equilibrium. Using the same
8.4 A note on inner-sphere reactions
77
arguments as in the Gouy-Chapman theory, the surface concentration cs
of a species with charge number z is:
!
"
ze0 φ2
s
c = c0 exp −
(8.16)
kT
where c0 is the bulk concentration, φ2 the potential at the reaction site,
and the potential in the bulk of the solution has been set to zero.
2. On application of an overpotential η, the Gibbs energy of the electrontransfer step changes by e0 [η−∆φ2 (η)], where ∆φ2 (η) is the corresponding
change in the potential φ2 at the reaction site. Consequently, η must be
replaced by [η − ∆φ2 (η)] in the Butler-Volmer equation (5.13).
These modifications are known as the Frumkin double-layer corrections.
They are useful when the electrolyte concentration is sufficiently low, so that
φ2 can be calculated from Gouy-Chapman theory, and the uncertainty in
the position of the reaction site is unimportant. Whenever possible, kinetic
investigations should be carried out with a high concentration of supporting
electrolyte, so that double-layer corrections can be avoided.
8.4 A note on inner-sphere reactions
There is no general law for the current-potential characteristics of inner-sphere
reactions. Depending on the system under consideration, various reaction
steps can determine the overall rate: adsorption of the reacting species, an
electron-transfer step, a preceding chemical reaction, coadsorption of a catalyst. If the rate-determining step is an outer-sphere reaction, the current will
obey the Butler-Volmer equation. A similar equation may hold if an innersphere electron transfer, for example, from an adsorbed species to the metal,
determines the rate. In this case, application of an overpotential η changes the
Gibbs energy of this step only by a fraction of F η; furthermore, the concentration of the adsorbed species will change with η. These effects may result in
phenomenological equations of the form:
!
"
αF η
βF η
kox = k0 exp
, kred = k0 exp −
(8.17)
RT
RT
with apparent transfer coefficients α and β, but α and β may depend on
temperature.
If the rate-determining step is the adsorption of an ion, the reaction obeys
the laws for ion-transfer reactions (see Chapter 12), and again a ButlerVolmer-type law will hold.
Problems
1.
Derive Eq. (5.13) from Eqs. (5.10) and (5.12).
78
2.
8 Phenomenological treatment of electron-transfer reactions
The reduced species of an outer-sphere electron-transfer reaction is generated
by a chemical reaction of the form:
A ! red
3.
Denote the forward and backward rate constants of this reaction by ka and
kb . When the reaction proceeds under stationary conditions, the rates of the
chemical and of the electron-transfer reaction are equal. Derive the currentpotential relationship for this case. Assume that the concentrations of A and
of the oxidized species are constant.
The Gibbs energy of activation in Eq. (5.4) can be split into an enthalpy and
†
†
. Define two transfer coefficients
− T ∆Sox
an entropy term: ∆G†ox = ∆Hox
αH = −
†
1 ∂∆Hox
,
F ∂(φ − φ00 )
αS =
†
1 ∂∆Sox
F ∂(φ − φ00 )
and derive the corresponding current-potential relations. Note: For outersphere electron-transfer reactions αS seems to be negligible; it has, however,
been used to explain a temperature dependence of the apparent transfer
coefficients in some inner-sphere reactions.