Chemistry Reference
In-Depth Information
Reorganisation of this equation for the application of the potential dif-
ference of D
E
results in:
-
a
F
R
nE
T
D
=
() ()
=
j
cn
Fk
e
[1.20]
c
O
c
0
D
E
0
where (k
c
)
D
E
= 0
is the rate constant at equilibrium and a the transfer coeffi-
cient (0 <a<1).
Similarly, an equation for the oxidation reaction can be deduced:
(
)
1
-
a
nE
T
F
D
=
() ()
=
[1.21]
j
cn
Fk
e
R
a
R
a
0
D
E
0
When the cathodic current density (
j
c
) is equal to the anodic current density
(
j
a
), the net current flowing across the electrode-solution interface is zero,
and the net flux of O and R is zero. For this condition, the current
densities represent the equilibrium-exchange current density (
j
0
), given
by:
jj
:
=
j
[1.22]
ca
0
which is associated with the equilibrium potential difference D
E
e
.The dif-
ference between this equilibrium potential D
E
e
and D
E
is called the over-
potential, h:
h=
D
E
[1.23]
e
Equations 1.22 and 1.23 can be combined to obtain an expression for the
net current when a potential (the overpotential) is applied, which is also
known as the Butler-Volmer (BV) equation
71
:
(
)
1 ah
-
n
T
F
-
ah
n
T
F
È
Í
˘
˙
j
=-=
j
j
j
e
-
e
[1.24]
R
R
a
c
0
Figure 1.12 illustrates the relationship between current density and applied
overpotential expressed in the previous equation.
The BV relation is often used in a form where the electrode potential is
referred to by the more accessible equilibrium potential,
E
e
, and where the
exchange current density,
j
0
, replaces the rate constant k
0¢
:
n
F
F
R
n
Ó
˛
(
)
(
)
(
)
1
-
a
DD
EE
-
-
a
DD
EE
-
e
e
j
=
j
e
-
e
R
T
T
[1.25]
0
¢
(
(
)
a
1
-
a
) (
)
j
=
n
Fk
0
c
•
c
•
[1.26]
0
R
O
where
c
•
represents a concentration in the middle of the solution.
(D
E
-D
E
e
), the overpotential, is indicated with the symbol h.To indicate
