Environmental Engineering Reference
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where A is the contact area and N
i
is the number of atoms (or molecules) of the ith
species, with m
i
(T, a
i
, f
i
) being the electrochemical potential of the corresponding
reservoir at temperature T, activity a
i
, and electrostatic potential f
i
(see above).
Because of the presence of an electrostatic potential, here we have to use electroche-
mical potentials instead of chemical potentials. The sum over i involves metal atoms
(me), excess electrons (e) at the electrodes, anions (a), cations (c), and neutral com-
pounds (n). Then, (5.14) can be written as
a
fg
, f
e
, f
S
)
¼
1
g(T,
½
GT,
ð
a
fg
,
N
fg
Þ
N
n
m
n
(T, a
n
, f
S
)
N
e
m
e
(f
e
)
A
N
me
m
me
(T, a
me
, f
e
)
N
a
m
a
(T, a
a
, f
S
)
N
c
m
c
(T, a
c
, f
S
)
(5
:
15)
In order to evaluate different approximations in the context of modeling electroche-
mical systems [Haftel and Rosen, 2003; Kitchin et al., 2004; Gunnarsson et al., 2004;
Feng et al., 2005; Rossmeisl et al., 2006; Taylor et al., 2006; Jacob, 2007a, b], in the
following, we shall discuss each term of (5.15) separately:
†
The term GT, a
fg
, N
f
ð Þ
is the Gibbs free energy of the full electrochemical
system (x
1
, x , x
2
in Fig. 5.4). It includes the electrode surface, which is influ-
enced by possible reconstructions, adsorption, and charging, and the part of the
electrolyte that deviates from the uniform ion distribution of the bulk electrolyte.
The importance of these requirements becomes evident if we consider the theor-
etical modeling. If the interface model is chosen too small, then the excess
charges on the electrode are not fully considered and/or, within the interface
only part of the total potential drop is included, resulting in an electrostatic poten-
tial value at x
¼
x
2
that differs from the required bulk electrolyte value f
S
.
However, if we constrain such a model to reproduce the electrostatic potential
f
e
at the electrode and f
S
at the far end of the electrolyte region, then the equi-
librated system might differ geometrically and electronically from the realistic
interface. Therefore, the modeled electrolyte region must be sufficiently large
that, without any constraints, a certain “buffer zone” (see Fig. 5.4) at x . x
2
of the electrolyte region self-consistently reaches f
S
. Since, besides temperature
and activities, the Gibbs energy also depends on the composition of the system
(including electrolyte ions and the excess charges on the electrode surfaces),
G implicitly depends on the electrode potential Df.
†
The term N
n
m
n
(T, a
n
, f
S
) accounts for all neutral species, such as water or mol-
ecules involved in an electrochemical reaction. Since these are related to the elec-
trolyte reservoir, the electrostatic potential that appears in m
n
is that of the bulk
electrolyte. Although these species still might have dipole or higher multipole
moments,
the
constancy
of
the
potential
within
the
reservoir
allows
m
n
(T, a
n
, f
S
)
to
be
replaced
by
the
corresponding
chemical
potential
m
n
(T, a
n
), which is potential-independent.
†
The term N
me
m
me
(T, a
me
, f
e
) gives the energy necessary to remove the metal
atoms required to build the electrode from its corresponding bulk electrode
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