Environmental Engineering Reference
In-Depth Information
q
A
!
q
!
q
B
. If the interaction is sufficiently strong, electron transfer takes place
every time the system reaches the intersection q
; in this case, the reaction is adiabatic.
On the other hand, if the interaction is weak, electron transfer will seldom occur, and
the reaction is said to be nonadiabatic. In the latter case, the last two terms in the elec-
tronic Hamiltonian (2.1) can be considered as small, and first-order perturbation
theory can be used to obtain the rate. However, if the interaction is strong, the adiabatic
energy surface can be calculated by Green function techniques [Davison and Sulston,
2006], and the rate obtained from Kramers [1940] theory.
In first-order perturbation theory, the rate can be calculated in a straightforward
manner. The rate for a transition from a metal state k to the reactant orbital a is,
after thermal averaging [Schmickler, 1996],
r
p
lk
B
T
v(k
!
a)
¼
j
V
k
j
2
h
exp
(1
a
1
k
)
2
4lk
B
T
(2
:
3)
where k
B
is Boltzmann's constant; a corresponding expression holds for the reverse
process. In order to obtain the total rate constant k
red
for the reduction of an oxidized
reactant one first has to fix the energy scale. For this purpose, the Fermi level of the
metal is usually taken as the energy zero. The energy scale in the solution is then
obtained by noting that at equilibrium the forward and backward rates to the Fermi
level must be equal. This gives 1
a
¼
lat equilibrium. Application of an overpoten-
tial h shifts the energy scales of the metal and the reactant by an amount e
0
h. The
coupling constants are usually taken as constant: V
k
¼
V
¼
const. The total rate is
then obtained by integrating over all occupied states 1
k
. This results in
r
p
lk
B
T
ð
d1 r
m
(1) f (1) exp
(l
þ
1
e
0
h)
2
4lk
B
T
k
red
¼
j
V
j
2
h
(2
:
4)
Here, f (1) denotes the Fermi - Dirac distribution, and the integral over 1
k
has been
converted into an integral over the energy 1 by means of the metal density of states
r
m
(1). The rate at equilibrium has an energy of activation
4
land is thus solely deter-
mined by the energy of reorganization. The interaction between the reactant and the
metal enters only into the pre-exponential factor, and the rate is proportional to the
square of the corresponding coupling constant, as is common for theories based on
first-order perturbation. On bare metals, the electronic interaction is typically too
strong for first-order perturbation theory to be valid. Therefore, (2.4) is usually applied
to electrodes covered by an insulating film such as oxides or organic layers.
When the electronic interaction is sufficiently strong, first-order perturbation theory
no longer appplies. The reaction proceeds adiabatically, and the system is in electronic
equilibrium for all solvent configurations. In order to calculate adiabatic potential
energy surfaces, the solvent coordinates q
n
are replaced by a single effective coordi-
nate q; this is permissible as long as all modes are classical. Further, it is convenient
to normalize q by the transformation: q
¼
q
=
g; the transformed variable has a simple
interpretation: a solvent configuration q would be in equilibrium with a reactant of
Search WWH ::
Custom Search