190
4 – Electrode reactions
or
F
i
D z
X
2
z 0
2
2
= −
=
`
j
and
z
X
FSD
I
2
z 0
2
2
= −
=
`
j
3. a. Given the relation X = X
0 + ∆X (z) e
jωt
, we start by differentiating X with
respect to time
t
X
j X e
(z)
j t
2
2
ω Δ
=
ω
D z
X
D
z
X e
2
2
2
2
(z) j t
2
2
2
2 Δ
=
ω
Applying Fick’s second law gives
t
X
D z
X
2
2
2
2
2
2
=
so we obtain
D
z
X
j X
2
2
(z)
(z)
2
2
ω
Δ
Δ
=
which we write in the form
z
X
D
j
X
0
2
2
(z)
(z)
2
2
ω
Δ
Δ
−
=
Integration of this differential equation gives
X
e
e
(z)
z
z
D
j
D
j
α
β
Δ
=
+
#
#
−
~
~
We show that β = 0. For z = 0, this gives
∆X (0) = α
Moreover, we have
z
X
z
X
e
(z)
j t
2
2
2
2Δ
=
ω
and
z
X
e
(z)
D
j
z
D
j
2
2
α
Δ
= −
#
ω −
~
which gives, for z = 0,
z
X
e
2FSD
I
z 0
D
j
j t
2
2
α
= −
= −
ω
ω
=
`
j
By replacing α by ∆X (z = 0) , we obtain
X
e
2FS j D
I
(z 0)
j t
ω
Δ
=
ω
=
Identification with the equation X = X
0 + ∆X (z) e
jωt
gives
X X
2FS j D
I
0
ω
=
+