Environmental Engineering Reference
In-Depth Information
Furthermore, one should consider the following thermodynamic relations
µ
1
|
Z
1
|µ
3
µ
M
and
(5.60)
µ
x
|
Z
2
|µ
3
µ
2
where
chemical potentials of cations, anions, electrons,
metal atoms, and oxidant atoms, respectively;
Z
1
and
Z
2
are valences of metal
and nonmetal atoms. Thereafter one may utilize the Gibbs-Duhem equation as
given by
µ
1
,
µ
2
,
µ
3
,
µ
M
,
µ
x
n
M
d
µ
M
n
X
d
µ
X
0
(5.61)
along with the stoichiometric relation:
n
M
n
X
|
Z
2
|
(5.62)
|
Z
1
|
where
n
M
and
n
X
represent number of kg atoms of metal and oxidant, respectively.
Finally, the parabolic rate equation as derived by Wagner takes the following
form in which the roles played by both ionic and electronic defects and their
corresponding mobilities become apparent:
(o)
X
µ
1
A
d
n
d
t
1
Z
2
|
1
ξ
k
r
ξ
(
t
1
t
2
)
t
3
σ
d
µ
X
(5.63)
F
2
|
(i)
X
µ
where
d
n
/d
t
the rate expressed in kg equivalent s
1
,
t
1
,
t
2
,
t
3
transport numbers of cations, anions, and electrons (or positive
holes), respectively,
A
area of the oxide perpendicular to the direction of the diffusing
species (or area of the sample, m
2
),
F
Faraday number
96,500
10
3
coulomb (kg equiv.)
1
,
d
µ
X
difference in chemical potentials of oxidant, expressed in Joules
(kg atom)
1
,
(o)
X
and
(i)
X
µ
µ
chemical potentials of nonmetal (oxidant) at the oxide-oxygen
and metal-oxide interfaces, respectively,
σ
total electrical conductivity of the scale or oxidation product, in
S
⋅
m
1
,
ξ
instantaneous thickness of the product scale, and the term in pa-
rentheses is the rational rate constant (
k
r
) expressed in kg equiva-
lent m
1
s
1
, i.e.,
