Environmental Engineering Reference
In-Depth Information
5.4.5 Rules for Writing Defect Reaction
In formulating any type of defect formation reaction, it is essential to have mass
balance and charge balance, i.e., the electroneutrality condition must be satisfied.
For evaluating the defect concentrations in a crystal, the defect equilibria are
to be treated in terms of the law of mass action. Use of this law implies the
applicability of Boltzmann statistics, i.e., random distribution of defects are con-
sidered to be valid. However, at very large concentration of defects this approach
breaks down. Moreover, at very low temperatures there will be every likelihood
of defect association and less probability for the presence of free defects.
Schottky Disorder Formation
This type of defect creation occurs as per the following reaction:
M
M
O
O
V
x
M
V
O
M
M
O
O
(at the surface of the oxide lattice)
The net reaction is represented by
Null
V
x
M
V
O
(5.7)
where null designates a perfect crystal of unit activity. Under the situation of
δ
1, Henry's law is applicable. However, it is convenient to use concentration
units in terms of number of defect per cubic meter of the crystal represented by
[V
x
M
] and [V
O
] (atom fraction can also be used). Then, one can write from Eq. 5.7:
∆
G
1
kT
[V
x
M
][V
O
]
K
S
K
1
exp
(5.8)
It is further known that
∆
G
1
RT
ln
K
S
∆
H
1
T
∆
S
1
Therefore,
∆
S
1
k
∆
H
1
kT
K
1
exp
exp
K
S
(5.9)
Frenkel Disorder Formation
This can be presented by either
MO
M
1
δ
O
δ
V
M
δ
M
i
(5.10)
or, simply,
M
M
M
i
V
x
M
