134
3 – Transport in ionic solids
c. The concentration of potassium in its normal position is
[K ]
M
K
ρ
=
#
where ρ and M denote the density and molar mass of KCl, respectively.
Inserting the corresponding values gives
[K ]
M
74.56
1.984
K
ρ
=
=
#
[ ]
.
K
m ol cm
2 66 10
K
2
3
#
=
−
−
#
Given that the concentration of potassium vacancies is equal to that of
barium, we obtain
[V ′
K ] = x # [K
#
K ]
Numerical evaluation gives
[V ′
K ] = 1.4 # 10
−4 # 2.66 # 10
−2
[V ′
K ] = 3.72 # 10
−6
mol cm
−3 = 3.72 # 10
−3
mol L
−1
2. At 440 °C (T = 713 K and 1/T [K] = 1.4 # 10
−3
), we obtain from figure 44
log σ K = − 5.5, so the conductivity σ K = 3.16 # 10
−6
S cm
−1
.
Given that
F u [V ]
K
2
V
K
σ =
#
#
l
u
the electrochemical mobility u V
u of a potassium vacancy is obtained from
the relation
F [V ]
u V
2
K
K
σ
=
l
u
with
u
z F
u
=
u
Numerical evaluation gives
.
.
u
96 480 3 72 10
3 16 10
V
2
6
6
#
#
=
−
−
#
u
.
u
J s mol cm
9 12 10
V
11 1 1
2
#
=
−
− −
u
Using this result in the Nernst-Einstein relation gives
D
u RT
V
V
= u
.
.
D
912 10
8 314 713
V
11
#
=
− #
#
.
D
c m s
5 4 10
V
7
2 1
#
=
−
−
3. a. At 440 °C the diffusion coefficient for the K
+
cation is
D
e
2 10
K
5
.
.
.
.
8 314 713
0 76 1 6 10
602 10
19
23
#
=
− −
#
#
#
#
#
−
*