Environmental Engineering Reference
In-Depth Information
where
N
i
is the number density of ions, and the electric field strength from an
individual ion is
E
D
e
/
r
2
.Fromthisweobtain
r
e
E
e
3/2
N
i
dE
E
5/2
2
π
N
1/3
i
P
(
E
)
dE
D
,
.
(1.18)
From this one can estimate a typical value
E
0
of the electric field strength:
eN
2/3
i
E
0
.
(1.19)
This value corresponds to the electric field strength at the mean distance between
ions.
We now determine the distribution function
P
(
E
) on the basis of the standard
method with the characteristic function
F
(
g
), that is,
Z
exp(
i
Eg
)
P
(
E
)
d
E
,
F
(
g
)
D
(1.20)
where
g
is a three-dimensional variable. Correspondingly, the inverse transforma-
tion gives
)
3
Z
exp(
1
P
(
E
)
D
i
Eg
)
F
(
g
)
d
g
.
(1.21)
(2
π
Introducing the probability
p
(
E
i
) of a given electric field strength created by the
i
th
ion, we have
Z
p
(
E
i
)
d
E
i
.
Y
P
(
E
)
D
i
Evidently, the probability
p
(
E
i
) is identical for different ions, and the characteristic
function is
Z
exp(
i
E
i
g
)
p
(
E
i
)
d
E
i
.
Y
F
(
g
)
D
f
i
(
g
),
f
i
(
g
)
D
(1.22)
i
Let
n
ions (
n
1) be located in a large volume
Ω
,so
d
r
i
Ω
p
(
E
i
)
d
E
i
D
,
and
d
r
i
is the volume element where the
i
th ion is located with the origin of the
frame of reference that is at a test point. For the partial characteristic function and
using
p
(
E
i
)
d
E
i
D
d
r
i
/
Ω
this gives
Z
exp(
i
E
i
g
)
d
r
i
Z
exp(
i
E
i
g
)
1
d
r
i
,
1
Ω
1
Ω
f
i
(
g
)
D
D
C
1
and this gives
Z
exp(
i
E
i
(
r
i
)
g
)
1
d
r
i
.
1
Ω
ln
f
i
(
g
)
D
