Environmental Engineering Reference
In-Depth Information
Surface potential,
Ψ
0
Stern layer
+
+
Diffuse layer,
Ψ
(
x
)
+
+
+
+
+
+
+
+
+
+
+
+
+
FIGURE 3.20
Distribution of charges in a solution near a negatively charged surface.
where
u
=
ze
Ψ
/kT
and
κ
is the inverse of the Debye length defined earlier. For very
small values of
Ψ
0
(
−
25 mV or less for a 1:1 electrolyte), that is,
u
0
1, we can
derive the simple relationship,
Ψ = Ψ
0
e
−κ
x
.
(3.86)
The significance of
κ
is easily seen in this equation. It is the value of
κ
at which
Ψ = Ψ
0
/e
, and hence 1
/
κ
is taken as the
effective double-layer thickness
.
If
u
0
1, that is,
Ψ
0
25 mV (for a 1:1 electrolyte), the result is
4
kT
ze
·
e
−κ
x
,
Ψ =
(3.87)
which tends to a value of
Ψ
0
=
4
kT/ze
as
x
→
1
/
κ
.
The expression for
Ψ
obtained above can be related to the surface charge density
σ
through the equation
2
n
0
ε
∞
sinh
u
0
2
.
d
2
σ =
4
Ψ
d
x
2
d
x
kT
=
(3.88)
π
π
0
For small values of
u
0
we get
σ =
(
εκ
/
4
π
)
Ψ
0
, which is similar to that for a charged
parallel plate condenser (remember undergraduate electrostatics!).
If the definition of Debye length is cast in terms of the following equation,
n
i
z
i
, then the above equation becomes generally applicable to
any type of electrolyte. Note that the ionic strength
I
is related to
2
e
2
/
κ
=
(
4
π
ε
kT)
Σ
n
i
z
i
, since
Σ
n
i
=
(N
A
C
i
/
1000
)
with
C
i
in mol/L. It is useful to understand how the value of
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