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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