Chemistry Reference
In-Depth Information
Sodium Ions
Chloride Ions
Distance
FIGure 4.10
Variation of Na
+
and Cl
-
ions near the charged surface.
C
s
Cl−
= C
Cl−
[exp + (εψ/kT)]
(4.19)
where C
Na+
and C
Cl−
are the number of sodium and chloride ions per milliliter, respec-
tively, in the bulk phase. The magnitude of ψ varies with x as shown in Figure 4.10,
from its maximum value, ψ
o
, at plane AA′. From the foregoing equations, we find
that the quantities C
s
Na+
and C
s
Cl−
will decrease and increase, respectively, as the dis-
tance x increases from the interface, until their values become equal to C
Na+
and C
Cl−
,
where ψ is zero. The variation of C
s
Na+
and C
s
Cl
− with x are given in Figure 4.10. The
extended region of x between AA′ and BB′ in Figure 4.10 may be termed the diffuse
or the Gouy-Chapman double layer.
The volume density of charge [per milliliter] at a position within AA′ BB′ may be
defined as equal to
ε [c
s+
− c
s−
]
(4.20)
can be expressed by the Poisson relation
d
2
ψ/d
2
x = −[4 π]/D
(4.21)
In this derivation, it is assumed that the interface is flat, such that it is sufficient to
consider only changes in ψ in the x direction normal to the surface plane.
The following relation can be derived from this:
[d5 ψ/d x5] = d[dψ/dx]/dx
(4.22)
= −[4 π N ε/1000 D] c
[e
−ε/kT
e
+εψ/kT
]
(4.23)
where c is the bulk concentration of the electrolyte.
In a circle of unit surface area on the charged plane A-A′, the negative charges
acquired by the adsorbed organic ions (amphiphiles) within this area represent the
surface charge density σ:
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