Colloidal Systems - Surface and Colloid Chemistry: Principles and Applications - page 147

hand, the oppositely charged ions, negative, will be strongly attracted toward the
surface. This produces the so-called Boltzmann distribution:

n − = n o e (z εe ψ/k T) (7.6)

n + = n o e −(z εe ψ/k T) (7.7)

This shows that positive ions are repelled, while negative ions are attracted to the
positively charged surface. At a reasonably far distance from the particle, n + = n − (as
required by the electroneutrality). Through some simple assumptions, one can obtain
an expression for ψ ( r ), as a function of distance, r , from the surface as

ψ ( r ) = z e/( D r ) εe − κ r (7.8)

where κ is related to the ion atmosphere around any ion. In any aqueous solution,
when an electrolyte, such as NaCl, is present, it dissociates into positive (Na + ) and
negative (Cl − ) ions. Because of requirement of electroneutrality (i.e., there must be
same positive and negative ions), each ion is surrounded by an appositively charged
ion at some distance. Obviously, this distance will decrease with increasing concen-
tration of the added electrolyte. The expression 1/κ is called the Debye length.
As expected, the D-H theory tells us that ions tend to cluster around the central
ion. A fundamental property of the counterion distribution is the thickness of the ion
atmosphere. This thickness is determined by the quantity Debye length or Debye
radius (1/κ). The magnitude of 1/κ has dimension in centimeters, as follows:

κ = ((8 N 2)/(1000 k B T)½ I 1/2 (7.9)

The values of k B = 1.38 10 −23 J/molecule K, e = 4.8 10 −10 esu. Thus, the quantity k B T/e
= 25.7 mV at 25°C. As an example, with a 1:1 ion (such as NaCl, KBr) with concen-
tration 0.001 M, one gets the value of 1/κ at 25°C (298 K):

1/κ = (78.3 1.38 10 −16 298)/(2 4 Π 6.023 10 17 )(4.8 10 −10 ) 2 ) 0.5

= 9.7 10 −7 cm

= 97 Å (7.10)

The expression, in equation, can be rewritten as

ψ (r) = ψ o (r) exp(−κ r ) (7.11)

which shows the change in ψ (r) with the distance between particles (r). At a dis-
tance 1/κ, the potential has dropped to ψ o . This is accepted as corresponding with
the thickness of the double layer. This is an important analysis since the particle-
particle interaction is dependent on the change in ψ (r). The decrease in ψ (r) at the
Debye length is different for different ionic strength (Figure 7.6).