Environmental Engineering Reference
In-Depth Information
where 1 /
κ
is the Debye length given by
kT
i
4p
ε
2
n i z i e 2 .
κ
=
(3.65)
Rewriting n i in terms of molality we have
1000
i
N A e 2
n i z i e 2
m i z i .
=
(3.66)
i
z i ,wehave
BI 1 / 2 where B
N A e 2 /
Since ionic strength I
=
( 1 / 2 )
Σ
m i ·
κ =
=
( 8
π
kT) 1 / 2 . Therefore we have the familiar equation given in Section 3.2.4 as
follows:
1000
ε
I 1 / 2 ,
log
γ ± =−
A(z + z )
·
(3.67)
(N A .e 2 )/( 2.303 2
where A
B . Note that A and B are functions of temperature
and most importantly the dielectric constant of the solvent. For water at 298 K, A
=[
ε
RT)
]
=
0.511 (L/mol) 1 / 2 and B
0.3291Å 1 (L/mol) 1 / 2 . The above equation is called the
Debye-Huckellimitinglaw and has been confirmed for a number of dilute solutions of
electrolytes. Many applications in environmental engineering involve dilute solutions
and this is an adequate equation for those situations. The theory is untenable at high
NaCl concentrations (i.e., high I values or small values of 1 /
=
κ
). Attempts have been
made to correct the Debye-Huckel law to higher ionic strengths.Without enumerating
the detailed discussion of these attempts, it is sufficient to summarize the final results
(see Section 3.2.4 and Table 3.3).
3.4.3.5
Molecular Theories of Solubility: An Overview
The molecular model for the dissolution of nonpolar solutes in water starts by con-
sidering the energy associated with the various stages of bringing a molecule from
another phase (gas or liquid) into water. The Gibbs free energy for the process has
two components (Figure 3.10). The first stage involves the formation of a cavity in
water capable of accommodating the solute. The solute is then placed in the cavity,
whereupon it establishes the requisite solute-solvent interaction. The water then rear-
ranges around the solute so as to maximize its favorable disposition with respect to the
solute.This concept of solubility was suggested by Uhlig (1937) and Eley (1939).The
free energy for solution
Δ G is the sum of the cavity term G c and the solute-solvent
interaction term G t . The work required to stretch a surface is the work done against
the surface tension of the solvent. Therefore, the work required to form a cavity in
water should be the product of the cavity area and surface tension of water. Thus G c
is directly proportional to surface tension. This was verified by Saylor and Battino
(1958) for a nonpolar solute, argon. Choi, Jhon, and Eyring (1970) showed that in
fact it is not the bulk surface tension of the solvent that should be considered but
the microscopic surface tension of the highly curved cavity. The surface tension of a
 
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