Page 31 - Partition & Adsorption of Organic Contaminants in Environmental Systems

22 FUNDAMENTALS OF THE SOLUTION THEORY
•
A = logg 1 (at x Æ ) 0 (2.19)
1
and
•
B = logg 2 (at x Æ ) 0 (2.20)
2
Thus, in the limit of x 2 Æ 0 (i.e., x 1 Æ 1),

logg 2 Æ logg 2 • and logg 1 Æ 0 (i.e., g 1 Æ 1) (2.21)

Similarly, as x 1 Æ 0 (i.e., x 2 Æ 1),

log g 1 Æ log g 1 • and log g 2 Æ 0 (i.e., g 2 Æ 1) (2.22)

Equation (2.21) or (2.22) is simply Raoult’s law, which must be satisfied. If x 2
is small (<<1),

•
logg 2 B logg 2 constant (2.23)
Therefore, for a substance at dilution, Henry’s law [i.e., Eq. (2.7)] is also
satisfied. If the x i of a substance is small in a solution, the logg i varies with
-2
(1 + x i ) according to Eqs. (2.17) and (2.18). In this case, the variation of g i
with x i will not be substantial. The van Laar equations are best suited for
systems that exhibit positive deviations from Raoult’s law. Whereas logg 1 and
logg 2 vary with the solution composition, their values cannot change in sign
with composition (i.e., the numerical value of g 1 or g 2 cannot undergo a tran-
sition from above 1 to less than 1).
As noted with the Flory–Huggins theory, the relations above hold
mainly for systems where components 1 and 2 have comparable molecular
sizes. For systems where the size disparity is large, the logg term should
be replaced by c/2.303 and the mole fraction (x) by the volume fraction
(f) in the van Laar equations. However, the conclusion regarding the vari-
ability of c with concentration would remain the same as that of logg with
concentration.

2.6 MOLAR HEAT OF SOLUTION

For solid and liquid solutes that have limited solubility in a given solvent, the
partial molar heat of solution for the solute may be calculated from the tem-
perature dependence of the solute solubility. To derive this property, we begin
with the partial molar free energy (i.e., the chemical potential) of the solute
in solution with respect to some standard state at constant pressure (P) and
temperature (T). One recalls from Eq. (1.27) that when a substance with two
or more phases is brought to equilibrium, the chemical potentials of the sub-