Handbook of Solvents - George Wypych - ChemTech - Ventech!

5.5 The phenomenological theory of solvent effects 299
But in the preceding paragraph no restriction has been placed on the identity of solvent
component 2, so the conclusion must apply to any cosolvent 2 combined with the common
solvent 1, which is water. This means that all mixed solvent systems in this study as described
should yield the same value of gA. But this is not observed. Indeed, the variation in
gA can be extreme, in a chemical sense; see Table 5.5.2. This constitutes a logical difficulty.
There is another anomaly to be considered. In nearly all of the nonelectrolyte solubility
data that have been subjected to analysis according to eq. [5.5.23] the solute solubility increases
as x2, the organic cosolvent concentration, increases, and gA is positive, the
physically reasonable result. But in the sucrose-water-ethanol system, the sucrose solubility
decreases as x2 increases, and gA is negative. There appears to be no physically reasonable
picture of a negative gA value.
A further discrepancy was noted in 5.5.3.2, where we saw that some of the solvation
constants evaluated from surface tension data did not agree closely with the corresponding
numbers found in solubility studies.
5.5.4.2 A modified derivation
Recognizing that the original condition that g and A are independent of composition was
unnecessarily restrictive, we replace eq. [5.5.18] with eq. [5.5.52], where the subscripts 1
and 2 indicate values in the pure solvents 1 and 2.
��γ = gAγ + ( g Aγ − gAγ ) f
[5.5.52]
1 1 1 2 2 2 1 1 1 2
It is important for the moment to maintain a distinction between gA in the original formulation,
a composition-independent quantity, and � in eq. [5.5.52], a composition-dependent
quantity. Eq. [5.5.52] combines the composition dependence of three entities into a
single grouping, �Âγ, which is probably an oversimplification, but it at least generates the
correct values at the limits of x 2 = 0 and x 2 = 1; and it avoids the unmanageable algebraic
complexity that would result from a detailed specification of the composition dependence
of the three entities separately. Eq. [5.5.14] now is written ΔG gen med = �Âγ and development
as before yields eq. [5.5.53] as the counterpart to eq. [5.5.23], where δ MgAγ =g 2A 2γ 2 -
g 1A 1γ 1.
δ
M
*
ΔGsoln =
( δM γ / − ln ) + ( δM γ − ln )
gA kT K K x x gA kT K K K K x
2 1 1 1 2 1 2 1 2
2
2
2
2
x1 + Kx 1 1x2 + KK 1 2x2 [5.5.53]
Comparison of eqs. [5.5.23] and [5.5.53] gives eq. [5.5.54], which constitutes a specification
of the meaning of gA in the original formulation in terms of the modified theory.
( )
gA γ − γ = g γ A − g γ A
[5.5.54]
2 1 2 2 2 1 1 1
Now, the right-hand side of eq. [5.5.54] is a constant for given solute and solvent system, so
the left-hand side is a constant. This shows why gA in the original theory (eq. [5.5.23]), is a
composition-independent parameter of the system. Of course, in the derivation of eq.
[5.5.23] gA had been assumed constant, and in effect this assumption led to any composition
dependence of gA being absorbed into γ. In the modified formulation we acknowledge