Handbook of Solvents - George Wypych - ChemTech - Ventech!

294 Kenneth A. Connors
the three quantities on the right-hand side of eq. [5.5.42]. Thus the problem is solved in
principle. 18
In practice, of course, there are difficulties. Each of the δ MΔG soln terms contains three
adjustable parameters, for nine in all, far too many for eq. [5.5.42] to be practicable in that
form. We therefore introduce simplifications in terms of some special cases. The first thing
to do is to adopt a 1-step model by setting K 2 = 0. This leaves a six-parameter equation,
which, though an approximation, will often be acceptable, especially when the experimental
study does not cover a wide range in solvent composition (as is usually the case). This
simplification gives eq. [5.5.43].
δ
C C C
S
gA γ kT K K x gA γ kT
Δ G
C
x K x
* S S
( ′ − ln ) ( ′ 1 1 2 − lnK
) K x
=
−
S
+
x + K x
M comp
1 1 2
L L L
( gA ′ −kT
ln K ) K x
−
L
x + K x
γ 1 1 2
1 1 2
1 1 2
1 1 2
[5.5.43]
Next, in what is labeled the full cancellation approximation, we assume K 1 C =K1 S =K1 L =
K 1and we write ΔgA=gA C -gA S -gA L . The result is
δ
ΔG
*
M comp
( ln + Δ γ′
)
kT K gA K x
=
x + K x
1 1 2
1 1 2
[5.5.44]
and we now have a 2-parameter model. The assumption of identical solvation constants is
actually quite reasonable; recall from the solubility studies that K 1 is not markedly sensitive
to the solute identity.
The particular example of cyclodextrin complexes led to the identification of another
special case as the partial cancellation approximation; in this case we assume K 1 C =K1 S <
K 1 L , and the result is, approximately. 19
δ
M comp
( − γ′
)
kT K gA K x
Δ G
x K x
* ln
=
+
1 1 2
1 1 2
[5.5.45]
Functionally eqs. [5.5.44] and [5.5.45] are identical; the distinction is made on the basis
of the magnitudes of the parameters found. Note that gA in eq. [5.5.45] is a positive
quantity whereas ΔgA in eq. [5.5.44] is a negative quantity. In eq. [5.5.45] it is understood
that gA and K 1 refer to L. Eqs. [5.5.44] and [5.5.45] both have the form
δ
M comp
( + γ′
)
kT K G K x
Δ G
x K x
* ln
=
+
1 1 2
1 1 2
[5.5.46]
where G = ΔgA in eq. [5.5.44] and G = -gA in eq. [5.5.45]. Table 5.5.5 shows G and K 1 values
obtained in studies of α-cyclodextrin complexes. 19,20 The assignments are made on the
basis of the magnitude of K 1; those values substantially higher than typical solubility K 1 values
suggest that the full cancellation condition is not satisfied. After the assignments are
made, G can be interpreted as either ΔgA (full cancellation) or -gA (partial cancellation).