Handbook of Solvents - George Wypych - ChemTech - Ventech!

5.5 The phenomenological theory of solvent effects 285
where γ′ = ( γ2 −γ1)/
2. Notice that the general medium and solvation effects are coupled
through the solvation constants K1 and K2. When x2 = 0, eq. [5.5.19] yields ΔGgen med =gAγ1. We interpret this as a quantitative expression
for the hydrophobic effect. In general, eq. [5.5.19] describes the solvophobic effect.
This is a phenomenological description, not a detailed structural description.
5.5.2.5 The total solvent effect
The solution free energy is now obtained by inserting eqs. [5.5.13] and [5.5.19] into eq.
[5.5.4]. We obtain
*
ΔGsoln ( x2) = gAγ1 + ΔGintersol + ΔGWW
+
( γ′ − ln ) + ( 2 γ′
− ln )
gA kT K K x x gA kT K K K K x
2
x1 + K1x1x2 2
+KK 1 2x2 When x 2 = 0, eq. [5.5.20] gives
( 0)
1 1 1 2 1 2 1 2
2
2
[5.5.20]
*
ΔGsoln x2 = = gAγ 1 + ΔGintersol + ΔGWW
[5.5.21]
With the Leffler-Grunwald delta operator symbolism 7 we define
* ( ) ( )
* *
δMΔG = ΔGsoln x2 − ΔGsoln
x2
= 0 [5.5.22]
which, applied to eqs. [5.5.20] and [5.5.21], gives our final result:
δ
MΔsoln ( γ′ − ) + ( 2 γ′
− )
G
gA kT lnK K x x gA kT lnK
K K K x
*
=
2
2
x + Kxx + KKx 1 1 1 2 1 2 1 2
2
2
1
1 1 2 1 2 2
[5.5.23]
The quantity δMΔ Gsoln
*
can be read “the solvent effect on the solution free energy.” Because
of eq. [5.5.1], δMΔ G
* is proportional to the “relative solubility,” log[( x3) x / ( x3)
x 0]
2 2 = , that
is, the logarithm of the solubility in the mixed solvent of composition x2 relative to the solubility
in pure water. The subtraction that yields eq. [5.5.23], a workable equation with just
three unknown parameters (gA, K1, and K2), has also prevented us from dealing with absolute
solubilities.
5.5.3 APPLICATIONS
5.5.3.1 Solubility
It will be no surprise that the first use of eq. [5.5.23] was to describe the equilibrium solubility
of solid nonelectrolytes in mixed aqueous-organic solvents. 1 Equilibrium solubility in
mol L -1 ,c3, is converted to mole fraction, x3, with eq. [5.5.24], where ρ is the saturated solution
density, w is the wt/wt percentage of organic cosolvent, and M1,M2,M3are the molecular
weights of water, cosolvent, and solute. 8
x
3
c3
=
⎛ w
c3( 1000ρ
− c3M3) ⎜
+
⎝M
( 1−w)
M
2 1
⎞
⎟
⎠
[5.5.24]