Handbook of Solvents - George Wypych - ChemTech - Ventech!

284 Kenneth A. Connors
A similar cycle yields eq. [5.5.12].
ΔG = ΔG − ΔG
=−kTln K K
[5.5.12]
2 MM WW
1 2
Combination of eqs. [5.5.8] - [5.5.12] then gives
ΔG
solv
( ln ) ( ln )
− kT K K x x + −kT
K K K K x
=
2
2
x + K x x + K K x
1 1 1 2 1 2 1 2
2
2
1
1 1 2 1
2 2
+Δ [5.5.13]
Obviously when x2 =0,ΔGsolv = ΔGWW. 5 Eq. [5.5.13] is the desired expression relating
the solvation energy to the solvent composition.
5.5.2.4 The general medium effect: solvent-solvent interactions
Here we make use of Uhlig’s model, 6 writing eq. [5.5.14] as the energy required to create a
molecular-sized cavity in the solvent.
G ww
ΔGgen med = gAγ
[5.5.14]
In eq. [5.5.14] g is a curvature correction factor, an empirical quantity that corrects the
conventional surface tension γ for the curvature of the cavity needed to contain a solute molecule.
A is the surface area of this cavity in � 2 molecule -1 ; in Sections 5.5.3 and 5.5.4 we
treat the meaning of A in more detail, but here we only make the assumption that it is essentially
constant, that is, independent of x 2.
There is a subtlety in assigning the value of γ, for implicit in our model (but treated
more fully in reference 1) is the condition that the γ of eq. [5.5.14] is the surface tension of
the cavity surface at its equilibrium composition. But this is the composition of the solvation
shell immediately adjacent to the molecule, and this is in general different from the composition
(x 1,x 2) of the bulk solvent mixture. Let f 1 and f 2 be the equilibrium mean fractional
concentrations of water and cosolvent, respectively, in the solvation shell, so f 1 +f 2=1.
These fractions are defined
1
f1( FWW FWM
)
2 2 = + [5.5.15]
1
f2= ( FWM 2
+ 2FMM
)
[5.5.16]
Now we define, for use in eq. [5.5.14],
γ = γ f + γ f [5.5.17]
11 2 2
( )
γ = γ + γ −γ
1 2 1 2
f [5.5.18]
where γ 1 and γ 2 are the surface tensions of pure component 1 (water) and 2, respectively.
Combining eqs. [5.5.14, 5.5.16, and 5.5.18] gives for the general medium effect
gAγ′ K x x + 2gAγ′
K K x
ΔGgen med = gAγ1
+
2
2
x + K x x + K K x
1
1 1 2 1 2
2
2
1 1 2 1
2 2
[5.5.19]