Chemistry Reference
In-Depth Information
4.3.3 P
reFerenTial
s
olvaTion
The availability of the local compositions allows one to calculate a quantity called
preferential solvation
. There is no generally established definition of preferential
solvation (see Section 3.1 in Chapter 3). For a binary mixture, the excess local
molar fraction has been usually considered synonymous with preferential solva-
tion: the more positive δ
x
ji
, the higher the preference of
i
to be solvated by
j
than by
the other species. However, in a ternary system, where the central particle can be
solvated by three different species, this might be insufficient; δ
x
ji
may be positive,
but less positive than say δ
x
ki
: in this case, the preference of
i
is
k
>
j
>
i
. On this
basis, as suggested in a previous work (Matteoli and Lepori 1995), a quantitative
measure of the preferential solvation, psi
i
j,k
, of solvaton
i
with respect to solvatant
species
j
and
k
can be assumed to be the difference of the two corresponding excess
local mole fractions,
ps
i
jk
,
=δδ
x
x
(4.26)
ji
ki
The more positive the value of psi
i
j,k
is, the larger the preference by
i
to be solvated
by
j
than by
k
. To apply Equations 4.25 and 4.26, a procedure to calculate
V
cor
must
be devised. This is illustrated in the next section. A procedure to obtain preferential
solvation in the form of a “limiting coefficient of preferential solvation” without the
need of calculating
V
cor
is illustrated in Section 1.3.4 in Chapter 1. Although we have
followed this procedure in previous work, in this chapter we have preferred to cal-
culate the excess mole fractions to show in a more realistic way what happens to the
microscopic structure of the mixtures considered here.
4.4
CALCULATION ALGORITHM
4.4.1 K
irKwood
-B
uFF
i
inTegrals
It is evident that even with the simplest expressions for
G
E
and
V
E
, like the Redlich-
Kister equation with only one parameter
p
for each binary and no ternary param-
eters, it is not a simple process to derive explicit analytic expressions for each
G
ij
=
f
(
x
i
,
p
i
). While it is possible to obtain the expressions for ln γ
i
from
G
E
using Equation
4.21, the subsequent steps, which concern the derivatives in Equations 4.20 and their
combinations in Equation 4.15 through Equation 4.18 and again in Equations 4.13
and 4.12, are so as to discourage following this route. In addition, the final equations
would depend on the type of the
G
E
and
V
E
expressions used to represent the experi-
mental data. For these reasons, we have developed a software application which,
taking as input the analytical forms and the parameters of the equations for
G
E
and
V
E
, calculates all six
G
ij
at any given values of
x
1
,
x
2
. The trick used to avoid algebraic
treatment of the many derivatives required is to approximate their exact value at
each
x
1
,
x
2
pair with the corresponding incremental ratio, ∂
f
/∂
x
≅ [
f
(
x
+ δ
x
) -
f
(
x
)]/δ
x
.
Using a particularly small value of the increment δ
x
, the error in the final
G
ij
values
is absolutely negligible. For the benefit of the reader who would like to write his/her
own program, the main steps of this application are now explained.
