Chemistry Reference
In-Depth Information
2.1 INTRODUCTION
Traditionally, the properties of mixtures were studied by examining the various
excess quantities, such as the excess free energy, excess entropy, enthalpy, volume,
and so forth (Prigogine with contributions from A. Bellemans and V. Mathot 1957;
Rowlinson and Swinton 1982). These quantities convey
global
properties of the mix-
tures. They are global in the sense that they convey information on the
macroscopic
properties
of the mixture. Recently, an alternative approach to the study of the prop-
erties of liquid mixtures at a
local
level has been suggested. This view is based on
the inversion of the Kirkwood-Buff (KB) theory of solutions (Kirkwood and Buff
1951; Ben-Naim 1977, 1987, 1988, 1989, 1990b, 1992, 2006) on one hand, and on
the solvation thermodynamics on the other hand. This approach is more informative,
since it provides
microscopic information
on the surroundings of a single molecule
in the mixture.
There are essentially three significant quantities that can be derived from the
inversion of the KB theory. The first is a measure of the extent of deviation from
symmetrical ideal (SI) solution behavior, Δ
AB
, defined below in the next section.
It also provides a necessary and sufficient condition for SI solution. The second is
a measure of the extent of
preferential solvation
(PS) around each molecule. In a
binary system of A and B, there are only two independent PS quantities; these mea-
sure the preference of, say, molecule A to be solvated by either A or B molecules.
Deviations from SI solution behavior can be expressed in terms of either the sum or
difference of these PS quantities. Finally, the Kirkwood-Buff integrals (KBIs) may
be obtained from the inversion of the KB theory. These provide information on the
affinities
between any two species; for instance, ρ
A
G
AA
measures the excess of the
average number of
A
particles around
A
relative to the average number of A particles
in the same region chosen at a random location in the mixture. All these quantities
can be obtained from the KB integrals.
The second group consists of the solvation thermodynamic quantities. These
quantities also convey local information on the average interaction free energy of
a particular molecule with its surroundings, the effect of the solute on the solvent
structure, and so forth (Ben-Naim 2009, 2011).
In the next section, the local quantities are defined and a brief discussion of the
procedure for calculating these quantities is described. In subsequent sections, we
present a complete set of results for a one-dimensional mixture of two components.
2.2 DEFINITIONS OF THE GLOBAL AND LOCAL QUANTITIES
The fundamental
global
quantity is the excess Gibbs energy of the entire mixture.
For a two-component system, this is defined by
E
G
NN
E
o
o
G
=
=
x
(
µµ
−
−
k Tx
ln)
+
x
(
µµ
kkT
−−
ln
x
)
(2.1)
m
AA
A
B
A
B
BB
B
B
(
+
)
A
B
where μ
A
and μ
o
A
are the chemical potentials of
A
in the mixture, and in the pure state
at some fixed temperature
T
, and pressure
p
. The excess is defined with respect to a
