Global and Local Properties of Mixtures - Fluctuation Theory of Solutions - page 37

symmetric ideal solution. There are other excess quantities defined with respect to
an ideal gas mixture, or with respect to ideal dilute solution of A in B (Ben-Naim
1992, 2006). Each of these excess quantities measures different properties of the
solution. In our case, G E measures the deviation from similarity , (Ben-Naim 1992,
2006) between A and B.
Once we have the excess Gibbs energy, either from experiments or from theo-
retical calculations, we can derive all other excess quantities from the relationships
provided in Section 1.1.1 in Chapter 1. Other quantities such as excess heat capacity,
excess compressibility, and so forth, can also be derived by standard methods (Ben-
Naim 1992, 2006).
We next turn to the local quantities. The most fundamental quantities to be stud-
ied are the KBIs (Kirkwood and Buff 1951; Ben-Naim 2006), G αβ , defined as

∞
∫ [

2

G = g ( )14 r − π r dr (2.2)

αβ αβ

0

where g αβ is the (angular averaged) pair correlation function and the integration is
extended over the entire macroscopic volume of the system. It should be stressed
that these functions are defined in the open system with respect to both A and B. In
a closed system, the normalization of the pair correlation functions is

closed

G αβ =− δρ (2.3)

αβ α

where ρ α is the number density of the species α. The significance of G αβ in Equation 2.2
as a local property stems from the following considerations: ρ α g αβ ( r )4π r 2 dr is the
average number of α particles in a spherical shell of radius r and width dr , around
a β particle, and ρ α 4π r 2 dr is the average number of α particles in the same volume
element 4π r 2 dr chosen at a random point in the system. Therefore, the integral,

R

cor

∫

2

ρ GR ( ) = ρ [ gr ( )] − 14 π rdr (2.4)

ααβ cor α αβ

0

provides a measure of the change in the average number of α particles in a spherical
region of radius R cor brought about by placing a β particle at the center of this region.
The significance of G αβ ( R cor ) as a measure of the local properties around a β par-
ticle follows the meaning of the pair correlation function. For most systems of inter-
est (excluding solid solutions or systems near the critical point), g αβ ( r ) approaches
unity (that is, no correlation) for r values of the order of a few molecular diameters;
hence, R cor in Equation 2.4 can be replaced by infinity, as in Equation 2.2. Thus,
the major contribution to G αβ arises from a small local region around the β particle.
Positive ρ α G αβ values indicates that when a β particle is placed at the center of
the correlation volume (i.e., the volume 4

3
π R cor / ), an excess of α particles will be
attracted around β, compared to the average number of particles in the same volume
chosen at a random point in the system. Because of this property and because G αβ

3