Global and Local Properties of Mixtures - Fluctuation Theory of Solutions - page 53

∞ for

for
for

<
≤< +
>+

r
r

σ

() = Ur ε σ σδ

σσ

(2.77)

0 r

We have seen that systems of hard rods form SI solutions. Therefore, all the excess
thermodynamic quantities are zero. We have already examined the dependence of the
local properties on the ratio of diameters in Section 2.6. Therefore, in this section, we
choose equal diameters for the particles σ AA = σ BB = 1, and explore the dependence
of the thermodynamic properties of the mixture on the ratio of the energy parameter
ε. In the succeeding calculation, we choose dimensionless parameters,

σσσ === 1

AA AB BB

ε =− 1 (2.78)

AA

ε = ε ε = ε ε

BB AB AA BB

and choose T = 1 and p = 1 for the numerical illustrations.
Figure 2.8 shows the volume per particle as a function of the mole fraction x A for
various values of the energy parameter ε. Since we have chosen equal diameters
for the particles, the case ε = -1 corresponds to a SI solution. Since T and p are kept
constant, the volume of the system increases or decreases according to the corre-
sponding increase or decrease of |ε|.
Figure 2.9 shows the excess Gibbs energy of the mixture for the same set of
molecular parameters at ε = −1; we have a SI solution and g E = 0. As |ε| either
increases or decreases, we find positive deviations from SI behavior.

1.8

2.00

ε = -1

ε = -0.001

1.7

1.95

1.6

ε = -3.5

ε = -0.25

1.90

1.5

ε = -0.5

1.4

1.85

ε = -6

ε = -0.75

1.3

1.80

1.2

ε = -1.0

1.75

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

x A x A

FIGURE 2.8 The volume of a system of A and B particles interacting via a square-well
potential (Equation 2.77) with parameters as in Equation 2.78 for different values of ε.