Page 29 - Partition & Adsorption of Organic Contaminants in Environmental Systems

20 FUNDAMENTALS OF THE SOLUTION THEORY

The inability of Raoult’s law to account for the solution behavior of a
macromolecular substance is illustrated in the next example. Consider a binary-
component system in which the two components (1 and 2) have similar struc-
tures and compositions (e.g., styrene and polystyrene) and their molecuclar
weights are 100 and 10,000 daltons, respectively. Assume that the solution is
made 10% by weight of component 1 and 90% by weight of component 2. The
mole fractions calculated for components 1 and 2 are thus x 1 = 0.917 and x 2 =
0.083. Since in this case x 1 is close to 1,Raoult’s law would suggest that g 1 should
approach 1 and thus P 1 should be close to P° 1. However, the measured P 1 in this
case is only about 0.1P° 1 instead. To reconcile this deviation on the basis of
Raoult’s law, one would be forced to assume that g 1 = 0.11 (i.e., g 1 < 1), but this
assumption cannot be justified because there is no evidence for a specific inter-
action between the two molecularly similar components.
Consider a binary solution consisting of a small molecule with a molar
.
volume of V 1 and a linear polymer of a molar volume of , in which V 2 >>V 1
V 2
By a statistical treatment of the number of spatial configurations that compo-
nent 2 may assume in mixing with component 1, Flory (1941) and Huggins
(1942) developed a thermodynamic expression for the free energy of mixing
between components 1 and 2. The chemical activities of components 1 and 2
are given as

lna 1 = lnf 1 + ( 1- V V 2 )f 2 + c f 2 2 (2.10)
1
1
and

lna 2 = lnf 2 - ( V V 1 - 1)f 1 + c 1 ( V V 1 )f 1 2 (2.11)
2
2
where a i is the activity of component i, f i is the volume fraction (where f 1 +
f 2 = 1), and c 1 is the Flory–Huggins interaction parameter for component 1
[i.e., the sum of its excess enthalpic (c H) and entropic (c S) contributions to its
incompatibility with component 2]. The c H term accounts for the heat of
mixing, similar to the lng term in Raoult’s law. For systems with completely
linear and flexible polymer segments, the entropy of mixing for components 1
and 2 is given by the first two terms to the right in Eqs. (2.10) and (2.11). The
c S term corrects for the entropy loss upon mixing when the polymer suffers
certain restriction on its orientation. Thus, c S is approximately 0 if the polymer
segments are highly flexible to adopt a large number of spatial orientations.
/ ) term may be viewed as the c 2 term for component
In Eq. (2.11), the c 1 (V 2 V 1
2. As seen, if there is no molecular-size disparity between the two components,
), Eqs. (2.10) and (2.11) are then reduced to Raoult’s law, since in
(i.e.,V 1 = V 2
this case x 1 f 1 , x 2 f 2 , and c= lng.
As seen later, Eq. (2.10) offers a more general account of the activity of
an organic solute with natural organic matter and biological lipids, where a
V 0, Eq. (2.10)
moderate-to-large molecular-size disparity is observed. If /V 21
then becomes