Handbook of Solvents - George Wypych - ChemTech - Ventech!

202 Christian Wohlfarth
where:
r size parameter (segment number)
M molar mass
In comparison with Equation [4.4.72], the size parameter remains explicit in the reduced
equation of state. Thus, a simple corresponding-states principle is not, in general, satisfied.
But, in principle, this equation of state is suitable for describing thermodynamic
properties of fluids over an extended range of external conditions from the ordinary liquid to
the gaseous state (it gives the correct ideal gas limit) and also to conditions above the critical
point where the fluid is supercritical. Equation of state parameters for many liquids and liquid/molten
polymers have recently been reported by Sanchez and Panayiotou 275 and for
polymers by Rodgers 262 and by Cho and Sanchez. 263 To extend the lattice fluid theory to
mixtures, appropriate mixing rules are needed. There is a fundamental difficulty here, because
the segment size of any component is not necessarily equal to that of another but the
molecular hard-core volume of a component must not change upon mixing. Consequently,
the segment number of a component in the mixture, ri, may differ from that for the pure
0 276
fluid, ri . But, following the arguments given by Panayiotou, the number of segments may
remain constant in the pure state and in the mixture. This assumption leads to a simpler formalism
without worsening the quantitative character of the model. Thus, the following mixing
rules may be applied:
* *
P* = P ψ + P ψ −ψ
ψ Χ [4.4.81a]
1 1 2 2 1 2 12
*
V* =ΣΣψψ V
[4.4.81b]
i j ij
1/ r =Σψ / r
[4.4.81c]
i i
* * *
where Vii = Viand
Vij provides an additional binary fitting parameter and Equation [4.4.80]
provides the mixing rule for T*. The final result for the residual solvent activity in a binary
polymer solution reads:
~
~ ⎛ ⎞ ~ ⎛ ⎞
residual ⎛ ⎞ ⎜V−1⎟
⎛ ⎞
lna1 = r1⎜V−1⎟ln r V
~ 1
1 1
⎝ ⎠ ⎜ ⎟
− −
⎜ − ⎟
⎜ ⎟
⎝ V
⎝ ⎠ ⎜ ⎟
⎠
⎝ ⎠
−
~
~
V 1 1 V
ln ln
~
~
V 1 V 1
r ⎛ ⎞ ⎛ ⎞
1
− ⎜ 1 1
− ⎟+
⎜
Χ12 r ⎟
1 ψ
~ ⎜ ~ ~ ⎟ ⎜ ~ ⎟
T1 ⎝V
V1
⎠ ⎝ V ⎠
2
2
~ ~ ~
rP
⎛
1
1 V−
⎞
⎜ V1⎟
⎝ ⎠
+ ~
T1
[4.4.82]
The last term in Equation (4.4.82) is again negligible at normal pressures. Various
other approximations were given in the literature. For example, one can assume random
mixing of contact sites rather than random mixing of segments, 277,278 as well as non-random
mixing. 277,279 The model is applicable to solutions of small molecules as well as to polymer
solutions. Like the Prigogine-Flory-Patterson equation of state, the lattice-fluid model and
its variations have been used to correlate the composition dependence of the residual solvent
activity. 277,279 These studies show that again entropic parameter Q 12 and interaction parameter
X 12 have to be fitted to experimental data of the mixture to provide better agreement