Handbook of Solvents - George Wypych - ChemTech - Ventech!

196 Christian Wohlfarth
From the historical point of view and also from the number of applications in the literature,
the common method is to use activity coefficients for the liquid phase, i.e., the polymer
solution, and a separate equation-of-state for the solvent vapor phase, in many cases the
truncated virial equation of state as for the data reduction of experimental measurements explained
above. To this group of theories and models also free-volume models and lattice-fluid
models will be added in this paper because they are usually applied within this
approach. The approach where fugacity coefficients are calculated from one equation of
state for both phases was applied to polymer solutions more recently, but it is the more
promising method if one has to extrapolate over larger temperature and pressure ranges.
Theories and models are presented below without going into details and without
claiming completeness, since this text is not dedicated to theoretical problems but will only
provide some help to calculate solvent activities.
4.4.4.1 Models for residual chemical potential and activity coefficient in
the liquid phase
Since polymer solutions in principle do not fulfill the rules of the ideal mixture but show
strong negative deviations from Raoult’s law due to the difference in molecular size, the
athermal Flory-Huggins mixture is usually applied as the reference mixture within polymer
solution thermodynamics. Starting from Equation [4.4.11] or from
where:
0 n mixG
RT lna1 = RT lnx
1 1 = 1 = 1 − 1 =
n1
⎛∂
Δ ⎞
γ Δμ
μ μ ⎜
⎟
⎝ ∂ ⎠
T, P, nj≠1
[4.4.67]
a1 activity of the solvent
x1 mole fraction of the solvent
γ1 μ1 0
μ 1
R
activity coefficient of the solvent in the liquid phase with activity a1 =x1γ1 chemical potential of the solvent
chemical potential of the solvent at standard state conditions
gas constant
T absolute temperature
n1 amount of substance (moles) of the solvent
n total amount of substance (moles) in the polymer solution
Δ mixG molar Gibbs free energy of mixing,
the classical Flory-Huggins theory 46,47 leads, for a truly athermal binary polymer solution,
to:
or
where:
athermal athermal
ln a1 1 / RT ln(
1 2) 1 2
r
1 ⎛ ⎞
= Δμ = − ϕ + ⎜ − ⎟ϕ
[4.4.68a]
⎝ ⎠
lnγ ln 1 1 ϕ ϕ
1
1 1
athermal ⎡ ⎛ ⎞ ⎤ ⎛ ⎞
= ⎢ −⎜ − ⎟
⎣ ⎝ r
⎥ + ⎜ − ⎟
⎠ ⎦ ⎝ r ⎠
1 2 2
ϕ 2
volume fraction of the polymer
[4.4.68b]