Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 151
or in the case of liquid-liquid equilibrium to
I I 0I II II 0II
γ xf = γ x f
[4.4.8]
i
i
i
i
i
i
If the standard state in both phases is the same, the standard fugacities cancel out in
Equation [4.4.8]. Equation [4.4.8] also holds for solid-liquid equilibria after choosing appropriate
standard conditions for the solid state, but they are of minor interest here.
All expressions given above are exact and can be applied to small molecules as well as
to macromolecules. The one difficulty is having accurate experiments to measure the necessary
thermodynamic data and the other is finding correct and accurate equations of state
and/or activity coefficient models to calculate them.
Since mole fractions are usually not the concentration variables chosen for polymer
solutions, one has to specify them in each case. The following three quantities are most frequently
used:
mass fractions wi = mi / ∑mk
[4.4.9a]
volume fractions ϕi nV i i nkVk = / ∑ [4.4.9b]
* *
segment (hard-core volume) fractions ψ i nV i i nkVk = / ∑ [4.4.9c]
where:
mi mass of component i
ni amount of substance (moles) of component i
Vi molar volume of component i
*
Vi molar hard-core (characteristic) volume of component i.
With the necessary care, all thermodynamic expressions given above can be formulated
with mass or volume or segment fractions as concentration variables instead of mole
fractions. This is the common practice within polymer solution thermodynamics. Applying
characteristic/hard-core volumes is the usual approach within most thermodynamic models
for polymer solutions. Mass fraction based activity coefficients are widely used in Equations
[4.4.7 and 4.4.8] which are related to activity by:
Ω i = ai / wi
[4.4.10]
where:
Ωi mass fraction based activity coefficient of component i
ai activity of component i
wi mass fraction of component i
Classical polymer solution thermodynamics often did not consider solvent activities
or solvent activity coefficients but usually a dimensionless quantity, the so-called
Flory-Huggins interaction parameter χ. 44,45 χ is not only a function of temperature (and
pressure), as was evident from its foundation, but it is also a function of composition and
polymer molecular mass. 5,7,8 As pointed out in many papers, it is more precise to call it
χ-function (what is in principle a residual solvent chemical potential function). Because of
its widespread use and its possible sources of mistakes and misinterpretations, the necessary
relations must be included here. Starting from Equation [4.4.1b], the difference between the
chemical potentials of the solvent in the mixture and in the standard state belongs to the first