Handbook of Solvents - George Wypych - ChemTech - Ventech!

150 Christian Wohlfarth
where:
μ i chemical potential of component i
R gas constant
P 0 and x 0 denote the standard state pressure and composition. For binary polymer solutions
the standard state is usually the pure liquid solvent at its saturation vapor pressure at T.
The standard state fugacity and the standard state chemical potential of any component i are
0 0
abbreviated in the following text by their symbols fi and μ i , respectively.
Phase equilibrium conditions between two multi-component phases I and II require
thermal equilibrium,
I II
T = T
[4.4.2a]
mechanical equilibrium,
I II
P = P
[4.4.2b]
and the chemical potential of each component i must be equal in both phases I and II.
I II
μ = μ
[4.4.3]
i
i
For Equation [4.4.3] to be satisfied, the fugacities of each component i must be equal
in both phases.
where:
I II
f = f
[4.4.4]
i
i
Applying fugacity coefficients, the isochemical potential expression leads to:
I I II II
φ x = φ x
[4.4.5]
i
i
i
i
φi fugacity coefficient of component i.
Fugacity coefficients can be calculated from an equation of state by:
∂P
RT
lnφi ln
RT ∂n
V i V TVn , ,
dV
∞
1
⎡⎛
⎞ ⎤
PV
= ∫ ⎢ ⎜
⎟
− ⎥ −
⎝ ⎠
RT
⎣
⎢
j ⎦
⎥
[4.4.6]
where a pressure explicit equation of state is required to use Equation [4.4.6]. Not all equations
of state for polymers and polymer solutions also are valid for the gaseous state (see
section in Subchapter 4.4.4), however, and a mixed gamma-phi approach is used by applying
Equation [4.4.7]. Applying activity coefficients in the liquid phase, the isochemical potential
expression leads, in the case of the vapor-liquid equilibrium (superscript V for the
vapor phase and superscript L for the liquid phase), to the following relation:
where:
V
L
φ yP= γ x f
i
i i i
0
i
[4.4.7]
yi mole fraction of component i in the vapor phase with partial pressure Pi =yiP γi activity coefficient of component i in the liquid phase with activity a
L
= x γ
i i
i