Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 171
applied to calculate the solvent vapor fugacity coefficient. This is done if dedicated equations
of state are applied for further modeling. However, in most cases it is common practice
to use the virial equation of state for the purpose of reducing primary VLE-data of polymer
solutions. This procedure is sufficient for vapor pressures in the low or medium pressure region
where most of the VLE-measurements are performed. The virial equation is truncated
usually after the second virial coefficient, and one obtains from Equation [4.4.6]:
where:
⎛
⎝
m
m m
lnφi = ⎜
2∑ i ij −∑∑
j=
1 i=
1 j=
1
Bii Bjj Bij yB yyB
i j ij
⎞ P
⎟
⎠RT
[4.4.19]
second virial coefficient of pure component i at temperature T
second virial coefficient of pure component j at temperature T
second virial coefficient corresponding to i-j interactions at temperature T.
In the case of a strictly binary polymer solutions Equation [4.4.19] reduces
simply to:
lnφ 1
11
= B P
RT
[4.4.20]
To calculate the standard state fugacity, we consider the pure solvent at temperature T
and saturation vapor pressure P s for being the standard conditions. The standard state
fugacity is then calculated as:
0
f = P
1
s
1
( )
L s s
⎡V1
P − P1 + B11P ⎤ 1
exp ⎢
⎥
[4.4.21]
RT
⎣⎢
⎦⎥
where:
s
P1 saturation vapor pressure of the pure liquid solvent 1 at temperature T
L
V1 molar volume of the pure liquid solvent 1 at temperature T
The so-called Poynting correction takes into account the difference between the chem-
s
ical potentials of the pure liquid solvent at pressure P and at saturation pressure P1 assuming
that the liquid molar volume does not vary with pressure. Combining Equations [4.4.7,
4.4.20 and 4.4.21] one obtains the following relations:
L s ( B11 −V1)( P −P1)
⎡
⎤
V 0
s
a1 = φ 1 y1P / f1 = ( P1 / P1)
exp ⎢
⎥ [4.4.22a]
RT
⎣⎢
⎦⎥
L L s
γ1 = a1 x1 = P1 x1P1 L s ( B11 −V1)( P −P1)
⎡
⎤
/ ( / ) exp ⎢
⎥ [4.4.22b]
⎣⎢
RT
⎦⎥