Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 175
or
ΔT
c
2
ΔT
c′
= + ′ ′ + ′ ′ +
2
st
st
2
RT V ⎡ 1 1
2 ⎤
= kVPO
Γ c Γ c
0 ⎢ + 2 2 + 3 2 + � ⎥
[4.4.32a]
Δ H M
LV 1 ⎣ 2
⎦
2
RT V ⎡
⎤
1 1
2
kVPO
⎢ Γ c Γ c
0
2 2 3 2 � ⎥
[4.4.32b]
Δ H M
LV 1 ⎣ 2
⎦
where:
c2 mass by volume concentration c2 =m2/ν c′ 2 mass by mass concentration c2 ′ =m2/m1 ν volume of the polymer solution
mi mass of component i
V1 molar volume of the solvent
M1 molar mass of the solvent
M2 molar mass of the polymer
Γ2,Γ3, ... second, third, ... VPO-virial coefficients based on g/cm 3 concentrations
Γ2′ , Γ3′
, .. second, third, ... VPO-virial coefficients based on g/g concentrations
In the dilute concentration region, these virial equations are usually truncated after the
second virial coefficient which leads to a linear relationship. These truncated forms of
Equation [4.4.32] are the basis for applying VPO to polymer characterization, which will
not be discussed here - please see Refs. 26,34,40 Solvent activities can be estimated from second
virial coefficients with some care regarding the necessary accuracy of all numerical values
included. The molar mass of the polymer, M2, is the number-average, Mn, if
polydisperse samples are investigated. Corresponding averages of the virial coefficients can
be introduced, too. The estimation of higher virial coefficients than the second one is difficult
and hardly leads to satisfying results, because measurements at high polymer concentrations
cause a lot of problems. In some cases, however, as in the above-mentioned paper
by Gaube et al., 126,127 precise measurements were done for polymer concentrations up to
30-40 wt% and second and third virial coefficients were obtained in good quality.
As pointed out above, there is another way VPO can be applied to measure activity differences
between two polymer solution drops that differ slightly in concentration (in the
same solvent, of course). In this case, VPO is quasi an isopiestic experiment and the unknown
activity can be determined by using reference solutions with known solvent activity
values: 125
a 1(T, w polymer) =a 1(T, w reference) [4.4.33]
Reference solutions can be made with the same organic solutes that are used for calibration.
In the case of water, NaCl or KCl solutions may be applied as it is done for many
isopiestic (isothermal distillation) measurements with aqueous solutions.
4.4.3.1.3 Comparison of experimental VLE-methods
The general aim of all experiments is to measure solvent activities in polymer solutions over
the complete concentration range and for all desired temperatures (and pressures). Additionally,
the dependence on molar mass of the polymer has to be taken into account. As is
clear from all explanations above, there is no really universal method to fulfill all purposes.