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Handbook of Solvents - George Wypych - ChemTech - Ventech!

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4.4 Measurement <strong>of</strong> solvent activity 173<br />

Ω 1<br />

⎛<br />

∞<br />

= ⎜<br />

⎜<br />

⎝<br />

TR<br />

0<br />

0 s<br />

VgM2P1 ( − )<br />

s L<br />

⎞ ⎡P1<br />

V1 B ⎤ 11<br />

⎟exp<br />

⎢<br />

⎥<br />

[4.4.26b]<br />

⎟<br />

⎠ ⎣⎢<br />

RT<br />

⎦⎥<br />

The standard state pressure P has to be specified. It is common practice by many authors<br />

to define here zero pressure as standard pressure since pressures are usually very low<br />

during GLC-measurements. Then, Equations [4.4.26a and b] change to:<br />

⎛<br />

∞<br />

= ⎜<br />

⎜<br />

⎝<br />

γ 1<br />

Ω 1<br />

TR<br />

0<br />

0 s<br />

VgM2P1 ⎛<br />

∞<br />

= ⎜<br />

TR<br />

⎜<br />

⎝V<br />

M P<br />

0<br />

0 s<br />

g 1 1<br />

( − )<br />

s L<br />

⎞ ⎡P1<br />

V1 B ⎤ 11<br />

⎟exp<br />

⎢<br />

⎥<br />

[4.4.27a]<br />

⎟<br />

⎠ ⎣⎢<br />

RT<br />

⎦⎥<br />

( − )<br />

s L<br />

⎞ ⎡P1<br />

V1 B ⎤ 11<br />

⎟exp<br />

⎢<br />

⎥<br />

[4.4.27b]<br />

⎟<br />

⎠ ⎣⎢<br />

RT<br />

⎦⎥<br />

One should keep in mind that mole fraction-based activity coefficients become very<br />

small values for common polymer solutions and reach the value <strong>of</strong> 0 for M 2→∞, which<br />

means a limited applicability to at least oligomer solutions. Therefore, the common literature<br />

provides only mass fraction-based activity coefficients for (high-molecular) polymer/(low-molecular)<br />

solvent pairs. Furthermore, the molar mass M 2 <strong>of</strong> the polymeric liquid<br />

is an average value according to the usual molar-mass distribution <strong>of</strong> polymers. Additionally,<br />

it is a second average if mixed stationary liquid phases are applied.<br />

Furthermore, thermodynamic VLE-data from GLC-measurements are provided in the<br />

literature as values for (P 1/w 1) ∞ , see Equation [4.4.25], i.e., classical mass fraction based<br />

Henry’s constants (if assuming ideal gas phase behavior):<br />

H<br />

12 , =⎛<br />

∞<br />

P ⎞ 1 TR 0<br />

⎜ ⎟<br />

⎜ L<br />

0<br />

⎝w<br />

⎟<br />

1 ⎠ VgM =<br />

Thus, Equation (4.4.27b) reduces to<br />

∞<br />

Ω1 =<br />

1<br />

( − )<br />

[4.4.28]<br />

s L<br />

H ⎡P<br />

12<br />

1 V ⎤<br />

1 B<br />

, 11<br />

exp ⎢<br />

⎥<br />

[4.4.29]<br />

s<br />

P1<br />

⎣⎢<br />

RT<br />

⎦⎥<br />

The data reduction for finite concentration IGC by elution on a plateau is more complicated<br />

than for infinite dilution IGC via Equations [4.4.24 to 26] and will not be explained<br />

here. A detailed analysis <strong>of</strong> the elution on plateau mode was made by Conder and<br />

Purnell. 108,109 For the determination <strong>of</strong> thermodynamic properties <strong>of</strong> polymer solutions by<br />

finite-concentration IGC the reader is referred to the paper by Price and Guillet 110 who provide<br />

a comprehensive derivation <strong>of</strong> all necessary equations.<br />

The data reduction <strong>of</strong> ebulliometric measurements can be made either by using Equations<br />

[4.4.22] or by applying the relation for the boiling point elevation <strong>of</strong> a binary mixture:

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