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Handbook of Solvents - George Wypych - ChemTech - Ventech!

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4.4 Measurement <strong>of</strong> solvent activity 205<br />

where:<br />

Xm group mole fraction for group m<br />

The group mole fraction Xm for group m in the solution is given by<br />

X<br />

m<br />

=<br />

∑ν<br />

j<br />

∑∑<br />

j<br />

p<br />

( j)<br />

m<br />

ν<br />

w / M<br />

( j)<br />

p<br />

j j<br />

w / M<br />

j j<br />

[4.4.89b]<br />

The residual activity coefficient <strong>of</strong> group k in reference solutions containing only<br />

(i)<br />

component i, Γk is similarly determined using Equations [4.4.88, 4.4.89], with the exception<br />

that the summation indices k, m, p refer only to the groups present in the pure component<br />

and the summations over each component j are calculated only for the single<br />

component present in the reference solution.<br />

The group interaction parameter pairs amn and anm result from the interaction between<br />

the groups m and n. These parameter are unsymmetric values that have to be fitted to experimental<br />

VLE-data <strong>of</strong> low-molecular mixtures. They can be taken from UNIFAC tables, e.g.,<br />

Refs. 2,284,286-289 and, additionally, they may be treated as temperature functions.<br />

The free-volume contribution, which is essential for nonpolar polymer solutions, follows,<br />

in principle, from Equation [4.4.74] with parameter X12 = 0 as applied by Raetzsch<br />

and Glindemann, 290 or in a modified form from Equation [4.4.90] as introduced by Oishi<br />

and Prausnitz and used also in Danner’s <strong>Handbook</strong>. 2<br />

⎛<br />

fv ⎜V<br />

lnQ1 = 3c1ln<br />

⎜<br />

⎝V<br />

~ / 13<br />

1<br />

~ / 13<br />

⎛<br />

⎞<br />

⎜ ~ ~<br />

−1⎟<br />

⎜ V1−V ⎟−c1⎜<br />

~ ~ /<br />

−1<br />

⎟ ⎛<br />

⎠<br />

⎜V⎜1−V<br />

1<br />

⎜ ⎜<br />

⎝ ⎝<br />

13<br />

⎞<br />

⎟<br />

⎟<br />

⎟<br />

⎞<br />

⎟⎟<br />

⎟⎟<br />

⎠⎠<br />

[4.4.90]<br />

where:<br />

c1 external degree <strong>of</strong> freedom parameter <strong>of</strong> the solvent 1, usually fixed = 1.1<br />

To get a predictive model, the reduced volumes and the external degree <strong>of</strong> freedom parameter<br />

are not calculated from Flory’s equation <strong>of</strong> state, Equation [4.4.72], but from some<br />

simple approximations as given by the following relations: 2<br />

where:<br />

~<br />

i<br />

V<br />

spez, iM i<br />

=<br />

. r<br />

υ<br />

0 01942<br />

∑<br />

υ w<br />

~<br />

spez, i i<br />

i<br />

V =<br />

0. 01942 rw / M<br />

υspez,i ri Mi i<br />

i<br />

∑<br />

i i i<br />

[4.4.91a]<br />

[4.4.91b]<br />

specific volume <strong>of</strong> component i in m 3 /kg<br />

segment number <strong>of</strong> component i based on Bondi’s van-der-Waals volumes<br />

molar mass <strong>of</strong> component i (for polymers the number average is recommended)

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