Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 209
number of extensions and applications also are summarized. Application to mixtures and
solutions needs mixing rules for the characteristic parameters and introduction of binary fitting
parameters 322,327,328 (details are not given here). Examples for applying PHC to polymer
solutions are given by Liu and Prausnitz 328 or Iwai, Arai and coworkers. 329-331
The chain-of-rotators (COR) equation of state was developed by Chao and
coworkers 332 as an improvement of the PHC theory. It introduces the non-spherical shape of
molecules into the hard-body reference term and describes the chain molecule as a chain of
rotators with the aim of an improved model for calculating fluid phase equilibria, PVT and
derived thermodynamic properties, at first only for low-molecular substances. Instead of
hard spheres, the COR-model uses hard dumbbells as reference fluid by combining the result
of Boublik and Nezbeda 333 with the Carnahan-Starling equation for a separate consideration
of rotational degrees of freedom; however, still in the sense of
Prigogine-Flory-Patterson regarding the chain-character of the molecules. It neglects the effect
of rotational motions on intermolecular attractions; however, the attractive portion of
the final equation of state has an empirical dependence on rotational degrees of freedom
given by the prefactor of the double sum. For the attractive perturbation term, a modified
Alder’s fourth-order perturbation result for square-well fluids was chosen, additionally improved
by an empirical temperature-function for the rotational part. The final COR equation
reads:
PV
RT
( y −1)
( )
3
( y −1)
2
2
4y −2y
⎛α
−1⎞3y
+ 3αy − α + 1
= 1+ + c⎜
⎟
3
⎝ 2 ⎠
⎛ c ⎧
⎫⎞
mA
+ ⎜1+
⎨B0
+ B1 / T + B2T⎬⎟∑∑ ⎝ 2⎩
⎭⎠
n m
V T
~ ~
~ ~
nm
m n
[4.4.95]
where:
05 .
y packing fraction with y = V/(V0τ) and τ = ( π / 6) 2 = 0. 7405 (please note that in a
number of original papers in the literature the definition of y within this kind of
equations is made by its reciprocal value, i.e., τV0/V) c degree of freedom parameter, related to one chain-molecule (not to one segment)
V0 hard-sphere volume for closest packing
Anm empirical coefficients from the attractive perturbation term
B0,B1,B2 empirical coefficients for the temperature dependence of the rotational part
α accounts for the deviations of the dumbbell geometry from a sphere
As can be seen from the structure of the COR equation of state, the Carnahan-Starling
term becomes very small with increasing chain length, i.e., with increasing c, and the rotational
part is the dominant hard-body term for polymers. The value of c is here a measure of
rotational degrees of freedom within the chain (and related to one chain-molecule and not to
one segment). It is different from the meaning of the c-value in the PHC equation. Its exact
value is not known a priori as chain molecules have a flexible structure. The value of α for
the various rotational modes is likewise not precisely known. Since α and c occur together
in the product c(α - 1), departure of real rotators from a fixed value of α is compensated for
by the c-parameter after any fitting procedure. As usual, the value of α is assigned a constant
value of 1.078 calculated according to the dumbbell for ethane as representative for the
rotating segments of a hydrocarbon chain. The coefficients Anm and the three parameters B0,