Handbook of Solvents - George Wypych - ChemTech - Ventech!

212 Christian Wohlfarth
wrong phase equilibrium calculations and, therefore, also to wrong solvent activities from
such calculations. Some ways to overcome this situation and to obtain reliable parameters
for phase equilibrium calculations are provided in Ref., 371 together with examples from the
literature that will not be repeated here.
The perturbed-hard-sphere-chain (PHSC) equation of state is a hard-sphere-chain theory
that is somewhat different to SAFT. It is based on a hard-sphere chain reference system
and a van der Waals-type perturbation term using a temperature-dependent attractive parameter
a(T) and a temperature-dependent co-volume parameter b(T). Song et al. 259,260 applied
it to polymer systems and extended the theory also to fluids consisting of
heteronuclear hard chain molecules. The final equation for pure liquids or polymers as derived
by Song et al. is constructed from three parts: the first term stems (as in PHC, COR or
SAFT) from the Carnahan-Starling hard-sphere monomer fluid, the second is the term due
to covalent chain-bonding of the hard-sphere reference chain and the third is a van der
Waals-like attraction term (more details are given also in Prausnitz’s book 49 ):
PV
RT
2
4η−2η = 1+
r + −r
( 1−η)
⎩⎪
⎧
⎪(
1−η/ 2)
( 1 ) ⎨
( 1−η)
3 3
()
⎫ 2
⎪ r a T
−1⎬
−
RTV
⎭⎪
[4.4.99]
where:
η reduced density or packing fraction
a attractive van der Waals-like parameter
r chain segment number
The reduced density or packing fraction η is related to an effective and temperature-dependent
co-volume b(T) by η = r b(T)ρ/4, with ρ being the number density, i.e., the
number of molecules per volume. However, PHSC-theory does not use an analytical
intermolecular potential to estimate the temperature dependence of a(T) and b(T). Instead,
empirical temperature functions are fitted to experimental data of argon and methane (see
also 49 ).
We note that the PHSC equation of state is again an equation where three parameters
have to be fitted to thermodynamic properties: σ, ε/k and r. These may be transformed into
macroscopic reducing parameters for the equation of state by the common relations T*=ε/k,
P*=3ε/2πσ 3 and V* = 2πrσ 3 /3. Parameter tables are given in Refs. 86,260,372-374 PHSC was successfully
applied to calculate solvent activities in polymer solutions, Gupta and Prausnitz. 86
Lambert et al. 374 found that it is necessary to adjust the characteristic parameters of the polymers
when liquid-liquid equilibria should correctly be calculated.
Even with simple cubic equations of state, a quantitative representation of solvent activities
for real polymer solutions can be achieved, as was shown by Tassios and coworkers.
375,376 Using generalized van der Waals theory, Sako et al. 377 obtained a three-parameter
cubic equation of state which was the first applied to polymer solutions:
where:
PV
RT
aT ()
( + )
V − b + bc
= −
V −b
RT V b
a attractive van der Waals-like parameter
b excluded volume van der Waals-like parameter
[4.4.100]