Page 83 - Separation process principles 2

48 Chapter 2 Thermodynamics of Separation Operations

especially suitable for use with a digital computer, which was expressed in terms of liquid molar volume and the latter in
widely used before the availability of the S-R-K and P-R terms of the enthalpy of vaporization. The resulting model is
equations. c
Simple models for the liquid-phase activity coefficient, @i@,(6i - 1 (2-61)
yiL, based only on properties of pure species, are not gener- i=l
ally accurate. However, for hydrocarbon mixtures, regular-
where @ is the volume fraction assuming additive molar
solution theory is convenient and widely applied. The theory
volumes, as given by
is based on the premise that nonideality is due to differences
in van der Waals forces of attraction among the different
molecules present. Regular solutions have an endothermic
heat of mixing, and all activity coefficients are greater than
one. These solutions are regular in the sense that molecules
are assumed to be randomly dispersed. Unequal attractive and 6 is the solubility parameter, which is defined in terms of
forces between like and unlike molecule pairs tend to cause the volumetric internal energy of vaporization as
segregation of molecules. However, for regular solutions the
species concentrations on a molecular level are identical to
overall solution concentrations. Therefore, excess entropy
due to segregation is zero and entropy of regular solutions is Values of the solubility parameter for many components can
identical to that of ideal solutions, in which the molecules be obtained from process simulation programs.
are randomly dispersed. Applying (2-59) to (2-61) gives an expression for the ac-
tivity coefficient in a regular solution:

Activity Coefficients from Gibbs Free Energy
Activity-coejficient equations often have their basis in Gibbs
free-energy models. For a nonideal solution, the molar
Gibbs free energy, g, is the sum of the molar free energy of Because In ylL varies almost inversely with absolute temper-
an ideal solution and an excess molar free energy gE for non- ature, v, L and 6, are frequently taken as constants at some
ideal effects. For a liquid solution, convenient reference temperature, such as 25°C. Thus, the
estimation of y~ by regular-solution theory requires only
the pure-species constants VL and 6. The latter parameter is
often treated as an empirical constant determined by back-
calculation from experimental data. For species with a criti-
cal temperature below 25"C, VL and 6 at 25°C are hypothet-
ical. However, they can be evaluated by back-calculation
where g = h-Ts and excess molar free energy is the sum of from phase-equilibria data.
the partial excess molar free energies. The partial excess When molecular-size differences, as reflected by liquid
molar free energy is related by classical thermodynamics to molar volumes, are appreciable, the following Flory-Huggins
the liquid-phase activity coefficient by size correction can be added to the regular-solution free-
energy contribution:

\- --I
Substitution of (2-65) into (2-59) gives
-
-
RT e "K 1 I P,T,x,
where j # i, r # k, k # i, and r # i.
The relationship between excess molar free energy and The complete expression for the activity coefficient of a
excess molar enthalpy and entropy is species in a regular solution, including the Flory-Huggins
c correction, is
gE = hE - TsE = Ex,(I;: - T~:) (2-60) 2
C
1=1 (::I ::] :
Regular-Solution Model YI L = exp +In - +I--

For a multicomponent, regular liquid solution, the excess i
molar free energy is based on nonideality due to differences L
in molecular size and intermolecular forces. The former are