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1.m LTICO C C J U ILI6 R IA THE WILSON EQUATION Wileon's e q u e d n provides b basis
I R. V. ORYE
for the estimation of multicomponent v&por-liquid date from parameters obklnab'e*Om
Wth
binary
the m''ticomponent
equilibria'
activity co-
efficients. equipment and processes can be designed more confidently.
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I I
H. PRAUSNITZ
xpcrimental data for vapor-liquid equilibria common for binary. systems but are quite sc . for ternary systems and nearly nonexistent for mixtures containine The amount d .> four or more comoonents. experimental work required to obtain vapor-liquid equilibria for a mixture rises very sharply as the number of components in the mixture increases; even for a ternary mixture the experimental effort required & almost. one order of magnitude larger than that needed for a binary mixture. Howcver, industrial separation operations are rarely limited to binary mixtures. More often than not it it
n
necessary to separate liquid mixtures with many components. For rational design of separation equipment, such as distillation columns, it is necessary to estimate the vapor-liquid equilibria of the mixture to be separated, since all the required experimental data are almost never available. Further, it is commonly not economical to make all the measurements necessary for a complete description of the desired equilibria. I n order to make the best possible estimate with a minimum of experimental data, it is efficient and useful to express the problem in thermodynamic terms and to utilize a reasonable molecular model which can yield the desired information with equations containing only a small number of physically significant parameters. Such an estimation procedure is presented here. The Excess Gibbs Energy
Thermodynamic techniques for the calculation of multicomponent solution properties were described many years ago by van Laar (23), and in more general and modern terms by Wohl (27). Extensions and modifications of Wohl's methods have been given by many authors, most notably by Black (5, 6). The useful thermodynamic concept for efficiently expressing the nonideality of a liquid mixture is the excess Gibbs energy g E which was originally introduced by Scatchard (79). The quantity g E (per mole of liquid mixture) is written as a function of the liquid composition (mole fractions) at constant temperature and pressure, and from this function the activity coefficient ys for any component k is found by the exact relation
where n, is the number of moles of component i and nT is the total number of moles. In the past it has been customary to follow Wohl's procedure and express gE as a polynomial series in the mole fractions (or volume fractions) of the components in the mixture. Second-power terms in this series represent deviations from ideal behavior due to interactions between two molecules; third-power terms, those due to interactions between three molecules; and so on. One of the disadvantages of this approach is that, at least in principle, it is not possible to predict multicomponent properties from binary data unless the series is truncated after the second-power term. Higherorder terms necessarily include interactions between three or more dissimilar components. Various semiempirical procedures have been given for circumventing this difficulty (27) and their effectiveness appears to vary from one liquid mixture to another (20), but in many cases it has been found that a good representation of the properties of a ternary mixture requires at least some experimental information on the ternary in addition to extensive binary data. A completely different expression for the excess Gibbs energy has recently been proposed by Wilson (26) who writes g E as a logarithmic function of the liquid
composition. Wilson's equation has a semitheoretical basis and requires only two parameters for a binary mixture. The extension of Wilson's model to mixtures containing more than two components follows in a straightforward way and does not require any additivity assumptions such as those used in previous treatments. We have tested Wilson's equation for a variety of binary and multicomponent systems and have come to the conclusion that it provides an excellent basis for the calculation of multicomponent vapor-liquid equilibria using only experimental data for binary mixtures. The Wilson Equation
The excess Gibbs energy consists of two parts, an excess enthalpy and an excess entropy, according to gE
= hE
-
TsE
(2)
The simplest assumption one can make about g E is to set it equal to zero; this leads to the equations for ideal solutions. The next simplest assumption is to set either sE or hE equal to zero. Most equations for the excess Gibbs energy were derived on the assumption that sE = 0 and that hE could be written as a polynomial expansion in the mole fractions or volume fractions. The condition sE = 0 leads to the concept of regular solutions. Van Laar, Hildebrand, Scatchard, and others have developed this approach. An alternate starting point is to assume that hE = 0 ; this leads to the concept of athermal solutions which has been developed by Flory (7), Huggins ( 8 ) )and others in the study of polymer solutions. Wilson's equation is a semiempirical extension of the theoretical equation of Flory and Huggins, but Wilson considers mixtures of molecules which not only differ in size but also differ in their intermolecular forces. The Flory-Huggins equation for athermal mixtures is N
a, _ gE - E x i In -
RT
i=i
(3)
XI
where 9 , is the volume fraction of i and x i is the mole fraction of i. These are related by
(4) i=i
where usL = molar liquid volume of pure i. For mixtures where all components have the same molar liquid volume, = x i , and the Flory-Huggins equation predicts ideal behavior. To derive the Wilson equation we first consider a binary solution of components 1 and 2. If we focus attention on a central molecule of type 1, the probability of finding a molecule of type 2 compared to finding a molecule of type 1 about this central molecule is defined by
(5) VOL. 5 7
NO. 5
MAY 1965
19
Equation 5 says that the ratio of the amounts of 2 and 1 molecules about a central 1 molecule is equal to the ratio of the mole fractions of 2 and 1 weighted statistically by the Boltzmann factors exp -(X12/R?') and exp - (X11/RT).A i 2 and A i 1 are, respectively, proportional to the 1-2 and 1-1 interaction energies. Analogous to Equation 5, the probability of finding a molecule of type 1 compared to the probability of finding a molecule of type 2 about a central molecule 2 is given by XZI
--
XI
xzz
x2
- [AIz/RTI -[WRTI
exp exp
___.____
(6)
Using the definitions embodied in Equations 5 and 6, Wilson has empirically redefined the volume fractions of the Flory-Huggins equation into what he terms the local volume fractions i 1 and E z .
'
x l v l L___exp - [Al,/RTl 1 = -- xlvlL exp - [X11/RT] x2uzL exp - [ A d R T I
+
(7)
and
(8)
If we define (9)
and ViL
A21
s z -- exp -[(Xi,
- X,z)/RT]
(10)
V ZL
and substitute local volume fractions E1 and E 2 for and @ z in the Flory-Huggins equation, we obtain the Wilson equation for the excess Gibbs energy of a binary solution : ______
__
__
. ~
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Advantages of Wilson's Equation
The Wilson equation has two features which make it particularly useful for engineering applications. First, it has a built-in temperature dependence which has at least approximate theoretical significance; to a good approximation one may consider the quantities (A,, Aii) and (Xi, - A,,) to be independent of temperature, at least over a modest temperature interval, which means that parameters obtained from data at one temperature may be used with reasonable confidence to predict activity coefficients at some other temperature not too far away. This is an important advantage in the design of isobaric distillation equipment where the temperature varies from plate to plate. The built-in temperature dependence of the excess Gibbs energy as expressed by the Wilson equation makes it possible to estimate the heat of mixing from isothermal Gibhs energy data by utilizing the exact relation
TABLE I . CALCULATED VAPOR COMPOSITIONS FROM FIT OF X - T DATA AT ONE ATMOSPHERE
[ETHANOL(1)-HEXANE(2)l
The generalization of this result to a solution containing any desired number of components is
where A,,
IL = Vexp --
[(A,,
- X,,)/RT]
(13)
VJL
and A,,
= u-- ,L exp - [(A,, -
X,,)/RT]
(14)
VjL
+
Note that, whereas A
