COM MUN I CAT I ON
PREDICTING ACTIVITY COEFFICIENTS FROM LIQUID PHASE SOLUBILITY LIMITS When the Margules equations are used to predict activity coefficients from liquid phase solubility limits, absurd predictions result when the solubility limits are highly asymmetric. In this case the predicted activity coefficients cross unity as the composition i s varied, and the prediction usually yields limiting activity coefficients for the two components which are on opposite sides of unity. The Scatchard-Homer equations behave in a similar fashion when there i s an imbalance between solubility limit asymmetry and molal volume asymmetry. In contrast, the Van Laar equations never show this absurd behavior because their mathematical form will not allow the activity coefficient to cross unity.
the activity coefficients for a partially miscible liquid system can be predicted from the liquid phase solubilities by use of any two-constant activity coefficient equations such as the Van Laar or Margules equations. This was pointed out by Carlson and Colburn ( 7 ) , a n d the procedure is also discussed and illustrated by other authors (4, 5), but no thorough study of the relative merits of different equations or of the accuracy to be expected for different systems has been reported. Nevertheless, the present writer has observed some aspects of the behavior of the Margules a n d Scatchard-Hamer equations which make these equations much less satisfactory than the Van Laar equations, and this note points out a n d illustrates this phenomenon. T h e behavior will be illustrated by comparing the activity coefficient predictions for the five systems shown in Table I. For each system, the Van Laar constants a n d the Margules constants were determined from the indicated solubility limits. Determination of the Scatchard-Hamer constants required the molal volumes in addition to the solubility limits. T h e symmetrical forms of these equations, as given by Carlson a n d Colburn ( I ) , have been used. I n these symmetrical forms, A is the common logarithm of 7 1 a t X I -+ 0, and B is the logarithm of yz a t xz + 0. Consider first the comparison between t h e behavior of the Margules equations and that of the Van Laar equations. For the propylene oxide-water system, the Margules constants in Table I agree well with the Van Laar constants; thus both methods predict essentially the same values for the end points of the activity coefficient curves. Indeed, the complete curves predicted by these two methods are in close agreement, as shown in Figure 1. T h e experimental data of Wickert, Tamplin, and Shank ( 6 ) are seen to be reasonably well approximated by both the Van Laar and the Margules predictions. For the other four systems shown in Table I, the Margules equations yield ridiculous predictions. For example, consider N PRINCIPLE,
I binary
Table 1.
System . -. -~ Cornpo Component 7 nent 2 T . "C.
Aniline Isobutylalcohol 1-Butanol Phenol Propylene oxide
100
Water Water Water Water Water
100
~~
Ref.
XI 0 01475 0.628
0.0213 00207 4 3 . 4 0,02105 36 3 0 166
90 90
l&EC FUNDAMENTALS
I
I
I
1
1
1
I
1
20-
x PO
Figure 1 .
Activity coefficients for propylene oxide-water at
36.3"C.
-_ _ ---_ W
.
Van Laar prediction Scathcard-Homer prediction Margules prediction Eperimental[6), 35-99'C.
the isobutyl alcohol-water system. T h e constants in Table I show that the Margules equations predict that one limiting activity coefficient is less than unity while the other one is greater than unity. T h e complete curves are shown in Figure 2, where it can be seen that the Margules equations predict activity coefficient variations which are absurd. T h e Van Laar prediction, on the other hand, is well behaved a3d is in reasonable agreement with the data of Colburn. Schoenborn, and Shilling (2). Reference to Table I shows that the Margules prediction exhibits this unrealistic behavior for the aniline-water, 1-
Constants Determined from Solubility limits
Solubility Limits
~~~
-0
0.4025 0.364 0.2675 0,625
(3)
(3) (3) (3) (6)
M o l a l Volumes, Cc. / M o l e VI V,
91.2 92 91.5 89.2 69.9
18 18 18 18 18
V a n Laar Constants B A
1.8337 1.6531 1.6477 1.6028 1.1103
0.6067 0.4020 03672 0.2872 0.7763
.Wargules Constants A B
1.5996 0.6193 0.2446 -0.1408 1.0743
4514 -3 0478 -4.1104 -8.2001 0.7046 -0
Scatchard-Hamar Constants A B
1.7726 1 6399 1.6433 1 5985 -0.0509
0.5760 0.3943 03645 0.2842 0.6047
Iyie i
20
and when predictions are based upon solubility limits alone the only constraints imposed are the requirements that the (ys) product for any component must be the same in both saturated phases. \Vhen x1 is small a n d X Z is considerably larger, the Margules equations accommodate these constraints by allokving to dip below unity, and absurd predictions result. T h e mathematical form of the Scatchard-Hamer equations does permit the activity coefficients to cross unity, but for the first four systems listed in Table I the Scatchard-Hamer predictions are \vel1 behaved and are in very close agreement with the \-an Laar predictions. Apparently the high values of the ratio VI V 2compensate for the low values of the ratio x1 /i2, and well behaved predictions result. I t seems, however, that a delicate balance is required between the molal volume ratio a n d the solubility limit ratio or else the Scatchard-Hamer predictions will show the same kind of unrealistic behavior found in the Slargules predictions. For example, consider the propylene oxide-water system. T h e value of T', b'2 is 3.9, and the value of X 1 ' x 2 is 0.44. This solubility limit ratio is sufficiently near unity that the Margules prediction is well behaved, but the molal volume ratio is too high for that solubilitl- limit ratio. T h e result is an absurd prediction by the Scatchard-Hamer equations, as shoivn in Table I and Figure 1. O n the other hand, if XI?:, ' is much less than unity and VI 172 is near unit);, the Scatchard-Hamer equations will also fail (since the)- are identical with the Margules equations \\.hen /
06-
\
-
\
\
0.4-
-
\
\ \ \\
0.2-
'.
-/
I
\/
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/\ \
0.1
-
/
I
I
I
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//
\
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-
\
I
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y$T
10;3
v1
= 1'2).
These considerations show that the Van Laar equations are generally far superior to the Margules and Scatchard-Hamer equations for predicting activity coefficients from liquid phase solubilitv limits alone. Nomenclature
butanol-Lvater, and phenol-water systems also. T h e complete curves are similar to those shown i n ' f i g u r e 2, but the unrealistic dips belo\v unity are less pronounced for the anilinewater system a n d are more pronounced for. the l-butanolwater a n d phenol-water systems. I n contrast? the V a n L a a r predictions for these systems are well behaved a n d are in reasonable agreement with the experimental values. Thus: for the systems chosen, rhe \'an Laar prediction is far superior to the Margules prediction. I t is well to inquire into the reason why the Margules prediction is good for the propylene oxide-water system but absurd for the other four systems. T h e complete activity coefficient prediction is a unique function of the solubility limits; no other properties of the system are used. To answer this question, Margules predictions were calculated for a number of fictitious systems with different solubility limits. T h e results showed that the hfargules prediction was absurd-i.e., y crossed unity-if the solubility limits were highll- asymmetrical ---that is, if x i was small while ~2 was relatively large. O n the other hand, the Margules prediction was well behaved when xl y 2 was near unity. n o matter how large or small x 1 and 1 2 were. When = Q2, ' 4 = B , a n d the hlargules prediction is identical with the \.an Laar prediction. I n contrast to this behavior, the \:an Laar prediction is well a n d i 2may be) because the mathebehaved no matter \vhat i1 matical form of the \'an Laar equations does not permit the activity coefficient to cross unity. If the solubility limits are highly asymmetric. the \-an Laar equations \vi11 just predict an asymmetric activity coefficient plot such as that sho\vn in Figure 2. O n the other hand, the mathematical form of the Margules equation doc=s not prevent y from crossing unity,
A . B = Van Laar. Margules, or Scatchard-Hamer constants, nomenclature of Carlson and Colburn ( 7 ) I' = molal volume of pure liquid (at room temperature). cc. mole x = mole fraction in liquid phase GREEh
y
= activity coefficient in liquid phase
SUBSCRIPTS 1 = component 1 (organic component) 2 = component 2 (i\aterl = isobutyl alcohol iB PO = propylene oxide \V = water SUPERSCRIPTS
-
= saturated organic phase = saturated aqueous phase
literature Cited
(1) Carlson. H . C.. Colburn. .4. P., Ind. En
