1679
Heats of Mixing of Globular Molecules
Heats of Mixing of Globular Molecules Differing in Size D. D. Deshpande’ and D. Patterson* Chemistry Department, McGill University, Montreal 101, Canada (Received October 31, 1972) Publication costs assisted by the National Researoh Council of Canada
The van der Waals combining rules, recently suggested by Rowlinson and collaborators, give a simple prediction for HE in terms of differences between the components in intermolecular energies (0) and molecular sizes (6) and deviations from the Berthelot rule ([): HE a Z(1- [) - 8$/2. Data for six new systems and six from the literature are in fair accord with this prediction.
Introduction Interest in the thermodynamics of mixtures of quasispherical molecules has been renewed by theoretical2 and experimentals work. According to the older average potential or random mixture theory, a difference of molecular size should lead to very large positive contributions to the excess functions, GE, HE, and VE. It is now agreed2 that this prediction is incorrect, and Rowlinson and collaborators propose the use of new “van der Waals” combining rules. We wish to point out in this note that the new theory gives an extremely simple prediction for HE. Data for systems containing globular molecules are in fair agreement, while Flory’s theory4 seems to give poor results. van der Waals Prediction for HE The systems selected for discussion, together with experimental values of HE at equimolar concentration and 25”, are found in Table I. We calculate the equimolar HE following Marsh, i. e., using the one fluid approximation and the van der Waals combining rules together with the van der Waals equation of state. (Use of the two-fluid approximation or a more complicated equation of state would not affect the qualitative conclusions presented here (cf. ref 5 and 6)). Thus
HE
= -/
+
xla,/Vl
+
~ z a Z / V z (1)
v, = (Ui/2RT)[l - (1 - 4b,RT/UL}”2] ( 2 ) where at and b, are the van der Waals constants and the averaged quantities are given by
=
=
+ blx12 + U,Xl2
+ azx; 2blZx1x2+ bzxz2 2a,zx,xz
(3)
with
Here the a / b correspond to interaction energies, e, and 5 is the empirical coefficient close to unity which takes into account any deviation from the geometric mean rule. The values of a and b were obtained from critical data.397 The energy and size difference parameters 0 and $ are given by 1 +
e
= (az/bz)l(a,/b1)=
1 +
4
TC,2/TC,1
= b d b , = V,,,/V,,,
(5)
Figure 1 shows values of the equimolar HE calculated for two values of E . Parallel smooth curves, almost straight
lines, are obtained when the HE are plotted against the product $4. The explanation of this simple result follows from an approximate theoretical expression of Rowlinson and collaborators for GE for a Lorentz-Berthelot mixture (eq 4.5 of ref Za), valid for $,$
