18 Polar and Quadrupolar Fluid Mixtures K. E . GUBBINS and C. H . TWU
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Cornell University, Ithaca, NY 14853
Phase e q u i l i b r i u m c a l c u l a t i o n s may be made by e i t h e r semie m p i r i c a l o r t h e o r e t i c a l methods. I n the s e m i e m p i r i c a l approach one uses an e m p i r i c a l equation o f s t a t e f o r one o r more of the phases i n v o l v e d ; f o r the l i q u i d phase one o f the e m p i r i c a l equat i o n s f o r the excess Gibbs energy (Wilson, van Laar, etc.) i s u s u a l l y used. The s e m i e m p i r i c a l approach gives good r e s u l t s provided that one has a s i g n i f i c a n t amount o f experimental data a v a i l a b l e f o r the mixture. However, such methods are b e t t e r s u i t e d t o i n t e r p o l a t i o n o f e x i s t i n g data than to e x t r a p o l a t i o n o r p r e d i c t i o n . T h e o r e t i c a l methods are based i n s t a t i s t i c a l thermodynamics, r e q u i r e l e s s mixture data, and should be more r e l i a b l e f o r p r e d i c t i o n . T h e o r e t i c a l l y - b a s e d methods that have found extensive use by chemical engineers i n c l u d e r e g u l a r s o l u t i o n theory ( 1 ) , corresponding s t a t e s methods (conformal s o l u t i o n theory) (2-4), and p e r t u r b a t i o n expansions based on a hard sphere f l u i d as reference system (3,_4) • mixtures of simple nonpolar molecules these methods g i v e good r e s u l t s , e s p e c i a l l y the c o r r e sponding s t a t e s and p e r t u r b a t i o n expansion t h e o r i e s ( 3 ) . However, a l l three t h e o r i e s are based on the assumption that the molecules are s p h e r i c a l , w i t h i n t e r m o l e c u l a r f o r c e s that are a f u n c t i o n only of the i n t e r m o l e c u l a r s e p a r a t i o n . This assumption i s s t r i c t l y v a l i d only f o r mixtures o f the i n e r t gases (Ar, Kr, Xe) and f o r c e r t a i n fused s a l t s and l i q u i d metals. I n s p i t e o f t h i s r e s t r i c t i o n the corresponding s t a t e s and p e r t u r b a t i o n methods have been a p p l i e d w i t h success to mixtures i n which the i n t e r m o l e c u l a r f o r c e s depend on the molecular o r i e n t a t i o n s , e.g., mixtures cont a i n i n g 02, N , l i g h t hydrocarbons, e t c . (see r e f . 4 f o r review o f a p p l i c a t i o n s up t o 1973). The extension to weakly n o n s p h e r i c a l molecules can be accomplished, f o r example, by i n t r o d u c i n g shape f a c t o r s as suggested by Leland and h i s colleagues ( 5 ) . These methods have been e x t e n s i v e l y e x p l o i t e d f o r both thermodynamic (2) and t r a n s p o r t (6) p r o p e r t i e s . They are p r e d i c t i v e only i f equat i o n s are a v a i l a b l e f o r the composition dependence o f the shape f a c t o r s . The e x i s t i n g methods f o r doing t h i s work w e l l f o r r e l a t i v e l y simple mixtures, but break down when c o n s t i t u e n t s w i t h F
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r
2
344
In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
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18.
345
Fluid Mixtures
GUBBINS A N D T W U
s t r o n g l y orientation-dependent forces are present (e.g., s t r o n g l y p o l a r o r quadrupolar c o n s t i t u e n t s ) . In t h i s paper we review a r e c e n t l y developed t h e o r e t i c a l method f o r phase e q u i l i b r i u m c a l c u l a t i o n which e x p l i c i t l y accounts f o r s t r o n g l y orientation-dependent f o r c e s . These a n i s o t r o p i c forces are taken i n t o account through a p e r t u r b a t i o n scheme i n which the reference f l u i d i s composed o f simple s p h e r i c a l molec u l e s ; i n p r a c t i c e , the known p r o p e r t i e s o f argon, o r those o f a Lennard-Jones f l u i d simulated on the computer, may be used. Such a p e r t u r b a t i o n scheme was f i r s t suggested by Pople (_7) more than twenty years ago, but was not immediately used f o r l i q u i d phase c a l c u l a t i o n s because the reference f l u i d p r o p e r t i e s were not s u f f i c i e n t l y w e l l known. Since 1972 the theory has been extended and improved, and i t s s u c c e s s f u l a p p l i c a t i o n to l i q u i d s o f s t r o n g l y p o l a r o r quadrupolar molecules dates from 1974 (_7) . The most s u c c e s s f u l form o f the theory i s b r i e f l y o u t l i n e d i n the Theory S e c t i o n . I n the S e c t i o n on R e s u l t s the theory i s used to c l a s s i f y mixture phase diagrams i n terms o f the i n t e r m o l e c u l a r forces i n v o l v e d , and a l s o to p r e d i c t v a p o r - l i q u i d e q u i l i b r i a f o r s e v e r a l b i n a r y and ternary mixtures. Intermolecular
Forces
The i n t e r m o l e c u l a r p a i r p o t e n t i a l u between two a x i a l l y symm e t r i c molecules o f components a and $ can be w r i t t e n as a sum o f parts u*
6
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( r Q ^ )
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where u ^ ^ , u d i , u , and U i j are m u l t i p o l a r ( e l e c t r o s t a t i c ) , a n i s o t r o p i c d i s p e r s i o n , a n i s o t r o p i c charge o v e r l a p , and i n d u c t i o n terms, r e s p e c t i v e l y . Here r i s the i n t e r m o l e c u l a r s e p a r a t i o n , (9ll) are the p o l a r angles g i v i n g the o r i e n t a t i o n of molecule i , the r_ d i r e c t i o n being taken as the p o l a r a x i s , and c|> = i - 2« We take the i s o t r o p i c c e n t r a l p o t e n t i a l u ( r ) to be an (n,6) model (8), s
o v
n c
Q
where eapN a g and n ^ are p o t e n t i a l parameters. The m u l t i p o l a r , d i s p e r s i o n , overlap and i n d u c t i o n p o t e n t i a l s are u s u a l l y a p p r o x i mated by the f i r s t few terms i n a s p h e r i c a l harmonic expansion.* a
a
+In p r a c t i c e not a l l o f these c o n t r i b u t i o n s are e q u a l l y important, and some may be neglected f o r p a r t i c u l a r f l u i d s . I n c a l c u l a t i n g thermodynamic p r o p e r t i e s f o r the f l u i d s considered i n t h i s review, the m u l t i p o l a r forces make the l a r g e s t a n i s o t r o p i c c o n t r i b u t i o n .
In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
P H A S E EQUILIBRIA A N D
346
F L U I D PROPERTIES IN
C H E M I C A L INDUSTRY
Equations f o r these terms are given i n s e v e r a l reviews (7-10). The l e a d i n g m u l t i p o l e term i n u i i s the d i p o l e - d i p o l e p o t e n t i a l i n the case of p o l a r f l u i d s , and the quadrupole-quadrupole potent i a l f o r f l u i d s of l i n e a r symmetrical molecules (C0 , N , e t c . ) . These are given by m u
t
2
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u
yy
=
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C
Q
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=
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c
c
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C
1
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S
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c)
(4)
where s^ = s i n 0-^, c-^ = cos 6^, c = cos , and y and Q are the d i p o l e and quadrupole moments. Experimental values of y and Q have been t a b u l a t e d by Stogryn and Stogryn (11). The a n i s o t r o p i c o v e r l a p p o t e n t i a l may be approximated f o r symmetrical l i n e a r molecules ( C 0 , N , B r , e t c . ) by (7) 2
aB U
ov
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2
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w h i l e f o r unsymmetrical ymmetrical l i n e a r molecules (HC1, NO, N ) , e t c . ) one has 2
aB , [ a 3 ] 1 2 . aS, = 4 e V j 6_ (c_ ov a3 \ r / 1 1 a
u
0
(6)
c ) 2
where n i n Eq. (2) i s taken to be 12, and where 6± and 6 are dimensionless o v e r l a p parameters t h a t must l i e w i t h i n the ranges -0.5 £ 61 2 ) . I n p r a c t i c e , the l i q u i d ranges o f the two components would not o v e r l a p i n such cases, so that l i q u i d l i q u i d i m m i s c i b i l i t y (and hence c l a s s I I b e h a v i o r ) would not be observed i n Lennard-Jones mixtures (the only e x c e p t i o n t o t h i s statement seems to be when the u n l i k e p a i r i n t e r a c t i o n i s improbab l y weak). Thus, the use of t h e o r i e s based on the Lennard-Jones or o t h e r i s o t r o p i c p o t e n t i a l models cannot be expected to g i v e good r e s u l t s f o r systems of c l a s s I I , and w i l l probably g i v e poor r e s u l t s f o r most systems of c l a s s e s I I I , IV and V a l s o . The polar/nonpolar mixtures s t u d i e d here e x h i b i t four of the s i x c l a s s e s of behavior shown i n F i g u r e 2. C l a s s V i s presumably present a l s o , but i s i n d i s t i n g u i s h a b l e from c l a s s IV because the l o c a t i o n of the s o l i d - f l u i d boundary i s not c a l c u l a t e d . For p o l a r / n o n p o l a r mixtures i n which the r e f e r e n c e system i s a weakly n o n i d e a l Lennard-Jones m i x t u r e , i n c r e a s i n g the d i p o l e moment o f the p o l a r molecule causes a continuous t r a n s i t i o n among the classes, 11
c
c
a
I -> I I -> (V) •> (IV) -> I I I
In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
356
P H A S E EQUILIBRIA
A N D F L U I D PROPERTIES
IN C H E M I C A L INDUSTRY
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