Chapter 8
A Statistical Mechanics Based Lattice Model Equation of State
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Applications to Mixtures with Supercritical Fluids Sanat K. Kumar, R. C. Reid, and U. W. Suter Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
A Statistical-Mechanics based Lattice-Model Equation of state (EOS) for modelling the phase behaviour of polymer-supercritical fluid mixtures is presented. The EOS can reproduce qualitatively all experimental trends observed, using a single, adjustable mixture parameter and in this aspect is better than classical cubic EOS. Simple mixtures of small molecules can also be quantitatively modelled, in most cases, with the use of a single, temperature independent adjustable parameter. During the last decade, increasing emphasis has been placed on the use of the equation of state (EOS) approach to model and correlate high-pressure phase equilibrium behaviour. More successful applications have employed some form of cubic EOS (1-3) although others (e.g., 4) have been proposed. However, as the types of systems studied have become more complex, the inherent weaknesses of a cubic EOS have become apparent. We, in particular, are interested in studying phase behaviour of systems comprising polymer molecules in the presence of a supercritical fluid. Here the size disparity of the component molecules can be large. One approach would have been to adopt the modified perturbed hard chain theory (5,6) which has been adapted for mixtures of large and small hydrocarbon molecules. We, however, elected to study whether lattice theory models could be of value for systems of our interest. Studies based on this approach have been attempted for different systems (7-15), and an interesting model has been proposed by Panayiotou and Vera (16.) . Our approach is similar in many respects to the last reference although significant differences appear in treating mixtures. 0097-6156/87/0329-0088$06.00/0 © 1987 American Chemical Society
Squires and Paulaitis; Supercritical Fluids ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
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Lattice Model Equation of State
KUMAR ET AL.
8.
Components
Theory. M o l e c u l e s a r e assumed t o " s i t " on a l a t t i c e o f c o o r d i n a t i o n number ζ and o f c e l l s i z e V J J . Each m o l e c u l e ( s p e c i e s 1) i s assumed to occupy r ^ s i t e s (where r ^ can be f r a c t i o n a l ) , and t h e l a t t i c e h a s empty s i t e s c a l l e d h o l e s . There a r e N Q h o l e s and m o l e c u l e s . To a c c o u n t f o r the c o n n e c t i v i t y o f the segments o f a m o l e c u l e , an e f f e c t i v e chain length i s defined as,
zqi = z r i - 2 r ^ + 2
1)
w h e r e i n i t has been assumed t h a t c h a i n s a r e n o t c y c l i c , z q ^ now r e p r e s e n t s the e f f e c t i v e number o f e x t e r n a l c o n t a c t s p e r m o l e c u l e . The i n t e r a c t i o n energy between segments o f m o l e c u l e s i s d e n o t e d by w h i l e t h e i n t e r a c t i o n energy o f any s p e c i e s w i t h a h o l e i s zero. Only n e a r e s t neighbour i n t e r a c t i o n s a r e c o n s i d e r e d , and p a i r w i s e a d d i t i v i t y i s assumed. The c a n o n i c a l p a r t i t i o n f u n c t i o n f o r t h i s ensemble c a n be f o r m a l l y r e p r e s e n t e d as
Ω =
I exp(-0E all s t a t e s {n}
{ n )
)
2)
w h e r e /3=l/kT. On t h e assumption o f random m i x i n g o f h o l e s and m o l e c u l e s , and f o l l o w i n g t h e approach o f P a n a y i o t o u and V e r a ( 1 6 ) , we o b t a i n an e x p r e s s i o n f o r Ω which i s v a l i d o u t s i d e t h e c r i t i c a l r e g i o n o f t h e pure component, i . e . ,
N
_ f j J
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f (N + N n
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z
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_N z
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where S i s t h e number o f i n t e r n a l arrangements o f a m o l e c u l e and σ a symmetry f a c t o r . U s i n g t h e f o l l o w i n g r e d u c i n g p a r a m e t e r s (z/2)e
=
n
P*v
= RT*
H
4)
and d e f i n i n g V, t h e t o t a l volume o f t h e system, V = v (N + H
0
r i
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5)
x
an EOS t h a t d e f i n e s t h e pure component i s o b t a i n e d , i . e . ,
2
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v
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ν
-
1
6)
J
Τ
Here ϋ i s t h e e f f e c t i v e s u r f a c e f r a c t i o n o f m o l e c u l e s and t h e t i l d e (~) denotes r e d u c e d v a r i a b l e s . A l l q u a n t i t i e s , e x c e p t ν i n t h e EOS, are r e d u c e d b y t h e parameters i n E q u a t i o n 4. The s p e c i f i c volume v , i s r e d u c e d by v * , t h e m o l e c u l a r h a r d - c o r e volume,
Squires and Paulaitis; Supercritical Fluids ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
SUPERCRITICAL FLUIDS
90
v
*
β
N
i
r
i
v
H
7)
E x p r e s s i o n s f o r t h e c h e m i c a l p o t e n t i a l o f a p u r e component c a n a l s o be d e r i v e d from E q u a t i o n 3 and s t a n d a r d thermodynamics ( 1 7 ) .
D e t e r m i n a t i o n o f p u r e component p a r a m e t e r s . I n order t o use the EOS t o model r e a l s u b s t a n c e s one needs t o o b t a i n € and v * . F o r a p u r e component below i t s c r i t i c a l p o i n t , a t e c h n i q u e s u g g e s t e d b y J o f f e e t a l . (18) was u s e d . T h i s i n v o l v e s t h e m a t c h i n g o f c h e m i c a l p o t e n t i a l s o f e a c h component i n t h e l i q u i d and t h e v a p o u r p h a s e s a t the v a p o u r p r e s s u r e o f t h e s u b s t a n c e . A l s o , t h e a c t u a l and p r e d i c t e d s a t u r a t e d l i q u i d d e n s i t i e s were matched. The s e t o f e q u a t i o n s so o b t a i n e d was s o l v e d by t h e u s e o f a s t a n d a r d Newton's method t o y i e l d t h e p u r e component p a r a m e t e r s . V a l u e s o f c and v* f o r e t h a n o l and water a t s e v e r a l temperatures a r e shown i n T a b l e 1. I n this calculation v and ζ were s e t t o 9.75 χ 1 0 " m m o l e " and 10, r e s p e c t i v e l y ( 1 6 ) . The c a p a b i l i t y o f t h e l a t t i c e EOS t o f i t p u r e component VLE was f o u n d t o be q u i t e i n s e n s i t i v e t o v a r i a t i o n s i n ζ (6
L a t t i c e c o o r d i n a t i o n numbers ( z ) and the c e l l volumes (vjj) f o r b o t h the p u r e components and m i x t u r e l a t t i c e s a r e assumed t o have the same v a l u e . The p a r t i t i o n f u n c t i o n f o r t h i s ensemble c a n be f o r m u l a t e d f o l l o w i n g E q u a t i o n 2. I t i s assumed now t h a t the p a r t i t i o n f u n c t i o n , f a r f r o m t h e b i n a r y c r i t i c a l p o i n t c a n be a p p r o x i m a t e d by i t s l a r g e s t t e r m . S i n c e m o l e c u l e segments and h o l e s c a n d i s t r i b u t e t h e m s e l v e s non-randomly, the p a r t i t i o n f u n c t i o n must i n c o r p o r a t e terms t o a c c o u n t f o r t h i s e f f e c t . The nonrandomness c o r r e c t i o n r ^ j
Squires and Paulaitis; Supercritical Fluids ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
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SUPERCRITICAL FLUIDS
a l l o w s f o r d i s t r i b u t i o n o f t h e segments o f s p e c i e s i about t h e segments o f s p e c i e s j o v e r t h e random v a l u e s o f such c o n t a c t s . I t i s d e f i n e d through the e q u a t i o n
Ν..
-
N9.
Γ..
13)
where N^j i s t h e a c t u a l number o f i - j c o n t a c t s and N§j i s t h e number o r i - j c o n t a c t s i n t h e c o m p l e t e l y random c a s e . E x p r e s s i o n s f o r t h e non-randomness c o r r e c t i o n must be o b t a i n e d t h r o u g h t h e s o l u t i o n o f t h e " q u a s i c h e m i c a l " e q u a t i o n s ( 2 0 ) . These e q u a t i o n s c a n be s o l v e d i n a c l o s e d a n a l y t i c form o n l y i n t h e c a s e o f a twocomponent system ( i n c l u d i n g h o l e s ) . I n o r d e r t o e n s u r e t h e mathematical t r a c t a b i l i t y o f t h e b i n a r y r e s u l t s , i t i s t h e r e f o r e assumed t h a t h o l e s d i s t r i b u t e randomly w h i l e m o l e c u l e s do n o t . The s o l u t i o n f o r t h e q u a s i c h e m i c a l e x p r e s s i o n s f o r t h e pseudo two-component system y i e l d s an e x p r e s s i o n f o r t h e nonrandomness c o r r e c t i o n T j j , w h i c h c a n be r e p r e s e n t e d m a t h e m a t i c a l l y a s ,
Γ
j J
1 +
[1 - 4
