15 Mixing Rules for Cubic Equations of State G. Ali Mansoori
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Department of Chemical Engineering, University of Illinois, Chicago, IL 60680
Through the application of conformal solution theory of statistical mechanics a coherent theory for the development of mixing rules is produced. This theory allows us to use different approximations for the mixture radial distribution functions for derivation of a variety of sets of conformal solution mixing rules some of which are density and temperature dependent. The resulting mixing rules are applied to the van der Waals, Redlich-Kwong, and Peng-Robinson equations of state as the three representative cubic equations of state.
T h e r e e x i s t s a wealth of i n f o r m a t i o n in t h e l i t e r a t u r e about cubic equations of s t a t e a p p l i c a b l e t o v a r i e t i e s of f l u i d s of chemical and engineering interestAlthough cubic equations of s t a t e a r e generally e m p i r i c a l m o d i f i c a t i o n s of the v a n d e r Waals equation of s t a t e , they have found widespread applications in p r o c e s s design calculations because of t h e i r s i m p l i c i t y - Extension of their a p p l i c a b i l i t y to m i x t u r e s i s generally acieved by introduction of m i x i n g rules f o r their parameters. Mixing rules are expressions r e l a t i n g p a r a m e t e r s of a m i x t u r e equation of s t a t e t o pure fluid parameters through, usually, some composition dependent expressions. Except f o r the van der Waals equation of s t a t e the m i x i n g r u l e s f o r cubic equations of state are empirical expressions. In t h e p r e s e n t r e p o r t w e introduce a statistical mechanical conformal solution technique through which we can d e r i v e v a r i e t i e s of s e t s of m i x i n g r u l e s a p p l i c a b l e t o cubic equations of s t a t e . This pressure, energy, and c o m p r e s s i b i l i t y equations of s t a t i s t i c a l m e c h a n i c s . In P a r t II o f t h e p r e s e n t r e p o r t w e introduce the conformal s o l u t i o n t h e o r y of polar fluid m i x t u r e s ( 1 ) and i t s r e l a t i o n s h i p t o t h e i d e a of m i x i n g r u l e s . In P a r t III we introduce the concept of the c o n f o r m a l 0097-6156/ 86/ 0300-0314506.00/ 0 © 1986 American Chemical Society
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
Mixing Rules for Cubic Equations of State
MANSOORI
15.
315
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s o l u t i o n m i x i n g r u l e s and we p r o d u c e d i f f e r e n t s e t s of m i x i n g r u l e s b a s e d on d i f f e r e n t a p p r o x i m a t i o n s f o r the mixture radial distribution functions. In P a r t IV w e r e v i e w the e x i s t i n g f o r m s of t h e c u b i c e q u a t i o n s of s t a t e f o r m i x t u r e s and the d e f i c i e n c i e s of t h e i r m i x i n g r u l e s and c o m b i n i n g r u l e s . F i n a l l y , i n P a r t IV w e i n t r o d u c e g u i d e l i n e s for t h e use of c o n f o r m a l solution mixing rules and c o m b i n i n g r u l e s in equations of s t a t e and we d e m o n s t r a t e a p p l i c a t i o n of such m i x i n g r u l e s and combining r u l e s f o r three r e p r e s e n t a t i v e cubic equations of s t a t e . II. C o n f o r m a l S o l u t i o n T h e o r y o f M i x t u r e s Conformal solutions refer to substances whose intermolecular potential energy function, are related to each other and t o those of a r e f e r e n c e fluid, designated by s u b - s c r i p t (oo), according to (1,2) *ij = ' i j ' o o f r / h i j " )
usually
(1)
3
For substances whose intermolecular potential energy f u n c t i o n c a n b e r e p r e s e n t e d b y an e q u a t i o n o f t h e f o r m *ij = i j [ ( i j E
L
/ r
>
n
"(l-ij/r) ]
(2)
m
and f o r w h i c h e x p o n e n t s m a n d n a r e t h e s a m e a s f o r t h e r e f e r e n c e s u b s t a n c e , c o n f o r m a l p a r a m e t e r s f^- a n d h|j w i l l be d e f i n e d b y t h e f o l l o w i n g r e l a t i o n s w i t h r e s p e c t t o t h e i n t e r m o l e c u l a r p o t e n t i a l e n e r g y p a r a m e t e r s E j j a n d L^y f..
hj
= F--/F ij oo» I 1
/ l l
n
= (I L --/i / L
h--
ij
^ ij oo^
(3)
}3
Thus t h e c o n f i g u r a t i o n a l thermodynamic properties of a pure s u b s t a n c e of t y p e (a) a r e r e l a t e d to those of t h e reference substance according to the following relations: F
a
( V , T) = f
P (V, a
a a
T) = ( f
F (V/h 0
a a
/h
a a
S ( V , T) = S ( V / h a
c
G ( P , T) = f a
a a
a a
, T/f
)P (V/h 0
,
T/f
a a
a a
a a
) - NkT£nh
, T/f
) + Nkenh
a a
G (Ph
a a
/f
a a
,T/f
a a
H (Ph
a a
/f
a a
, S )
0
a a
a a
a
(4)
a
) a
(5) (6)
a
) - NkTfcnh
a
a
(7)
and H ( P , S) = f a
w h e r e F, pressure,
0
0
(8)
P, S , G, a n d H a r e t h e H e l m h o l t z f r e e energy, entropy, Gibbs free energy, and e n t h a l p y ,
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
316
respectively. A c c o r d i n g to the above equations, a l l the thermodynamic properties of substance (a) c a n be e x p r e s s e d i n t e r m s o f t h e p r o p e r t i e s of a r e f e r e n c e p u r e substance (o) through the conformal p a r a m e t e r s f and h . The c o n f o r m a l s o l u t i o n t r e a t m e n t of f l u i d s c o m p o s e d of p o l a r m o l e c u l e s i s m o r e c o m p l i c a t e d than f o r n o n - p o l a r fluids. This i s m a i n l y due t o e l e c t r o s t a t i c i n t e r a c t i o n s which cause a d e p a r t u r e of the i n t e r m o l e c u l a r potential from spherical symmetry. The e l e c t r o s t a t i c potential between two otherwise neutral molecules arises from permanent a s y m m e t r y in the charge distribution within the molecules. For any p a i r of l o c a l i z e d charge d i s t r i b u t i o n , the mutual e l e c t r o s t a t i c i n t e r a c t i o n energy can be w r i t t e n in t e r m s o f an i n f i n i t e s e r i e s of i n v e r s e powers of s e p a r a t i o n of any t w o p o i n t s . F o r no o v e r l a p b e t w e e n t h e charge distributions the s e r i e s c o n v e r g e s ( l ) . Thus t h e t r u e p a i l — p o t e n t i a l of p o l a r m o l e c u l e s i s o r i e n t a t i o n - d e p e n d e n t and i s t h e s u m o f d i s p e r s i o n f o r c e a s w e l l a s e l e c t r o s t a t i c interactions. In o r d e r t o e x t e n t u t i l i t y of t h e above f o r m u l a t i o n of the c o n f o r m a l s o l u t i o n t h e o r y t o p o l a r f l u i d s we have p r o p o s e d the f o l l o w i n g a n g l e - a v e r a g e d potential function f o r polar molecular interactions which r e p r e s e n t s the f i r s t o r d e r c o n t r i b u t i o n to the anisotropic f o r c e s ( 1 ) a a
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a a
* (r,T) = K€ [(o t j
i j
1 J
/ r ) - (o^/r)™]
+ 7|i Mj /[450(kT) r i
4
4
3
- Q| Qj /(1.4kTr 2
-
n
2
1 0
1 2
] -
n
t
2
Hj /(3kTr ) 2
6
(M Qj +vij Q i
2
2
2
) - (oc |ij +oc |i )/r 1
2
j
2
1
1
2
)/(2kTr ) 8
(9)
6
where K = [n/(n-m and w h e r e | i p Q j , a n d OCJ a r e the dipole moment, quadrupole moment, and p o l a r i z a b i l i t y of m o l e c u l e i , r e s p e c t i v e l y . For a polar fluid, whose intermolecular potential energy function c a n be r e p r e s e n t e d by e q . 9 the c o n f o r m a l p a r a m e t e r s f and h a a
a
a
w i l l have the following f o r m s : f
aa = Eaa< > > oo< > >
where
T
r
/ E
T
r
E,j(T,r) = K e A i j
LijCT.r) =
0
h
1 j
a a = a a< > >
(T,r)[H (T,r)] i j
ij[ ij< > >]~ H
T
r
i j
i j
7M
i
4
ii
j
4
r
/ L
oo( > > J T
r
n / m
, / m
HtjCT.r) = [ C ( T , r ) / A ( T , r ) ] Ajj(T,r) = 1 +
T
a
m /
^ n
m )
/[1800(kT) r 3
, 2
- o m
i j
n
Ke ] j j
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
< > 10
15.
MANSOORI
and
Mixing Rules for Cubic Equations of State
C|j(T r) = 1 + ii Mj /[12lcTr f
1
2
2
6
"" o m
m
1 j
317
K€ j] 1
+ (7/20)Q Q /[kTr ^ " ' " o y ^ C y i
+ (n +
Q
j
(oc ii
j
1
2
1
2
2
j
2
+y
2
+oc |i
j
Q
2
i
j
2
1
)/[8kTr - o 8
2
)/[4r - a 6
m
m
m
1 j
1 J
m
K6| ] j
Ke ] i j
The b a s i c c o n c e p t o f t h e CST of m i x t u r e s i s t h e s a m e a s f o r pure fluids, except that f and h i n e q s . 4 - 8 s h o u l d be a a
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replaced
with
parameters, f
x x = x x < ij> f
f
f
and
x x
a a
h
,
x x
the
mixture
conformal
as g i v e n b e l o w h
ij' i x
)
h
xx = xx( h
Uy
h
ij» i>
< >
x
11
Eqs.ll are c a l l e d the conformal solution mixing rules. F u n c t i o n a l f o r m s of t h e s e m i x i n g r u l e s w i l l be d i f f e r e n t f o r different theories of m i x t u r e s as it will be demonstrated l a t e r in this r e p o r t . In t h e f o r m u l a t i o n of a m i x t u r e t h e o r y w e a l s o n e e d t o know t h e c o m b i n i n g r u l e s for unlike-interaction potential parameters which are u s u a l l y e x p r e s s e d by the f o l l o w i n g e x p r e s s i o n s ffj = ( 1 - k i j ) ( f i t f j j >
1 / Z
;
h i j = C l - *tj)[( h
t i
1 / 3
+hjj
1 / 3
)/2]
3
(12)
w h e r e k j j and fcjj a r e a d j u s t a b l e p a r a m e t e r s . III. S t a t i s t i c a l M e c h a n i c a l T h e o r y of l i i x i n a R u l e s The m o s t i m p o r t a n t r e q u i r e m e n t i n t h e d e v e l o p m e n t o f t h e CST o f m i x t u r e s are mixing rules. In t h e d i s c u s s i o n p r e s e n t e d h e r e we have i n t r o d u c e d a new t e c h n i q u e to r e - d e r i v e t h e e x i s t i n g m i x i n g r u l e s and d e r i v e a n u m b e r of new mixing rules some of which are densityand temperature-dependent. A c c o r d i n g to s t a t i s t i c a l mechanics the m a c r o s c o p i c t h e r m o d y n a m i c p r o p e r t i e s of a p u r e f l u i d a r e r e l a t e d to i t s m i c r o s c o p i c m o l e c u l a r c h a r a c t e r i s t i c s by the f o l l o w i n g t h r e e equations (5,4) oo
(13)
u = Ujg + 2 n p j 0 ( r ) g ( r ) r d r 0 2
oo
P = pRT + ( 2 / 3 ) n p J r * r ( r ) g ( r ) r d r 0 2
(14)
oo (15) (4TT/RT)J[g(r)-l]r dr 0 w h e r e u i s t h e i n t e r n a l e n e r g y , P i s t h e p r e s s u r e and KJ i s the isothermal compressibility, 0(r) is the pair intermolecular potential energy f u n c t i o n , and g ( r ) i s t h e
k
t
= 1/pRT -
2
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
318
radial (or pair) distribution function. Eqs.13-15 are commonly called the energy equation, the virial (or pressure) equation, and the c o m p r e s s i b i l i t y equation, respectivelyFor a multicomponent mixture these equations assume the following forms ( 5 - 5 ) oo
u = u
1 g
+ 2np2iSjX Xjj0 j(r)g (r)r dr 0 i
i
(16)
2
1 j
oo
P = pRT + ( 2 / 3 ) n p 2 2 j X x J r ^ ( r ) g 0 ic = ( l / p R D l B l / S i S j X i X j l B ^ j i
i
j
1 j
| J
(r)r dr
(17)
2
(18)
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T
In t h e a b o v e e q u a t i o n s s u m m a t i o n s a r e o v e r a l l t h e ( c ) c o m p o n e n t s o f t h e m i x t u r e , Xj a n d xj a r e t h e m o l e f r a c t i o n s , and |B| i s a c x c d e t e r m i n a n t w i t h i t s r e p r e s e n t a t i v e t e r m s i n the f o l l o w i n g f o r m oo
ij
B
=
x
i ij 6
+
x
i jP ij x
G
G
ij
=
4nJ[g j(r)-1 ] r d r 0 2
1
w h e r e 6,-j i s t h e K r o n e e k e r d e l t a , a n d IBIJJ i s t h e c o f a c t o r o f t e r m B y i n d e t e r m i n a n t |B|. E q s . 1 3 - 1 8 c a n b e u s e d i n t h e manner p r e s e n t e d below in o r d e r t o d e r i v e m i x i n g r u l e s b a s e d on d i f f e r e n t m i x t u r e t h e o r y a p p r o x i m a t i o n s : III.1. O n e - F l u i d T h e o r y o f M i x i n g R u l e s : F o r t h e d e v e l o p m e n t of o n e - f l u i d m i x i n g r u l e s we i n t r o d u c e a p s e u d o - p u r e f l u i d which can r e p r e s e n t the configurational p r o p e r t i e s of a m i x t u r e p r o v i d e d that the p s e u d o - p u r e f l u i d and the m i x t u r e m o l e c u l a r i n t e r a c t i o n s o b e y e q . 1. By r e p l a c i n g eq.1 in e q s . 1 3 , 1 4 , 1 7 , a n d 18 a n d t h e n e q u a t i n g c o n f i g u r a t i o n a l i n t e r n a l e n e r g y , p r e s s u r e , and i s o t h e r m a l c o m p r e s s i b i l i t y of t h e p s e u d o - p u r e f l u i d and t h e m i x t u r e w e w i l l o b t a i n t h e following equations f
xx xx^oo(y)%o(y)y
f
xx xx/y^ oo(y)goo(y>Y
h
h
2 d
y 2i2jX x f h jj0 =
,
1
2 d
j
i j
1
o o
( y ) g ( y ) y d y (19)
y 2i2jX x f h Jy0' =
1
j
i j
j j
2
i j
o o
(y)g (y)y dy i j
2
(20) {1-4nph J[g x x
0 0
(y)-1]y dy}-1=2i2 x x |B| /|B| 2
j
i
j
1 J
(21)
It s h o u l d b e p o i n t e d o u t t h a t f o r t h e c a s e o f t h e h a r d - s p h e r e fluid eq.19 vanishes, eq.21 remains the same, while eq.20 reduces to the following form h
x x 9 o o < ) = 2 i 2 j X x h g ( 1) 1
i
j
i j
i j
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
(22)
MANSOORI
15.
Mixing Rules for Cubic Equations of State
Solution of eqs.19-21 should produce the t w o n e c e s s a r y expressions (mixing rules) relating f and h of t h e p s e u d o - p u r e f l u i d t o f j j a n d h|j o f c o m p o n e n t s o f t h e mixture. F o r t h i s p u r p o s e w e s h o u l d u s e an a p p r o x i m a t i o n technique r e l a t i n g t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s (RDF) i n t h e m i x t u r e t o t h e p u r e r e f e r e n c e f l u i d RDF. H o w e v e r , a t a f i r s t g l a n c e i t s e e m s t h a t w e h a v e i n o u r hand t h r e e e q u a t i o n s and t w o unknowns. A s i t w i l l be d e m o n s t r a t e d b e l o w f o r m o s t o f t h e a p p r o x i m a t i o n s o f t h e m i x t u r e RDFs which a r e used here these t h r e e equations produce t w o mixing rules. In t h e p r e v i o u s i n v e s t i g a t i o n s for the d e v e l o p m e n t of m i x i n g r u l e s (5-11) a l l t h e i n v e s t i g a t o r s have used only eq.19 and/or eq.20. Our s t u d i e s i n d i c a t e t h a t w h i l e e q s . 1 9 and 20 a r e e s s e n t i a l i n t h e d e v e l o p m e n t o f m i x i n g r u l e s , eq.21 can add a new dimension which could be s i g n i f i c a n t i n t h e c a l c u l a t i o n o f p r o p e r t i e s o f m i x t u r e s . In what f o l l o w s d i f f e r e n t a p p r o x i m a t i o n s w i l l be used f o r relating gy to g in order to derive different sets of mixing rules. x
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319
O
x
x
x
Q
III. 1 . i . Random M i x i n g A p p r o x i m a t i o n ( R M A ) f o r M i x t u r e RDFs: In t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t t h e n o n - s c a l e d RDF o f a l l t h e c o m p o n e n t s o f t h e m i x t u r e a n d t h e i n t e r a c t i o n RDFs a r e i d e n t i c a l (5_), i . e . Q\](r)
=g
2 2
(r) = ... = g^(r)= ...
(23)
When t h i s a p p r o x i m a t i o n i s r e p l a c e d in e q s . 1 9 - 2 1 , eq.21 w i l l v a n i s h a n d e q . 1 9 a n d 20 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xx< > = 2 i S j X x ^ ( r ) r
i
*'xx< > r
=
j
(24)
i j
SiSjXiXjf'ijCr)
(25)
For e x a m p l e , in the case of the L e n n a r d - J o n e s (12-6) intermolecular potential function we will derive the f o l l o w i n g m i x i n g r u l e s (12) f r o m e q s . 1 3 and 14. 'xx^xx f
xx xx h
2
= SiSjXiXjfijhij
4
= SiSjXiXjfjjhjj
2
4
(
2
6
)
(27)
For a h a r d - s p h e r e potential we will d e r i v e only one m i x i n g r u l e t h r o u g h the RMA and that i s d e r i v e d by r e p l a c i n g e q . 2 3 in 2 2 . The r e s u l t i n g m i x i n g r u l e w i l l be
h
x x
, / 3
= ZiZjXiXjh^'S
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
( 2 8
)
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
320
l l l . l . i i . Conformal Solution Approximation (CSA) f o r Mixture RPFs: This a p p r o x i m a t i o n technique seems m o r e logical f o r use in the d e v e l o p m e n t of m i x i n g r u l e s than R M A . A c c o r d i n g t o t h i s a p p r o x i m a t i o n t h e s c a l e d RDFs i n a m i x t u r e a r e a l l identical (5J, i-egii(y)=
9zz(y) = - - - = g j j ( y ) = . . .
(29)
When we u s e t h i s a p p r o x i m a t i o n in e q s . 1 9 and 20 t h e y both produce the same m i x i n g r u l e which i s
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f
XX XX h
= SlSjXiXjf^hy
(
N o w , b y r e p l a c i n g e q . 2 9 i n 21 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g |B*|/pRTic
Txx
= SiSjX|Xj|B*||j
3
0
)
will
(31)
where |B*| j = x [ 6 j + X j ( h j / h ) ( p R T i c - 1 ) ] . Eq.30 is actually the second van d e r Waals mixing rule which i s well known, but e q . 3 1 i s a new m i x i n g r u l e f o r h which i s replacing the f i r s t van d e r Waals mixing rule. This new mixing rule, in p r i n c i p l e , i s a c o m p o s i t i o n - , t e m p e r a t u r e - , and d e n s i t y - d e p e n d e n t m i x i n g r u l e . This i s because K y w h i c h a p p e a r s i n t h e r i g h t a n d l e f t hand s i d e s o f t h i s equation i s g e n e r a l l y t e m p e r a t u r e - and d e n s i t y - d e p e n d e n t . For example, f o r a binary m i x t u r e eq.31 can be w r i t t e n in t h e f o l l o w i n g f o r m (5.) t
i
1
1
x x
T x x
x
x
x
h
x
x
= {liSjXjXjhjj + x x ( h 1
{1+x
2
1 1
h
2 2
1 2
2
)(pRTic
By u s i n g t h e h a r d - s p h e r e p o t e n t i a l ( b y r e p l a c i n g 22) w e w i l l d e r i v e t h e f o l l o w i n g m i x i n g r u l e
e q . 2 9 in
h
x
x
2 2
1 2
)(pRTK
- 1 ) }/ (31-1)
n
- 2h
T x x
-1)}
l X 2
(h +h
-h
x
T x x
= SiSjXtXjhjj
(32)
This mixing rule i s the f i r s t van d e r Waals m i x i n g rule which, in conjunction with eq.30 is usually used f o r calculation of mixture thermodynamic properties ( 7 , 8 . 1 0 , 1 1 ). It s h o u l d b e p o i n t e d o u t t h a t e q . 3 2 c o n s t i t u t e s another mixing rule f o r hard-sphere m i x t u r e s . A s a result, while the CSA approximation produces two mixing rules f o r potential functions with two parameters, it also produces two mixing rules f o r a h a r d - s p h e r e potential which i s a one-parameter potential function.
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
MANSOORI
15.
Mixing Rules for Cubic Equations of State
321
MLI.iii, H a r d - S p h e r e Expansion (HSE) A p p r o x i m a t i o n f o r M i x t u r e RDFs: It i s d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d ( x ) can be expanded around t h e h a r d - s p h e r e ( h s ) RDF in t h e f o r m ( 3 ) g
x x
(y)= g
(y)+ (f x
h s
X
/ T
o*^l
+
< xx o*> 92 f
/ T
2
+
—
< > 33
Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e h a r d - s p h e r e m i x t u r e RDFs as t h e f o l l o w i n g
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gij= 9 i j
h s
(y)
(fij/T *)g!(y)
+
0
+
(fij/To*) g (y) 2
2
+
—
(
3 4
>
The j u s t i f i c a t i o n b e h i n d t h i s e x p a n s i o n i s g i v e n e l s e w h e r e ( 6 9 ) . Now b y r e p l a c i n g e q s . 3 3 a n d 3 4 i n e i t h e r o f e q s . 1 9 o r 20 w e w i l l b e a b l e t o d e r i v e t h e f o l l o w i n g t w o m i x i n g r u l e s by e q u a t i n g the coefficients of the s e c o n d and t h i r d o r d e r i n v e r s e t e m p e r a t u r e t e r m s of the r e s u l t i n g e x p r e s s i o n . T
f
XX XX = h
SiSjX^jfyhy
' x x ^ x x = 2i2jX x f 2h l
J
1 J
(
3
5
)
(36)
1 J
These m i x i n g r u l e s a r e used f o r c a l c u l a t i o n of e x c e s s p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e m i x t u r e (13) at t h e s a m e t h e r m o d y n a m i c c o n d i t i o n s ( 9 ) . A p p l i c a t i o n o f t h e HSE a p p r o x i m a t i o n in eq.21 w i l l not produce any additional mixing rule. I l l . l . i v . D e n s i t y E x p a n s i o n (DEX) A p p r o x i m a t i o n f o r M i x t u r e R D F s : i t h a s b e e n d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d can b e e x p a n d e d a r o u n d t h e d i l u t e g a s RDF, e x p [ - 0 ( r ) / k T ] , i n the f o r m (14) g
x x
( y ) = [1 + F
x x
( y ) ] exp[-0 (r)/kT]
(37)
x x
Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e d i l u t e g a s m i x t u r e RDFs a s the f o l l o w i n g g i j ( y ) = [1 + F
( y ) ] exp[-0 (r)/kT]
x x
(38)
1 j
Now b y r e p l a c i n g e q s . 3 7 a n d 3 8 i n e q . 1 9 a n d a f t e r a n u m b e r of a l g e b r a i c m a n i p u l a t i o n s w e w i l l d e r i v e t h e f o l l o w i n g mixing rule fxx xx=2 2 x x f h
1
j
1
j
1 j
h {1-(f /f i j
i j
x x
-1)[u-u
+T(C -C v
v i g
1 g
)/kT
)/(u-u
1 g
)]}
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
(39)
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
322
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The l a t t e r m i x i n g r u l e c a n b e u s e d , j o i n e d w i t h a n o t h e r m i x i n g r u l e , f o r c a l c u l a t i o n of m i x t u r e p r o p e r t i e s . Similar a p p r o x i m a t i o n s c a n be u s e d i n o r d e r t o d e r i v e o t h e r m i x i n g r u l e s f r o m the v i r i a l and c o m p r e s s i b i l i t y e q u a t i o n s . III.2. M u l t i - F l u i d T h e o r y o f M i x i n g R u l e s : The b a s i c a s s u m p t i o n i n d e v e l o p i n g t h e m u l t i - f l u i d m i x i n g r u l e s i s the same as the o n e - f l u i d a p p r o a c h e x c e p t that in this case we w i l l s e a r c h f o r a h y p o t h e t i c a l multicomponent ideal m i x t u r e which could r e p r e s e n t the configurational p r o p e r t i e s of a multicomponent r e a l m i x t u r e , both w i t h the s a m e number of c o m p o n e n t s and at the same t h e r m o d y n a m i c conditions. In t h i s c a s e e q s . 1 9 - 2 1 w i l l be r e p l a c e d b y t h e f o l l o w i n g s e t of e q u a t i o n s f
xi xi^oo(y)9oo(y^V dy=2jXjf h jj0
f
h
x 1
2
h
x 1
Jy0-
o o
(y)g
o o
i j
1
(y)y dy=2jX f h 2
j
i j
1 j
o o
(y)g j(y)y dy
Jy^'
o o
(40)
2
i
(y)g (y)y dy i j
2
{l-4nph J[g (y)-1]y2dy}-1=2 X3|B| /|B| x i
0 0
j
i j
(41) (42)
E x p r e s s i o n s f o r Bjj and G|j w i l l b e t h e s a m e a s i n e q . 1 8 . In the c a s e of the h a r d - s p h e r e f l u i d e q . 4 0 w i l l r e d u c e to the following form h
xigoo
eq.41
h s
( )=2jX h g 1
j
i j
j j
h s
(1),
(40-1)
w i l l v a n i s h and e q . 4 3 w i l l r e m a i n t h e s a m e .
Ml.2.i. A v e r a g e Potential Model ( A P M ) f o r M i x t u r e RDFs: t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t (5_), g ( r ) = [ g ^ r ) + gjj(r)]/2 l j
Q (r) * gjj(r)
In (43)
t 1
When this a p p r o x i m a t i o n i s r e p l a c e d in e q s . 4 0 - 4 2 , eq.42 w i l l v a n i s h a n d e q . 4 0 and 41 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xi< > = 2 j X j 0 ( r )
(44)
*'xi(r) = SjXj^'i^r)
(45)
r
i j
For e x a m p l e , in the c a s e of t h e intermolecular potential function f o l l o w i n g m i x i n g rules(JL2) . f
h
2
= 7-x-f-h-
Lennard-Jones (12-6) we w i l l derive the
2
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
(
4
6
)
15.
MANSOORI
Mixing Rules for Cubic Equations of State
323
'xihxi = SjXjfjjh^
(47)
4
For a h a r d - s p h e r e rule hx i
1
/
= 2jXjhij
3
p o t e n t i a l we w i l l d e r i v e only one m i x i n g
(*»>
1 / 3
Ml.2.ii. Multi-fluid CSA Approximation for Mixture A c c o r d i n g to this approximation t h e s c a l e d RDFs m i x t u r e a r e r e l a t e d as the following
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gij(y) = tgij(y) + gjj(y)]/2
RDFs: in a
g (r) * gjj(y)
(49)
t 1
W h e n w e u s e t h i s a p p r o x i m a t i o n i n e q s . 4 0 a n d 41 t h e y b o t h produce the same m i x i n g rule which is f -h =
Y x f - h -
( °) 5
N o w , b y r e p l a c i n g e q . 4 9 i n 42 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g
will
|B*|/ RTK
(51)
P
= SjXjlB*^
T x 1
where |B*|
tj
= x {6 i
i j
+ (x h /2)[(pRTK j
1 j
T x i
-1)/h +(pRTic j-1)/h j]} x 1
T x
X
Eq.50 i s actually the second van d e r Waals m u l t i - f l u i d m i x i n g r u l e , but eq.51 i s a new m i x i n g r u l e f o r h j . By using the h a r d - s p h e r e p o t e n t i a l (by r e p l a c i n g e q . 4 9 in 4 0 - 1 ) we will derive the following mixing rule X
h
= ZjXjhfj
x j
(52)
This mixing rule is the f i r s t m u l t i - f l u i d m i x i n g rule which, in conjunction with eq.50 for calculation of m i x t u r e thermodynamic should be p o i n t e d out that eq.52 c o n s t i t u t e s rule for hard-sphere mixtures. I l l . 2 . i i i . M u l t i - F l u i d HSE M i x i n g R u l e s : as t h e o n e - f l u i d c a s e w e c a n d e r i v e rules f
van der Waals i s usually used properties. It another mixing
In a s i m i l a r m a n n e r the following mixing
h •= Y x f - h -
(53)
f' x i - "h x i• = Y xJ ifj hi j 2
These
2
mixing
rules
(
are
used
for
calculation
of
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
5
4
excess
)
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
324
p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e
mixture.
III. 2 . i v , M u l t i - F l u i d DEX M i x i n g R u l e s : In a s i m i l a r m a n n e r a s the o n e - f l u i d case we can d e r i v e the f o l l o w i n g m i x i n g rule f
xi xi h
=
SjXjfijh^d-Cf^/fxi-DEupUig)/^ +T(C
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This mixing rule can be used, rule, f o r calculation of m i x t u r e
v 1
-C
v 1 g
)/(u u
joined with properties.
r
1 g
)]}
another
(55) mixing
IV. A p p l i c a t i o n o f M i x i n g R u l e s f o r C u b i c E o u a t i o n s o f S t a t e In o r d e r t o a p p l y t h e v a r i e t i e s o f t h e c o n f o r m a l solution mixing rules which a r e introduced here f o r cubic and other equations of s t a t e t h e f o l l o w i n g c o n s i d e r a t i o n s should be taken into account: (i) Conformal solution mixing rules a r e f o r the molecular conformal volume parameter, h, and the m o l e c u l a r c o n f o r m a l e n e r g y p a r a m e t e r , f. (ii) Conformal solution mixing rules a r e applicable for c o n s t a n t s o f an e q u a t i o n o f s t a t e o n l y . B e f o r e u s i n g a s e t o f m i x i n g r u l e s f o r an e q u a t i o n o f s t a t e one h a s t o e x p r e s s t h e p a r a m e t e r s of t h e equation of s t a t e with r e s p e c t t o the m o l e c u l a r c o n f o r m a l p a r a m e t e r s h and f. This w i l l then make i t p o s s i b l e t o w r i t e t h e c o m b i n i n g r u l e s a n d m i x i n g r u l e s f o r t h e e q u a t i o n o f s t a t e . In w h a t f o l l o w s m i x i n g r u l e s and combining rules f o r three representative cubic e q u a t i o n s of s t a t e a r e d e r i v e d and t a b u l a t e d . IV.1. M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e : The v a n d e r W a a l s e q u a t i o n o f s t a t e ( 1 5 ) c a n b e w r i t t e n i n the f o l l o w i n g f o r m Z = Pv/RT = v / ( v - b ) - a/vRT
(56)
P a r a m e t e r b of this equation of s t a t e i s p r o p o r t i o n a l to m o l e c u l a r v o l u m e ( b « h ) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume)(molecular energy (aocfh). Then, in order to apply the mixing rules introduced in this r e p o r t f o r the van d e r W a a l s equation of s t a t e we must r e p l a c e h with b and f w i t h a/b i n a l l t h e m i x i n g r u l e s . In T a b l e I m i x i n g r u l e s f o r t h e v a n d e r W a a l s e q u a t i o n o f s t a t e b a s e d on d i f f e r e n t t h e o r i e s of m i x t u r e s a r e r e p o r t e d . The combining rules f o r a ^ and b y (i*j) of this equation of s t a t e , c o n s i s t e n t w i t h eqs.12 w i l l be
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
MANSOORI
15.
Mixing Rules for Cubic Equations of State
325
T a b l e I: M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e One-Fluid
Mixing
Rules
a RMA
[2i2jX Xja jb j]^/2 i
i
i
2 2jX x a b 3]1/2
/ [
i
i
j
i j
i j
Theory{
^ , Theory{
vdW
^ i ^ i ^ ^ j ^ i j l i i . ^ i ^ j ^ i i ^iJ_
]
a = SiSiXiXiQii
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b = SiSjXiXjbij
HSE
a = SiSjXiXja^j
Theory{
b =
[2i2jX Xja j] /2 2jX Xja j /b j i
a = [a DEX
v d W
1
2
i
+(b/vRT)
2
i
i
i
2i2jX Xja j /b i
2
i
]/[ 1+a
i j
v
d
W
/vRT]
Theory{ b = SiSjXiXjbij a = SjSjXjXjaij
CSA
Theory{ 1+A
-|B*|/ ZiSjXiXjlB*^;
X X
A PuMl t i -TFhl u e iodr y {M i x i n g M b
B*
i j
=x (6 i
i j
+x A J
x x
b j/b) i
Rules
= SjXjaij vdW
Theory{ b
=
SjXjbij
= SjXjay HSE
Theory{ b
a
DEX
i
=
la j
=
A
X
X
I
V C
i
j
j
i
2
i
j V Y " K i / v R T ) SjXjajj^/b^j]/[ 1 +a i
bi
= SjXjbij
b
d W
V
/
v
R
T
^
ai
= SjXjaij
Theory{ 1+A
X
j
Theory{
CSA
A
[SjX a j]2/2 x a j /b j
=
PRTK
= pRTK
T
T
x
x
x
i
-1 -l
x i
« | B * | / 2jXj|B*lij;
B*i =x [6 j+x b j(A i/bi+A j/b )/2] j
i
i
= [2a(v-b)^-RTb(2v-b)]/[RTv
j
i
X
X
^-2a(v-b)^l
= [2ai(v-bi) -RTbi(2v-bi)]/[RTv -2ai(v-bi) l 2
2
2
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
j
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
326
a
ij = < - ij) ij( ii jj/t> 1
k
b
a
a
1 i
b )
1 / 2
;
)/2]
3
j j
(57) b
t j
= (1-£
1 j
)[(b
1 1
1 / 3
+b
j j
1 / 3
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IV.2. M i x i n g R u l e s f o r the R e d l i c h - K w o n g E q u a t i o n of S t a t e : The R e d l i c h - K w o n g equation of s t a t e (16) which i s an e m p i r i c a l m o d i f i c a t i o n o f t h e v a n d e r W a a l s e q u a t i o n c a n be w r i t t e n in the f o l l o w i n g f o r m
Z = Pv/RT = v / ( v - b ) -
a/[RT
3 / 2
(v+b)]
(58)
P a r a m e t e r b of t h i s e q u a t i o n o f s t a t e i s p r o p o r t i o n a l t o m o l e c u l a r v o l u m e (boch) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume) (molecular energy) or (ai2jX x a j(a/b)^3( _ i
DEX
j
i
1
.
/ 2 i 2 j X
/
[ ( a i j
4/3
X j a i j
p2/3(
b i
b / a
b i r
l/3
)273_|]^
Theory{ b = SiSjXjXjbij _ _ _ _ _ _ _ ^
CSA
Theory{ 1+A =|B*|/ S i S j X i X j l B * ^ ;
B*ij=
X X
Multi-Fluid
Mixing ai -
APM
XiCeij+xjAxxbij/b)
Rules [2jX a j
1 j
2/3b
1 j
4/3]5/2
/ 2 j X j a i j
2/3
b l j
10/3
Theorvl
,
ai =
[2jX
vdW Theory{
J
J
j
a
J
i
j
2/3
b
i
j
1/3]3/2
/
[
2
j
X
j
b
|
j
]
l/2
1 J
bi = SjXjbij
_______ HSE
Theory{
^
^ ^ ^ . z / S b y ^ S j Z ^ x j ^ ^ / S b y - l / S
=
ai = _ > j X a i j ( a / b ) ^ 3 { - ( j
DEX
1
[
a i j
/
)2"/3(
b i j
/
b i
a i
)2/3_
1 ] C i
}
Theory{ bi = SjXjbij
CSA
Theory{ 1+__
C Ci
x i
= |B*|/ _>jXj|B*lij ;
B*ij= x ( 6 j + x b j [ A i
i
j
i
x i
/b +A I
x j
/bj])
= (3/2)(a/bR)T "V/Untv/Cv+b)]-l/2; = C3/2)(ai/biR)T- £n[v/(v+b i ) ] - l / 2 3 / 2
A
X
A
X
X
I
= =
PRTK PRTK
T
x
T x 1
x
-1
-1
=-l+RT /2( 2_ 2)2/[ 3
=
v
b
-l+RT3/2( 2v
b l
2)2/[
R T
R T
{ ( v
{ ( v
v + b
))
v + b i
2_ (
)}2-
a
a i
2 v
+b)(v-b)
2
1
(2v+bi)(v-bi) ]
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
2
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
328
and T and P a r e the c r i t i c a l point t e m p e r a t u r e and p r e s s u r e , r e s p e c t i v e l y ; and u> i s t h e a c e n t r i c f a c t o r . In order to utilize the statistical mechanical mixing rules f o r the P e n g - R o b i n s o n equation of s t a t e we must f i r s t s e p a r a t e t h e r m o d y n a m i c v a r i a b l e s f r o m constants of this equation of s t a t e . F o r t h i s p u r p o s e we m a y w r i t e t h i s e q u a t i o n of s t a t e in t h e f o l l o w i n g f o r m c
c
Z = Pv/RT = v / ( v - b ) -
[(A/RT+C-2(AC/RT)
1 / 2
]/
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[(v+b)+(b/v)(v-b)]
(61)
w h e r e A = a ( l + 8 ) and C = a 6 / R T . This new f o r m of the P e n g - R o b i n s o n equation of s t a t e i n d i c a t e s that t h e r e e x i s t three independent constant p a r a m e t e r s in this equation w h i c h a r e A , b, a n d C. P a r a m e t e r s b and C a r e p r o p o r t i o n a l t o t h e m o l e c u l a r v o l u m e (boch a n d C « h ) w h i l e p a r a m e t e r A i s proportional to (molecular volume)(molecular energy) or (Aocfh). B a s e d on d i f f e r e n t t h e o r i e s o f m i x t u r e s m i x i n g r u l e s f o r this new f o r m of t h e Peng-Robinson equation of s t a t e a r e r e p o r t e d i n T a b l e III. The combining r u l e s f o r t h e unlike i n t e r a c t i o n p a r a m e t e r s of this equation of s t a t e a r e as t h e f o l l o w i n g 2
c
A
ij = ^- ij) ij( ii jj k
b
A
b
ij = (1-^ij)[( ii
c
ij =
b
A
c
/ b
1 / 3 + b
2
c
(62)
ii jj>
jj
b
1 / 3
)/2]
3
^ - ^ n i c ^ ^ c ^ ^ / z ] ^
(63)
(64)
Similar procedures t o those d e m o n s t r a t e d above can be used f o r d e r i v a t i o n of conformal solution mixing r u l e s f o r o t h e r cubic equations of s t a t e .
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
15.
MANSOORI
Mixing Rules for Cubic Equations of State
T a b l e HI: One-Fluid
329
M i x i n g Rules f o r the P e n a - R o b i n s o n Eg. of S t a t e Mixing Rules
A = [SiSjXiXjAijb^/^^jXiXjA^byS]!/? RMA Theory{ b = t Z i S j X i X j A i j b i j f / S i S j X i X j A i j b i j l J ^ C = [_> iSjXiXjAijCij^/SiSjXiXjAijCij]' ^ 7
A = 2i2jXiXjAjj _Ei_EjXiXjbij vdW Theory! b = 2i_£jXjXjCij C =
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A HSE Theory!
2i_!iXiXj Ajj - CSiSjXiXjAij^/SiSjXiXjA^/bij = [S i_>j x i Xj Aj j ] ^ /2 i2j x i Xj Ai j - /C i j C =
DEX Theory!
A - SiSjXiXjAijd-lCAij/bijXb/A)-!!^ = Si-SjX-jXjb-jj C SiSjXjXjCij
=
b
b
=
CSA Theory!
A = SiSiXjXjAjj xx=lB*I/ 2i_>jX |B*hj; C = 2i_!jXiXjCij 1+A
B* =x (6 j+x A b /b);
lXj
ij
i
i
j
xx
ij
Multi-Fluid Mixing Rules A APM Theory! b - [SjXjAijbijJ/^jXjAijbij]Ml C = C__ x A jC 3/2: x A C ]1/2 j
i
ij
j
j
ij
ij
SjXjAtj SjXjbij ZjXjCij
A vdW Theory! b C HSE Theory!
j
Ai = SjXjAij i 2j j ij] /_>jXjAij /bij Ci [ZjXjAylZ/SjXjAijZ/Cy
b
=
[
x
A
2
2
Ai = 2jXjAij{1-[CAij/bij)(b/A)-1]^ i) DEX Theory! i = 2 j j i j Ci - SjXjCij b
x
b
Ai = SjXjAij CSA Theory { 1+A = |B*[V __jX |B*| ; Ci = SjXjCij xi
j
B* =Xi[6ij+(x b /2)(A /bi+A /bj)]
ij
ij
j
ij
xi
A = P R T K - 1 = -HRT/{RTv v-b)^-2Av^/( v ^ + b ^ } A = p R T < - l = -HRT/{RTv /( v - b i ) - 2 A i V / ( v + b i ) } I = {[A-Y(ACRT)]/(2bRTv 2)}en[CY+b-b^2)/(v+b+bv 2)] + ^(ACRT)/!2[^(ACRT)-A]) Ci= {[Ai-v^(AiCiRT)]/(2biRTV2)}£n[(v+b bjV2)/(v+b i+bjV-)] +V(AiCiRT)/{2[V(A iCiRT)-Ail} x x
x i
T x x
2
T x i
2
3
r
2
2
2
r
r
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
xj
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS
330
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Acknowledgment: The a u t h o r t h a n k s P r o f e s s o r C a r o l H a l l f o r h e r h e l p f u l c o m m e n t s and c o r r e c t i o n s . This r e s e a r c h i s s u p p o r t e d b y t h e U.S. D e p a r t m e n t of E n e r g y G r a n t N o . DE-FG02-84ER13229.
Literature Cited 1. Massih, A.R.; Mansoori, G.A. Fluid Phase Equilibria 1983, 10, 57. 2. Brown, W.B. Proc. Roy. Soc. London Series A, 1957, 240;Phil. Trans. Roy. Soc. London Series A, 1957, 250. 3. Hill, T.L. "Statistical Mechanics" McGraw-Hill, New York, N.Y. 1956. 4. Kirkwood, J.G.; Buff, F. J. Chem. Phys. 1951, 19, 774. 5. Mansoori, G.A.; Ely, J. F. Fluid Phase Equilibria 1985, 22, 253. 6. Lan, S.S.; Mansoori, G.A. Int. J. Eng. Science 1977, 15, 323. 7. Leach, J.W.; Chappelear, P.S.; Leland, T.W. AIChE J. 1968, 14, 568; Proc. Am. Petrol. Inst. Series III, 1966, 46, 223. 8. Leland, T.W. Adv. Cryogenic Eng. 1976, 21, 466. 9. Mansoori, G.A.; Leland, T.W. J. Chem. Soc., Faraday Trans.II 1972, 68, 320. 10. Mansoori, G.A. J. Chem. Phys. 1972, 57, 198. 11. Rowlinson, J.S.; Swinton, F.L. "Liquids and Liquid Mixtures" 3rd Ed., Butterworths, Wolborn, Mass. 1982. 12. Scott, R.L. J. Chem. Phys. 1956, 25, 193. 13. Mansoori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W. J. Chem. Phys. 1971, 54, 1523. 14. Mansoori, G.A.; Ely, J.F. J. Chem. Phys. 1985, 82, 406. 15. Van der Waals, J.D. "Over de continuiteit van den gasen vloeistoftoestand" Leiden, 1873. 16. Redlich. O.; Kwong, J.N.S. Chem. Rev. 1949, 44, 233. 17. Peng, D.Y.; Robinson, D.B. Ind. Eng. Chem. Fundam. 1976, 15, 59. RECEIVED November
5, 1985
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.