12
Application of a New Local Composition Model in the Solution Thermodynamics of Polar and Nonpolar Fluids 1
2
Downloaded by MICHIGAN STATE UNIV on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012
M. H. Li, F. T. H. Chung , C.-K. So , L. L. Lee, and Κ. E. Starling School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019
A local composition model in solution thermodynamics is developed for the calculation of vapor-liquid equilibria of mixtures of molecules vastly different in size, polarity and strength of interaction. The concepts of nearest neighbor numbers, coordination shell, and pair interaction energies are interpreted in terms of modern liquid theory. For broad ranged applications, an accurate equation of state is introduced in the spirit of the mean density approximation by differentiation of the Helmholtz free energy. Calculations of the vapor-liquid equilibria of 83 binary and ternary systems, including nonpolar hydrocarbons, hydrogen-bonding alcohols, water, ammonia, and carbon dioxide show good agreement with experimental data. S i n c e i t s i n t r o d u c t i o n , the l o c a l c o m p o s i t i o n model (LCM) f o r the e x c e s s f r e e energy has been u s e d i n s o l u t i o n thermodynamics f o r the c a l c u l a t i o n o f v a p o r l i q u i d e q u i l i b r i a of h i g h l y n o n i d e a l m i x t u r e s . The systems i n c l u d e d p o l a r f l u i d s ( s u c h as a l c o h o l s , water and a c e t o n e ) (1-3) . The f o r m u l a t i o n was i n a form s u i t a b l e f o r c a l c u l a t i o n of a c t i v i t y c o e f f i c i e n t s . However, the o r i g i n a l method was n o t u s e f u l f o r the c a l c u l a t i o n of d e n s i t i e s . I t a l s o d i d n o t a p p l y to t h e near c r i t i c a l r e g i o n . On t h e o t h e r hand, m i x t u r e 'Current address: NIPER, PO Box 2128, Bartlesville, OK 74005 Current address: Aspen Tech, 251 Vassar St., Cambridge, MA 02139
2
0097-6156/86/0300-0250S08.75/0 © 1986 American Chemical Society
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
Downloaded by MICHIGAN STATE UNIV on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012
12.
LI ET A L .
New Local Composition Model of Fluids
251
p r o p e r t i e s have been p r e d i c t e d by u s i n g e q u a t i o n s o f s t a t e w i t h m i x i n g r u l e s f o r the e q u a t i o n p a r a m e t e r s ( 4 ) . The p r e d i c t e d p r o p e r t i e s i n c l u d e d d e n s i t y , e n t h a l p y and v a p o r - l i q u i d e q u i l i b r i u m (VLE). The o v e r l a p p i n g a p p l i c a t i o n s o f t h e s e methods have c o e x i s t e d f o r many y e a r s w h i l e t h e two a p p r o a c h e s remained s e p a r a t e . Only r e c e n t l y have t h e r e been e f f o r t s t o a s s i m i l a t e the u s e f u l f e a t u r e s o f the l o c a l c o m p o s i t i o n a c t i v i t y c o e f f i c i e n t model i n t o e q u a t i o n s o f state(5.). The t h e o r e t i c a l groundwork c o n n e c t i n g b o t h a p p r o a c h e s was l a i d by L e e , et a l . (6) where a m o l e c u l a r t h e o r y was e s t a b l i s h e d f o r the LCM. In t h i s work, we d e m o n s t r a t e the f e a s i b i l i t y of the a p p r o a c h by c a l c u l a t i n g f o r the v a p o r - l i q u i d e q u i l i b r i a as w e l l as the d e n s i t i e s o f h i g h l y n o n i d e a l m i x t u r e s . We f i n d t h a t l o c a l c o m p o s i t i o n m i x i n g r u l e s p e r f o r m b e t t e r than c o n f o r m a i s o l u t i o n m i x i n g r u l e s f o r most of the systems studied.
Thfvry New F r e e E n e r g y M o d e l . The c o n v e n t i o n a l p r a c t i c e i n s o l u t i o n thermodynamics i s t o employ e q u a t i o n s o f s t a t e to d e s c r i b e the gaseous s t a t e of m i x t u r e s w h i l e u s i n g the s o - c a l l e d a c t i v i t y c o e f f i c i e n t m o d e l s ( e . g . , van L a a r , M a r g u l e s , R e d l i c h - K i s t e r ) t o d e s c r i b e the l i q u i d mixtures. We a t t e m p t a s y n t h e s i s h e r e by c o m b i n i n g t h e a c t i v i t y c o e f f i c i e n t model w i t h the e q u a t i o n o f s t a t e . The b a s i c a p p r o a c h i s o u t l i n e d below, and d i s c u s s i o n s t h a t f o l l o w w i l l g i v e the d e t a i l s . Outline
of T h e o r e t i c a l
Steps
1.
L o c a l c o m p o s i t i o n e x p r e s s i o n o f energy, (by i n t e g r a t i o n : A/NkT = / dp (U/N))
U *
2.
H e l m h o l t z f r e e energy (H.F.E.) (by the r e l a t i o n : Ρ = - 3 A / d V l )
A *
The pre s s u r e
Ρ
T
3.
We f i r s t g i v e a s t a t i s t i c a l m e c h a n i c a l definition of t h e l o c a l c o m p o s i t i o n s . F o r s i m p l i c i t y , we c o n s i d e r a b i n a r y m i x t u r e of s p h e r i c a l m o l e c u l e s o f t y p e s A and B. The number of n e i g h b o r i n g Β m o l e c u l e s s u r r o u n d i n g a c e n t r a l A m o l e c u l e i s g i v e n i n terms o f the r a d i a l d i s t r i b u t i o n function, 8βΑ^ ^' L Γ
r
BA - PB V BA where L i s t h e range ( r a d i u s ) o f t h e c o o r d i n a t i o n . S i m i l a r l y , t h e number of n e i g h b o r s A s u r r o u n d i n g t h e center A i s n
( L )
4
n
r
g
( r )
(
1
)
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
252
n
EQUATIONS OF STATE: THEORIES AND APPLICATIONS
r
« PA V
L
AA< >
4
2
"
g
(
AA
2
)
We c a l l d n e a r e s t n e i g h b o r numbers. The c o o r d i n a t i o n number of the c e n t e r A i s t h e n t h e sum of i t s A n e i g h b o r s and Β n e i g h b o r s , a
Z
n
A " AA
+
n
n
Π
β
Α
t
n
e
(
BA
3
)
As a consequence of d e f i n i t i o n s E q u a t i o n s 1-3, ' l o c a l c o m p o s i t i o n s ' of Β m o l e c u l e s and A m o l e c u l e s s u r r o u n d i n g a c e n t e r of type A a r e
Downloaded by MICHIGAN STATE UNIV on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012
X
B A
=
B A
N
/
Z
A
-
B A
N
/
(
N
A A
+
N
BA>
(
4
the
)
and X
n
AA
" AA
/ z
n
A
n
- AA/< AA
+
n
S u b s t i t u t i n g the i n t e g r a l and 5, we have
X
B A
=
'S
BAdr
P B
Jo
[PA
P
+
X
A A
A
[PA
P
+
If X
A
=
χ
Λ
r
r
"
AAdr
Β ΒΑ
^0
/ ( χ
N
R
A A d r
^0
B
4
2
* A A
]
/
A A < ' )
" «BA] 2
through Α
* B A
e
B BA *P ( χ
X
α
Α/ Α>
0
1
9
)
Under weak c o n d i t i o n s ( 6 ) , one can show t h a t E q u a t i o n i s c o n s i s t e n t w i t h E q u a t i o n 16 t h r o u g h the G i b b s Helmholtz r e l a t i o n . T h i s e q u a t i o n s a t i s f i e s the pure f l u i d l i m i t s of A°/NkT =
(l/a )
f°A
A
/ k T
(
2
0
19
)
E q u a t i o n 19 s e r v e s as a f r e e energy model f o r future calculations. Next we s h a l l l o o k a t the m i x i n g r ul e s · N - F l n i d T h e o r i e s and P r e u j a r e s . When an e q u a t i o n of s t a t e d e v e l o p e d f o r a pure s u b s t a n c e i s e x t e n d e d t o m i x t u r e s , one of the q u e s t i o n s i s the c o m p o s i t i o n dependence of the new e q u a t i o n . T h i s dependence i s u s u a l l y i n c o r p o r a t e d t h r o u g h m i x i n g r u l e s a p p l i e d t o the s t a t e v a r i a b l e s ( d e p e n d e n t , i n d e p e n d e n t or b o t h ) and/or the p a r a m e t e r s of the e q u a t i o n . A fundamental u n d e r s t a n d i n g o f c o m p o s i t i o n dependence can be g a i n e d t h r o u g h s t u d y i n g some model m i x t u r e s i n s t a t i s t i c a l mechani c s . The v i r i a l p r e s s u r e e q u a t i o n f o r a b i n a r y m i x t u r e of m o l e c u l e s o f type A and Ng m o l e c u l e s o f type Β i s g i v e n i n terms o f the p a i r p o t e n t i a l s and p a i r c o r r e l a t i o n f u n c t i o n s ( p c f ) as P/(pkT) = 1 - βρ/6
χ
2$»i îi ~ x
1
This
equation
d
r ( a u
3 3
serves
ii 3
/ d r
i i > 8 i i ( r ; Τ , ρ,χ) i , j = A,B
as our
3
3
starting
point
(21)
of
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
Downloaded by MICHIGAN STATE UNIV on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012
12. LI ET A L .
255
New Local Composition Model of Fluids
e x a m i n a t i o n of the m i x i n g r u l e s . We note t h a t the c o m p o s i t i o n dependence has an ' e x p l i c i t ' p a r t , under the d o u b l e summation ( u n d e r l i n e d by ^ and an ' i m p l i c i t ' p a r t , c o n t a i n e d i n the f u n c t i o n a l dependence of the p a i r c o r r e l a t i o n f u n c t i o n s , Sjj» A t t e m p t s have been made to a p p r o x i m a t e E q u a t i o n 21 oy the s o - c a l l e d van der Waals η-fluid t h e o r i e s ( i . e . o n e - f l u i d , twof l u i d and t h r e e - f l u i d t h e o r i e s ) where the c o m p o s i t i o n dependence i s s i m p l i f i e d and the p c f ' s a r e e v a l u a t e d a t r e d u c e d s t a t e s c h a r a c t e r i z e d by the c o r r e s p o n d i n g energy and s i z e p a r a m e t e r s . In the f o l l o w i n g we s h a l l examine f i r s t the e f f e c t s of m o l e c u l a r s i z e s on m i x i n g r u l e s i n the absence of a t t r a c t i v e e n e r g i e s ( e . g . i n h a r d sphere m i x t u r e s ) , and t h e n of a t t r a c t i v e e n e r g i e s i n a d d i t i o n to the s i z e d i f f e r e n c e s ( e . g . i n m i x t u r e s of L e n n a r d Jones m o l e c u l e s ) . The f o r m u l a t i o n due to H e n d e r s o n and L e o n a r d (9) f o r m u l t i f l u i d t h e o r i e s i s summarized below. Hard Sphere M i x t u r e s . For h a r d s p h e r e m i x t u r e s , the difference i n species i s manifested i n size. We s h a l l denote the b i g sph e r e s as t h o s e w i t h d i a m e t e r d ^ and s m a l l s p h e r e s w i t h d i a m e t e r ά" . The c r o s s i n t e r a c t i o n i s c h a r a c t e r i z e d by the d i a m e t e r d ^ ( (
x )
Ξ
8o< BBiP BB>
AB< AB'P'*>
Z
«o< AB'P AB>
AA< AA-P'X> BB
( d
;
BB P'
d
d
(
2
4
)
d
d
(
2
5
)
d
d
(
2
6
)
and g
d
In the made, (d
two
fluid
x
*AA AA-P' >
Ξ
theory,
d
d
the f o l l o w i n g
*o< xA-P xA>
assumptions
(
are
2
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
7
)
256
g
EQUATIONS OF STATE: THEORIES AND APPLICATIONS
( d
;
Z
BB BB P'*>
*o
( d
;
d
xB P xB>
(
2
8
)
(
2
9
)
and 8
AB< AB'P'»)
=
D
8 B B ] /
+
[*AA
2
where
=Σ
*h and
Downloaded by MICHIGAN STATE UNIV on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch012
d
xj
djA
(3°>
Σ xj
*jB
< >
i
xB
=
Finall^, g
i j
(
d
the
;
i j P ' *
)
31
one-fluid S
g
d
theory
i s simply
given
d
by (
o< x'P x>
3
2
)
where
d
x
2*i*J* iJ 3
-
33
< >
We ca examine the v a l i d i t y of a l l t h r e e m o d e l s by c o m p a r i n g the p r e s s u r e v a l u e s o b t a i n e d above w i t h known simulation r e s u l t s f o r hard spheres. For example, the c o n t a c t v a l u e s a r e a c c u r a t e l y g i v e n by the C a r n a h a n S t a r l i n g (C-S) f o r m u l a , g
+
]
4
1
)
In this e q u a t i o n the approximation i s made t h a t f o r mixtures the potentials of mean force are set proportional to the component H e l m h o l t z free energies (see Hill(i4)). To obtain the pressure, we d i f f e r e n t i a t e A' w i t h r e s p e c t to volume, a c c o r d i n g t o Ρ
-ΘΑ/aV
=
thus P
=
X
I
(42)
T
obtaining P
A
+
Χ
Β
U -ri
M to
a
M
d
-M d
o -ri
03 O
o
M 1
rd +*
P« M to
co
Ο
to
tH
Qi
u
>
M w Ζ
o o fH
αϊ
Ο
d Ρ< d at ο 03
U U U Μ Λ to « M 03 to 1 © -M P i -Μ -M 04 M +·> es*-» •M 1 •M 03 Ο o) to I ©
0k
o d
co co
fH
o •o
o
1
d d © CO
co
*d
•H d d κ o o C0 «M
«
Ο
1 d 1 «H OS fH Ο ο d ο d
d o o co co •4·» •M O O
fH
ι j
jQ '
a
O
+> CO
1 1 C1
j
fH M
fH «n 00
j
CD
00 oo d d •H •H H
•H 10 o o
a fH o 03 o a
Ο
•i-4
Ο
d o •ri •P
•M -M d •H fH CO O
I
n ο © ο Ό 1 «Η d
