Thermodynamics of Aqueous Systems with ... - ACS Publications

3 Phase Equilibria in Aqueous Electrolyte Solutions A. E . M A T H E R

1

and R. D. DESHMUKH

Downloaded by UNIV QUEENSLAND on May 26, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch003

University of Alberta, Edmonton, Alberta, Canada

Many of the undesirable substances present in gaseous or liquid streams form volatile weak electrolytes in aqueous solution. These compounds include ammonia, hydrogen sulfide, carbon dioxide and sulfur dioxide. The design and analysis of separation processes involving aqueous solutions of these materials require accurate representation of the phase equilibria between the solution and the vapor phase. Relatively few studies of these types of systems have been published concerning solutions of weak electrolytes. This paper will review the methods that have been used for such solutions and, as an example, consider the alkanolamine solutions used for the removal of the acid gases (HS and CO) from gas streams. In general, the formulation of the problem of vapor-liquid equilibria in these systems is not difficult. One has the mass balances, dissociation equilibria in the solution, the equation of electroneutrality and the expressions for the vapor-liquid equilibrium of each molecular species (equality of activities). The result is a system of non-linear equations which must be solved. The main thermodynamic problem is the relation of the activities of the species to be measurable properties, such as pressure and composition. In order to do this a model is needed and the parameters in the model are usually obtained from experimental data on the mixtures involved. Calculations of this type are well-known in geological systems (1) where the vapor-liquid equilibria are usually neglected. 2

al.

2

S o l u t i o n s o f Weak E l e c t r o l y t e s (2) m e a s u r e d t h e v a p o r p r e s s u r e s

Van Krevelen o f aqueous

et

C u r r e n t address: L e h r s t u h l f u r P h y s i k a l i s c h e Chemie I I , R u h r - U n i v e r s i t a t B o c h u m , 463o Bochum 1, W . - G e r m a n y 0-8412-0569-8/80/47-133-049$05.00/0 © 1980 American Chemical Society

In Thermodynamics of Aqueous Systems with Industrial Applications; Newman, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

50

THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS

s o l u t i o n s o f NH3+co2/ NH3+h2s and t h e t e r n a r y m i x t u r e NH3+co2+H2S, a t temperatures between 20 and 60 °C. The e x p e r i m e n t a l d a t a were o b t a i n e d i n NH3-rich s o l u t i o n s and d i d n o t extend t o t h e d i l u t e r e g i o n o f i n t e r e s t i n p o l l u t i o n c o n t r o l . In t h e i r t h e o r e t i c a l work, Van K r e v e l e n e t a l . used p s e u d o - e q u i l i b r i u m c o n s t a n t s , d e f i n e d as f o l l o w s : 0

γ

Η 8

%H

2

Κ

1

hs" "nhJ (1)

Ύ

HS Downloaded by UNIV QUEENSLAND on May 26, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch003

m 3

=Κ -

Ύ

+ ΝΗ„ 1

From t h e i r e x p e r i m e n t a l d a t a , v a l u e s o f K f o r t h e v a r i o u s e q u i l i b r i a were o b t a i n e d . I n some cases t h e v a l u e s were almost c o n s t a n t a t a g i v e n temperature, w h i l e i n o t h e r cases l o g Κ' was found t o be a l i n e a r function o f the i o n i c strength:

i In t h e i r model Van K r e v e l e n e t a l . n e g l e c t e d t h e second i o n i z a t i o n o f H S and t h e c o n c e n t r a t i o n o f OH" and H i o n s . While t h i s method was a b l e t o reproduce t h e ex p e r i m e n t a l d a t a , t h e model does n o t l e n d i t s e l f t o ex t r a p o l a t i o n t o r e g i o n s where d a t a were n o t o b t a i n e d and i t i s n o t u s e f u l f o r d i l u t e s o l u t i o n s as t h e a c c u r a c y o f t h e p r e d i c t i o n s i s poor. Van K r e v e l e n e t a l . used m o l a r i t y i n s t e a d o f m o l a l i t y as t h e measure o f t h e i r c o n c e n t r a t i o n s . The use o f m o l a l i t y here does n o t a l t e r t h e e s s e n t i a l f e a t u r e s o f t h e method. Dankwerts and M c N e i l (30 have employed t h e method o f Van K r e v e l e n e t a l . t o p r e d i c t t h e p a r t i a l p r e s s u r e o f carbon d i o x i d e over c a r b o n a t e d a l k a n o l a m i n e s o l u t i o n s . The c e n t r a l f e a t u r e o f t h i s model i s t h e use o f p s e u d o - e q u i l i b r i u m c o n s t a n t s and t h e i r dependence on i o n i c s t r e n g t h . The r a t i o o f t h e p s e u d o - e q u i l i b r i u m constant a t a c e r t a i n i o n i c s t r e n g t h t o t h a t a t zero i o n i c s t r e n g t h has been termed t h e " i o n i c c h a r a c t e r i z a t i o n f a c t o r " . However, i o n i c s t r e n g t h a l o n e i s i n s u f f i c i e n t t o determine t h e i o n i c c h a r a c t e r i z a t i o n f a c t o r s . As w e l l t h e i o n i c c h a r a c t e r i z a t i o n f a c t o r s a r e sometimes n o t a s i m p l e l i n e a r f u n c t i o n o f i o n i c strength. Lemkowitz e t a l . (4) used a s i m i l a r model t o t h a t proposed by Van K r e v e l e n e t a l . They c o r r e l a t e d t h e e q u i l i b r i a i n t h e C0 +NH3+Urea+H 0 system. The pseudoe q u i l i b r i u m c o n s t a n t f o r u r e a f o r m a t i o n , as w e l l as +

2

2

2


3.

MATHER AND DESHMUKH

51

Phase Equilibria

the v a p o r p r e s s u r e o f ΝΗ3 and the Henry's c o n s t a n t f o r CO2 were t r e a t e d as parameters and were determined by u s i n g the model t o p r e d i c t bubble p o i n t p r e s s u r e s , Kent and E i s e n b e r g (_5) a l s o c o r r e l a t e d s o l u b i l i t y d a t a i n the system H S+C02+alkanolamines+H20 u s i n g p s e u d o - e q u i l i b r i u m c o n s t a n t s based on m o l a r i t y . I n s t e a d o f u s i n g i o n i c c h a r a c t e r i z a t i o n f a c t o r s , they a c c e p t e d p u b l i s h e d v a l u e s o f a l l but two pseudoe q u i l i b r i u m c o n s t a n t s and found t h e s e by f i t t i n g d a t a f o r MEA and DEA s o l u t i o n s . They were a b l e t o o b t a i n e x c e l l e n t f i t s by t h i s approach and a l s o d i s c o v e r e d t h a t the f i t t e d p s e u d o - e q u i l i b r i u m c o n s t a n t s showed an A r r h e n i u s dependence on temperature. T h i s procedure o f lumping a l l n o n - i d e a l i t i e s i n t o a few a d j u s t a b l e parameters i s u n s a t i s f a c t o r y f o r many r e a s o n s . Thermodynamic r i g o r i s l o s t i f e x p e r i m e n t a l l y determined d i s s o c i a t i o n c o n s t a n t s o r vapor p r e s s u r e s are d i s r e g a r d e d . A l s o the parameters determined i n t h i s way a r e a c c u r a t e o n l y over the range o f v a r i a b l e s f i t t e d and u s u a l l y the model cannot be used f o r e x t r a p o l a t i o n t o o t h e r c o n d i t i o n s . The a t t r a c t i v e f e a t u r e o f t h e s e models i n the p a s t was t h e i r need f o r l i t t l e i n p u t i n f o r m a t i o n and the s i m p l e e q u a t i o n s c o u l d o f t e n be s o l v e d a l g e b r a i c a l l y . The f i r s t r i g o r o u s method f o r weak e l e c t r o l y t e s o l u t i o n s was t h a t o f Edwards e t a l . (5) . Because comparisons w i t h the models o f o t h e r workers w i l l be made, the thermodynamic framework w i l l be o u t l i n e d and the assumptions t h a t were made s t a t e d . F o r a s i n g l e s o l u t e which d i s s o c i a t e s o n l y i n the aqueous s o l u t i o n , the model i s based on f o u r p r i n c i p l e s : 1. Mass b a l a n c e on the e l e c t r o l y t e i n the l i q u i d phase. 2. The r a t i o o f the m o l e c u l a r t o the i o n i c c o n c e n t r a t i o n s o f the e l e c t r o l y t e i s d e t e r mined by the d i s s o c i a t i o n c o n s t a n t K. The a c t i v i t y i s r e l a t e d t o the m o l a l i t y through the a c t i v i t y c o e f f i c i e n t γ^:


2

a. = γ.πι.

(3)

where γ. -> 1 as ^m-> 0. The s u b s c r i p t j r e f e r s to a l l s o l u t e s p e c i e s . 3. Bulk e l e c t r o n e u t r a l i t y o f the l i q u i d phase. 4. F o r t h e m o l e c u l a r s o l u t e , e q u i l i b r i u m between the v a p o r phase and the l i q u i d phase i s g i v e n by: φ y Ρ a-*a Y

=

γ m Η 'a a a

(4)


52

THERMODYNAMICS

OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS

F o r t h e s o l v e n t , water, t h e v a p o r - l i q u i d e q u i l i b r i u m i s g i v e n ^ by: by ( P - P

S

)

(5)

s

φ y Ρ = a P % exp V w www ^ Y

RT

L

Edwards e t a l . (6) made t h e assumption t h a t was e q u a l t o a t t h e same p r e s s u r e and temperatâre. F u r t h e r they used t h e v i r i a l e q u a t i o n , t r u n c a t e d a f t e r the second term t o e s t i m a t e Ρpure a* These assumptions are s a t i s f a c t o r y when t h e t o t a l p r e s s u r e i s low o r when t h e mole f r a c t i o n o f the s o l u t e i n t h e vapor phase i s near u n i t y . F o r t h e water, t h e assumption was made t h a t φ , φ^, and t h e e x p o n e n t i a l term were u n i t y . These assumptions a r e v a l i d when t h e s o l u t i o n c o n s i s t s m o s t l y o f water and t h e t o t a l p r e s s u r e i s low. The a c t i v i t y c o e f f i c i e n t o f t h e e l e c t r o l y t e was c a l c u l a t e d u s i n g t h e extended Debye-Huckel t h e o r y :


pure a

ν

-A, ζ In γ . =

IF

+

2

1

L>

6

ik

^

(6)

kPw

In a p p l y i n g t h i s e q u a t i o n t o m u l t i - s o l u t e systems, t h e i o n i c c o n c e n t r a t i o n s a r e o f s u f f i c i e n t magnitude t h a t m o l e c u l e - i o n and i o n - i o n i n t e r a c t i o n s must be c o n s i d e r e d . Edwards e t a l . (β) used a method p r o p o s e d by Bromley Ç7) f o r t h e e s t i m a t i o n o f t h e β p a r a m e t e r s . The model was found t o be u s e f u l f o r t h e c a l c u l a t i o n o f m u l t i - s o l u t e e q u i l i b r i a i n t h e NH3+H9S+H2O and NH3+CO2+H2O systems. However, because o f t h e assumptions r e g a r d i n g t h e a c t i v i t y o f t h e water and the use o f o n l y two-body i n t e r a c t i o n parameters, t h e model i s s u i t a b l e o n l y up t o m o l e c u l a r c o n c e n t r a t i o n s o f about 2 m o l a l . As w e l l t h e temperature was r e s t r i c t e d t o t h e range 0 ° t o 100 oc because o f t h e e q u a t i o n s used f o r t h e Henry's c o n s t a n t s and t h e d i s s o c i a t i o n c o n s t a n t s . I n a l a t e r s t u d y , Edwards e t a l . (8) extended t h e c o r r e l a t i o n t o h i g h e r concen t r a t i o n s (up t o 10 - 20 molal) and h i g h e r tempera t u r e s (0° t o 170 ° C ) . I n t h i s work t h e a c t i v i t y c o e f f i c i e n t s o f t h e e l e c t r o l y t e s were c a l c u l a t e d from an e x p r e s s i o n due t o P i t z e r (9):

In γ. 1 1

/Γ Φ 1

[1+1.2 /τ

+ — ln(1+1.2 Γϊ) 1.2 '

(1)

+ 2 1

3) 3

-

J

F

1-Π+2 >/ï)exp(-2/l)j

L


3.

MATHER

A N D DESHMUKH

53

Phase Equilibria

" 7 ? ζ S "ϋ * co c o 2

M

RR'NH

C0

" W n C O O

m

" W ' N H

2

M

S= "Ή*

m

+

H

S =

7

Y

m

/

(19) 2

H S ~ "hS"

Y

C0= H+ C 0 = ' HCO" "HiCO-

=

H

co

YuH Sq ? = H J I

2

Y

co

2

m

co

γ

H

2

(21)

(22) 2

(23)

m ν

s

(20)

'2"

b y Ρ = a P Φ, , exp w w w w w w K

C0

-

(P-pS)

8

r

RT

w

(24)


56

THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL

APPLICATIONS

V a l u e s o f t h e d i s s o c i a t i o n c o n s t a n t s and Henry's c o n s t a n t s were d e t e r m i n e d from t h e l i t e r a t u r e . The f u g a c i t y c o e f f i c i e n t s , ^, were c a l c u l a t e d u s i n g t h e Peng-Robinson (J_9) e q u a t i o n o f s t a t e . The a c t i v i t y o f the s o l v e n t , w a t e r , was s e t e q u a l t o i t s mole f r a c t i o n . A l s o t h e f u g a c i t y c o e f f i c i e n t o f water a t i t s v a p o r p r e s s u r e , φ** and t h e P o y n t i n g c o r r e c t i o n were assumed t o be u n i t y . The model i s hence r e s t r i c t e d t o r e l a t i v e l y d i l u t e s o l u t i o n s , b u t t h i s r e s t r i c t i o n c a n be r e moved by d e t e r m i n i n g t h e e x p r e s s i o n f o r a u s i n g t h e Gibbs-Duhem e q u a t i o n , as shown by Edwards e t a l . ( 8 ) . The a c t i v i t y c o e f f i c i e n t s o f t h e s o l u t e s p e c i e s have been d e t e r m i n e d from t h e e x t e n d e d Debye-Huckel ex p r e s s i o n g i v e n by Guggenheim ( 2 0 ) , E q u a t i o n ( 6 ) . T h i s e q u a t i o n was used by Edwards e t a l . (6). The major problem i n a p p l y i n g i t t o a l k a n o l a m i n e s o l u t i o n s i s the e s t i m a t i o n o f t h e B's s i n c e t h e p r o c e d u r e o f Bromley cannot be used as t h e i n p u t parameters - t h e i o n i c e n t r o p i e s o r s a l t i n g - o u t parameters have n o t been d e t e r m i n e d f o r ethanolammonium o r carbamate i o n s . The f o l l o w i n g b a l a n c e e q u a t i o n s f o r t h e r e a c t i n g s p e c i e s c a n be formed


w

Electroneutrality "rr'NHJ

+

'V

=

m

0H~

+

m

HS-

+

^ 0 3

+

"rr'NCOO"

+

2

+

2

m

S=

co=

m

( 2 5 )

Mass b a l a n c e s m

A

m

A

m

=

"rr'NH

°C0

a

= 2

m

+

Itl

C0

RR'NH+ +

2

" W n C O O "

"ΉΟΟ-

2

A H S » "^S

+

+

+

+

m

C0=

+

m

(

RR'NC00-

m

2

(

"HS" S=

(

2

7

)

2

6

8

)

)

Here OLQQ and CXHOS a r e t h e mole r a t i o s i n t h e l i q u i d phase xcarbon t o n i t r o g e n and s u l f u r t o n i t r o g e n ) and a r e t h e e x p e r i m e n t a l l y measured c o n c e n t r a t i o n s . The m a t h e m a t i c a l p r o b l e m i s t o s o l v e E q u a t i o n s (15) t o ( 2 8 ) . Twelve s p e c i e s e x i s t : H S , C 0 , RR'NH, 2

2


3.

MATHER AND DESHMUKH

C 0

57

Phase Equilibria

R R , N H

HS~, S , HCO3, 3' 9 ' RR'NCOO , Η , OH and H 0 . Hence t h e r e a r e t w e n t y - t h r e e unknowns (n^ and Yi f o r a l l s p e c i e s e x c e p t water p l u s x ~ ) . To s o l v e f o r the unknowns t h e r e a r e t w e n t y - t h r e e independent e q u a t i o n s : Seven c h e m i c a l e q u i l i b r i a , t h r e e mass b a l a n c e s , e l e c t r o n e u t r a l i t y , t h e use o f E q u a t i o n (6) f o r t h e e l e v e n a c t i v i t y c o e f f i c i e n t s and t h e phase e q u i l i b r i u m f o r x . The problem i s one o f s o l v i n g a system o f non l i n e a r a l g e b r a i c e q u a t i o n s . Brown's method (2J_, 22) was used f o r t h i s purpose. I t i s an e f f i c i e n t p r o c e d u r e , based on a p a r t i a l p i v o t i n g t e c h n i q u e , and i s a n a l o gous t o G a u s s i a n e l i m i n a t i o n i n l i n e a r systems o f equations. The a p p l i c a t i o n o f t h i s model t o a l k a n o l a m i n e s o l u t i o n s i s not p o s s i b l e d i r e c t l y since the s p e c i f i c i n t e r a c t i o n parameters (B's) f o r alkanolammonium i o n s and carbamate i o n s a r e n o t a v a i l a b l e . A l s o t h e d i s s o c i a t i o n c o n s t a n t f o r t h e s i m p l e s t amines (MEA, DEA, TEA) i s known o n l y o v e r t h e range o f temperatures between 0 ° and 50 °C and t h e e q u i l i b r i u m c o n s t a n t f o r carbamate f o r m a t i o n i s known o n l y a t 18 °C f o r MEA and DEA. In monoethanolamine s o l u t i o n s t h e unknown i n t e r a c t i o n parameters and e q u i l i b r i u m c o n s t a n t s were determined by f i t t i n g t h e model t o d a t a f o r t h e t h r e e component systems C0 +MEA+H 0 and HjS+MEAH^O. The agreement o f t h e f i t t e d model w i t h t h e d a t a was found t o be good. The parameters o b t a i n e d i n t h i s way were then used t o p r e d i c t t h e p a r t i a l p r e s s u r e s o f m i x t u r e s o f H^S and C 0 over aqueous MEA s o l u t i o n s . The p r e d i c t i o n s were i n good agreement w i t h e x p e r i m e n t a l d a t a , except a t the higher p a r t i a l pressures. T h i s p r o c e d u r e c o u l d n o t be employed f o r d i i s o p r o panolamine (DIPA) s o l u t i o n s s i n c e d a t a were a v a i l a b l e o n l y f o r one amine c o n c e n t r a t i o n a t two temperatures. I n t h i s case d a t a f o r m i x t u r e s o f H S+C0 +DIPA+H90 were used t o g e t h e r w i t h t h e d a t a f o r H S+DIPA+H 0 and C0 +DIPA+H 0 t o o b t a i n t h e i n t e r a c t i o n parameters and e q u i l i b r i u m c o n s t a n t s . The r e s u l t s a r e shown i n F i g u r e s 1 and 2 t o be i n good agreement w i t h t h e ex p e r i m e n t a l d a t a (23). I n t h i s c a s e , however, i n con t r a s t t o t h e case o f MEA, t h e p r e d i c t i o n s use p a r a meters e v a l u a t e d from d a t a f o r t h e f o u r component system. 2


w

2

2

2

2

2

2

2

2

2

Conclusions The c o r r e 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 i n aqueous s o l u t i o n s o f weak e l e c t r o l y t e s i s important f o r the s e p a r a t i o n o f u n d e s i r a b l e components from gases and l i q u i d s . The major problem i n such c o r r e l a t i o n s i s t h e e s t i m a t i o n o f t h e a c t i v i t y



THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS

Figure 2.

Effect of H S on the solubility of C0 (( ) experimental (23); ( 2

2

in 2.5N DIPA solutions at 100°C ) predicted


3.

MATHER AND DESHMUKH

59

Phase Equilibria

c o e f f i c i e n t s o f the i o n i c s p e c i e s and a l t h o u g h a number o f models have been proposed, the d e t e r m i n a t i o n o f the parameters i n a new case i s not a s i m p l e m a t t e r As w e l l d i s s o c i a t i o n c o n s t a n t s and Henry's c o n s t a n t s f o r a s p e c i e s must be a v a i l a b l e over the temperature range o f i n t e r e s t . Both t h e s e problems o c c u r i n the a p p l i c a t i o n o f the fundamental thermodynamics t o a l k a n o l a m i n e s o l u t i o n s c o n t a i n i n g H S and C 0 . However, by u s i n g l i m i t e d e x p e r i m e n t a l d a t a , the parameters i n the model may be o b t a i n e d and the r e p r e s e n t a t i o n o f the e q u i l i b r i a i s good o v e r the range o f importance i n i n d u s t r i a l processes. Downloaded by UNIV QUEENSLAND on May 26, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch003

2

a = Αψ = Η = I = Κ = m = Ρ = R = Τ = ν = y = =

2

Nomenclature activity Debye-Huckel parameter Henry's c o n s t a n t f o r m o l e c u l a r s o l u t e i o n i c s t r e n g t h = o.5 /Hiti^z^ e q u i l i b r i u m constant m o l a l i t y , mole k g " p r e s s u r e , Pa _^ gas c o n s t a n t , J mol Κ temperature, Κ 3 „1 p a r t i a l molar volume, cm mol vapor phase mole f r a c t i o n i o n i c charge on s p e c i e s i 1

Greek α β, β'°', φ

letters = mole r a t i o i n the l i q u i d phase, mole/mole amine βΠ) = i n t e r a c t i o n parameters, kg mol = vapor phase f u g a c i t y c o e f f i c i e n t

Superscripts saturation

s

=

'

=

a A i,

S u b s c r i p t s= = j, k =

w 1c, 2c 1y, 2y

pseudo-equilibrium

= = =

constant

molecular species carbamate e q u i l i b r i a , amine s p e c i e s o r component, amine equilibria water carbonic acid e q u i l i b r i a hydrogen s u l f i d e e q u i l i b r i a


60

THERMODYNAMICS

OF

AQUEOUS

SYSTEMS

WITH

INDUSTRIAL

APPLICATIONS


Literature cited

1. Crerar, D.A. Geochim. Cosmochim. Acta, 1975, 39, 1375. 2. Van Krevelen, D.W.; Hoftijzer, P.J.; Huntjens, F.J. Rec. Trav. Chim., 1949, 68, 191. 3. Dankwerts, P.V.; McNeil, K.M. Trans. Inst. Chem. Eng., 1967, 45, T32. 4. Lemkowitz, S.M.; de Cooker, M.G.R.T.; Van den Berg, P.J. J. Appl. Chem. Biotechnol., 1973, 23, 63. 5. Kent, R.L.; Eisenberg, B. Hydrocarbon Processing, 1976, 55, (2), 87. 6. Edwards, T.J.; Newman, J.; Prausnitz, J.M. Α.I.Ch.E.J., 1975, 21, 248. 7. Bromley, L.A. J. Chem. Thermo., 1972, 4, 669. 8. Edwards, T.J.; Maurer, G.; Newman, J.; Prausnitz, J.M. Α.I.Ch.E.J., 1978, 24, 966. 9. Pitzer, K.S. J. Phys. Chem., 1973, 77, 268. 10. Nakamura, R.; Breedveld, G.J.F.; Prausnitz, J.M. Ind. Eng. Chem. Proc. Des. Dev., 1976, 15, 557. 11. Beutier, D.; Renon, H. Ind. Eng. Chem. Proc. Pes. Dev., 1978, 17, 22o. 12. Edwards, T.J.; Newman, J.; Prausnitz, J.M. Ind. Eng. Chem. Fundam., 1978, 17, 264. 13. Cruz, J.-L.; Renon, Η. Α.I.Ch.E.J., 1978, 24, 817. 14. Cruz, J.-L.; Renon, H. Ind. Eng. Chem. Fundam., 1979, 18, 168. 15. Atwood, K.; Arnold, M.R., Kindrick, R.C. Ind. Eng. Chem., 1957, 49, 1439. 16. Klyamer, S.D.; Kolesnikova, T.L. Zhur. Fiz. Khim., 1972, 46, 1o56. 17. Klyamer, S.D.; Kolesnikova, T.L.; Rodin, Yu.A. Gazov. Prom., 1973, 18(2), 44. 18. Deshmukh, R.D.; Mather, A.E. Chem. Eng. Sci. (in press). 19. Peng, D.-Y.; Robinson, D.B. Ind. Eng. Chem. Fundam., 1976, 15, 59. 20. Guggenheim, E.A. Phil. Mag., 1935, 21, 588. 21. Brown, K.M. SIAM J. Numer. Anal., 1969, 6, 56o. 22. Brown, K.M. in Numerical Solution of Nonlinear Algebraic Equations, ed. by G.D. Byrne and C.A. Hall, Academic Press, 1973, p. 281. 23. Isaacs, E.E.; Otto, F.D.; Mather, A.E. Can. J. Chem. Eng., 1977, 55, 21o. RECEIVED

January 31, 1980.


Thermodynamics of Aqueous Systems with ... - ACS Publications

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