Activity Coefficients at High Concentrations in Multicomponent Salt

37 Activity Coefficients at High Concentrations in Multicomponent Salt Systems 1

Downloaded by CORNELL UNIV on May 29, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch037

EDWARD W. FUNK Corporate Research—Science Laboratories, Exxon Research and Engineering Company, Linden, NJ 07036

Phase behavior in concentrated aqueous electrolyte systems is of interest for a variety of applications such as separation processes for complex salts, hydrometallurgical extraction of metals, interpretation of geological data and development of high energy density batteries. Our interest in developing simple thermodynamic correlations for concentrated salt systems was motivated by the need to interpret the complex solid-liquid equilibria which occur in the extraction of sodium nitrate from complex salt mixtures which occur in Northern Chile (Chilean saltpeter). However, we believe the thermodynamic approach can also be applied to other areas of technological interest. Understanding of phase behavior in concentrated salts systems requires liquid-phase activity coefficients for the electrolytes and for water in the multicomponent system. Although there is a large number of experimental data (]_,2_,_3) for ternary aqueous electrolyte systems, few equations are available to correlate the activity coefficients of these systems in the concentrated region. The most successful present techniques are those dis cussed by Meissner and co-workers (4,5) and Bromley (£) Our approach is different from previous methods in two basic aspects. First, we define our standard state as the saturated solution and, second, we define our activity coefficients in a way similar to that commonly used for nonelectrolytes. This approach allows a simple thermodynamic treatment of the concentrated region although the approach is not appropriate, either practically or theoretically for the highly dilute region. To illustrate the use of our thermodynamic treatment for concentrated systems, we have selected the system HCI-NaCl-hLO where there are complete and precise thermodynamic data for the ternary system and the constituent binaries from low molality 1

Work done at Universidad Técnica del Estado, Santiago, Chile. 0-8412-0569-8/80/47-133-717$05.75/0 © 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.

718

T H E R M O D Y N A M I C S OF AQUEOUS SYSTEMS W I T H INDUSTRIAL

APPLICATIONS


up t o h i g h l y c o n c e n t r a t e d s o l u t i o n s . F i g u r e 1 shows t h e mean i o n i c a c t i v i t y c o e f f i c i e n t s o f HCI a n d NaCl a s f u n c t i o n s o f t h e t o t a l m o l a l i t y a t 2 5 ° C ; t h i s f i g u r e shows t h a t 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 show a l a r g e change w i t h composition i n the concentrated region. We p r e s e n t r e s u l t s d e s c r i b i n g t h e s o l i d - l i q u i d and t h e v a p o r - l i q u i d e q u i l i b r i a i n t h e NaCl-HCI-HpO s y s t e m . In t h e f i r s t p a r t , p u r e l y e m p i r i c a l r e l a t i o n s a r e used t o d e s c r i b e t h e a c t i v i t y c o e f f i c i e n t s and t h e s e c o n d p a r t i n c l u d e s use o f a s e m i - e m p i r i c a l model (2) t o d e s c r i b e t h e c o m p o s i t i o n a l dependence o f t h e a c t i v i t y c o e f f i c i e n t s . The f i n a l s e c t i o n o f t h e p a p e r d i s c u s s e s a r e a s o f a p p l i c a t i o n s o f t h i s thermodynamic t e c h n i q u e s t o systems o f p r a c t i c a l i n t e r e s t and a l s o l i m i t a t i o n s o f o u r a p p r o a c h . Thermodynamics o f D i l u t e

Solutions

The a c t i v i t y c o e f f i c i e n t s i n d i l u t e aqueous s o l u t i o n s ( m o l a l i t i e s l e s s t h a n 0 . 2 ) can be d e s c r i b e d u s i n g t h e B r o n s t e d Guggenheim t h e o r y The e q u a t i o n s f o r t h e a c t i v i t y c o e f f i c i e n t s a r e d e r i v e d by f i r s t d e f i n i n g t h e e x c e s s G i b b s e n e r g y as t h e d i f f e r e n c e between t h e G i b b s e n e r g y o f t h e s o l u t i o n and t h a t when each i o n i s i n i t s s t a n d a r d s t a t e of i n f i n i t e d i l u t i o n . For convenience, t h i s e x p r e s s i o n i s d i v i d e d i n t o two p a r t s ; t h e f i r s t l e a d s t o t h e Debye-Huckel l i m i t i n g l a w , and t h e second i s t h e c o r r e c t i o n t o t h e DebyeHuckel t h e o r y . The f i r s t p a r t o f t h e e x c e s s G i b b s e n e r g y i s n o t w r i t t e n e x p l i c i t l y , s i n c e i t i s n o t p o s s i b l e t o form a s i m p l e , i n t u i t i v e expression that gives, a f t e r appropriate d i f f e r e n t i a t i o n , the Debye-Huckel l i m i t i n g l a w . However, t h e s e c o n d p a r t o f t h e e x c e s s G i b b s e n e r g y i s v e r y s i m i l a r t o t h e Wohl e x p a n s i o n u s e d f o r n o n - e l e c t r o l y t e s o l u t i o n s (1). Considering a ternary aqueous s o l u t i o n o f i o n s j , k and a common i o n i , t h e e x c e s s Gibbs energy ( g ) i s : E

g

= g

E

E

(Debye-Huckel)

+ 2A^ .XjX.

+ 2A

i k

x.x

k

0)

where i o n j b e l o n g s t o e l e c t r o l y t e 2 and k t o e l e c t r o l y t e 3. x . i s t h e i o n i c mole f r a c t i o n o f i o n i , and t h e p a r a m e t e r s Α . . , Α.. c h a r a c t e r i z e t h e i n t e r a c t i o n s between p a i r s o f i o n s . A $ s u g g e s t e d by t h e B r o n s t e d r u l e , no t e r m s a p p e a r f o r t h e i n t e r a c t i o n between i o n s o f l i k e s i g n . Equation 1 i s d i f f e r e n t i a t e d to c a l c u l a t e the i o n i c a c t i v i t y c o e f f i c i e n t s u s i n g t h e same t e c h n i q u e a s f o r n o n - e l e c t r o l y t e s . C o m b i n i n g t h e s e i o n i c a c t i v i t y c o e f f i c i e n t s t o f o r m t h e mean i o n i c a c t i v i t y c o e f f i c i e n t s , γ.., we o b t a i n t h e B r o n s t e d Guggenheim e q u a t i o n s f o r two 1-1 e l e c t r o l y t e s w i t h a common ion : J

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

FUNK

Multicomponent Salt Systems

719


37.

Figure 1.

Mean ionic activity coefficients for the HCl-NaCl-H 0 2

system at 25°C


720

T H E R M O D Y N A M I C S O F AQUEOUS SYSTEMS W I T H INDUSTRIAL

- c l ^ / d

1ηγ

2

+ il/2) +

1ηγ =-ciV2/(i +

+

3

2

A

^

+ ( +

(

A

«

+A ^ ) ^

APPLICATIONS

(2)

+

(3)

where c i s t h e Debye-Huckel c o n s t a n t , n^m- a r e t h e m o l a l i t i e s o f t h e e l e c t r o l y t e s , and I i s t h e i o n i c s t r e n g t h . The i n t e r a c t i o n p a r a m e t e r s o f E q u a t i o n s 2 and 3 c a n be r e a d i l y c a l c u l a t e d u s i n g o n l y d a t a f o r t h e b i n a r y s o l u t i o n s . However, w i t h o u t f u r t h e r i n t e r a c t i o n s t e r m s , t h e Bronsted-Guggenheim e q u a t i o n s are l i m i t e d t o m o l a l i t i e s b e l o w 0.2.


Harned*s

Rule

A c t i v i t y c o e f f i c i e n t s i n concentrated solutions are often d e s c r i b e d u s i n g Harned s r u l e (l_). T h i s r u l e s t a t e s t h a t for a ternary solution at constant total m o l a l i t y t h e l o g a r i t h m o f t h e a c t i v i t y c o e f f i c i e n t o f each e l e c t r o l y t e i s proportional to the m o l a l i t y o f the other electrolyte. The e x p r e s s i o n s f o r t h e a c t i v i t y c o e f f i c i e n t s a r e written : 1

Tog γ log γ

2

3

= log γ = log γ

2

(

0

)

3

(

0

)

^

Λ

(4)

- α ^

(5)

where s u b s c r i p t 2 r e f e r s t o HCI and 3 t o N a C l . The a c t i v i t y coefficient Y (n) * calculated at the total molality o f the s o l u t i o n and a s s u m i n g t h e a b s e n c e o f 3; t h e r e i s a s i m i l a r d e f i n i t i o n f o r YO/QV p a r a m e t e r s ouo and α~ c h a r a c t e r i z e t h e i n t e r a c t i o n s ô c c u r i n g between e l e c t r o l y t e s 2 and 3. Harned's r u l e c o r r e l a t e s t h e e x p e r i m e n t a l a c t i v i t y c o e f f i c i e n t s f o r most t e r n a r y aqueous e l e c t r o l y t e s s o l u t i o n s . For d i l u t e s o l u t i o n s , E q u a t i o n s 4 and 5 r e d u c e t o t h e B r o n s t e d - G u g g e n h e i m e q u a t i o n s , and t h e p a r a m e t e r s ou- and a ~ can be e x p r e s s e d i n t e r m s o f t h e i n t e r a c t i o n p a r a m e t e r s o f t n e Bronsted-Guggenheim t h e o r y . For concentrated s o l u t i o n s , Harned s r u l e i s a s i m p l e e m p i r i c a l e x t e n s i o n o f t h e Bronsted-Guggenheim theory. T h u s , i t i s s u r p r i s i n g how w e l l t h e r u l e d e s c r i b e s a c t i v i t y c o e f f i c i e n t s in highly concentrated s o l u t i o n s . F i g u r e 2 p r e s e n t s t h e p a r a m e t e r s o f E q u a t i o n s 4 and 5 as f u n c t i o n s o f t e m p e r a t u r e and t o t a l m o l a l i t y . The e x p e r i m e n t a l d a t a c o m p i l e d and d i s c u s s e d by Harned and Owen (]_) were used t o c a l c u l a t e t h e e x p e r i m e n t a l parameters shown i n F i g u r e 2. The p a r a m e t e r c u c h a n g e s s i g n i f i c a n t l y w i t h t e m p e r a t u r e and m o l a l i t y , however becomes i n d e p e n d e n t o f m o l a l i t y a t h i g h molalities. On t h e o t h e r h a n d , o u v a r i e s i n a c o m p l i c a t e d manner w i t h t e m p e r a t u r e and m o l a l i t y . The r e s u l t s o f F i g u r e 2 show t h a t t h e p a r a m e t e r s o f H a r n e d ' s r u l e c a n n o t be r e l i a b l y s

2

T

h

e

2

2

1

3

?


FUNK


111


37.

Figure 2. Interaction parameters for Harned's Rule as a function of temperature and total molality


722


APPLICATIONS

e x t r a p o l a t e d t o u n s t u d i e d c o n d i t i o n s o f t e m p e r a t u r e and t o t a l s o l u t i o n mol a l i t y . The s u c c e s s o f H a r n e d ' s r u l e f o r t e r n a r y s o l u t i o n s i s l a r g e l y f o r t u i t o u s , and t h e r u l e has no t h e o r e t i c a l b a s i s t o e x p e c t t h a t i t would be u s e f u l f o r s o l u t i o n s c o n t a i n i n g more t h a n two e l e c t r o l y t e s . Furthermore, f o r high concentrations o f s e v e r a l e l e c t r o l y t e s , a c t i v i t y c o e f f i c i e n t s such as YO/QN hypothetical. T h e r e a r e , u n f o r t u n a t e l y , few expérimentai d a t a a v a i l a b l e to t e s t Harned's r u l e f o r concentrated s o l u t i o n s o f t h r e e o r more e l e c t r o l y t e s . A


Thermodynamics o f C o n c e n t r a t e d

R

E

Solutions

The B r o n s t e d - G u g g e n h e i m e q u a t i o n s p r o v i d e a h i g h l y s a t i s factory description o f the a c t i v i t y c o e f f i c i e n t s in d i l u t e solut i o n s ; however, t h e i r e m p i r i c a l e x t e n s i o n t o c o n c e n t r a t e d s o l u t i o n s (Harned's r u l e ) introduces several serious problems. For c o n c e n t r a t e d s o l u t i o n s , t h e a c t i v i t y c o e f f i c i e n t o f an e l e c t r o l y t e i s c o n v e n i e n t l y d e f i n e d as t h o u g h i t were a n o n electrolyte. This i s a practical d e f i n i t i o n f o r the description o f phase e q u i l i b r i a i n v o l v i n g e l e c t r o l y t e s . T h i s new a c t i v i t y c o e f f i c i e n t f \ can be r e l a t e d t o t h e mean i o n i c a c t i v i t y c o e f f i c i e n t by e q u a t i n g e x p r e s s i o n s f o r t h e l i q u i d - p h a s e f u g a c i t y w r i t t e n i n t e r m s o f each o f t h e a c t i v i t y c o e f f i c i e n t s . F o r any 1-1 e l e c t r o l y t e , t h e r e l a t i o n i s :

x.r.f? = Π , ^ Η . where t h e a c t i v i t y c o e f f i c i e n t o f e l e c t r o l y t e P ? , i s n o r m a l i z e d such t h a t : as

(6) i alone in s o l u t i o n , (7)

The s t a n d a r d s t a t e f o r t h e mean i o n i c a c t i v i t y c o e f f i c i e n t i s Henry's constant H., f ? i s the standard-state fugacity f o r the a c t i v i t y c o e f f i c i e n t Γ.> and x . i s t h e mole f r a c t i o n o f e l e c t r o l y t e i c a l c u l a t e d as tftough the e l e c t r o l y t e s d i d n o t d i s s o c i a t e in s o l u t i o n . The a c t i v i t y c o e f f i c i e n t Γ° i s n o r m a l i z e d such t h a t i t becomes u n i t y a t some mole f r a c t i o n x * . F o r N a C l , x i i s c o n v e n i e n t l y t a k e n as t h e s a t u r a t i o n p o i n t . Thus Γ? "is u n i t y a t 25°C f o r t h e s a t u r a t i o n m o l a l i t y o f 6 . 0 5 . The a c t i v i t y c o e f f i c i e n t o f HCI i s n o r m a l i z e d t o be u n i t y a t an HCI m o l a l i t y of 10.0 f o r a l l temperatures. These s t a n d a r d s t a t e s have been chosen t o be c l o s e t o c o n d i t i o n s o f i n t e r e s t i n phase e q u i l i bria. F i g u r e 3 shows t h e a c t i v i t y c o e f f i c i e n t o f HCI i n aqueous solution, » as a f u n c t i o n o f l i q u i d - p h a s e c o m p o s i t i o n f o r 1 0 , 25 and 5 0 ° C . Experimental a c t i v i t y - c o e f f i c i e n t data given by Harned and Owen (]_) were used i n c o n j u n c t i o n w i t h


FUNK



37.


723

724


APPLICATIONS

Equation 6 to c a l c u l a t e experimental values o f ; the ratio o f s t a n d a r d s t a t e s a p p e a r i n g i n E q u a t i o n 6 , f ? / H . , was c a l c u l a t e d u s i n g e x p e r i m e n t a l d a t a f o r t h e s t a n d a r d - s t a t e c o m p o s i t i o n and t h e n o r m a l i z a t i o n c o n d i t i o n , E q u a t i o n 7. The a c t i v i t y c o e f f i c i e n t shown i n F i g u r e 3 changes s m o o t h l y w i t h c o m p o s i t i o n and does n o t have a minimum s u c h as t h e mean i o n i c a c t i v i t y c o e f f i c i e n t o f HCI shown i n F i g u r e 1. F i g u r e 4 shows t h e a c t i v i t y c o e f f i c i e n t o f N a C l , Vy as a f u n c t i o n o f X3-X3 f o r v a r i o u s t e m p e r a t u r e s f r o m 0 to 50*0. A g a i n t h e mean i o n i c a c t i v i t y c o e f f i c i e n t d a t a c o m p i l e d by Harned and Owen were t r a n s f o r m e d u s i n g E q u a t i o n 6 t o o b t a i n t h e TS a c t i v i t y c o e f f i c i e n t s shown i n F i g u r e 4 . F o r b o t h HCI and NaCl t h e a c t i v i t y c o e f f i c i e n t Γ? c h a n g e s r a p i d l y w i t h l i q u i d - p h a s e mole f r a c t i o n o n l y i n d i l u t e s o l u t i o n s . These concentrations are not o f present i n t e r e s t , since they are already w e l l d e s c r i b e d by t h e B r o n s t e d - G u g g e n h e i m e q u a t i o n s .


2

The a c t i v i t y c o e f f i c i e n t s o f H C I ( 2 ) and N a C l ( 3 ) a t m o l a l i t i e s above 0 . 2 i n t h e i r r e s p e c t i v e b i n a r y s o l u t i o n s can be c a l c u l a t e d by:

ln

= ( - 5 9 . 5 6 + 0.1 7 9 t )

[0.1525 - x ]

In Γ 3 = - 2 8 . 0 5 [ x * - g ] - 0 . 0 4 3 6 X

(8)

2

e

5

0

(

0

-

0

5

9

4

"

X

3

}

(9)

f r o m 0 t o 50°C and where t h e change o f t h e s a t u r a t i o n mole f r a c t i o n , x * , with temperature accounts f o r the temperature v a r i a t i o n o f the a c t i v i t y c o e f f i c i e n t o f NaCl. The d a t a o f S e i d e l l (3) were used t o e x p r e s s x ^ a s : x* = 0.0990 + 0.00028 (t/10)

(10)

from 0 t o 50°C where t i s t h e t e m p e r a t u r e i n ° C . F o r t e r n a r y aqueous s o l u t i o n s o f HCI and N a C l , t h e f o l l o w i n g s e m i e m p i r i c a l equations a r e proposed t o d e s c r i b e 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 the e l e c t r o l y t e s : In C

2

In C

3

= In P*

+A

= In P

+A

3

2 3

3 2

x x

3

2

(11 ) (12)

where A and A a r e p a r a m e t e r s used t o c h a r a c t e r i z e t h e i n t e r a c t i o n s between t h e two d i f f e r e n t e l e c t r o l y t e s a n d , l i k e t h e p a r a m e t e r s i n H a r n e d ' s r u l e , must be c a l c u l a t e d f r o m d a t a f o r t h e t e r n a r y m i x t u r e . The a c t i v i t y c o e f f i c i e n t p? i s c a l c u l a t e d at the m o l a l i t y o f e l e c t r o l y t e i , not at the t o t a l m o l a l i t y . S i n c e f? i s c a l c u l a t e d u s i n g E q u a t i o n 8 o r 9 , E q u a t i o n s 11 and 12 a r e l i m i t e d t o t h e c a l c u l a t i o n o f ft a t e l e c t r o l y t e m o l a l i t i e s , m.., above 0 . 2 . 2

3

3 2



Figure 4.

Activity coefficient of NaCl as a function of temperature and composition (experimental points: C) 0°C; (O) 10°C; (V)20°C; Ο 30°C; ( + )40°C; (A) 50°C


726


APPLICATIONS

The f ° . terms o f E q u a t i o n s 11 and 12 g i v e t h e c o n t r i b u t i o n t o t h e e x c e i s G i b b s e n e r g y due t o t h e e l e c t r o l y t e s n o t b e i n g a t t h e i r respective standard-state concentrations. The t e r m s c o n t a i n i n g t h e i n t e r a c t i o n parameters give t h e c o n t r i b u t i o n t o t h e e x c e s s G i b b s e n e r g y due t o i n t e r a c t i o n s between e l e c t r o l y t e s 2 and 3. T h i s s e c o n d p a r t o f t h e e x c e s s G i b b s e n e r g y h a s t h e same form u s e d i n t h e B r o n s t e d - G u g g e n h e i m t h e o r y , H a r n e d s r u l e , and n o n - e l e c t r o l y t e s o l u t i o n s . E q u a t i o n s 11 a n d 12 can be e a s i l y e x t e n d e d t o s o l u t i o n s c o n t a i n i n g more t h a n two e l e c t r o l y t e s . E q u a t i o n s 11 and 12 were f i t t o t h e e x p e r i m e n t a l a c t i v i t y c o e f f i c i e n t s o f HCI and NaCl as d e s c r i b e d by H a r n e d ' s r u l e . F i g u r e 2 was used t o c a l c u l a t e t h e p a r a m e t e r s f o r H a r n e d ' s r u l e f r o m 10 t o 40°C and f o r t o t a l m o l a l i t i e s 0 . 2 t o a p p r o x i m a t e l y 6 . The i n t e r a c t i o n p a r a m e t e r A i s independent o f the t o t a l m o l a l i t y ; t h e p a r a m e t e r Aô d e c r e a s e s w i t h t o t a l m o l a l i t y b u t a p p e a r s t o reach a c o n s t a n t value a t h i g h m o l a l i t i e s . Both parameters a r e weak f u n c t i o n s o f t e m p e r a t u r e and c a n be e x p r e s s e d b y : 1


2

A A

2 3

3 2

3

= 48.50 - 0.70 (t/10)

(13)

= 6 8 . 8 0 - 1 . 3 3 ( t / 1 0 ) - 3 . 2 0 m.

(14)

from 0 t o 50°C and i n t h e t o t a l m o l a l i t y r a n g e 0 . 2 t o a p p r o x i m a t e l y 10 m. A c t i v i t y c o e f f i c i e n t s o f HCI c a l c u l a t e d u s i n g E q u a t i o n s 11 and 1 3 a r e w i t h i n + 1% o f t h e e x p e r i m e n t a l v a l u e s . F o r N a C l , d e v i a t i o n s between e x p e r i m e n t a l and c a l c u l a t e d a c t i v i t y c o e f f i c i e n t s a r e l e s s t h a n + 3%. E s t i m a t e s o f t h e a c t i v i t y c o e f f i c i e n t s a t t o t a l m o l a l i t i e s above 10 a r e p r o b a b l y not r e l i a b l e . A c t i v i t y C o e f f i c i e n t o f Water i n C o n c e n t r a t e d

Solutions

E q u a t i o n s 8 and 9 c a n be used w i t h t h e Gibbs-Duhem e q u a t i o n t o c a l c u l a t e ΓΊ°(.Μ> t h e a c t i v i t y c o e f f i c i e n t o f w a t e r , f o r e a c h o f the binary systems. The Gibbs-Duhem e q u a t i o n f o r a b i n a r y aqueous e l e c t r o l y t e s o l u t i o n i s w r i t t e n :

l n

(15)

Γο·) •

Γ

where t h e a c t i v i t y c o e f f i c i e n t o f w a t e r i s n o r m a l i z e d such t h a t i t becomes u n i t y when t h e 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 is unity. S u b s t i t u t i o n o f E q u a t i o n 8 i n t o E q u a t i o n 15 g i v e s : In

ff

( 2 )

= (-59.56 + 0 . 1 7 9 t ) [ ( χ * - χ ) + Ι „ £

(16)


37.

FUNK

121



f o r t h e a c t i v i t y c o e f f i c i e n t o f w a t e r i n t h e HCl-hÔ s y s t e m , where x * = 0 . 1 5 2 5 . F i g u r e 5 shows a c o m p a r i s o n between p r e d i c t e d and e x p e r i m e n t a l a c t i v i t y c o e f f i c i e n t s o f w a t e r i n t h e H C I - H O systçm a t 1 0 , 2 5 and 5 0 ° C ; E q u a t i o n 16 p r e d i c t s a c c u r a t e v a l u e s of f l ( 2 ) except i n very d i l u t e s o l u t i o n s . F o r such s o l u t i o n s , t h e B r o n s t e d - G u g g e n h e i m t h e o r y c a n be used t o c a l c u l a t e t h e a c t i v i t y c o e f f i c i e n t o f water. The a c t i v i t y c o e f f i c i e n t o f w a t e r i n t h e NaCl-HÔ s y s t e m c a n be w e l l d e s c r i b e d b y s u b s t i t u t i o n o f o n l y t h e f i r s t t e r m o f E q u a t i o n 9 i n t o E q u a t i o n 1 5 . The r e s u l t i n g e x p r e s s i o n f o r t h e a c t i v i t y c o e f f i c i e n t o f water i s :

1ηΓ

1

(

3

= . 2 8 . 0 5 C(x§ - x )

)

3

+in

(17)

The s e c o n d t e r m o f E q u a t i o n 9 i s o n l y i m p o r t a n t f o r t h e c a l c u l a t i o n o f t h e a c t i v i t y c o e f f i c i e n t o f NaCl a t l o w c o n c e n t r a t i o n s , and makes l i t t l e c o n t r i b u t i o n t o t h e i n t e g r a l i n E q u a t i o n 15. Equation 17 p r e d i c t s 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 water w i t h i n 1% o f t h e e x p e r i m e n t a l v a l u e s f o r m o l a l i t i e s above 0 . 2 . F o r t h e t e r n a r y s o l u t i o n , t h e Gibbs-Duhem e q u a t i o n c a n be e a s i l y i n t e g r a t e d t o c a l c u l a t e t h e a c t i v i t y c o e f f i c i e n t o f water when t h e e x p r e s s i o n s f o r 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 are written at constant m o l a l i t y . For Harned's r u l e , i n t e g r a t i o n o f t h e Gibbs-Duhem e q u a t i o n g i v e s t h e a c t i v i t y o f water a s : -55.51 v m• ,r

Ί

l

o

„ 9

a

w 3 , , \ ο f v =n\ Q (ou- + o u ) - 2 oua (Y =0) • ' 3 2 3 3 2 ' 23 ( Y

)

=

a

n*\ (18)

v

Y

v u

q

u

9

u u

3

f o r a c o n s t a n t t o t a l m o l a l i t y o f m. The f r a c t i o n o f t o t a l e l e c t r o l y t e i n s o l u t i o n a s i i s e x p r e s s e d by Y . ; a ( Y ~ ) i s t h e a c t i v i t y o f w a t e r f o r a g i v e n v a l u e o f Y - . The s t a n d a r d s t a t e f o r w a t e r i s p u r e w a t e r a t t h e t e m p e r a t u r e o f t h e s y s t e m . The B r o n s t e d - G u g g e n h e i m e q u a t i o n s can a l s o be s u b s t i t u t e d i n t o t h e Gibbs-Duhem e q u a t i o n t o c a l c u l a t e t h e a c t i v i t y o f w a t e r i n d i l u t e solutions. Harned and R o b i n s o n (J3) g i v e t h e r e s u l t a n d a d e t a i l e d d i s c u s s i o n o f t h e Bronsted-Guggenheim e q u a t i o n s . E q u a t i o n s 11 and 12 a r e n o t w r i t t e n f o r c o n s t a n t m o l a l i t y , and c a n n o t be e a s i l y used w i t h the* Gibbs-Duhem e q u a t i o n t o o b t a i n an a n a l y t i c a l e x p r e s s i o n f o r t h e a c t i v i t y o f w a t e r i n t h e ternary solution. However, i t i s p o s s i b l e t o p r o p o s e a s e p a r a t e equation f o r the a c t i v i t y c o e f f i c i e n t o f water that i s c o n s i s t e n t w i t h t h e p r o p o s e d model o f c o n c e n t r a t e d s o l u t i o n s . w

The a c t i v i t y Π (2 3 ) '

l

s

e

s

t

l

'

m

a

c o e f f i c i e n t o f water i n t h e t e r n a r y t

e

solution

d by:



APPLICATIONS


728

(Ό.1525 — Χ * ) Figure 5.

Activity

coefficients

of water in the HCI-H 0 (A), and 50°C (%) 2

system at 10° (O),

25°


37.

1 Η

FUNK

Γ

1 ( 2

τ*ί(2

, ) 3

=

Υ

2

Ί

η

^ ( 2 )

+

Υ

3

1

η

Γ

1

0

( 3 )

(

The s t a n d a r d - s t a t e f u g a c i t y o f w a t e r i n t h e t e r n a r y i s expressed s i m i l a r l y b y :

In f «


729


( 2 f 3 )

= Y In f » 2

( 2 )+

Y

3

In f °

(

3

)

1

9

>

solution,

(20)

where f ? / . > i s t h e s t a n d a r d - s t a t e f u g a c i t y o f w a t e r i n t h e b i n a r y s o l u t i o n ^ o r e l e c t r o l y t e i . The a c t i v i t y c o e f f i c i e n t Γ?/*\ i s c a l c u l a t e d u s i n g t h e mole f r a c t i o n o f i i n t h e t e r n a r y s o T u t i o n and e i t h e r E q u a t i o n 1 6 o r 1 7 . The s t a n d a r d - s t a t e f u g a c i t y o f w a t e r i n t h e t e r n a r y s o l u t i o n changes w i t h Y. s i n c e t h e s t a n d a r d s t a t e f u g a c i t y o f w a t e r i s d i f f e r e n t i n each b i n a r y s y s t e m . E q u a t i o n 2 0 i s s i m i l a r t o t h e e x p r e s s i o n d e r i v e d by O ' C o n n e l l and P r a u s n i t z (KO f o r t h e c o m p o s i t i o n dependence o f H e n r y ' s constant i n a mixed s o l v e n t . Equation 19 estimates t h e a c t i v i t y c o e f f i c i e n t o f water i n t h e t e r n a r y s o l u t i o n using o n l y data f o r t h e b i n a r y m i x t u r e s ; t h e r e f o r e , i t c a n n o t be e x p e c t e d t o g i v e very precise r e s u l t s . V a p o r - l i q u i d e q u i l i b r i u m d a t a f o r t h e two b i n a r y s y s t e m s 0]_) were used t o c a l c u l a t e t h e s t a n d a r d - s t a t e f u g a c i t i e s r e q u i r e d i n E q u a t i o n s 6 and 2 0 . In t h e t e m p e r a t u r e r a n g e 0-50°C, t h e r e f u g a c i t i e s can be e x p r e s s e d b y : ln

f | = 1.332 + 0.781

In

f ^

In

ffjgj

2

)

(t/10)

(21)

= 0.885 + 0.610

(t/10)

(22)

= 1.485 + 0.591

(t/10)

(23)

where t h e f u g a c i t y i s i n mm. H g . F i g u r e 6 compares e x p e r i m e n t a l and c a l c u l a t e d a c t i v i t y c o e f f i c i e n t s o f w a t e r i n t h e t e r n a r y s y s t e m a t 25°C and a t o t a l m o l a l i t y o f 3.0. E q u a t i o n 18 was used t o e x p r e s s t h e e x p e r i m e n t a l activity coefficients. Agreement between e x p e r i m e n t a l and c a l c u l a t e d v a l u e s i s s u r p r i s i n g l y good c o n s i d e r i n g t h a t E q u a t i o n 1 9 c o n t a i n s no t e r n a r y p a r a m e t e r s . The a c t i v i t y c o e f f i c i e n t o f w a t e r i n t h e HCI-NaCl-HÔ s y s t e m i s n o t a s t r o n g f u n c t i o n o f c o m p o s i t i o n , and E q u a t i o n T9 p r o v i d e s an a d e q u a t e d e s c r i p t i o n o f the a c t i v i t y c o e f f i c i e n t s . Vapor-Liquid E q u i l i b r i a E q u a t i o n s 11 and 1 9 e x p r e s s t h e n e c e s s a r y l i q u i d - p h a s e activity coefficients for the calculation o f vapor-liquid e q u i l i b r i a i n t h e HCI-NaCl-H 0 system. E q u a t i o n 11 i s v e r y c o n v e n i e n t ?



APPLICATIONS


730

Figure

6.

Activity

coefficients of water in the HCl-NaCl-H 0 and total molality of 3.0 ((%) experimental) 2

system at

25°C


37.

FUNK

731


f o r v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s , s i n c e HCI i s t r e a t e d as a n o n - e l e c t r o l y t e i n b o t h l i q u i d and v a p o r p h a s e s . This a v o i d s t h e c u s t o m a r y e q u a l i t y ( 1 2 ) between t h e f u g a c i t y o f HCI v a p o r and t h e mean i o n i c T u g a c i t y o f H C I . At moderate p r e s s u r e s , t h e v i r i a l e q u a t i o n o f s t a t e , t r u n c a t e d a f t e r t h e s e c o n d v i r i a l c o e f f i c i e n t , can be used t o d e s c r i b e the vapor phase. As s u g g e s t e d b y H i r s c h f e l d e r , e t . a l . (1 3) t h e t e m p e r a t u r e dependence o f t h e v i r i a l c o e f f i c i e n t s i s e x p r e s s e d : B


B

n

2 2

= 49.85 [ 1 . 0 - 0.328

e

= 77.43 [ 1 . 0 - 0.704

e

1

2

8

8

/

4 0 0 / T

]

T

(24)

]

(25)

where Τ i s t h e t e m p e r a t u r e i n ° K . The c o r r e l a t i o n o f P i t z e r ( 1 4 ) was used t o c a l c u l a t e t h e s e c o n d v i r i a l c o e f f i c i e n t s o f H C I , and t h e e x p e r i m e n t a l d a t a o f 0 ' C o n n e l l and P r a u s n i t z (1_5) were used t o c a l c u l a t e Β-μ f o r w a t e r . The c r o s s v i r i a l c o e f f i c i e n t s are estimated by: B

1 2

= 6 3 . 6 9 [ 1 . 0 - 0.516

e

7

3

5

/

T

]

(26)

where s i m p l e m i x i n g r u l e s were used f o r t h e t h r e e p a r a m e t e r s i n E q u a t i o n s 24 a n d 2 5 . F i g u r e 7 shows t h e p r e d i c t e d v a p o r - p h a s e mole f r a c t i o n s o f HCI a t 25°C a s a f u n c t i o n o f t h e l i q u i d - p h a s e m o l a l i t y o f HCI f o r a c o n s t a n t NaCl m o l a l i t y o f 3. A l s o i n c l u d e d a r e p r e d i c t e d v a p o r phase mole f r a c t i o n s o f HCI when t h e i n t e r a c t i o n p a r a m e t e r A i s taken as z e r o . T h e r e a r e u n f o r t u n a t e l y no e x p e r i m e n t a l v a p o r l i q u i d e q u i l i b r i u m data a v a i l a b l e f o r t h e HCI-NaCl-H 0 system; however, c o n s i d e r i n g t h e e x c e l l e n t d e s c r i p t i o n o f t h e l i q u i d phase a c t i v i t y c o e f f i c i e n t s and t h e l o w t o t a l p r e s s u r e s , i t i s e x p e c t e d t h a t p r e d i c t e d mole f r a c t i o n s w o u l d be w i t h i n 2-3% o f t h e experimental values. 2

3

2

Solid-Liquid

Equilibria

The s o l i d - l i q u i d e q u i l i b r i u m f o r NaCl pressed b y : f

S

3

χ

ff " Γ 3

3

i s rigorously ex

(27)

where f | i s t h e f u g a c i t y o f s o l i d s o d i u m c h l o r i d e . The r a t i o o f f u g a c i t i e s i n E q u a t i o n 2 7 i s t h e s o l u b i l i t y p r o d u c t o f NaCl and c a n be d e t e r m i n e d u s i n g s o l u b i l i t y d a t a . At saturation i n t h e NaCl-HpO s y s t e m , Ç° i s u n i t y and t h e s a t u r a t i o n mole f r a c -


THERMODYNAMICS


732

0.01

I 5

I

OF

I 5

AQUEOUS SYSTEMS

I

I

I

7 MOLALITY OF

W I T H INDUSTRIAL

I 9

I

APPLICATIONS

ι

t 11

I 13

MCI

Figure 7. Predicted vapor-liquid equilibria in the HCl-NaCl-H O for NaCl molality of 3.0 z

system at 25°C


37.

FUNK

733


t i o n , x ^ , i s equal t o t h e s o l u b i l i t y p r o d u c t . T h e r e f o r e , Equa t i o n 1 0 c a n be used t o e x p r e s s t h e s o l u b i l i t y p r o d u c t o f NaCl as a f u n c t i o n o f t e m p e r a t u r e . The s e l e c t e d n o r m a l i z a t i o n o f t h e NaCl a c t i v i t y c o e f f i c i e n t has two p a r t i c u l a r a d v a n t a g e s f o r s o l i d - l i q u i d e q u i l i b r i a . First, the s o l u b i l i t y product i s c a l c u l a t e d d i r e c t l y from a v a i l a b l e s o l u b i l i t y d a t a ; no a c t i v i t y - c o e f f i c i e n t d a t a a r e r e q u i r e d . S e c o n d , t h e a c t i v i t y c o e f f i c i e n t o f NaCl h a s a c l e a r i n t e r p r e t a t i o n ; i t p r o v i d e s a q u a n t i t a t i v e measure o f how HCI c h a n g e s t h e s o l u b i l i t y o f NaCl f r o m i t s s t a n d a r d - s t a t e v a l u e o f x i S o l i d - l i q u i d e q u i l i b r i u m d a t a (]6) f o r t h e HCI-NaCl H 0 s y s t e m a t 25°C were used w i t h E q u a t i o n 27 t o c a l c u l a t e experimental a c t i v i t y c o e f f i c i e n t s o f NaCl. Table 1 shows a c o m p a r i s o n between t h e e x p e r i m e n t a l a c t i v i t y c o e f f i c i e n t s and t h o s e c a l c u l a t e d u s i n g E q u a t i o n 1 2 . The agreement between e x p e r i m e n t a l and c a l c u l a t e d a c t i v i t y c o e f f i c i e n t s i s v e r y g o o d , and E q u a t i o n 12 s h o u l d be u s e f u l f o r p r e d i c t i o n s o f s o l i d - l i q u i d e q u i l i b r i a at other temperatures. E q u a t i o n 27 i s s i m i l a r t o t h e s o l i d - l i q u i d e q u i l i b r i u m r e l a t i o n used f o r n o n - e l e c t r o l y t e s . As i n t h e c a s e o f 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 r e l a t i o n f o r HCI, t h e s o l i d - l i q u i d e q u i l i b r i u m e x p r e s s i o n f o r NaCl i s s i m p l e s i n c e t h e e l e c t r o l y t e i s t r e a t e d t h e r m o d y n a m i c a l l y t h e same i n b o t h p h a s e s .


2

E x t e n s i o n o f T e c h n i q u e by V e r a and Co-Workers V e r a and c o - w o r k e r s (7,1_7,1JB) have e x t e n d e d t h e t h e r m o d y n a m i c c o r r e l a t i o n and macTe two a d d i t i o n s . F i r s t , t h e y have developed a semi-empirical expression f o r t h e excess Gibbs energy i n place o f t h e simple e m p i r i c a l e q u a t i o n s o r i g i n a l l y used ( E q u a t i o n s 8 and 9 ) . A l s o , w h i l e t h e y use a s t a n d a r d s t a t e o f t h e e l e c t r o l y t e o f a s a t u r a t e d s o l u t i o n , t h e y change t h e s t a n d a r d s t a t e o f w a t e r back t o t h e c o n v e n t i o n a l one o f p u r e w a t e r . The e x p r e s s i o n f o r t h e e x c e s s G i b b s e n e r g y s u g g e s t e d b y C o r r e a and V e r a i s : g^./RT = Ax I n χ + B x

1 / 2

+ Kx + c x

3 / 2

+ Dx + . . . 2

(28)

t o d e s c r i b e a b i n a r y s y s t e m o f e l e c t r o l y t e i i n aqueous s o l u t i o n and where A , B, C , D, . . . a r e a d j u s t a b l e p a r a m e t e r s . Κ i s not an i n d e p e n d e n t p a r a m e t e r and i s d e t e r m i n e d by t h e n o r m a l i z a t i o n condition f o rthea c t i v i t y c o e f f i c i e n t . A p p l i c a t i o n o f standard thermodynamics l e a d s t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e a c t i v i t y c o e f f i c i e n t o f water: l n P, = a x + b x

1 / 2

+cx

3 / 2

+dx + 2

(29)



734


TABLE

APPLICATIONS

I

CALCULATED ACTIVITY COEFFICIENTS OF NaCl AND EXPERIMENTAL VALUES OBTAINED FROM SOLID-LIQUID EQUILIBRIUM DATA m

3

T (Exp.)

P (Cal.)

3

3

0.00

6.126

1 .00

1.00

1.00

5.096

1 .20

1.24

2.00

4.054

1.51

1.53

3.00

3.100

1 .97

2.07

5.00

1.884

3.27

3.27

6.00

1.020

6.09

5.83

7.50

0.496

12.73

12.10

8.50

0.306

20.69

20.10


37.

FUNK

Multicomponent

Salt

735

Systems

where A = - a , B - 2 b , C=-2c, and D=-d. Use o f t h e Gibbs-Duhem e q u a t i o n and t h e n o r m a l i z a t i o n c o n d i t i o n l e a d s t o t h e e x p r e s s i o n for the a c t i v i t y c o e f f i c i e n t o f the e l e c t r o l y t e : ln Γ

= (K-a) - a l n χ + b x "

1

1 / 2

+ c^x + c y

Activity Coefficients at High Concentrations in Multicomponent Salt

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