Two New Activity Coefficient Models for the Vapor-Liquid Equilibrium

4 Two New Activity Coefficient Models for the Vapor-Liquid Equilibrium of Electrolyte Systems CHAU-CHYUN CHEN, HERBERT I. BRITT, JOSEPH F. BOSTON, and LAWRENCE B. EVANS

Downloaded by CORNELL UNIV on November 24, 2012 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch004

Department of Chemical Engineering and Energy Laboratory, Massachusetts Institute of Technology, Cambridge, M A 02139

The use of modern process simulators for the analysis and design of processes involving electrolytes has been greatly limited by the lack of adequate correlations for electrolyte thermodynamics. For most systems of industrial importance, empirical correlations are applicable only to one particular system, over a limited range of conditions. The empirical correlations do not provide a framework for treating new systems or for extending the range of existing data, because the nonidealities have not been accounted for in a general and consistent manner. As in the nonelectrolyte case, the problem of representing the thermodynamic properties of electrolyte solutions is best regarded as that of finding a suitable expression for the non-ideal part of the chemical potential, or the excess Gibbs energy, as a function of composition, temperature, dielectric constant and any other relevant variables. Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (3), Pitzer and co-workers (4,5,6), and Cruz and Renon (7), presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. In t h i s p a p e r , two new m o d e l 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 i o n i c and m o l e c u l a r s p e c i e s i n e l e c t r o l y t e systems a r e p r e s e n t e d . The f i r s t i s an e x t e n s i o n o f t h e P i t z e r e q u a t i o n and i s c o v e r e d i n more d e t a i l i n C h e n , e t a l . ( 1 1 ) . The s e c o n d i s based on t h e l o c a l c o m p o s i t i o n c o n c e p t and r e p r e s e n t s work i n p r o g r e s s .

0-8412-0569-8/80/47-133-061$07.25/0 © 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.

62

THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS


Nature of E l e c t r o l y t e

Systems

The t h e r m o d y n a m i c p r o p e r t i e s o f a m i x t u r e depend on t h e f o r c e s w h i c h o p e r a t e between t h e s p e c i e s o f t h e m i x t u r e . E l e c t r o l y t e s y s t e m s a r e c h a r a c t e r i z e d by t h e p r e s e n c e o f b o t h m o l e c u l a r s p e c i e s and i o n i c s p e c i e s , r e s u l t i n g i n t h r e e d i f f e r e n t types of i n t e r a c t i o n . They a r e i o n - i o n i n t e r a c t i o n , m o l e c u l e - m o l e c u l e i n t e r a c t i o n , and i o n - m o l e c u l e i n t e r a c t i o n . The f o r c e s i n v o l v e d i n each i n t e r a c t i o n a r e b r i e f l y d i s c u s s e d in the f o l l o w i n g paragraphs. The i o n - i o n 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 e l e c t r o s t a t i c f o r c e s between i o n s . These e l e c t r o s t a t i c f o r c e s a r e i n v e r s e l y p r o p o r t i o n a l t o t h e s q u a r e o f t h e s e p a r a t i o n d i s t a n c e and t h e r e f o r e have a much g r e a t e r r a n g e t h a n i n t e r m o l e c u l a r f o r c e s w h i c h depend on h i g h e r powers o f t h e r e c i p r o c a l d i s t a n c e . E x c e p t at s h o r t - r a n g e , o t h e r f o r c e s a r e r e l a t i v e l y i n s i g n i f i c a n t compared t o t h e i n t e r i o n i c e l e c t r o s t a t i c f o r c e . Many d i f f e r e n t t y p e s o f f o r c e s a r i s e f r o m m o l e c u l e molecule i n t e r a c t i o n . They may be e l e c t r o s t a t i c f o r c e s between permanent d i p o l e s , i n d u c t i o n f o r c e s between a permanent d i p o l e and i n d u c e d d i p o l e s , o r d i s p e r s i o n f o r c e s between n o n - p o l a r molecules, etc. (Prausnitz, U 2 ) ) . Forces involved in moleculem o l e c u l e i n t e r a c t i o n a r e known t o be s h o r t - r a n g e i n n a t u r e . The f o r c e s i n v o l v e d i n i o n - m o l e c u l e i n t e r a c t i o n a r e a l s o short-range in nature. The d o m i n a n t f o r c e s a r e e l e c t r o s t a t i c f o r c e s between i o n s and permanent d i p o l e s . As d i s c u s s e d by R o b i n s o n and S t o k e s (11) r e g a r d i n g aqueous e l e c t r o l y t e s y s t e m s , i t seems l i k e l y t h a t t h e i o n - m o l e c u l e i n t e r a c t i o n e n e r g i e s o f t h e w a t e r m o l e c u l e s i n t h e f i r s t l a y e r about a monatomic i o n w o u l d be l a r g e compared w i t h t h e t h e r m a l e n e r g y ( R T ) , and t h e s e c o n d l a y e r o f w a t e r m o l e c u l e s w i l l be much l e s s s t r o n g l y bound t o t h e i o n t h a n t h e f i r s t . It is probably only with p o l y v a l e n t monatomic i o n s o f s m a l l s i z e t h a t t h e i n t e r a c t i o n e n e r g i e s o f t h e w a t e r m o l e c u l e s i n t h e s e c o n d l a y e r w o u l d be comparable to the thermal energy. The e x c e s s G i b b s e n e r g y o f e l e c t r o l y t e s y s t e m s can be c o n s i d e r e d as t h e sum o f two t e r m s , one r e l a t e d t o l o n g - r a n g e f o r c e s between i o n s and t h e o t h e r t o s h o r t - r a n g e f o r c e s between a l l the s p e c i e s . As d i s c u s s e d by R o b i n s o n and S t o k e s ( 1 3 ) , long-range f o r c e s dominate i n the r e g i o n of d i l u t e e l e c t r o l y t e c o n c e n t r a t i o n and s h o r t - r a n g e f o r c e s d o m i n a t e i n t h e r e g i o n o f high e l e c t r o l y t e concentration. It i s the long-range nature of t h e e l e c t r o s t a t i c f o r c e s between i o n s t h a t have no c o u n t e r p a r t in n o n e l e c t r o l y t e systems. The P i t z e r

Equation

In a s e r i e s o f p a p e r s . , P i t z e r and h i s c o - w o r k e r s (4,5,6.) proposed a very u s e f u l s e m i e m p i r i c a l equation f o r the


CHEN ET AL.

4.

63

Activity Coefficient Models

u n s y r c n e t r i c e x c e s s G i b b s f r e e e n e r g y o f aqueous systems. The b a s i c e q u a t i o n i s G

electrolyte

ex*

— ^ ( Π + Σ Σ λ , ^ ΐ ^ ^ + Σ Σ Σ μ ^ ^ ^ n RT ij ijk

(1)

w


The f u n c t i o n f ( I ) e x p r e s s e s t h e e f f e c t o f l o n g - r a n g e e l e c t r o s t a t i c f o r c e s between i o n s . It is a function of ionic s t r e n g t h , t e m p e r a t u r e and s o l v e n t p r o p e r t i e s . The e m p i r i c a l f o r m chosen by P i t z e r f o r f ( I ) i s f(I)

41 =-Α —1η(1+1.2/Γ) 1.2

(2)

φ

The p a r a m e t e r s λ^· a r 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 g i v i n g t h e e f f e c t o f s h o r t - r a n g e f o r c e s between s o l u t e s i and j ; t h e p a r a m e t e r s y-jjk a r e c o r r e s p o n d i n g t h i r d v i r i a l c o e f f i c i e n t s f o r t h e i n t e r a c t i o n o f t h r e e s o l u t e s i , j , and k. The s e c o n d v i r i a l c o e f f i c i e n t s are a f u n c t i o n o f i o n i c s t r e n g t h . Dependence o f t h e t h i r d v i r i a l c o e f f i c i e n t s on i o n i c s t r e n g t h is neglected. The λ and y m a t r i c e s a r e t a k e n t o be s y m m e t r i c . To make t h e b a s i c P i t z e r e q u a t i o n more u s e f u l f o r d a t a c o r r e l a t i o n o f aqueous s t r o n g e l e c t r o l y t e s y s t e m s , P i t z e r m o d i f i e d i t by d e f i n i n g a new s e t o f more d i r e c t l y o b s e r v a b l e p a r a m e t e r s r e p r e s e n t i n g c e r t a i n c o m b i n a t i o n s o f t h e s e c o n d and third virial coefficients. The m o d i f i e d P i t z e r e q u a t i o n i s G

ex*

=f(I)+^m m ,(0 ,+Zm^ ce' a +2E^ m [B (l)+(Em Z C ca c c

n RT w

c

a

c

c a

c c

c

c

c c

. )+EEm m ,(6 aa )//Z Z ] a

a

a

1

c a

c

a a

,+Em^ c

a a

. ) c

(3)

a

E s s e n t i a l l y , t h e new p a r a m e t e r s Β and θ a r e b i n a r y i o n - i o n p a r a m e t e r s and C and Ψ t e r n a r y i o n - i o n p a r a m e t e r s . The i o n - i o n i n t e r a c t i o n p a r a m e t e r s , Β and C , a r e c h a r a c t e r i s t i c o f each aqueous s i n g l e - e l e c t r o l y t e s y s t e m . The i o n - i o n d i f f e r e n c e p a r a m e t e r s , θ and Ψ , a r e c h a r a c t e r i s t i c o f each aqueous m i x e d - e l e c t r o l y t e system. R e c o g n i z i n g t h e i o n i c s t r e n g t h dependence o f t h e e f f e c t o f s h o r t range f o r c e s i n b i n a r y i n t e r a c t i o n s , P i t z e r was a b l e t o d e v e l o p an e m p i r i c a l r e l a t i o n f o r B ( I ) . The e x p r e s s i o n f o r s y s t e m s c o n t a i n i n g s t r o n g e l e c t r o l y t e s w i t h one o r b o t h i o n s univalent i s ç a

B

c a

(I)

= Bj^+B^tl-d+a/TJexpi-a/T)]^!


(4)

64


parameters in the m o d i f i e d P i t z e r e q u a t i o n a r e β Ι ) , β ΐ ) , C, θ , and Ψ. The m o d i f i e d P i t z e r e q u a t i o n has been s u c c e s s f u l l y a p p l i e d t o t h e a v a i l a b l e d a t a f o r many p u r e e l e c t r o l y t e s ( P i t z e r and M a y o r g a , (!5)) and m i x e d aqueous e l e c t r o l y t e s ( P i t z e r and K i m , ( 6 J ) . The"~fit t o the experimental data i s w i t h i n the probable experimental e r r o r up t o m o l a l i t i e s o f 6 . H o w e v e r , i n most aqueous e l e c t r o l y t e s y s t e m s o f i n d u s t r i a l i n t e r e s t , not o n l y s t r o n g e l e c t r o l y t e s b u t a l s o weak e l e c t r o l y t e s and m o l e c u l a r n o n e l e c t r o l y t e s a r e p r e s e n t . While the m o d i f i e d P i t z e r e q u a t i o n a p p e a r s t o be a u s e f u l t o o l f o r t h e r e p r e s e n t a t i o n o f aqueous s t r o n g e l e c t r o l y t e s i n c l u d i n g m i x e d e l e c t r o l y t e s , i t c a n n o t be u s e d i n t h e f o r m j u s t p r e s e n t e d t o r e p r e s e n t the i m p o r t a n t case o f systems c o n t a i n i n g m o l e c u l a r solutes. A u n i f i e d t h e r m o d y n a m i c model f o r b o t h i o n i c s o l u t e s and m o l e c u l a r s o l u t e s i s r e q u i r e d t o model t h e s e k i n d s o f systems.


ϋ

1

P r e v i o u s A p p l i c a t i o n s of the P i t z e r Equation Electrolytes

t o Weak

R e c e n t l y , t h e P i t z e r e q u a t i o n has been a p p l i e d t o model weak e l e c t r o l y t e s y s t e m s by B e u t i e r and Renon (9.) and E d w a r d s , et a l . (10). B e u t i e r and Renon used a s i m p l i f i e d P i t z e r equation f o r the ion-ion i n t e r a c t i o n c o n t r i b u t i o n , applied D e b y e - M c A u l a y ' s e l e c t r o s t a t i c t h e o r y ( H a r n e d and Owen, ( 1 4 ) ) f o r t h e i o n - m o l e c u l e i n t e r a c t i o n c o n t r i b u t i o n , and adopted" M a r g u l e s t y p e t e r m s f o r m o l e c u l e - m o l e c u l e i n t e r a c t i o n s between t h e same m o l e c u l a r s o l u t e s . Edwards, et a l . a p p l i e d the P i t z e r e q u a t i o n d i r e c t l y , w i t h o u t d e f i n i n g any new t e r m s , f o r a l l i n t e r a c t i o n s ( i o n - i o n , i o n - m o l e c u l e , and m o l e c u l e - m o l e c u l e ) while neglecting a l l ternary parameters. B r o m l e y ' s {I) ideas on a d d i t i v i t y o f i n t e r a c t i o n p a r a m e t e r s o f i n d i v i d u a l i o n s and c o r r e l a t i o n between i n d i v i d u a l i o n and p a r t i a l m o l a r e n t r o p y o f i o n s at i n f i n i t e d i l u t i o n were a d o p t e d i n b o t h s t u d i e s . In a d d i t i o n , they both n e g l e c t e d c o n t r i b u t i o n s from i n t e r a c t i o n s among i o n s o f t h e same s i g n . T h e r e a r e d r a w b a c k s w i t h t h e a p p r o a c h e s t a k e n by b o t h B e u t i e r and R e n o n , and E d w a r d s , e t a l . F i r s t , the P i t z e r e q u a t i o n i s a v i r i a l - e x p a n s i o n t y p e e q u a t i o n and s e m i - e m p i r i c a l in nature. E s t i m a t i n g i n t e r a c t i o n parameters using Bromley's a p p r o a c h i s based on an i n t e r p r e t a t i o n o f t h e p a r a m e t e r s t h a t i s u n c e r t a i n at b e s t and i n any c a s e i s n o t v a l i d a t h i g h i o n i c strength. Second, t e r n a r y parameters i n the P i t z e r equation can be s i g n i f i c a n t f o r s y s t e m s o f h i g h i o n i c s t r e n g t h . These p a r a m e t e r s s h o u l d n o t be n e g l e c t e d i n a model c o v e r i n g h i g h i o n i c strength e l e c t r o l y t e systems. Third, Bronsted's p r i n c i p l e of s p e c i f i c ion i n t e r a c t i o n i s the b a s i s f o r assuming t h a t i n t e r a c t i o n s between i o n s o f t h e same s i g n can be neglected. However, B r o n s t e d ' s p r i n c i p l e of s p e c i f i c i o n


4.

CHEN ET AL.

65


i n t e r a c t i o n i s n o t a l w a y s v a l i d , as d i s c u s s e d i n P i t z e r * s p a p e r (£). Fourth, t h e o r e t i c a l aspects o f the physical chemistry o f s a l t i n g - o u t e f f e c t s on n o n - e l e c t r o l y t e s a r e s t i l l i n a development s t a g e . W h i l e many t h e o r i e s have been p r o p o s e d i n t h e l i t e r a t u r e , none i s q u a n t i t a t i v e l y s a t i s f a c t o r y , i n c l u d i n g Debye-McAulay's e l e c t r o s t a t i c t h e o r y .


Extension of t h e P i t z e r

Equation

In t h i s s t u d y t h e P i t z e r e q u a t i o n i s a l s o u s e d , b u t a d i f f e r e n t , more s t r a i g h t f o r w a r d a p p r o a c h i s a d o p t e d i n w h i c h t h e drawbacks j u s t d i s c u s s e d do n o t a r i s e . F i r s t , terms a r e added t o t h e b a s i c v i r i a l f o r m o f t h e P i t z e r e q u a t i o n t o a c c o u n t f o r m o l e c u l e - i o n and m o l e c u l e - m o l e c u l e i n t e r a c t i o n s . Then, f o l l o w i n g P i t z e r , a s e t o f new, more o b s e r v a b l e p a r a m e t e r s a r e defined that are functions of the v i r i a l c o e f f i c i e n t s . Thus, the P i t z e r equation i s extended, rather than m o d i f i e d , t o account f o r t h e presence o f m o l e c u l a r s o l u t e s . The i n t e r p r e t a t i o n o f t h e terms and p a r a m e t e r s o f t h e o r i g i n a l P i t z e r e q u a t i o n i s unchanged. The r e s u l t i n g e x t e n d e d P i t z e r e q u a t i o n i s G

ex* =f(I)*Zm m .(e .+Zm^ . )+ZZm m .(e ce' a aa' c

n RT w

c

c c

c c

a

a

a

+2EDn m [B (I)+(zm Z C )//Z^7] c

ca

a

ca

c

c

c

ca

a a

.+Em^ c

a a

. ) c

(5)

%"\η' mm' Σ(E&ca,n¥a- ^cc ' , m ¥ c ') mm ma c'

+ Σ Σ

x

+

1

The p a r a m e t e r s D a r e b i n a r y parameters r e p r e s e n t i n g t h e i n t e r a c t i o n s between s a l t c a and m o l e c u l a r s o l u t e m i n an aqueous s i n g l e s a l t , s i n g l e m o l e c u l a r s o l u t e s y s t e m . Binary parameters ςο',Γη * ^ a ' m represent the differences between t h e i n t e r a c t i o n s o f a s p e c i f i c m o l e c u l a r s o l u t e w i t h two u n l i k e s a l t s s h a r i n g one common a n i o n o r c a t i o n . Ternary molecule-ion v i r i a l c o e f f i c i e n t s are neglected i n t h i s study t o simplify the extension. I t i s i n t e r e s t i n g t o note t h a t t h e molecule-ion i n t e r a c t i o n contribution i n equation (5) i s consistent with t h e w e l l - k n o w n Setschenow e q u a t i o n . The Setschenow e q u a t i o n i s used t o r e p r e s e n t t h e s a l t i n g - o u t e f f e c t o f s a l t s on m o l e c u l a r n o n e l e c t r o l y t e s o l u t e s , when t h e s o l u b i l i t i e s o f t h e l a t t e r a r e small (Gordon, ( 1 5 ) ) . The Setschenow e q u a t i o n i s c a m

ω

l n

Y*m

= k

a n (

s,m s

(6)

m

where k ^ i s t h e Setschenow c o n s t a n t ( a s a l t - m o l e c u l e i n t e r a c t i o n p a r a m e t e r ) and m i s t h e m o l a l i t y o f t h e s a l t . The D ' s a r e e q u i v a l e n t t o t h e Setschenow c o n s t a n t s and t h e ω s a r e e q u i v a l e n t t o d i f f e r e n c e s between Setschenow c o n s t a n t s . s

m

s

1



66


The t h i r d v i r i a l c o e f f i c i e n t s f o r m o l e c u l e - m o l e c u l e i n t e r a c t i o n s can be t a k e n as z e r o f o r aqueous s y s t e m s c o n t a i n i n g m o l e c u l a r s o l u t e s a t low c o n c e n t r a t i o n . The r e m a i n i n g term f o r the m o l e c u l e - m o l e c u l e i n t e r a c t i o n c o n t r i b u t i o n i s e q u i v a l e n t to the unsymmetric t w o - s u f f i x Margules model. S u m m a r i z i n g t h e r e s u l t s o f t h i s s e c t i o n , an e x c e s s G i b b s f r e e e n e r g y model f o r b o t h i o n i c and m o l e c u l a r s o l u t e s i n aqueous e l e c t r o l y t e s y s t e m s i s o b t a i n e d by e x t e n d i n g t h e P i t z e r equation i n order to account f o r the presence of m o l e c u l a r solutes. Model p a r a m e t e r s i n c l u d e b i n a r y i o n - i o n i n t e r a c t i o n and d i f f e r e n c e p a r a m e t e r s , t e r n a r y i o n - i o n i n t e r a c t i o n and d i f f e r e n c e parameters, s a l t - m o l e c u l e i n t e r a c t i o n parameters or Setschenow c o n s t a n t s , s a l t - s a l t d i f f e r e n c e parameters f o r m o l e c u l a r s o l u t e s a l t i n g , and u n s y m m e t r i c M a r g u l e s p a r a m e t e r s f o r molecule-molecule i n t e r a c t i o n s . L i k e the P i t z e r e q u a t i o n , t h e model i s d e s i g n e d f o r c o n v e n i e n t and a c c u r a t e r e p r e s e n t a t i o n o f aqueous e l e c t r o l y t e s y s t e m s , i n c l u d i n g m i x t u r e s w i t h any number o f m o l e c u l a r and i o n i c s o l u t e s . A p p l i c a t i o n of the Extended P i t z e r

Equation

To t e s t t h e v a l i d i t y o f t h e e x t e n d e d P i t z e r e q u a t i o n , c o r r e l a t i o n s o f 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 were c a r r i e d o u t f o r three systems. Since the extended P i t z e r equation reduces t o t h e P i t z e r e q u a t i o n f o r aqueous s t r o n g e l e c t r o l y t e s y s t e m s , and i s c o n s i s t e n t w i t h t h e S e t s c h e n o w e q u a t i o n f o r m o l e c u l a r n o n - e l e c t r o l y t e s i n aqueous e l e c t r o l y t e s y s t e m s , t h e main i n t e r e s t h e r e i s aqueous s y s t e m s w i t h weak e l e c t r o l y t e s o r partially dissociated electrolytes. The t h r e e s y s t e m s c o n s i d e r e d a r e : t h e h y d r o c h l o r i c a c i d aqueous s o l u t i o n a t 298.15°K and c o n c e n t r a t i o n s up t o 18 m o l a l ; t h e N H 3 - C O 2 aqueous s o l u t i o n a t 2 9 3 . 1 5 ° K ; and t h e K 2 C O 3 - C O 2 aqueous s o l u t i o n o f t h e Hot C a r b o n a t e P r o c e s s . In each c a s e , t h e c h e m i c a l e q u i l i b r i u m between a l l s p e c i e s has been t a k e n i n t o a c c o u n t d i r e c t l y as l i q u i d phase c o n s t r a i n t s . Significant p a r a m e t e r s i n t h e model f o r each s y s t e m were i d e n t i f i e d by a p r e l i m i n a r y o r d e r o f m a g n i t u d e a n a l y s i s and a d j u s t e d i n 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 data c o r r e l a t i o n . Detailed discusions and v a l u e s o f p h y s i c a l c o n s t a n t s , s u c h as H e n r y ' s c o n s t a n t s and c h e m i c a l e q u i l i b r i u m c o n s t a n t s , a r e g i v e n i n Chen e t a l . ( 1 1 ) . T-P-x-y d a t a f o r h y d r o c h l o r i c a c i d c o n c e n t r a t i o n up t o 18 m o l a l were o b t a i n e d f r o m Vega and V e r a ( 1 6 J . The f o l l o w i n g r e a c t i o n s occur i n the l i q u i d phase. HC1

H 0 2

% H * H

+ +

+ CT + 0H-

The l e a s t s q u a r e s d a t a c o r r e l a t i o n was c a r r i e d o u t on H C 1 v a p o r m o l e f r a c t i o n and t o t a l


CHEN ET AL.

4.


D H C l , H C " h Η Π HC1» * H e n r y ' s c o n s t a n t f o r hydrogen c h l o r i d e as a d j u s t a b l e p a r a m e t e r s . F i g u r e 1 shows e x p e r i m e n t a l d a t a and c o r r e l a t i o n r e s u l t s . The a v e r a g e p e r c e n t a g e d e v i a t i o n f o r t o t a l p r e s s u r e i s 0 . 4 4 , and t h a t f o r HC1 v a p o r f r a c t i o n i s 0.35. The same d a t a was p r e v i o u s l y c o r r e l a t e d w i t h t h e same o b j e c t i v e f u n c t i o n by C r u z and Renon ( 7 J . T h e i r r e s u l t s were 0 . 9 9 p e r c e n t d e v i a t i o n f o r t o t a l p r e s s u r e and 0 . 3 4 p e r c e n t d e v i a t i o n f o r HC1 v a p o r f r a c t i o n . The d a t a r e p o r t e d by van K r e v e l e n , e t a l . ( J 7 ) a t 2 9 3 . 1 5 ° K were used f o r d a t a c o r r e l a t i o n o f t h e N H 3 - C O 2 aqueous s o l u t i o n s y s t e m . The f o l l o w i n g r e a c t i o n s o c c u r i n t h e l i q u i d phase.


λ

NH C0

a n c

+ H 0 * N H 4 + 0H" + H 0 * H + HCO3

3

2

2

2

HC05 NH3

*

+

=

CO3 + H

+ HCO3

H 0 * H 2

*

+

NH9COO-

+ H?0

+ 0H-

+

The l e a s t pressures

s q u a r e s d a t a c o r r e l a t i o n was c a r r i e d o u t on p a r t i a l o f N H 3 and C 0 w i t h e l ) N H H C 0 , 3 ( 0 ) ( N H ) C 0 , 0

2

B(0)NH4NH C00,

DNH HC03,NH3

2

4

a

D

4

3

4

(NH4)2C03,NH , 3

a n d

D

2

3

NHdNH?C00,NH 2

and

adjustable'parameters. The a v e r a g e p e r c e n t d e v i a t i o n o f c a l c u l a t e d v e r s u s measured p a r t i a l p r e s s u r e o f

*K

C0

+

2

?

HC0 ~^C03~ 3

at 3 8 3 . 1 5 0 K

a

s

i s 1 1 . 5 % and f o r H 0 i s 2

10.5%.




HCI Concentration

(Molality)

Figure 1. Total vapor pressure and vapor phase HCI mole fraction of the HCI aqueous solution at 298.15 Κ (( ) calculated; (O, A) data from Ref. 16)




CHEN ET AL.

Figure 2. Partial pressures of NH and C0 of the NH -C0 aqueous solution at 293.15 K (experimental data ( Ί 7 Λ Ό s (2N NH ); f | J NH (IN NH ); (O) CO, (2N NH ); (Φ) C0 (IN NH ); ( ) calculated) 3

2

N

3

3

3

2

2

H

3

3

3


70



L i m i t a t i o n s o f the P i t z e r

Equation

The P i t z e r e q u a t i o n j u s t p r e s e n t e d i s s u b j e c t t o a l l o f the l i m i t a t i o n s of a v i r i a l - e x p a n s i o n type e q u a t i o n . The equation parameters, denoting the short-range i n t e r a c t i o n s between and among s o l u t e s p e c i e s , a r e a r b i t r a r y , h i g h l y t e m p e r a t u r e - d e p e n d e n t and a r e c h a r a c t e r i s t i c o f t h e s o l v e n t . B i n a r y p a r a m e t e r s a r e e x p r e s s e d as e m p i r i c a l f u n c t i o n s o f i o n i c strength. D i f f e r e n t e m p i r i c a l f u n c t i o n s are proposed f o r d i f f e r e n t types of e l e c t r o l y t e s . Ternary parameters are r e q u i r e d f o r e l e c t r o l y t e systems at h i g h e r i o n i c s t r e n g t h . F u r t h e r m o r e , f o r the i n d u s t r i a l l y i m p o r t a n t c l a s s o f mixed s o l v e n t e l e c t r o l y t e systems, the P i t z e r equation i s not a p p l i c a b l e b e c a u s e i t s p a r a m e t e r s a r e unknown f u n c t i o n s o f s o l v e n t c o m p o s i t i o n and b e c a u s e t h e e m p i r i c a l e x p r e s s i o n s r e q u i r e d are a v a i l a b l e o n l y f o r w a t e r . T h e r e f o r e , although the P i t z e r e q u a t i o n h a s been shown t o be a c o n v e n i e n t and a c c u r a t e r e p r e s e n t a t i o n o f aqueous e l e c t r o l y t e s y s t e m s , a more v e r s a t i l e model i s needed t o c o v e r a w i d e r v a r i e t y o f e l e c t r o l y t e s y s t e m s . The L o c a l

C o m p o s i t i o n Model

N o n e l e c t r o l y t e s y s t e m s , w h i c h a r e c h a r a c t e r i z e d by s h o r t - r a n g e f o r c e s between m o l e c u l e s , h a v e f r e q u e n t l y been s t u d i e d using the l o c a l composition concept. M o d e l s s u c h as W i l s o n ( 1 9 ) , NRTL (Renon and P r a u s n i t z , ( 2 0 ) ) , and UNIQUAC (Abrams and P r a u s n i t z , ( 2 1 ) ) have r e s u l t e d . Such m o d e l s have p r o v e n t o be a g r e a t advancement o v e r o l d e r m o d e l s b a s e d on a l g e b r a i c e x p a n s i o n s o f m o l e f r a c t i o n , s u c h as t h e M a r g u l e s model. In t h i s s t u d y t h e l o c a l c o m p o s i t i o n c o n c e p t i s a p p l i e d to the short range i n t e r a c t i o n f o r c e s o c c u r r i n g i n e l e c t r o l y t e s y s t e m s w i t h t h e hope t h a t a s i m i l a r a d v a n c e o v e r t h e P i t z e r model w i l l r e s u l t . However, i t must be e m p h a s i z e d t h a t t h e l o c a l c o m p o s i t i o n c o n c e p t i s i n no s e n s e r i g o r o u s . I t i s used to develop c o r r e l a t i n g e x p r e s s i o n s , with adjustable parameters, f o r experimental data. The p u r p o s e i n a d o p t i n g a q u a s i t h e o r e t i c a l approach i s t o develop e x p r e s s i o n s w i t h a s m a l l number o f p a r a m e t e r s , t h a t a p p l y o v e r w i d e c o n c e n t r a t i o n r a n g e s , may be e x p r e s s e d as s i m p l e f u n c t i o n s o f t e m p e r a t u r e , and may be u s e d t o p r e d i c t t h e b e h a v i o r o f m u l t i c o m p o n e n t systems. The v a l i d i t y o f t h i s a p p r o a c h may be d e t e r m i n e d o n l y by i t s e m p i r i c a l s u c c e s s o r f a i l u r e . We b e l i e v e t h a t t h e model t o be p r e s e n t e d i n t h i s s t u d y i s q u i t e s u c c e s s f u l by t h e s e c r i t e r i a , as d e m o n s t r a t e d by t h e e x a m p l e s shown i n t h i s p a p e r and a d d i t i o n a l work we have done w i t h m u l t i c o m p o n e n t s y s t e m s i n v o l v i n g weak e l e c t r o l y t e s . A f u n d a m e n t a l d i f f e r e n c e between e l e c t r o l y t e s y s t e m s and n o n e l e c t r o l y t e systems i s the presence o f long range i o n - i o n e l e c t r o s t a t i c f o r c e s in e l e c t r o l y t e systems. No a t t e m p t was made t o d e v e l o p a l o n g - r a n g e c o n t r i b u t i o n model b a s e d on t h e



4.

CHEN ET AL.


71

l o c a l compostion concept. I n s t e a d , t h e Debye-Huckel f o r m u l a as p r o p o s e d by F o w l e r and Guggenheim ( 2 2 ) was used w i t h o u t m o d i f i c a t i o n t o r e p r e s e n t t h e unsymmetric excess Gibbs energy c o n t r i b u t i o n a r i s i n g from t h e long-range ion-ion e l e c t r o s t a t i c forces. N e v e r t h e l e s s , the l o c a l composition concept i s c o n s i s t e n t w i t h t h e Debye-Huckel f o r m u l a i n t h e s e n s e t h a t t h e B o l t z m a n n d i s t r i b u t i o n l a w i s assumed i n b o t h m o d e l s . The Debye-Huckel f o r m u l a i s a f u n c t i o n o f s o l v e n t d e n s i t y , d i e l e c t r i c c o n s t a n t , and i o n i c s t r e n g t h . I t i s known t o c o r r e c t l y account f o r the ion-ion e l e c t r o s t a t i c c o n t r i b u t i o n i n the l i m i t o f i n f i n i t e d i l u t i o n . When e l e c t r o l y t e c o n c e n t r a t i o n i n c r e a s e s , s h o r t - r a n g e f o r c e s s t a r t t o p l a y a r o l e and f i n a l l y dominate i n t h e r e g i o n o f high e l e c t r o l y t e c o n c e n t r a t i o n ( R o b i n s o n and S t o k e s , ( 1 3 ) ) . The g e n e r a l a p p r o a c h t a k e n i n t h e p r e s e n t s t u d y i s as follows. The D e b e y e - H u c k e l f o r m u l a i s u s e d t o r e p r e s e n t long-range ion-ion i n t e r a c t i o n s w h i l e t h e l o c a l composition concept i s used t o r e p r e s e n t s h o r t range i n t e r a c t i o n s o f a l l kinds. The l o c a l c o m p o s i t i o n model i s b a s e d on two f u n d a m e n t a l a s s u m p t i o n s ; 1) t h a t t h e l o c a l c o m p o s i t i o n o f a n i o n s a r o u n d a n i o n s i s z e r o , and s i m i l a r l y f o r c a t i o n s , w h i c h i s e q u i v a l e n t t o a s s u m i n g t h a t r e p u l s i v e f o r c e s between i o n s o f l i k e c h a r g e are l a r g e , 2) t h a t t h e d i s t r i b u t i o n o f a n i o n s and c a t i o n s around s o l v e n t m o l e c u l e s i s such t h a t t h e n e t i o n i c charge i s zero. The l a t t e r a s s u m p t i o n we r e f e r t o a s l o c a l electroneutrality. The l o c a l c o m p o s t i o n model i s d e v e l o p e d as a s y m m e t r i c m o d e l , b a s e d o n p u r e s o l v e n t and h y p o t h e t i c a l p u r e c o m p l e t e l y dissociated liquid electrolyte. T h i s model i s t h e n n o r m a l i z e d by i n f i n i t e d i l u t i o n a c t i v i t y c o e f f i c i e n t s i n o r d e r t o o b t a i n an u n s y m m e t r i c l o c a l c o m p o s i t i o n m o d e l . F i n a l l y t h e unsymmetric D e b y e - H u c k e l a n d 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 s a r e added t o y i e l d t h e excess Gibbs energy e x p r e s s i o n proposed i n t h i s s t u d y . Development o f t h e L o c a l


Among t h e v a r i o u s m o d e l s i n c o r p o r a t i n g t h e l o c a l c o m p o s i t i o n c o n c e p t f o r s h o r t - r a n g e i n t e r a c t i o n s , t h e NRTL equation i s adopted i n t h i s s t u d y . E l e c t r o l y t e systems a r e c h a r a c t e r i z e d by e x t r a o r d i n a r i l y l a r g e h e a t s o f m i x i n g . Compared t o t h e h e a t o f m i x i n g , t h e n o n i d e a l e n t r o p y o f m i x i n g i s n e g l i g i b l e , which i s c o n s i s t e n t w i t h t h e b a s i c assumption b e h i n d t h e NRTL e q u a t i o n . In a d d i t i o n , t h e NRTL e q u a t i o n i s a l g e b r a i c a l l y simple w h i l e a p p l i c a b l e t o mixtures which e x h i b i t phase s p l i t t i n g . No s p e c i f i c volume o r a r e a d a t a a r e r e q u i r e d . In t h e NRTL m o d e l , t h e l o c a l m o l e f r a c t i o n s XJ-J and x-ji o f s p e c i e s j and i , r e s p e c t i v e l y , i n t h e i m m e d i a t e neighborhood o f a c e n t r a l molecule o f species i a r e r e l a t e d by


THERMODYNAMICS

72

x

ji/ ii

OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS

=( j/ i)Gji

x

x

(7)

x

where Gji ji

=exp(-otTji) (gji-9ii)/RT


T

=

The q u a n t i t i e s gj-j and g-j-j a r e , r e s p e c t i v e l y , e n e r g i e s o f i n t e r a c t i o n between j - i and i - i p a i r s o f s p e c i e s , and a r e i n h e r e n t l y symmetric ( g j i = g i j ) . The nonrandomness f a c t o r , a, was f i x e d a t a v a l u e o f 0 . 2 i n t h i s s t u d y . F o r c o n v e n i e n c e i n r e p r e s e n t i n g o t h e r l o c a l mole f r a c t i o n r a t i o s , we i n t r o d u c e a d d i t i o n a l n o t a t i o n as f o l l o w s : x

ji/*ki

=( j/ k)Gji,ki x

(8)

x

where Gji,ki T

=exp(-cxTji

ji,ki

=(gji-9ki)/

R T

W h i l e t h e d e r i v a t i o n t h a t f o l l o w s may be g e n e r a l i z e d t o handle a l l types o f e l e c t r o l y t e s y s t e m s , f o r the sake o f s i m p l i c i t y , t h e d e r i v a t i o n w i l l be b a s e d on a s i n g l e c o m p l e t e l y - d i s s o c i a t e d e l e c t r o l y t e , s i n g l e solvent system. In a binary mixture of s i n g l e completely-dissociated e l e c t r o l y t e and s i n g l e s o l v e n t , we assume t h a t t h e r e a r e t h r e e t y p e s o f cells. One t y p e c o n s i s t s o f a c e n t r a l s o l v e n t m o l e c u l e w i t h s o l v e n t m o l e c u l e s , a n i o n s and c a t i o n s i n t h e i m m e d i a t e neighborhood. The o t h e r two t y p e s have e i t h e r an a n i o n o r c a t i o n as t h e c e n t r a l s p e c i e s , and an i m m e d i a t e n e i g h b o r h o o d c o n s i s t i n g o f s o l v e n t m o l e c u l e s and o p p o s i t e l y - c h a r g e d i o n s , b u t no i o n s o f l i k e c h a r g e ( i . e . , X c c a a ^ ) l° l mole f r a c t i o n s a r e r e l a t e d by: = x

x x x

cm am mm= 1 mc ac = 1 ma ca + x

+ x

(central (central (central

1

+ x

+ x

=

T h e

c a

solvent cells) cation cells) anion c e l l s )

Among t h e t h r e e t y p e s o f c e l l s t h e r e a r ef o u r d i s t i n c t l o c a l m o l e f r a c t i o n r a t i o s : Xcm/ mm> a m / m m > x / x , and x / caI* notable that the assumption that X c c a a 0 equivalent t o t h e assumption that g and g a r e much g r e a t e r t h a n t h e o t h e r interaction energies. X

x

m c

a c

x

x

i s

m a

= x

c

c

a

=

1 S

a

By combining e q u a t i o n s ( 7 ) , (8) a n d ( 9 ) , t h e f o l l o w i n g expressions f o r t h e local mole f r a c t i o n s i n terms o f o v e r a l l mole f r a c t i o n s may be d e r i v e d :


(9)

CHEN ET AL.

4.

xim iîm/( aâm cW m nm) a c = a/ a mGmc,ac ca ~ c'( c m ma,ca' = x

x

x

x

x

x

x

73


+ x

x

Π

G

= c

> > ) a

,

m

+ x

x

+ x

1

Λ

λ

00)

G

In o r d e r t o o b t a i n an 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 , we f i r s t d e f i n e g ( ) , g ^ ' , and g ( ) as t h e r e s i d u a l G i b b s e n e r g i e s p e r mole o f c e l l s o f c e n t r a l a n i o n , c e n t r a l c a t i o n and c e n t r a l s o l v e n t m o l e c u l e , r e s p e c t i v e l y . These G i b b s e n e r g i e s a r e r e l a t e d t o t h e l o c a l mole f r a c t i o n s as f o l l o w s : a

9

=

i?i


9 ^ 9

c

m

^ S ma9ma+ ca9ca) a

x

x

m

= Zc( m gmc acgac) x

'

=

x

,

+ x

C

amgam cmgcm iTmgnfni +x

+x

We t h e n a d o p t t h e p u r e s o l v e n t as t h e r e f e r e n c e s t a t e f o r t h e s o l v e n t , and a h y p o t h e t i c a l p u r e c o m p l e t e l y - d i s s o c i a t e d l i q u i d e l e c t r o l y t e as t h e r e f e r e n c e s t a t e f o r t h e e l e c t r o l y t e . The r e f e r e n c e G i b b s e n e r g i e s p e r mole a r e t h e n : ^ref

"

9ref

= Z

g a

( m

]

Z

câc

a9ca

(12)

= 9

ref

^nfîïi

In b o t h e q u a t i o n s ( 1 1 ) and ( 1 2 ) t h e c h a r g e number Z and Z a r e introduced t o account f o r t h e f a c t that t h e r a t i o of t h e c o o r d i n a t i o n number o f c e n t r a l a n i o n c e l l s t o t h a t o f c e n t r a l c a t i o n c e l l s must be e q u a l t o t h e c o r r e s p o n d i n g r a t i o o f c h a r g e n u m b e r s . The m o l a r e x c e s s G i b b s e n e r g y may now be d e r i v e d b y sumning t h e changes i n r e s i d u a l G i b b s e n e r g y r e s u l t i n g when x moles o f s o l v e n t a r e t r a n s f e r e d from t h e s o l v e n t r e f e r e n c e s t a t e t o t h e i r c e l l s i n t h e m i x t u r e , and when x moles o f a n i o n s and x moles of c a t i o n s a r e t r a n s f e r e d from t h e e l e c t r o l y t e r e f e r e n c e s t a t e t o their respective c e l l s in themixture. The e x p r e s s i o n i s : ç

a

m

a

S u b s t i t u t i n g equations g /RT

x

+ x

x

am a z

( 1 3 ) we o b t a i n

=Xm cm^cm m am am

e x

cells

( 1 1 ) and ( 1 2 ) i n t o e q u a t i o n

c

+x

x

c mc c mc,ac x

z

T

i:

+ x

a ma a ma,ca x

z

T

The a s s u m p t i o n o f l o c a l e l e c t r o n e u t r a l i t y a p p l i e d t o t h e o f c e n t r a l s o l v e n t m o l e c u l e s may be s t a t e d as = x

cm c z


74


Substituting equation following equality:

(7)

into t h i s r e l a t i o n s h i p leads to

the (16)

9am ~9cm S i n c e the i n t e r a c t i o n e n e r g i e s are i n f e r r e d from t h i s r e s u l t t h a t : T

am

T

mc,ac " m a , c a

= T

cm

i t may be (17)

~ ca,m T

T

= T

m,ca

(18)

The b i n a r y p a r a m e t e r s T and T t h e n become t h e o n l y two i n d e p e n d e n t a d j u s t a b l e p a r a m e t e r s f o r a s i n g l e c o m p l e t e l y d i s s o c i a t e d e l e c t r o l y t e , s i n g l e solvent system. In o r d e r t o combine e q u a t i o n ( 1 4 ) w i t h t h e D e b y e - H u c k e l f o r m u l a , which accounts f o r the long-range f o r c e c o n t r i b u t i o n , i t i s necessary to normalize to the i n f i n i t e d i l u t i o n reference s t a t e f o r the i o n s : C


symmetric,

g */RT e x

A

J

M

M

c

a

= g / R T - x l n Y g > -x 1nY§> ex

c

(19)

a

A f t e r e m p l o y i n g e q u a t i o n (14) t o o b t a i n I n l a n d l n y f a n d s u b s t i t u t i n g back i n t o e q u a t i o n ( 1 9 ) , t h e f i n a l r e s u l t i s : g

e x

/RT

=Xm( cm am) ca,m c mc c m,ca a ma a m,ca ~ c( c m,ca Gcm ca,m) ~ a( a m,ca âm' ca,m) x

+ x

x

x

x

+x

z

z

T

z

T

T

+ x

+

T

+

x

z

i:

The e q u a t i o n s f o r b i n a r y s y s t e m s j u s t p r e s e n t e d can g e n e r a l i z e d t o m u l t i c o m p o n e n t s y s t e m s c o n s i s t i n g o f any c o m b i n a t i o n o f weak and s t r o n g e l e c t r o l y t e s , m o l e c u l a r s o l v e n t s , and m o l e c u l a r s o l u t e s . D i s c u s s i o n of the Local

(20)

T

T

be


The l o c a l c o m p o s i t i o n model makes i t p o s s i b l e t o s t u d y e l e c t r o l y t e thermodynamics over a wide range o f c o m p o s i t i o n s . I t assumes t h a t t h e D e b y e - H u c k e l f o r m u l a i s a d e q u a t e t o r e p r e s e n t t h e l o n g - r a n g e i o n - i o n e l e c t r o s t a t i c c o n t r i b u t i o n and t h e l o c a l c o m p o s i t i o n model can a c c o u n t f o r t h e s h o r t - r a n g e i n t e r a c t i o n s among a l l s p e c i e s . While the v a l i d i t y of the Debye-Huckel f o r m u l a a t h i g h i o n i c s t r e n g t h i s q u e s t i o n a b l e , i t i s hoped t h a t t h e s h o r t - r a n g e c o n t r i b u t i o n w i l l d o m i n a t e a t high i o n i c s t r e n g t h , so t h a t accounting f o r the long-range i o n - i o n e l e c t r o s t a t i c c o n t r i b u t i o n a c c u r a t e l y i s not c r i t i c a l . S y s t e m s w i t h weak e l e c t r o l y t e s , o r p a r t i a l l y d i s s o c i a t e d e l e c t r o l y t e s , c a n be s t u d i e d i f c h e m i c a l e q u i l i b r i u m among i o n i c s p e c i e s and m o l e c u l a r s p e c i e s i s c o n s i d e r e d . Multi-



4.

CHEN ET AL.


75

s o l v e n t s y s t e m s c a n be i n v e s t i g a t e d w i t h t h e k n o w l e d g e o f m i x e d - s o l v e n t d i e l e c t r i c c o n s t a n t and d e n s i t y w h i c h a r e r e q u i r e d f o r t h e Debye-Huckel f o r m u l a . The h y p o t h e t i c a l p u r e c o m p l e t e l y - d i s s o c i a t e d l i q u i d e l e c t r o l y t e model h a s n o t h i n g t o do w i t h s o l i d s a l t c r y s t a l s . However, s a l t p r e c i p i t a t i o n c a n a l s o be s t u d i e d i f s o l u b i l i t y p r o d u c t c o n s t a n t s a r e known. I t s h o u l d be n o t e d t h a t t h e l o c a l c o m p o s i t i o n model i s n o t c o n s i s t e n t w i t h t h e commonly a c c e p t e d s o l v a t i o n t h e o r y . According t o the solvation theory, i o n i c species are completely s o l v a t e d by s o l v e n t m o l e c u l e s . In o t h e r w o r d s , t h e l o c a l mole f r a c t i o n o f s o l v e n t m o l e c u l e s around a c e n t r a l i o n i s u n i t y . T h i s becomes u n r e a l i s t i c when a p p l i e d t o h i g h c o n c e n t r a t i o n e l e c t r o l y t e s y s t e m s s i n c e t h e number o f s o l v e n t m o l e c u l e s w i l l be i n s u f f i c i e n t t o c o m p l e t e l y s o l v a t e i o n s . W i t h t h e l o c a l composition model, a l l ions a r e , e f f e c t i v e l y , completely surrounded by s o l v e n t m o l e c u l e s i n d i l u t e e l e c t r o l y t e systems and o n l y p a r t i a l l y s u r r o u n d e d by s o l v e n t m o l e c u l e s i n h i g h concentration e l e c t r o l y t e systems. Therefore, the local c o m p o s i t i o n model i s b e l i e v e d t o be c l o s e r t o t h e p h y s i c a l r e a l i t y than t h e s o l v a t i o n t h e o r y . A p p l i c a t i o n o f the Local


A w i d e v a r i e t y o f d a t a f o r 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 , osmotic c o e f f i c i e n t s , vapor pressure d e p r e s s i o n , and v a p o r - l i q u i d e q u i l i b r i u m o f b i n a r y and t e r n a r y e l e c t r o l y t e s y s t e m s have been c o r r e l a t e d s u c c e s s f u l l y by t h e l o c a l composition model. Some r e s u l t s a r e shown i n T a b l e 1 t o T a b l e 10 and F i g u r e 3 t o F i g u r e 7. In each c a s e , t h e c h e m i c a l e q u i l i b r i u m b e t w e e n t h e s p e c i e s h a s been i g n o r e d . That i s , c o m p l e t e d i s s o c i a t i o n o f s t r o n g e l e c t r o l y t e s h a s been a s s u m e d . T h i s a s s u m p t i o n i s n o t r e q u i r e d by t h e l o c a l c o m p o s i t i o n model b u t h a s been made h e r e i n o r d e r t o s i m p l i f y t h e s y s t e m s t r e a t e d . In g e n e r a l , d a t a a r e f i t q u i t e w e l l w i t h t h e m o d e l . F o r e x a m p l e , w i t h o n l y two b i n a r y p a r a m e t e r s , t h e a v e r a g e s t a n d a r d d e v i a t i o n o f c a l c u l a t e d Ι η γ * v e r s u s m e a s u r e d ΙηΎ* o f t h e 50 u n i - u n i v a l e n t aqueous s i n g l e e l e c t r o l y t e s y s t e m s l i s t e d i n Table 1 i s only 0.009. A l t h o u g h t h e f i t i s n o t a s good a s t h e P i t z e r e q u a t i o n , w h i c h a p p l i e s o n l y t o aqueous e l e c t r o l y t e s y s t e m s , w i t h two b i n a r y p a r a m e t e r s and o n e t e r n a r y p a r a m e t e r ( P i t z e r , ( 5 ) ) , i t i s q u i t e s a t i s f a c t o r y and b e t t e r t h a n t h a t o f Bromley's equation ( J J . Data c o r r e l a t i o n r e s u l t s f o r s i n g l e - s a l t , s i n g l e - s o l v e n t b i n a r y s y s t e m s a r e shown i n T a b l e 1 t o T a b l e 6 and F i g u r e 3 t o F i g u r e 6 . T h e r e i s an o b v i o u s t r e n d between ,ca s t a n d a r d d e v i a t i o n o f c a l c u l a t e d Ιηγ* v e r s u s measured Ι η γ * . When t h e a b s o l u t e v a l u e o f x increases, standard deviation also increases. This i s c o n s i s t e n t with t h e physical meaning o f T . The l a r g e r t h e a b s o l u t e v a l u e o f , t h e s t r o n g e r t h e i n t e r a c t i o n between c a t i o n and a n i o n . ' In T

m

Mc

a

c

m

a n c l

a

T

m

c

a


76


Table

1.

Electrolyte

No. o f d a t a * Points

23 18 21 23 17 12 CSNO3 KAc 18 Κ Br 22 KC1 20 KCNS 21 KF 19 KH m a I o n a t e 21 KH s u c c i n a t e 20 KH2PO4 14 KI 20 KNO3 18 Κ OH 23 LiAc 19 L i Br 23 L i CI 23 19 LiC104 Lil 17 LÏN03 23 LiOH 19 LiTol 20 NaAc 18 NaBr 19 16 NaBr03 Na b u t y r a t e 18 NaCI 23 18 NaC103 23 NaC104 NaCNS 19 NaF 10 Na f o r m a t e 18 NaH2P04 23 Nal 18 Na m a l o n a t e 21 23 NaN03 NaOH 23 Na p r o p i o n a t e 17 Na s u c c i n a t e 21 NH4CI 23 AgNQ3 CsAc CsBr CsCl Csl


D a t a and R e s u l t s o f F i t f o r Aqueous S o l u t i o n s o f u n i - u n i v a l e n t e l e c t r o l y t e a t 298.15°K - 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 molality 0.1· - 6 . 0 0 . Ι -3.5 Ο. Ι - 5 . 0 Ο. 1·- 6 . 0 0.1-- 3 . 0 0.1· -1.4 0.1· -3.5 0.1· - 5 . 5 0.1 -4.5 0.1-- 5 . 0 0.1 - 4 . 0 0.1· - 5 . 0 0.1 -4.5 0.1· - 1 . 8 0.1 -4.5 0.1· - 3 . 5 0.1 - 6 . 0 0.1· - 4 . 0 0.1 - 6 . 0 0.1· - 6 . 0 0.1· - 4 . 0 0.1 - 3 . 0 0.1· - 6 . 0 0.1 - 4 . 0 0.1· - 4 . 5 0.1 - 3 . 5 0.1-- 4 . 0 0.1· - 2 . 5 0.1 - 3 . 5 0.1· - 6 . 0 0.1· - 3 . 5 0.1 - 6 . 0 0.1 - 4 . 0 0.1 - 1 . 0 0.1 - 3 . 5 0.1 - 6 . 0 0.1 - 3 . 5 0.1 - 5 . 0 0.1 - 6 . 0 0.1 - 6 . 0 0.1 - 3 . 0 0.1 - 5 . 0 0.1 - 6 . 0

T

m,ca

7.295 8.462 8.381 8.368 8.280 8.988 8.459 7.901 7.917 7.319 8.679 7.338 7.982 8.924 7.620 7.642 9.733 8.304 10.331 9.900 9.464 9.157 8.804 8.920 7.396 8.257 8.672 7.587 7.230 8.715 7.128 7.799 7.770 7.517 7.295 8.138 8.752 7.527 7.071 9.225 8.277 8.075 7.614

T

ca,m

-3.059 -4.500 -4.034 -4.043 -3.963 -4.057 -4.476 -3.962 -3.944 -3.644 -4.373 -3.462 -3.861 -4.017 -3.892 -3.327 -4.945 -4.278 -5.251 -5.046 -4.986 -4.889 -4.562 -4.275 -3.710 -4.356 -4.435 -3.589 -4.104 -4.400 -3.527 -3.937 -4.078 -3.677 -3.776 -3.711 -4.535 -3.659 -3.381 -4.647 -4.435 -3.968 -3.800

σ

1ηγ* 0.012 0.009 0.008 0.009 0.007 0.003 0.007 0.002 0.002 0.001 0.004 0.006 0.002 0.003 0.004 0.008 0.019 0.004 0.046 0.036 0.022 0.024 0.010 0.027 0.012 0.006 0.008 0.001 0.008 0.01,4 0.004 0.007 0.009 0.001 0.003 0.002 0.009 0.003 0.003 0.026 0.006 0.002 0.002


4.


CHEN ET AL.

Table 1. RbAc RbBr RbCl Rbl RbNQ3 TIAc

*(Robinson

17 23 23 17 17

HCIO4

HI

HNO3

*(Robinson

Electrolyte

N1SO4 CUSO4

ZnS04 CdS04 UO2SO4

*(Robinson

0.008 0.004 0.003 0.004 0.013 0.014

Highest Molality

m,ca

ca,m

3.0 6.0 6.0 3.0 3.0

9.742 9.957 10.488 9.483 8.327

- 5 . 087 - 5 . 106 - 5 . 328 -5.059 -4.341

σ

1ηγ*

0.014 0.031 0.058 0.017 0.008

and S t o k e s , ( 1 3 ) )

Table 3.

BeS04 MgSC4 MnS04

0.014

D a t a and R e s u l t s o f F i t f o r A c i d s a t 298.15°K Assuming Complete D i s s o c i a t i o n - 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

No. o f D a t a * Points

HBr HCI

-3.295 -4.545 -3.891 -3.983 -3.949 -3.287 -3.618

and S t o k e s , ( 1 3 ) )

Table 2.

Acid

7.170 8.602 7.920 8.086 8.052 7.648 7.683

0.1-6.0 0.1-3.5 0.1-5.0 0.1-5.0 0.1-5.0 0.1-4.5 0.1-6.0

23 18 21 21 21 20 23

NH4N03


Continued

D a t a and R e s u l t s o f F i t f o r Aqueous S o l u t i o n s o f B i - b i v a l e n t E l e c t r o l y t e s a t 298.15°K - 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

No. o f D a t a * Molality Points 18 0.2-4.0 0.2-3.5 16 0.2-4.0 18 0.2-2.5 15 0.2-1.4 11 0.2-3.5 17 0.2-3.5 17 0.2-6.0 22

T

m,ca

11.728 11.623 11.499 11.704 12.128 11.693 11.481 11.316

T

ca,m

-6.905 -6.827 -6.732 -6.826 -7.043 -6.827 -6.704 -6.646

σ

1ηγ*

0.049 0.047 0.046 0.042 0.043 0.046 0.053 0.078

and S t o k e s , ( 1 3 ) )



78

Table 4


Electrolyte

D a t a and R e s u l t s o f F i t f o r U n i - u n i v a l e n t E l e c t r o l y t e E f f e c t on t h e Vapor P r e s s u r e o f M e t h a n o l a t 298.05°K No. o f D a t a * Points

LiCl NaBr NaOH Nal KI

9 9 9 16 9

*(Bixon et

al.,

Table 5

T°K

5.3554 1.556 5.9413 4.5200 1.1219

11.783 10.717 10.372 9.716 10.765

28 28 28 28 28

Table 6

Molality

^

0.05-6.0 0.05-6.0 0.05-6.0 0.05-6.0 0.05-6.0

*(Robinson

14 14 14 14 14

m

>

c

Molality

8.831 8.744 8.629 8.510 8.420

0.1-4.0 0.1-4.0 0.1-4.0 0.1-4.0 0.1-4.0

and S t o k e s ,

T

m,ca

7.860 7.831 7.773 7.769 7.760

Fit

σ

a

-4.406 -4.409 -4.380 -4.334 -4.288

1 η γ

(23))

T

ca,m

-3.994 -3.989 -3.970 -3.971 -3.969

*

0.018 0.014 0.011 0.008 0.005

T e m p e r a t u r e E f f e c t on D a t a and R e s u l t s o f f o r Aqueous KBr S o l u t i o n s - 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


333.15 343.15 353.15 363.15 373.15

0.034 0.002 0.019 0.011 0.002

T e m p e r a t u r e E f f e c t on D a t a and R e s u l t s o f f o r Aqueous NaCl S o l u t i o n s - 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

* s m o o t h e d d a t a ( S i l v e s t e r and P i t z e r ,

T°K

-5.562 -5.176 -5.633 -5.186 -5.138

(25))


273.15 298.15 323.15 348.15 373.15

Highest Molality

σ

Fit

1ηγ* 0.001 0.001 0.001 0.002 0.002

(13))


4.

CHEN ET AL.


T a b l e 7.


Acid HCI HCI HCI HBr HBr HBr

Salt

D a t a and R e s u l t s o f F i t 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 s o f HCL and HBr i n H a l i d e S o l u t i o n s a t 298.15°K A c i d C o n c e n t r a t i o n = 0.01M ( a p p r o x i m a t e d a s 0.01m)

No. o f D a t a * Point 13 9 11 12 11 15

KC1 NaCl L i CI KBr NaBr L i Br

*(Harned

X

T

C09

9 5

HoO =

H 0,C09 2

T

M X , H X ( = • ΗΧ,ΜΧ) Τ

-1.633 -1.129 -0.106 -1.464 -1.113 -0.307

σ

1η *

=

Highest τ Molality 5.732 3.942

5.733 10.414

τ

σ

*

] T

-4.115 -6.109

1

m

0.014 0.008

-1.644917+(0.320488D-1)*(T-273.15) T

C0 ,H 0*(2.442172) 2

γ

0.020 0.008 0.022 0.009 0.007 0.027

D a t a and R e s u l t s o f F i t on S o l u b i l i t y o f C a r b o n D i o x i d e i n Aqueous S o l u t i o n s a t 298.15°K

No. o f Data* Points

NaCl KC1

Highest S a l t Molality 3.5 3.0 4.0 3.0 3.0 3.0

and Owen, ( 2 4 ) )

Table 8.

Salt

79

2

b i n a r y d a t a o b t a i n e d f r o m H o u g h t o n , G . , A . M. M c L e a n , and P. D. R i t c h i e , ( 2 6 ) ) (CO2-H26

+ ( Y a s u n i s h i , A . and F. Y o s h i d a , ( 2 4 ) ) + o r i g i n a l d a t a were e x p r e s s e d i n t e r m s o f m o l a r i t y and t h e Ostwald c o e f f i c i e n t


80


Table 9.

x '


2

0.148 0.148 0.148 0.148 0.292 0.292 0.292 0.292 0.500 0.500 0.500 0.700 0.700 0.700 0.900 0.900

V a p o r - L i q u i d E q u i l i b r i u m Data C o r r e l a t i o n f o r M e t h a n o l - W a t e r - N a B r s y s t e m a t 298.15°K

Salt Molality

0.852 0.852 0.852 0.852 0.708 0.708 0.708 0.708 0.500 0.500 0.500 0.300 0.300 0.300 0.100 0.100

Expt. p**

1 2 4 7. 1 1 2 4 5. 7 1 2 4 1 2 2. 8 1 1. 9

49.8 50.1 51.1 50.4 68.0 68.9 68.6 65.2 85.1 84.35 80.4 99.5 95.7 91.4 114.2 107.3

Data* yi 0.603 0.627 0.675 0.756 0.742 0.762 0.798 0.820 0.850 0.860 0.884 0.920 0.926 0.932 0.977 0.979

C a l c . Value p** yi 50.4 52.7 53.2 47.7 68.7 70.9 69.3 64.2 86.9 86.7 79.9 100.4 97.2 92.7 113.6 107.1

diff Ρ

0.622 0.655 0.686 0.693 0.766 0.783 0.792 0.788 0.859 0.862 0.853 0.916 0.911 0.906 0.969 0.966 mean dev~

cliffy

+0.6 +0.019 +2.6 + 0 . 0 2 8 +2.1 +0.011 -2.7 -0.063 +0.7 +0.024 +2.0 +0.021 +0.7 - 0 . 0 0 6 -1.0 -0.032 + 1.8 +0.009 +2.35 +0.002 - 0 . 5 -0.031 +0.9 -0.-004 +1.5 - 0 . 0 1 5 +1.3 - 0 . 0 2 6 -0.6 -0.008 -0.2 -0.013 1.35 0 . 0 1 9 5

Η ? 0 NaBr =8.672+0.244*(78.48-D)/(78.48-32.66) N a B r H?0 =" .435+0.244*(78.48-D)/(78.48-32.66) CH?0H,NaBr =10.717-3.493*(32.66-D)/(78.48-32.66) NaBr,CHi0H =-5.176-3.493*(32.66-0)/(78.48-32.66) H 0,CH 0H =0.2944 CH 0H,H 0 =0.1936 ( m e t h a n o l - w a t e r b i n a r y d a t a o b t a i n e d f r o m G m e h l i n g , J . and U. Onken, (27)) τ

T

4

T T

T

2

T

3

3

d

2

=0.9971+(-0.163939D-2)*(x ')+(0.1701563D-5)*(xi') -(0.6285073D-7)*(X!') 1

2

3

D

χ x

ΐ' 2

=78.48-(0.4233608)*(xi')-(0.33070470-3)*(xi')2 -(0.3434429D-7)*(x!')3 =

=

x

CH OH/( CH OH H O) H 0/( CH 0H H 07 x

3

X

2

X

+x

3

3

+ X

2

2

* ( C i p a r i s , (28)) * * ( u n i t : mirtigT


4.

CHEN E T AL.

81



T a b l e 10. V a p o r - L i q u i d E q u i l i b r i u m Data C o r r e l a t i o n f o r M e t h a n o l - W a t e r - L i C l s y s t e m a t 298.15°K

X 1

"

Molality

0.152 0.298 0.470 0.700 0.958

0.848 0.702 0.530 0.300 0.042

Η?0 L i C l l i d ' HoO

XH-JOH L i C l

" ρ** P

t

47.3 65.3 80.0 96.3 115.3

1 1 1 1 1

°

a

t

* yi a

0.605 0.765 0.860 0.930 0.993

ρ**

C a l c

ν

47.7 65.2 81.0 98.6 117.4

diff P

Ί

* "6 yi

+0.4 0.597 0.751 -0.1 0.840 +1.0 0.917 +2.3 0.988 +2.1 mean d e v . 1.2

diff y -0.008 -0.014 -0.020 -0.013 -0.005 0.012

-Π.783-1.853*(32.66-D)/(78.48-32.66)

\iCl,CH30H

=-5.562-1.853*(32.66-D)/(78.48-32.66)

CH 0H,H 0 3

X

=9.900-0.2239*(78.48-D)/(78.48-32.66) =-5.046-0.2239*(78.48-D)/(78.48-32.66)

τ

T

E

=0.1936

?

H 0,CH 6H =0-2944 ( m e t h a n o l - w a t e r b i n a r y d a t a o b t a i n e d f r o m G m e h l i n g , J . and U. Onken, ( 2 7 ) ) T

2

d

3

=0.9971+(-0.163939D-2)*(x ')+(0.1701563D-5)*(x ') -(0.6285073D-7)*(xi')3 1

1

2

D =78.48-(0.4233608)*(χι')-(0.3307047ϋ-3)*(χ ')2 -(0.3434429D-7)*(xi') Ί

3

χ

ι'

=

χ

2'

= H 0/( CH 0H XH 07

X

X

CH OH/(XCH OH XH O) 3

2

+

3

x

3

+

2

2

* ( C i p a r i s , (28)) * * ( u n i t : mmHgT


THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS ι

ι

ι

I

I

I

icient ο

20

-

-

ο ο

^

HCI

>%

:> ο

Two New Activity Coefficient Models for the Vapor-Liquid Equilibrium

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