Thermodynamics of Aqueous Systems with ... - ACS Publications

23 Thermodynamics of Aqueous Electrolytes at Various Temperatures, Pressures, and Compositions KENNETH S. PITZER

Downloaded by UNIV LAVAL on May 26, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch023

Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720

Early in this century there was great interest in the ap parently anomalous properties of aqueous electrolytes. The anomaly concerned the limiting behavior at low concentration. While several investigators contributed substantially to the re solution of this problem, it was the classic work of Debye and Hückel (1) which provided a simple yet adequate explanation of the effect on thermodynamic properties of the long-range electrostatic forces between ions in solution. The experimental work of that era tended to emphasize dilute solutions at room temperature. While Debye and Hückel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical me chanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electro static energy was included--not the direct effect of the hard core on thermodynamic properties. As is often the case, after the intense activity of the 1920's, the investigation of aqueous electrolytes proceeded at a more relaxed pace. But careful and systematic experimental re search continued in this area and was summarized by Harned and Owen (2) and by Robinson and Stokes (3) in their excellent mono graphs. The latter volume contains in the appendix a comprehen sive set of tables of the osmotic and activity coefficients of the common inorganic solutes at 25°C and at concentrations up to 6 M in most cases. Subsequently major t h e o r e t i c a l advances were made, p r i n c i p a l l y by Mayer, i n c r e a t i n g an adequate s t a t i s t i c a l mechanical theory i n which both long-range e l e c t r o s t a t i c f o r c e s and s h o r t range f o r c e s of whatever o r i g i n were p r o p e r l y considered. Friedman has c o n t r i b u t e d g r e a t l y to f u r t h e r t h e o r e t i c a l advances and w i l l d i s c u s s recent work i n t h i s symposium. A l s o important are the Monte Carlo c a l c u l a t i o n s of Card and V a l l e a u (4). The w r i t e r has published (5) an elementary review of these t h e o r e t i c a l advances; more advanced reviews are a v a i l a b l e by Friedman (6) and by Andersen (7). The r e s u l t i s that the p r o p e r t i e s o f u n i v a l e n t aqueous e l e c t r o l y t e s based on the hard core and other simple 0-8412-0569-8/80/47-133-451$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.


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models (8) are now known from r e l i a b l e theory up to a concentrat i o n of 1 or 2 M. While the a c t u a l i n t e r i o n i c p o t e n t i a l s of mean f o r c e doubtless d i f f e r somewhat from the models t r e a t e d , the r e s u l t i n g thermodynamic p r o p e r t i e s w i l l be s i m i l a r . Thus we have good t h e o r e t i c a l guidance i n s e l e c t i n g forms of equations f o r s e m i - e m p i r i c a l use. For more h i g h l y charged ions or a s o l v e n t w i t h s u b s t a n t i a l l y lower d i e l e c t r i c constant the s i t u a t i o n i s somewhat more complex, but the theory i s reasonably s a t i s f a c t o r y and i s being f u r t h e r improved. From the thermodynamic v i e w p o i n t , the b a s i c s t a t i s t i c a l theory i s s t i l l too complex to provide u s e f u l working equations, but i t does suggest forms of equations w i t h some p u r e l y t h e o r e t i c a l terms, and other terms i n c l u d i n g parameters to be evaluated e m p i r i c a l l y . In g e n e r a l , the t h e o r e t i c a l terms a r i s e from the e l e c t r o s t a t i c i n t e r a c t i o n s which are simple and well-known w h i l e the e m p i r i c a l , terms r e l a t e to short-range i n t e r i o n i c f o r c e s whose c h a r a c t e r i s t i c s are q u a l i t a t i v e l y but not q u a n t i t a t i v e l y known from independent sources. But, as we s h a l l see, t h i s d i v i s i o n i s not complete - there are i n t e r a c t i o n s between the two c a t e g o r i e s . In recent years there has been a resurgence of experimental research on aqueous i n o r g a n i c e l e c t r o l y t e s emphasizing the broader domain of h i g h temperatures or h i g h pressures or both. A l s o many organic s o l u t e s have been i n v e s t i g a t e d a t room temperature. Thus most of the pure aqueous e l e c t r o l y t e s l i k e l y to be of engineering i n t e r e s t have been i n v e s t i g a t e d at room temperature, a s u b s t a n t i a l number have been s t u d i e d over the 0 - 50°C range, and a s m a l l e r but i n c r e a s i n g number at h i g h pressures and at temperatures to 300°C or o c c a s i o n a l l y h i g h e r . Most p r a c t i c a l systems, however, are mixtures r a t h e r than pure e l e c t r o l y t e s . The experimental measurement of a wide v a r i e t y of mixtures over c l o s e l y spaced g r i d s of composition would be very burdensome. I t i s here, f o r mixed e l e c t r o l y t e s , that theory, confirmed by a l i m i t e d number of experiments, i s p a r t i c u l a r l y v a l u a b l e . The e l e c t r o s t a t i c i n t e r a c t i o n s are a l l simply d e f i n e d ; a l s o the short-range f o r c e s between a p a i r of ions of d i f f e r e n t s i g n are the same i n a mixed e l e c t r o l y t e as i n the pure e l e c t r o l y t e comprising that p a i r of i o n s . Thus a proper d e f i n i t i o n of terms w i l l a l l o w the e v a l u a t i o n of a l l of these e f f e c t s i n mixtures from i n f o r m a t i o n on the v a r i o u s pure e l e c t r o l y t e s . I t i s f o r the e f f e c t of short-range f o r c e s between ions of the same s i g n that new terms a r i s e f o r mixtures. But ions of the same s i g n r e p e l one another and are u n l i k e l y to be so c l o s e together that t h e i r s h o r t range f o r c e s have a l a r g e e f f e c t . Indeed Bronsted (9) p o s t u l a t e d that these d i f f e r e n c e s among short-range i n t e r a c t i o n s among ions of the same s i g n could be ignored and Guggenheim (10) developed d e t a i l e d equations on that b a s i s . Kim and the w r i t e r (11) found that such d i f f e r e n c e s were not completely n e g l i g i b l e f o r mixtures of s i n g l y charged ions or f o r 2-1 charged mixtures but that they were very s m a l l . A l s o these d i f f e r e n c e terms can be evaluated from e x i s t i n g measurements on simple mixtures of the most



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important i o n s . T h e r e a f t e r , f o r complex mixtures of p r a c t i c a l importance, a l l of the important terms are known and only very s m a l l terms must be neglected from l a c k of i n f o r m a t i o n . From the preceding paragraphs i t i s c l e a r that the c a p a c i t y to c a l c u l a t e the p r o p e r t i e s of a v a r i e t y of mixed e l e c t r o l y t e s depends on an adequate t h e o r e t i c a l s t r u c t u r e w i t h i n which the a v a i l a b l e experimental data can be organized. Thus the primary emphasis f o r the remainder of t h i s paper w i l l be the d e s c r i p t i o n of t h i s s t r u c t u r e of s e m i - e m p i r i c a l equations. The a r r a y of sub stances f o r which experimental data are a v a i l a b l e w i l l be d e s c r i b ed i n general terms but there i s not s u f f i c i e n t space to l i s t r e s u l t s i n d e t a i l . A l s o a severe t e s t of p r e d i c t i o n s f o r mixed e l e c t r o l y t e s w i l l be r e p o r t e d . But before t u r n i n g to the d e t a i l e d c o n s i d e r a t i o n of e l e c t r o l y t e s of moderate c o n c e n t r a t i o n , i t i s i n t e r e s t i n g to note the p r o p e r t i e s of a few systems which e x i s t as l i q u i d s from pure fused s a l t s to d i l u t e aqueous s o l u t i o n s . Miscible Electrolytes There are two systems f o r which the vapor pressure and t h e r e by the a c t i v i t y of water has been measured over the f u l l range o f composition from fused s a l t to d i l u t e s o l u t i o n i n water. In each case the s a l t i s a simple mixture of approximately equal m o l a l p r o p o r t i o n s . The system (Ag,T£)N03 was measured at 98 C by T r u d e l l e , Abraham, and Sangster (12) w h i l e (Li,K)N03 was measured i n the most concentrated range at 119°C by Tripp and B r a u n s t e i n (13) and over the remainder of the range at 100°C by B r a u n s t e i n and B r a u n s t e i n (14). These r e s u l t s are shown on Figure 1 which a l s o i n c l u d e s s i m i l a r data f o r s e v e r a l systems of l a r g e but l i m i t e d s o l u b i l i t y . The composition v a r i a b l e i s the mole f r a c t i o n on an i o n i z e d b a s i s , i . e . , x^ = n^/(n-^ + v ^ ) where n^ and Ώ.2 are moles of water and s a l t , r e s p e c t i v e l y , and ν i s the number of ions i n the s a l t . On t h i s b a s i s Raoult's law a p p l i e s i n the very d i l u t e range, w i t h the Debye-Huckel c o r r e c t i o n a p p l i c a b l e as the concentration increases. The s i m i l a r i t y of the curves on Figure 1 to those f o r non e l e c t r o l y t e s o l u t i o n s i s s t r i k i n g . The dashed l i n e r e p r e s e n t i n g a i = X«L can be c a l l e d " i d e a l - s o l u t i o n behavior" f o r these systems, as i t i s f o r n o n e l e c t r o l y t e s , but i t i s r e a l i z e d that a s t a t i s t i c a l model y i e l d i n g that r e s u l t would be more complex f o r the i o n i c case. A l s o the Debye-Huckel e f f e c t i s a departure from t h i s i d e a l behavior. N e v e r t h e l e s s , i t seems worthwhile to explore the use f o r these systems o f the simple equations f o r n o n e l e c t r o l y t e s . One of the s i m p l e s t and most s u c c e s s f u l had i t s o r i g i n i n the work of van Laar (15) and has been w i d e l y used s i n c e . P r a u s n i t z (16) d i s c u s s e s t h i s and r e l a t e d equations as w e l l as the c o n t r i b u t i o n s of Margules, H i l d e b r a n d , Scatchard, Guggenheim, and others to t h i s t o p i c . For the a c t i v i t y of e i t h e r component, referenced to the pure l i q u i d , one has


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2 In a

= In x

1

+ w^

1

(la)

2

2 ln a Z

l

z W

= ln x

2

=

i ^

n

+

i

v

(lb)

n (b /b )] 2

2

(lc)

1

= vn /[n (b /b ) + vn ]

2

2

=

2

n

+ w,^

2

1

1

2

2

i^ 2^ i^'

w

h

(Id) ()

h

le

Note f i r s t that i f ( b ^ / b ^ i s u n i t y , z. and z reduce to the mole f r a c t i o n s x^ and X2« Then one has the even simpler equations


2

2 l n a^ = l n

+ wx

(2a)

2

2 l n a = l n x + wx^ . (2b) In e i t h e r equations (1) or (2) the n o n - i d e a l i t y parameter w (sometimes w r i t t e n w/RT) a r i s e s from the d i f f e r e n c e between the i n t e r molecular a t t r a c t i o n of u n l i k e species as compared to the mean of the i n t e r m o l e c u l a r a t t r a c t i o n f o r p a i r s of l i k e s p e c i e s . The second parameter i n equation ( 1 ) , ( b ^ / b ^ , i s sometimes a s c r i b e d to the r a t i o of the volumes o f the molecules o r t o the r a t i o of m o l a l volumes i n the l i q u i d , although i n some systems, e s p e c i a l l y m e t a l l i c s o l u t i o n s , equation (1) i s s t i l l q u i t e s a t i s f a c t o r y but (b^/b ) departs g r e a t l y from the r a t i o of m o l a l or atomic volumes. For fused s a l t - w a t e r mixtures i t seems best t o regard (b^/b2) as a f r e e l y a d j u s t a b l e parameter and subsequently t o compare the values w i t h r a t i o s of m o l a l volumes. Equation (1) was f i t t e d t o the two systems remaining l i q u i d over the f u l l range of composition w i t h the r e s u l t s w^ = 1.02, 0>i/b2) = 0.50 f o r (Ag,T£)N0 -H 0 and w = -0.89, Q>i/b2) = 1-2 f o r (Li,K)N03~H20. Water i s component 1 and the s a l t component 2. For the l a t t e r system the simpler equation (2) serves almost as w e l l w i t h w = -0.80 ( t h i s i m p l i e s b /b2 = 1.0). The c a l c u l a t e d curves based on equation (1) are compared w i t h the experimental data i n F i g u r e 2. These r e s u l t s shown i n Figures 1 and 2 demonstrate the s i m i l a r i t y of the e f f e c t s of short-range f o r c e s on the p r o p e r t i e s of n o n e l e c t r o l y t e s and concentrated e l e c t r o l y t e s . One f i n d s both p o s i t i v e and negative d e v i a t i o n s from i d e a l i t y and these e f f e c t s may be a s c r i b e d t o the d i f f e r e n c e between the i n t e r m o l e c u l a r pot e n t i a l energy o f a t t r a c t i o n of u n l i k e s p e c i e s t o the mean of the corresponding p o t e n t i a l s f o r p a i r s of l i k e molecules. Previous d i s c u s s i o n of these systems has focused on the h y d r a t i o n of the p o s i t i v e i o n as the dominant e f f e c t , but we see i n F i g u r e 1 that 2

2

2

3

2

x

1



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Figure 1.

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Activity of water for water-salt solutions over a very wide range of composition

(Ae,TI)NO -H O s

t

_

|

Vl,K)N0 -H 0 ©Π9·ς 3

2

·ιοο·ο

0

1

0.2

I 06

I

0.4

;

x

Figure 2.

I 0.8

1.0

l

Comparison of the calculated and experimental activity of water for water-salt solutions over the full range of composition


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the s h i f t from n i t r a t e t o c h l o r i d e i s comparably important. J u s t as f o r n o n e l e c t r o l y t e s , one must consider a l l i n t e r m o l e c u l a r forces i n e l e c t r o l y t e s . With experimental data f o r the a c t i v i t y of water, one can, of course, i n t e g r a t e the Gibbs-Duhem equation t o o b t a i n the a c t i v i t y of the s a l t , over the same range i n composition, without the use of any model or s e m i - e m p i r i c a l equation. But equation (1) appears to f i t so w e l l that i t s use i s very convenient. As presented, the constant o f i n t e g r a t i o n i s evaluated f o r the p u r e - l i q u i d reference s t a t e f o r each component. Thus equation ( l b ) gives the a c t i v i t y of the s a l t i n r e l a t i o n t o the pure fused s a l t . Since t h i s form i s obtained by i n t e g r a t i n g the Gibbs-Duhem equation over composit i o n t o the fused s a l t , x^ = 0, x = 1, the Debye-Huckel range i s avoided and no e r r o r from that source i s introduced. I f the fused s a l t does not e x i s t a t the temperature of i n t e r est, one normally uses the i n f i n i t e l y d i l u t e s o l u t e standard s t a t e . While these equations can e a s i l y be converted t o that b a s i s , the r e s u l t s are not immediately u s e f u l f o r two reasons: (1) Debye-Hiickel e f f e c t s are s i g n i f i c a n t i n the d i l u t e range and are not considered, and (2) the u s u a l composition s c a l e f o r the s o l u t e standard s t a t e i s m o l a l i t y r a t h e r than mole f r a c t i o n . Both of these problems have been overcome, and the more complex r e l a t i o n s h i p s are being presented elsewhere (17). However, f o r most purposes, the v i r i a l c o e f f i c i e n t equations f o r e l e c t r o l y t e s a r e more convenient and have been w i d e l y used. Hence our primary pres e n t a t i o n w i l l be i n those terms.


2

V i r i a l C o e f f i c i e n t Equations f o r E l e c t r o l y t e s A very e f f e c t i v e method of r e p r e s e n t i n g the p r o p e r t i e s of non-ideal gases i s by use of a s e r i e s i n i n c r e a s i n g powers of d e n s i t y o r c o n c e n t r a t i o n s . The c o e f f i c i e n t s , c a l l e d v i r i a l coe f f i c i e n t s , are unambiguously r e l a t e d t o a p a r t i c u l a r number of molecules. Thus the f i r s t term r e l a t e s to i n d i v i d u a l molecules and i s the i d e a l gas law. The second v i r i a l c o e f f i c i e n t a r i s e s from b i n a r y i n t e r m o l e c u l a r f o r c e s and may be e i t h e r p o s i t i v e or negative as r e p u l s i v e or a t t r a c t i v e f o r c e s predominate. The t h i r d v i r i a l c o e f f i c i e n t a r i s e s from t r i p l e i n t e r a c t i o n s , e t c . The MacMillan-Mayer (18) s o l u t i o n theory e s t a b l i s h e d that a f o r m a l l y s i m i l a r treatment a p p l i e d t o s o l u t e s i n a s o l v e n t provided the i n t e r m o l e c u l a r p o t e n t i a l s are replaced by p o t e n t i a l s of mean f o r c e i n that s o l v e n t . For e l e c t r o l y t e s one must recognize the l o n g range character of coulombic f o r c e s which prevents t h e i r i n c l u s i o n i n the v i r i a l s e r i e s . But as Mayer (19) and others have shown, one may combine a Debye-Hiickel term f o r e l e c t r o s t a t i c e f f e c t s w i t h a v i r i a l s e r i e s f o r the e f f e c t s of short range f o r c e s . I n t h i s case, however, the v i r i a l c o e f f i c i e n t s depend on the i o n i c s t r e n g t h as w e l l as the temperature and other p r o p e r t i e s . These t h e o r e t i c a l p r i n c i p l e s were used by the w r i t e r (20) to e s t a b l i s h the form of an equation f o r e l e c t r o l y t e p r o p e r t i e s i n which the


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v i r i a l c o e f f i c i e n t s a r e evaluated e m p i r i c a l l y . V i r i a l type equa t i o n s were used e a r l i e r f o r e l e c t r o l y t e s (10, 21, 22) but without r e c o g n i t i o n of the i o n i c s t r e n g t h dependence of the second v i r i a l c o e f f i c i e n t . The b a s i c equation i s p o s t u l a t e d f o r the excess Gibbs energy from which other f u n c t i o n s can be obtained from appropriate derivatives. G /n RT = f (I) + W λ ( I ) m m + \ H μ i j i j k eX

w

J

J

J

mm m^

eX


(3)

J

Here G /n i s the excess Gibbs energy per k i l o g r a m of s o l v e n t and m^, mj, e t c . , a r e the m o l a l i t i e s of the v a r i o u s ions o r n e u t r a l s o l u t e s present. The long-range e l e c t r o s t a t i c f o r c e s l e a d t o the Debye-Hiickel term f ( I ) where I i s the i o n i c s t r e n g t h . Short-range i n t e r p a r t i c l e - p o t e n t i a l e f f e c t s a r e taken i n t o account by the v i r i a l coefficients f o r binary i n t e r a c t i o n s , Uijk f o r ternary, etc. As noted above, e l e c t r o s t a t i c e f f e c t s l e a d to an i o n i c s t r e n g t h dependence on λ f o r i o n i c i n t e r a c t i o n s . For μ t h i s i s neglected; a l s o μ i s omitted i f a l l ions are of the same s i g n . While f o u r t h v i r i a l c o e f f i c i e n t s could be added, they do not ap pear to be needed f o r most a p p l i c a t i o n s . Indeed the t h i r d v i r i a l c o e f f i c i e n t s are so s m a l l that they can o f t e n be omitted a t moder ate c o n c e n t r a t i o n ( I up to about 2 ) . The d e r i v a t i v e equations f o r osmotic and a c t i v i t y c o e f f i c i e n t s , which are presented below, were a p p l i e d t o the experimen t a l data f o r wide v a r i e t y of pure aqueous e l e c t r o l y t e s a t 25 C by P i t z e r and Mayorga (23) and to mixtures by P i t z e r and Kim (11). L a t e r work (24-28) considered s p e c i a l groups of s o l u t e s and cases where an a s s o c i a t i o n e q u i l i b r i u m was present (H^PO^ and H^SO^). While there was no attempt i n these papers t o i n c l u d e a l l s o l u t e s for which experimental data e x i s t , n e a r l y 300 pure e l e c t r o l y t e s and 70 mixed systems were considered and the r e s u l t i n g parameters reported. This represents the most e x t e n s i v e survey of aqueous e l e c t r o l y t e thermodynamics, although i t was not as thorough i n some r e s p e c t s as the e a r l i e r e v a l u a t i o n of Robinson and Stokes (3). I n some cases where data from s e v e r a l sources a r e of compar able accuracy, a new c r i t i c a l e v a l u a t i o n was made, but i n other cases the t a b l e s o f Robinson and Stokes were accepted. In a d d i t i o n to the a c t i v i t y and osmotic c o e f f i c i e n t s a t room temperature, the f i r s t temperature d e r i v a t i v e s and the r e l a t e d enthalpy of d i l u t i o n data were considered f o r over 100 e l e c t r o l y t e s (26, 29). The data f o r e l e c t r o l y t e s a t higher temperatures become "progressively more sparse. Quite a few s o l u t e s have been measured up to about 50°C (and down to 0°C). A l s o , over t h i s range, the equations u s i n g j u s t f i r s t temperature d e r i v a t i v e s have some v a l i d i t y f o r rough estimates i n other cases. But the e f f e c t s of the second d e r i v a t i v e (or the heat c a p a c i t y ) on a c t i v i t y co e f f i c i e n t s a t h i g h e r temperatures i s very s u b s t a n t i a l .



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SYSTEMS W I T H INDUSTRIAL

APPLICATIONS

Sodium c h l o r i d e has been s t u d i e d much more thoroughly at high temperature than any other e l e c t r o l y t e . The osmotic c o e f f i c i e n t measurements of L i u and Lindsay (30) and v a r i o u s types of measurements of Federov and a s s o c i a t e s (31) are p a r t i c u l a r l y noteworthy. A p r e l i m i n a r y e f f o r t to represent a l l of the data on sodium c h l o r i d e by v i r i a l c o e f f i c i e n t equations has been p u b l i s h e d (32) and a r e v i s i o n i s being completed (33). Over the wide range of temperature to 300°C and c o n c e n t r a t i o n to 10 M, sodium c h l o r i d e shows only very moderate and slow changes i n i t s p r o p e r t i e s ; the p r i n c i p a l change i s the i n c r e a s e i n the Debye-Huckel parameter (34) which lowers both a c t i v i t y and osmotic c o e f f i c i e n t s . The osmotic c o e f f i c i e n t f o r NaCl at high temperature i s shown on F i g u r e 3; the curves were c a l c u l a t e d by the v i r i a l c o e f f i c i e n t equation (33). The behavior of a t y p i c a l 2-1 e l e c t r o l y t e , MgCl^, i s shown on F i g u r e 4, which gives both the experimental values of Holmes, Baes, and Mesmer (35) and t h e i r curves from a v i r i a l c o e f f i c i e n t equation. Only a few a d d i t i o n a l s a l t s have been s t u d i e d e x t e n s i v e l y at h i g h temperatures, although s o l u b i l i t y i n f o r m a t i o n y i e l d s l e s s a complete p i c t u r e f o r s e v e r a l others (36). However, the general p a t t e r n s of behavior are simple enough that one can make estimates under some circumstances based on d e t a i l e d data a t room temperature f o r the s o l u t e of i n t e r e s t and high temperature data f o r other s o l u t e s of the same valence type. The importance of the v i r i a l - c o e f f i c i e n t equations i s e s p e c i a l l y great f o r mixed e l e c t r o l y t e s . Of the needed v i r i a l coe f f i c i e n t s f o r a complex mixture such as sea water, most are determined by the pure e l e c t r o l y t e measurements and a l l the others of any s i g n i f i c a n c e are determined from data on simple mixtures such as NaCl-KCl, NaCl-MgCl , NaCl-Na^O^, e t c . , which have been measured. The e f f e c t of the terms obtained from mixtures i s very s m a l l i n any case and these terms can be ignored f o r a l l but the most abundant s p e c i e s . A very severe t e s t of these v i r i a l - c o e f f i c i e n t equations f o r the sea-water-related Na-K-Mg-Ca-Cl-SO,-H 0 system has been made by Harvie and Weare (37) who c a l c u l a t e d the s o l u b i l i t y r e l a t i o n s h i p s f o r most of the s o l i d s which can a r i s e from t h i s complex system. There are 13 i n v a r i a n t p o i n t s w i t h four s o l i d s present i n the system Na-K-Mg-Cl-SO^-^O and the p r e d i c t e d s o l u t i o n composit i o n s i n a l l 13 cases agree w i t h the experimental values of B r a i t s c h (38) s u b s t a n t i a l l y w i t h i n the estimated e r r o r of measurement. In p a r t i c u l a r , Harvie and Weare found t h a t f o u r t h v i r i a l c o e f f i c i e n t s were not r e q u i r e d even i n the most concentrated s o l u t i o n s . They d i d make a few s m a l l adjustments i n t h i r d v i r i a l c o e f f i c i e n t s which had not p r e v i o u s l y been measured a c c u r a t e l y , but otherwise they used the p r e v i o u s l y p u b l i s h e d parameters. There are a l s o many l e s s severe t e s t s (11) of p r e d i c t i o n s f o r mixed e l e c t r o l y t e s which i l l u s t r a t e the accuracy to be expected i n v a r i o u s cases. Thus i t i s w e l l - e s t a b l i s h e d t h a t the v i r i a l coe f f i c i e n t equations f o r e l e c t r o l y t e s y i e l d r e l i a b l e p r e d i c t i o n s of 2

?



23.

PITZER

459


12.0

m (molality)

Figure 3. Osmotic coefficient for sodium chloride solutions at various temperatures

0.7 !

OS

I

I

1.0

I

1.5

I

2.0

I

LI

25 3.0

3.5

m/mol kg"

1

Journal of Chemical Thermodynamics

Figure 4.

Osmotic coefficient for magnesium chloride solution at various temperatures (35)


460

THERMODYNAMICS

OF AQUEOUS SYSTEMS

W I T H INDUSTRIAL

APPLICATIONS


m i x e d - e l e c t r o l y t e p r o p e r t i e s provided the c o e f f i c i e n t s measurable f o r the pure e l e c t r o l y t e components a r e known. The p r e d i c t i o n s are more accurate i f the a d d i t i o n a l c o e f f i c i e n t s measurable from simple mixtures a r e a l s o known but t h e i r e f f e c t i s u s u a l l y very small. The working equations f o r osmotic and a c t i v i t y c o e f f i c i e n t s , d e r i v e d from equation ( 3 ) a r e given as equations ( 4 ) and ( 5 ) , r e s p e c t i v e l y . The v a r i o u s secondary r e l a t i o n s h i p s a r e d e f i n e d i n s e v e r a l a d d i t i o n a l equations s t a t e d and b r i e f l y d e s c r i b e d t h e r e a f t e r . A d d i t i o n a l d e t a i l s and d e r i v a t i o n s of equations f o r the entropy, the heat c a p a c i t y , and other r e l a t e d f u n c t i o n s can be found i n v a r i o u s p u b l i s h e d papers ( 1 1 , 2 0 , 2 3 - 2 9 , 3 2 - 3 4 ) .

A

3/2

- 2 — z r n r + L ΥL Τ m m

-

(Φ-1)

I

_l/2 I+bl T l l

+ i y y

c a

c a

(Β

+ ZC

φ

ca

m m (Θ , + £ m ψ , ) + c c ce L a cc'a

2 £

1

J"

4

Φ

f f

)

ca

y

I

o o»

2

m m , (Θ* , a

a

a

a

+ Im ψ , ) L

(4)

f

c aa c I

Although one wishes a c t i v i t y c o e f f i c i e n t s f o r n e u t r a l combin a t i o n s of i o n s , i t i s convenient to use equations f o r s i n g l e - i o n a c t i v i t y c o e f f i c i e n t s which can then be combined a p p r o p r i a t e l y . l n γ - z F + Y m ( 2 Β ^ + ZC ) + 1 m ( 2 Θ . . + ï m ψ M M a Ma Ma c Me a Mca a c a L

+ \ II a a ln γ

= 4

χ

L

«η ,Ψ , Λ

ω

F + l m (2B c

+ k II

c X

Μ

L

+ |z | l l m m C c a M

c

a

c X

z

11 c a

)

(5a)

ca

+ ZC ) + i ^ 2 0 ^

»Λ.*^.γ + l vl

Μ

+ I

(5b)

vc »a ca a

c

Here m i s the m o l a l i t y of c a t i o n c w i t h charge z and correspon d i n g l y f o r anion a. Sums over c o r a cover a l l c a t i o n s o r anions, r e s p e c t i v e l y . B s and 0 s are measurable combinations of λ s whereas C s and ψ s a r e combinations of the y's i n Equation ( 3 ) . Note that the 0 s and φ s a r e zero and these terms disappear f o r pure e l e c t r o l y t e s . f

T

f

1

1

f

1


23.

PITZER

461


The e l e c t r o s t a t i c f u n c t i o n f must c o n t a i n the Debye-Hiickel l i m i t i n g law w i t h the parameter A, = (1/3)(2πΝ d / 1 0 0 0 ) φ o w

1 / 2

2

(e /DkT)

3 / 2

but i t proves e m p i r i c a l l y advantageous t o take an extended form. Among a l t e r n a t i v e s , the form found best was f (I) = ^ I b "

1

ln(l-bl

1 / 2

)

(6)


Φ

w i t h b = 1.2 chosen f o r a l l e l e c t r o l y t e s i n water. At 25°C the value of Αφ i s 0.391. The a p p r o p r i a t e d e r i v a t i v e gives the term i n Equation (4) f o r φ. For the a c t i v i t y c o e f f i c i e n t i t i s conven ient to define F = -ATl φ

1 / 2

/(l+bI

1 / 2

) + (2/b)ln(l+bI

ce

1 / 2

f

)] + Τ 7 m m B £ c a ca b a L

a a

which i n c l u d e s both the Debye-Hiickel term w i t h A and c e r t a i n de r i v a t i v e s o f the second v i r i a l terms. The second v i r i a l c o e f f i c i e n t s , Β^χ» a r e f u n c t i o n s of i o n i c s t r e n g t h . Again an e m p i r i c a l choice was made among t h e o r e t i c a l l y p l a u s i b l e forms f o r Β^χ and the f o l l o w i n g was chosen w i t h β'°' and Ê ' ' parameters f i t t e d t o the data f o r each s o l u t e . 1

12/

·* • «£' • C> *

Thermodynamics of Aqueous Systems with ... - ACS Publications

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