Prediction of Activity Coefficients of Strong Electrolytes in Aqueous

25 Prediction of Activity Coefficients of Strong Electrolytes in Aqueous Systems H. P. MEISSNER

Downloaded by CORNELL UNIV on June 4, 2012 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch025

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

In planning the processing of an aqueous solution containing more than one strong electrolyte, typical questions are: a) upon evaporation, which electrolyte will precipitate first, b) at what concentration will precipitation start, c) how much precipitation will occur as evaporation proceeds before a second electrolyte starts to co-precipitate, and d) what is the effect of temperature on this behavior. Answers require information on the activity coefficients of the solution components over the composition and temerature ranges of interest. The object here is to review a simple empirical method for predicting these activity coefficients, which involves the use of a single charac teristic constant for each cation-anion combination present. This constant is unchanged by the presence of other electrolytes, and is readily derived from experimental measurements. Electrolyte Characterization. The generalized dissociation of a strong electrolyte is as follows: (1) where subscripts i and j designate the cation A and the anion Β respectively, with z and z being the ion charges, andv and i

j

i

v the s t o i c h i o m e t r i c c o e f f i c i e n t s , whose sum ( v . + v . ) i s d e s i g n J ι J nated as ν... The i o n charges ζ. and ζ. are used t o c h a r a c t e r i z e ij J ι e l e c t r o l y t e s ; thus NaCl, NH^NO^ e t c are 1:1 e l e c t r o l y t e s , MgCl^ i s a 2:1 e l e c t r o l y t e , e t c . E l e c t r o l y t e s l i k e MgC^ forming an j

ion having a charge g r e a t e r than u n i t y are c a l l e d h i g h e r e l e c t r o l y t e s . Values of ζ and ν f o r t y p i c a l e l e c t r o l y t e s are l i s t e d i n Table 1. In a s o l u t i o n of a s i n g l e e l e c t r o l y t e , here c a l l e d a "pure s o l u t i o n " , s u b s c r i p t s 1 and 2 are used t o designate c a t i o n s and anions r e s p e c t i v e l y . When more than two ions are p r e s e n t , then 0-8412-0569-8/80/47-133-495$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.

THERMODYNAMICS

496

OF

AQUEOUS SYSTEMS W I T H INDUSTRIAL

APPLICATIONS

the s o l u t i o n i s c a l l e d a "mixed" s o l u t i o n , w i t h c a t i o n s i d e n t i f i e d by the odd i n t e g e r s 1,3,5, e t c . , and anions by the even i n t e g e r s 2,4,6, e t c . Concentrations are expressed both i n m o l a l i t i e s and i n i o n i c s t r e n g t h u n i t s , r e l a t e d as f o l l o w s f o r i o n s : 2

I . = O.Sim.z. ); The

2 jZj

)

t o t a l i o n i c s t r e n g t h of s o l u t i o n s I


1. = 0 . 5 ( m

T

= 0.5(m z 1

2 1

+ m z 2

2 2

+ m^z^

2

(2) i s defined as I , where T

+ ...)

(3)

A s u p e r s c r i p t " ° " w i l l be used t o designate a pure s o l u t i o n , f o r which I t h e r e f o r e becomes I ° . I t i s e a s i l y shown (1_) that f o r T

2

m

Z

Z

V

T

u s t r a t e

a pure s o l u t i o n , I ° i s equal t o 0 . ^ ^ 2 l 2 1 2 " ° ^ ^ a p p l i c a t i o n of these equations, values of m o l a l i t i e s and i o n i c strengths f o r pure s o l u t i o n s of t y p i c a l e l e c t r o l y t e s having a m o l a l i t y of u n i t y are shown i n Table 2. 2

A c t i v i t y C o e f f i c i e n t s . The 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 of a c a t i o n - a n i o n p a i r (here shortened t o " a c t i v i t y c o e f f i c i e n t " f o r convenience) are d i r e c t l y measureable i n pure and o c c a s i o n a l l y i n mixed s o l u t i o n s . The 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 designated as γ?. i n pure s o l u t i o n and γ.. i n mixed s o l u t i o n . In t h i s development, a t t e n t i o n i s focused e x c l u s i v e l y on a c t i v i ty c o e f f i c i e n t s of c a t i o n - a n i o n p a i r s , w i t h no use being made of a c t i v i t y c o e f f i c i e n t s of i n d i v i d u a l ions. E l e c t r o l y t e s i n Pure S o l u t i o n . I t has been found t h a t i n pure s o l u t i o n (2) at any constant temperature, e x p e r i m e n t a l l y 1/z ζ determined values of the term ( Ύ ^ ) 1 2 , when p l o t t e d against I° f o r v a r i o u s e l e c t r o l y t e s , form the curve f a m i l y of f i g u r e 1. 2

„ . χ 1 / ζ ζ i s designated as Γ° and c a l l e d the For convenience, (γ ) 1 2 12 "reduced a c t i v i t y c o e f f i c i e n t " . The data p o i n t s a t 25°C f o r s i x d i f f e r e n t e l e c t r o l y t e s are p l o t t e d on F i g . 1 t o show t y p i c a l agreement w i t h the isotherms presented. These curves are drawn from an a n a l y t i c a l equation which was f i t t e d t o p u b l i s h e d a c t i v i t y c o e f f i c i e n t data i n pure s o l u t i o n s a t 25°C f o r about 100 d i f f e r e n t e l e c t r o l y t e s (3). By t h i s equation, γ ° a t a given /

Ο

1

0

&

2

temperature i s uniquely determined by two f a c t o r s , namely the t o t a l i o n i c s t r e n g t h I ° and the q u a n t i t y q° « The term q° i s a 2

2

constant f o r each curve i n f i g u r e 1, but d i f f e r s from one curve to the next. Each e l e c t r o l y t e , t h e r e f o r e , can be c h a r a c t e r i z e d by i t s q° value a t 25°C The a n a l y t i c a l equation i n v o l v e d i s presented below, and i s a p p l i e d t o pure s o l u t i o n s by u s i n g 1°.


25.

MEISSNER

497

Activity Coefficients of Strong Electrolytes

Table I


I l l u s t r a t i v e Values of ν and ζ V. 1

NaCl CaCl

2

(NH ) S0 4

MgS0

2

4

4

A1 (S0 ) 2

4

3

ν.

ζ.

_J_

1

ζ. _J_

ζ . :ζ. .1 1

1 1

1 2

2 3

1 2

1 1

1:1 2:1

2

1

3

1

2

1:2

1

1

2

2

2

2:2

2

3

5

3

2

3:2

Table I I Values of 1° i n Pure S o l u t i o n f o r a M o l a l i t y (m. . ) of U n i t y _ i l NaCl CaCl

2

(NH ) S0 4

MgS0

2

4

4

A1 (S0 ) 2

Na.PO.

4

3

_

i

l

_ J L

_

!

1 1

1 3

1 1 / 2 1 2

1

3

2

1

4

1

1

15

2

1

6

3

1 2 9

_J_ 1 2 1

_J_ 1/2 1 2

1

2

3

6

11/2 1

4 1/2


THERMODYNAMICS OF AQUEOUS SYSTEMS WITH INDUSTRIAL APPLICATIONS


498

Ionic

Strength , I American Institute of Chemical Engineering

Figure 1. Typical isotherms for various electrolytes of the reduced activity coefficients vs. the total ionic strength (data points presented at 25°C): (0) LiBr; Ο HCI; (%) LiCl; (U) CaCl,; (*) Pb(ClO h; (O) Ca(N0 ) ; (+ ) ΝΗ>Ν0 ; (Q)AgNO, (3) h

3

2


3

25.

MEISSNER

for

I , and

499


f o r w,mix 12 23 34

...

(13)

where Ν

R

12 12 " N + N + N_, + 12 23 34 1 0

R

O Q

Ν _ 23 2 3 N- + N _ + N . + 12 23 34

(

=

0

1

4

)

e t C

Q

'

Values o f A . a t the d e s i r e d t o t a l i o n i c s t r e n g t h I a r e 12 ,mix m obtained as u s u a l from equation (6) a f t e r s u b s t i t u t i n g A ^ 1 0

&

m

a

f o r ( ^)-L2 ^

nt n

i

s

equation, or from Figure 2 p l u s equation ( 7 ) .

To i l l u s t r a t e a p p l i c a t i o n of equation (13), c o n s i d e r a mixed (unsaturated) s o l u t i o n at 25°C, 4.11 m o l a l i n NaCl (q° i s 2.23) and 8.55 m o l a l i n NaNO^ (q° i s -0.39), making I equal t o 12.66. T

By equation ( 8 ) , q by equation ( 6 ) , (13),

a^

i s 1.35 and q

N a C 1 > m i x

A N a C

^ i s 0.57 and

i s 0.04, hence

K C 1 > m i x

i s 0.67.

From equation

as p r e d i c t e d equals the experimental v a l u e of 0.64

s o l u t i o n , by equation ( 8 ) , q . is LâoU^,mix sa

w

2 Ι " · ' ' ' 4 2 When I i s 6.31, Γ . i s 0.64 by equation ( 4 ) , making Τ Gyp,mix 0/

m

Y

Gyp mix

3.69

J

e c

*

χ 10"

5

u a l

t c

0.168.

n

S u b s t i t u t i n g i n t o equation (11): 2

= m^ (0.168) (0.77)

2

yp

S o l v i n g Gypsum's s o l u b i l i t y m

i s found to be 0.047 m o l a l , Gyp


25.

MEISSNER

509


versus an experimental value of 0.048 ( 6 ) . 2. In 0.178 m o l a l K^SO^. Over t h i s s o l u t i o n , s i n c e q

„ i s -0.25 and I i s 0.534, a° i s 0.997 by equation ( 6 ) . K b0. Τ w 2 4 Gypsum s o l u b i l i t y i s again minute, hence iç ++ mix i s negligible, =/ i s 2/3 and I + / I i s 1/3, making v

n

o

a

I

I

S 0

T

K

T

4

2 1 q^ . equal t o (-0.25x=- - O.lSx^) or -0.22 by equation ( 8 ) . Gyp,mix ^ 3 3 ^ trace Using t h i s q v a l u e , when I i s 0.534, Γ ™ . i s 0.594 Τ ' CaSO^mix trace 4 by equation ( 4 ) , making γ . equal t o (0.594) o r 0.124. n

J

m

n

0

uabu^,mix


By equation (11): 5

2

2

3.69 χ 1 0 " = in (πι + 0.178) ( 0 . 1 2 4 ) ( 0 . 9 9 7 ) G G S o l v i n g , m i s 0.013, versus an experimental m o l a l i t y of G 0.011 ( 7 ) . 3. I n 3.0 m o l a l (NH^^SO^. C a l c u l a t i n g as above, Gypsum s o l u b i l i t y i n t h i s s o l u t i o n i d found t o be 0.043 m o l a l , v e r sus an experimental v a l u e of 0.039 ( 7 ) . D i s c u s s i o n . The approach t o mixtures d i s c u s s e d here i s based on the f i n d i n g that f o r a s o l u t i o n of constant dry composi t i o n , a l l curves of Γ.„ . , Γ . , e t c . versus the t o t a l i o n i c 12,mix 32,mix s t r e n g t h f a l l i n t o the curve f a m i l y of F i g u r e 2, and t h e r e f o r e a l s o conform to equation ( 4 ) . This i s a d i r e c t consequence of the form of equations (8) and (9) of r e f e r e n c e ( 8 ) , showing the r e l a t i o n between Γ . , and Γ° , Γ° , e t c . and I . These i z ,mix 1Z ΖJ l e a r l i e r equations, of course, do not i n v o l v e " ^ ^ χ " their 0 0

Ί Ο

m

a

n

d

replacement by equaiton (4) f o r l o c a t i n g the mixture curve i s l a r g e l y a matter of convenience. Equation (13) f o r c a l c u l a t i n g a . over a mixed s o l u t i o n w,mix i s d e r i v e d from the modified Gibbs equation f o r a mixture ( 7 ) , namely n

&

-55.5 d i n a

W j m i x

= v^dm^ + v^dn^... + m ^ v ^ d l n y ^ + n. v dlnY3 2 3

2 3

2 m

+ ...

By combination w i t h equation ( 3 ) , terms i n v o l v i n g m and ν a r e replaced by terms i n v o l v i n g I and Z. Recognizing t h a t Γ 1/z

equals

ζ

IZ,mix

-j[^> then f o r a mixture of constant dry composition,

m

equation (13) f o l l o w s d i r e c t l y . I t i s evident t h a t the examples presented here show c o n s i d e r able success. S i m i l a r r e s u l t s have been obtained w i t h other


510

THERMODYNAMICS

OF

AQUEOUS

SYSTEMS W I T H INDUSTRIAL

APPLICATIONS

systems, i n c l u d i n g some i n v o l v i n g double s a l t s (9,10). S u l f u r i c a c i d and some cadmium and z i n c s a l t s , however, so do not conform t o these r e l a t i o n s . I n view of the many u n c e r t a i n t i e s i n v o l v e d , t h e r e f o r e , c a l c u l a t e d compositions of saturated s o l u t i o n s should be t r e a t e d w i t h c a u t i o n and, where p o s s i b l e , checked by experiment. Nomenclature A.. 1 J

H y p o t h e t i c a l water vapor a c t i v i t y f o r a h y p o t h e t i c a l s o l u t i o n of e l e c t r o l y t e i j , f o r which ( q ° j ) i

pure

h y p o t h e t i c a l

i s assumed equal t o q.. . f o r the mixture under c o n s i d e r a ble ,mix tion. ' a c t i v i t y (a°^) f ° e l e c t r o l y t e i j i n pure s o l u t i o n , J


a^j

i s

r

(a.. . ) i s f o r t h i s e l e c t r o l y t e i n mixed s o l u t i o n , (a° ) i j ,mix w i s f o r water i n a pure s o l u t i o n , a . i s f o r water i n a w,mix mixed s o l u t i o n . J

r

I

i o n i c s t r e n g t h , equations (2) and ( 3 ) ; I

m N_. η

T

i s the t o t a l i o n i c r

strength

f o r mixed s o l u t i o n s , and equals 1 ° ^ f ° pure

solutions. m o l a l i t y ; gram moles per 1000 grams water. gram moles of e l e c t r o l y t e i j added t o water t o make up a s o l u t i water o n system. moles of h y d r a t i o n per formula weight of an e l e c t r o l y t e : Α Β · nH 0 ν . ν. 2 o

q^.

c h a r a c t e r i s t i c constant as i n equations ( 4 ) , ( 5 ) , and ( 8 ) .

R t

see equation (14) °C

ζ

i o n charge Greek.

ν γ., Γ 1 J

s t o c h i o m e t r i c c o e f f i c i e n t , equation (1). 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 of i j i n s o l u t i o n l 2. "Reduced a c t i v i t y c o e f f i c i e n t , namely γ 1 / z

z

11

Subscripts

and S u p e r s c r i p t s .

° i j mix sat Τ

denotes a "pure" s o l u t i o n r e f e r s t o c a t i o n i , stands f o r odd i n t e g e r s 1,3,5 e t c . r e f e r s t o c a t i o n j , stands f o r even i n t e g e r s 2,4,6 e t c . denotes a mixed s o l u t i o n denotes a saturated s o l u t i o n r e f e r s t o " t o t a l " s o l u t i o n , as i n I

w

r e f e r s t o water i n the s o l u t i o n .

T


25.

MEISSNER


511


Literature Cited

1. Kusik, C.L.; Meissner, H.P., Ind. Eng. Chem., Proc. Des. Dev. 1973, 12, 112. 2. Meissner, H.P., Tester, J.P. Ind. Eng. Chem., Proc. Des. Dev. 1972, 11, 128. 3. Kusik, C.L.; Meissner, H.P., AIChE Symp. Ser. 1978, 173, 74 14. 4. Meissner, H.P.; Kusik, C.L., AIChE J1. 1972, 18, 294. 5. D'Ans, J . , "Die Lösungsgleichewichte der Systeme der Salze Ozeanischer Salzablagerungen," Ver. für Ackerbau, Berlin, 1933. 6. International Critical Tables, McGraw Hill Book Co., NY 1933, III end IV. 7. Seidell, Α.; Linke, W.F., "Solubilities of Inorganic and Metal Organic Compounds", 4th Ed. van Nostrand & Co., Princeton, NJ, 1958. 8. Meissner, H.P.; Kusik, C.L., Ind. Eng. Chem., Proc. Des. Pev 1973, 12, 205. 9. Meissner, H.P.; Kusik, C.L. Ind. Eng. Chem, Proc. Des. Pev. 1979, 18, 391. 10. Kusik, C.L.; Meissner, H.P.; Field, E.L., Ind. Eng. Chem., Proc. Pes. Pev., 1979, 18. 11. Meissner, H.P.; Peppas, N.A., AIChE J1. 1973, 19, 806. 12. Smithsonian Tables, 9th Rev. Ed. Washington, 1954. 13. Harned, H.S. and Owen, H.H., "Physical Chemistry of Electro lyte Solutions," 3rd Ed. Rheinhold, NY, 1958. RECEIVED January 31,

1980.


Prediction of Activity Coefficients of Strong Electrolytes in Aqueous

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