Solubilities in Multicomponent Systems - American Chemical Society


Clinogonial Projection. Clinogonial projection belongs to the empirical methods enabling us to obtain .... Graphical solution by the clinogonial proje...
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Chapter 3

Solubilities in Multicomponent Systems Jaroslav Nývlt and J i ř íStávek

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Institute of Inorganic Chemistry, Czechoslovak Academy of Sciences, Majakovského 24, 16600 Prague 6, Czechoslovakia

The basic problem in determining phase equilibria in multicomponent systems is the existence of a large number of variables, necessitating extensive experim­ ental work. If ten measurements are considered satis­ factory for acceptable characterization of the solu­ bility in a two-component system in a particular tem­ perature range, then the attainment of the same reli­ ability with a three-component system requires as many as one hundred measurements. Therefore, a reliable correlation method permitting a decrease in the num­ ber of measurements would be extremely useful. Two dif­ ferent methods - the first of them based on geometrical considerations, and the second on thermodynamic con­ dition of phase equilibria - are presented and their use is demonstrated on worked examples.

The basic problem i n determining phase e q u i l i b r i a i n multicomponent systems i s the existence of a large number of variables, necessitat i n g extensive experimental work. I f ten measurements are considered satisfactory f o r acceptable characterization of the s o l u b i l i t y i n a two-component system i n a p a r t i c u l a r temperature range, then the a t tainment of the same r e l i a b i l i t y with a three-component system r e quires as many as one hundred measurements;with more complicated systems, the necessary number of measurements w i l l be several orders of magnitude higher. Therefore, a r e l i a b l e correlation method permitt i n g decrease i n the number of measurements would be extremely useful. Methods that more or less comply with these requirements can be c l a s s i f i e d into two groups : purely empirical methods based on cert a i n geometrical concepts, and methods derived from thermodynamic descriptions of phase e q u i l i b r i a , which replace unknown quantities by an empirical function. Two t y p i c a l methods w i l l be introduced.

O097-615^AM38-O035$06.00/0 © 1990 American Chemical Society

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

CRYSTALLIZATION AS A SEPARATIONS PROCESS

36 Clinogonial Projection

Clinogonial projection belongs to the empirical methods enabling us to obtain the s o l u b i l i t y i n a three component system from binary data. I t i s based on the assumption that the plane corresponding to the s o l u b i l i t y of one of the components can be obtained by geometr i c a l projection of the binary s o l u b i l i t y (or melting) curve f o r the mixture with the other component from the apex corresponding to the solvent. The p r i n c i p l e of the method i s explained in Figure 1". A three-component system A + B + C, represented in"Figure 1 % i s considered. The quantity monitored, e.g. the saturation temperature i n a given point of the system, generally has t^g, t ^ and t g ^ values i n the corresponding binary systems. A number of planes are constructed at r i g h t angles to t r i a n g l e ABC, so that they pass through the same coordinate i n binary systems AC and BC, i . e . = t . Another system of planes perpendicular to triangle ABC passes through point C. The required value of the saturation temperature at point S, corresponding to tg i s given by the t^ value corrected f o r projection of the difference appropriate t^g value from l i n e a r dependence i n the binary system AB, i . e . :

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ff

f i c

A

* s " V

where

A t

t

1

A B

X

( S ) ]

- C

(

1

(

A B = *1B " **AB

**AB = *0B

+

~ X

~ A

+

X

(

B

)

t

)

*0A ~ 0B

(

'

2

3

)

)

On substitution into "Equation 1", the r e s u l t i n g r e l a t i o n s h i p i s obtained : (4) t

= t

s

+

[ 1 - x (S)] [ t c

A

B

- —

( X

A

t +

Q

- t

A

Q B

) -t

Q B

]

*B

The c a l c u l a t i o n according to "Equation 4" i s very simple and the whole procedure can be mechanized by assembling a suitable table : t * (S) t A/( A B^ ^A^OB products t. x

1

C

A

X

u

+ X

s

e

f

u

l

Q

B

Very frequently, one isotherm i n a ternary rather than the shape of function t^gj thus the i n "Figure 1" i s known and the shape of another determined. On the basis of "Equation 1" and an t

T

= t

2

+ At

A B

diagram i s known shape of curve t ^ isotherm has to be analogous "Equation 5"

[1 - x ( T ) ] c

(5)

written f o r an a r b i t r a r y point T located on the same straight l i n e with / ( XgJt "the r e l a t i o n can be written : X

X

A

+

A

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

3.

Solubilities in Multicomponent Systems

NYVLT &STAVEK

t

- t

s

t

1

T

- t

= 1 - x(S)

37

1 - x (T) c

= A t » u f o r X . / C X . + X T . ) = const. ** A A B

(6)

The "Equation 6" expresses a l i n e a r dependence between tp and Xg(T). The other dependence between t and x^, i s obtained d i r e c t l y from t e r nary diagram. For example, the following corresponding pairs of values can be read d i r e c t l y from "Figure 1" : x ( R ) ... t ; XQ(S) ... t . ; XQ(T) ... t ^ etc. Both the dependencies are depicted schematically i n "Figure 2". F i r s t l y , the t ( x ) values corresponding to the intercepts of various connecting l i n e s t ^ = t g ^ with the straight l i n e x./(x.+Xg) are found from the triangular diagram. Further, the t g value i s plotted on the graph on the v e r t i c a l l i n e constructed through the point x = 1. The l i n e connecting points t„ with point t , located on trie v e r t i c a l l i n e passing through point x = x ( S ) , determines the length of segment KL = t^g. An a r b i t r a r y t,j, value i s selected and the appropriate segment i s plotted on the v e r t i c a l l i n e passing through point x~ = 1 , a l i n e i s drawn p a r a l l e l to straight l i n e t g - L and the intercept of t h i s l i n e with curve t ( x ) y i e l d s the x (T) value, corresponding to the isotherm i n "Figure 1". As an example, the c a l c u l a t i o n i s shown f o r the system ammonium sulphite - ammonium sulphate - water. The isotherm of the ammonium sulphite s o l u b i l i t y curve at 20° C i s drawn i n "Figure 3 and the points representing aqueous solutions of ammonium sulphite and ammonium sulphate saturated a t 0, 20, 50 and 70° C are also given. For an a r b i t r a r y chosen straight l i n e "a", the x^ values corresponding to the intercepts o f t h i s l i n e with 2 dashed connecting l i n e s f o r i d e n t i c a l temperatures i n the binary mixtures are read from t h i s diagram. The procedure described above leads to the construction of diagram i n "Figure 4". Open c i r c l e s represent the dependence tCH^O) read from "Figure 3 , calculated points represented by f u l l c i r c l e s are redrawn back into "Figure 3 » The whole procedure repeated f o r other a r b i t r a r y l i n e s "a" would give another set of points corresponding to i n d i v i d u a l isotherms. The method described i s very rapid and y i e l d s s a t i s f a c t o r y r e s u l t s unless the extrapolations are carried out over too wide a range. c

Q

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c

r

1

C

c

p

p

U

11

Q

ff

f,

Expansion of Relative A c t i v i t y Coefficients A three component system consisting of a solvent (0) and two further components (1 and 2) can be considered. The phase equilibrium between the s o l i d (s) and l i q u i d (1) phases i s characterized by equality of the chemical potentials of a given component i n the two phases. Supposing that the component are completely immiscible i n the s o l i d phase we obtain from the condition of equality of chemical potentials : log . = - logI\ = ^ [T, P, sat] (7) Xj

where x. i s the r e l a t i v e molality of component i and T. i s i t s r e lative activity coefficient : X

i

=

n

i

/

n

0 i

r

i

=

f

i

/

f

o i



In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

(

8

)

CRYSTALLIZATION AS A SEPARATIONS PROCESS

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38

Figure 2. Graphical solution by the clinogonial projection method. (Reproduced with permissionfromref. 1. Copyright 1977 Academia.)

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

3. NYVLT & STAVEK

Solubilities in Multicomponent Systems

39

(NH^)S0; 2

•C

50%

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20 0

\

\\\\ \

\

\

\

V \ HaP

5050% 70*C

20

Figure 3. Example of construction by the clinogonial projection method. (Reproduced with permissionfromref. 1. Copyright 1977 Academia.)

~f70

60

SO

AO

ho 20 10

o!T

0.7

0.8

0.9

1.0

Figure 4. An auxiliary diagram. (Reproduced with permissionfromref. 1. Copyright 1977 Academia.) In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

40

CRYSTALLIZATION AS A SEPARATIONS PROCESS

In these equations, subscript i denotes the substance which i s i n equilibrium and subscript Oi denotes i t s saturated solution i n pure solvent 0 at given temperature. I f the compound 1 i s dissociated and ions M and X i n solution do not originate from any other component, i . e . : M X n+ n-

=

n

M

+

z +

z

+

n X ~ -

(9)

"Equation 7" can be rewritten to give : 1

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log ( S ^ S ^ ) ^

=

log

= •

X l

(10)

1

where and S^Q are the a n a l y t i c a l s o l u b i l i t y products of component 1 i n the multicomponent system and i n the pure solvent, respectively, and n^ = n + n i s the t o t a l number of p a r t i c l e s dissociated from one molecule of"the substance. I f the presence i n solution of two electrolytes with a common cation i s assumed, then holds : X

n

i

x

" i +1

+

+

n

x

i

+

B

/m

2 20 10 (12)

x. = n. x. ll - l and from the basic equation we obtain : n In k n m i+/ i 1 0 log [ " . ( Z x )1+ 1 ] = i=1 n m n

1

X

i

1

n

+

l

1 +

^

(13)

1Q

and a similar equation holds also f o r a system containing two elect r o l y t e s with a common anion (only the + and - signs i n subscripts are interchanged). For a l l possible combinations of mono- and d i valent ions, a common equation i s obtained : ^

=— log [ a +e

x*

3

. (x + F B ) ]

(U)

t

where B

X

= 2 •

a = n

1 +

m

/m

20 W

F

n

- 2 >1+

3

_

= n

(

1

5

)

+

1 - +

(the f i r s t sign i n subscripts i s v a l i d f o r a common anion and the second sign holds f o r a common cation). For example, i n the case of s o l u b i l i t y of CuSO. i n the system CuSO. - K S0. - H 0 i t holds a = 1 ;g = 2 and F = 1. On the left-hand side of "Equation H i s , of course, s t i l l an unknown function involving the r e l a t i v e l y a c t i v i t y c o e f f i c i e n t . I t s value must, i n general, depend on the composition, temperature and pressure of the system : o

4

4

2

4

o

2

w

*

1

= *

1

(m

1f

m, 2

my

... n^, T, P) .

(16)

As, however, the composition of the condensed two-phase system i s unambigously determined by (k - 1) concentration values, t h i s general

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

3.

Solubilities in Multicomponent Systems

N W L T & STAVEK

41

functional dependence can be rewritten i n the form : $

1

= $

1

(m

2>

my

. . . m)

(T, P = const.).

k

(17)

This form has the advantage of not containing concentration value m^ and thus permits the e x p l i c i t expression of x.. from the basic equat i o n . The expansion of the general function given by "Equation 17" into the MacLaurin series with respects to m o l a l i t i e s m ^ yields the equation : k k k $ = Z Q m + Z Z Q m.m. + (18) i=2 ' i=2 j=2 Downloaded by UNIV LAVAL on November 17, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch003

J

J

or, f o r a three-component system : *1

= Q

12

, n

2

+ Q

122

, n

+

( 1 9 )

2 ---

The adjustable i n t e r a c t i o n constants Q can be evaluated from the experimental data f o r three-component systems; these constants can then be employed f o r concentration of temperature interpolations and also f o r c a l c u l a t i o n of phase e q u i l i b r i a i n multicomponent systems. Moreover, the constants Q usually depend very l i t t l e on temperature, as the r e l a t i v e m o l a l i t i e s , related to the s o l u b i l i t y of the substance i n the pure solvent, are employed; hence calculations of other i s o therms can be carried out e a s i l y . For example, values of Q f o r the system NaNO^ - NaCl - H 0 i n the temperature range 0 to 100° C are (higher terms are zero) : t [° C] «1 12

0 25 30

AO 50 75 91 100

2

-0.025 -0.020 -0.022 -0.019 -0.017 -0.015 -0.008 -0.008

As an example, s o l u b i l i t y i n the system KC1 - NaCl - H 0 at 20° C has been calculated. Necessary data are given as follows : 58.443 a= 3= F = 1 Solubit Q = -0.015 ; m = 4.586 S o l u b i l i t y of NaCl Q = 0.000 ; Q = 0.0007 ; m = 6.135. 2

1Q

12

21

211

2Q

l i n e s correspond to i d e a l system data ( a l l Q = 0). I t can be seen that even small value of the i n t e r a c t i o n constants Q can s i g n i f i c a n t l y move the r e s u l t i n g equilibrium curve. As the interaction constants are usually rather small, a rough informat i o n can be drawn even from the supposition of a l l Q = 0, i . e . without any previous knowledge of the ternary system.

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

CRYSTALLIZATION AS A SEPARATIONS PROCESS

42

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AMI

Figure 5. Comparison of real and ideal behavior in idalized system NaCl-KCl-H 0. (Reproduced with permissionfromref. 1. Copyright 1977 Academia.) 2

Conclusions Two d i f f e r e n t methods have been presented i n t h i s contribution f o r correlation and/or prediction of phase e q u i l i b r i a i n ternary or multicomponent systems. The f i r s t method, the clinogonial projection, has one disadvantage : i t i s not based on concrete concepts of the system but assumes, to a certain extent, a d d i t i v i t y of the properties of i n d i v i d u a l components and attempts to express deviations from add i t i v i t y of the properties of i n d i v i d u a l components and attempts to express deviations from a d d i t i v i t y by using geometrical constructions. Hence t h i s method, although simple and quick, needs not necessarily y i e l d correct results i n a l l the cases. For t h i s reason, the other method based on the thermodynamic description of phase e q u i l i b r i a , r e l i a b l y describes the behaviour of the system. Of cource, the theory of concentrated i o n i c solutions does not permit a p r i o r i c a l c u l a t i o n of the behaviour of the system from the thermodynamic properties of pure components; however, i f a s a t i s f a c t o r y equation i s obtained from the theory and i s modified to express concrete systems by using few adjustable parameters, the results thus obtained are s t i l l substant i a l l y more r e l i a b l e than r e s u l t s correlated merely on the basis of geometric s i m i l a r i t y . Both of the methods shown here can be e a s i l y adapted f o r the description of multicomponent systems.

Literature Cited 1.

Nývlt, J . Solid - Liquid Equilibria; Academia, Prague, 1977

RECEIVED May 12, 1990

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.