Binary common-ion alkali halide mixtures thermodynamic analysis of solid-liquid phase diagrams

CALPHAD vo1.6, No.1, pp. 11-24, Printedin the USA.

1982

0364-5916/82~010011-14~03.00/0 (c) 1982 PergamonPress Ltd.

BINARY COMMON-ION ALKALI HALIDE MIXTURES THERMODYNAMIC ANALYSIS OF SOLID-LIQUID PHASE DIAGRAMS II. SYSTEMS WITH COMPLETE SOLID MISCIBILITY APPLICATION OF THE LIQFIT METHOD (presented as a poster at CALPHAD X, Vienna, 1981) Joke A. Bouwstra and Harry A-J. Oonk General Chemistry Labosatory, Chemical Thermodynamics Group University of Utxecht, Padualaan 8, Utrecht, The Netherlands

ABSTRACT

LIQFIT is an iterative procedure - with intermediate ohase diagram calculations - in which the calculated liquidus is made to run through the experimental lfquidus points. The computations yield a reliable solidus and in addition the Redlich-Kister coefficients of the AGE function. The latter is the difference between the excess Gibbs energies of the liquid and solid mixtures. The calculations are based on the equal-G curve (EGC), i.e. the projection on the TX olane of the,intersection of the GTX surfaces of the liquid and solid mixtures. The EGC is the link between the phase diagram and the AGE function. Results are presented for I.8 common-ion alkali halide systems which show complete miscibility in the solid state.

Introduction Recently we published two methods for deriving thermodynamic excess functions from TX phase diagrams. The two methods are referred to as the EXTXD/SIVAMIN method (1,2,3,4,5) and the LIQFIT method (6,5). The EXTXD/SIVAMIN method is a statistical one which operates with two sets of function values derived from the phase diagram. In the case of eutectic equilibria one set of function values is derived from the left-hand liquidus and the other set from the right-hand liquidus. In part I (7) results were presented for 15 eutectic binary common-ion alkali halide mixtures showing negligible solid miscibility. In the case of equilibria between mixed solid and mixed liquid phases the EXTXD/SIVAMIN method is foredoomed to failure as a result of the fact that one of the equilibrium curves - the solidus - cannot be located with sufficient accuracy by current experimental techniques. Fortunately, the other equilibrium curve - the liquidus - can be located with high precision. In the LIQFIT method, which is an iterative procedure based on the equal-G curve, the calculated liquidus is made to run through the experimental liquidus points. The computations yield a thermodynamically sound solidus and in addition the coefficients of the excess Gibbs energy difference function, i.e., the difference between the excess Gibbs energies of the liquid and solid mixtures. In this part we present the results of LIQFIT computations on 18 alkali halide mixtures showing complete solid miscibility. First we give a short description gf,~~~_~~~~f9Trmlation and the computational procedure. Received 21 December 1981

11

12

J.A. Bouwstraand H.A.J. Ckmk

Thermodynamic formulation Under isobaric conditions the expressions for the molar Gibbs energies of the liquid(liq) and solid (sol) mixtures expressed in the variables temperature,T, and mole-fraction-of-the-second-component, X, are Gliq(T,X)

GSOl

=

1 1

(l-X)G;liq(T) + XG;liq (T) +RT (l-X)ln(l-X) + XlnX [

+ GEliq(T,X)

(T,X) = (l-X)G:sol(T) + XG;"l (T) +RT (1-X)ln(l-X) + XlnX c

+ GEsol(T,X)

(1) (2)

* where R is the gas constant and refers to pure component: the subscript 1 refers to the first component and the subscript 2 to the second (in this paper the second component is the one with the larger non-common ion). At constant temperature the two functions correspond to two curves in the GX plane. When for a given temperature the two curves intersect, there will be equilibrium between a solid and a liquid phase, the compositions of which are given by the abscissae of the points of contact of the common tangent drawn to the two functions. The two sets of points of contact (the positions of the points of contact as a function of temperature) constitute the solidus and the liquidus curve in the TX plane. A third curve which can be drawn in the TX plane is the equal-G curve (EGC). It represents the set of points of intersection of the G-curves, i.e., it is the solution of IIG

=

Glib

_ GSol =

o

.

(3)

Substitution of (1) and (2) in (3) gives

(1-x) AGT(T) + X AG:(T) +

_IG~(T,x) = o ,

(4)

which can also be written as (we neglect heat-capacity influences) (l-x) (AH:

- T.ZS:) + X( 2H; - TAS;)

+

AHE

- T 4SE(X) = 0 ,

(5)

where H stands for enthalpy and S for entropy; AH* is the enthalpy of melting and AS* the entropy of melting. For given X, the left-hand side of (5) becomes zero for T = TEGC :

(I-X) ( AH:-TEGC AS:) + x( AH:-TE~C As:) + In this equation

AH~(x)-T~~C,\S~(X)

= o

.

AHE(T

EGC ASE(X) can be replaced by _4GiGC(X) which is the difference between the excess Gibbs energies of the liquid and solid mixtures along the EGC : AC&(x)

=

AHE

- TEGC ASP

.

It is now clear, from equations (6) and (7), that the EGC is related to the function as

(7) AGE

13

BINARYCOMMON-IONALKALIHALIDEMIXTURES

TEGC (X)

= TzERo(W

+

AGiGC (Xl

(8)

’

(1-X) As: + X6$

(1-X) AH; + XAH; where the "zero line" is given by

T

ZERO(') =

(9) (1-X) is; + x3s;

*

A phase diagram with EGC and zero line is shown in figure 3. The distance from zero line to EGC is the excess Gibbs energy difference divided by the (mean) entropy of melting (of the components).

Computational procedure The computational procedure, the flow-diagram for which is represented by figure 1, is as follows.

EXPERIMENTAL LIQUIDUS

X(IMPROVED EGC IN CYCLE j+l) - X(EGC IN CYCLE j) = X~EXPERIMENTAL LIQUIDUS) X(CALCULATED LIQUIDUS IN CYCLE j)

EXCESS GIBBS ENERGY DIFFERENCE FUNCTION h CALCULATION OF PHASE D1AGRAM

CALCULATED LIQUIDUS THROUGH

FIG 1 Flow-diagram of LIQFIT.

First an estimate is made of the EGC oosition. From the estimated EGC the difference function

AGEGC is calculated, see equations (8) and (9).

The difference-function values are fitted to the following four-parameter Redlich-Kister expression.

.\G;GC(X)

=

X(1-X)

AGl+ AG2( 1-2X)+ AGj (l-2)02+ AG4 (1-2X)3 3 I:

.

(10)

J.A. Bouwstraand H.A.J. Oonk

14

The AG coefficients are composed of the coefficients of the Redlich-Kister expressions for the excess enthalpies and excess entropies of the liquid and solid phases. For the i-th coefficient , where

(11)

-IHi = Htiq - Hrol

, and

(12)

ASi

.

(13)

.jGi =

AHi - T ASi

= Siiq

-

ST”

Next a phase-diagram calculation is made based on the computed difference function. If the individual excess functions of the phases are unknown, i.e. if there is no information about the diverse coefficients on the right-hand sides liq and Siiq of (12) and (131, calculations are made putting GEliq=O (i.e. all Hi sol Es01 is put equal independent of temperature (Gi are made zero) and taking G to

- AGi derived from the EGC). If, on the other hand, there is information on

one or more of the individual excess functions, that information is included in the phase-diagram calculation. For example, if HEliq is the only individual Eliq= HEliq and excess function known, calculations are made with G GESol(independent of temperature) = HEliq - AGE,

where

.iGE is the difference

function derived from the EGC. After the phase diagram has been calculated, a new cycle starts by improvement of the EGC position. The EGC is shifted in the direction in which the difference between calculated liquidus and experimental liquidus will be reduced. For a given temperature cross-section the EGC shift is obtained from (X stands for mole fraction) X(improved EGC in cycle j+l) - X(EGC in cycle j) = f

X(experimenta1 liquidus) - X(calculated liquidus in cycle j) 1 3

(14) .

The use of the damping factor f is recommended in order to avoid oscillations. The procedure is repeated until further iteration does not reduce the difference between the calculated liquidus and the experimental liquidus points. As a measure for the disagreement between calculated liquidus and experimental liquidus points we usehe mean mole-fraction difference, which is defined as n Ax = $

abs(Xiiq - X,fiq ) talc exp i=l r(

(15)

where n stands for the number of experimental liquidus points, i.e. the number of temperature cross-sections for which the mole-fraction difference is taken. In most instances we use in our computations a dummy set of experimental liquidus points instead of the original experimental points themselves. The dummy points are read from the curve drawn by eye through the original liquidus points.

15

BINARYCOMMON-ION ALKALI HALIDEMIXTURES

ALKALI HALIDE MIXTURES WITH COMPLETE SOLID MISCIBILITY Systems The binary systems for which calculations were made are indicated in figure 2, see also part I (7).

SOLID STATE MISCIBILITY

q •ZI l.m lKl

complete,this paper limited conflictingdata negligible, part I negligible, calculations to be published

completed,

FIG 2 Solid state miscibilityin common-ionalkalihalidemixtures.The substancesare set out in order of increasingsum of ionic radii. Melting Droperties The entropies of melting of the oure substances are summarized in Table 1. The sources of the experimental data are given in ref.-i. TABLE 1 Entropyof MeltingValuesdividedby the Gas Constant substance LiCl LiBr NaCl NaBr NaI KF KC1 KBr

A S*/R

2.71 2.58 3.14 3.08 3.04 3.14 3.03 3.02

substance KI F&F RbCl RbBr F&I fsC1 CSBX CSI

dS*/R 3.03 2.90 2.87 2.90 2.85 2.65 3.12 3.16

J.A. Bouwstraand H.A.J. Oonk

Binary data A survey of the sources of the phase diagrams and of the information on the individual excess functions used in the calculations is given in Table 2. In all cases we used a dummy set of exnerimental liquidus points.

TABLE 2 Sourcesof phase-diagram data. Surveyof excessfunctionstaken into accountin the calculations, with their sources. system

ref diagram

SEliq

ref

HEso ref

(Na,K) Cl (Na,K) Br (Na,K)I

(81

f + +

(9)

+

(10) (12)

(9) (13)

+ +

(K,Rb)F (K,Rb)Cl (K,Rb)Br (K,Rb)I

(14) (16) (181 (19)

+ + + +

(15) (9) (9) (9)

fK,Cs)Cl

(K,Cs)Br

(201 (21)

+ +

(9) (9)

(Rb,Cs)CL

(22)

+

(9)

Li(C1,Br) Na(Cl,Br) K(Cl,Br)

(23) (25) (27)

+ + +

(24) (9) (241

Rb(Cl,I)

(29)

+

(241

Na(Br,I) K(Br,I) Rb(Br,Il Cs(Br,I)

(301 (8) (31) (21)

+ + + +

(241 (24) (24) (24)

(7) (11) (11)

SEsol ref (7)

+

(191

(17) (17) (17)

+ +

(26) [28)

117)

+ + + t

(11) (11) (32) (32)

Results In this section we first give detailed results for the systems (Na,KfCl and Na(Cl,Br). the NaCl + KC1 system of all the experimental phase diagrams published for this system we preferred the diagram published by Luova and Muurinen (8). Below we compare the results with those obtained for Coleman and Lacy's (33) phase diagram, which we used in ref.6. The Redlich-Kister coefficients of the individual excess functions which we used in the intermediate phase-diagram calculations are given in Table 3. For this system we did calculations making varying use of the experimental information on the individual excess functions. The results of the various calculations on this system are shown in Table 4 , which also includes the results of one type of calculation based on Coleman and Lacy's phase diagram. The calculated phase diagram corresponding to the first row of Table 4 is shown in figure 3.

BINARY

COMMON-ION

17

ALKALI HALIDE MIXTURES

800

0

1

-X FIG 3

The system NaCl + KCl. Calculated phase diagram with equal-G curve and zero line. Shaded circles : experimental liquidus points ; open circles : experimental solidus points.

TABLE 3 The system NaCl f KCl. Hedlich-Kistercoefficients of the excz?ss enthalpies of solid and liquid mixtures, expressed in k.I1mol -1' mol . and of the solid state excess entropy, expressed in J K excess quantity

ref.

Redlich-Kistercoefficients z 2 S2 z3

ES01 H HEliq Es01 S

17.96

8.36

-2.19

-0.14

6.27

9.10

4.37

0

(7)

0

0

(9)

4.01

0

(71

TABLE 4 The NaCl + KC1 system. Calculated coefficients CI~the excess Gibbs energy difference function, expressed in kJ mol . phase diagram

HEliq HEsol SEsol + means included

Luova Luova Luova

+ f +

Coleman

+

+ +

+

&G bG2

AG3

-13.02 -12.90 -12.86

-1.87 -2.44 -2.41

-1.44 -2.76 -2.88

1.22 0.08 0.02

-14.75

-1.93

-1.99

-2.81

1

AG*

J.A. Bouwstraand H.A.J.Oonk

18

the NaCl + NaBr system The phase-diagram data used for the calculations are those reported by Gromakov and Gromakova (25). We made two runs of calculations. In the first we used the experimental information on the excess enthalpies of the solid and liquid states given in Table 5. In the other we ignored the information on the individual excess functions. The results of the two runs of calculation are given in Table 6. The calculated phase diagram corresponding with the first row of Table 6 is shown in figure 4.

800

0

1

-X FIG 4

The system NaCl + NaBr. Calculated phase diagram. Shaded circles : experimental liquidus points; open circles : experimental solidus points. TABLE 5 The NaCl + NaBr system. Redlich-Kister coefficients of the excess enthalpies of solid and liquid mixtures, expressed in kJ mol-1. Z_

Z_

Z.

ref.

5.59

0.33

0.20

0.14

(26)

0.35

0.05

0

0

excess quantity Es01 H BEliq

TABLE 6 The NaCl + NaBr system. Calculated coefficients of the excess Gibbs energy difference function, expressed in kJ mol-'. HEliq Es01 ,B

AG1

included

-2.29

-0.07

-0.46

0.31

ignored

-2.28

-0.05

-0.48

0.22

AG2

AG3

AG4

19)

BINARY COMMON-ION ALKALI HALIDE MIXTURES

19

the 18 alkali halide systems A survey of the results of calculations made on 18 common-ion alkali halide systems are given in Table 7. The results were obtained using the experimental information on the individual excess quantities as indicated in Table 2. The mean temperature difference for which values are given in the last column of the table is defined as

‘T

=$

xi

abs (Ti - Ti 1, talc exp

and Ti

Ti

being taken for the talc

exp

same mole fraction, and where n is the number of experimental points for which the suaunation is made. TABLE

7

Survey of results of LIQFIT calculations on 18 alkalihalidesystems. The four excess Gibbs energy difference function coefficients, expressed in kJ mol-1; the mean mole-fraction difference with respect to the dummy liquidus points; the mean mole-fraction difference with respect to the original liquidus points and the mean temperature difference with respect to the original liquidus points, expressed in K. system

.tG4

'X

AK

AT

*G2

AG3

(Na,K)Cl -13.02 (Na,K)Br -10.17 (Na,K)I -8.81

-1.87 0.21 -1.48

-1.44 0.24 -1.10

1.22 -0.96 0.31

0.003 0.002 0.003

0.005 0.014 0.005

1.46 2.77 0.52

-2.22 -3.10 -2.36 -1.47

-0.43 -0.30 1.03 -0.40

1.12 1.13 -0.90 0.13

0.04 -0.32 -0.69 -0.65

0.007 0.002 0.004 0.002

0.015 0.023 0.013 0.015

0.80 0.61 0.83 0.62

(K,Cs)Cl -8.49 (K,Cs)Br -10.19

-2.24 -0.08

-2.43 -0.71

-1.27 -1.39

0.005 0.002

0.008 0.004

0.73 0.83

(Bb,Cs)Cl -2.77

0.98

-1.90

0.79

0.002

0.027

2.61

0.44 -0.07 -0.55

-2.12 -0.46 -0.05

0.89

0.005 0.003 0.005

0.008 0.002 0.021

0.89

0.31 0.22

Bb(Cl,I) -11.16

-1.73

-3.72

0.80

0.006

0.008

1.93

Na(Br,Il K(Br,I) Rb(Br,I) Cs(Br,I)

-0.34 -0.44 -1.71 -0.43

-1.82 -0.15

-1.67 -0.32 -0.66 0.59

0.005 0.004 0.005 0.003

0.007 0.006 0.013 0.022

0.51 0.53 1.08 2.20

(K,Bb)F (K,Rb)Cl (K,Bb)Br (K,Bb)I

lG1

Li(Cl,Br) -4.07 Na(Cl,Br)

-2.29

K(Cl,Br)

-2.05

-5.23 -4.25 -4.07 -6.52

-0.01 -0.56

dummy

orig.

orig.

0.12 0.76

The positions of the calculated liquidus and solidus curves are given in Table 8.

20

J.A. Bouwstra and H.A.J. Oonk

TABLE 8 Calculated liquidus and solidus temperatures, expressed in K. The headings of the columns are mole-fraction values. system

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(Na,K)Cl

liq sol

1074

(Na,K)Br

WI

1022

997.5 972.3 948.1 928.5 918.0 923.9 943.8 967.0 990.a 966.7 941.7 927.8 920.3 917.6 918.8 924.0 936.3 962.8

1013

933

909.9 886.9 867.5 859.5 864.8 878.8 898.5 920.2 943.7 884.0 868.0 861.7 859.5 861.1 866.3 876.2 891.4 916.7

966

SO1

1042.4 1007.8 973.8 945.3 932.6 944.5 965.2 989.3 1015.7 1043 984.9 951.4 938.7 933.8 932.6 933.7 938.2 949.7 976.2

(Na,K)I

liq sol

(K,afF

liq sol

1053 1053.8 1054.5 1056.7 1061.9 1070.5 1081.7 1094.0 1106.6 1118.5 1129 1053.7 1054.4 1056.2 1059.5 1065.0 1073.0 1084.0 1098.0 1114.1

(K,W)Cl

lQ4

1049

1036.1 1023.2 1010.7 999.8 991.5 987.3 987.4 990.1 993.3 1030.2 1014.3 1002.7 994.5 989.4 987.1 987.1 989.5 993.1

995

1013 liq SO1

991.4 981.5 912.7 964.7 957.0 949.7 944.5 942.9 946.1 987.7 977.2 968.3 960.2 952.9 947.2 943.8 942.8 945.3

953

950.3 942.9 937.2 932.4 928.8 926.6 948.5 941.0 935.3 931.2 928.3 926.4

926

sol (K,Bb)Br fK,Rbl

I

1%

960

SO1 (K,Cs)Cl

liq

1043

1013.3 983.3 953.0 923.5 902.1 891.3 888.7 894.7 906.1 941.5 919.2 908.0 899.6 893.0 889.6 888.6 891.6 900.8

1018

989.9 961.2 932.1 903.4 877.7 860.1 864.2 878.8 895.9 924.8 892.2 875.8 866.0 861.4 859.0 859.8 864.6 880.2 *

913

999

984.3 970.5 958.0 946.2 935.4 925.0 915.2 908.0 909.0 976.7 961.0 948.3 937.4 927.5 918.5 911.2 907.1 908.0

921

880

861.0 843.3 828.3 816.8 808.2 801.1 795.5 794.6 804.7 846.4 829.3 818.7 810.7 803.9 798.4 794.7 794.1 798.6

823

SOL

(K,Cs)Br

1.Q sol

fSJ,C&l

liq sol

Li(CI,Br) liq sol Na(Cl,Br)

u sol

nearly horizontal

920

1073 1061.0 1049.7 1039.6 1030.7 1023.5 1017?8 1013.7 1011.6 1oll.a 1015 1056.3 1043.7 1034.2 1026.5 1020.8 1016.3 1013.1 1011.5 1011.7

K(Cl,Br)

liq sol

1044

RbfCl,If

Iiq sol

990

959.2 928.2 896.0 864.0 837.5 835.4 845.8 865.9 890.0 862.8 844.7 840.7 837.4 835.2 834.7 837.1 843.1 856.1

915

NafBr,I)

liq sol

1021

997.0 974.8 956.0 941.8 930.3 921.8 917.5 919.0 925.9 972.2 954.2 941.7 931.8 924.1 919.0 917.5 918.1 923.8

935

K(Br,I)

liq

1009

991.3 975.3 960.7 948.9 940.6 936.7 937.1 940.5 946.6 979.8 962.2 950.7 943.3 938.7 936.5 936.7 939.4 945.2

954

SO1

1034.0 1025.0 1017.3 1011.8 1008.5 1007.3 1007.0 1008.2 1010.0 1013 1030.5 1021.2 1014.9 1010.6 1008.0 1007.0 1006.9 1007.9 1009.8

BbfBr,I)

liq sol

954

934.8 917.8 904.2 895.7 892.4 894.2 898.7 904.5 910.3 921.1 906.3 898.2 894.1 892.4 893.4 896.7 902.4 909.3

915

Cs(Br,I)

Iiq sol

913

895.8 878.9 863.8 853.4 850.5 855.6 865.8 879.6 895.7 880.2 863.2 855.1 851.3 850.4 852.3 857.3 866.1 881,9

913

BINARYCOMMON-IONALKALIHALIDEMIXTURES

21

Discussion from Gibbs enerav to ohase diaaram The position of the two-phase region in the TX plane is determined in the first place by the position of the equal-G curve, i.e. in view of the equations (8) and (9), by the positions of the melting points of the pure components, the magnitude of the entropy of melting and the excess Gibbs energy difference function ,AG*. Eliq In this context the role of the individual excess functions G and GEsol is rather limited : after AGE has been defined, the individual functions have a relatively small influence on the courses of the liquidus and the solidus curves. Expressed in another way, with fixed AGE the courses of the liquidus and solidus curves will change substantially only if both (both, in view of the fixed AGE) the individual excess functions are changed substantially. In practice such substantial changes (still in both states simultaneously) are precluded by the circumstance that systems in which the components are miscible in the solid state usually show small deviations from ideality in the liquid state. from phase diagram to difference function These considerations imply that an experimental phase diagram is capable of providing the correct excess Gibbs energy difference function even if its solidus is unknown and additional information on the individual excess functions is partly or completely lacking. This statement is supported by the results shown in Table 6 for NaCl + NaBr and by the results obtained within the same data set for NaCl + KCl, see Table 4. We observe from these tables that there is not much difference between the results of calculations made with and made without the inclusion of information on the individual excess functions. In general, these differences are smaller than the differences shown by the results of calculations on the same system but based on phase-diagram data from different (even equally valid, see Table 4) sources. All this does not mean that one could just as well ignore information on the excess functions : the difference function is not the only goal of the computations. the calculated solidus Another purpose of our calculations is to provide the correct solidus curve. In cases where the difference function is rather small (and therewith the excess Gibbs energy of the solid state) the solidus curve is rather insensitive to the inclusion of additional experimental information. For the two cases considered in Table 6, the NaCl + NaBr system, the average mole-fraction difference between the calculated solidi is less than 0.003, which is, at any rate, negligible with respect to the reliability of the experimental liquidus. In cases where the excess Gibbs energy of the solid state is large, a lack of information on the excess functions may become critical owing to the vicinity of the region of demixing in the solid state. For instance, in the NaCl + KC1 system, ignorance of all the available excess functions leads to a calculated phase diagram in which the solid-liauid equilibrium interferes with the region of demixing in the solid state (only in the case of Coleman and Lacy's diagram). Calculated solidus temperatures are given in Table 8. It should be emphasized that the calculated solidi owe their status completely to the experimental liquidi. This means that the calculated solidi are as reliable as the experimental liquidi (The quality of the latter has not been discussed in this paper. On the other hand, in each case we used the data set which we thought was the best). It is obvious that one should keep these considerations in mind when using the information given in Table 8.

22

J.A. Bouwstraand H.A.J.Oonk

the excess Gibbs energy difference function The excess Gibbs energy difference function the coefficients of which are given in Table 7 can be used in a number of ways. We mention the following applications. In the first place the information on the difference function - which is valid along the EGC! - can be combined with other information on the system, either to evaluate the coefficients of an excess function for which experimental data are not available, or to check the mutual consistency of the diverse excess functions within one system. Secondly, the results of this investigation can be used as a starting point for the calculation (prediction) of multi-component equilibria. In a number of cases such a prediction can reliably be made using, apart from the transition properties of the pure components, nothing more than the results of this investigation. (We mean those cases where the region of demixing is far away, so in those calculations the liquid state can be taken as an ideal mixture, and the solid state excess Gibbs energy can be taken as being independent of temperature and with coefficients opposite to those of the difference function). As a last application we mention the calculation of the equilibrium distribution coefficient (kg). The latter is related to the initial slope of the excess Gibbs energy difference function (34,5) :

$Oo2 In kz

-

\GE/dX)x_o

To1 1 + (d

=

.

(16)

RTOl

In this formula the index 1 refers to the solvent and the index 2 to the solute; To stands for melting point of pure component. With respect to the convention followed above (smaller non-common ion defines the first component), the initial slope is (d AGE/dX)X_O

=

,IG1 +

1G2

+

AG3 +

AG4

(17)

when the component with the smaller non-common ion is the solvent, and the initial slope is (d AGE/dX)X_O

=

2G1 -

AG2 +

.iG3 -

AG4

when the component with the larger non-common ion is the solvent. For example, in very dilute solutions of KC1 in NaCl, the quotient of the equilibrium mole fractions of KC1 is calculated as

(xzol/x;iq)

x_o

=

kz

=

0.17

.

(18)

BINARYCOMMON-IONALKALIHALIDEMIXTURES

23

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