sivamin method

CALPHAD ~01.5,No.1, pp.55-74. PergamonPress Ltd. 1981.Printedin Great Britain.

0364-5916/81/010055-20$02.00/O

BINARY COMMON-ION ALKALI HALIDE MIXTURES THERMODYNAMIC ANALYSIS OF SOLID-LIQUID PHASE DIAGRAMS I. SYSTEMS WITH NEGLIGIBLE SOLID MISCIBILITY APPLICATION OF THE EXTXD/SIVAMIN METHOD Harry Oonk, Koos Blok, Ben van de Koot and Nice Brouwer General Chemistry Laboratory, Chemical Thermodynamics Group University of Utrecht, Padualaan 8, Utrecht,The Netherlands

ABSTRACT Solid-liquid phase diagrams of 15 binary common-ion alkali halide mixtures, showing negligible solid miscibility, have been analyzed using the EXTXD/SIVAMIN method. The principles of and the philosophy behind the method, which permits the simultaneous derivation of the excess enthalpy and excess entropy functions from a single TX phase diagram, are discussed. The significance of the results is investigated by recalculating the phase diagrams and by comparing the excess enthalpy functions with experimental heats of mixing. Introduction Recently we published two methods for deriving thermodynamic excess functions from TX phase diagrams. The two methods are referred to as the EXTXD/SIV~IN method (1,2,3) and the LIQFIT method (4). The EXTXD/SIV~IN method is a statistical one and operates with two sets of function values derived from the phase diagram. One set is derived from the first of the two equilibrium conditions, the other from the second equilibrium condition. The method is characterized by the precision with which it reproduces the experimental diagram, i.e. by the high extent of agreement between experimental diagram and the phase diagram calculated with the computed functions. It is obvious that such a sensitive method requires highly accurate phase-diagram data. Highly accurate phase-diagram data are available for eutectic equilibria, for regions of demixing and for liquid-vapour equilibria, but not, as a rule, for the equilibria between mixed-crystalline solid state and liquid mixture. For the latter the liquidus curve can be determined with high precision, the solidus curve, however, can not (at least not by the current non-equilibrium methods). For these equilibria reliable solidus curves can be obtained by the LIQFIT method, i.e. by an iterative procedure of phase-diagram calculations in which the calculated liquidus curve is made to run through the experimental liquidus points. The method not only yields a reliable phase diagram, it also provides information on the excess properties of the system. As a further check on the significance of the two methods we decided to investigate the class of the binary common-ion alkali halide systems. In part I we consider the EXTXD/SIVAMIN method and its application to alkali halide mixtures with negligible solid miscibility. This part,therefore, is to be regarded as the written version of our contribution to CALPHAD IX, Montreal 1980. In part II we consider the LIQFIT method and its application to alkali halide systems with complete solid miscibility. Alkali halide systems with limited solid miscibility will be considered in part III. Before going into the method and the results we consider the excess Gibbs energy function and make some observations concerning its derivation from TX phase diagrams. 55

H. Oonk et al.

56

The Excess Gibbs Enersv Function Under isobaric conditions the thermodynamicmixing properties of a binary mixture are known completely, if its excess Gibbs energy is known as a function of temperature and mole fraction (X denotes the mole fraction of the second component). The general form of the excess Gibbs energy is .

GR(T,X) = X(1-X) f(T,X)

(1)

For the function of temperature and mole fraction on the right-hand side we adopt the so-called Redlich-Rister expression (S), which is f(T,X) = Gl(T) + G2(T)(l-2X) + G3(T) (1-2X)2 + . . .

l

(2)

The temperature dependence of the coefficients can be given by the following expression, which can be derived by Taylor's series expansion around the reference temperature 6i (see also 61, and which is (T - S - T In: 1 + C;(0)(#S2- #T2+ Te ln$ ) Gi(T) = Hi(S) - T S,(e) + C,,(e) + .....

(3)

C and C' are the coefficients of the Redlich-Kisterexpressions where H for theiLx%s &nthalp*, the excess entropy the excess heat capacity and the derivative of the latter with respect to t&perature. At the reference temperature itself the i-th excess Gibbs energy coefficient is given by GiW

=

H,(e) - e site)

.

(4)

The excess-function coefficients can be arranged in the following "scheme of coefficients". 5

H2

H3

*4

-

In general, the excess properties of a real mixture are such that the dominating coefficients are found in the upper left-hand corner of the scheme. The more coefficients that are known, the better the mixing properties of the system are known.

Observations lo

from excess Gibbs energy to phase diagram (and vice versa ?)

We now consider two different excess functions and the regions of demixing to which they gixe rise. The first function is defined by the following set of coefficients : H3 = 4.37 s3 = 4.01 The region of demixing calculated from this set of coefficients is shown in figure 1 along with the experimental values of Bunk and Tichelaar (7) for the system KC1 + NaCl. function I

Xl

= 17.96 % Sl = 6.27

H2 = - 8.36 S2 = - 9.10

-1 Throughout this paper the H a$ G ppefficients are expressed in kJ mol and the S coefficients in J K mol .

BINARYCOLON-ION ALKALIHALIDEMI.Xl?.IEES

57

FIG 2

FIG 1 Calculated region of demixing corresponding to function I. The calculated curve runs through the data points obtained for the system KC1 + NaCl.

Calculated region of demixing corresponding to function II. The calcul.atedcurve runs through the data points obtained for the system KCL + NaCl.

The second function is defined by the set function II

Hl = 29.10

H2 = - 7.37

= 23.48

S2 = - 5.82

s1

and its corresponding region of demixing is shown in figure 2, again with the experimental data for the KC1 + NaCl system. Inspection of figures 1 and 2 reveals that, although the phase diagram stands in a unique relation to the excess function from which it is calculated, the reverse is not necessarily the case. This means that one needs other information in order to find out which of the two functions gives the best description of the excess properties far outside the phase-equilibrium region. 2O

function values along a curve in the TX plane

For some hvnothetical mixture let the excess Gibbs energy, in arbitrary units, be defined-by GEfT,X) = X(1-X) (1500 - 0.5 T/K)

.

(5)

If we now suppcse that for this mixture excess Gibbs energy values can be determined (in some way or other) along the parabola T/K = 500 - 500 (1-2X)2, then it is obvious that the function values along the parabola are correctly given by GE along pariX) = X(1-X) j1250 + 250(1-2X)2 1 .

(61

In general, a set of function values f(T,X) along a curve T(X) can be represented by an "interpolation formula" f'(X), as well as by an interpolation formula f"(T), because the curve in the TX plane can also be given as

58

H. Oonk et ah

X(T). Or, in terms of the scheme of coefficients, interpolation formulae for a set of function values along a curve can be composed only of coefficients of the first row as well as only of coefficients of the first column. Moreover, several other intermediate formulae can be constructed to describe the values along the curve. Only one of these formulae has the rank of a real excess Gibbs energy function. function values DERIVED from the phase diagram

3O

We now consider the procedure of a simple statistical analysis of a TX phase diagram, and as an example we take the region of demixing where for a given T the equilibrium compositions of the two phases are X' and Y'. From the equilibrium conditions ,&1(phase I) =

,&1(phase

II)

and

/dl2(phase I) =

/Ct2(phase II)

(7a,b)

it follows that Y(X') /clF(X') -

(8a)

(y') = RT ln(l-Y') - RT ln(l-X')

pE(Y')

(8b)

= RT 1nY' - RT lnX'

After subtraction the following equation is obtained (9) The right-hand side of this equation represents a numerical value which can be derived from the phase diagram. The left-hand side can be used as the regression model for the excess coefficients. For instance,if one wants to calculate the coefficients H1 and S1, equation (9) reads (I-Y')X' 2(x'-Y')(H~-TS~) = RT ln(l_x,)y,

(10)

.

And if one wants to calculate the three coefficients H1, S1 and H2 the "regression equation'* becomes 2(X'-Y')(H1- TS1) - 6(Y'- X'+ Xf2- Yt2) H2 = RT ln~:I~~{~:

.

(11)

If we denote the left-hand sides of (10) and (11) by f and the righthand sides by f then, in the statistical procedu%%he following sum over n data trip?& GX'Y' has to be minimized by adapting the values of the coefficients H and S. SUM =

P i=l

(fobsi -

fcalc ) i

(12)

As an example we consider the region of demixing in the solid state of AgCl + NaCl. For this system we replaced the original nine data triples TX'Y' (8) by a dummy set of 30 triples I which we read from the "eye fit" of the original points. Calculations based on regression equation (10) gave the following result. -2 final value SUM = 137~10~ J'mol = 8.80 s1 = 1.97 *1 Calculations based on (11) gave final value SUM = 12.3~10~ J2mol-2. =10.25 S1 = 5.98 H2 =-2.74 % The regions of demixing calculated from the two sets of coefficients are shown in figure 3. We observe that, although the combination of two H coefficients and one S coefficient yields a considerably lower value for the sum of the squares of the residuals (final value SUM), its calculated phase diagram displays a much greater disagreement with the original data set than does the diagram for the combinationof one H and one S coefficient.

BINARYCO~~-IO~ ALEALIHALIDEMIXTURES

59

FIG 3 Regionof &mixing in the systemAgCl + NaCl. Circles : experimental data points. Curves : regionsof demixingcalculatedfrom the coefficients obtainedby the simple statistical method ; l/l refersto the combinationof one H and one S coefficient: 2/l refersto the combinationof two H coefficients and one S coefficient.

The EXTXD/SIVAMIN method The third observation shows that it can be dangerous to use for the interpretation of the phase diagram a set of function values derived from that phase diagram, What one should do in fact is to follow a procedure in which the difference between the experimental diagram and the calculated diagram is minimized. On the other hand, the procedure followed above in the AgCl + NaCl example does not distinguish itself by making exhaustive use of the possibilities, as follows from the following reasoning. The two equations f8a) and (8b), which can be written as Fl(excess coefficients)

=

numerical value Nl

F2(excess coefficients)

=

numerical value N2

and ,

(13a) (13b)

were subtracted to give equation (Q), which is of the form Fl - F2

=

Nl - N2

.

(14)

The consequence of such a procedure is that two independent sets of data are reduced to one set (loss of information) and, what is more, two regression equations (two different constraints on the coefficients) are reduced to one regression equation. In that situation we realized that the two data sets should be used simultaneously in one and the same procedure operating with two regression equations. We were fortunate that the statistical procedure we were looking for was available at the computer centre of our university ; we mean the computer subroutine SIVAMIN, which is a slightly modified version of the Golub and Reinsch method (91, written by J. van der Star. As another improvement we incorporated in a modified effective-variance method of Clutton-Brock (10) the use of mathematically defined weights. In the effective-variance method the following sum is minimized

where fobs represents the right-hand sides of the equations (13a) and (13b) the left-hand sides of these equations. %z f%%lation of the weights the reciprocals of the variances of fobsand fcalc are f talc' is rather complicated b&ause the errors in fobs

H. Oonk et aik

60

highly dependent and, moreover, depend on the as yet unknown coefficients. In the EXTXD/SIV~IN method combinations of the coefficients (the parameters to be adjusted) are selected in an iterative procedure, the weights being adjusted at every step. The weights are calculated by correlated-error analysis (2). In applying the EXTXD/SIVAMINmethod (of which a closer description is given in the appendix) to real systems we found that it had the potential of a "non-derived"method, in that it reproduced the experimental phase diagram with great accuracy. In fact, the coefficients from which the phase diagrams shown in figures 1 and 2 were calculated were obtained by applying the EXTXD/ SIVAMIN method to the region of demixing in the system KC1 + NaCl. As griteria for the measure of (dis)agreementbetween the experimental and the calculated phase diagram we use the disagreement index D, the mean temperature and the mean mole-fraction difference Ax. difference A The disagreemTnt index D is defined as D

I Z&o

- Tcalcl

(16)

L Tew

where the summation is made over the dummy data points, i.e. the points read from the curve drawn "by eye" through the original experimental points. The mean temperature and the mean mole-fraction differences are taken with respect to the original data points themselves and are defined as

AT

=-

’ n

A X =- n'

1/

~~~~

-

'exp

- 'calcl

!ccalc

K

I

and

’

(17) (18)

respectively, and where n stands for the number of original TX data points. As an example the diagrams shown in figures 1 and 2 have n (calculatedfor all eleven data points except the point at t#e%i;'of 0.0056 and 0.0053, respectively,while the investigators state (7) that the uncertainties in the X values are 0.015. Choice of coefficients to be adjusted In the the~~~n~ic analysis of phase diagrams the coefficients Hi, Si, Ci, etc. appear as parameters, the values of which have to be found in the computational procedure. First we have to deal with the problem of the most proper choice of coefficients, that is to say in cases where the only available information is one single TX phase diagram. In comparable studies, such as vapour pressure measurements, where there is only one parameter for Ii,one for S, and so on, the experimental values are usually fitted so that e , equation (3), is given the mid-range value. In these cases the data usually do not permit more than the evaluation of H(0) and S(e). For TX phase diagrams the situation is quite similar and the best thing to do is to confine oneself to the parameters H. and S and to realize that the computed values are valid for a temperatu&e 8, which is the mean temperature of the phase-equilibriumregion. Accordingly, the excess Gibbs energy function to be derived from the phase diagram - compare the equations (3) and (4) - consists of coefficients . CL(T) = Hi(e) -TSi(e) (19) This is a function with coefficients of the upper two rows of the scheme of coefficients, in which those with index 1 (HI and S1) dominate. In practice-calculationsare made with an increasing number of adjustable parameters. In our calculations we follow the route

BINARY COMMON-ION ALKALI

"combination"

coefficients

l/O

I%

111

Bl

s1

2/l

H1

Si

HALIDE

MIXTURES

61

(parameters to be adjusted)

H2

2/2 312

In those cases where the number of adjustable parameters is odd, the last parameter Ii. in fact represents G (e). And if one ghould consider makingicalculations with only H parameters (this is optional in our program), one should realize that in fact G.(B) values are obtained.In that case the excess function calculated, of co&se, represents the excess Gibbs energy for the mean temperature 9, and not the excess enthalpy. In our view such a limitation should not be made a priori. If it appears a Posterfori that owing to the strong correlation between the Ii and the S parameters (see the above observations) a satisfactory separation is not possible, one can always consider dropping the S parameters (in addition, EXTXD/SIVAMIN computations based on data sets read from phase diagrams that had been calculated with B coefficients only - regular solution approximation - yielded the correct H values and for the S parameters values that did not differe significantly from zero; this means that if the S coefficients are really zero the method is capable of demonstrating that). Each of the combinations gives rise to a "solution", the significance of which is examined in terms of the "internal criteria". These are the standard deviation of the fit, the F-test value, the disagreement index D and the mean temperature difference A . The first two of these criteria refer to the function values derived from thg phase diagram, while D and AT refer to the phase diagram itself. we do not consider the A criterion further as the eutectic alkali halide phase diagrams have been dgtermined by method; in which the liquidus temperature is measured for fixed X. The F-test value is the square of the quotient of the value calculated for the last coefficient and its standard deviation. A large value for F indicates that the combination gives a considerably better fit than the foregoing one. The first observation made above indicates that situations might arise where the internal citeria fail to distinguish between diverging solutions. EUTECTIC ALKALI HALIDE SYSTEMS WITH NEGLGIBLE SOLID MISCIBILITY The regression equations In the case of eutectic equilibria with negligible solid miscibility the numerical values, equations (13a) and f13b), are given by %=

Tf 01

N2 =

P

AS:(T)

dT - RT ln(l-X')

AS;(T)

dT - RT 1nY'

.

(20a

(20b‘1

02

represents the entropy of melting (asterisks are In these equations AS" used to denote pure components), T is the melting point of the pure first component and To2 is the melting p8fnt of the pure second component. of the excess coefficients, see also equations (13af and The functions (13b), are

62

H. Oonk et al.

k' -T

E j=l

S. (*j-l-*jX')(l-*Xl)j-* 3

(*la)

k (l-Y')*

1 i=l

Hi(1-2iY')(l-2Y1)i-2 k' Sj(l-*jY')(l-*Y')j-*

-T

(*lb)

where

k' is either equal to k or to L=i TX' are the data points read from the-left-hand liquidus and TY' are the data points read from the right-hand liquidus. Both X' and Y' are mole fractions of the second, i.e. the right-hand component. Svstems

The systems for which we present the results in this paper are all common-ion alkali halide mixtures that show negligible solid miscibility. The systems were selected with the help of figure 4 where the substances are arranged in order of increasing sum of ionic radii. Complete miscibility in the solid state was deduced from phase diagrams and, when necessary, verified by the LIQFIT method (4).

@okIstate miscibility

INaF tiNaF

Lll KF Li% ma RbF

H5s!!e!! Lie NaCl RbF

q

IiiNd

1 nrgliiibk:n&ctad

KEr CacY Rb& KI CSL Rbl Cd

FIG 4 solid statemiscibilityin common-ionalkalihalidemixtures.The substancesare set out in order of increasingsum of ionic radii.

63

BINARY COMMON-ION ALKALI HALIDE MIXTURES

Limited miscibility has been demonstrated for a number of systems in a number of ways. For example in the case of (Li,Na)F and (Na,Rb)Cl from interdiffusion experiments (11); for (Li,Na)F and. (Na,K)F by thermal analysis (12); for (Li,Na)Cl by phase-diagram calculation (13); etc; Negligible has been reported, e.g. for (Li,K)F, (Li,K)Cl and (Li,K)Br (14), for Li(F,Cl) (15) and for Na(F,Cl) (16). Systems for which solid miscibility can be neglected are to be found in figure 4 towards the lower left-hand corner. The systems which we selected for the computations are (Li,K)F, (Na,Rb)F, fNa,Cs)F, (Li,KfCl, (Li,Rb)Cl., (Li,K)Br, (Li,Rb)Br, (Na,Rb)Br, (Li,K)I, Li(F,Cl), NafF,Cl), Na(F,Br), Na(F,I), K(F,Cl) and K(F,I). Properties of the pure components The entropies of melting (AS*) and the heat-capacity changes on melting (AC*) needed for the computations were derived from-several sources. The p values are assembled in Table 1. TABLE 1 Values for the Entropy and Heat-CapacityChanges on Melting used for the Calculations, Divided by the Gas Constant. In some cases Estimated Values are used. subst

f&*/R

ref.

ACz/R = a f b T/K a b 2.37x10-3 -1.43

LiF

2.89

(18,191

LiCl

2.71

(17)

0

LiBr

2.58

(17)

3.3

LiI NaF

2.38 3.15

(17) f18,22,23)

1.0 (est.) -4.8x10-5 0.06

NaCl

3.14

(17)

3.23

NaBr

3.08

(17)

1.0 (est.)

NaI

3.04

(17)

1.0 (est.)

ref.

(18) (20,211

-2.55~10-~

(18) (24)

KF

3.14

(18)

0.48

(18)

KC1

3.03

(17,25,26)

1.0

(20,261

KBr

3.02

(17,21,25f

-2.0

(20,211

KI

3.03

(17)

0

RbF

2.90

(17)

0.45

(18)

RbCl

2.87

(17)

0.65

(20)

F&Br

2.90

(17)

0.5 (est.)

CSF

2.68

(171

0*47

(18)(est.)

Phase diagram data In all cases calculations were made with data points which we read from liquidus curves that had been drawn

"by eye" through the experimental

liquidus points. The initial slopes of the liquidus curves were calculated from the entropies of melting with the help of Van 't Hoff's law. Dummy data points were read from the liquidus curves at regular temperature intervals. For example in the case of Na(F,Cl) we used intervals of 15 R, which yielded nine points on the NaCl side and 21 points on the NaF side.

3.38t.44)

-3.44(,50)

-6.09(1.7)

-4.71(1.6)

-14.04(.97)

-14.13f.95)

-13.83(.77)

-13.26t3.0)

-27.541.81)

-27.3Oc.66)

-26.85(2.4)

%

-27.47l.82)

-15.64c.18)

5

3.04t.38)

3.66c.431

-9*27(1.2)

-7.81(1.2)

-2.13(.50)

The Eutectic in Table 2.

3.09(.401

3,32(.69)

-13.80(.71)

-12.25(.70)

1.99X.46)

2.00(.44)

0.081.05)

H2

2.18(.51)

2.18(.50)

s2

3

0.08(.39)

=3

s3

34.7

34.3

42.2

43.5

0.4

0.4

0.5

0.7

0.9

1.9

% 1.61

F

.0017

0 .0013

19 -0013

2 .0016

210

.0044

D

BT

-14.32

-14.22

-14.19

-14.20

-15.64

Gl

as

1.75

1.51

2.01

2.38

2.12

have the Same Meaning

115.9 7873

s.d.

945 K. The Symbols

TABLE

5.4

-0.35(.93)

3 .00016

20 .00054 350 .00016

5.5

22 .00079

.0029

21.2

242

D

332 .00097

F

26.3

77.4

.d.

17.5

-1.26c.75

s3

1.17(.06)

H3

System Li + KF. Mean TWnperature

3.15(.71)

-0.26(.05)

-11.46t.63)

1.61(.10)

-10.03(.64)

s2

s1

s1

I12

TABLE2 -1 -1 The Sutectic System NaF + NaCl. Calculated H values inkJ mOl and S values in J K-lmol with Calculated Standard Deviations in parentheses; Standard Deviation (s.d.1 of the Fit;' (C$ and G values for F-test value (F); Disagreement Index CD); Mean Temperature Difference the Mean Temperature, 1060 K.

-0.07

-0.06

0.08

G2

0.16

0.38

0.02

-0.26

0.08

G3

0.98

1.17

R w .

‘cc

m

&

8

D: .

BINARYCOON-ION

ALKALI

65

HALIDE MIXTURES

Result& this section we first give detailed results for the systems Na(F,Cl), (Li,K)F and (Li,KtCl. the NafF,Cl) system The experimental data on this system were obtained by Grjotheim et al. (16) by means of the cooling-curve technique. The results of our calculations are given in Table 2. We observe that for this system, which displays moderate deviation from idealmixing behaviour, there are three solutions with mean temperature differences of 0.5 K or less ! We also observe that the separation of the G coefficients into the Ii and Si sion coefficients is indeed a delicate matter. This follows from the dispe). in the calculated H and Sl values, whereas the dispersion in the Gl values is considerably les&. The correlation between Ii and S is also demonstrated by the calculated uncertainties : the calculated &certainty in an H coefficient is about 8 times the calculated uncertainty in the corresponding S coefficient. putting all the criteria together we can say that there are two excellent solutions between which it is difficult to distinguish. These are the 312 and the 3/3 solutions (owing to the calculated uncertainties and the high F-test value, we have a slight preference for the 3/2 solution). the (Li,K)F system The experimental data were obtained by Aukrust et al. (14), by means of thermal analysis, by high-temperaturefiltration and also by visual observation of the onset of crystallization. The dummy data set used for the calculations consisted of 18 points from the left-hand liquidus and 18 points from the right-hand liquidus : on both sides the eutectlc point and 17 points at 20 K intervals from 780 to 1100 K. The results o the calculations are given in Table 3, which does not include values for g because, unfortunately, the experimental data were only available In t% form of the plotted diagram (of which we used a photographic enlargement). influence of errors in melting properties Owing to the fact that for this system the calculated coefficients do not change much from solution to solution, the (Li,K)F system is suited for investigating the influence of experimental errors in aSw and AC* on the P values of the computed coefficients. Calculations made with modified values for these quantities (the modifications being representative for the current, not inconsiderable,dispersion in alkali halide literature values) revealed the following picture for the uncertainties in the coefficients : In

HI (2.5)

Sl(2.6)

Gl(l.0)

H2(0.9)

S2(0.8)

G2(0.1)

.

We observe, again, that the uncertainties in the R coefficients are about 8 times the uncertainties of the S coefficients. calculations with fixed coefficients For the (Li,K)F system we also give the results of calculations with two sets of fixed coefficients. In the first of these two sets we made calculations with all Hi= 0, with the exception of Hl. The value used for the latter was obtained by extrapolating the experimental values for 1360 K (27)‘ for 1181 K (28) afd for 1176 K (27) to T = 6 = 945 K. The extrapolated value is -23.0 kJ mol and we estimate its uncertainty to be 2.0 . In the second set we fixed all Hi coefficients to zero. The results of calculations with fixed coefficients are assembled in Table 4, next page.

66

H. Conk et aZ.

TABLE

4

Results of Calculations on the LiF+KF System with Two Sets of Fixed Values for the H Coefficients. For Comparison,the Results of the Normal Calculation are Included.

0.00

0.00

0.00

fixed

18.60 18.60

0.05

20.55

0.27

-15.64 -14.04

-27.47

-4.12

103D

Gl

2.3

-14.69

2.2

-14.69

0.05

1.7

-15.05

0.01

8.8

-17.58

7.9

-17.58 -0.04

3.2

-19.42 -0.23

4.4

-15.64

1.7

-14.20

G2

-27.54

0.08

-14.13

1.6

-14.19

-27.30

2.00

-13.63

2.18

1.3

-14.22

-0.06

-26.85

1.99

-13.26

2.18

1.3

-14.32

-0.07

we observe is allowed

0.08

G3

0.66

3.89

0.08

0.08

that the best results are obtained (see D values) when the program to adjust the I-I as well as the S coefficients.

Incidentely, the results of calculations with all Si= 0 (regular-solution approximation) were even worse than those with all Hi= 0. the (Li,K)Cl

system

(Li,K)Cl system makes it possible to compare (for fixed properties of the pure components) the results of computations on different experimental. data sets for the same system. The melting points, reported by the investigators for the pure substances, lie in a range of about 10 K. The distances between the experimental liguidus curves go up to about twice that value. For each of the experimental data sets calculations were based on dummy data points read from the eye-fitted liguidus curves. The results are given in Table 5 for reasons of comparability for the same combination of coefficients. If we put_fhe dispersion displayed by the G1 values in Table 5 at 1.0 (still in kJ mol ), then the following picture is obtained for the dispersions In the computed coefficients : The

G2 (0.7) H2W5) S2(2.7) H1(5.W Sl(7.W G+'?) We see, again, that the dispersions of H and S and of Ii and S2 are coupled only of by the mean temperature. These dispersio A s are &he result' not experimental inaccuracies in the liguidus points, but also of some kind of subjectivity in the construction of the eye-fitted curves. Especially with regard to the first data set of Table 5, it should be remarked that a criterion like A loses part of its significance if the data points do not cover the whole m&e-fraction range in a regular manner.

BINARY

COMMON-ION

TABLE Results of Calculations system LiCl+KCl. ref.

ALU1

5

Based on Phase Diagrams

H2

$2

67

HALIDE MIXTURES

from Different

800D

AT

SOurCeS

Gl(800 K)

for the

G2(800

H1

s1

(29)

-26.8

-18.3

0.9

1.0

1.2

1.9

-12.1

0.1

(301

-19.9

-6.6

-1.6

-1.4

3.1

3.2

-14.6

-0.1

(31)

-21.5

-9.9

-5.5

-7.3

2.5

1.7

-13.6

0.4

(32)

-19.6

-6.6

-4.4

-3.7

1.5

6.1

-14.3

-1.5

(33)

-32.0

-24.2

-3.3

-3.5

1.1

3.8

-12.6

-0.5

(14)

-23.1

-11.5

-0.4

0.0

1.6

-13.9

-0.4

K)

The results obtained for the complete set of the systems considered are summarized in Table 6. The Table gives for each system the coefficients of the best fit (i.e. in terms of the internal criteria), the value of the disagreement index D, the mean temperature difference n and the mean temperature 8 (the disagreement index multiplied by 8 gives thtfmean temperature difference with respect to the dummy data set).

TABLE 6 Survey of Results gf Calculations on 15 Alkali Halide Systems. H values in kJ mol ; S values in J K-lmol-1; Disagreement Index Mean Temperature Difference (AT) and Mean Temperature (8). system

Iref.)

Hl

s1

H2

s2

H3

s3

I

10%

AT

1.3

945 1.3

1050

1.3

5.3

1000

1.4 (14)

-26.23

-20.45

fLi,RbfBr

(31)

-30.07

(Na,Fb)Br

(31)

3.89

(Li,K)I

(35)

Li(F,Clf

B/K

0.4

2.0

(Li,KfBr

(D);

800 0.8

750

5.26

6.75

-0.82

0.8

760

-24.53

10.56

12.78

-1.80

1.5

2.2

700

10.00

-1.67

1.4

1.7

880

-23.77

-16.54

1.83

3.3

(15)

4.97

7.26

3.70

Na(F,Clf

(161

-2.13

-3.44

Na@',Br)

(36)

5.54

5.50

Na(F,I)

(37)

32.52

28.12

39.25

30.31

KW,CI)

(38)

-13.82

-14.48

-2.02

K(F,I)

(37)

-20.39

-21.65

-7.02

--9.95

690

2.79

0.59

0.4

1.5

890

3.66

3.10

1.17

0.2

0.4

1060

-2.67

-4.50

0.5

2.0

1030

6.11

1.2

1.7

990

1.6

1.8

980

-0.89

0.7

3.1

930

68

H. Oonket al.

Discussion reproduction of phase diagram The most important characteristicof the EXTXD/SIVAMIN method is the accuracy with which it reproduces the experimental phase diagrams. This is reflected, for instance, by figures 1 ani 2 for the region of demixing in the KC1 + NaCl system and by the very low T values displayed by the high quality eutectic data sets such as Na(F,Cl). The importance of the accuracy of reproduction lies in the fact that it permits one to draw a number of pertinent conclusions. uncertainties in the calculated coefficients due to experimental errors If one leaves aside the problem of their separation, there are uncertainties in the calculated H and S coefficients due to the uncertainties in the experimental data. The latter are experimental erros in the transition properties of the pure components and errors in the experimental sets of liquidus points. The influence of errors in the transition quantities - investigated for the (Li,R)F system - and the influence of errors in the phase diagram - investigated for the (Li,X)Cl system - can be combined in the following statement. "An accurate eutectic phase diagram of an alkali halide mixture consists of data points regularly spaced over the whole mole-fraction range and is capable of giving 'solutions'with A values less than about 1.5, the remaining uncertainties in the H and TG coefficients being Gl(l.5) Hl(5) R3(3) G3(l) H2(3) G2(1) while the uncertainties in the S coefficients are related to those in the N coefficients through the mean temperature." the separation of H and S coefficients The separation of the H and S coefficients is indeed a delicate matter, as follows from the observation that the uncertainties calculated and the dispersions found for the H coefficients invariably are related, through the mean temperature, to the uncertainties and dispersions of the S coefficients. In view of this it is surprising that the calculations do not lead to nonsensical H values. Or, expressed positively, taking into account the uncertainties due to experimental errors, the calculated H coefficients are in unexpected accordance with the experimentallyobtained values. The calculated and experimental values for the dominating coefficient HI are for compared in figure 5 which includes the results obtained earlier (2) regions of demixing. The following comments can be made about this comparison. The coefficients derived from the phase diagrams hold for the lower side of the regions of existence of the liquid mixtures. This implies that the experimental values, which preferably are measuted over the whole composition range, refer to temperatures that are considerably higher. Recently it has become clear that for a number of alkali halide mixtures (27) the heats of mixing change considerably with temperature. The influence of temperature is known for the (Li,K)F system (271, and we observe from figure 5 that extrapolation into the solid-liquid equilibrium region supports the calculated values. Similar changes can also be expected for the other systems at the bottom of figure 5. Another point is that the comparison is made with values from one experimental group (the inclusion of results from other groups would increase the significance of the calculated values !I.

BINARY COMMON-IONALXALIHALIDEMIXTURES

69

EXPERIMEBTAL H1 VALUES

HI VALUES CALCULA!I'ED

system, temperature, state (ref.)

equilibrium, system, temperature

+30

ROD, (Na,KfCl,685 two tyoss of solutions, corresoondinq with ficures 1 and 2

+20

(Na,K}Cl,298,sol~~enched) (7)

ROD, (Na,K)Cl,685

Bl two data sets I

(Na,Ag)Cl,623,sol(41) (Na,Ag)Br,623,sol (41)

*lo

ROD, (Na,Ag)Cl,426

1 ROD, (Na,Ag)Br,455 ROD, (Na,Ag)Br,415

i EUT,Li(F,Cl),lUSU

i Na(F,C1),1287,liq~Na,Rb~F,1281,liq t4U Li(F,C1),1152,liqf401

'

EUT,(Na,RbJF,lUSU

EUT,Na(F,C1),1060

EUT,NafF,C1),1060 two types of solutions

-il (27) fLi,K~Br,1018,1~~~~~~~~F,l~60,~~~ 1Li,K)C1,1013,1iq~~~Qk~p 1181fiq

uncertainty in values EUT

(28)

'Li,x'F,1176,1iqtLi,~~=~,~O~~,~iq(39) Li,K)F,945,liq,extrapoLated- - - E

-20 EUT,fli,K)Cl,80G s -30

EUT, (Li,K)Br,760 ----

EUT,(Li,Rb)Cl,750 --EUT(Li,KfF,Q45

FIG. 5 Comparison of experimental and calculated Hi values of the opinion that if more accurate values for the transition quantities were available the EXTXD/SIV~IN method would be capable of finding H coefficients which could bear comparison with the values derived from heat of mixing experiments. Accordingly, the H values calculated for the systems which display AT values of less than about 1.5 are realistic indications for the heats of mixing at the lower side of the regions of existence of the liquid mixtures. We are therefore

eye fit of experimental

data

Calculations based on dummy data sets read from eye-fitted curves are to be preferred to calculations based on the original data points, as they provide more accurate coefficients and , moreover, give calculated phase diagrams that are in closer agreement even with the original points themselves (for more details the reader is referred to 2 and 42).

70

H. Oonk et

a$.

the excess Gibbs energy coefficients The calculated G coefficients reflect in a number of ways the fact that heterogeneous equilibria are governed by the Gibbs energy. The G coefficients calculated in one run lie close together and their values are less dependent on the accuracies of the transition quantities than are the Ii and S coefficients. The G coefficients still make sense when calculations are made with irrealistic sets of adjustable parameters, (see Table 4, second group). Therefore the G coefficients have a higher rank than the H and S coefficients. They are more than "indications", in that they have to be considered as coefficients of the real excess Gibbs energy function. The G coefficients obtained for the alkali halide systems are assembled in Table 7 and accompanied by realistic estimates of their uncertainties.

TABLE

7

-1 Excess Gibbs Energy Coefficients Expressed in W mol for Liquid Binary Alkali Halide Mixtures with Estimated Uncertainties in parentheses and Valid for the Mean Temperature 8 .

8ystem

B/K

=1

G2

=3

945

-14.2f1.2)

-0.1(0.8)

U?a,Bb)F

1050

l.lf0.8)

0.5tO.8)

(Na,Cs)F

1000

0.5(2.01

0.4(1.2)

ILi,K)Cl

800

-13.9(1.2)

-0.4(0.8)

(Li,Eb)Cl

750

-17.9(1.6)

-1.9(0.8) -1.0(0.8)

(Li,K)Br

760

-10.7(1.2)

O.l(O.8) -0.8CO.8)

tLi,KfF

(Li,Bb)Br

700

-12.9(2.0)

(Na,Bb)Br

880

-4.9(2.0)

-1.711.2)

(Li,K)I

690

-12.4f2.5)

1.8(1.6)

Li(F,Cl)

890

-1.5(1.2)

1.2lO.8)

Na(F,CI)

1060

l.S(O.4)

1030

-0.1f1.6)

Na(F.1)

990

4.7t2.0)

K(F,Cl)

980

0.4C2.0) -2.0(0.8)

x(F,I)

930

Na(F,Br)

-0.3f2.0)

l.S(O.8)

1.6cl.2) -1.812.0)

0.6f0.8)

0.4f0.8) 1.2CO.41 2.4CO.8) 9.2C2.0) 6.1t2.0)

2.2l1.2) -0.9C2.0)

choice of parameters to be adjusted From the point of view of phase-diagram analysis the most reliable results are obtained when calculations are made with both the H and S coefficients as adjustable parameters and with a stepwise increase of their number on condition that the number of S parameters is as close as possible to the number of H parameters. Even calculations with E coefficients fixed to the best available experimental values lag behind the results that can be obtained by the normal procedure.

71

BINARY COMMON-ION ALKALI HALIDE MIXTURES

Calculations with H coefficients fixed to the experimental values, are of greater importance for the evaluation of real S coefficients than for the improvement of the G coefficients. As an example, the coefficient SI obtained for (Li,K)F by fixing the HL -8.8 coefficient to the experimental value (see Table 4) has the value of with an uncertainty of 3, whereas the value of Sl found in the normal calculation is -13.3 with an uncertainty of 5. types of solutions In some cases it happens that diverging solutions give equally good results, i.e. they cannot be distinguished by the criteria. A striking example is the region of demixing in the XC1 f NaCl system, where three types of solutions give comparably low values for the mean mole-fraction difference, the mean temperature difference and the disagreement index. The two extremes of these three types are dealt with in figures 1 and 2 and in figure 5. The diverging types even appear when calculations are based on the the calculated phase diagram (dry data set consisting of exact T values and mole-fraction values calculated to three decimal places) corresponding to one of the types of solutions. In cases like this additional information is needed in order to indicate the type of solution which is the highest in rank from the point of view of real excess Gibbs energy function. In the KC1 + NaCl case the type of solution with H of about 18 kJ mol-1 gives an excellent descriwtion of the mixed solid &tate in its whole region of existence (2,42). A References 1.

N. Brouwer, H.A.J. Oonk, 2. Phys. Chem. N.F., 105, 113 (1977).

2.

N. Brouwer, H.A.J. Oonk, Z. Phys. Chem. N.F., 117,

3.

N. Brouwer, H.A.J. Oonk, 2. Phys. Chem. N.F.,

4.

J.A. Bouwstra, N. Brouwer, A.C.G. van Genderen, H.A.J. Oonk, Thermochim. Acta, 38, 97 (1980).

5.

0. Redlich, A.T. Rister, Ind. Eng. Chem., 40, 345 (1948).

6.

E.C.W. Clarke, D.N. Glew, Trans. Faraday Sot., 62_, 539 (1966).

7.

A.J.H. Bunk, G.W. Tichelaar, K. Ned. Akad. Wet. Proc., Ser. B 56, 375

8.

C. Sinistri, R. Riccardi, C. Margheritis, P. Tittarelli, 2. Naturforsch. Teil A, 72, 142 (1972).

9.

G.H. Golub, C. Reinsch, Contribution I/l0 in : J.H. Wilkinson, C. Reinsch I Linear Algebra, Springer Verlag, Berlin-Heidelberg-New York, 134 (1971).

121,

55 (1979). 131

(1980).

(1953).

10.

M. Clutton-Brock, Technometrics, 2, 261 (1967).

11.

J.M. Short, R-Roy, J. Am. Ceram. Sot., 47,

12.

J.L. Holm, Acta Chem. Stand., 19, 638 (1965).

13.

P.-L. Lin, A.D. Pelton, C.W. Bale, J. Am. Ceram, Sot., 62, 414 (1979).

14.

E. Aukrust, B. Bjtirge, H. Flood, T. FBrland, Ann. N.Y. Acad. Sci., 79_, 830 (1960).

15.

H-M. Haendler, P.S. Sennett, C.M. Wheeler Jr., J. Electrochem. Sot., 106 I 265 (1959).

149 (1964).

72

H, oonk et az.

16.

K. Grjotheim, T. Halvorsen, J.L. Helm, Acta Chem. Stand., 21, 2300 (196

17.

A.S. Dworkin, M.A. Bredig, J. Phys. Chem., ti 269 (1960).

18.

A.C. Macleod, Y. Chem. Sot., Faraday Trans. 1, 2,

19.

T-B. Douglas, J.L. Dever, J. Am. Chem. Sot., 76, 4826 (1954).

20.

K.K. Kelley, U.S. Bureau of Mines, Bulletin 584 (1960).

21.

I. Bibas, J. Leonardi, C.R. Acad. Sci. Paris, Ser. C, 268, 877 (1969).

22.

W-B. Frank, J. Phys. Chem., 65, 2082 (1961).

23.

C.J. O'Brien, K.H. Kelley, J. Am. Chem. Sot., 2,

24.

R.A. Robie, B.S. Hemingway, J.R. Fisher, U.S. Geological Survey, Bulletin 1452 (1978).

25.

D.I. Marchidan, M. Gambino, Bull. SOC. Chim. France, 1954 (1966).

26.

W.T. Thompson, S.N. Flengas, Can. J. Chem., 49, 1550 (1971).

27.

K.C. Hong, O.J. Kleppa, J. Chem. Thermodyn., 8, 31 (1976).

28.

J.L. Helm, O.J. Kleppa, J. Chem. Phys., E1

29.

T.W. Richards, W.B. Meldrun, J. Am. Chem. Sot., 39, 1816 (1917).

30.

H. Keitel,

31.

S.D. Gromakov, L.M. Gromakova, Zh. Fir. Khim., 27, 1545 (1953).

32.

I.G. Murgulescu, S.Sternberg, Z. Phys. Chem., 2.&, 114 (1962).

33.

S. Sternberg, I. Adorian, Rev. Roum. Chim., 9,

945 (1973).

34.

D.L. Deadmore, 3.5. Machin, J. Phys. Chem., 3,

824 (1960).

35.

D-B. Leiser, O.J. Whittemore Jr., J. Am. Ceram. Sot., 2,

36.

N.S. Dombrovskaya, Z.A. Koloskova, Izv. Sekt. Fiz. Khim. Anal., g, (1938).

37.

A.G. Bergman, F.O. Platonov, Izv. Sekt. Fiz. Khim. Anal., lo, 253 (1938

38.

W. Plato, Z. Phys. Chem., E,

39.

L.S. Hersh, O.J. Kleppa, J. Chem. Phys. 42, 1309 (1965).

40.

O.J. Kleppa, M.E. Melnichak, 4th Intern. Conference on chemical Thermodynamics, Montpellier, France, August 1975, book III, contr. 111-26.

41.

O.J. Kleppa, S.V. Meschel, J. Phys. Chem.,

42.

H.A.J. Oonk, Phase Theory : The Thermodynamics of Heterogeneous Equilibria, Elsevier Sci. Publ. Comp., Amsterdam, in press.

2026 (1973).

5616 (1957).

2425 (1968).

Neues Jahrb. Mineral. Geol., Abt. A, 52, 378 (1925).

60 (1967). 211

364 (1907).

2,

3531 (1965).

73

BINARYCOMMON-IONALKALIHALIDEMIXTURES

Appendix the HXTXD/SIVAMIN method principle The principle of the method is the simultaneous calculation of the coefficients of the excess enthalpy and excess entropy functions (heat-capacity coefficients can easily be included). The calculation of the desired coefficients is realized by suitable parameter representations of the thermodynamic equilibrium conditions. For binary systems there are two equilibrium conditions, which for n temperatures give rise to 2n regression equations. A closer study of these equations reveals that in the case of eutectic systems these equations are really independent. In other cases correlation coefficients must be taken into account. In the past we carried out our calculations by the familiar least-squares method with the normal equations in matrix notation, but we soon discovered that the inverse matrix was highly unstable, owing to the finite machine precision. computational procedure The actual calculation method we use is a variant of the Gauss-Newton method (non-linear parameter models can be used as well as linear ones). The over-determined (linear) parameter system, which arises from the equilibrium conditions , (13a) and (13b), is solved by the singular value decomposition method, followed by the QR algorithm combined with Householder transformations, as described by Golub and Reinsch (9). It turned out that considerable improvements could be made by introducing weights as the reciprocals of the effective-variances of fobs- fcalc, sum (15) calculated by correlated-error analysis taking into &count the experimental uncertainties. Our computer program EXTXD/SIVAMIN is written in ALGOL 60. In the near future a FORTRAN version will be available as well. The program consists of three main parts. One part defines the desired regression equations with the wanted excess parameters H and S. In one part the function-residuals fobs - fcalc are calculated. Then there is the part in which the subroutine SIVAMIN is defined and called. The name of the subroutine (procedure in ALGOL) must be seen as an identifier, which has a list of parameters placed between brackets. SIVAMIN is called by SIVAMIN[k,n,co,psi,func,base,grad,toll,tol2,nit,imax,calls]

;

The input parameters are : k n

psi base to11 to12

imax

type:

integer

the number of (unknown)coefficients

integer

the number of function-residuals (contributions in sum (15) array [l:k] initial values of the wanted coefficients (e.g. all zero) procedure problem-dependent procedure which defines n function residuals, func 1:n and n weights w l:n real a tolerance, e.g. 10-8 meant as a stop for the minimization of sum (1;) -8 real a tolerance, e.g. 10 for the difference between the calculated coefficients in two successive runs, likewise meant as a stop if toll is also satisfied integer

the maximum number of iterations allowed, e.g. 50

74

H. Oonk et al.

The output parameters are : co

array b:k,l:k)

the covariance matrix of the coefficients calculated as the inverse of J T.J , where J is the approximate Jacobian

Psi func

array b:kJ

found values of the coefficients

array [1:x+

func[i‘j contains the value of the i-th function-residual for the found coefficients, defined by the procedure base

grad

array [l:kj

the gradient vector for the location of the minimum of sum (15)

nit

integer

the number of iterations performed by the process

conv

boolean

a criterion for convergence: if conv is "true" convergence is achieved within the criteria toll and to12; if conv is "false" convergence is not achieved within imax iterations

calls

integer

the number of times that base is called by SIVAMIN

sivamin

procedure identifier

contains the value of the sum (15) for the output values of the coefficients psi[l] ,... .. .. psi[kl

During our calculations we found SIVAMIN to be an excellent optimization procedure, rather sensitive to bad data sets (in those cases dependency between the model parameters is frequently discovered and SIVAMIN prints a message reporting in which iteration step and between which parameters the dependency exists; the process then continues with a reduced set of parameters for the current iteration step ; with improved data sets or through the input of more realistic experimental uncertainties the parameter dependency often disappears). All our calculations were carried out on a large computer of the Control Data Corporation, type Cyber 175, compiler version ALGOL-60, level 498.