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Ind. Eng. Chem. Fundam. 1985. 2 4 , 147-152
Stolcos, T.; Eckert, C. A. Ind. Eng. C h m . Furnham. 1985, following article In this Issue. Tamakl, S.; Shiota, I. Phys. Kondens. Mater. 1988, 7 , 383-389. Tomlska, J. Cslphad. 1980, 4, 83-81. Umar, I . H.; Meyer, A.; Watabe, M.; Young, W. H. J . Phys. f : Metal Phys. 1974, 4, 1691-1708. Wilson, J. R. Metall. Rev. 1965, IO, 381-390.
Prausnitr, J. M.; Anderson, T. F.; Grems, E. A.; Eckert, C. A.; Hsleh, R.; O’Connell, J. P. “Computer Calculations for Multlcomponent Vapor-Liquid and Llquid-Liquid Equlllbrla”; Prentlce-Hail: Englewood Cliffs, NJ, 1980; Chapters 4 and 6. Predel, B.; Sandlg, H. Mater. Scl. €ng. 1989, 4, 49-57. Prlgogine, I. “Molecular Theory of Solutions“; North-Hoiland Publlshlng Co.: Amsterdam, The Netherlands. 1957; p 28. Rattl. V. K.; Zlman, J. M. J . Phys. f : Metal Phys. 1974, 4 , 1684-1890. scott, R. L. J . chem. im,25, 193-205. Sherwood, A. E.; De Rocco, A. G.; Mason, E. A. J. Chem. Phys. 1968, 44, 2984-2994. Shlmoji, M. “Llquld Metals”; Academic Press: New York, 1977.
mp.
Received f o r review July 7, 1982 Revised manuscript received July 2, 1984 Accepted September 21, 1984
Nonideal Behavior in Liquid Metal Solutions. 2. Chemical-Physical Theory Model Thomas Stolcos and Charles A. Eckert” Department of Chemical Engineering, Universe of Illlnols, Urbana, Illinois 6 180 1
The “physicai” model developed in part 1 is coupled with a “chemical” model to account for the very strong negative deviations from Raouit’s law in systems forming intermetallic compounds. The resulting chemicai-physical model represents well both Gibbs energies and enthalpies of highly solvated metal systems. New data are presented for two titanium systems, for which only a limited composition range is experimentally accessible. The theory permits facile extension of such limited data.
Introduction
icant, are kept to a minimum. This model is shown to correlate successfully many results on strongly solvated liquid metal solutions. In addition, the method is applied to systems where only limited data are available and it is used to extend such data to experimentally inaccessible regions. Evidence of Existence of Intermetallics in t h e Liquid State Substantial evidence has been presented indicating that in many liquid alloys, the atomic configuration deviates from random mixing and intermetallic compounds are formed. The magnesium-tin binary is considered as an example. The phase diagram of this system, depicted in Figure 1, clearly shows the existence of a solid compound with the empirical chemical formula Mg2Sn,which melts congruently at 1051 K. The stoichiometry of the intermetallic compound is deduced from the location of the maximum of the liquidus curve, which occurs at 66.7 at. % Mg. Viscosity measurements (Gebhardt et al., 1955) a few degrees above the melting temperature of the solid compound shown in Figure 2 indicate that this structure-sensitive quantity reaches a relatively broad maximum at the same position in the concentration range as that corresponding to the stoichiometric composition of the compound. The viscosity increases due to an increase in the binding forces and the formation of structural units with directional bonds. The electrical resistivity also displays a similar maximum at the same composition (Steeb and Entress, 1966). From the shape of these curves, it appears that local order persists in the liquid phase not only in the very narrow homogeneity region of the solid intermetallic but over a large portion of the composition range. Results of X-ray structure studies for the same alloy by the latter authors are also attributed to the fact that part of the melt is solvated. In addition to the physical properties which provide evidence about the existence of intermetallic compounds,
Many binary liquid alloys have chemical and physical properties with pronounced minima or maxima, and such behavior cannot be explained in terms of weak and nonspecific interactions between the components. Experimental results show that such anomalies usually occur at characteristic compositions where stable intermetallic compounds exist in the solid state and they show very large negative deviations from Raoult’s law. These observations have led many investigators to propose the existence of metallic clusters in the melt and incorporate them in solution models. Dolezalek (1908) first used the concept of compound formation and attempted to interpret solution nonidealities solely in terms of chemical forces. This idea was applied to metallic systems by Hildebrand and Eastman (1915), who assumed the presence of the compound HgT12 in thallium amalgams in order to explain deviations of these solutions from ideal behavior. In order to account for both chemical forces which lead to the formation of intermetallic compounds, and the nonspecific physical forces between the different species present in a mixture, several authors have developed chemical-physical models. Jordan (1970) has proposed a regular associated model based on the theory of Prigogine and Defay (1954). His model assumes that the free atoms in solution interact equally strongly with the AB type compounds. Osamura and Predel(l977) have followed a similar analysis, but their formulation accounts for unequal interactions. Cox (1979) has also modeled the combined effects of both chemical and physical contributions. This paper is devoted to the development of a solution model involving both chemical and physical forces. This is achieved by combining the physical theory model used in part 1 (Stoicos and Eckert, 1985) with the appropriate expressions associated with compound formation. The number of parameters, all of which are physically signif0
1985 American Chemical Society
148
-
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 900
1000
Mole Fraction of Sn
Figure 1. Phase diagram for magnesium-tin alloys.
"'I
I O
Mole Fraction of Mq !
I
o Gebhordt e;u/. (1955)
I
Figure 3. Excess stability function for magnesium-antimony alloys at 1073 K.
general method). Here we shall deal only with systems where no more than two compounds suffice to describe the alloy properties. The stoichiometry of the compounds is usually evident from the phase diagram. We may consider the chemical reactions I , Ot2
A+B==AB mA + nB 5 A,B,
I
0,4 0,6 0.8
1.0
Mole Fraction of Sn
Figure 2. Viscosity of liquid magnesium-tin alloys at 1073 K.
variation of thermodynamic functions may also give useful information for the same purpose. For example, reduction of the entropy of a system is often related to compound formation. Darken (1967) has defined the excess stability of a system as excess stability =
(5)
(1)
T,P
He has shown that this function exhibits pronounced maxima a t compositions where intermetallic compounds exist. This happens because when a mixture is close to compound formation, any fluctuation which drives the system away from the concentration of the compound requires a large Gibbs energy change, whereas a composition fluctuation causes no significant change in excess Gibbs energy in a random distribction. The excess stability function can be calculated by differentiating the experimental activity coefficient data numericdy. Figure 3 depicts the results of such a procedure performed for the magnesium-antimony system, where the excess stability function shows a sharp peak at the composition corresponding to the Mg3Sb2compound. Figure 3 depicts the results of such a procedure performed for the magnesium-antimony system, where the excess stability function shows a sharp peak at the composition corresponding to the Mg3Sb2compound. Alger and Eckert (1983) have carried out analytical calculations of this quantity and have discussed the subject in a more extensive manner. Detailed presentations of evidence supporting the occurrence of intermetallic compounds in the liquid phase have also been offered by Jordan (1979) and Predel(1979). Model Equations For this work, we assume that in a binary alloy the interaction between the constituent metals leads to the formation of no more than two intermetallic compounds, with chemical formulae AB and A,B,. Certainly formulations are possible with more than two compounds and in fact in principle any number of compounds can be treated (see, for example, Eckert et al., 1982, for a more
(2)
(3) where the equilibrium constants are written in terms of the activities of the reactants and products and are given by
(4)
Thus, the A-B mixture actually contains four chemical species, A, B, AB, and A,B,. The properties associated with the components A and B, as mixed together, are designated as bulk or apparent and constitute measurable quantities, while those which refer to the actual chemical species in solution AB, A,B,, and monomeric A and B, are called true properties. Prigogine and Defay (1954) have shown that under equilibrium conditions, the bulk chemical potentials of the components A and B are equal to the chemical potentials of the uncombined species A and B in the actual solvated solution. Then, in terms of the activity coefficients and mole fractions, the equality for A becomes XAYA = Z A ~ A (6) In this expression zA is the true mole fraction of A and CYA is the true activity coefficient of A. In order to obtain the bulk activity coefficient yAfor a given overall composition xA, expressions are needed to provide the corresponding true quantities. The true activity coefficient characterizes the interaction between the species after equilibration has occurred. Since such interactions are expected to be weak, the physical model discussed in part 1 can be employed. The true mole fractions depend on the extent of compound formation. The equilibrium expressions are
(7)
and two more relationships are needed, since there are four unknown true mole fractions. Considering the conservation of the species and a mass balance of the components.
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
two additional equations become available ZA
+ Z B + ZAB + ZA,B, ZA
.%A =
ZA
=1
+ ZAB + ~ Z A , B ,
+ Z B + 2 Z A B + ( m + n)ZA,B,
,Ahmino Knudren Cells
Graphite Block-
(9)
140
(10)
Pt/Pt, 13% Rh
Ceromic Support -
Simultaneous solution of eq 7-10 specifies all the true mole fractions. If the mixture were to contain A,B, compound (where m, n both > 1)after rearranging eq 6 and substituting xA, one obtains
Thermocouples
MolyWcnum
Figure 4. Crucible assembly for evaporation studies by the Knudsen effusion method. Water -Cooled Flange1
where
-
[Heat
Shields (Mo) Cooling Water Coils lonizotim
--
-
Taking the limit of eq 11as ZB 0 and ZA --, 1,then YA/‘YA 1. Also when ZA 0 and ZB 1, then YA/CYA 1. Thus YA/‘YA would go to unity at both ends of the concentration scale, and a similar result can be derived for yB/ag. Such a result, called the “hooking” problem, is true for any chemical theory where m # 1 or n # 1. This problem has been discussed by Cox (1979) and Eckert et al. (1982); they show that no solvated liquid metal system has ever been found which “hooks”, and the values of y m for higher solvated systems are always much less than unity. This evidence demands the existence of one or more compounds where n = 1and/or m = i; thus for conciseness we have introduced the simplest possible second compound, AB. In order to calculate ?A, first the chemical theory eq 7-10 must be solved to obtain a value for zk From zA,the true activity coefficient CYA can be evaluated by the physical theory model. We assume that the intermetallic compounds are neutral and do not interact with any species; charge transfer occurs only between the dissimilar, unsolvated metal atoms. Since the equilibrium constants are not measurable quantities, they have to be determined along with the charge transfer parameter by fitting the experimental data. Most often the data are from EMF measurements or effusion techniques and yield activity data for only one species. Activities for the other species are developed from the Gibbs-Duhem equation. The details of the numerical technique for data fitting are presented by Stoicos (1982). +
Experimental Section Activity coefficients in liquid metal systems are usually measured by either an EMF technique or by some sort of vapor pressure method. Each has advantages and limitations, but the experimental method chosen here was the Knudsen effusion technique. This method, applicable at higher temperatures than EMF measurements, involves the effusion through a very small knife-edged hole into a vacuum, of a vapor in equilibrium with a liquid. An expression based on the kinetic theory of dilute gases was originally derived by Knudsen (1909) and relates the flux from the orifice to the pressure of the vapor. In this study the flux is measured as the weight loss of the container, and the relative volatilities of the components are sufficiently different that the effusion beam contains (almost)
Coils Cells To Diffusion Pump Heot Shields (Mol
RP
Cooxiol Cable To Induction Furnace
Thermocouple Feedt hrough
Figure 5. Vacuum and heating section assembly.
only atoms of the more volatile material. The Knudsen cells were made of Aremcolox machinable ceramic from Aremco Products. This material was chemically inert to all metals studied. The lid of each cell had a small effusion hole, about 0.03 cm in diameter, and four such cells were held firmly in a graphite block, as shown in Figure 4. The block was supported in a high-vacuum chamber internally heated by an induction furnace, as shown in Figure 5. Radiation losses were minimized by multiple molybdenum heat shields. Temperatures were monitored at various points by Pt/Pt-13% Rh calibrated thermocouples, and their signals were used to control the furnace. The maximum differences in temperature at various points, or at various times throughout a run, was *2 K. The vaccum was maintained at a pressure less than Pa at all times, so that the mean free path of the effusing molecules is much greater than the dimensions of the hole. Lead metal sticks with a purity of 99.99% and copper pellets at a purity of 99.9% were obtained from Fisher Scientific. Tin metal sticks (Baker Chemical) had a purity of 99.9%. Titanium metal cylindrical rods, 99.9% pure, were used (CERAC). In each run the more volatile metal was loaded pure into one Knudsen cell, and alloys at various compositions into the other three. After the vacuum was established, the cells in the graphite block were brought quickly to temperature and held for about 1h. Then the power was cut and the cells cooled. Heating and cooling times were about 5 min, and the effusion rate is lower at lower temperatures. Moreover, to a large extent the uncertainties introduced by the transient temperature are normalized out by the pure metal reference cell.
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Table I. Experimental Results for the Cu-Pb System % change XPba T, K YPb in compnb 0.077 1276 6.79 3.3 0.147 1276 5.43 4.9 0.298 1276 3.38 6.3 0.475 1276 1.89 5.4 0.607 1276 1.34 4.9 0.731 1276 1.16 5.1 0.803 1276 1.09 5.1 0.950 1276 1.10 5.6 0.109 0.220 0.346 0.492 0.595 0.695 0.794 0.898
1373 1373 1373 1373 1373 1373 1373 1373
5.15 3.73 2.47 1.57 1.30 1.29 1.21 1.08
4.4 6.1 6.4 5.3 5.6 6.3 6.4 6.4
nAverage mole fraction before and after each run. *Percent change in mole fraction during each run.
Table 11. Experimental Results for the Ti-Sn System at 1776 K %no YSn % change in compnb 0.321 0.087 0.1 0.453 0.175 0.4 0.532 0.311 1.1 0.568 0.353 0.8 0.612 0.383 1.4 0.660 0.403 1.6 0.684 0.497 2.2 0.758 0.637 3.2 0.790 0.643 3.9 0.843 0.861 4.9 0.906 0.822 5.9 "Average composition before and after each run. change in composition during each run.
*Percent
The weight losses correspond to only a slight depletion of the material in the cell, but appropriate corrections were made for the composition changes (of only a fraction of 1%). The data were analyzed by considering the equilibrium requirements and the Knudsen equation. The activity coefficient of the more volatile component i is then given by
Table 111. Experimental Results for the Ti-Cu System at 1280 K YCU % change in compn* 0.357 0.076 0.2 0.419 0.076 0.2 0.497 0.312 1.1 0.558 0.601 2.7 0.629 0.648 3.3 0.700 0.728 4.1 a Average composition before and after each run. change in composition during each run.
Table IV. Chemical-Physical Theory Model for Selected Liquid Alloys alloy (A-B) AZa compd K Sn-Ti 0.0 TiSn 38.34 Cu-Ti 0.00673 TiCu 57.26 Mg-Sn 0.00448 MgSn 142.3 Mg2Sn 735.2 Mg-Pb 0.0179 MgPb 4.30 X Mg,Pb 53.65 Mg-Pb 0.0179 MgPb 1.06 x Mg2Pb 40.19 Mg-Sb 0.00279 MgSb 4.74 X Mg3Sb2 4.80 X Mg-Bi 0.00326 MgBi 4.36 X Mg3Biz 3.24 X In-Cu 0.0112 CuIn 0.0217 Cu21n 1.32 A1-Sb 0.00769 AlSb 4.80 X In-Sb 0.00251 InSb 2.808
*Percent
Parameters
T. K
10-3 lo3 10' lo2 lo7
lo-*
1776 1280 1073 1073 923 923 973 973 1073 1073 973 973 1073 1073 1300 900
* Charge transferred from uncomplexed metal A to uncomplexed metal B. I
f'p
0 Daia of Irwin (1976) -Chemical-Physical Theory Madel
O"7rsl
-I
u
-50
0.2
0.4
0.6
0,8
1.0
Mole Fraction of Mg
Data for the system copper-lead were taken at two temperatures and are shown in Table I. These data are in excellent agreement with existing data at other temperatures (Hultgren et al., 1973), as shown in Figure 6 of part 1, and demonstrate the validity of the technique. This system shows only positive deviations from Raoult's law. Our purpose was to show how limited data could be extended by appropriate thermodynamic models, and for this purpose we chose two titanium systems, Ti-Sn at 1776 K and Ti-Cu at 1280 K. In both cases the range of data was limited by the formation of solids. For the Ti-Sn system liquid data could be taken only for xsn > 0.32, and in the Ti-Cu system the range was 0.700 > xcu > 0.357. The results are shown in Tables I1 and 111, and both systems show relatively strong negative deviations from Raoult's law. Further details of the experimental techniques and results are given by Stoicos (1982). Applications and Discussion The chemical-physical theory model has been applied to a number of liquid metal systems that exhibit pro-
Figure 6. Chemical-physical theory applied to magnesium activity coefficient data in magnesium-antimony solutions at 1073 K.
nounced negative deviations from Raoult's law, for which there is strong evidence of compound formation. The estimated parameters for a number of such systems are compiled in Table IV. In calculating the equilibrium constants, the true activity coefficient of each compound is normalized to unity at the composition corresponding to its chemical formula, where its true mole fraction reaches a maximum. Figure 6 represents a fit of the magnesium activity coefficient data for the magnesium-antimony system, where the most stable compound is Mg3Sbz,and of course the MbSb compound is assumed as well. Figure 7 represents the fit of copper activities in the system Cu-In, using the compounds CuzIn and CuIn. In both cases shown, as well as in all of the other systems investigated, the experimental results are well represented by the chemicalphysical theory. The copper-indium system was included to show that the theory can handle both negative and positive deviations from Raoult's law within the same alloy.
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 ' I d -
0
dolo of Hullpren bo/. (1973) Model
-Chemicol- Physical Theory -
151
Ideal Chemical Theory
h
-0
0.2
0.4
06
0.8
LO
-5
0
Mole Froction of Cu
Figure 7. Chemical-physical theory applied to copper activity coefficient data in copper-indium solutions a t 1073 K.
Mole Froclion of I n
0.4
0.6
0.8
1.0
Mole Froction of Mq
Figure 10. Comparison of the chemical-physical theory with the ideal chemical theory (using the same values for K ) for the magnesium-antimony system at 1073 K.
-Chemicol-Physicol Theory
x
Figure 8. True mole fractions in the indium-antimony system at 900 K.
02
-1,5
0
0.2
0,4
OS6
Model
Oa8
1.0
Mole Froction of Sn
Figure 11. Fit of the chemical-physical theory to limited data for tin activity Coefficients in the tin-titanium system at 1776 K.
Mole Froclion of Mg
Figure 9. True mole fractions in the magnesium-antimony system a t 1073 K.
The ideal chemical theory in which the chemical forces dominate over the physical forces and the true solution is treated as ideal can predict and correlate only negative deviations and does not represent these data well. The equilibrium constants can be used in calculating the composition dependence of the various species present in a compound-formingsystem. Figure 8 shows the real mole fractions present in the In-Sb system, where the only compound formed is the 1-1 adduct, and compound formation is relatively weak, never exceeding 20% at the maximum ( x = 0.5). A similar calculation was carried out for the same system by Osamura and Predel (1977). On the other hand, Figure 9 represents the Mg-Sb system which shows much stronger solvation, and the two compounds present are taken as Mg3Sb2and MgSb, both with relatively high equilibrium constants. The true mole fraction of MgSb reaches a maximum at 50 at. % Mg with
a 36 mol % conversion. The maximum solvation occurs at 60 at. 90Mg where the true mole fraction of Mg,Sb, reaches a maximum with conversion over 84 mol % . The presence of this compound is so dominant that it depletes most of the solution constituents and causes a highly asymmetric ZMgSb curve. Figure 10 shows the influence that the charge-transfer parameter has in the determination of the activity coefficients for the magnesium-antimony system. The dashed line represents the calculated results assuming there are no physical interactions in the mixture. In this case, it can be shown that the activity coefficients of magnesium and antimony have the same values at infinite dilution
with K,, = 1 and KMgSb = 894.4. The discrepancies between &e two curves indicate that the inclusion of the charge-transfer parameter in the model affects the activity coefficients in the regions of the concentration range where compound formation is not predominant. One distinct advantage of any sort of model of this type is the ability to extrapolate very limited experimental data. Because the parameters have physical significance, one can use quite fragmentary data to evaluate them, with a good expectation that behavior can be predicted outside the range of the data. An example of such a situation is provided by the two titanium systems for which experimental data are presented. Because of freezing, only part of the composition range is experimentally accessible at reasonable temperatures. However, the data taken are quite sufficient to
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Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985 n
T h l s Work
-Chemical-Physical
-3
0
0.2
0.4
Theory Model
06
0,8
IO
Mole Fraction of Cu
F i g u r e 12. Fit o f t h e chemical-physical theory t o l i m i t e d data for copper activity coefficients in t h e copper-titanium system a t 1280
K.
Mole F r o c l i o n of Mg
F i g u r e 13. Application o f t h e chemical-physical theory t o t h e ent h a l p y of mixing.
evaluate the model parameters, and the results are shown in Figures 11 and 12. The chemical-physical approach can in principle also be used to represent enthalpy data. The physical theory parameter A 2 has been shown to be temperature independent, but the chemical equilibrium constants vary with temperature, depending on the AH associated with each K , as in the Gibbs-Helmholtz expression AH = R T l ( d TIn) K (15) The enthalpy effects for each K can be evaluated from either calorimetric data or from Gibbs energy data at multiple temperatures. Evaluation is not straightforward because of coupling between the chemical and physical effects. As an example, the K s as a function of temperature have been evaluated for the Mg-Pb system from the EMF data of Lantratov (1959), and the resulting physical-chemical prediction of the excess enthalpy is compared with calorimetric data (Sommer et al., 1980) in Figure 13. The agreement is reasonable, and the division of Gibbs energy into the separate energy and enthalpy portions is considered a good test of a solution theory.
Compared to other chemical-physical models (Harris and Prausnitz, 1969; Jordan, 1970),the present formalism is characterized by a greater degree of generality because it can account for more than one intermetallic, with independently evaluated equilibrium constants. Furthermore, the nonspecific interactions are described by a model with a physically significant rather than an empirical parameter. Acknowledgment The authors are grateful for the financial support of the Standard Oil Company (Indiana). Nomenclature a = thermodynamic activity (dw/dt)io = weight loss rate of pure component i (dw/dt)i = weight loss rate of component i from a mixture gE = excess molar Gibbs energy A H = enthalpy for compound formation K i= equilibrium constant for the formation reaction of compound i K,, = equilibrium constant of the true activity coefficients for the formation reaction of compound i m, n = stoichiometric coefficients in an intermetallic compound R = gas constant x = bulk mole fraction z = true mole fraction Greek Letters LY = true activity coefficient 7' = bulk activity coefficient Registry No. Tin 32.1-90.6,titanium 9.4-67.9(atomic)alloy, 95217-09-7; c o p p e r 35.7-70, titanium 30-64.3(atomic) alloy, 87931-81-5; copper 6-92.3, lead 7.7-95(atomic)alloy, 95217-10-0. Literature Cited Alger, M. M.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1983, 22, 249-258. Cox, K. R. Ph.D. Thesis, Unlversity of Illinois, Urbana-Champaign, 1979. Darken, L. S. Trans. Mefall.SOC.AIM€ 1967, 239, 80-89. Dolezalek, F. 2.Phys. Chem. 1908, 6 4 , 727-747. Eckert, C. A.; Smith, J. S., Irwin, R. B.; Cox, K. R. AIChE J . 1982, 28, 325-333. Gebhardt. E.; Becker, M.; Sebastian, H.; 2. Mefalkd. 1955, 46, 669-672. Harris, H. G.; Prausnitz, J. M. Ind. Eng. Chern. Fundam. 1989, 8 , 180-188. Hildebrand. J. H.; Eastman, E. D. J. Am. Chem. SOC.1915, 37, 2452-2459. Hultgren, R.; Desai, P. D.; Hawkins, D. T.; Gleiser, M.; Kelley, K. K., Ed.; "SelectedValues of the Thermodynamic Properties of Binary Alloys"; American Society for Metals: Metals Park, OH, 1973. Irwin, R. B. Ph.D. Thesls, University of Illinois, Urbana-Champaign, 1978. Jordan, A. S. I n "Calculation of Phase Diagrams and Thermochemistry of Alloy Phases"; Chang, Y. A.; Smith, J. G., Ed.; The Metallurgical Society of AIME: Warrendale, PA, 1979; p 145. Jordan, A. S. Met. Trans. 1970, I , 239-249. Knudsen, M. Ann. fhys. 1909, 28, 999. Lantratov, M. F. Russ. J . Inorg. Chem. 1959, 4 , 636-638. Osamura, K.; Predel, B. Trans. J . I . M . 1977, 18, 765-774. Predel. B. I n "Calculation of Phase Diagrams and Thermochemistry of Alloy Phases"; Chang, Y. A.; Smith, J. G., Ed.; The Metallurgical Society of AIME: Warrendale, PA, 1979; p 72. Prigogine, I.; Defay, R. "Chemical Thermodynamics"; translated by Everett, D. H.; Longmans, Green and Co.: New York, 1954. Sommer. F.; Lee, J. J.; Predel, 6.; Z . Metallkd. 1980. 71, 818-821. Steeb, S.; Entress, H. 2.Metallkd. 1988, 5 7 , 803-807. Stoicos, T. Ph.D. Thesis, University of Illinois, Urbana-Champaign, 1982. Stolcos, T.; Eckert, C.A. Ind. Eng. Chern. Fundam. 1985, preceding article in this issue. Received f o r r e v i e w July 7, 1982 R e v i s e d m a n u s c r i p t r e c e i v e d July 2, 1984 A c c e p t e d S e p t e m b e r 21, 1984
