IMRICH KLEIN,'
IRVIN M. KRIEGER, and H. BENNE KENDALL
Department of Chemistry and Chemical Engineering, Case Institute of Technology, University Circle, Cleveland, Ohio
Analytical Representation of M uI tic0 mponent Eutectic Syste ms This method has been used to calculate ternary phase diagrams for known systems of fused salts, organic compounds, and metals T H E FUNCTIONAL RELATIONSHIPS between freezing point and composition in a ternary eutectic system, customarily represented by surfaces in a threedimensional diagram. can be expressed bv quadratic and cubic functions. This type of representation can be used to predict the isothermals and the Janecke projection for a three-component system from experimental data on the three binary systems. The accuracy of the predictions using quadratic representations depends upon the extent of binary and ternary interactions; when these interactions are small, as in the hydrocarbon system studied here, the predicted behavior accords well with that observed experimentally. For those cases where the quadratic representation proved inaccurate, the use of a cubic function improved accuracy of prediction of the higher temperature isotherms but did not significantly improve prediction of the Janecke projection. When data on the ternary system were used together n i t h the binary data to fit quadratic representations to the solubility surfaces, a reasonably accurate representation of the entire
3
Figure 1. Typical condensed ternary polythermal phase diagram. Binary and ternary eutectic points are shown, together with the Janecke projection of the three-phase univariant lines on the base plane Present address, Textile Fibers Department, E. I. du Pont de Nemours & Co.: Inc., Richmond, Va.
diagram was obtained, including isothermals, Janecke projection, and the ternary eutectic point. This was true even where predictions from binary data alone were not satisfactory. Thus it is concluded that the quadratic representation is adequate for predicting behavior near the ternary eutectic when the binary systems are nearly ideal; if appreciable interactions are present, the predictions are inaccurate even when cubic functions are used. If, however, ternary data are available to fit the quadratic function, a representation which is highly satisfactory for interpolation and extrapolation is obtainable. These methods can be readily extended to systems of four or more components. Hoivever, prediction of quaternary system behavior from binary system data probably would be inaccurate, and prediction from ternary data would be satisfactory only for systems without strong interactions among the components. When quaternary system data are available to fit the quadratic functions, the representation obtained should be reasonably accurate. A condensed ternary polythermal phase diagram (Figure 1) can be represented by the equations of its surfaces, giving temperature as a function of composition variables ( 5 ) . But differential relationships derived are too involved to permit ready integration. The analysis was extended to more complicated systems where congruently melting binary compounds are formed (7-3). Recently, a simplified approximation proposed previously (6, 7) was adopted (4, where the surfaces for an n-compo-
nent system are represented by planes in n-dimensional space. When applied to a three-component system, the method involves passing a plane through the melting point of a pure component and the two adjacent binary eutectic points. The equations of the three planes thus determined are readily formulated and can be used to determine analytically the isothermals, the univariant lines formed by intersections of two planes, and the invariant ternary eutectic point. Although the use of plane surfaces is a very rough approximation, the ease with which the method can be extended to systems of more than three components makes it potentially useful in predicting the behavior of (n 1)-component systems. When applied to ternary eutectic mixtures, however, the method is overly simplified because the diagrams for these systems are curved. Also, the method does not use valuable experimental data on binary solubility, which are usually available where the binary eutectic point is known. Such data together with quadratic surfaces rather than planes to represent the curved surfaces should give more accurate estimates of ternary system behavior. Consider a ternary system containing mole fractions X I , xg,, and x3 having respective melting points TP, T ! , and T!. Further, consider only those cases in which the pure components crystallize from the melt. The equation for one of the solubility surfaces may be expressed in Taylor series form:
+
T I = T,O
+
a2ye
+
03x3
aqqxgxg
+
+ ne?x: + + ... ai&
(1)
AVAILABLE FOR ONE DOLLAR
The complete manuscript from which this article was condensed containing all equations and additional figures and text
Clip and mail coupon on reverse side VOL. 52, NO. 7
JULY 1960
619
....
Figure 2. Naphthalene-phenanthreneacenaphthene system’
__
Calculated projection, mole fraction basis. Calculated projection, weight fraction basis. Experimental Janecke projection, both mole and weight fraction basis
---_
The other two surfaces may be represented by similar equations, and if third and higher terms are ignored, the equations represent quadratic surfaces. In the quadratic relationships, all coefficients except the cross termse.g., u23-are determined from binary equilibrium diagrams. T h e freezing point depression in dilute solution suggests that an estimate of the cross term may be obtained from the relationship: =
ap3
a22
+
(2)
a33
If desired, concentrations in the ternary system may be expressed in weight fractions wl,w2,and w g . The quadratic surface equations are then : T1 =
r: + LYILL‘?
+
LYJZCQ
+
a??W;
+
+
0 ( 2 3 ~ 2 ~ ’ 3 a33LL’;
(3)
Application of the Method LVhen only binary data are available, the experimental points whose compositions lie between the pure component, i, and i j . eutectic composition may be fitted by least squares to
-
(T,-
rp)/x, =
‘ I ;
f
Uji Xi
(4)
The entire procedure was formulated as a program for the IBM 650 digital computer, so that with the data for the three binary systems as input. the ma-
Figure 3. Bismuth-tin-cadmiumsystem. Predicted Janecke projection and 205” isothermal i s obtained by fitting third order surfaces to binary system data
- - - - Janecke projection
_---Experimental values - . - . - . 205’ isothermal as
computed from second order surface
chine would punch out the coefficients in the equations for the rhree surfaces, and also the Janecke projection coordinates point by point. An auxiliary program was written to give point by point the isothermals at any desired temperature. Coordinates of the ternary eutectic points were determined graphically rather than by machine calculation. The method was applied to the systems, naphthalene-phenanthrene-acenaphthene (Figure 2 ) , and potassium-lithium-thallium nitrates. An attempt to apply the method to the ternary alloy system, bismuth-cadmiumlead, was unsuccessful. I t was thought that improved precision could be obtained by extending the Taylor series expansions to include third order terms. A digital computer program was written to obtain the coefficients of the third order surfaces and to compute the Janecke projection. The results of using this procedure, applied to the bismuth-tin-cadmium system, are shown in Figure 3. The third order program gives a distinctly better representation of the isothermal than does the second order program. The third order technique was also applied to the system of potassium-lithium-thallium nitrates, with no appreciable improvement over results with second order equations. I
I
ORDER COUPON TO: EDITOR, I/EC 1 155 Sixteenth St., N.W. Washington 6, D. C. For complete manuscript; Klein, Krieger, and Kendall Enclosed Name and title Address.......
620
cash
Figure 4. Isothermals and Janecke projection are obtained b y fitting second order equations to experimental data on the ternary system
0 check
0 money order
................................................
...............................................
INDUSTRIAL AND ENGINEERINGCHEMISTRY
I I I I I I I I I I I I I I
Ternary Systems Data \%’hen data on the ternary system are available in addition to binary solubilities, they should be utilized to determine the coefficients of the fitted surfaces. The set of solution compositions from which component 1 can crystallize, with their respective initial freezing points, can be fitted to Equations 2 or 3 by the least squares technique which leads to a set of six simultaneous linear equations. A program for the IBM 650 computer was written to find the coefficients in the linear equations. At least six points are required, and at least one point must be a ternary rather than binary mixture. The fitting procedure was applied to data (8) for the bismuth-tin-cadmium system (Figure 4). Nomenclature a, a = coefficients in eq. of surfaces T = temperature TO = melting point temperature x = mole fraction zt = weight fraction Subscripts 1. 2, 3, i , j = components in systems
ternary
literature Cited (1) Grigoriev, A. T., Doklady Akad. Nauk S.S.S.R. 83,231-4 (1952). (2) Zbid.,84, 989-92 (1952). (3) Itid., 85, 1281-4 (1952). (4) Palatnik, L. S., Kopeliovich, I. M., J. Phys. Chem. (U.S.S.R.) 30, 1948-52 (1956j. (5) Roozeboom, H. W. B., “Die Heterogenen Gleichgewichte,” vol. 111, Sect. .2, Friedrich Vieweg Br Sohn, Braunschwelg, Germany, 1913. (6) Saurel, P., J . Phys. Chem. 6 , 261-4 (1902). (7) Zbid., pp. 313-20. (8) Stoffel, A , , Z . anorg. Chem. 53, 167, (1907). RECEIVED for review August 14, 1959 ACCEPTEDMarch 9, 1960 Work sponsored financially by the North American Coal Co.
