THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ACTIVITIES IN DILUTE MOLTEN ALLOYS John M. yealy Robert D. Pehlke December, 1961 IP-543

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TABLE OF CONTENTS Page LIST OF TABLES.......................................... iv INTRODUCTION........................................................ 1 INTERACTION PARAMETER AND INFLUENCE OF TEMPERATURE.................. INTERACTION PARAMETERS IN NONFERROUS ALLOYS...................... 5 EFFECT OF TEMPERATURE............................................... 8 APPLICATIONS OF SOLUTION MODELS..................................... 8 SUMMARY..........1.........1..................................... 11 ACKNOWLEDGEMENT........................................... 11 BIBLIOGRAPHY................................. 12 REFERENCES FOR INTERACTION PARAMETERS.............................. 13 iii

LIST OF TABLES Table Page I NONFERROUS INTERACTION PARAMETERS..................... 15 II ALLOY ENTHALPY DERIVATIVES CALCULATED FROM e VS 1/T SLOPES...................................... 20 III COMPARISON OF EXPERIMENTAL RESULTS WITH VALUES CALCULATED FROM WAGNER'S RELATIONSHIP..................... 21 Figure 1 VARIATION OF E WITH THE RECIPROCAL OF T............... 22 iv

INTRODUCTION A problem which has been of considerable interest in the field of metallurgy is the prediction of the effect of one alloying element present in dilute concentration on the thermodynamic properties of another dilute alloying element. Ultimately, the prediction may be accomplished entirely from statistical mechanical considerations. At the present, however, the problem remains unsolved since liquid solution models which are simple enough for analytical treatment are inadequate for such purposes. Thus, we desire some means of organizing the information on dilute systems which will serve as a guide for further study on interactions in dilute alloys, and also provide a basis for the utilization of thermodynamic data in the engineering design of alloys. INTERACTION PARAMETER AND INFLUENCE OF TEMPERATURE Probably the best basis for the organization of experimental data is the interaction parameter concept devised by Wagner(l) This parameter is the coefficient in a Taylor series expansion for the logarithm of the activity coefficient. o in yi n Yi nn yi = nn yi + xi xi xi ++... Xi = ~ xj = (1) + ri xi(2 f2 7i) +Xi Xj (2ei ) + * +. xi Xi Xoi dax x = ~0 Xj xiO x j = -1 -

-2 - If all except the first order terms are neglected, the expression for the logarithm of the activity coefficient becomes linear with respect to the mole fractions of the solutes present in the dilute alloy. i j k ~n ni 7 +n i xi i + + xk ei +.0 (2) The interaction parameter is thus expressed as: E Qn i _l) =0 (3) xi j= (an Yi (4) xj /i =xj= 0 The experimental determination of the parameter e~, which represents the influence of alloying element i on its own activity coefficient in dilute solution, can be accompli.shed from thermodynamic studies of the dilute binary alloy, The parameter ci represents the effect' of additions 1.. of component j on the activity coefficient of component i when both are very dilute. Thermodynamic data on the dilute ternary alloy are required to compute this parameter. The inclusion of second order terms would seem to permit a better representation of-the data since a binomial expression would be fitted to the activity data rather than a linear one. Indeed, for a ternary system such a representation can be extended somewhat further away from the infinitely dilute solution than the form suggested by Wagnero For a multicomponent system, however, we must evaluate experimentally terms of the type:

-3 - r2 in yi 1 l ax8 xk J xi = Xj xK = 0 This would require quarternary data and thus defeats the engineering advantage of the simpler expression. It can be easily demonstrated from the Gibbs-Duhem equation that: EJ - e. (5) ^- J This reciprocity relationship is an important advantage of the Wagner representation. A slight modification of this representation has been proposed(2) which involves the use of weight percent as a concentration variable rather than mole fraction, and also common logarithms rather than natural ones: log fi = log f i (i)+ e i+ j) +. (6) where: ai fi =j log Fi = i <o% j i = %j =0 The reciprocal relationship for this parameter is: j M. i e. = e. (M = molecular weight) (8) i Mj 8J The relationship between the two parameters is: j M solvent j ) i (2.303)(100)Mj i

An important but previously undiscussed aspect of the use of interaction coefficients is their variation with temperature. It can easily be shown from thermodynamic relationships that this variation is given by: a EJ ^ 1 J ( _aH ) (10) 6T RT2 V x i X xj Xi j = 0 The derivative of the molar enthalpy is obtainable from heat of solution data. When the derivative of enthalpy with respect to mole fractions of the two dilute solutes is not a strong function of temperature, an extrapolation to temperatures other than those in the experimental range can be accomplished directly. If Equation (10) is rewritten in the form: 1d i= ~ (82H ') (11) xlx) Ri \xj i = 0 then a linear variation with the reciprocal of the absolute temperature is indicated. For a binary, strictly regular solution, we have: an 7i = (1-xi)2 W (12) RT where W is the solvent-solute molar interaction energy. The interaction parameter predicted from this model is: in Yi 2W G( n 7ni ) = ei = 2- - (13) Txi= 0 RT If W is a function of temperature, the effect of temperature on the parameter is: i F d R _ 2 l dW W (1 dT R T IdT T21

-5 - which can be rewritten as: i d ci =2 IT dW W () d.E 2[l.W ]w (15) dA R dT If, however, W is only a slight function of temperature, or if the extrapolation is over a small temperature interval, a proportionality with the reciprocal temperature is indicated. On the basis of these considerations, a linear l/T relationship is recommended for interpolation or extrapolation to temperatures not directly measured by experiment. The application of this relationship to the tabulated parameters will be discussed later. INTERACTION PARAMETERS IN NONFERROUS ALLOYS In 1955, Chipman(2) discussed the application of the Wagner representation to alloy steel thermochemistry and presented values for several interaction coefficients in liquid iron at 1600~C. A more complete list has recently been prepared for liquid iron alloys(3). In Table I, interaction parameters are presented for dilute liquid nonferrous solvents as calculated from experimental data in the literature. Several experimental methods have been used to measure the thermodynamic properties of liquid metallic solutions. Except for gas solubility measurements, very few of the data extended into the highly dilute region. For purposes of calculating an interaction parameter, the extrapolation to infinite dilution was attempted only where the experimental data included solutions at least as dilute as 10 mole percent. This is far from an ideal

-6 - situation, and indeed, the results represent only the authors' best estimate from available sources. Where data at more than one temperature were available, all values are given. The interaction parameter was computed as the slope of a plot of In yi versus xi in the case of binary systems to determine ei. or versus xj at xi = 0 to determine Ei in the case of ternary systems. The limiting slope of the free energy of mixing with respect to composition was also utilized to compute the parameter since data are often presented in this form. The partial molar free energy of mixing is: AGi = Gi - Gi = RT En ai (16) In the case of a binary alloy: n yi = RT [i- RT n x] (17) where the bracketed term is recognized as the excess molar free energy of mixing: 1 -ixs in 7i = T G (18) The interaction parameter is then given by: - XS - ~i (Ua n Y- im ~~lim an Gi ~~(19) i = J 1 i AGi ] ii ( i a1Txi =0 RT x O axi Extending this development to the case of ternary data, and noting that xi and xj are independent: - 0 i (20) xj-0 o

-7 - Gas solubilities are usually given as cc of gas (STP) per 100 gms of alloy versus pressure at constant temperature. In all cases where gas solubility data were analysed, the gas was diatomic, and the data followed Sievert's law. This means that the gas dissolves atomically and that the dissolved gas follows Henry's law. Of course the self interaction of any material which follows Henry's law is zero. Consequently, to get e. one need not,..,,. /S ton 7i V i take the limit as xi -O0 but can use xji Txix = at any constant value of xi within the Henry's Law region. In the case of hydrogen, for example, if the gas phase is ideal, one has P = f (= fugacity) (21) H2 H2 thus: Ej _1 i n PH2 (22) H 2 axj xXj = O,xH where "H" represents atomic hydrogen in solution. If the data are given at constant pressure we can use: a /in xH H \ x TP,xj = (23) or: ej _ L ~log (wt. H) j (24) H L 6(wt.% j) I T, P,(% j)0

EFFECT OF TEMPERATURE In several instances, data at three or more temperatures were available. Several examples are presented in Figure 1 showing the relationship between e and 1/T. Straight lines were drawn through the points in accordance with the conclusions above. The slopes of these lines were used to compute an approximate value for the second derivative of the alloy enthalpy with respect to the mole fractions of the dilute components. These values, calculated by means of Equation (11), are given in Table II. APPLICATIONS OF SOLUTION MODELS Although many solution models have been proposed, none of these has been found satisfactory for quantitative predictions of interaction parameters because of conceptual or mathematical limitations(4). Actually, our demands are reasonably simple; we do not wish to predict absolute values of the ordinary thermodynamic functions but only the effect of added components on the partial molar free energy of the others. Only two theories have as yet been applied to this specific problem. One is the free electron theory and the other is based on a nearest neighbor lattice model. Studies of gas solubilities in metal alloys led Wagner to the formulation of a free electron theory(5). The basis of the theory is that when an added element dissolves as protons and electrons, if the ion-electron interactions are much stronger than the ion-ion interactions, then the primary variable of importance is the free energy of the electrons and the effect of added elements thereon. Based on this idea Wagner has derived a relationship between ej and the self-interaction parameters, ei and e. This 1 i J -8 -

-9 -relationship is: ~ + -' z+ (25) A corollary of this result is that for a given solvent, all self-interaction parameters should have the same sign. This has'been found to be true only when deviations from ideality are large. Wagner verified his equation for the case of several alkali amalgams(6). His comparisons are included in Table III. In general, we can hope to make a reasonable prediction by means of Wagner's equation whenever the self-interaction parameters are very large and have the same sign. Comparisons between values predicted from this theory and experimental values are made in Table III. A reasonably simple model for solutions and one which is especially attractive both because of our quasi-crystalline picture of a liquid and because of the simplicity of the basic concepts involved is the chemical bond theory based on the neighbor lattice model first devised by Ising. The model was applied by Hildebrand who hypothesised the "regular solution". This development proved to be somewhat too general for accurate analysis and a later simplification, the "strictly regular solution" was discussed by Fowler and Guggenheim(7). The only case examined in the literature for such models is that of a binary non-electrolyte. The problem is somewhat more complicated when we deal with a multicomponent system. Let us consider for the moment a dilute solution of A and B in a solvent C. Possible types of pairs are: A-A, A-B, B-B, A-C, B-C, and C-C. We are first tempted to assume that the latter three types of pairs would predominate and that there would be few of the first three types of

-10 - bond. This means that nearly all solute molecules A and B are surrounded by C and that there is no direct A-B interaction. Thus, under the assumptions of the regular solution we would have no effect of A on B or of B on A. Alcock and Richardson(8 9), however, have applied a similar model to the case of a ternary alloy in which one solute is somewhat more dilute than the other. Strictly speaking, their problem is somewhat different from the one at hand, for we are interested in the interactions at infinite dilution of all of the solutes. In the development of a lattice model for this problem, all types of bonds must be taken into account. In the case of a ternary this would be 6 different ones whose relative magnitudes would be expected to determine the extent and type of clustering in the model lattice. In view of this complexity, and the assumptions implicit in the model, the obtaining of dependable quantitative results with this approach is precluded at present. Several difficulties which arise immediately and are not easily treated theoretically include the variation of the interaction energies with composition, the non-normality of the solution, the influence of composition on more than one degree of freedom, and the case where the coordination numbers of the solutes are different from that of the solvent. Hence, no attempt is being made at present to derive a suitable model on this basis. Finally, the attempt of Gokcen and Ohtani(10) to establish an empirical correlation for interaction parameters in iron on the basis of atomic number should be mentioned. The results are interesting, but without a definite theoretical base. The general utility of this approach appears to be quite limited.

SUMMARY Wagner's expression for predicting activities of dilute solutes in multicomponent systems is a simple and promising quantification of a complex metallurgical problem. The data available for its use are presently limited. Table I of this paper has summarized these parameters for nonferrous metallic solutions. The interaction parameter can be extrapolated or interpolated to temperatures other than those measured by experiment if a linear 1/T relationship is assumed. Such a procedure is indicated by theoretical considerations. The direct prediction of parameters from statistical mechanics appears to exist somewhere in the distant future. Wagner, however, has attempted to predict interaction coefficients from the self interaction parameters, for the case of certain types of solutes. A comparison of data on several systems indicates that this is a reasonable estimate whenever the values of the self interactions are large and of the same sign. ACKNOWLEDGEMENT This work was supported in part by the United States Atomic Energy Commission under Contract No. AEC-AT(ll-1)-979. -11 -

BIBLIOGRAPHY 1. Carl Wagner., Thermodynamics of Alloys, Addison-Wesley Press, Inc., Cambridge, 1952. 2. John Chipman, "Atomic Interaction in Molten Alloy Steels," Journal of the Iron and Steel Institute, June 1955, 97. 3. John F. Elliott and Molly Gleiser, Thermochemistry for Steel Making, Addison Wesley Publishing Company, In press. 4. R. A. Oriani, "Thermodynamics and Models of Metallic Solutions," The Physical Chemistry of Metallic Solutions and Intermetallic Compounds, I, (1959) H. M. Stationery Office, London. 5. Carl Wagner, "On the Solubility of Hydrogen in Paladium Alloys," Zeitschrift fiir Physikalische Chemie, 193, (1944), 407. 6. Carl Wagner, "Thermodynamic Investigations on Ternary Amalgams," Journal of Chemical Physics, 19, (1951) 626. 7. R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Macmillan Co., New York, 1939. 8. C. B. Alcock and F. D. Richardson, "Dilute Solutions in Molten Metals and Alloys," Acta Metallurgica, 6, (1959), 385. 9. C. B. Alcock and F. D. Richardson, "Dilute Solutions in Alloys," Acta Metallurgica, 8 (1960), 882. LO. N. A. Gokcen and M. Ohtani, "Thermodynamic Interaction Parameters of Elements in Liquid Iron," Transactions of the Metallurgical Society of the A.I.M.E., 218, (1960), 533. -12 -

REFERENCES FOR INTERACTION PARAMETERS 1. C. B. Alcock and F. D Richardson, "Dilute Solutions in Molten Metals and Alloys," Acta Metallurgica, 6, (1958), 385. 2. H. E. Bent and E. S. Gilfillan, "The Activity of Potassium in Dilute Potassium Amalgams," J.A.C.S., 55, (1953), 3989. 3. H. E. Bent and J. H. Hildebrand, "The Vapor Pressure of Sodium and Cesium Amalgams," J.A.C.S., 49, (1927), 3011. 4. M. B. Bever and C, F Floe, "Solubility of Hydrogen in Molten Copper-Tin Alloys," Transactions of the A.I.M.E., 156, (1944), 149. 5. R. G. Blossey, Unpublished experimental data, Department of Chemical and Metallurgical Engineering, University of Michigan, November, 1961. 6. E. Burmeister and K. Jellinek, "Vapor Pressures and Activities of a Binary Alloy," Zeitschrift Fir Physikalische Chemie, 165, (1933), 121. 7. T. Busch and R. A. Dodd, "The Solubility of Hydrogen and Nitrogen in Liquid Alloys of Iron, Nickel and Cobalt", Transactions of the Metallurgical Society of the A.I.M.E., 218, (1960), 488. 8. John Chipman, "Activities in Liquid Metallic Solutions," Discussions of the Faraday Society, (The Physical Chemistry of Process Metallurgy), (1948), 25. 9. John Chipman and J. F. Elliott, "The Thermodynamics of Liquid Metallic Solutions," Thermodynamics in Physical Metallurgy, A.S.M., 1950. 10. J J. Egan, "Thermodynamics of Liquid Magnesium-Bismuth Alloys," Acta Metallurgica, 7, (1959), 560. 11. J. F. Elliott and John Chipman, "The Thermodynamic Properties of Liquid Cadmium Solutions," J.A.C.S., 73, (1951), 2682. 12. H. Frauenschill and F. Halla, "Activities in the Ternary Liquid System Na-Cd-Hg," Zeitschrift Ffr ElectrochemieS, 53, (1949), 144. 13. F. Halla and R. Herdy, "Activities in the Liquid Ternary System NaPb-Hg," Zeitschrift ffr Electrochemie, 56, (1952), 213. 14. J. Humbert and J. F. Elliott, "The Solubility of Nitrogen in Liquid Fe-Cr-Ni Alloys", Sc.D. Thesis, M.I.T., 1960. -13 -

-14 - 15. K. Jellinek and Gustav Rosner, "On the Vapor Pressure and Activities of a Liquid Component in Binary Alloys at.Elevated Temperatures," Zeitschrift fflr Physikalische Chemie, A152, (1930), 67. 16. 0. J. Kleppa, "A Thermodynamic Study of Liquid Metallic Solutions, I, The System Lead-Gold", J.A.C.S., 71, (1949), 3275. 17. 0. J. Kleppa," ---, II, The System Tin-Gold," J.A.C.S., 72, (1950), 3346. 18. 0. J. Kleppa, " ---, III, The Systems Bismuth-Gold and Thallium-Gold," J.A.C.S., 73, (1951), 385. 19. J. Kleppa, " ---, IV, The Systems Zinc-Bismuth and Zinc-Lead," J.A.C.S., 74, (1952), 6052. 20. 0. Kubaschewski and J. A. Catterall, Thermodynamic Data of Alloys, Pergamon Press, 1956. 21. G. N. Lewis and M. Randall, "The Thermodynamic Treatment of Concentrated Solution, and Applications to.Thallium Amalgams," J.A.C.S., 43, (1921), 223. 22. W. R. Opie and J. J. Grant, "Hydrogen Solubility in Aluminum and Some Aluminum Alloys," Transactions of the A.I.M.E., 188, (1950), 1237. 23. J. S. Pedder and S. Barratt, "The Determination of the Vapor Pressures of Amalgams by the Dynamic Method," Journal of the Chemical Society (London), (1933), 537. 24. G. Spiegel and H. Ulich, "Lithium-Amalgam Electrodes in Non-Aqueous Solutions, Zeitschrift fir Physikalische Chemie, A78, (1957), 187. 25. Carl Wagner and G. Engelhardt, "Contribution to the Knowledge of Thermodynamic Activities in Binary Alloys," Zeitschrift fidr Physikalische Chemie, 159, (1932), 241. 26. Carl Wagner, "Thermodynamic Investigations on Ternary Amalgams," Journal of Chemical Physics, 19, (1951), 626. 27. M. Weinstein, "The Thermodynamic Properties of the Liquid ManganeseLead and Solid Manganese-Iron System," M.S. Thesis, M.I.T., 1959. 28. T. C. Wilder, "The Thermodynamic Properties of the Liquid AluminumSilver and Aluminum-Lead-Silver Systems," M.S. Thesis, M.I.T., 1959. 29. Henry A. Wriedt and John Chipman, "Solubility of 'Oxygen in Liquid Nickel and Fe-Ni Alloys," Transactions of the A.I.M.E., 203, (1955), 477.

-15 - TABLE I NONFERROUS INTERACTION PARAMETERS Solvent i j Temperature, ~C Ref. Aluminum Antimony Ag Cu H H H H H H H H H Si Cd Cd Pb Pb Sn Ag H Cu Cu Cu Cu H Si Si Si Si H Cd Pb Cd Pb Sn -3.1 see cu 39.0 16.6 20.1 4.3 0 11.5 6.2 4.2 1.8, see eH 1.5 2.8 2.8 0.59 2.2 1000 700-1000 700 800 900 1000 700-1000 700 800 900 1000 700-1000 500 500 500 500 905 8 * 22 22 22 22 22 22 22 22 22 *X 11 11 * 11 20 Bismuth Au Cd Cd Hg Pb Pb Pb Pb Pb Sn T1 T1 Zn Zn Au Cd Pb Hg Mg Cd Pb Pb Pb Sn Tl Tl Zn 2.1 -1.22 1.62 -0.33 0.85 1.62 1.44 0.75 0.91 -0.056 3.22 2.14 -3.3 700 500 500 321 700 500 500 665 475 330 270 480 600 18 11 11 20 10 * 11 25 25 25 27 25 19

-16 -TABLE I (CONT 'D) Solvent Cadmium i Bi Bi Hg Hg Hg Na Na Na Pb Pb Pb Pb Sb Sb Sn Sn Zn Cr Fe H H N N N Ni Si S V J i Temperature, ~C Bi. Pb Hg Hg Na Hg Na Na Bi Pb Sb Sn Pb Sb Pb Sn Zn N H Fe Ni Cr Si V H N Cu N -6.5 -3.2 3.36 2.5 -6 -6 17.8 15.8 -3.2 -4.56 0 2.86 0 -6.5 2.86 -1.5 -1.78 -19.5.53.55.09 -19.5 80 -72.09 80 -4(+2) -72 500 500 327 350 350 350 350 395 500 500 500 500 500 500 500 500 682 1600 1600 i6oo 1600 1600 1600 1600 1500 1600 11 11 20 23 12 *X 12 20 11 11 11 11 11 6 11 6 Ref. Cobalt 5* * 7 7 5 5 5 ** * Copper H S S S S S S Sn Sn Au Co Fe Ni Pt Si H 5+.3 6.9(+1) -4.8(+.8) -7.4(+.5) -6.6(+1) 9.2(+0.2) 6.9(+2) 5+.3 1100-1300 1115 1300 1300 1300 1200 1200 1100-1300 4 ** ** **x ** ** ** *

-17 - TABLE I (CONT'D) Solvent i i i Temperature, ~C Ref. Copper (cont 'd) Gold Lead Zn Zn Zn Zn Bi Pb Sn T1 Ag Ag Au Bi Bi Cd Cd Cd Cd Hg Mg Mn Na Na Sb Sb Sn Sn T1 Zn Ba Cd Ce Zn 0.58 Zn 0.72 Zn 1.185 Zn 1.40 Bi 0.1 Pb 5 Sn 38.9 T1 0.1 Ag -0.6 Ag -0.92 Au 3.2 Bi 2.6 Cd -0.86 Bi -0.86 Cd -2.6 Sb -1.6 Sn 1.55 Na -5.25 Mg 0.6 Mn 0 Hg -5.25 Na 3.6 Cd -1.6 Sb 0.16 Cd 1.55 Sn -1.17 T1 0.1 Zn -5.3 Ba 127 Na 6.0 Ce 126.9 66.0 61.0 57.0 51.0 K 73 802 727 653 604 700 600 600 700 1085 1000 600 500 500 500 500 500 500 400 855 750-1000 4oo 400 500 500 500 500 450 653 15 15 15 15 18 16 17 18 20 8 16 11 11 * 11 11 11 * 20 27 15 15 * 11 * 11 20 19 Mercury 298 550 25 281 301 527 578 0 3 * 3 3 3 2 2 K

TABLE I (CONT'D) Solvent i J i I Temperature, ~C Ref. Mercury (cont'd) K K Li Li Na Na Na Na Na Pb T1 T1 T1 T1 T1 Na T1 Li T1 Cd K Na Pb T1 Na K Li Na T1 T1 65.3 61.6 67 54.4 50.3 47.0 43.8 46.5 30.8 26.5 17.5 6.0 46.5 46.1 30.7 29.6 23.4 36.0 43.0 6 22.1 6 30.8 17.5 22.1 6.0 13.1 25 50? 200 250 310 390?? 25? 350? 25 335.5 375 350? 400 400? 400??? 325? 2 2 26 3 3 3 3 * 26 24 26 12 26 3 3 3 12 26 13 13 26 * * * * 21 26 Nickel Co Cr Fe Fe Fe H H N N 0 S H N H N 0 Co Fe Cr Fe Fe Cu 2.58 -23.6.009 -3.25 132 2.58.009 -23.6 -3.25 132 0(+1) 1600 1600 1600 1600 1594 1600oo 1600 1600 1600 1594 1600 * * * * * 7 7 14 14 29 **X

-19 - TA:BLE. I (CONT'D) Solvent i j Temperature, ~C Ref. Silver Al Al 6.4 700 28 5.6 800 28 4.6 900 28 4.22 1000 8 Cd Cd 1.47 827 20 Pb Pb -0.4 1000 8 Sodium Cd Hg 4.2 350 Hg Cd 4.2 350 9 Hg Hg 2.0 350 9 Thalium Au Au 2.6 700 18 Hg Hg 0.091 325 21 Sn Sn -1.7 325 8 Sn Sn -0.8 414 8 Sn Sn 0 478 8 Tin Au Au 5.8 600 17 Cd Cd -1.05 500 11 Cd Cd -2.36 700 20 Cd Pb 0 500 11 Cd Zn -1.65 700 Cu H see ECu 1000-1300 H Cu -9. 5 1000 4 H Cu -8.05 1100 4 H Cu -5.64 1200 4 H Cu -4.45 1300 4 Pb Cd 0 500 Pb Pb 0.65 500 11 T1 T1 -3.2 352 8 T1 T1 -1.78 414 8 T1 T1 -0.475 478 8 Zn Cd -1.65 700 20 Zn Zn -1.0 437 20 Zn Zn -0.6 684 6 Zinc Cd Cd -.33 682 6 * Calculated from reciprocity relationship. ** Values taken from (2) where x not zero.

-20 - TABLE II ALLOY ENTHALPY DERIVATIVES CALCULATED FROM e VS I/T SLOPES Solvent Element i /2H x Element -- ai jji xj i=o xj=O Ag Al Al Bi Cu Hg Hg Hg Sn Sn Tl Al H H Pb Zn Ce K Na H Tl Sn Al Co Si Pb Zn Ce K Na Cu Tl Sn 1.9 8 1.8 4.0 1.0 1.1 3.8 2.3 -3.9 -1.2 -7.5 x 104 x 104 x 104 x 103 x 104 x 105 x 104 x 104 x 103 x 104 x 103

-21 - TABLE III COMPARISON OF EXPERIMENTAL RESULTS WITH VALUES CALCULATED FROM WAGNER'S RELATIONSHIP* Solvent Metal Metal i Metal J Sb Cd Cd Pb Hg Hg Hg Hg Sn Cd Sb Bi Sn Na Na K Li Zn Pb Pb Pb Cd T1 K Tl Tl Cd i j i j (Experimental) 1.5 0.59 -6.5 -4.56 -6.5 -4.56 -1.17 -2.6 55 13 35 55 55 13 26.5 13 -2.4 -1 i (Exp) 2.8 0 -3.2 1.355 22 46.5 30.8 17.5 -1.6 ej 1Ca _. (Calc) 0.8 -5 -5 1.7 21 44 27 18.6 -1.6 * Experimental data from Table I.

II 10 9 8 7 6 5 4 3 2 0 i j Solv. Ref. 0 H Cu Sn 4 l- - — f ORDINATE FOR THESE POINTS IS ( E) * EXAMPLE DATA FROM TABLE I r r\ I 0.6 0.7 0.8 0.9 1.0 1.1 1.2 ITO ToK 1 10 FIGURE I, VARIATION OF E WITH THE RECIPROCAL OF T

UNIVERSITY OF MICHIGAN 3 9015 02844 06371111111 3 9015 02844 0637