THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING SOLUTE INTERACTIONS WITH ZINC IN DILUTE SOLUTION WITH MOLTEN BISMUTH S i_ ^;.j ~ t,t,'C''' " ",.::. Jeremy VG' k'c * *,,; A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1965 May, 1965 IP-707

ACKNOWLEDGEMENTS The author would like to acknowledge with gratitude all those who directly or indirectly aided in the conduct of the research and the preparation of this thesis. Special thanks are offered to the following: The Doctoral Committee: Professor Ro Do Pehlke, Chairman and Director of the sponsoring research project, for offering the opportunity to perform this research, for helpful suggestions, and for critical discussions during the investigation. Professor Eo Eo Hucke for his tutelage and valuable discussions of thermodynamics in metallurgy. Professors R, Eo Balzhiser, Mo Jo Sinnott, and E. Fo Westrum for their advice, cooperation, and service on the committee. Dr, Do Vo Ragone for encouragement of these graduate studies. Mr. P. Do Goodell for many valuable discussions and for invaluable assistance during the hectic moments at the start of each experimental run. Mro Weldon Daines and Mr. Robert Moore for spectrographic analyseso Dro Co Wo Phillips of the Ford Motor Co. for arranging donation of goldo Mro Wo M, Boorstein for basic construction of the experimental apparatus My fellow graduate students for helpful and interesting discussionso ii

The Atomic Energy Commission for financial support under Contract AT(1-1)1352. The Industry Program of the College of Engineering, University ~of Michigan, for reproducing this thesiso My wife and children for patience and encouragement throughout the graduate studies and experimental program, iii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS o o o o o o o o o o o o o o o o o oo o o o o o o o o ii LIST OF TABLES o o o. o.. o o o o o o o o... o.. o.. o o o, vi LIST OF FIGURESo o o o.,.o..o o.o.............o....o.o... vii NOMENCLATURE,,,......o o o..... o ooo.ooo...,........ x xii ABSTRACT.....oo oo.. o o....... oo......................o........o ix I. INTRODUCTION o.................................. o....... ooo. 1 II, REVIEW OF THE LITERATURE....................... 4o A, Concept and Experimental Development of the Interaction Parameter,.,......o....oo.oo... o..o.o......o.oo oooo o.. 4 Bo Theoretical Studies of Solute Interactions.,.o.,,,....... 9 C, Experimental Methods. o............ o.................o.. 12 Do Previous Studies of Binary and Ternary Systems Involving Zinc and Bismuth,...........o................ O.......... 13 III. EXPERIMENTAL PROGRAM..........o.............................o 16 A, Experimental Design...o ooo............................ 16 Bo Experimental Procedures.................................. 49 IVo EXPERIMENTAL RESULTS.....o0......0 0000...0..........000 0.... 75 A, Zinc-Bismuth Binary System o....,................oo 75 B, Zinc-Bismuth-j Ternary Systems...o..o.....o o. o. o o. o o 90 C. Prediction of Zinc Activity in Multi-Component Solutionso...........O................o.o.....o. 13 4 Do Confirmation of Basic Assumptions.....................o..o 151 V o DISCUSSION, o o. o o o...., OO o. o o o... o o o o o o o o o 166 Ao Rationale for Observed Interactions in Ternary Alloys.... 166 B, Prediction of Ternary Interactions from Simple Solution Models........OOOOOOOOOOOOOOO.OOO..OO..OOOO..OOO.OO...O.. 217 C, Validity of Wagner's Prediction Model for Multi-Component Solutions.....o o o o o o o o o o o o o o o o o o o o a o o......o o o 2 9 Do Limitations of Experimental Results, o o, o oo o......, oo 246 E o Suggested Further Research,.o o.o, o.o o....oooooO........o 247 iv

TABLE OF CONTENTS (CONT'D) Page VI. SUMMARY AND CONCLUSIONSo................ o o.............. 250 APPENDIX A - Experimental Data for Multi-Component Alloy Studies,..o. 00...0000.. 0....000 00..................... 255 APPENDIX B - Alternate Derivations of the Temperature Dependence of Interaction Parameters............................. 257 APPENDIX C - Interaction Parameter Determination by Linear Regression Technique.o..... 0....................... 2 o.o. o o 263 REFERENCES o oo..,,0 ooo o.oo 00000............. oo0oo00 0.ooo00.. o.o 271 V

LIST OF TABLES Table Page Io Sources of Error in Galvanic Cell Determination of Activityo..........oo oooooo... 000000000......... 00000 30 II. Standard Free Energy of Formation for Various Metal Chloridesoo..o.o.oo.o.o.oo.o.oO..oo.oo.o.o.o.o.o.o.o.o.o.o.o 40 IIIo Minimum Values of AG~ or AG~ for One Per Cent Dis placement Error in Measured El;trode Potentials........... 43 IV. Section of Periodic Table Incorporating Suitable Solutes and Solvents for Galvanic Cell Studies of Activity at 450 to 650~C 00000000000... 000000.000000 00000000000000 00 a 0 45 V. Experimental Materials........... 60 V o Experimental Materials.000000000000000,00000000000,000000000 60 VIo Experimental Data for Binary Bismuth-Zinc Alloys............ 77 VIIo Summary of Binary Electrode Potentials and Activity Coefficients for Bismuth-Zinc Alloys........0o.o.000.0.0.000.0000. 88 VIIIo Experimental Data for Bismuth-Zinc-j Ternary Alloysoo....oooo 92 IXo Summary of Interactions with Zinc in Dilute Solution with Molten Bismutho.. o. o 00. o o o o o o.0 o o o 96 X, Temperature Dependence Constants for Interactions with Zinc in Dilute Solution with Molten Bismuth in the Range 450 to 650~C.oo000 oooooOoOOo.oo00000000000000000oo.00000000o.oo o o o 127 XIo Results of Multi-Component Solution Studies of Additivity Hypothesis.................................................. 143 Hypothesis, 0000000.00000000000.00000000000 00 0 0 0 00 0 0 00 000 1 0 0 0 0 0 XII. Results of Chemical Analyses of Alloy Electrodeso o..o o o o 162 XIIIo Electronegativity Values for Elements Interacting with Zinc in Bismuth,.. o o o o o o.oO 182 XIVo Factors for Thermodynamic Evaluation of Zinc-j Binary Systems, 00 o000 0 00 ^.O. 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 o0 0 0 197 XVo Comparison of Observed and Predicted Ternary Interaction Parameters Using Simple Solution Models. 0...o00000000.0.0,. 224 XVIo Quantitative Comparisons of Multi-Component Interaction Prediction Models0.....o, o oooo oo oooooo o.oo.o.oo.oooo oo 242 vi

LIST OF FIGURES Figure Page 1o Methods of Evaluating Interaction Parameters;..............0o 17 2o Binary Phase Diagrams of the Bismuth-Zinc and Lead-Zinc Systems in the Zinc-Dilute Region............................ 46 3o Galvanic Cell Apparatus.............................o...... 50 4o Schematic Diagram of Multi-Electrode Galvanic Cell.......... 52 5o Schematic Diagram of Temperature Control and Potential Measurement Circuits......................................... 55 6, Schematic Diagram of Atmosphere System....................... 57 70 Activity of Zinc versus Mole Fraction of Zinc in Bismuth at 550~C................................................... o 79 8, Activity of Zinc versus Mole Fraction of Zinc in Bismuth at 600 ~C................................................ O. 80 9o Slope of EMF-Temperature Curves versus Mole Fraction of Zinc in Bismuth,.. 81 10o Determination of Zinc Self-Interaction in Dilute Solution with Molten Bismuth........................................ 84 11. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Copper - for Indicated Constant Mole Fractions of Zinc....................................................... 106 12, Determination of Cu-Zn Interactions in Molten Bismuth..o..o. 107 13 Temperature Dependence of First-Order Cu-Zn Interaction Parameter.................................................. 107 14, Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Gallium - for Indicated Constant Mole Fractions of Zinc.................o.................. 108 15 Determination of Ga-Zn Interactions in Molten Bismuth,...... 109 16o Temperature Dependence of First-Order Ga-Zn Interaction Parameter........................................... 109 vii

LIST OF FIGURES (CONT'D) Figure Page 17o Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Silver - for Indicated Constant Mole Fractions of Zinc....................................... 110 18. Determination of Ag-Zn Interactions in Molten Bismuth,...... 111 19o Temperature Dependence of First-Order Ag-Zn Interaction Parameter....................................o 111 20, Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Cadmium - for Indicated Constant Mole Fractions of Zinc................................... 112 21, Determination of Cd-Zn Interactions in Molten Bismuth....... 113 22. Temperature Dependence of First-Order Cd-Zn Interaction Parameter....................................... 113 23. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Indium - for Indicated Constant Mole Fraction of Zinc........................................................ 114 24. Determination of In-Zn Interactions in Molten Bismuth,....... 115 25o Temperature Dependence of First-Order In-Zn Interaction Parameter................................................... 115 26. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Tin - for Indicated Constant Mole Fractions of Zinc......................................... 116 27. Determination of Sn-Zn Interactions in Molten Bismuth,.....,, 117 28. Temperature Dependence of First-Order Sn-Zn Interaction Parameter........................,.................... o. 117 29, Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Antimony - for Indicated Constant Mole Fractions of Zinc.......o.......o......................................... 118 300 Determination of Sb-Zn Interactions in Molten Bismutho...... 119 31o Temperature Dependence of First-Order Sb-Zn Interaction Parameter,......o o.......o...o.o.o.....o.o.............. o 119 viii

LIST OF FIGURES (CONT'D) Figure Page 32. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Gold - for Indicated Constant Mole Fraction of Zinc...............O.... o............. 120 33. Determination of Au-Zn Interactions in Molten Bismuth....... 121 34, Temperature Dependence of First-Order Au-Zn Interaction Parameter................................o........., 121 35. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Mercury - for Indicated Constant Mole Fractions of Zinc O o...........OO. o.................O O.. o 122 36. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Lead - for Indicated Constant Mole Fractions of Zinc oooo..o,.,..O.o.oooo..oo.... o.00.o.o.o. o...o...o.o.oo.... 123 370 Determination of Pb-Zn Interactions in Molten Bismutho,,,,.,e 124 38, Temperature Dependence of First-Order Pb-Zn Interaction Parameter. o o o o O o o o... o. o o o o o o. o o.o o o. o. o 124 39. Summary Plot of Temperature Dependence of Interaction Parameters.. o.oo.o.oo.o.o.o.o..o.o....oooo.ooo 126 40. Electrode Potentials versus Temperature for Bi-Zn-Au Alloyso, 130 41. Isotherms for Bi-Zn-Au System Showing Liquidus Boundaries in the Bismuth Corner.....o.......o..o.o.o................ooo 131 42, Electrode Potentials versus Temperature for Bi-Zn-Cu Alloys.. 132 43, Isotherms for Bi-Zn-Cu System Showing Liquidus Boundaries in the Bismuth Corner.........,.o.......o. o.,.,....o...... 133 44. Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quaternary Alloys of Bi-Zn-AgSb - Comparison of Observed and Predicted Values.............. 144 45o Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quaternary Alloys of Bi-Zn-InSb and Bi-Zn-Ag-Pb - Comparison of Observed and Predicted Value s.................................................., 145 ix

LIST OF FIGURES (CONT'D) Figure Page 46. Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quinary Alloys Based on Bi-Zn - Comparison of Observed and Predicted Values......... 148 47. Schematic Diagram of Electrode Potential Behavior During the Faraday Yield Experiment............................. 156 48. First-Order Zinc-j Interaction Parameters versus Atomic Number of j - Correlation by Period........oo................. 168 49. First-Order Zinc-j Interaction Parameters versus Atomic Numof j - Correlation by Sub-Group........................... 169 50. Second-Order Zinc-j Interaction Parameters versus Atomic Number of j - Correlation by Period...................o 170 51. First-Order Zinc-j Interaction Parameters versus Second-Order Zinc-j Interaction Parameters......................... 171 52. First-Order Interaction Parameters for Zinc-j versus Electronegativities of the Added Solute j................. 183 53. First-Order Zinc-j Interaction Parameters versus Atomic Volume of Added Solute j................................... 187 54. First-Order Zinc-j Interaction Parameters versus Atomic Radius of Added Solute j.................................. 188 55~ First-Order Zinc-j Interaction Parameters versus Ionic Radius of Added Solute j..................................... 189 56. Darken-Gurry Plot of Atomic Radius and Electronegativity for Zinc, Bismuth, and Added Solutes............................ 190 57. Electronegativity-Size Factor Correlation Attempts for FirstOrder Zinc-j Interaction Parameters....................... 192 58. First-Order Zinc-j Interaction Parameters versus Solubility Parameter of Added Solute j................................ 195 59. First-Order Zinc-j Interaction Parameters versus Excess Partial Molal Free Energy of Mixing of Zinc for Equimolar Mixture of Zinc and j.................................... 200 x

LIST OF FIGURES (CONT'D) Figure Page 60. First-Order Zinc-j Interaction Parameters versus Partial Molal Heat of Mixing of Zinc in Equimolar Mixture of Zinc and j................................................ 201 61. First-Order Zinc-j Interaction Parameters versus First and Second Ionization Potentials of Bismuth, Zinc, and Added Solutes j.............................. 204 62. First-Order Zinc-j Interaction Parameters versus Work Function for Bismuth, Zinc, and Added Solutes j................. 206 63. First-Order Zinc-j Interaction Parameters versus Standard Free Energy of Formation of Chlorides at 550~C............... 207 xi

NOMENCLATURE A, A Intercept in slope-intercept equations, B' Slope in slope-intercept equations Cp Heat capacity at constant pressure Observed electrode potential difference between alloy and standard electrodes Jr Faraday's constant, 23,060 calories per volt G Gibbs free energy H H - TS re. C- Molar excess free energy of mixing G Excess partial molar free energy of mixing of i Gi - Gi 4_ "Extra" excess partial molal free energy of mixing of i due 6(j) to addition of j OG Partial molal free energy of i ZGo Partial molal free energy of i at high dilution of i C7o GL' Molal free energy of pure liquid i Adz+ Standard free energy of formation /No X Minimum difference in standard free energy of formation to limit displacement error to one per cent of measured potential 1J4 H j~ Enthalpy and related enthalpy quantities defined similar to 11 vJ those for G Activity coefficient factor appearing in Equation (68) Gas constant, 1,987 cal/mole/~K S S) etcEntropy and related entropy quantities defined similar to those i for G T Temperature, ~K 2V Molar volume of i xii

M Average molar volume of mixture WVj Interchange energy between i and j X Electronegativity 7 Coordination number in solution QL Thermodynamic activity of component i e Electron ai ~L Activity coefficient of component i (ii) (weight) I (as subscript) Primary dilute solute j (as subscript) Added dilute solute k (as subscript) Solvent At Natural logarithm k 9 Common logarithm rm (as running index) Number of additional solutes besides primary In Number of electrochemical equivalents, also valence n (as subscript) Additional solutes n Number of bonds between two components in solution Mole fraction Valence electrons jPp Exponent, e L J aj Binary solution of i and j; i in j i (jk)(J4k)Ternary solution; i dilute in j and k A Difference a. 6 Activity coefficient: - x.i v?~ Activity coefficient at infinite dilution, the Henry's Law " Constant xiii

S Solubility parameter Volume-weighted solubility parameter in ternary solution Self or binary interaction parameter ( < 6 Ternary interaction parameter (first order) ~~ j - Ternary interaction parameter (second order) ac; / [ "Extra" excess partial molar enthalpy of i "f Extra" excess partial molar entropy of i: First-order contribution to q ~j First-order contribution to a I Second-order contribution to q ds Second-order contribution to a ( Volume fraction Me Chemical potential of electrons /71 Chemical potential of ions 1- / (as subscript) First order effect T[ 2- (as subscript) Second order effect xiv

ABSTRACT An investigation was made of the effect of small amounts of added solutes on the thermodynamic activity of zinc in dilute solution with molten bismuth in the temperature range 450 to 650~C. The investigation was designed to test two hypotheses: that interaction effects in ternary systems are periodic with the atomic number of the added solute; and that ternary effects may be used to predict the activity in higher-order solutions. The experimental measurements of activity were made in a multi-electrode galvanic cell apparatus using a fused LiCl-KCl electrolyte, The initial measurements defined the activity of zinc in binary solution with bismuth, Thermodynamic interactions with zinc were determined for ten Group-B elements from the 4th, 5th, and 6th periods of the Periodic Tableo The activity of zinc was then measured in higher-order solutions through septenary, made by alloying various combinations of these solute elements to the basic solution of zinc in bismuth. The prediction of the ternary effects was considered on the basis of periodicity, alloying criteria, thermodynamic factors, and the application of solution theories. The activity of zinc in the binary solution obeyed Henry's Law to at least o050 mole fraction. The addition of lead, gallium, or indium increased the activity of zinc, Cadmium or tin additions slightly xv

decreased zinc activity. Copper, mercury, silver, or antimony caused moderate decreases while gold strongly decreased the zinc activity. The interaction effects for copper, gold, silver, and antimony had a linear dependence on reciprocal absolute temperature, while the effects caused by the other elements were essentially independent of temperature. The temperature dependence is related to additional contributions to zinc entropy and enthalpy caused by the solute addition. The ternary effects were periodic except for tin and antimonyo Consequently, the hypothesis of purely periodic behavior was rejected. A semi-quantitative explanation of the effects was possible with a freeelectron model where relative electronegativities are used to express changes in the electron/atom ratio caused by the solute additions. The thermodynamic behavior of the binary solution between zinc and the added solute also accounted for much of the ternary effect. The regular solution model correctly predicted half of the ternary interactions while random solution and quasi-chemical models correctly predicted almost all cases to which they could be appliedo The prediction of zinc activity in quaternary and higher-order solutions was found to be excellent using a phenomenological model based on the ternary effects. The linearity of the results for the ternaries permitted these predictions to be carried to relatively concentrated solutions with fair successo The position of the liquidus phase boundaries was determined in a portion of the bismuth corner of the systems Bi-Zn-Cu and Bi-Zn-Au, xvi

I, INTRODUCTION The formulation of quantitative thermodynamic relations for metallurgical processes such as alloying, refining, electrolysis, or diffusion, or for experimental situations where a solid or liquid metal alloy is affected by its environment, rests on defining the free energy change. This may be in explicit terms, such as a reaction constant or a chemical potential, or implicitly, such as a solubility. The general procedure is to formulate an expression in terms of the standard free energies plus a means to "correct" or account for the fact that the reacting species may be present in some condition other than their standard state. The corrections are made using the thermodynamic activities of the products and reactants so defined that the activity in the standard state is unity, However, it was observed experimentally that constituents not directly concerned in a reaction may also affect its equilibrium or kineticso For example, to take two cases from ferrous metallurgy, manganese affects the deoxidizing power of silicon in molten iron or chromium may affect the carbon-oxygen reaction. Recognition of such effects helped give impetus to an intensive study by a number of experimenters of means of representing the thermodynamic properties of multi-component systems and, in particular, the interactions between elements present at dilute concentrations. A great amount of experimental data has been obtained for ferrous systemso -1

-2The practical uses of such formulations in ferrous and nonferrous netallurgical practice are manifest: the physical chemistry of steelmaking, the refining of ores, corrosion involving liquid metal coolants, multi-component diffusion, etc; however, such information is also necessary for quantitative understanding of alloy phase diagrams and the physical structure or constitution of metallic phases and solid or liquid solutions. The approaches to obtaining such information may be phenomenological, ie., "what mathematical function adequately describes observed behavior and permits extrapolation to new situations?", or theoretical, such as an explicit model for solution behavior. Ideally, both approaches should eventually convergeo The most accepted and useful means for organizing interaction behavior in dilute solutions is Wagner's concept of the interaction parameter. From the development of this concept a phenomenolgical model has arisen in the literature with two major hypotheses that have not as yet been fully and systematically investigated. To summarize them briefly: (1) The activity of a given solute in a multi-component system may be obtained by summing effects of interactions with each of the additional solute elements taken as individualso This arose from Wagner's proposal that a Taylor series be used to represent In yi, where yi a. 1 is the activity coefficient defined as The interaction parameters Xi are the coefficients of the series, (2) For interactions occurring with a solute i in a given solvent k, the direction and magnitude of the interactions caused by an additional solute j may be a periodic function of the atomic number of j o

-3The first hypothesis can be termed "additivity" and the second termed "periodicity". The purpose of this investigation was to test these hypotheses in a non-ferrous system where all the dilute solutes were metallic. The experimental alloys are not particularly useful in a practical sense but were chosen for experimental convenience and the requirement that periodic variations of the added solute element must be possible. The quantity measured was the activity of zinc in dilute solution with molten bismuth in the temperature range 450-650~C, A multi-electrode galvanic cell apparatus was used. The hypothesis of periodicity was investigated by studies of ternary additions of 10 Group-B elements from the 4th, 5th, and 6th periods in the Periodic Table, Following the determination of the ternary interactions, the activity of zinc was measured in higherorder solutions through septenary, made by alloying various combinations of the previously studied solute elements added to the basic solution of zinc plus bismuth, The hypothesis of additivity was investigated by comparing the observed activity coefficients with those calculated by summing the effects of the individual solute elements; The experimental results from ternary alloys were studied in terms of the periodicity, thermodynamic factors, and alloying considerations as related to Wagner's electron model for solution interactions. The applicability of several simple solution theories and models of interaction behavior was testedo In particular, attention was given to the methods of Alcock and Richardson (17,27) and Wada and Saito,(15)

IIo REVIEW OF THE LITERATURE The literature is reviewed in terms of the historical development of the interaction parameter concept and the available experimental studies. The theoretical background of solution thermodynamics and experimental methods is reviewed, A summary is given of previous thermodynamic studies of the alloy systems studied in this investigation. Ao Concept and Experimental Development of the Interaction Parameter The question of solute interactions in ternary liquid metallic solutions has been a matter of specific experimental and practical concern for approximately 25 years. Richardson(l) attributed the first experimental studies (carbon and oxygen in liquid iron) to Marshall and Chipman's 1942 studies, (2) although he noted that Sievert's 1910 measurements on gas solubilities were made in ternary solutions but no attempt was made to express the results in dilute alloys in terms of the interactions between solutes, Chipman and co-workers continued experimental and other studies of ternary interactions. In 1949, Chipman and Elliott(3) showed that Korber's work on reactions of molten iron-manganese alloys with silicate slags could be interpreted to show the effects of various alloying elements on the activity of silicon. By 1951, Chipman and a series of coworkers had accumulated enough of their own data, supplemented by the results of others, to present relations for the effects of several elements on the activity coefficient of sulfur or oxygen in liquid iron,(4) -4

-5fi The method used was to plot log - versus,j, where fi is the f activity coefficient of i in the ternary with j and fl the activity coefficient in the corresponding binary alloy of the same content of 1 0 The accepted present-day concepts of accounting for solute interactions stem from Wagner's suggestion in 1952, (5) that a Taylor series could be used to express the partial molar free energy of a solute or else the logarithm of the activity coefficient of the solute. If the series for the logarithm is expanded about the point xi = 0, the following expression is obtained: 2X(X) ( /( ),, a,.... (. + — (1) (all derivatives at infinite dilution with respect to solutes) The partial differential coefficients of the series thereby explicitly express the effects of the various additional solute elements on the activity coefficient of the primary solute. If, as the expression is usually formulated, the second-order and higher terms are neglected, the equation becomes a linear function of the mole fraction of the various solutes: Y p l' XJ +, X t,., + XJ T T,'L expressi o s rh-ofdi(2) This expression summarizes the hypothesis of additivityo

-6The partial differential coefficients of the truncated series are termed the interaction parameters and are defined as: =6 se/Cf /rnte-ra.ta parct te. r a i -t-Ij (2a) (C. {= lrk ltraCTO pcd e~r_ ( a-X )- (2b) Since the excess free energy of mixing of component i is.-xs _Xs RT~nyi(76)the interaction given by the thermodynamic relation AGi. RTlny 7 the interaction parameters can directly show the change in excess free energy occasioned by the addition of the various solutes. In the region where Henry's Law applies and the activity is directly proportional to concentration, 0 i lnyi = lny = constant and hence E - 0. Wagner further showed from the definition of the partial molar free energy, Gi = i) nj, that the Maxwell relations of thermoani nj,.^ dynamics could be used to show in infinitely dilute solution that 4. E- e (3) 6 J This expression has been termed the reciprocity relation and has the advantage that the effects of i on j and j on i are determined from the same set of data.

-7In 1955 Chipman(6) showed that his earlier concepts were equivalent to Wagner's approach and presented an extensive determination and compilation of interaction parameters between various metals and carbon, oxygen, nitrogen, silicon, and sulfur in liquid irono As more data became available for these and other elements, this compilation for a liquid iron solvent has been revised and up-dated. The latest form was presented by Elliott, Gleiser, and Ramakrishna(7) in 1963o A considerable amount of experimental effort has been expended towards defining interaction effects in molten irono Although this listing is by no means all-inclusive, some of the more important investigators have been Chipman, Elliott, and co-workers in the United States, Turkdogan and co-workers in Great Britain in the 1950's, Schenck, Neumann, et al in Germany, and a number of researchers in Japano Additional summaries and theoretical interpretations of experimental results in liquid iron have been made by Ohtani and Gokcen(8) in 1960 and Wada(9) in 1964o Evidence for periodic variation of the interaction effects in liquid iron with the atomic number of the added solute was cited by Turkdogan, (10) Wada, (9) Fuwa and Chipman, (11) Neumann, Schenck and Patterson, (12) and Ohtani and Gokcen(8) among others. Schenck, Frohberg, and Steinmetz(l3) and Daines and Phelke(l4) noted a comparable effect for ternary metallic additions on carbon dissolved in molten cobalto However, all these authors were concerned with cases where the primary solute was carbon, sulfur, nitrogen, oxygen, or hydrogen - all gases or small-sized non-metallic elements, Wada and Saito(15)

-8and Wada(9) collected the limited data for cases where two metal solutes were dissolved in liquid iron, but the data were too sparse to indicate if consistently periodic behavior existed. Whereas a considerable amount of data are available for ferrous systems, the available information for non-ferrous systems is more limited, This is not to say that the thermodynamic properties of nonferrous systems have not been investigated; there are a great many such studies reported in the literatureo Compilations and critical evaluation of binary data and a limited amount of ternary data have been made by Kubaschewski and Catterall, (53) Kubaschewski and Evans(40) and Hultgren et alo(54) However, in most cases the data are unsuitable for the calculation of interaction parameters since the dilute solution region was not adequately covered. Dealy and Pehlke(l6) presented in 1963 a compilation of non-ferrous interaction parameters for those cases where data in the literature were complete enough to justify extrapolation to infinite dilution. Only experimental data were used that included solutions at least as dilute as 10 mole per cent and it was commented, "This is far from an ideal situation, and indeed, the results represent only the authors' best estimate from available sources," Values for only about 25 ternary systems were given together with a much greater number of values for self-interaction parameterso Experimental studies specifically aimed at determining ternary interactions in non-ferrous systems at elevated temperatures have been quite limitedo Wagner(23) studied interactions in mercury amalgams, but these were conducted at low temperatures. At higher temperatures,

-9Alcock and Richardson(17) studied interactions with sulfur in several non-ferrous binary systems. Balzhiser(l8) investigated third-component interactions with the uranium-bismuth system. Obenchain(l9) studied third element interactions with bismuth-aluminum and lead-aluminum alloyso Wilder and Elliott(20) reported interactions in the aluminum-bismuth-lead system. Pehlke and co-workers(21,63) have recently conducted studies of interactions in ternary non-ferrous systems. A limited test of the additivity hypothesis was recently made by Okajima and pehlke(73) for multi-component additions to a solution of cadmium in lead. The ternary interaction parameters used to predict the multi-component effects were taken from the compilation of Dealy and Pehlke(l6) and it was stated that a quantitative evaluation of the additivity hypothesis in a non-ferrous system could not be made because of the limited accuracy of the available ternary interaction parameters. It was mentioned that a few tests were reported in the literature for ferrous systems involving sulfur, carbon, or nitrogen as the primary solute, Bo Theoretical Studies of Solute Interactions Paralleling the phenomenological and experimental studies of solute interactions has been an extensive theoretical effort aimed at understanding the nature of the interactions and developing means for their prediction. Much of this has necessarily rested on the general development of thermodynamic theories of solution behavior. Although other parameters have been proposed, the most commonly accepted ones are Wagner's which derive from the truncated Taylor series

-10for the natural logarithm of the activity coefficient. Schenck, Frohberg and Steinmetz(l3) proposed an "efficiency" parameter and Turkdogan proposed a solubility difference function, (10) both of which find use where the interaction is measured as a solubility effect. Ohtani and Gokcen(8) proposed parameters which are not limited to the case where all solutes are dilute. The various parameters can be shown to be related by appropriate correction terms(ll'3) and most reduce to Wagner's parameters in the case where all solutes are dilute. The temperature dependence of interaction parameters has recently been discussed by Dealy and Pehlke(l6) and by Chipman and Corrigan.(22) The basis for theoretical study of the interaction parameters is the fact that excess free energy is given by the equation AGi = RTlnyi. Hence, any thermodynamic formulation to explain partial solution behavior of component i should be differentiable with respect to the mole fraction of j to obtain the interaction parameter. Conversely, the adequacy of any solution theory can be tested by comparing observed interaction effects with the predicted effects. The theoretical approaches to the interaction effects may thus be divided into cases where the interaction is considered directly or where it follows as a consequence of some solution model. Specific discussion of interaction theory on a physical basis has been made by Wagner(5'23) and Himmler(24) in terms of free electrons. The activity coefficient of a solute is expected to be increased by an additional solute if both of them change the electron/atom ratio in

-11solution in the same direction. If the electron/atom ratio is changed in opposite directions, the interaction is expected to be negativeo Periodicity of interaction behavior was discussed by Schenck, Frohberg, and Steinmetz(l3) in terms of valency electrons and the competition between the solutes and solvent for electrons to fill their unfilled inner shells. Other recent discussions of solution behavior in terms of electronic constitution and physical parameters have been made by Kleppa(25) and by Laurie and Pratt(26) Many of these concepts follow the lead of Wagner(5'23) from his series of papers and his classic book, "Thermodynamics of Alloys)" Somewhat more consideration has been given to the interaction consequences of solution models, Alcock and Richardson(l7'27) and Wada and Saito(l5) have derived specific expressions for the calculation of interaction parameters from statistical or chemical approaches to pairinteractions in solutions. Bonnier and co-workers(29'30) have considered similar models for solution interactions. The general applicability of some simple models to metallurgical solutions was recently discussed by Oriani(30) and Oriani and Alcock31) Additional general comments on common solution theories have been made by Richardson.(1'32) Although the general question of solution thermodynamics and exact solution models continues to be a question occupying metallurgists, physicists and physical chemists (as evidenced by theoretical discussions in such journals as Journal of Chemical Physics, Acta Metallurgica, Journal of Physical Chemistry, etco), the application to the metallurgical field continues to be in terms of the less sophisticated (and more practically useful)

-12solution modelso These consist of the regular solution discussed by Hildebrand and Scott,(33,3 ) the sub-regular solution model of Hardy(35) which has been applied to ternary systems by Yokokawa, Doi, and Niwa,(36) and the quasi-chemical model of Guggenheim,(37) This latter formed the basis for Wada and Saito's and Alcock and Richardson's approaches, Lumsden(38) also examined the thermodynamics of metallic alloys in terms of their experimental behavior and the statistical mechanics of liquid solutions The status and prospects for solution thermodynamics has been discussed periodically in conferences and symposiao In particular, the 1948 Discussion of the Faraday Society, the 1949 Seminar on Thermodynamics in Physical Metallurgy of the American Society for Metals, the 1958 Symposium on The Physical Chemistry of Metallic Solutions and Intermetallic Compounds of the National Physical Laboratory (Great Britain), the 1961 International Symposium on the Physical Chemistry of Process Metallurgy (Pittsburgh), and the 1964 Conference on Applications of Fundamental Thermodynamic Principles to Metallurgical Processes (University of Pittsburgh), have yielded valuable information on the uses and limitations of solution theories and their application to dilute solute interactions, C. Experimental Methods The determination of interaction effects necessarily requires that activity measurements be made in dilute solutions, General summaries and critical discussions of the available experimental methods have

-1 - been made by Chipman and Elliott, (3) Chipman, Elliott, and Averbach, (39) Kubaschewski and Evans, (40) Wagner, (5) and Lumsden(38) The most important experimental methods are those involving equilibrium distribution of the solutes, static or dynamic vapor pressure measurements, or galvanic cell measurements,( The galvanic cell method, which was used in this investigation, has been utilized by a number of workers, Wagner(5) mentions a number of the experimental studies. Elliott and Chipman(41) attribute the first (42) elevated temperature galvanic cell studies to Taylor() in 1923o A comprehensive bibliography of galvanic cell studies was given by Elliott and Chipman in 1949,5) while additional studies since that time are too numerous to summarize hereo The general limitations of the galvanic cell method for thermodynamics studies have been considered by Chipman, Elliott, and Averbach(39) and Wagner(5) and Wagner and Werner(43) Dealy and (44) Pehlke() recently discussed the specific limitations of the galvanic cell method in the determination of interaction parameterso Do Previous Studies of Binary and Ternary Systems Involving Zinc and Bismuth The basic system on which the interaction measurements were made was the dilute solution of zinc in bismuth. Although previous studies have been conducted on this binary system and several of the ternary systems used on the present investigation, no systematic activity data are available in the literature that cover the dilute solution range below o052 mole fraction zinc.

-14Galvanic cell studies of the binary system zinc-bismuth were made at 600~C by Kleppa(44) and by Kleppa and Thalmayero(46) Lantratov and Tsarenko(47) also used galvanic cell measurements to cover approximately the same composition range at temperatures between 420 and 800~Co Yokokawa, Doi, and Niwa(36) used a dynamic vapor pressure method to determine the activity of zinc at 352~C at three compositions between o052 and ol51 mole fraction zinc, Wittig, Muller, and Schilling(48) calorimetrically studied the heats of mixing in the bismuth-zinc system at 470~Co An activity investigation of the zinc-bismuth system was also reported by Oleari, Fiorani, and Valenti(49) but the reference was unavailable. Critical evaluations of binary data for the zinc-bismuth system (primarily based on Kleppa's results) were made by Kubaschews1ij and Catterall(53) and Hultgren et alo(54) Activity measurements of zinc in ternary systems involving bismuth were made by Valenti, Oleari, and Fiorani(50) for Zn-Bi-Pb at 440 and 520~C, by Fiorani and Oleari(51) for Au-Cd-Bi at 450 to 550~C, and by Oleari and Fiorani(52) on Zn-Sn-Bi at 450 to 550~C, all by the galvanic cell method. Dynamic vapor pressure measurements were used by Yokokawa et al(36) to determine zinc activities in the Sn-Zn-Bi and In-Zn-Bi ternaries at 352~CC All of these investigations were conducted at constant mole ratios of bismuth to the secondary solute and none of the composition ranges approached the region where the solution was dilute with respect to both solutes, Iso-activity plots presented by Oleari and co-workers gave qualitative indication of the expected direction of the dilute solution interaction as did the data of Yokokawa et al,

-15The indications were that cadmium would have a very slight effect on the activity of zinc but the direction was uncertain, Indium and lead apparently increased the zinc activity, while tin decreased ito Phase diagrams were available for several of the experimental systems. The zinc-bismuth binary system is discussed in Hansen's compilation of binary diagramso(55) A ternary diagram for the Bi-Zn-Cu system was presented by Henglein and Koster, (56) while Geurtler et alo(57) summarized the available ternary information for the Bi-Zn-Ag, Bi-Zn-Sn, Bi-Zn-Sb, and Bi-Zn-Pb systems. Oleari and co-workers(50-52) presented information on the liquidus surface boundaries for the Bi-Zn-Cd, Bi-Zn-Sn, and Bi-Zn-Pb systems.

IIIo EXPERIMENTAL PROGRAM The experimental program was planned so that a simultaneous study could be made of the hypotheses of periodicity and additivity of interaction effects in a system where all the components were non-ferrous metalso The interaction parameters were determined for a series of thirdelement additions to a given solute-solvent system, zinc in bismuth. The choice of the third elements allowed the test of the periodicity hypothesis. Once the ternary systems were studied, the added solutes were combined to form higher-order solutions for the test of the additivity hypothesis. In the sections which follow, the experimental design for interaction parameter determination is considered. The advantages and limitations of the various experimental methods are discussed and the basis for the selection of the zinc-bismuth system is reviewed. Finally, the experimental equipment, procedures, and treatment of the data are discussed. A. EXPERIMENTAL DESIGN 1o Methods of Evaluating Interaction Parameters Before discussing the choice of the experimental method, the experimental design for the determination of interaction parameters should be consideredo i 61 bny The binary or self-interaction parameter, ci ( ~lnz xi= is evaluated as the limiting slope of a plot of lnyi versus x. and is obtained from activity measurements made on dilute binary alloyso (See Figure 1) -16

xi. 0= ei slope = (Mlri) i k X xi 1. Determine ai for values of xi 3. Plot Ini versus xi 2. Calculate Xi __ i4. Limiting slope is interaction i xi parameter. a) Determination of Self-Interaction Parameter. k \ vary xj at constant xi X Binary c i - k x =const alloys Binary a ry x. with i - k x \ const i/g \ \ alloy alloys i, ii j Method I Method II b) Experimental Compositions for Interaction Parameter Measurements Using Runs at Constant Mole Fraction of One Solute. Binary Alloy xi = coast. intercept = x x Xi 0O Ternary xj = 0 n^7i n ^j Alloy8 Slope = ^n ci\ on _ ( —- a Ixi - k\ xj / xi onst. Xj Xi = 0 xJi x' 0 Xj -~ —- Xi 1. Plot data from each run, then 2. Extrapolate slopes to xi = 0; slope determine slope. of this plot is 2nd-order parameter. c) Method I. Runs at Constant x, Varying x xi - const. This point from binary data intercept (2nli) xj - const. I nnyiylpei "i ( n7i) xi =-0.. xj ) xj = const. Slope \ dxi ) xT = 0 x =0 x = 0 xi Xj 1. Plot data from each run, extrapolate 2. Plot extrapolated values versus xj, to xi -0. limiting slope is a? i xji const. ope npi\ Xi 0 Slope xix j7 xi o XJ 5. Determine 2nd-order parameter from separate plot of slopes of extrapolation lines used in Step 1. d) Method II. Run t Constnt x, Varying xi Figure 1. Methods of Evaluating Interaction Parameters.

-18k from binary x A; const. n71 data - t =s )loonst. X data x S, rxk con t. cost.X~~O X _ Slop \ J / xi - 0 xi/xk. const. i JX xj 1. Conduct runs along constant 2. Plot data from each run, then determine slope. mole ratio lines. Intercept ( — nyi) xi - 0o I ~ ~/ ^^ Slope of this line is (n \ n7i i 2nd-order parameter aXj xJ 0 NOTE: i 0 is the j-k binary side of Xi "k of the isotherm, hence i. 0 - const. Xk Xi Xk 3. Plot slopes versus x, intercept is 1 xi e) Method III. Runs at Constant -, Varying x k X. -= const. XC1 xk X const. - C / Si p \ / / y \ ni <^^ \ / Z//^ o-n'i\ ~ __ — Intercept is (Inyi) xi O /^ f^ \ ________|__Or___ xi - l) i Xi 1 + Cl 1. Conduct runs along constant 2. For each constant x, extrapolate mole ratio lines. Nk ~~mole ratio lines. ^~n7i to xi = 0; Note that at xi 0, j reduces to binary case and xj + C Xk + Cl From binary data ('n7i).o-T x i =0/a ( Cn i)T - Slope - ( 7i) - 0 xi - 0 Xj 3. Plot extrapolated vaiues versus xj, limiting slope is e. Xj f) Alternate Method III. Runs at Constant -, Varying xi Figure 1 (Cont'd)

-19The definitive expression for the first-order ternary parameter, i = (-. — xi=O, requires that the activity coefficient of constituent x.=O J i be determined to vanishingly small concentrations of both solute elements Several methods by which this can be accomplished are indicated schematically in Figure 1. First, a systematic series of experimental runs could be made in which the concentration of the third element or added solute is varied while holding constant the concentration of primary solute. A second method would be one in which the third element concentration is held constant and the primary solute concentration is varied, Finally, it is possible to hold constant the ratio of either solute to the solvent while the concentration of the other solute is varied. By suitable manipulation of plots of lnyi versus concentration, the interaction parameter is found as the appropriate slope or intercept at zero concentration of the two solutes, The various methods will be discussed below in greater detail. Method I, which was used in the present investigation, requires that the activity coefficient of constituent i be determined as a function of the third element content, x., at several constant values of ( bl ny. xi o The limiting slope of a plot of lnyi versus x. is -_i xi=con ~1 )J \ dx /.=O J By plotting the slope values versus x. and then extrapolating to xi = O, [ ny, 1 the intercept is I J xi=O, the desired parameter. The limiting slope xjO J

-20of this plot is, in turn, ( i, a second-order interaction \ axjxi J xi=0 X0=O xj=0 parameter. It is desirable that as many data points as possible be available to allow a reasonable decision regarding the limiting slope. For this purpose Method I is of advantage since the end-point of the relation between lnyi and xj at xj = 0 is known independently from binary data. Thus, if n data points are known from the ternary alloys, an n + lth point is available to aid in constructing the intermediate plot. Method II requires the same amount of experimental data as Method I, however, in this case the amount of the third element, xj, is held constant and the amount of the primary solute, xi, is varied systematically. The intercept at xi = 0 of a plot of lnyi versus Xi is the value (lni) xi=). The limiting slope of a plot of xj=const. / \. piny (flni) xi=O versus xj is the desired parameter, ~i = aj x xj=const s21ny i The second-order parameter xixj X is found as the limiting slope O dln.xi=O (lny xjxj0 l n of a plot of 6^ — versus x.. The values of — i i X.=Const J 6xi Xj=const xi=o O are previously determined as the limiting slopes of the plots of lnzi versus xi, but are subject to some error because the limiting portion of the plot is extrapolated. In using Method II, the binary data are of no help in constructing the intermediate plots for the determination

-21of eJ, however, they do provide the value of lny which is the endpoint of the final plot. Method III requires that several runs be made where the ratio of one solute to the solvent is held constant while the amount of the xi other solute is varied. That is, either - = const. and x. is varied, xk J x. or a_ = const. and x is varied. Then, for example, the valuesof Xk lny. are plotted versus x.. The limiting slopes of these plots x xi A k = const. onyi. xi/xk=const. ln.1 x/kcnt are plotted versus the ratio at which they were ob6x. \ / )Xj=0 V J / xj=O Xi Xi tained. The intercept of this plot at -= 0 corresponds to the xk Xk 6nyi condition x =0, xk = 1 and is the desired parameter i 6 j xi. xj=O /21nyi The limiting slope of this plot is and is the secondxj=O J order parameter. (This will be discussed later in detail). It is preferable that the'data employed in each intermediate plot (where the mole fraction of one solute is held constant) be obtained under comparable experimental conditions. Consequently, the use of a multi-alloy experimental apparatus is highly desirable. Of the various methods described, a sequence of experiments following Method I is preferable since this has the advantage of clearly showing the nature of the interaction as the data are being obtained. Thus, in each run at constant x., the effect of the additions, xj, is directly evident.

-22Furthermore, as noted previously, data from binary alloys provide an independent end-point to the relation between lnzi and xi) and finally another advantage of Method I is that the second-order parameter is easily obtainedo By using the same sequence of compositions for both x. and xj i, e,, if a regular matrix of compositions is employed, 1 J a set of data can be analyzed by Method I and cross-checked by Method IIo This was done in several cases and virtually identical results were obtained, Method I is slightly preferred since the final parameter is obtained by extrapolation of the slopes taken from the intermediate plots. Each of these slopes has the same mathematical form as the final interaction parameter and is an intermediate parameter describing the effect at some finite compositiono Statistical studies on a representative set of data showed that closer confidence limits could be placed on the firstorder interaction parameter if Method I were used rather than Method IIo (Appendix C) By proper selection of the compositions to be studied, Method III would also permit cross-checks between its two variations, however, the advantage would be lost of varying xj at constant x. and thus clearly showing the interaction independent of the value of xi o The only advantage of Method III is that it would provide data at constant mole ratio which is required by some methods of ternary integration of the GibbsDuhem equation (3 60) However, if needed, these values could be obtained by interpolation of data obtained by Method I if a regular matrix of compositions were usedo Since the principal purpose of this investigation

-23was to determine the interaction parameters, Method III was of no advantageo 2o Experimental Methods for Determining Activity Although a number of experimental methods are available for the measurement of activities and activity coefficients it has been pointed out(3) that fundamentally they reduce to two cases. First, a solution may be equilibrated with another phase in which the activity is known or may be calculated or, second, a partial equilibrium may be established and then the potential determined to bring the system to complete equilibrium. For liquid metallic solutions the available methods are those related to either static or dynamic vapor pressure measurements, the equilibrium distribution of a solute between immiscible solvents, the deduction of the activity from the equilibrium constant of a chemical reaction provided that the activities of the other constituents are known or can be fixed, or by means of electrode potential (galvanic cell) measurements Chipman and Elliott(3) state that, where applicable, the electrode potential method is one of the most precise methods for determining activities, Kubaschewski and Evans(40) reported that the electrode potential method is considered slightly superior to other equilibrium methods

-24In the present investigation it was necessary that a large number of alloys be investigated under reproducible conditions and, in order to make best use of Method I for the evaluation of interaction parameters, preferably in groups where the concentration of one solute could be held constant and that of the other varied systematicallyo The use of the equilibrium distribution method would limit the investigation to systems where multiphase equilibrium is expected and would be dependent on the ability to deduce or fix the activity in at least one of the phaseso The precision of the method depends on the ability to analyze chemically for small differences in concentration of the constituent of interest while in the presence of other alloying elements. Vapor pressure measurements require that only the constituent of interest have an appreciable vapor pressure or that a multi-component effluent vapor must then be analyzed chemically or that a radioactive isotope of the constituent of interest be usedo Norman, Winchell, and Staley(58) used a mass spectrometer to analyze the effluent vapor from liquid In-Sb-Zn alloys, The reported accuracy of such measurements is 90 per cent or less and has the disadvantage that the alloy is depleted of the volatile component in order to make a measurement, The available vapor pressure methods are in general not readily adaptable to multi-alloy experimental runs, In the case of electrode potential or galvanic cell measurements, a direct measurement of the activity is possible through the free

-25energy relationships and thus the problems of deducing activity from chemical or other analyses can be avoided. Furthermore, the yield of data per experiment can be increased substantially by using a multi-electrode apparatus since a series of compositions can be investigated at one time, using a common reference electrode, without interference from each othero The galvanic cell is thus ideally suited for the application of Method I for the evaluation of the interaction parametero In order to use the electrode potential method at elevated temperatures, concentration cells of the following type are set up:(3) ]'~ e/ecir/ (~', ~,o),.. _' s,~. ('.Fs ka/.'^ he electde reacti s )e w en The electrode reactions are written as: Anode: ( pur (i ) + n e. (5) Cathode: (/j ( / (6) The net reaction of the cell is: (L pare) l=~ (L /, r//.Iy) (7) The alloy might consist of merely metal i plus its solvent or might include the additional solutes, At elevated temperatures,

-26electrolytes of fused alkali chlorides have been successfully used by a number of investigatorso(5 20 21, 39 etc.) At constant temperature and pressure, the free energy change of the reaction (Equation (7)) is given (60,76) by: G,-;G- -~', -n.. R.f7~-_Zl_ (8) 0t (,sUrI id/ 1dj If the standard state is taken as pure liquid i and the activity in the standard state is taken as one, then Equation (8) becomes: -^^ 3f; T r.., (9) By definition, the activity coefficient is L L (10) Thus, Equation (9) can be rewritten as: Or in equivalent form Or in equivalent form: A I // 9___~ 2 _ T N e - ~^; (12) (c dssolec( /l lJ/oJ

-27The assumptions inherent in the use of the measured potentials in these equations are the following: 1, The cell is reversibleo 2o The electrolyte exhibits only ionic conductanceo 3o The net cell reaction is the only one that occurs to a significant extent. 4o The alloy compositions are as stated, are single-phase, or if multi-phase, their compositions are known. The interaction parameters are then obtained by suitable manipulation of the logarithm of the activity coefficient as discussed in the preceding sectiono When the experimental runs are conducted by Method I (constant Xi while varying xj), the interaction parameters can be calculated directly from the electrode potentials by virtue of the following relationships, Since, by definition, ( - = ( ) 0 (2a) LI yj 0 Equation (11) can be differentiated with respect to xj, while holding P, T, and x constant. i 0 D- (I ( ):ETL (13) J~ /WT hXr \) Q ^PT

-28The limiting slope of a plot of E. versus x (at constant P, T, and xi) is Xj/o Xj 0 P, T P Then by Method I, cross-plotting j versus x. \ xj/ xi=const 1 xj=0 and extrapolating to xi - 0, the intercept is (___' = _ __ (ti) P.- T, or C - _ _ r J __ (16)!-',,'T - J-o J AT (XJ TXJ P. T 30 Sources of Error in Galvanic Cell Measurements The application of the galvanic cell method requires more than that alloys of proper composition be coupled with a fused alkali-chloride electrolyte, and a potential measured across a pair of leads extending into the alloyso The method depends on the realization in practice of the assumptions listed on po 27, and furthermore, as aptly stated by Chipman et al, (39) "because of very subtle conditions that may be present

-29in a given cell, the unwary experimentalist may be deluded into thinking that the results may have high accuracy and precision, whereas they may have only high precision," It is essential that the various sources of error that might affect the measured potentials be recognized and their affects alleviatedo The means available for this may rest in the design and construction of the experimental apparatus, the care and consistency in the experimental procedures, or in the choice of the alloy systems studiedo In some cases where errors are unavoidable, their extent should be estimated and minimized, if possible, by means of the experimental design. A summary of the various sources of error that might conceivably be encountered in an investigation such as the present one has been made in Table Io(395739) For convenience, the error sources have been classified in three major categories~ 1o Deviations in electrode compositions from physical or chemical causse S 2, Effects occurring within the electrolyte. 30 Effects related to the cell operationo The various errors are loosely classified as systematic or random, This distinction is intended to indicate that if such an error occurs it may affect the measured cell potential consistently in one direction or in either directiono At first glance. Table I indicates that an imposing list of pitfalls may be encountered in galvanic cell studies, however, closer examination reveals that many of the sources of error can be alleviated

-30TABLE I SOURCES OF ERROR IN GALVANIC CELL DETERMINATION OF ACTIVITY Classification Possible Effect on Observed EMF Remedy Deviation in Electrode Compositions Physical Effects Initial Composition Random, but probably slight Care in calculations and weighings; be certain alloy is single phase Loss of Constituents From System a. Solubility on Electrolyte Systematic; if active element, increases Use excess of metal chloride in electrolyte; maintain b. Volatilization EMF; if third element, decreases inter- almost static atmosphere at increased pressure action Change of Composition Within System (Material Transfer) a. Current Flow Generally random; redistribution by Avoid lengthy closing of circuit and possible short b. Diffusion diffusion can decrease observed EMF circuiting; improve cell geometry; make running time c. Spillage or Mixing short; use care in handling assembled cell Chemical Effects (Side Reactions) Reactions With Cell Materials Random Use fresh, clean refractories; be sure lead materials do not dissolve Reaction With Metal Chlorides in Electrolyte a. With Solvent Systematic; decreases EMF since active Restrict systems studied to those meeting criteria b. With Third Element element is displaced from electrolyte for minimum difference standard free energy of into alloy chloride formation (see Tables II-and III) Effects Within the Electrolyte Non-Ionic Conductivity Systematic, increases EMF Use alkali-chloride electrolyte with only a small amount of the active metal chloride Multi-valency of Ions Systematic, increases EMF due to oxidation Keep concentration of active metal chloride small, confirm and change of "n" in activity calculations valence by Faraday's Law Moisture or Oxides Systematic, decreases EMF due to reduction Dry electrolyte materials and use care in handling; of active metal chloride Maintain dry, inert atmosphere above cell Liquid Junction Potentials Random, opposes desired EMF by super- Allow time for diffusion in electrolyte before starting imposition readings; agitate electrode leads; refer measurements to a reference binary electrode replicated from cell to cell Concentration Gradients Random, enhances or opposes desired EMF by a superimposed concentration cell Effects Related to Cell Operation Irreversibility Systematic, reduces EMF by unwanted Be certain cell is at equilibrium for readings; check material transfer by changes of electrode position Thermal Equilibrium Systematic, opposes or increases EMF Allow time between temperature changes to insure thermal equilibrium Polarization Random, may affect EMF either direction Close circuit for readings only momentarily; allow time depending on current flow for diffusion between readings; use liquid electrodes for faster diffusion Temperature Measurement a. Thermocouple Calibration Variable effect on EMF, not predictable Check calibration of thermocouples and temperature b. Thermocouple Placement distribution in cell Thermal Potentials a. Tenrperature Gradients Variable effect on EMF, not predictable Check temperature distribution; agitate electrolyte b. Thermocouple Effect From to minimize gradients; be sure leads are homogeneous Dissimilar Leads or else measure thermocouple effect separately and c. Thermocouple Effect From compensate readings; use liquid electrodes Inhomogeneous Electrodes

-31by careful experimental procedures, while others can be eliminated in the experimental design. The principal source of errors whose effects are always in one direction are the possibilities of compositional changes - either gains or losseso The loss of an alloy constituent may occur by volatilization or by solution in the electrolyte, while an increase in the concentration of the primary solute may occur from side reactions taking place at the electrolyte/alloy interface. The initial compositions also could be in error by virtue of purely mechanical errors in charge calculations or weighings. Furthermore, there are processes which could tend to redistribute the alloying elements within the cello This could occur if the circuit is closed for an excessive time during the measurement of the potentials or at some other time by a short circuit across the leads. The redistribution might also take place by a diffusional process. If the cell is handled roughly or the electrodes agitated too severely, it is even possible to transfer material by spilling. Any process which causes a decrease in xo will result in an increase in the measured potential, while any loss of the secondary solute j'reduces the extent of the interaction in whatever direction it occurs. When the electrodes are solid, there is the possibility that compositional inhomogeneities or residual strains might give rise to extraneous potentialso Purely chemical effects are termed "side"-reactions and are those reactions which can occur in addition to the net cell reactiono

-32These can be subdivided into reactions that might take place with the cell materials and those arising from the electrolyte. The first type can be controlled by using only fresh, clean materials in assembling the cell, using care to insure that no impurities are introduced and by being certain that the lead and containing materials are unattacked by the alloys or the electrolyte, The side reactions which occur at the alloy/electrolyte interface are unavoidable but can be minimized by judicious choice of the systems to be investigated. These reactions are also termed displacement reactions since they can result in an unwanted reduction or displacement of the i-metal-chloride in the melt by either the solvent or the second soluteo(5 43944) The two types of displacement reactions are written as: Type I Displacement: ( IL / ()CL —- ( j I 17) Type Ii Displacement: (i/)J I ( ) — CIL + L —'L (18) II JY' In c i''J where ni, nj, nk are the valences in the electrolyte of the metal ions of the solutes i and j and the solvent k The result of either reaction is that the concentration of i increases in the alloy and concentration gradients also result in both the electrolyte and the alloy, thus causing overpotentials, The

-33 - estimation and control of these reactions has been discussed by Wagner and Werner(43) and by Dealy and Pehlke,(44) By assuming chemical equilibrium at the interface, and that ai = xi7i, ak 1, 7kl -= 1 a thermodynamic analysis can be made of the expected error in potential due to either reaction. For Type I reactions the expression for the error takes the form: relative error -' T,i ( ( (19) t L JVX, y, / (19) where AG0 is the standard free energy change for the reaction and is the difference in the standard free energies of formation for the appropriate metal chlorideso For Type II reactions, the expression has a slightly different form: relative error "' i _J _ <;_ l \ ^/ RT ja X L l (20) It can be seen that the error in measured potential increases as x. is increased, increases as xi is decreased, and increases as x is increased. It should also be noted that the chloride of the iCl metal i whose activity is to be measured must have a free energy of formation more negative than those of the solvent or the other soluteso If an acceptable level of error is fixed in advance and the minimum values of xi and xic1 and maximum value of x. are also fixed, it is then possible to compute the minimum AGI or AGi which I II

-34meets these criteria. Thus, the feasibility of galvanic cell investigation of a proposed alloy system may be easily determined. The calculated values become less as the valence of the metals increase, Dealy and Pehlke(44) considered the case where the tolerable error was one per cent at 5500C and xi, x. and xic. were oOl. A tabulation was made of the minimum AGo for both types of reaction as a function of n. and n The values of AG" can be used in conjunction with the free j 1,II energies of formation of various metal chlorides to aid in the selection of systems for studyo It is also useful to convert the calculated minimun AGi to corresponding potential differences which can be utilized 1, II for similar selection criteria with the electromotive force series determined by Laitinen and Liu(59) for the fused LiCl-KCl electrolyteo Thus, in the case of displacement reactions, the unavoidable error can be minimized in advance by means of the experimental designo Errors arising from effects within the electrolyte are those due to the failure of the assumptions regarding the conductivity of the electrolyte or the valency of the i-ions, to the presence of impurities, or the existence of concentration gradients or junction potentials. It is generally held that the use of an electrolyte containing a small concentration of the i-metal chloride dissolved in an alkali-metal chloride, ioeo, the LiCl-KCl eutectic, will minimize both the effects of nonionicity of conduction and the possibility that the (i)n+ ions may be present in more than one valency state (39%42) This is the major reason why the alkali-metal chlorides were developed in preference to merely

-35using the pure fused i-metal chloride as the electrolyte. If multivalent ions were present in significant amounts, current could flow by their oxidation to the more positive state at the cathodeo Displacement reactions similar to the types discussed previously may also be caused by the presence of oxidizing or reducing agents in the electrolyte. Hence, it is essential that the electrolyte be prepared as free as possible from moisture, that an inert atmosphere be maintained above it, and that the electrodes be kept as free as possible of oxidation during their preparation. Further possibilities for error are that concentration gradients in the electrolyte might give rise to a liquid junction potential at the interface between regions of differing concentration or that a separate concentration cell may be superimposed on the desired cello The operation of the cell may also result in errors unless reasonable care is taken. The electrodes must have sufficient time for diffusion so that constitutional equilibrium is reached and a thermal equilibrium must also be established before meaningful readings can be madeo In taking readings, the circuit must be closed only momentarily so that mass transfer and polarization are avoided. A null-balance potentiometer greatly alleviates this possibility since the current flow is slight and in random directions as the balance is attained. Polarization is minimized by allowing sufficient time for diffusion between readings,

-36Absolute errors in interpreting the data may also result from the temperature measurement practice. The thermocouples should be checked and their placement verified to confirm that the true cell temperature is being measured. It is also necessary to eliminate the possibility of thermal potentials due to temperature gradientso Another thermal effect is the imposition by a thermocouple effect of a potential on the leads if they happen to be dissimilar or non-homogeneous. If it is necessary to use dissimilar leads the effect can be compensated for by making a separate evaluation of the thermocouple formed by the leads and then using these results to correct the measured cell potentials (70) 4o Alleviation of Errors Fundamentally there are only a few ways for dealing with the various sources of error that can occur in a galvanic cell investigation, The error source may be eliminated completely or made negligibly small by means of careful experimental technique and design, there may be reasonable grounds for assuming that it does not exist in a particular system, or a means can be introduced to compensate for the erroro The previous discussion of error sources included comments on the means of eliminating or reducing many of them. Primarily this consists of using care in the preparation and assembly of the cells, in the minimization of moisture and impurities in the electrolyte, and in confining measurements to systems where displacement reactions are negligibleO In studying liquid cells at high temperatures, the diffusion rates are usually high enough to correct concentration gradients that may occur

-37from some transient effect such as current flow or a thermal gradiento The use of an electrolyte composed of only a few mole per cent of the i-metal chloride dissolved in eutectic LiCl-KCl has a number of advantageso Not only does this reduce the tendency for side reactions, but it inhibits the solution of the electrode metals in the electrolyte and minimizes the liquid junction potential.(39 2) The reversibility of the cell may be inferred from the stability of the potentials and their reproducibility with timeo The valency and the reversibility can also be confirmed by auxiliary experimental procedures in which known amounts of current are passed through the cell in both directions, Nevertheless, despite taking all reasonable precautions in the experimental design and procedure, the large variety of possible error sources means that there still may be some random error possible from the cumulative effects of marginal sources. The utilization of a multielectrode apparatus does provide a means for compensating these effects, and if Method I is used for the experimental design, it is possible to normalize the interaction parameter calculations to the same base point, Method I requires that the mole fraction of the active element be held constant in each run while the concentration of the third element is varied in the other electrodes. By making one of the electrodes in each cell a binary alloy, it is then possible to directly evaluate the parameter from this base point since the parameter, a derivative, is a relative quantityo The experimental data from two such cells may be compared more accurately by normalizing the potentials from the ternary

-38alloys relative to the same value for the binary alloys. By this means, compensation can be made for effects operating on all the electrodes of a cello Such effects might consist of the unavoidable residual impurities in the electrolyte or the standard or to a junction potential at the boundary between the standard and the electrolyteo By running a number of such binary alloys, and assuming that cumulative errors occur randomly, it is possible to make a statistical analysis of the inherent or "cell factor" errors systematically affecting the absolute value of the potential of each electrode and to establish standard values of potential for the binary alloys of each composition. By referring the interaction parameter calculations to such standard values or base points, the remaining scatter in the data can be inferred as primarily related to "third element" or ternary factorso It is thus possible, through a combination of experimental technique and design, to realize in practice the assumptions on which rest the use of the equation for activity determinationo 5o Choice of Alloy Systems The choice of the alloy system to be studied was based on the desire to test hypotheses regarding the periodicity and the additivity of the interactions with added soluteso It was thus necessary to choose a binary system permitting dilute solute additions taken in distinct sequences in the Periodic Table and also that would be expected to form single-phase liquid solutions in both ternary and multi-component systems. However, a number of other requirements had to be meto In order to form

-39the galvanic cells the primary solute had to be substantially more active in the electromotive force series than many other metals. The single phase liquid solutions had to exist at temperatures within the range of the materials used for cell construction and the alloys should attack neither the containers nor the electrode leads. The necessity of minimizing side reactions also governed the choice of the solute and the solvento Finally, potential health hazards had to be considered with the use of certain materials The starting point for the selection of the alloy systems was to consider the possible cells from the standpoint of experimental convenience and the likelihood of displacement reactionso A preferable upper temperature limit for the studies was 650-700~Co The basis for the preliminary selection was the standard free energy of formation of the metal chlorides and calculations which provided criteria for assessing the possibilities of displacement reactionso These were based on the minimum difference in free energy of chloride formation so as to result in only one per cent error in the measured potentials assuming the worst cases where the mole fractions of the active metal and its chloride in the electrolyte were both 01 and the mole fraction of the third element was large. Table II presents the AGf per gram equivalent at 550~C for the formation of a number of metal chlorideso The values summarized in this table were taken from several sources: Dealy and Pehlke(44) who made their calculations from values given in Pitzer and Brewer;(60) Yamagishi and emoto and K a ellogg (2) The table also includes

-40TABLE II STANDARD FREE ENERGY OF FORMATION FOR VARIOUS METAI CHLORIDES EMF of Metal Relative to Pt -ZGf for Chlorides at 550~C -kcal/g.atom in LiCl-KCl at 4500C -volts Dealy and Yamagishi and Laitinen and Metal Valence Pehlke(44) Kamemoto(61) Kellogg(62) Liu59) K 1 85.7 85.7 - Na 1 80.0 -- 80.0 - Ca 2 76.5 - Li 1 -- 80.6 -- -3.41 Gd 3 - 67.7 -- Mg 2 61.0 60.8 61.0 -2.58 Mn 2 44.8 42.9 44.5 -1.85 A1 3 42.1 -- 46.0 -1.80 T1 3 37.0 -- -- -1.37 Zn 2 36.8 36.2 36.5 -1.57 Cr+2 2 -- 36.0 -- -1.42 Cd 2 31.4 33.0 31.5 -1.32 Cr+3 3 29.0 29.3 -- -0.63 In 3 -- 29.0 -- -0.84 Pb 2 28.8 -- 29.0 -1.10 Fe+2 2 25.1 28.8 28.5 -1.17 Sn2 2 26.6 24.0 27.0 -1.08 Ga 3 -- -- -- -0.88 Cu+l 1 22.8 23.2 -- -0.85 Sb 3 19.2 23.1 -- -0.67 Co 2 24.2 23.0 -- -0.99 Sn+4 4 -- 22.9 -- -- Ag 1 20.5 20.4 20.2 -0.64 Fe+3 3 19.7 20.3 -- - Bi 3 19.2 19.1 19.0 -0.59 As 3 -- 18.4 -- -- Hg 2 17.8 -- 15 -0.50 Cu+ 2 -- 12.7 - +0.04 Te 2 -- 8.5 -- Pt 2 -- 0 -- 0 Au 1 - -- +0.5 +0.31

-41values taken from Laitinen and Liu's(59) electromotive force series for LiCl-KC1 electrolyte at 450~C since the displacement criteria may be interpreted as AGI or AEI o Wagner and Werner(3) suggested as an approximation that AEs values taken from Laitinen and Liu's series could also be used at considerably higher temperatureso Without yet making recourse to the actual minimum AG~ values, Table II was used for the preliminary selections by the following line of reasoning: In order to form feasible cells and to minimize Type I displacement, it was desirable to use a solvent whose AG- for the chloride was quite low on the scale with respect to the primary soluteo That is, the solvent should be a relatively noble metal, Starting from the bottom of Table II the elements that might be useful as solvents included mercury, antimony, bismuth, silver, nickel, copper, tin, lead,chromium, or cadmium. Mercury was eliminated because of its high vapor pressure and health hazardo The melting points for antimony, silver, nickel, copper, or chromium were all relatively high (600~C or more). This left lead, cadmium, tin, or bismuth as possible solvents since they were relatively low-melting, easily obtainable materialso Leaving the solvent for a moment, the choice of the active metal was considered, In order to form a large number of cells, the active element should fall high in Table IIo Since an alkali-metal chloride would form the major part of the electrolyte, the active metal had to be below the alkali metalso Again it was necessary to consider

-42the melting points since it was highly desirable that the standard electrode be liquid at the cell operating temperatureso On this basis, the first feasible active metals were thallium, zinc, or cadmium. Thallium was eliminated from consideration by virtue of its toxicity and ready oxidation. Of the choice between zinc and cadmium, zinc was preferable since it might be possible to use cadmium as a third-element addition to a zinc solutiono Cadmium also remained as a possible solvento Assuming that zinc was chosen as the active metal it was then possible to apply the quantitative criteria for displacement reaction possibilities. Table III (which was extended from the calculations of Dealy and Pehlke(44) gives minimum values of AGI and AGi for one per cent displacement error for xi = Xil = o01 and several mole fractions of the added solute j All solutions were assumed Raoultiano Assuming that zinc (valence 2) is the active metal, the AG~ between zinc chloride and cadmium chloride was 5,400, Table III shows that cadmium (valence 2) could not be used as a solvent with zinc, but it would be a feasible third element addition up to approximately five mole per cent. All other metals lying below cadmium in Table II would also be feasible third element additions from the standpoint of avoiding Type II displacemento The table also shows that lead would not be a feasible solvent if cadmium were the active metal whereas if zinc were the active metal, lead just meets the criterion for avoiding substantial displacement effects. All metals below lead were feasible solventso The metals remaining for practical consideration as solvents were bismuth, lead, or tin,

-43TABLE III MINIMUM VALUES OF AG~ OR AGO i FOR ONE PERCENT DISPLACEMENT ERROR IN MEASURED ELECTRODE POTENTIALS (Calculated From Equations (19 and 20)) 0 0o Valence G minfor AGI fo Displacement By Displacement By Solvent* Added Solute* Active Metal Solvent or Added Solute XiCl =.01 ni nj or nk Xi =.01 Xj =.01.05.10 1 1 16,250 8,700 11,310 12,480 2 8,640 4,920 6,230 6,810 3 5,990 3,500 4,370 4,760 2 1 15,100 7,550 10,200 11,310 2 8,110 4,350 5,650 6,230 3 5,640 3,120 3,990 4,370 3 1 14,500 6,900 9,500 10,670 2 7,710 4,000 5,330 5,900 3 5,420 2,900 3,770 4,150 * AG, is difference of standard free energy of formation of chlorides taken from Table II, k-cal/g. atom concentrations in mole fraction.

-44Next considered was the position of the various elements in the Periodic Table, a pertinent section of which is shown in Table IV. By comparing the displacement reaction criteria with this table it was noted that if either bismuth or lead were used as the solvent an uninterrupted sequence of third-element additions might be made with the Group-B elements of the 5th Period. If tin were the solvent, this sequence would be broken. Similarly, in the 6th Period, by using bismuth as the solvent in preference to lead, another almost complete sequence would be developed, although it would be broken since thallium did not meet the displacement criteria. Another reason for preferring bismuth to lead was based on a comparison of the phase diagrams for binary and ternary systems based on these elements. When the binary diagrams for Bi-Zn and Pb-Zn(55) were superimposed (Figure 2) it was noted for the zinc-poor compositions that the single-phase liquid region for Zn-Pb was much more restricted than the one for Zn-Bi. It was expected, therefore, that there would be a greater likelihood of finding single-phase liquid regions over wider concentration ranges at lower temperatures in multi-component solutions based on Zn-Bi. Further evidence to support this conclusion was gained from comparisons of the isotherms given by Guertler et al(57) for the Zn-Ag-Pb and Zn-Ag-Bi systems and was later confirmed by Okajima's(63) observations in the Zn-Pb-Ag system. The information available for the otherternary systems based on zinc plus bismuth was that wide single-phase liquid regions would be found in the bismuth corner at temperatures above 300-400oC.(50 52, 57)

TABLE IV SECTION OF PERIODIC TABLE INCORPORATING SUITABLE SOLUTES AND SOLVENTS FOR GALVANIC CELL STUDIES OF ACTIVITY AT 450 TO 650~C Group I II III IV V VI VII VIII Period A B A B A B A BA B A B A BA (K) (Ca) Sc? Ti? V? [(Cr)] (Mn) [Fe] Co Ni 9??9 4 Cu Ga [Ge'?As?Se Br Rb? Sr? Y? Zr? Nb? Mo? Tc? Ru Rh Pd??? 5 Ag (I In Sb?Te I Cs? Ba? Rare Earths Hf? [Ta] [W] Re? Os, [Ir] Pt 9 Au Hg (T1) |^l Po At ( ) Not feasible as third element additions with Zn or Cd due to displacement possibilities [ ] Not feasible as third element additions due to lack of solubility in Bi O Feasible as active elements 7i Feasible as solvents Feasible as third element additions by displacement criteria? Insufficient information to determine suitability as third element, or solubility questionable

-46Bi-Zn Syttem - Pb-Zn System 798~ - -' —~_ Zn-Pb ToC ^"- Zn-Bi \ 418~,4160 *\.400 318~ _ 254 ~ ___S^^ 300 2540 200,,,.,,... 100 30 40 50 60 70 80 90 100 MOL PCT Bi or Pb Figure 2. Binary Phase Diagrams of the Bismuth-Zinc and Lead-Zinc Systems in the Zinc-Dilute Region.(55)

-47Based on the above considerations, the following systems were chosen for study: Active Element: Zn Solvent: Bi Added Solutes: Cu, Ga, Ag, Cd, In, Sn, Sb, Au, Hg, and Pb The elements enclosed in parentheses in Table IV were deemed unsuitable as added solutes for the following reasons: For Group-B elements: No information could be found to assess the displacement possibilities for actinium, germanium, or selenium. Arsenic presents a health hazard and its chloride is gaseous at the proposed operating temperatureso The chloride of tellurium is likewise gaseous. Thallium did not meet the displacement criteriao Bromine and iodine are not metallic and polonium is radioactiveo No suitable sequences of the Group-A elements were possibleo Either the element lay above zinc in the electromotive force series or there was no information for assessing displacement possibilitieso Furthermore, the binary phase diagrams of a number of the Group-A elements indicated negligible solubility in bismuth at the temperatures of investigationO Beyond normal care in the handling of antimony, the only significant health problem was associated with mercuryo The volatility of mercury made it both a health problem and a problem of retaining it in solutiono It was decided to attempt only enough runs with mercury to establish the direction of its interaction with zinc, if any, and to estimate its extent. The runs would be made with extreme precautions to maintain the system closed to avoid mercury loss and the working area would be well ventilated.

-48In summary, the selection of the listed ternary additions to the basic system of zinc in dilute solution with molten bismuth was a logical process based on considerations of the hypotheses to be studied, the need for liquid, single-phase solutions at the temperatures of investigation, and the necessity of avoiding extraneous displacement reactions 60 Choice of Composition Ranges With the experimental scheme, method, and alloy system specified, the only remaining choice was the composition ranges to be studiedo The choice of compositions depended on two competing factors. First, it was desirable that the alloys be as dilute as possible with respect to the solutes, and yet a sufficient amount of the third element must be used to insure that its interaction was properly evaluated, On the other hand, the minimization of side reactions required that the concentration of the active element not be too low, nor that of the third element be too high. The available crucibles and containing tube materials for cell construction were of such sizes that a multi-electrode cell having five alloy electrodes and one pure zinc standard electrode was possibleo One of these electrodes was used primarily as a reference binary, thus leaving space for four ternary alloys per runo In order to allow some factor of safety with respect to displacement reactions, o015 mole fraction zinc was chosen as the lowest concentration to be studied. Similarly, O050 mole fraction was taken as

-49the upper limit of the third-element concentration. To allow the data to be evaluated by either Method I or Method II, (see pp. 19-21 ) a 4 x 4 matrix of the following levels of both solutes was adopted as the starting point for the studies: o015, o025, ~0375, and,050 mole fraction. The binary alloys of zinc in bismuth consisted of the same mole fractions plus an additional alloy at o075 mole fraction so as to overlap the pre(45) vious investigation of Kleppa(5) who had carried his studies down to 0639 mole fraction. It was expected that in some cases where the interaction might be strong, or that single-phase solubility was limited, the amount of third element addition could be reduced below o015, however, the attempts of other investigators(21) to use active-metal concentrations of.010 mole fraction suggested that cell stability deteriorated rapidly if the concentration was too low. The composition ranges for the multi-component solution studies were expected to be in the same general ranges. The zinc mole fraction would be,015 or,050, while the initial amounts of the third, fourth, fifth, etc. elements would also be o015 mole fraction eacho Depending on the results of the initial studies, the total added solute concentration would then be increased in order to investigate the region of validity of the results. B, Experimental Procedures lo Equipment The electrode potentials were determined in the multi-electrode galvanic cell apparatus which is shown in Figure 3. Apparatus of this

-::::::::::-::;::::- —: -;~0:,i~~i::i-ii~~~:8::::: ii:: -~;i::-_: i~ ~ ~ ~ ~:::::-:: -:-:-::::::~:::_:_::::::_:_::-:i —i:: -::i ~ ~ ~ ~ ~ ~ ~ ~ E Wi I~ ~ q ~ ~ ~ iiii-;,k:-ia~~~~~~~~~~~~~~~~tmNER~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i-:: HPIM~~~~~~~; Ai~:: _ ~~I~~ c -a:mmw 3: — ~~~~~~~~~~~~~~~~~mm* I R IW ~~~~~~~~~~~.~~~~~~~:i:~~~~~~Aln Ao: Fiur Glvni CllApartu'

-51type have been used previously by a number of investigators, The design of the present equipment, which was constructed by Boorstein(21) and modified somewhat for this investigation, was adapted from those used by Wilder(64) and Weinstein (65) The apparatus consisted of the cell itself, the furnace, the atmosphere system, the potential measuring circuits, and the auxiliary equipment for electrolyte preparationo a. Galvanic Cell The multi-electrode cell is illustrated schematically in Figure 4, The five individual alloy electrodes were contained in 15 mm diameter by 26 mm high recrystallized alumina crucibles arranged in a circle at the bottom of a 50 mm diameter by 90 mm high recrystallized alumina cell crucibleo The reference standard of pure zinc was contained in a 10 mm diameter by 26 mm high recrystallized alumina crucible at the center of the circle of alloy electrodes. The entire set of electrodes was immersed in the molten salt electrolyte, which filled the large cell crucible to about one inch above the small crucibleso The large crucible rested on a bed of porcelain saddles at the bottom of the containing cell tube. The electrode leads were 0,040 or 0,025-inch diameter tantalum wires approximately four feet long, protected from the molten salt at their lower ends by recrystallized alumina insulators extending into the alloy. The ends of the leads were bent into a circle in order to increase the area of contact with the liquid alloy.

-52Tygon Tubing Clamp Tantalum Lead inside Pyrex Cell Head Vycor Sheath Heu/ Int\ VPorcelain ThermoHelium Inlet Vacuum _ -- couple Sheath Outlet -- Clamp - Tygon Tubing Stop cock'- - - 64mm O.D. Vycor Cell Tube Chromel wound ll Cover Resistance -— _ __ A1203 Lead ProFurnace Funace --- tection Sheath Electrolyte -- - 1I. 2"o.D.x 3! Al2O3 Crucible Alloy —- - 1 A203 Crucible Tantalum Lead \' /- Zinc — \ S- Standard Porcelain Saddles Thermocouple (Only two of six leads and alloys shown) FIGURE 4 MULTI-ELECTRODE GALVANIC CELL

-53Above the alumina insulators, the leads were enclosed in 6 mm diameter Vycor tubes, which protected their upper ends from salt fumes to some extent, acted as insulators, and guided them to the head fo the cell tube. The shielded leads were held in position by an alundum spacing disc that also served as the crucible covero The entire assembly was contained in a 64 mm ID Vycor cell tube about 25 inches long, extending into the furnace below the side arms about 19 inches. The top of the tube was closed by a tapered-joint connection to a Pyrex head, which provided six openings for the electrode leads and one for the thermocouple, The openings extended slightly above the surface of the head for the attachment of Tygon tubing, sealed to the head with Glyptal painto The leads then passed through the Tygon tubing, which was closed by pinch clamps; this tubing was frequently renewed to avoid the possibility of leakage through cuts that could result from repeated clampingso A 6 mm diameter Vycor tube, sealed at the bottom, served as the thermocouple well. The well extended to the bottom of the large cell crucible at the periphery of the circle of electrodes, The side arms to the Vycor cell tube provided access to an atmosphere system for the evacuation of the system and the introduction and maintenance of a helium atmosphereo A stopcock was located in the inlet arm, while the outlet arm was permanently attached to a Tygon line which could be closed with a pinch clampo A ball-joint was used to connect the inlet arm to the atmosphere system, Wire springs which stretched

-54from the side arms to ears on the Pyrex head were used to seal the system against positive internal pressureo bo Furnace The cell tube was clamped vertically and extended into a chromel wire-wound resistance furnace, shown schematically in Figure 4~ The resistance coil was 18 gauge Chromel wire that had been coiled on a 1/4-inch mandrel and then wound. about a 3-inch ID alumina inner furnace tubeo The coil was sealed to the furnace tube with alundum cement and the space between the coil and shell insulated with vermiculite, with the coil spacing varied on the furnace tube so that a level temperature zone could be maintained. The total furnace resistance was 22 ohms. A chromel-alumel thermocouple in an alundum liner was butted to the furnace tube for use as the temperature control coupleo Temperature control was maintained by the "high=low" circuit shown schematically in Figure 5o The furnace power was obtained from a 230volt Variac autotra.nsformer wired so that two unequal voltages could be tapped, and regardless of the voltage setting, the tapped voltages were 36 volts apart. Either high or low voltage was fed to the furnace coil by a relay connected to a Foxboro electro-mechanical temperature controller. To obtain a desired cell temperature, it was necessary to set both the controller and the Variac to predetermined levels. In order to maintain temperatures in the 450-650~C range, "high't voltages of 98-133 volts were required. Depending on the ambient temperature, it was normally possible to maintain the furnace temperature to about + 1~Co

-55 - Copper Copper,jA Cell in Vycor Cell Tube - -I D [0 9 Q BC B Ice Bath Junction Ttalum Chromel Copper C Junction Block,~,. -lAlumel Ji| —I D Ten Pole Switches Copper E Leeds and Northrup Type K-3 Potentiometer., _ L^ — s _ ~Standard Cell, Working B O Battery and Electronic A@ 1 NNull Detector o E Jo ooo0 0 Cell Potential Measuring Circuit 230v Source Fuses Foxboro Controller Variac Relay Switch Volt Meter -- Ammeter (; Furnace Control | Couple ____ FIGURE 5 SCHEMATIC DIAGRAM OF TEMPERATURE CONTROL AND POTENTIAL MEASUREMENT CIRCUITS

-56co Atmosphere System The atmosphere system was designed so that the cell tube could be evacuated and flushed before establishing the helium atmosphere which was maintained throughout the experimental runso A line diagram of the system is shown in Figure 60 A Cenco "Hyvac 2" mechanical vacuum pump was used for evacuating the system, with an ace-;one-dry ice cold trap placed ahead of the pump. Pressure measurement was by means of a mercury filled U-tube manometer, The helium, which was obtained in purified form from the University of Michigan Plant Department, was passed through a drying train before entering the cell tube. The train consisted of a tube containing "indicating" anhydrous calcium sulfate, a tube filled with copper gauze that was maintained at 500~C in a small auxiliary resistance furnace, and a second calcium sulfate drying tube. Fore- and back-bubblers were used to maintain a helium pressure of 35 mm of mercury throughout the cell operationo do Potential and Temperature Measuring Equipment The circuit used for the electrode potential and temperature measurements is shown in Figure 5~ The tantalum leads from the electrodes were connected through a junction block to a pair of inter-connected tenpole gang switcheso The thermocouple was attached to compensating leads, which led to an ice-bath cold junction; copper leads then connected the cold junction to a junction point on the unit which also led to the gang switches. Common leads from the gang switches to the potentiometer completed the circuitry. The arrangement allowed rapid measurement of both

Stopcocks Helium in Towers Manometer Cell Tube and Meter Stick Copper Gauze Furnace ~~~~~-Trap To Vacuum Pump FIGURE 6 ATMOSPHERE SYSTEM

-58the temperature and electrode potential since any alloy electrode could be compared with the standard electrode, with alternate switching to the thermocouple if desiredo For the first 20 runs, a Leeds and Northrup No. 8687 Volt Potentiometer was used for the potential measurementso A Leeds and Northrup Type K-3 Potentiometer with a fresh standard cell then became available and was used in the remaining 65 runs. An electronic null detector was used with the Type K-3 instrument. The Volt Potentiometer was standardized against the Type K-3 Potentiometer and was found to have been in excellent agreement with the K-3o All thermocouples used in the galvanic cell runs were made from the same spools of 26 gauge Chromel and Alumel wire. The wires were insulated with two-hole ceramic tubingo The thermocouples were checked for calibration against the melting points of pure bismuth and zinc. Both virgin and used thermocouples checked in this manner were found to be accurate to within l'C. e Electrolyte Preparation Equipment Prior to its use, the electrolyte was fused, with continuous evacuation for several hourso This process was carried out in a separate il0ovolt powered auxiliary resistance furnace similar in construction to the main-unit furnaceo "On-off" temperature control was maintained by a Pyro-Vane Electronic Temperature Controllero The electrolyte was melted in a 40 mm diameter Vycor tube that was lowered to the bottom of a 64 mm diameter Vycor tube extending into

-59the furnace. The large Vycor tube was similar to the main-unit cell tube but had only one side arm, connected by a ball joint to a U-tube filled with glass woolo In turn, the U-tube was attached by heavy-wall rubber tubing to a tapered joint connection with the vacuum pumpo An acetonedry ice cold trap was placed ahead of the pump. The large Vycor tube was closed at the top by a Pyrex glass blind head which mated to a tapered joint. 20 Materials High-purity metals were used for the alloy and standard electrodeso The source of each metal and its quoted purity are summarized in Table V. The various alloying elements were cut into convenient-sized pieces with thoroughly cleaned wire cutters or shears, and any surface oxidation was removed with appropriate etchants. The tantalum lead wires were obtained from the Fansteel Metallurgical Corpo in 25 meter spools of 99~9+ per cent pure annealed wire. Both O040- and.025- inch diameter wires were used during the investigationo The recrystallized alumina crucibles and insulators were obtained from Morganite, Inc. as their Triangle RR Grade, while the Norton Company supplied the alundum sheet used for the crucible lid. Commercially prepared Reagent Grade chemicals were used for the electrolyteo

-6oTABLE V EXPERIMENTAL MATERIALS Quoted Form as Material Source Grade Purity-% Received Metals Zn ASARCO* A-58 99.999+ Splatters Bi ASARCO A-59 99.999+ Shot Ag ASARCO A-59 99.999+ Splatters Au Donated by -- 99.995+ Powder Ford Motor Co. Cu ASARCO A-58 99.999+ Rod Cd ASARCO A-60 99.999+ Splatters Ga Fisher Scientific Electronic 99.9999 Liquid Co. Hg t Instrument 99.999+ Liquid In ASARCO A-58 99.999+ Extruded Stick Pb ASARCO A-59 99.999+ Extruded Bar Sb ASARCO A-60 99.999+ Fragments Sn Vulcan Detinning Extra Pure 99.998+ Cast Bars Co. Chemicals ZnC12 Baker & Adamson Reagent 95 (min) LiCl J. T. Baker Reagent 99.0 KC1 Baker & Adamson Reagent 99.5 *ASARCO - American Smelting and Refining Company

3, Cell Operation Procedures The operation of the galvanic cell was a two-step procedure. The first step consisted of the preparation of the alloy, the standard electrodes, and electrolyte. The second step was the operation of the cell to obtain electromotive force values for the alloys over a range of temperatures. Approximately one week per run was required for the preparation and operation. ao Electrode Preparation Both the alloy and standard electrodes were pre-melted under a helium atmosphere in the absence of the electrolyte in order to freeze the tantalum leads in place for ease in subsequent handling, to insure proper positioning, and to homogenize the alloyso The total charge per electrode, approximately 17-18 grams, was dictated primarily by the amount of the elongated bismuth shot that could be packed into the electrode crucibles. When melted and solidified, the crucibles were about 60 per cent filled, The charge calculations were based on about 17,5 grams of bismuth per electrode and were normalized in terms of the most difficulty divisible constituent, Thus, the first weighing was of the element most difficult to divide into proper-size small pieces, ieo, copper, silver, or zinc, A convenient amount was weighed to +,0001 gram, and the weights of the other constituents required to obtain the desired mole fractions were then calculated from this base, The additional elements were also weighed to +,0001 gram, but it should be realized that while the first weight was that of an actual piece of material, the succeeding weights

-62represented the best attempt to add material to a balance pan to match a desired weight. This rather subtle point has been discussed in more detail by Acton (66) All the weighings were made on an Ainsworth Type LC Analytical Balance using Class S weightso The electrode compositions calculated from the weighed-in amounts were generally within o0001 mole fraction of the desired valueso Fresh crucibles and insulators were used for each run, The ends of the tantalum leads were cleaned with silicon carbide abrasive paper and washed in acetone prior to inserting them into the insulators and bending the tip to the circular shapeo Following the weighings, the cell was pre-assembled outside the tube but with the leads extending through the head, using a stainless steel cup in place of the large recrystalized alumina crucible, A metal fixture was wired to the top of the cup to hold the assembly together and to ensure correct lead positioningo Finally, titanium shavings were placed on top of the positioning fixture to serve as a "getter" during the lead-sinking operation. The entire assembly was lowered into the Vycor cell tube by a hook through a wire loop at the top of the positioning fixture. The pinch clamps at the cell head were then closed and the tube was placed in the cold furnace and connected to the atmosphere system. The tube was evacuated and flushed several times with dried helium before turning on the furnace, and several additional flushing cycles were conducted after the furnace temperature had risen slightly. During the flushing sequence,

-63the pinch clamp on the tube outlet line was kept closed. Finally the clamp was opened and a slightly flowing helium atmosphere at 35 mm of mercury was established. The furnace temperature was then brought to 450~C in about 2-3 hours, after which the gas flow rate was increased and the pinch clamps were released and retightened individually to permit the leads to be pushed down into the now-molten electrodes. The leads were oscillated about 50 times to aid in mixing the alloy and wetting the leads, but care was taken that the final position of the leads was such that they did not touch the crucible bottoms. After all the leads had been positioned, the cell was held for an additional hour at 450~C for further homogenizationo The cell tube stop-cock was then closed and the gas flow reversed to provide a static atmosphere. The tube was removed from the furnace and air-cooled (while maintaining the helium atmosphere within) in order to freeze the leads in place. The pre-sunk electrodes and head assembly were then removed from the cell tube, inspected, and the cell was re-assembled - this time in the large recrystallized alumina crucible with the alundum spacing disc as the cover. The 6 mm diameter Vycor insulating tubes were exchanged for those cut to the proper length for the actual running of the cello Additional pieces of zinc were added to the standard electrode crucible to fill the space made available by the melting and solidification of the initial zinc charge. This was done so that the volume of the zinc

standard was as large as possible, thus minimizing effects of impurities that might be introduced during lead sinkingo Since the diameter of the standard crucibles was 10 mm, the alloy electrodes in their 15 mm diameter crucibles were considerably greater in volumeo The total weight of the small zinc standards was about 5.5 grams - in contrast to the 17418 grams of the alloy electrodeso In several runs one of the alloy electrodes was replaced by an additional pure zinc standard in a 15 mm diameter crucibleo No difference was found between electromotive force measurements using this larger standard or the normal 10 mm diameter standard as a referenceo Finally, the Vycor thermocouple well was placed in position and the entire cell and head assembly was mounted on a ring stand adjacent to the cell tubeo b. Electrolyte Preparation While the electrode leads were being pre-sunk, the electrolyte was under preparation in the auxiliary furnace. The electrolyte consisted of 165 grams of the eutectic mixture of LiCl and KC1 (42 mole per cent LiC1 and 58 mole per cent KCl)plus 1o5 mole per cent of ZnC!. Previously, the electrolyte had been mixed in a large batch and evacuated continuously at room temperature in a vacuum dessicator for at least three days to remove moistureo The pre-treated electrolyte was then stored prior to use in a tightly closed jar in a dessicatoro This procedure was adapted from the method of Laitinen, Ferguson, and Osteryoung 67) The zinc chloride was vacuum dried separately for 4-6 hours at 250~C and then stored in a closed jar in a separate dessicator,

-65Approximately 1.5 mole per cent zinc chloride was added to the electrolyte in two ways during the experimental program. Initially, in about half of the runs, the zinc chloride was added to the molten electrolyte as it was being poured into the cell crucible. Later the procedure was changed and the dried zinc chloride was added to the eutectic mixture prior to its fusion. The entire mixture was then fused under continuous evacuation for four hours at 600~C. The purpose of the prolonged vacuum treatment of the electrolyte was to further remove any absorbed water. The vacuum treatment caused no significant loss of any of the electrolyte constituents, since a dummy batch carried through the entire preparation procedure was found unchanged in weighto c, Starting of Runs Before starting the run, the leads and the head were carefully checked to insure that each was in proper position and that the Vycor insulating tubes properly indexed with the openings in the head. The entire atmosphere supply system was purged with flowing helium, A Vycor funnel was positioned so the electrolyte could be poured into the large crucible, When everything was in place, the vacuum pump was disconnected from the auxiliary salt purification equipment and connected to the main unit, The Vycor funnel was heated with a gas-oxygen welding torch, and the molten electrolyte mixture was removed from the auxiliary unit and poured through the heated funnelo For those runs where the zinc chloride was not added at the start of the electrolyte preparation, the zinc

-66 chloride was melted in the Vycor funnel and allowed to run into the large crucible with the LiCl-KIl eutectico When all the electrolyte had been added to the crucible, the alundum disc was pushed down in its place as the crucible cover. Some of the electrolyte immediately solidified in contact with the large crucible and the electrode crucibleso This served to "cement" the electrode crucibles in position and allowed the entire assembly to be lifted by the lead wires and lowered into the cell tube. The cell tube was then placed in the furnace, sealed, and connected to the atmosphere system. In some of the earlier runs, the furnace was started coldo Later it was found more convenient to have the furnace temperature at about 550~C at the start of the run, which insured that the electrolyte was molten during the initial flushing operations, thus facilitating the removal of air bubbles that might have been trapped during the electrolyte transfero In addition, there was less chance for moisture and oxygen absorption at the electrolyte surface, The total time between the st.art of the electrolyte transfer and the sealing of the system was usually less than 30 secondso The cell tube was evacuated and flushed at least six times before the positive helium atmosphere was finally established, By this time the alloy and standard electrodes were molten and therefore the lead positions could be rechecked and adjusted so that the electrodes did not contact the crucible bottoms, When the system was thoroughly flushed, the pinch clamp on the he outlet tube was opened and a slight flow of helium was established against the fore-bubblero The internal pressure during

-67the runs was maintained at about 35 mm of mercury. The system was then allowed to equilibrate for several hours. do Operation During Experimental Runs Before meaningful data could be obtained it was necessary that compositional equilibrium be established both in the electrodes and the electrolyte, and that thermal equilibrium be achieved in the entire system, For those runs where the furnace was started cold this required from 24 to 48 hours, When the procedure was changed so that the furnace was almost at the operating temperature when the run was started, the initial equilibrium period was considerably reduced. Depending on the alloy system under investigation, this was as little as 4 to 6 hours and generally within 12 to 16 hourso The criterion adopted for satisfactory equilibrium was that the electrode potentials should vary only randomly over a range of about o2 mv in four or five sets of readings made during the course of an houro If the variation was within this range but was systematic rather than random, it was assumed that equilibrium was still slowly being established, Once equilibrium had been attained, the cell responded only to temperature fluctuations which were normally within a range of one or two degrees, In a few cases, a systematic drift was noted in the potentials for a more prolonged periodo In these instances, the potentials were measured with respect to the reference binary alloy composition included in most runs, It was found in virtually all such cases that the potential differences (ioe,, the interaction effect) between a series of alloy electrodes (of

-68constant zinc content and variable amounts of added solutes) were constant and independent of the absolute level of the potentialso The drift in the absolute potentials was assumed to be related to some extraneous reaction involving the standard and the electrolyte, and perhaps represented the effects of residual amounts of moisture and oxides in the electrolyteo (This point is discussed in greater detail in the section on Alleviation of Errors, p. 37 )o When it was determined that the cell had reached equilibrium or that a state of "dynamic" equilibrium could be assumed, the potential measurements were started. The electrode potentials were determined at a series of from four to eight temperatures within the range 4500C to 650~C. An effort was made to establish thermal equilibrium temperatures at close to even 50~ temperature incrementso The sequence of temperature measurements was random, some points being approached from above and some from belowo In some instances, a set of readings would be made at a sequence of rising temperatures such as 500~, 550~, 6000, and 625~Co Following this, a similar sequence of falling temperatures would be investigated at the intermediate values, ioeo, 575', 525~, and 475~C. A set of readings was accepted if it met the general criterion of less than.2 mv variation over one houro When the furnace temperature was changed, the establishment of a new thermal equilibrium required about two hourso In making a set of potential readings, the cell temperature was read and then the individual electrode potentials were read in sequenceo The cell temperature was then re-read and the average between the initial

-69and final temperatures was used as the temperature of the particular set of readingso The variation in indicated temperature, if any, was invariably in the range o2 to o4~C, During the potential readings, care was taken that the circuit was not completed for more than a fraction of a secondo Since the readings were approached from both directions, mass transport in the cells was assumed negligibleo The total operating time of each cell was from three to six days, depending mainly on the time to reach initial equilibrium. It was generally possible to investigate five or six temperatures in a day of runningo Although an effort was made to vary the pattern of the runs, since it was necessary to leave the cell unattended during the late night and early morning hours, the usual procedure was to take the highest temperature readings late during the evening and then set the temperature to a low value and allow it to cool during the nighto In any event, repeated excursions to a given temperature were made during the course of a run, Throughout each run, a running plot of temperature versus potential was made and a linear relationship was normally foundo Reversibility in a given run was inferred from the linearity and the low standard deviation of the emf-temperature relations and the general stability of the potential readings with time, A special experiment was also conducted to confirm the reversibility of the cells,

-70e, Conclusion of the Run When a sufficient amount of data had been obtained, or it became evident that the cell was no longer stable, the run was concluded by quenching the cell with watero It was necessary to adopt this procedure because of the corrosive nature of the electrolyteo During the cell operation, electrolyte occasionally collected at the bottom of the Vycor cell tube either from condensation on the tube walls or because of leakage through cracks that could develop in the large electrolyte crucibleo If the molten electrolyte was allowed to solidify in contact with the Vycor tube, this invariably caused the tube bottom to cracko A procedure was adopted to minimize this risko The thermocouple and its protection tube were withdrawn from the cello The head of the cell and the tantalum leads in their Vycor protection tubes were withdrawn as a unit and quenched in water. The alundum crucible cover was removed with a hook. A long stainlessssteel rod was then immersed in the electrolyteo The Vycor tube was removed from the furnace, clamped in place and allowed to cool slightly, water was then poured down the inner walls of the Vycor tube to dissolve any spilled electrolyte before it had a chance to solidify in contact with the wallso The quenching froze the rod into the salt remaining in the cell crucible, and it was then possible to withdraw the crucibleo Hot water was used to dissolve the salt away from the alloy electrodes, while all components of the cell were cleaned by boiling in hot HClo The portions of the leads that had been in contact with the alloys were clipped off and the remaining parts of the leads were cleaned with silicon carbide abrasive papero For the

-71next run, the leads were reversed and the "fresh" ends of the leads were used as the portion that contacted the alloyso In the initial runs,,040-inch diameter tantalum wire was used for the leads. However, it was soon noted that after several runs, the portion of the leads exposed to the vapor above the salt melt became severely embrittled. Consequently, the change was made to,025-inch lead wire, the wires were reversed after the first run, and then the leads were usually discarded after only two runs. The smaller wire diameter also allowed better closure of the Tygon head tubes with the pinch clamps, 4, Treatment of Data a, Reduction of Raw Data Once it had been established graphically that the emf-temperature relationship of the electrodes was linear, equations of slope-intercept form were fitted to the data by the method of least squares (68) The electrode potentials at the five integral temperatures at intervals of 50~C over the range from 450~ to 650~C were determined from the regression equation. All computations were made on an IBM 7090 computer using a program that had been written in MAD language. The program averaged the raw data at each temperature, if desired, and determined the slope and intercept of the least squares regression line, The standard error of estimate of the actual data was computed, together with the correlation coefficient, The standard deviation of the slope of the line was also computed,

-72From the interpolated potentials, the program then calculated the activity of zinc, the activity coefficient, and the natural logarithm of the activity coefficient of zinc in each alloy at each of the five integral temperatureso The program also calculated from the weighed-in values the mole fraction of each of the solute elements. Since the data consisted of sets of several readings taken at relatively fixed values of temperatures, two procedures were tried in the calculations. In one, the computer was instructed to average the temperature and potential data and then proceed with the average values for the further calculationso That is, the four sets of readings made, for example, at 499,6, 499o5, 500,1, and 500o2~C were averaged and then treated as one data point at 499o8~Co The later procedure adopted was to treat each point separately as an independent reading, as indeed it was. Data for several runs were calculated in both ways and the interpolated potentials were identicalo The statistical estimates of data scatter were somewhat improved by the second technique since the sample size had been considerably increased. The use of the individual data points also made it unnecessary to take an equal number of readings at each temperature, As noted previously (pa 28 ), suitable manipulation of the acti= vity equation allows the logarithm of the activity coefficient to be obtained directly from the electrode potentials with no need of computing the activities and activity coefficients as intermediate stepso A second computer program was written for this purpose to be used with the data

-73that had been corrected to the standard potentials of the reference binary alloysO A linear regression program was also written that could be used for the determination of the self-interaction and ternary interactions if it appeared that the data were linear. b, Determination of Interaction Parameters Once the natural logarithm of the zinc activity coefficient had been obtained for each alloy electrode, the interaction parameter was usually determined graphically by Method I (p 19 ). For each temperature and each constant mole fraction of zinc, a plot was made of the in activity coefficient versus the mole fraction of added solute. Weighed-in values were used for all compositions. The limiting slopes of these plots were the values The set of values determined for the third element at limiting dilution were then cross-plotted versus the mole fraction of zinc and extrapolated to xZn = O, where the intercept is the ternary interaction parameter Xi (_l_ _) (2a)' J'- - /i ^ -' o

-74The limiting slope from this plot is the second-order interaction parameter X =i o In addition to graphical determination, a computer program was written to determine the parameters by least squares analysis of the matrix of activity coefficient data. The program performed the analysis by Methods I and II, and it was found that identical values of the interaction parameter were obtained by either method, however, statistical estimates of the confidence level of the parameters were better for the Method I calculations (Appendix C)o The least squares method had the disadvantage that equal weight was placed on each data point. Since the binary data were known from repeated runs to better accuracy than the ternary data, it was desired to place more reliance on them in fitting the relation to determine the intermediate slopeso Consequently, the graphical method was used to determine the parameters reported in the next section.

IV. EXPERIMENTAL RESULTS The experimental results of this investigation and the calculation of the interaction parameters are discussed in the section which follows. The order of presentation is: the zinc-bismuth binary system, ternary systems based on zinc-bismuth, and the test of the additivity hypothesis by means of multi-component solution studies. This section also includes the experiments confirming the assumptions on which the galvanic cell method is based and an estimation of the errors involved. The correlation of the results and the rationale for the observed interactions is reserved for the following Discussion Section. A. Zinc-Bismuth Binary System The binary alloys of zinc in bismuth were studied primarily to determine the self-interaction, if any, of zinc and to provide a basis for assessing multi-component interaction effects. However, the experiments also provided a verification of the experimental technique and a means for estimating the experimental uncertainties. Consequently, these results are discussed in this section in detail. A total Of 63 successful binary electrodes were run at five concentration levels of zinc. In one run (Run 3) only binary alloys were studied, while in most of the remaining runs a binary alloy was included as an additional reference electrode with the ternary or higher-order alloys. -75

-76The replication of compositions was the following: XZn-Mole Fraction No. of Electrodes Run.015 25.025 14.0375 10.050 13.075 1 Previous studies of the zinc-busmuth system at 600~C have been made by Kleppa(4,546)and Lantratov and Tsarenko.(47) The most dilute solutions studied by Kleppa were.0639 and.1284 mole fraction zinc, while Lantratov and Tsarenko's most dilute compositions were.144 and.2611 mole fraction zinc. Therefore, the composition of.075 mole fraction zinc was included in this investigation in order to overlap Kleppa's range and to provide an indication of the consistency of the results of the three studies. The binary electrode results are summarized in Table VI. The values of the electromotive force (emf) were obtained by least squares analysis of the experimental data(68)(see pp. 71-72 ). The results presented are the interpolated potential at 550~C, the slope of the least squares line relating the emf and temperature, the standard error of estimate of the emf's from the least squares line, and the natural logarithm of the activity coefficient of zinc at 550~C. The standard error of estimate provides an indication of the scatter of data within an individual run, while the other quantities are affected by differences between runs.

-77TABLE VI EXPERIMENTAL DATA FOR BINARY BISMUTH-ZINC ALLOYS EMFat E at EMF at Run No ny Run No. 55000~C Rt a Iny X Zn =.015 X Zn =.0375 3 109.36.221.16 1.116 9 75.78.184.06 1.147 18 109.05.218.20 1.125 19 77.84.182.08 1.089 25 109.92.214.39 1.100 34 76.48.179.18 1.127 37 109.63.212.29 1.108 42 76.80.179.18 1.118 38 110.51.217.21 1.084 44 75.66.174.42 1.150 41 110.04.216.21 1.097 45 76.16.175.06 1.136 47 107.66.219.15 1.164 46 75.85.174.13 1.145 48 109.34.216.08 1.117 54 75.67.175.07 1.150 52 109.72.220.04 1.106 55 75.17.175.32 1.164 57 108.23.214.37 1.148 70 75.24.175.16 1.162 58 108.75.219.25 1.133 Average 76.6.177.17 1.140 64 110.48.219.25 1.084 65 109.13.219.10 1.123 Zn 68 111.96.229.07 1.043 71 110'.03.213.24 1.097 3 67.50.164.20 1.092 72 108.10.225.24 1.152 17 67.38.170.23 1.096 73 108.33.217.18 1.145 23 66,17.165.06 1.130 75 109.38.225.27 1.116 28 66.24.165.81 1.128 76 110.44.220.25 1.086 29 66.63.179.37 1.117 77 109.52.219.08 1.112 39 66.47.163.14 1.122 78 110.45.227.20 1.084 50 66.18.160.16 1.130 79 108.61.210.12 1.137 51 66.12.167.09 1.132 80 108.32.220.19 1.145 56 66.61.161.15 1.118 83 109.71.230.20 1.106 69 66.03.161.10 1.134 85 111.53.220.38 1.055 81 65.76.157.44 1.142 Average 109.53.219.20 1.111 8B 65.96.162.15 1.136 84 66.62.162.14 1.117 X Zn =.025 Average 66.43.164.23 1.120 3 92.08.197.20 1.092 n 75 16 94.58.199.34 1.022 20 92.22.194.57 1.088 3 52.82.143.18 1.101 27 91.63.197.32 1.105 32 93.04.198.24 1.065 33 91.20.198.11 1.117 40 91.61.194.20 1.106 43 89.62.191.16 1.156 49 90.23.194.16 1.145 53 89.86.192.27 1.155 58 90.62.196.12 1.134 61 90.95.187.28 1.124 66 92.20.190.38 1.089 67 91.84.189.46 1.099 Average 91.56.194.27 1.107 Note: EMF, millivolts ( ), millivolts/~C at a, standard error of estimate of emf values from least-squares line, my. XZn mole fraction

-78The average potential at 550~C and average emf-versus-temperature slope were calculated from the replicate data for each composition. (Table VI). To test the consistency with the other investigators, the activity of zinc was calculated at 550~C and 600~C from the average values. In addition, the 600~C data of Kleppa and Lantratov and Tsarenko were extrapolated to 550~C using their emf-temperature slopes. The results of these comparisons are shown in Figures 7 and 8 as plots of zinc activity versus zinc mole fraction at 550 and 600~C respectively. Figure 9 presents a plot of aE/6 t versus mole fraction zinc. In all cases, a smooth curve encompasses the data and shows that the present dilute solution results are consistent with the data taken on more concentrated solutions. This agreement, together with the general stability of the cell behavior, was taken as evidence that the present experimental apparatus was reliable and that there was apparently no significant alteration of zinc content during the runs. Figures 7 and 8 show that the zinc-bismuth system exhibits a strong positive deviation from Raoult's Law. Examination of Figures 7 and 8 and the data at other temperatures also revealed that the activity-versus-concentration relation was linear in the region below approximately.075 mole fraction, thus indicating that Henry's Law applies. The limiting slope of the 550~C line is 3.04 and is the Henry's Law constant or, the activity coefficient at infinite dilution. Therefore, on this basis it can be concluded that the self-interaction coefficient of zinc is zero.

0.7 0.6 T=550't HL 0 T s 0. Data of 5-ntrto n 0.4O0 0.05 0.10. 0.20 0.25 0.30 O This investigation 0.2- Fig Data of Kleppa (extrapolated)(45) Data of Lantratov and Tsarenko (interpolated)(47) 0.1' 0 0.05 0.10 0.15 0.20 0.25 0.30 XZn- MOLE FRACTION Figure 7. Activity of Zinc versus Mole Fraction of Zinc in Bismuth at 550~C

0.6 T a 600~C 0.5 0.4C N 0.3 0 This investigation (interpolated /e a( value) o A Data of Iaeppa(45) ~0.2 Elo/ n Data of LantrtoY and Tsarenko t7 0.1 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 XZn -MOLE FRACTION Znr Figure 8. Activity of Zinc versus Mole Fraction of Zinc in Bismuth at 6000C.

-810.25 i i, 0.20 0 E 0.15 0.10 0 This investigation 0.05- A Data of Kleppa(45) D Data of Lantratov and Tsarenko(47) 0 0.05 0.10 0.15 0.20 0.25 0.30 Xn — MOLE FRACTION Figure 9. Slope of EMF-Temperature Curves (aE/oT) versus Mole Fraction of Zinc in Bismuth.

-82However, a more sensitive test is to consider the behavior of either the activity coefficient or the logarithm of the activity coefficient, since both of these quantities must be constant in the Henry's Law region. Figure 10 presents a plot of the natural logarithm of the activity coefficient versus the mole fraction of zinc, with the values calculated from the average potentials indicated as circles, and the scatter in the actual log values indicated as superimposed frequency distribution bar graphs for each composition level. A horizontal line is also drawn at ln7 = 1.110, the value corresponding to the Henry's Law constant taken from Figure 7. The average values appeared to be in reasonable agreement with the horizontal line with the exception of the data for.0375 mole fraction. In addition, the experimental values appeared to be fairly randomly distributed about the average values, again with the exception of the.0375 mole fraction results. Here it appeared that experimental results might be skewed towards high values. In order to further study these results, several linear regression analyses were run on the experimental values of lni. A computer program was employed to determine the slope of the regression line, its intercept, the standard error of estimate, the standard deviation of the slope, and to calculate the value of "t" for the hypothesis that the slope is not significantly different from zero.(68) The results of the calculations are summarized below:

STATISTICAL ANALYSIS OF BINARY DATA AT 550~C For hypothesis For Linear Relation Between that slope 0 lnYZn and xzn (95% confidence) Std. Error (From Table Data Considered Slope Intercept alope of Est t in Ref. 68) Slope Intercept a________slope o f Est. t____calcpt. 9, n All 64 data points.275 1108o.239.029 1.151 2.000 54 points (excludesoo data for xn =.0375).140 1108.242.029 o578 2.005 w 52 points (excludes data for xZn =.0375,.064,.075) o337 1.104.280.029 l 203 2.005 39 points (only includes the data for XZn =.015,.025) - -439 1.118 1.072.032 - 410 2.020

I I I I I -I I I 1.41.31.2.I — I a j j Line for 7=3.04, taken from limitIi I ing slope n - c5' - - t r Figure 7 - 99% confidence Data of limits based on Kleppa a o.2 mv for EM 1.0 versus temperature Code * Average Value o0.9~~ gQ~~~~.L~~ -~ Bar Graph Showing Relative Frequency Distribution of Individual Data Points 0.8 I I I I I I 0O 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Xzn- MOLE FRACTION Figure 10. Determination of Zinc Self-Interaction in Dilute Solution with Molten Bismuth.

-85The tabulation shows that although the various least squares lines have slight slopes (io.e, the interaction parameter might not be zero), in no case was this slope significantly different from zero. Therefore, it was concluded that the self-interaction of zinc is essentially zero, The extent of the Henry's Law region depends on what interpretation is placed on the data at ~0375 mole fraction zinc. It appears most reasonable to conclude that these data are skewed, since if the data are accepted at face value, the activity-versus-concentration curve calculated from average potentials would have a point of inflection between.025 and ~0375 mole fraction zinc, an occurence that would be possible but unlikely. Since the average potentials for o015, o025, and.050 mole fraction yield Iny values that are in close agreement, the conclusion reached is that the Henry's Law region extends to at least 5050 mole fraction and might go to.075 mole fraction, although at this point some curvature does appear to be developing in the activityconcentration line. A further point to be considered is the within-run scatter as opposed to the between-run scattero Without using the formal technique of analysis of variance, it is still possible to assess these effects by a statistical argument(68) Table VI shows that the standard error of estimate is approximately the same for all compositions at an average value of.20 millivoltso Strictly speaking, this cannot be considered as a standard deviation, since by using linear regression, the confidence

-86limit placed on the value for the emf at a given temperature depends on where that temperature is in relation to the entire range of temperatures studied. For a point at the middle of the range, which is the case for the 550~ data, the variance of estimate is the standard error of estimate divided by the square root of the number of observationso(68) However, for the present discussion, it would be desirable to have a number that would express the variance of potential for a run as a whole, and as a conservative value, the standard error of estimate can be used. Assuming that o20 millivolts is a reasonable figure, then the application of 99 per cent confidence limits to the average potential for XZn o015 would suggest the range of Iny at 550~C from 1093 to 1127 as reflecting all the expected variation due to temperature and other uncertainties within a run, When these limits are applied to Figure 10, it is seen that some experimental values for iny fall outside this range and that a portion of the variation shown by the bar graphs must be due to differences between runso These differences are most likely caused by slight compositional differences and residual impurities and/or moisture in the electrolyte, since in the ternary runs the potentials of all electrodes appeared to be affected to the same extento On this basis, the majority of the scatter shown in the bar graphs of Figure 10 has been assigned to a "cell factor". Corrections for the "cell factor" were made in the following manner. Based on the regression analysis and the limiting slope drawn on Figure 7, lny = oll0 was adopted as the "best" or "most probable" value for the zinc-bismuth

-87system at 5500C. The potential corresponding to this value was calculated for each of the compositions studied. Extrapolations to the other temperatures, 450, 500, 600, and 650~C were made using the average values of the emf-temperature slopes. The result was a set of "standard" electrode potentials for binary alloys which are summarized in Table VII together with the average values from the original data. In most cases, the adjustment was quite small, ranging from a few hundredths of a millivolt to approximately one millivolt for the.0375 mole fraction data. The implications of this procedure are that all the uncertainties between electrodes have been combined in one correction factor. By using average values of emf-versus-temperature slopes, compositional variation, if any, has been subordinated to the absolute factors such as residual impurities or moisture in the electrolyte or junction potentials at the standard/electrolyte interface, etc. By applying the "cell factor", which is determined from the binary electrode results, to the absolute potentials of the multi-component electrodes in the same cell the results of several runs can be compared on a common basis. Since the interaction parameters are relative quantities, this merely means that the comparisons are made to a common base point, and that (hopefully) the remaining variation in the multi-component electrode results is due to factors associated with the added solutes and not to the factors resulting from the operation of a particular cell. The statistical analysis was confined to the 550~C data, since the confidence in interpolated potentials from least squares analysis is best at the mid-point of the temperature range studied.

TABLE VII SUMMARY OF BINARY ELECTRODE POTENTIALS AND ACTIVITY COEFFICIENTS FOR BISMUTH-ZINC ALLOYS "Average Values" "Standard Values" XZn-Mole Fraction XZn-Mole Fraction Temp. C.015.025.0375.050.015.025.0375.050 t. 450 87.63 72.16 58.36 50.03 87.67 72.05 59.39 50.48 500 98.58 81.86 67.21 58.23 98.62 81.75 68.24 58.68 550 109.53 91.56 76.06 66.43 109.57 91.45 77.09 66.88 c) 6oo 120.48 101.26 84.91 74.63 120.52 101.15 85.94 75.08 650 131.43 110.96 93.76 82.83 131.47 110.85 94.74 83.28 (6Ec/t)p.219.194.177.164.219.194.177.164 CD 450 y 3.96 = 3.96.Zn Zn > 500 3.44 3.42 * v 550 3.02 3.04 s 2600 2.70 2.72 ~ l650 2.43 2.46 * By visual fit of slope to plots of azn vs XZn **Value corresponding to ln7 calculated for XZn =.015 a, Standard values obtained by normal izing 550 emfs to be equivalent to ln = 1.110 (obtained from 64-point linear regression) and then applying (L)p values from "average values." b) Electrode potential, millivolts

-89If,20 millivolts is used as a conservative estimate of the standard devation of the potential within a run, then it can be said at the 99 per cent confidence level that uncertainties within a run are in the order of + o5 to 1,0 per cent of the absolute potential, The "cell factors" used to correct the data in multi-component runs depend on the temperature under consideration and thus it is difficult to fix an over-all estimate of the uncertainties removed by using the standard potentials. However, in most cases, the corrections were less than one per cent of the absolute level of the potential, and in only a very few cases was the correction much larger than one per cent. The Henry's Law constants for the binary alloys were determined for both the average potentials and the standard potentials for the range 450 to 650~C and are summarized in Table VII, The values for the average potentials were obtained by visual fit of the best slope to large scale plots of activity versus concentration, with less weight placed on the average data for o0375 mole fraction zinc than on the,015,.025, and o050 mole fraction data. The values for the standard potentials correspond to Iny for xzn -.015 o The differences between the two sets of values are slight, although it is felt that a slight uncertainty exists in the second decimal place. The preferred values are those obtained from the standard potentials. Since the interaction parameters for the ternary studies are determined from lny values, three decimal places were carried forward to the lny calculations.

-90The temperature dependence of the Henry's Law constant was found to fit the following type of relationship as suggested by Dealy and Pehlke(86) i = - 079' * i ) Using this equation, yO at 352~C is 5.6 The value of y7 calculated from the vapor pressure data of Yokokawa et al(61) at this temperature was 5o4o The detailed analysis of the binary alloy results thus showed that the self-interaction of zinc in bismuth is essentially zero, that the Henry's Law region extends to at least ~050 mole fraction zinc, and that the system exhibits a strong positive deviation from Raoult's Lawo Also, the experimental procedures have been verified, and a basis has been laid for assessing the confidence to be placed on the results from multi-component runs. Bo Zinc-Bismuth-j Ternary Systems The zinc-bismuth-j ternary systems were studied to determine the interaction parameter between the solute elements, Zn and j, and to investigate the possible periodicity of such interactions. The data are presented separately for each system studied. The presentation of the results in this section is primarily factual; the interpretation of the effects and the significance of temperature dependence are discussed in a later section on the Rationale of the Observed Interactions (pp.166-176)o

-91The order of presentation of the systems studied is by increasing atomic number of the solute j o The data are presented in form of tabulations of the electrode potential at 550~C, the slope of the emf-versus-temperature relation, the standard error of estimate for the least squares relation between emf and temperature, and the natural logarithm of the activity coefficient at 550~Co Before calculating the activity coefficient the electrode potential was corrected by the "cell factor", where the cell factor is the difference between the measured and standard values for the potential of the binary electrode included in the cello The interaction parameters were determined graphically by Method I (ppl19-21)o They were obtained by cross-plotting the limiting slopes of plots of ln7zn versus xj for various constant contents of zinc. The calculations were carried out at 450, 500, 550, 600, and 650~Co For each system, a plot is presented defining the temperature dependence of the interaction parameterso The intermediate slopes and the final values for the first-and second-order interaction parameters are summarized in Table IXo 1o Bismuth-Zinc-Copper System The results showing the effect of copper on the activity of zinc are presented in Table VIII and Figures 11 through 13. Additional data defining the single-phase liquidus boundary in the bismuth-rich corner of the system are presented in a later section (po128). Here it will be mentioned that in the first runs on this system, a marked

TABLE VIII EXPERIMENTAL DATA FOR BISMUTH-ZINC-j TERNARY ALLOYS x i 0.0075.015.025.0375.050 STD EMF EMFMF at (e EMF at )e EKN at EEM at 06 EMF at )EM at 01 le( (0 (;f7 Run No Xz for Binary- 550~C a n7 550C at lan7 550~C t a In7 550~C at Tln 550.C at a In 550~C (t l In Bi-Zn-Cu Alloys j - Cu.015 109.57 5 N - - - N - - - 109.66.208.37 1.108 110.77.205.56 1.076 119.92 e.208.46 1.044 114.83 e.218.39.962 68 C 112.49.223.58 1.110 113.16.219.49 1.091 114.49.219.36 1.054 115.41.215.19 1.028 115.09 e.213.23 1.037 U - - -.025 91.45 7 - - - U - - - 93.06 e.179.09 1.065 95.11 e.188.25 1.007 95.86 e.206.33.986 22 N - - - N 97.41.202.05.992 98.71 e.193.19.905 99.51 e.201.06 - U - - 66 C 92.20.190.38 1.110 N - - - 93.24.187.26 1.081 95.25 e.185.26 1.024 97.13 e.185.19.971 96.80 e.200.06.979 67 C 91.84.189.46 1.110 N - - 93.77.186.29 1.056 94.90 e.183.26 1.024 97.02 e.180.25.964 98.68 e.173.18.917.0375 77.09 8 N - - - N - - - 78.38.172.06 1.074 82.06 e.163.08.970 83.84 e.159.62.920 83.32 e.154.96.934 9 C 75.78.184.06 1.110 N - - - U - - - 83.55 e.167.13.852 82.08 e.168.66.993 U - - - 22 N - - - N N - - - 80.72 e.173.09 1.008 N - - - N - - - 70 C 75.074.174.36 1.110 75.99.171.31 1.083 77.00.169.28 1.065 72.33 e.151.30 1.186 78.72 e.180.26 1.002 N - - -.050 66.88 2 N - - - N - - - 69.05 e.158.06 1.049 70.71 e.155.11 1.002 72.12 e.159.10.962 73.88 e.155.10.613 69 C 66.25.159.29 1.110 67.09.157.26 1.086 68.31 e.155.26 1.052 69.68 e.152.26 1.013 71.39 e.155.39.965 N - Bi-Zn-Ga Alloys j - Ga.015 109.57 55A C i09.96.220.04 1.109 N - - - 107.09.216.05 1.192 106.56.221.11 1.207 106.08.213.04 1.220 106.88.213.04 1.198 52B C 109.48.221.04 1.109 N - - - 107.08.211.05 1.178 107.05.207.09 1.179 107.47.200.20 1.167 108.13.201.16 1.148 58 C 108.75.219.16 1.109 N - - - 107.42.214.14 1.148 107.53.215.09 1.145 107.62.213.18 1.142 N -.025 91.45 58 90.62.196.12 1.109 N - - - N - - - - - - 53 C 89.86.192.27 1.110 N - - - 89.26.190.15 1.127 89.47.189.11 1.121 89.96.188.11 1.107 U - - - 57 C (90.11) - - - 89.41.191.36 1.130 89.48.190.22 1.128 N - - - N - - - N - 61 C 90.90.187.28 1.109 N - - 89.56.186.24 1.148 89.69.185.22 1.144 90.06.185.19 1.134 N - - - 61 c 90.86.187.28 1.111 N - - N - - - - - - N - - -.0375 77.09 55 C 75.17.175.32 1.110 N - - 75.08.175.35 1.112 75.83.176.38 1.092 75.74.175.40 1.094 72.05.169.42 1.199.050 66.88 56 C 66.61.161.15 1.110 N - - 66.21.162.09 1.121 U - - - 67.19.159.10 1.094 67.93.159.09 1.073

TABLE VIII (CONT'D) x* 0.015.025.0375.050 STD EMFat EMF at EEF at g EMF at E Run No XE* foar B5 ry 550~C (t ff 1y 550~C ny 5C at a l ny Run No Xzn* for Binary 5%oc c( 100 550() Bi-Zn-Ag Alloys J. Ag.015 109.57 18 C 109.05.218.20 1.110 111.22.218.14 1.049 111.22.216.15 1.049 112.7?.215.08 1.006 113.92.214.12.973.025 91.45 16 94.64.199.34 - 93.64.196.28 1.048 94.82.193.48 1.014 97.49.197.45.938 97.23.190.41.945 21 N - - - 94.30.191.18 1.028 96.52.190.18.967 N - - -.0375 77.09 19 C 77.84.182.08 1.111 80.24.181.10 1.043 84.21.183.09.931 84.83.182.16.914 83.80.178.08.943 21 N - - 79.09.170.06 1.057 82.38.173.07.965 N - - -.050 66.88 17 C 67.38.170.23 1.110 69.22.168.24 1.058 71.17.169.20 1.003 71.58.167.25.991 73.37.166.24.941 Bi-Zn-Cd Alloys j - Cd.015 109.57 i 12 N - - - 109.74.227.05 1.105 110.39.224.01 1.087 U - - - u 15 C 111.89.221.23 1.109 N - - - N - - - 109.35.216.09 1.174 113.38.222.08 1.o68 65 C 109.06.221.10 1.109 U - - - U - - - 109.18.218,14 1.112 109.21.217.17 1.113.025 91.45 10 N - - - 92.96.201.13 1.068 93.10.199.05 1.064 91.09.196.08 1.120 91.09.195.06 1.120 14 N - - - 92.54.198.14 1.074 92.29.197.12 1.087 90.31.194.10 1.142 92.91.197.06 1.069.0375 77.09 13 N - - - 77.32.178.16 1.104 77.57.178.12 1.097 77.81.178.17 1.090 78.08.178.09 1.082.050 66.88 11 N - - - 68.06.167.11 1.077 67.94.166.06 1.080 67.58.164.09 1.090 66.68.163.04 1.116 15 N - - - N N - - 69.01.166.10 1.115 68.41.166.07 1.132 Bi-Zn-In Alloys j - In.015 109.57 48 C 109.13.226.32 1.110 107.79.223.33 1.148 107.22.219.34 1.164 107.08.215.33 1.168 106.54.211.32 1.183.025 91.45 43 C 89.82.191.16.110 U - - - - - - 87.95.191.26 1.163 87.91.191.28 1.164 49 C 90.41.191.25 1.110 89.69.190.22 1.131 89.52.190.21 1.135 89.72.191.16 1.130 90.09.190.14 1.149.0375 77.09 44 C 75.66.174.42 1.110 76.08.175.26 1.100 75.85.175.21 1.106 75.84.175.12 1.106 75.73.175.08 1.110 54A C 75.45.175.05 1.110 75.94.176.05 1.096 75.77.176.05 1.101 75.60.175.05 1.106 75.27.175.05 1.115 54B C 75.89.175.09 1.110 75.92.174.07 1.143 75.90.175.07 1.110 75.56.175.06 1.119 75.06.174.05 1.134.050 66.88 50 C 66.18.160.16 1.110 65.86.159.13 1.119 - - - - - - 64.32.158.19 1.162 51 c 66.12 - -.110 N - - - 65.76.168.16 1.120 65.43.168.20 1.130 N -.- -

TABLE VIII (CONT'D) 0.015.025.0375.050 STD EMF EMF at,e EMro at Se_ EMF at SCE EMF at 3eE EMF at de Run No X^n for Binar 550C t a T C Ty 550CC t a 1n1 550C t Iny 550C In 50C I 550C 1 a Iny 31-Zn-Sn Alloys j = Sn.015 109.57 25 C,D 109.92 - - 1.110 110.66 - 1.089 110.22 - - 1.102 111.62 - - 1.062 111.52 - - 1.065 47 C 108.01.216.18 1.110 108.64.216.18 1.092 108.91.216.18 1.085 109.52.216.18 1.068 110.16.216.18 1.049.025 91.45 20 C 92.21.194.57 1.110 95.85.197.62 1.007 95.39.196.58 1.023 94.64.195.69 1.042 94.40.194.57 1.098 27 c 91.63.197.32 1.110 92.03.196.23 1.099 91.49.193.58 1.114 93.97.198.41 1.044 92.14.195.35 1.096.0375 77.09 26 c 77.84.183.03 1.110 U - - u - - - 79.58.183.06 1.061 80.40.183.04 1.038 45 C 76.16.175.06 1.110 76.57.175.06 1.098 U - U - - - U - - - 46 c 75.58.174.15 1.110 76.08.174.13 1.096 76.21.174.14 1.092 76.61.174.14 1.081 U - - -.050 66.88 23 C 66.17.165.06 1.110 67.08.165.08 1.084 68.91.167.09 1.033 U - - - 69.03.166.12 1.029 24 c 67.26.164.21 1.110 67.28.164.18 1.109 67.54.164.18 1.102 67.83.164.18 1.094 67.96.165.22 1.090 Bi-Zn-Sb Alloys 3j Sb.015 109.57 36 c N - - - 111.54.217.34 1.055 114.73.217.30.965 N - - - N - - -. 37 C 109.63.212.29 1.110l 111.15.210.28 1.067 113.82.210.29.922 115.88.209.30.934 117.66.209.27.884.025 91.45 32 93.04.198.24 - 93.69.194.20 1.046 95.81.195.16.986 97.71.194.22.933 99.76.194.20.875.0375 77.09 34 76.48.179.18 l.110o 79.29.178.18 1.030 80.68.176.12.991 82.49.176.12.940 84.38.175.12.887.050 66.88 28 66.64.1589.97 1.109 U - - - 70.70.161.21.994 71.85.160.24.962 75.18.161.28.868 36 N - - - 68.89.162.20 1.053 N - - - 72.57.6.24.950 N - - - Bi-Zn-Hg Alloys j Hg.015 109.57 59 C 103.52.209.37 1.110 104.39.210.32 1.086 103.78.209.36 1.103 U -. - - U - - -.025 91.45 60 C,D 91.61.194 - 1.110 92.57 - - 1.083 93.57 - -.055 94.38 - - 1.032 95.35 - - 1.005 Bi-Zn-Pb Alloys j u Pb.015 109.57 38 c 110.51.217.21 1.110 110.68.218.18 1.105 110.35.218.19 1.115 107.87.215.25 1.185 112.19.221.26 1.063 64 C 110.48.219.25 1.110 109.80.218.37 1.129 109.29.217.42 1.144 108.64.217.38 1.162 108.73.219.30 1.159.025 91.45 40 c 91.6l.194.20 1.110o 91.14.194.24 1.123 90.47.194.26 1.142 90.04.193.27 1.154 89.22.193.19 1.178.0375 77.09 42 77.07.176.34 1.110 76.27.175.34 1.132 75.51.174.23 1.154 75.69.175.38 1.149 75.04.175.36 1.167.050 66.88 39 66.47.163.14 1.110 65.68.162.09 1.132 65.10.162.18 1.149 64.98.162.14 1.152 64.68.162.19 1.160

TABLE VIII (CONT'D) X 0.005.010.015.020 STD EMF EMF at (8 EMF at (8 EMF at 8) EMF at (8e) EMF at () Run No X for Binary 550~C at a ln7 550~C at oa n11 550~C at a ln 550~C at a 7 550"C at n7 Bi-Zn-Au Alloys j - Au.015 109.51 31 CD 110.00.225 - 1.110 N - - - N - - - 121.92 e.221.04.774 N 35 N - - 114.49.228.05.971 116.75.223.o8.908 119.95 e.230.09.818 N.025 91.45 33 C 91.20.198.11 1.110 93.84.196.16 1.036 98.71 e.190.31.927 102.94 e.181.26.779 104.89 e.185.10.724.050 66.88 29 C 66.66.179.37 1.110- - - N - - - 78.36 e.148.10.780 N 35 N - - - 70.60.155.29 1.005 74.15 e.14.08.905 N N - N.025.0375.050 STD EMF EMF at E EMF at aE EMF at ae Run No XZn for Binary 550~C at a ln7 550~C 6t a lny 550~C ot a lny Bi-Zn-Au Alloys. = Au (cont'd).015 109.51 31 C,D U 127.41e.225.18.619 133.66 e.222.08.443 35 N - - N - - - N - - -.025 91.45 33 C N -N - - N - -.050 66.88 29 84.07e.129.29.619 U - U 35 N - - - N N - * Mole Fraction weighed-in basis EMF = my; (t) = mv/~C for range 450~-650~C; a = Standard error of estimate of EMF data from least squares line, mv ln7 log activity coefficient, based on corrected EMF if so indicated in second column C = All EMF's corrected using difference of binary EMF from standard EMF as the cell correction factor D = Unstable standard, EMF obtained as difference from standard value assumed for binary N = Composition not run U = Discarded because EMF's were unstable or grossly in error e = Extrapolated value, obtained from data taken in single phase region at higher temperatures

-96TABLE IX SUMMARY OF INTERACTIONS WITH ZINC IN DILUTE SOLUTION WITH MOLTEN BISMUTH 7ZnC [ --- X -~ j In3zn Zn 2 n zn Solute-j Temp. C XZn*.015.025.0375.050 X — O Zn = Zn Cu 450 -3.2 -3.4 -3.7 -4.3 -2.95 -19 500 -3.0 -3.2 -3.4 -4.0 -2.7 -19 550 -2.7 -2.9 -3.1 -3.6 -2.4 -19 600 -2.5 -2.7 -2.9 -3.3 -2.2 -19 650 -2.3 -2.57 -2.9 -2.05 -18 Ga 450-650 +3.6 +0.6 +2.7 +.4 +0.7 +0.5 +5.0 +.7 -100 +20 Ag 450 -3.5 -3.7 -4.0 -4.2 -3.3 -17 500 -3.1 -3.4 -3.6 -3.8 -2.8 -20 550 -2.8 -3.0 -3.2 -3.5 -2.5 -20 600 -2.5 -2.8 -3.0 -3.2 -2.2 -21 650 -2.2 -2.5 -2.7 -3.0 -1.9 -22 Cd 450-650 -0.3 -0.3 -0.5 0 -0.3 0 In 450-650 +1.7 +1.0 +0.3 +0.5 +2.2 -40 Sn 450-650 -0.8 -0.7 -1.1 -1.0 +0.6 -0.5 -12 Sb 450 -5.4 -5.4 -5.5 -5.3 -5.4 0 500 -4.9 -5.1 -4.9 -4.9 -4.9 0 550 -4.4 -4.5 -4.4 -4.5 -4.45 0 600 -4.2 -4.4 -4.1 -4.3 -4.2 0 650 -4.0 -3.8 -3.8 -4.0 -3.9 0 Au 450 -22 -25 - -32 -19 -220 500 -21 -22 - -27 -18.5 -140 550 -19 -19.5 - -21 -18 -50 600 -18 -17.5 - -17 -17.5 0 650 -16 -15 - -13 -17 +80 Hg 550 -.5 to -2.0 -2.0 - - -1 to -2? Pb 450 +1.4 +1.5 +lo4 +1.4 +1.4 0 550 +1.3 +3 +1.3 +1.3 +1.3 0 650 +1.2 +1.2 +1.2 +1.1 +1.2 0 *Mole fraction

-97discontinuity was noted in the emf-versus-temperature relations between 500 and 600~C for some of the compositions studied. These discontinuities were the temperatures of transition from two- or three-phase regions to the single phase liquid regiono(56) The emf's reported in Table VIII where 550~C is below the transition point were obtained by extrapolation from the single phase regiono The potentials thus determined apply to the so-called "super-cooled" liquid. In the single-phase region, the slopes of the emf-temperature relations were decreased slightly by the addition of copper, thus indicating that the entropy of mixing was decreased. This system was the first one studied and the results of several runs were questionableo In addition, the reference binary electrode was not routinely included with each cell at the time of these runs. Consequently, the system was later reinvestigated and most reliance is placed on the results of the later runs. The individual plots of lnyzn versus xCu (Figure 11) are quite consistent, In all cases, the addition of copper increased the electrode potential at constant zinc content, thus causing a decrease in both the activity and activity coefficient, Over the range studied to.050 mole fraction copper, the effect on lnyzn was linear with the mole fraction of coppero The limiting slopes from these plots were a linear function of the mole fraction of zinc in the region below.0375 mole fraction. (Figure 12) The interaction parameters ranged from about -2o1 to -2.9 and

-98varied linearly with the reciprocal of the absolute temperature (Figure 13)o The second-order interaction parameter (the limiting slope of the lines in Figure 12) appeared to be almost constant with temperature and was -19o 2o Bismuth-Zinc-Gallium System The experimental results for the bismuth zinc gallium system are presented in Table VIII and Figures 14 through 16, The emf-versus-temperature relations were linear throughout the entire temperature range studied and their slopes were generally unchanged by the gallium additiono Thus, a single-phase liquid was present at all temperatures studied. The small additions of gallium increased the activity of zinc when the zinc content was low. However, the effect of the gallium addition became less as both the zinc and gallium contents were increased. In contrast to the bismuth-zinc-copper system, the effect of the added element was not linearo Replicate runs performed at o015 mole fraction zinc were not in complete agreement, During Run 52, the effect due to gallium appeared to have become less at lower temperatures during the latter portion of the run, This occurred during one late-night period when the apparatus was left unattended, Consequently, the run was divided into parts A and B at this point and considered as two separate determinations, The two parts of Run 52 were in good agreement at the higher temperatures, but the agreement was poorer as the temperature was decreased, The agreement with the replicate run, Run 58, was variable, depending on the temperature consideredo

-99Three determinations were made at.025 mole fraction zinc (Figure 14) and the agreement between runs was better. For the higher concentrations of zinc, the effect of gallium was very slight. The interaction parameter was determined as 5.0 +.7 from the median limiting, slopes indicated on Figure 15. There was no conclusive evidence for temperature dependence of the interaction parameter. The estimated second-order parameter was -100 + 20. 3. Bismuth-Zinc-Silver System The experimental results for the bismuth-zinc-silver system are presented in Table VIII and Figures 17 through 19. Published phase diagrams for this system(57) indicated that a wide-range single phase liquid existed at the bismuth-rich corner. The experimental observations of linear, continuous emf-temperature relations confirmed that the alloys were single phase. All the electrodes were well-behaved The addition of silver raised the electrode potential, thus decreasing the activity and activity coefficient of zinc. The effect on lnyZn was linear with the mole fraction of added silver. The entropy of mixing of zinc was decreased slightly by the silver additionsO The interaction parameters ranged from -1,8 to -3,3 and varied linearly with the reciprocal absolute temperature (Figure 19). The second-order interaction parameter at 550~C was determined as -20 and appeared to have a linear dependence on reciprocal absolute temperature.

-10041 Bismuth-Zinc-Cadmium System The experimental results for the bismuth-zinc-cadmium system are presented in Table VIII and Figures 20 through 2.2 The interaction between cadmium and zinc was very slight and at the border-line of the region of experimental accuracy The most consistent interpretation of the plots of lnyZn versus xCd was that the interaction was very slightly negativeo The first-order parameter was estimated as -O3 and the second-order parameter was zero~ No temperature dependence could be detectedo The emf-temperature relations were linear and continuous and the slopes were little changed by the cadmium additions. All the solutions studied were single phaseo(51) The very slight negative interaction was consistent with the iso-activity data presented by Oleari and Fiorani(5l) for higher concentrations of zinc, The most probable cause of the scatter in the experimental results for this system was the fact that the zinc-cadmium combination has free energies of chloride formation whose difference is close to the minimum value of AG~ for Type iL displacement. Chemical and spectroII graphic analysis showed that the zinc content in the ternary alloys did not change measureably during the runs, however, the accuracy of the analyses may not have been sufficiently good to detect slight changes in zinc content that could account for the scatter,

-1015. Bismuth-Zinc-Indium System The results for the bismuth-zinc-indium system are presented in Table VIII and Figures 23 through 25. The emf-temperature relations were well-behaved throughout the whole temperature range and the alloys were all single phase. The addition of indium increased the activity and activity coefficient for all compositions studied. However, the effect was diminished as the concentration of zinc was increased. For.015 mole fraction zinc, the effect of the indium addition was not linear over the entire range to.050 mole fraction indium. For the higher concentrations of zinc, the effect did appear to be linear. No measurable temperature dependence was found for the interaction parameter which was determined as +2,2. The second-order parameter was -40 and was also appeared independent of temperature. The agreement between replicate runs was generally quite good. In Run 54, at.0375 mole fraction zinc, there was a shift in the potential of the binary reference electrode during one over-night period, The run was separated at this point and treated as two determinations. When the "cell factor" was applied to the two sets of data the corrected emf's of the ternary electrodes were found in good agreement. This emphasized the value of the "cell factor" for referring the interactions to a common base pointo The positive first-order interaction parameter was consistent with the observations of Yokokawa et a.(36) that indium additions increased the activity of zinc at higher concentrations of zinc,

-10260 Bismuth-Zinc-Tin System The results of the study of the effect of tin on the activity of zinc in molten bismuth are summarized in Table VIII and Figures 26 through 28o The published data(52) and the experimental observations were that a single-phase liquid existed at all temperatures and compositions studied in this investigationo The effect of the zinc additions was to increase the electrode potential, thus decreasing the activity and activity coefficient of zinc. The effect appeared to be linear with the mole fraction of added tino In the determination of the interaction parameter, most reliance was placed on the data taken at o015 and o0375 mole fraction zinc, The replicate runs for these compositions were in good agreement, while data taken at the other zinc contents exhibited some scattero There was no significant temperature dependence to the interactionso The first-order interaction parameter was fixed at -o5 and the second order parameter at 12o The experimental results were qualitatively consistent with the observations of Yokokawa et al(36) and Oleari and Fiorani(52) for more concentrated solutionso 7. Bismuth-Zinc-Antimony System The experimental results for the bismuth-zinc-antimony system are summarized in Table VIII and Figures 29 through 31o All the compositions studied were single-phase throughout the experimental temperature range,(57) The addition of antimony decreased

-103the activity and activity coefficient and the effect on lnyzn was linear. The agreement between replicate runs was excellent. For.025 mole fraction zinc (Run 32), the cell factor correction was not applied since it was found that the values of lnyzn calculated from the uncorrected emf's for the ternary alloys fell on a line intersecting the ordinate (Xsb = 0) at the standard value of l1nzn for XZn =.025. Apparently in this case, the binary electrode may have contained less zinc than intended, since its emf at 550~C and emf-versustemperature slope were slightly high compared to the normal expectation. The interaction parameters ranged from -3.9 to -5~4 and were linear with the reciprocal absolute temperature. Second-order effects in this system were negligible. 8. Bismuth-Zinc-Gold System The results for the bismuth-zinc-gold system are summarized in Table VIII and Figures32 through 34, The addition of gold was found to cause a marked decrease in the activity of zinc, In addition, discontinuities were observed in the emf-temperature relations, thus indicating that the transition to the single-phase liquid solution occurred within the normal range of compositions studied. By reducing the gold additions to the range from.005 to.020 mole fraction it was possible to obtain sufficient data in the single-phase-region in the temperature range from 450 to 6500~C to establish the limiting relations between lnyzn and xAu. Estimates were also made of the position of the liquidus surface at the bismuth-rich corner of this system (ppo128-134).

-104 - The interaction parameter ranged from -17 to -19 and appeared to be linear with reciprocal absolute temperature. The second-order parameters ranged from +80 at 650~C to -220 at 450~C and also appeared to be linear with reciprocal absolute temperature, Since the interaction effects were well-defined by the data obtained at o015,,025, and ~050 mole fraction zinc, no runs were made at O0375 mole fraction zinc. 90 Bismuth-Zinc-Mercury System The results obtained for the bismuth-zinc-mercury system are summarized in Table VIII and Figure 35. Due to the volatility of mercury and the potential health hazard, it had been decided to make only enough runs on this system to define the interaction effect and9 if possible9 estimate its extent, The experimental system was maintained closed to avoid mercury loss and the runs were discontinued as soon as the trend of the mercury additions became evidento In Run 59 (,015 mole fraction zinc),> the electrodes containing o015 and,025 mole fraction mercury had increased potentials, while the results for the electrodes containing o0375 and,050 mole fraction mercury were rejected due to instabilityo In Run 60 (o025 mole fraction zinc) a consistent effect was found over the entire composition range studied, A linear decrease in lnyZn was observed as xHg was increasedo

-105The interaction parameter was estimated as -1 to -2 with most reliance placed on the data from Run 60. There appeared to be no temperature dependence to the interaction. The second-order effects, if any, were uncertain. From the two runs, the evidence was that mercury is a moderate negative interactor with zinc. The study of the system was discontinued at that point. 10o Bismuth-Zinc-Lead System The results obtained on the bismuth-zinc-lead system are summarized in Table VIII and Figures 36 through 38, The published information and the experimental observations of emf-versus-temperature behavior were that this system was composed of a single-phase liquid at all temperatures and compositions studied in the present investigation(50) The addition of lead caused a decrease in the electrode potentials, thus raising the activity and activity coefficient of zinc. The effect of lead on lnyzn was to cause a linear increase with increasing mole fraction of leado Only a very slight temperature dependence was found for the first-order interaction parameter and the second-order effects were negligibleo The interaction parameter ranged from +1.2 at 650to -+-1o4 at 450~C. (50) The iso-activity plot of Valenti et al 5 indicated that lead tended to increase the activity of zinc, The present results in dilute solution were consistent with those observationso

-10.9 r.u, 015 KZ -.025 T< erzture'C t^u ^ ~ Tewperstlls5 *C so< " 0C t;LlC 0 0.? - soo - 3.o e Run 6 0.7 50 -2.7 5 550 -2. 9 65o -2.3 6oo -2.7 305 -2. 0 0.01 0.02 0.03 0.04 0.05 0.01 0.02 093 0.04 0.05 CCucu- MOLE RACTIO N. 1.. - -— Ti --—' *~575 1 | t c ( 050D 50.-.7 5 55050 - 6 600 -2.5-55 -23. 6550 -2. 650 -2.7 650 -2.5 0.6 ----- --------— 1 ---- I ---— 0.6 - 0 0.01 0.02 0.03 0.0 0.05 o 0.01 0.02 lO3 0.04 0.05 XC-M -MOLE FRACTION XCU - MOLE FRACTION 1.3 - o3 Figure 11. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Copper - for Indicated50 Constant Mole Frac s of k.2C o1 -~.? 500 SC I I ^Sct c 11. ^ ^ ^ > ^ 550C -~. o C 0.9 0.9 600 IC ^ —^c 6~00 -. 650 /UL ^^7^ 0.60- Ba 70 500 500 > 600 -3.3 O0-. 650 -20.04 0.50 001. 0.0 0.04 0.05 f -MLEc FHACTcNXC —MOLE FRACTION Figure 11. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Copper - for Indicated Constant Mole Fractions of Zinc.

0'.'0 I I * I -I -I o' 0 X X x -2X -2 I -3 U-3 650~N S -4 -5, Ii\ -5., 0 0.01 0.02 0.03 0.04 0.05 0.OO10 0.0011 0.0012 0.0013 0.0014 XZn -MOLE FRACTION I/T - ~K-' Figure 12. Determination of Cu-Zn Interactions Figure 13. Temperature Dependence of Firstin Molten Bismuth. Order Cu-Zn Interaction Parameter.

~~~~~~~1.24~ ~ __.- 4 —. 1.2 X^Q^~~~~n~X -..1. iaO A R^ —un-_ _ ^ _ -- 40 C 5.0.8 0 09~o Run~ 1 t d N(^te —r aure dependence: XG- MOLE FRACTION X__.-MOLE FRACTION (X~n)' ~375~ _.5~ Ru0n 0 t56 1.4 ] O] g^ -0 —-- 0 Run- 55 ______ __. 1.- 1.0 600C 9 6'C5 —0t- u500 -------— moa [ No temperature dependence: No temperature dependence: +0.9 7 0.9 _ 5 C - 5 0 -MOLE FRACTION MOLE FRACTION 025 n r I ndi d Consant Me F ions of Zinc. Minimum +3.0 0.7, I I I, 0.7, I i11,, 0 0.01 0.02 0.03 0.04 0.05 0 0.01o 0.02o 0.03 0.04 0.05 XGa- MOLE FRACTION Xa-MOLE FRACTION 1.5 1.5 xZn = *0375 - xn -.050 1.4 - 1.4-.3 500 IC 1.2 - 500 C 1.2 - e c 1. xa 0 Run 55 08 ^O 001 002 003 004 0.05 0 t 0.01 0n02 0. 03 0.04 0.05 X R- MOLE FRACTION X5- MOLE FRACTION Figure l4. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Gallium - for Indicated Constant Mole Fractions of Zinc. +0.7 +0.5~~~~~~~~~~~~

O 6. — I I I 6 I I 0 o K c 2 - 2-. 0.01 0.02 0.03 0.04 O.5 0.00 I0 0.00124 Xn — MOLE FRACTION I/T- o K Figure 15. Determination of Ga-Zn Interactions of in Molten Bismuth. Figure 16. Temperature Dependence of FirstOrder Ga-Zn Interaction Parameter.

-1101.4 1 I 1.,,,,, 1.4 Temperature ~C XAg 0Temperature C 0.7 - "| -0.750 -3.57 550 -2.8 ORun 18500 -3.8 650 -2.2 650 -2.5 ORun 21 0.6, 0.6 _,_,_,_,,,0.6 * _________i_______ 1.4 G 1.4' (<^ AXzQn = M.375 Xzn =. 05 0.8 - 0.0 003 0. -0s - 0.02 0.03 0.0 )lx- 050 \( Ag.5 XAg xZn.025O A fo I cTemperatureaTemperature ~C Xn= 0 Temperature ~C ---- so 55 -2 0.7 450 -3.5 0.7 450 -3.5 500 -3.6 550 -2.8 Run 18 500 350 -3.2 Run 19 0600 -3.2 0 R un 1 7 60 -2.2 550 - 3. 0 650 -3.5 0 Run 21 650 -2.7 0.6...0.6 i _______ 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 004 0.05 Xa- MOLE FRACTION X - MOLE FRACTION Figure 17. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Silver - for Indicated Constant Mole Fractions of Zinc.

— I -I- - -I — I.o 0 0.01 0.02 0.03 0.04 0.05 0.000 0.001 0.0012 0.0013 0.0014 X- MOLE -FRACTION H/T- ~K10 -4 _ _ _ _ _ _ _ _ _ _ _ _ I I I - 5 i I J 0 0.01 0.02 0.03 0.04 0.05 0.0010 0.0011 0.0012 0.0013 0.0014 XZn- MOLE - FRACTION I/T - OK-' Figuzre 18. Determination of Ag-Zn Interactions Figure 19. Temperature Dependence of Firstin Molten Bismuth. Order Ag-Zn Interaction Parameter.

-112XZn =.015 1.4 450C 0 0 1.3 1.3 0 A 500 C 0. 1.2 1.2 A 550-C 1.1' 1.0 A 6000C.0 _ 0-1 ~ ~ ~ 650 0.9 A' 650~C 0.9 0 0 0 0.8 a XZn= 0015 o Run 12 0.8 ad Zn Run) 1 0 Run J ^XCd = 0 ~ i 4 temperature dependence: un 65 No temperature dependence: X Std.Values approx. -0.3 approx. -0.3 0.07 0.704 0.0 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.0 0.03 0.04 0 XCd- MOLE FRACTION X - MOLE FRACTION Zn =.037 Xn 050 1.4 * 1.4 450"C 1.3 - I C _50 0 0 500I C 0c 6 IOox —-- -- O.9X —-O I.O.X —---- 6000C 0.8 Znn 0. XZn - 050 0 Run 11 x-Od ) n0 N unl _ 0 I. N n~e No temperature dependence: X Std. Values No temperature dependence: X Std. Values approx. nil -0.5 07 0 0.01 0.02 0.03 4 0.05 0 0.01 0.02 0.03 0.04 005 X 0.01-0.0....03 0.04 0. X -MOLE FRACTI...ON X - MOLE FRACTIONCd Figure 20. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Cadmium - for Indicated Constant Mole Fractions of Zinc.

I I I l I z j 0 - 0 C 0 N 0^~~~~ UN. - H xO 5.j < -2 -2 3,1 1 -3 1 I I 0 0.01 0.02 0.03 0.04 0.05 0.0010 0.0011 0.0012 0.0013 0.0014 XZn- MOLE- FRACTION I/T -~K-I Figure 21. Determination of Cd-Zn Interactions Figure 22. Temperature Dependence of Firstin Molten Bismuth. Order Cd-Zn Interaction Parameter.

-1141.5.,., 1.5 XZn 015 xZn =.025 0.7. - ______ _ 45)_C 0_ 4_.7 1.4 0 2 3 4 0 500~ C I.- 1.3 50 1.2 5 (ln7Zn I Zn /X\ / XZn =.015 X8In 0~ 0 Run 48 In xZnI02O Run 43 0.8 No temperature dependence: 0.8 No temp e Run 49 No temperature.+ 1 dependence +1.0 0.7, 0.7 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Xn- MOLE FRACTION Xn- MOLE FRACTION 1.53 -, 1.53 1.4 1.2 450 - {~ 1.1 ~ o niaeaCntn oeFrcino ic1.. 1 iil F5006C 0; 1.2i 1.2 - o.9< --- — g —— ri650*C o.9 —---- - | rln7zn).% | |-(-SXIn XZn'050 0 Run 50 0.8 x/n XZn 0.375 0 Run 44 0.8 XINo temperatre Run 51 XIn=O m Run 54A dependence No temperature V Run 54kB +0.5 depe~ndnce 0.7,.....,,. 0.7 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Xrn- MOLE FRACTION Xzn- MOLE FRACTION Figure 23. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Indium - for Indicated Constant Mole Fraction of Zinc.

3 rllll 31''' II -A Nc 0 2-! Ma 0 0 0 0.01 0.02 0.03 0.04 0.05 0.000..0012 00013 00014 XZn-MOLE FRACTION I/T-~K'I Figure 24. Determination of In-Zn Interactions Figure 25. Temperature Dependence of Firstin Molten Bismuth. Order In-Zn Interaction Parameter,

1.3- — 3. 550- 1.1 1. —------ ---- -0 ~00___ *~C -0 0 10 0.8 - (xsn2C / n ^- 0.8 ( Sn ) xzn - 025 dependence: O Run 25 -No temperature dependence: ~ Run 27 0 -0.8 o Run 47_Q60 OC O 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Xg -MOLE FRACTION XSn - MOLE FRACTION 1.3 - 1 1 1.4 1 ^ 5 ~ -- D - - - - - - ^ _ __ _ 6 o ~1.20.5e~C n__.- - os 6090 0C XSn XSn a 0 0.8 - XXn. 0575 0.8 ( f-' negligible temperature e Run 25 0 Run 27 Negligible temperature dependence: Nu46 6No temperature dependence: O Run 24 0.7- - 0.07 - | -1-1 -| ~07 Median -1.0 ~~~~~~~~0.9~6.6~~Maximm 1.6 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 004 0.05 Xsn- MOLE FRACTION Xn - MOLE FRACTION Xzn.0375 XZn 50 Figure 26. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Tin - for Indicated Constant Mole Fractions of Zinc. 1.0 1.0 for Indicated Constant Mole Fractions of Zinc.

Iz (ton I I z 0 C N AC 0 _.1 -I, 0 0.01 0.02 0.03 0.04 0.05 0.0010 0.0011 0.0012 0.0013 0.0014 XZn- MOLE FRACTION l/T -~K-' Figure 27. Determination of Sn-Zn Interactions in Figure 28. Temperature Dependence on FirstMolten Bismuth. Order Sn-Zn Interaction Parameter.

-1181.4 I |.4 45xZ.015 X _5. 025 1.3 1.3 1.2 - 1.2 1.0 0, o "C.. 0.80 b l n 7 3 ~ ~ > ~ - 1 0.8 ( 7Zn _ "C ( 5xgb ) xzn -.015 650 CTemperature'C _____ \_0 C| 0.7 - Temperature'C XSb 0 ~C 0.7 - 450 5 4 5 450 -5.4 500 -5.1 500 -4.9 0 Run 57 550 -4.5 550 -4.4 Run 600 -4.4 600 -40 650 -3.8 X Std.Values 0.6 - - 4-0- - - 0.6 - 0 0.01 0.02 0.03 0.04 OD5 0 0.01 0.02 0.03 0.04 0.05 XSb- MOLE FRACTION XSb- MOLE FRACTION 1.4 — 1 --------— 1.4 1 (l^^xn, -n.0375 XZn " x050 50 O -4.9 Run 7 5500 -4.5 Figure 29. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Antimony - 1.7 -for Inicatei0 Constant Mole Fractions of Zinc.~ 650 -5o 0 I 0 0.01 0.02 0.035 0.04 5Temperature0 0 0 0.02 0.03 0.04 0.05 0Figu Temperature'C XSbNa a 0a7 o Z c t -5.3,550 -4.4 Consa M -4.3 o, 650 -3.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_ 0.6 0.6~~~~~~ 1.31 00.3 00 X ~ O 00.2 00.4 00 l~ Xb OEFATONXb OEFATO Figue2.NtrlLgrtmo icAtvt ofiin essMl rcinAtmn for Idicaed Cnstat Moe Frctios ofZinQ

-2 i I I -2 1 - -3 - -3 Z CO z 0 0 0 0 650 0 C c -4 ~~~N _ ~0 ~600 ~o0C n s h x O' ~'o a G 550 0C o0 0 cou)~~~~~~~ ~500 C -5 — 5;,T 1~~a ~~450 C c 0 XJ x -61 I-6 0 0.01 0.02 0.03 0.04 0.05 0.0010 0.0011 0.0012 0.0013 0.0014 X zMOLE FRACTION I/T - 0K-' zn Figure 31. Temperature Dei:etidence of FirstFigure 30. Determination of Sb-Zn Interactions Order Sb-Zn Interaction Parameter. in Molten Bismuth.

1.4 1 I.4 A 1.4 -Xzn.015 XZn *.025 *.050 1.3 1.3.3 Au -\Zn -.015 0 xx3 a ja Zn - " -.025A Z.050 Tagepetur. C -0 Tertu C XAu 0T epartr C Xu - 0 ~~e~~~c~~4~~~c ~ ~ Tmperature ~C Cl 450 -22 500 -21 450 -25 450 32 1.2 550 -19 1.2 500 -22 1.2 \500 -27 600 -18 550 -19.5 550 -21 650 -16 600 -17.5 600 -17 0 0 60 650 -15 650 - 5 \ \ \' \ \ -\ 1.1 10. 50 *C ~ 0 \ 1.0 D\ \ 6 \\ 0 \\~\ 0.. 0 00 \D \ D\ \ \ 0 Run 35^'KV^V~~~~~~~~~~~~~~~~~~~~~~~1 \ 0.9 500 " C ~~~~~ ~~~0.9 0.9 \ l ~ 5\ 450 Sc a 1.0 ~ ~~~~~~~~~~~~~~ 0 0.8 D550 RC 0.8 0 500 Ca0.80 0 0 \dl ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 Run 3 0.750 0 550C 0.7 0.7 ~~~~~~~~~~~~~~~~~~~~~~~~~0.7 oRn2 0.7 ~~~~~ ~~~600'C 0u Rw31 \0s oRun3 5 0 \ \ 450C 05u35 650 _ _ _ _ 50 6C 60 0 5 0.6_________ Io C _______ 06________________\ gS C ________ I.__________________. \_60.~_________________ 0.6 0.6 I I 0.6 6 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0 2 0.03 0.04 0.05 0 001 002 0 003 004 005 XAu - MOLE FRACTION XAu - MOLE FRACTION XAu- MOLE FRACTION Figure 32. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Gold - for Indicated Constant Mole Fraction of Zinc.

0 O -— I0 -5 Z -10- l -10 0pg ~~~~10~~ 650 ~C N -20 C15 - ^Cx~~~ ^ " — 450 C - C H-30 = -20 1 -400 0.01 0.02 0.03 0.04 0.05 -25 0.0010 0.0011 0.0012 0.0013 0.0014 X -MOLE FRACTION /T ~K Zn i/T- ~K-I Figure 33. Determination of An-Zn Interaction Figure 34. Temperature Dependence of Firstin Molten Bismuth. Order Au-Zn Interaction Parameter.

-1221.4 -! I 1.3 1.2 OM;i MM^Run 59,t:- = —= XZn =. 015 Temperature - 550~C uxn 6 05 1.0- (lnyZn\ - XHg XZn = const. XZn ___ XHg = 0.015 -.5 to -2.025 -2 Apparently no 0.9 O- temperature dependence 0.8 - 2 " 0 0.01 0.02 0.03 X Z- MOLE FRACTION 0 0.01 0.02 0.03 0.04 0.05 0.6 —-I-I -I 0 0.01 0.02 0.03 0.04 0.05 Xg - MOLE FRACTION Hg Figure 35. Natural Logarithm of Zinc Activity Coefficient versus Mole Fraction Mercury —for indicated constant mole fractions of zinc.

-1231 5 1.5 0 (5r lXXzn = 005 Xzn =.025 450 OC 1.34 = 4 1.4 5+ 1.3).00 OC 1,3 —500 -CI cd 1.2 5 - 1.2+ - 0 650.0 —-1 +1.2. 0 ^^ 5S o600 0c 0. 8 ~erature *~C X, o 0 un Te6perat4re ~C - >- = 0 O Run 40 - 1 650 OC 1 0 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 V,H XPb -MOLE FRACTION X-MOLE FRACTION. x~n = -:J 550 +1.54 550 +1.3 +1.521 2 ~ ~0,P 13 650 +1.2 550 + —-1 - - 0.71 ( 0.7 0 5 0 ~c_> -- 55-650 +1.1 __________________________..__ —.> — r o o_.. _ 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 Xpx-MOLE FRACTION XZn -MOLE FRACTION 4e50 ~C 45o =C Temperature ~C Xpb = 0 (3 Run 40 1~.4 FRACTIO. - N1.4 + -L F 500 +lC 550 +13C P 1.3 6 1.3 - I. l, l,,1, H Xpn =.037 5 xn.5 60'~s - -r-{ o 0.0.0.0.0. q_ Xpb-MOLEFRACTIONXpb-MOLEFRACTIo

3L I I I I 3i I II — 4 IC) 0 Ic 2 X B. c - _ p 450 C 550 C r. ~ 6504C x - 0 00 0 0.01 0.02 0.03 0.04 0.05 00010 0.0011 0.0012 00013 00014 X Zn- MOLE FRACTION I/T - KFigure 37. Determination of Pb-Zn Interactions in Figure 38. Temperature Dependence of FirstMolten Bismuth. Order Pb-Zn Interaction Parameter.

-12511. Summary of Ternary Interaction Results The experimental results for the ten ternary systems studied are summarized in Table IX. The parameters included in this table are the intermediate slopes and the firsts and second-order interaction parameters between zinc and the added solute at temperatures between 4500 and 650~Co The elements having positive interaction parameters, that is, those whose additions caused an increase in the activity of zinc were gallium, indium, and leado Cadmium and tin produced slightly negative interaction parameters. Silver, copper, mercury, and antimony caused moderately negative effects, while gold had a strongly negative effecto Where a significant temperature dependence was found for the first-order interaction parameter (copper, silver, gold, antimony, or lead), the variation was linear with respect to reciprocal absolute temperature over the range studiedo A summary plot of these relations is given in Figure 39o The first-order parameter could be represented by analytical expressions of slope-intercept form, e A + B./To All the secondorder interation parameters were found to be essentially independent of temperature except those for silver or gold. For these additions, a similar linear relation with reciprocal absolute temperature was found. The constants for the temperature dependence relations are summarized in Table X. The significance of the temperature dependence and the slopes and intercepts of the relations is discussed in detail on ppo208-217, It is sufficient to mention here that they represent the changes in the entropy and enthalpy of zinc due to the ternary interactiono

-126+10 I. _ + 5 Go In Pb 0 Zn n E (Bi Zn Ag Cu - 5 Sb -10 -15 Au -20 650 C 600C 550C 500C 450 C -25 0.0010 0.0011 0.0012 0.0013 0.0014 0.0015 I/T -K- Figure 39. Summary Plot of Temperature Dependence of Interaction Parameters.

-127TABLE X TEMPERATURE DEPENDENCE CONSTANTS FOR INTERACTIONS WITH ZINC IN DILUTE SOLUTION WITH MOLTEN BISMUTH IN THE RANGE 450 TO 650~C B. Temperature Dependence Equation: e = Aj + T(~K) Element First-rder Interaction, Second-Order Interaction, eJn Firstcond-Order Interaction, ~Zn Aj Bj Aj B'J Cu 1.292 -3070 -19 \0 Ga 5.0 +0.7 0 -100 +:20 0 Ag 3.012 -4543 -31.2 14,872 Cd -0.3 00 0 In +2.2 00 0 Sn -0.5 0 -12 0 Sb 1.473 -4945 0 0 Au -9.886 -6640 1140 -1,000,000 Hg -1 to -2 0 - Pb 0.483 666 0 0 Note: See page 213. a -. 2 * _3s Aj =- R~ where el= aXiaX Aj _- R where a2 = X j B. = where = 2 H B' =, where R) j -,X26X j R' 1xiiXj -X R W2 XX.'.2 i j

-12812, Confidence Limits for Interaction Parameters The standard error of estimate of potential within a given run was about.2 millivolts for cells run from 50 to 150 hours at temperatures in the range from 450-650~C. The difference between the absolute potentials of replicate cells was usually less than one per cent, A previous discussion (p. 37) covered the use of the reference binary electrode for determining a "cell factor" for the correction of the multicomponent alloy results to the same base points. It is difficult to fix precise confidence limits for the interaction parameters because of the several steps involved in their determination. A formal statistical analysis was run on data for the Bi-Zn-Sb system since the relations between lnYzn and xSb appeared to be well defined and the points could be given equal weight.(68) At the 90 per cent confidence level it was possible to fix the limits for the first-order interaction parameter at approximately +10 per cent of its absolute value (see Appendix C). Visual examination of the data for the other systems suggested that closer limits could be placed on some, while the same or greater limits could be applied to other systems, However, with the exception of the systems involving mercury or gallium, it can be said with reasonable certainty that the accuracy of the measured values of the interaction parameters is about +7-10 per cent of their absolute values. 13. Determination of Phase Boundaries for Ternaries with Cu or Au In the experimental runs on the Bi-Zn-Cu and Bi-Zn-Au systems marked discontinuities were observed in the emf-versus-temperature

-129relations for a number of the experimental alloys. Above the point of discontinuity the slopes of the relations were very similar to those of the binary alloy of the same mole fraction zinc, Below the discontinuity, the slopes were markedly different and varied according to the amount of third element present. The occurence of such discontinuities is interpreted as the transition between the single phase liquid region, and underlying twoor three-phase regions. A number of investigators(636971) have used such observations to determine liquidus boundaries. Figure 40 shows emf-temperature relations for the alloys of Bi-Zn-Au and Figure 42 shows relations for the Bi-Zn-Cu alloys. The lower parts of Figures 40 and 42 are plots of the temperature of discontinuity in the emf-temperature relations versus the mole fraction of solute j for which they were observed, with the zinc content as the parameter. A cross-plot from these curves at constant temperature permitted the construction of the liquidus surface isotherms shown in Figures 41 and 43, The complete isotherms for these systems at intermediate temperatures cannot be constructed from the limited experimental data, however, their general features may be readily deduced (Figure 41)o At both the bismuth and zinc-rich corners, small regions of single phase liquids exist. These are denoted as L1 and L2, The remainder of the isotherm consists of a succession of two and three-phase regions which radiate from the liquidus surface at the bismuth corner towards the Zn-Au or Zn-Cu side of the diagram. The two-phase regions correspond to the various solid solutions and/or intermetallic phases which occur in the Zn-Cu or Zn-Au binary systems ( /

-150170,, EMF versus Temperature for Bi-Zn-Au 160 XAu *.050 ^ — ~ 0 xAu =.0375 xzn=.015 xAu =.015 xAu =.010 140 | ^ ^ xAu =.005 XAu = 0 120_ xAu =.010 xZn=.025 xAu =.005 X3fu = 0 0 0- 80 5'0 I I 0.30- MF versus Temperature =.025 - 0.02450 500 550 600 650 TEMPERATURE- t Figure ^40. Electrode Potentials versus Temperature for Bi-Zn-Au Alloys. Fpigure 40. Electrode Potentials versus Temperature for Bi-Zn-Au Alloys.

-131Au Succession of 2 and 3,phase regions Bi Zn LL L2 Idealized schematic view of isotherms in the range 450-650~C TO X =0.64 Au o/0 / Enlarged view of isotherm f / / in bismuth corner, showing liquidus lines? O TO X =0.58 f o ~4 I Au 650 C T XAu= 0.55 0/ ci( -- - 6000C -- -— TO X 0.80 Zn Li * 550~C -- -TO X 0.60 Bi O / 0 0.02 0.04 0.06 0.08 0.10 XZn Figure 41. Isotherms for Bi-Zn-Au System Showing Liquidus Boundaries in the Bismuth Corner.

-132-...... I I I' EF versus Temperature for Bi-Zn-Cu } x,.015 130 050 120 x } 375X.025 ~ 60/ ~O50 7D F XC0 OP) }x,=. 05010 - -. 025 Xfu U015 5701 ~ I90 l ~XZn-X =.015 = 05 0.05 Temperature of Discontinuity of / YE 0 EMF versus Temperature Z^ Relation / X X 0 0.02- 4-_.0 2 x0 - 0.025 - __.02 70 > ~ 1- L = 0375 0~ _ 60 0 1 _________________ —-— n. I lI-050450 500 550 600 650 TEMPERATURE- C Figure Eectrode Ptential veru Temperature for Bi-Zn-Cu Alloy.050 0.05 - Temperature of Discontinuity of EVF versus Temperature Z Relation _ 0.04 - cr_0.03 -J 0 02.-"" xZn =.025

-/ of/ o / V( 0 Enlarged view of isotherms in Bismuth oorner, showing liquidus lines I / 600 *C L' 550 C'/. * 500 ~C Bi Bi''' /, /, /, / / 0 0.02 0.04 0.06 0.08 0.10 Xzn Figure 43. Isotherms for Bi-Zn-Cu System Showing Liquidus Boundaries in the Bismuth Corner.

-134The extent of the L1 region increases as the temperature increases, In the present study, the experimental temperatures coincided with limited single-phase area for these two systemso If the experimental temperatures had been lower, similar discontinuities might have been observed for some of the other ternary systems studied. In three dimensions, the single-phase liquid area extends upward and outward from the melting point of pure bismuth (271~C), Co Prediction of Zinc Activity in Multi-Component Solutions With the establishment of the interaction parameters describing the various third element effects on the activity of zinc in dilute solution with molten bismuth, the next portion of the investigation was designed to test the applicability of these parameters in the prediction of zinc activity in higher-order solutions, A previous semi-quantitative study with limited success had been made by Okajima and Pehlke(73) for several solute additions to cadmium in lead, The Taylor series expansion suggested by Wagner(5) for the logarithm of the activity coefficient may be applied to any order solution, providing the expansion is made about the point of infinite dilution with respect to the solutes, By ignoring effects beyond the first order, the usual foirmof the expansion is X^u - -X [ L r0 (3) J,

-135f;/ ^ /~ 46 X, E 21 7 J + (21) u'yI J t1 The inclusion of the second-order effects for a quaternary (72) system increases the number of terms in the expansion(72) ~ ~ ~ " \ [C - )d ~ ) Ad; I)^. )jy X uyjl aL (22) (all derivatives at infinite dilution of solutes)o In the general case, the inclusion of the second order effects would require knowledge of all the additional terms. However, in the case of the zinc-bismuth system the self-interaction parameter is zero ( ln7? (2ln7zn and hence the terms _ and - are eliminated (Even if xZn axZn2 they were not zero, they could be evaluated from the binary data.) Furthermore, in evaluating the first-order interactions for ternary alloys based on the zinc-bismuth system, it was noted in most cases that the relations between lnyzn and xj were linear. Thus, terms above the first order involving x; would vanish (that is, a xj is zero)o Therefore, for ternary systems of bismuth-zinc-j obeying this condition, the expression for the logarithm of the activity coefficient reduces to

-; t = x 6 t j (23), =evt,LO + XJ EJ + L J G (24) For a quaternary system9 that is9 zinc in bismuth plus two additional solutes, the Taylor series expansion throtgh the second order terms becomes: f\4o " o 7 \ x^/ JI2 Lil i (25) From ternary data it is relatively easy to evaluate the parapln7 \ / 21nny Zn meters ( a~ and ( - Z and both of these terms may be readily xin / \ ~n "' Zn included in the calculation of the activity coefficiento However, the 21n^ Zn \ term l -- ) cannot be evaluated unless data are taken on the quaternary systemo In the present case where four alloy electrodes are included per run, a minimum of four runs are needede to evaluate the firstand second-order coefficients for a ternary alloyo In a quaternary

-137alloy,'16 runs would be required to similarly determine the term I lnZn -XjlX2- I o Therefore, for a highly complex alloy, the number of runs \ J! ^~j2 to define all the second order terms would be prohibitiveo Nevertheless, the second-order terms from the ternary studies appear in the expression for higher order solutions and their inclusion should improve the calculation of the logaritrhm of the activity coefficiento The expressions available for calculation of activity coefficients in multi-component alloys may be termed the first-order and second-order solution interaction modelso The first-order model is the familiar equation of Wagner(5) 41g>-U. t X. (21) where m is the number of additional solutes besides Zno (21nyZn e The second-order model adds the cross-partial term I x x J \ xd oZn and is written as B^ -jR~ r~in~, iof 4; (24) For ternary systems where lnyi is a linear function of x, and the self-interaction parameter of i is zero, the second-order model is rigorouso For solutions of quaternary or higher order, the secondorder model can no longer be rigorous but should nevertheless provide a better representation for Inyi than the first-order model,

1.38 The assumption implicit in both models is that the interactions affecting the activity of a given solute in a dilute solution are primarily those between that solute and the others present, The crossinteractions between the e.xtra solutes are taken as negligible. In order to test the hypothesis of the additivity of the interaction effects and to compare the utility of the two solution interaction models, a series of experiments were run on quaternary, quinary, hexadic, and septenary solutions based on the zine-bismuth system. The aims of these experiments were as followso,o To test the validity of the prediction models for increasing numbers of solute elements, 2, To test the validity of the prediction models for various combinations of positively and negatively interacting solute S 3o To test the applicability of the prediction models as solute concentrations increased away from the dilute regiono The experimental procedure was to determine the activity coefficient of zinc in a given multi-component solution over the temperature range from 450~ to 650~C and -hen to compare the observed value with the activity coefficient calculated from either of the prediction modelso The base point for the comparison was the value of lnyn corresponding to the standard potential for that mole fraction of zinc, A reference binary electrode weas included wi-th each multi-component run so that a "cell factor" could be determined to bring the observed potentials to the same value as used in the prediction modelo

-139= The emf-versus-temperature relation for the multi-component alloy was compared with the one for the reference binary alloy in order to confirm that the solution was single phase in the experimental temperature regiono In multi-component solutions containing gold or copper, marked discontinuities were observed similar to the case of the ternary alloys (Figures 41 and 43)o Above the temperature of discontinuity, the slopes for the binary and multi-component alloys were very similaro Electrode potentials for the temperatures below the discontinuity point were obtained by extrapolation as the "super cooled" liquido All the other multi-component alloys were single phase in the experimental region, as evidenced by the similarity for the temperature dependence of potentialo Neither gallium nor mercury were used in these studies since some uncertainty existed in the measured interaction parameters with zinc for these elementso Since the possible number of alloys and compositions that could be studied was quite large, the experiments were simplified as follows: lo Equal mole fraction additions of the extra solutes were usedo 2o Most of the experiments were run with xZn = o015, the most dilute compositions used in the binary and ternary studies. A few runs were made with xZn- O050 in order to check the validity of the models over the Henry's Law region for zinc.

-14o3o Representative combinations of interactors were used; not all combinations and compositions were attempted. By using the equal mole fraction additions, the equations for the solution interaction models could be reduced further to the following forms: First Order Prediction Model: I ^ ^-j ^ J Bn t} = t 1X L 6 ()(jt-: X (26) Second Order Prediction Model:,'-K > ZI,Q ~; X\'(27),~-j 2 ^ i, _: n;2 )+ (28) The base point for making the comparisons were the following activity coefficient values for the zinc-bismuth binary system: Temperature C 0n In n _Zn Zn 450 4o00 ol386 550 3o03 lo110 650 2,46 o90O The interaction parameter values were taken from Table IXo

For the purpose of selecting the various combinations of interacting solutes, the elements were classified as follows: Type of Interaction Element Symbol for Interaction with Zinc Positive In + Pb + Slight Negative Cd Sn Moderate Negative Cu Ag Sb Strong Negative Au The final results of these studies are summarized in Table XI, while the experimental data are included in Appendix Ao The tabulated results are grouped according to the type of solution, the type of interactors present, and the level of the zinc and added solute concentrationso The observed activity coefficient is then compared with activity coefficients calculated using the First-Order or the Second-Order Solution Interaction Prediction Model. The comparisons are made at 450, 550, and 650O~C In those cases where a range of compositions were studied for a given alloy system, the results are plotted to show the region of validity of the interaction modelo The sections which follow are devoted to factual presentation of the resultso A more general discussion of the multi-component alloy studies is reserved for the Discussion section (ppo239-246)o

=1421o Quaternary Alloys A total of eleven different combinations of two added solutes were studied in the investigation of quaternary alloys, All alloys were studied in the "dilute" condition, ie,, where the mole fraction of zinc and both solutes was o015o In addition. three combinations were studied under "non-dilute" conditions, ioeo, where the mole fraction of zinc was O050, or where the mole fraction of zinc was o015 and the total added solute concentration was increased up to o215 mole fractiono In addition, replicate runs were made on six of the dilute compositions. The agreement between the measured and predicted activity coefficients was very good in all cases where the solutes were dilute. The agreement became better as the temperature was increasedo In almost all the cases, the second-order model yielded a slightly improved prediction over the first-order modelo No significant influence due to the particular combination of the interacting elements could be detected with certainty in the dilute solution resultso (Table XI) However, when the mole fraction of zinc was held constant and the total added solute concentration increased beyond o030 mole fraction, the prediction models were completely successful in only one of the three cases studied, For clarity these results are plotted in Figures 44 and 45 as lnyZn versus Exj since (under the conditions of these studies) Equations (26) and (28) predicted a linear relation with lnZZn. For the solutions containing antimony and silver (both negative interactors with zinc) the experimental and predicted values agreed excellently

-143TABLE XI RESULTS OF MULTI-COMPONENT SOLUTION STUDIES OF ADDITIVITY HYPOTHESIS Activity Coefficient of Zinc Activity Coefficient of Zinc Activity Coefficient of Zinc at 450~C at 550~C at 650~C Observed Predicted Observed Predicted Observed Predicted Type of Types of** Added Ycalc. 7calc. 7calc. 7calc. 7calc. 7calc. Solution Interactor xzn xj XBi Solute Elements Yobserved Model I Model II Yobserved Model I Model II 7observed Model I Model II Binary (Basis for Comparison for All Following Alloys) 4.00 3.04 2.46 Quaternary + +.015.015.955 Pb In 4.32 424 4.19.21 20.17 2.56 2.59 2.59.015.015.955 Pb In 4.12 3.14 2.55.015.015.955 Cd Sn 4.20 3.06 00 2.99 2.47 2.93 2.42.015.015.955 Cd Sn 3.96 3.02 2.45 + -.015.015.955 Pb Sb 3.76 3.76 2.82 2.27 2 2.56 5.76 5.76 2.84 2.90 2.902.36 2.36.015.015.955 Pb Sb 3.70 2.84 2.29 +.015.015.955 Pb Sn 4.06 405 4.04 3.08 3.07.05 2.49 49 2.48.015.015.955 Pb Sn 3.96 2.99 2.46.015.015.955 Sn Ag 5.78 35.77 3.76 2.90 2.90 2.88 2.49 2.57 2.56.015.015.955 Sn Ag 3.76 2.88 2.47 - -.015.015.955 Cu Ag 3.74 3.62 3.61 2.80 2.82 2.79 2.31 2.32 2.50 - =.015.015.955 Sb Au 2.61 2.75 2.65 2.17 2.17 2.15 1.84 1.80 1.85 - =.015.015.955 Ag Au 2.60 2.86 2.72 2.19 2.23 2.20 1.85 1.85 1.87 Quaternary - -.015.015.955 Ag Sb 3.72 35.51 35.50 2.77 2.73 2.72 2.27 2.25 2.24.015.025.935 Ag Sb 3.25 3.21 3.20 2.58 2.55 2.53 2.15 2.12 2.11.015.0375.910 Ag Sb 2.86 2.88 2.86 2.28 2.55 2.51 1.92 1.98 1.95.015.050.885 Ag Sb 2.70 2.58 2.56 2.20.14 2.11 1.88 1.84 1.81.015.100.785 Ag Sb 1.73 1.67 1.64 1.51 1.48 1.57 1.57 1.55.050.015.920 Ag Sb 3.49 3.48 3.46 2.74 2.73 2.70 2.26 2.27 2.22.050.050.850 Ag Sb 2.58 2.53 2.48 2 13 2.14 2.06 1.84 1.85 1.92 + -.015.015.955 Pb Ag 3.84 2.89 98 2.96 236 2.42 5.88 3.88 2.98 2.96 2.6 2.4.015.015.955 Pb Ag 3.82 2.90 2.55.015.025.935 Pb Ag 3.68 3.81 3.80 2.83 2.94 2.92 2.52 2.42 2.49.015.0375.910 Pb Ag 3.58 3.72 3.69 2.72 2.90 2.87 2.25 2.49 2.56.015.050.885 Pb Ag 3.60 3.62 3.60 2.80 2.86 2.82 2.52 2.7 2.55.015.100.785 Pb Ag 3.17 3.29 3.24 2.53 2.69 2.62 2.17 2.28 2.20 + -.015.015.955 In Sb 5.70 3.81 3.78 2.87 2.93 2.90 2.57 2.59 2.57.015.025.935 In Sb 3.46 3.70 3.64 2.73 2.86 2.82 2.28 2.46 2.352.015.050.885 In Sb 2.98 3.41 5.51 2.45 2.70 2.64 2.11 2.26 2.19.015.100.785 In Sb 2.41 2.90 2.74 2.10 2.41 2.29 1.90 2.15 1.95.050.015.920 In Sb 3.69 3.77 5.70 2.94 2.9 2.86 2.55 2.41 2.55.050.050.850 In Sb 5.14 5.58 5.56 2.50 2.70 2.49 2.09 2.28 2.26 Quinary - - -.015.015.940 Cu Ag Sb 5.24 5.56 5.54 2.59 2.63 2.62 2.17 2.18 2.17.015.015.940 Cd Sn Sb 3.67 3. 3.65 2.85 2.3429 5.65 5.65 2.82 2.80 2.5 2.29 2.29.015.015.940 Cd Sn Sb 3.88 2.90 2.45 +.015.015.940 Pb Cd Sn 3.96 4.4 4.05 02 5.06.05 2.44 2.46 44.015.015.940 Pb Cd Sn 4.02 3.042.44 + +.015.015.940 In Pb Ag 3.92 3.99 5.92 5.04 5.08.0o4 2.49 2.52 2.48.015.050.895 In Pb Ag 4.08 3.96 3.86 5.11 3.12 3.05 2.52 2.56 2.50.015.050.835 In Pb Ag 4.25 3.94 35.77 5.24 5.19 5.06 2.64 2.65 2.55.050.050.860 In Pb Ag 3.72 3.96 5.64 2.91 3.12 2.87 2.40 2.57 2.59 -.015.015.90 Cd Sn Ag.96 3.76 3.73 2.96 2.89 2.87 2.4o 2.56 2.55.015.050.895 Cd Sn Ag.54 3.54 35.49 2.75 2.75 2.72 2.24 2.27 2.25 +.-.015.015.940 Pb Cd Ag 3.94 3.88 3.85 3.00 2.96 2.96 2.4 2.42 2.41.015.050.895 Pb Cd Ag 5.653.74 3.72 2.79 2.90 2.88 2.28 2.59 2.56.050.050.860 Pb Cd Ag 3.62 3.74 5.65 2.83 2.90 2.82 2.3355 2.59 2.51 Hexadic + + +. -.015.015.925 In Pb Sn Sb 3.84 5.87 5.82 2.96 2.97 2.96 2.40 2.42 2.59 - -.015.015.925 Cd Sn Ag Sb 3.54 3.47 5.44 2.74 2.70 2.68 2.24 2.25 2.21 + + -.015.015.925 In Pb Ag Sb 5.77 5.70 3.66 2.86 2.88 2.84 2.50 2.57 2.54 + --.015.015.925 In Cd Ag Sb 5.62 5.62 5.57 2.84 2.81 2.78 2.55 2.54 2.29.050.015.890o In Cd Ag Sb 3.46.62 3.46 2.72 2.812 2.78 2.5 2.2 2.21 + +...015.015.925 In Pb Cd Sn 4.12 4.16 4.11 5.15 3.17 3.14 2.56 2.56 2.52.050.015.890 In Pb Cd Sn 5.92 4.16 4.00.02.17.05 2.45 2.56 2.46 Septenary +.. —.015.015.910 In Cd Sn Sb Ag 3.96 3.58 3.52 5.02 2.79 2.74 2.44 2.30 2.27 + + -.015.015.910 In Pb Sb Ag Cu 5.80 3.54 5.48 2.95 2.78 2.7 2.42 2.50 2.26 + +.-.015.015.910 In Fb Cd Sn Sb 5.80 5.85 3.80 2.87 2.96 2.92 2.51 2.41 2.58 * Equal mole fraction addition of extra solute. ** See p 1141 for classification of interactions.

-144Solvent: Bismuth 1.4- Solutes: Zinc + Antimony + Silver (XAg = xSb) 1st order model 1.3 - - 2nd order model - xZn =.015. \ - ____. 2nd order model - xzn =.050 ~1.2-~E 0 Experimental Results - XZn =.015 lD Experimental Results - XZn =.050.16 0.9 0.8 0.7 0.6 Q.5- 450 0C 0.5 0.3- 650 ~C 0.2 I I 0 0.10 0.20 ZXj-MOLE FRACTION Figure 44. Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quaternary Alloys of Bi-Zn-Ag-Sb —comparison of observed and predicted values.

-1451.5,,,, Solutes: Zinc + Lead + Silver (xAg =Pb) 1.4 1.2 -.. - 450~C I.I-J ~ ~ ~ ~550~C 0.9 Solutes: Zinc + Indium + Antivuiuy (x =n xSb) l A -.I-n Sb 1.01 ---- XZn =.015 — I-E —-XZn =.050 1st Order Model |^ --— Xk ~_~ ~2nd Order Model, XZn =.015 1. 2 - 2 -2nd Order Model, xZn =.050 4 6500C -- 0.7'~" ___ ~"- | ~6500C 0.6-, I I 0 0.10 0.20 EXj-MOLE FRACTION Figure 45. Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quaternary Alloys of Bi-Zn-In-Sb and Bi-Zn-Ag-Pb - Comparison of Observed and Predicted Values.

even out to a total solute concentration of o215o This is considerably away from the dilute region in which the ternary parameters were measured, however, it will be recalled that many of the ternary results were linear up to the highest concentration of solute studied (.10 mole fraction). The agreement was not so good for additional total solute concentrations beyond.03 mole fraction for the solutions containing lead plus silver or antimony plus indium. In both these cases a positive interactor was combined with a negative interactoro Although the agreement became better as the temperature was increased, the experimental values of lnyzn were always less than the predicted valueso The second-order model was closer to the experimental results, but only began to achieve fairly good agreement at 650~Co However, in both cases, the effect of the combination of solutes was less than if the negative interactor alone had been present at the same total concentrationo For example, the value of lnyzn for the case where indium and antimony were both present was greater than if antimony had been present alone at the same mole fraction and considerably less than if indium alone had been presento Therefore, the positive interactor moderated the effect of the negative interactor, yet the net effect on lnyzn in these cases was not the algebraic sum of the individual interactionso An increase in the zinc content to o050 mole fraction did not result in significantly different effects in the quaternary solutions,

-1472, Quinary Alloys Six compositions were investigated where three additional solutes were added to the zinc-bismuth system. A limited study of "nondilute" compositions was made on three alloys, while replicate runs were made on two of the dilute compositions. For the case where each of the solutes were present at o015 mole fraction the predicted values agreed quite well with the experimental resultso The second-order model produced slightly better results than the first-order model. The agreement between the predicted values and the measured values was better as the temperature was increased. (Table XI) For the "non-dilute" solutions the results were mixed. In'the system where indium, lead, and silver were the added elements, Figure 46 shows that the experimental results for xZn = o015 agreed well with the first-order model but not the second-order model. However, for the one alloy where xZn = o050, the second-order model yielded very good agreement with the experimental result. For the two systems containing cadmium, tin, and silver or lead, cadmium, and silver the predictions from either model were in fair agreement with the experimental resultso 3o Hexadic Alloys Five examples of alloys were studied where four solute elements were added to the zinc-bismuth system. For two of the compositions, zinc mole fractions of o015 and 0050 were studied. In the other systems, the zinc was held at o015 mole fraction. In all alloys, the added solutes were kept at o015 mole fraction eacho

-1481.2 I. 00 0 I)t- Xzn0 =.015 L -*-"-^'Added Solutes: I-Q:~~~~~~~ ~~ I-_n, In~ + Pb + Ag I. I xN * -' O XZn = ~050 I } ^^^~+ — 2nd Order Model 1.0 I-o._.. I I. 0 0 LO <QF~ |'^sI <XZn =.015 Added Solutes: e~C~~~.t~~~~~ <^~ ^Cd + Sn + Ag XN 1.0 I.I I o XZn =.015 Added Solutes: | 2nU Order M Pb + Cd + Ag < | o - ^n = *050. c XZn =.015 XN c 0 Observed -J --- st Order Model -- 2nd Order Model, XZn=.015 0 —--- 2nd Order Model xZn=.OQ5 0.9 0 0.10 0.20 ]Xj- MOLE FRACTION Figure 46. Logarithm of Zinc Activity Coefficient versus Total Mole Fraction of Added Solutes for Quinary Alloys Based on Bi-Zn - comparison of observed and predicted values.

-149In virtually all cases, the observed and predicted values were in very good agreement at all temperatures considered. The second-order model yielded better predictions for most of the alloys, Increasing the mole fraction of zinc did not appear to affect the validity of the predictionso (Table XI) 4. Septenary Alloys Three alloys were studied where five solute elements were added to the zinc-bismuth system. All the solute concentrations were held at o015 mole fraction. For one alloy the measured and predicted values were in fairly good agreement over the whole range of temperatures considered. However, in the other two cases the observed values for the activity coefficient were distinctly higher than the predicted values, although the difference was less at 650~C than at the lower temperatures. The disagreements were associated with the alloys having two or three negative interactors, while the alloy producing fair agreement had only one negative interactoro (Table XI) 5 Summary of Multi-Component Interaction Effects The experimental results of the multi-component solution studies may be summarized as follows: 1) Test of the effect of increased number of solute elements: a) For xzn - o015 and X Bi >.9 For solutions up to hexadic, the agreement between observed and predicted values was excellent at all temperatureso For the septenary alloys, the

-150agreement was only fair in two cases, and moderately good in the other case. b) For xZn -.05 and XBi = 9: The agreement was good to excellent for the quaternaries, and fair to good for the quinaries and hexadics. 2) Test of the effect of the departure from the "dilute" solution region: a) For Xzn -- o015 and ZXo -> o20: The agreement in the J quaternary solution for silver plus antimony was excellent at all temperatureso For the systems with lead plus silver or indium plus antimony, the observed activity coefficients were significantly less than the predicted values, however, the positively interacting component did moderate the effect of the negative interactoro The agreement improved somewhat as the temperature was increased but only became fair at best. b) In the quinary alloys, the results were fairly good in two out of three cases, c) For the hexadic alloys the limited evidence was that the agreement became slightly poorer as XZn was increased at a constant level of the total solute concentrationo 3) Test of the effect of various combinations of positive and negative on the activity coefficient of zinc: In the dilute solutions, (xBi = ~9) the effect of different combinations of interacting elements was not significant for any of the

-151solution types studiedo In the non-dilute quaternaries, two systems where positive and negative interactors were combined resulted in lower than expected values for 7Zn o In the case where both the interactors were negative, the predicted and observed activity coefficients were in excellent agreement throughout the entire range of compositions studied. The second-order model yielded improved predictions but did not completely account for the observed effects. The divergence between observed and predicted values became less at the highesttemperature studied, Do Confirmation of Basic Assumptions In the discussion of the experimental method, mention was made of the basic assumptions inherent in utilizing the galvanic cell method for activity determination. These included the reversibility of the cell, the ionic conductance of the electrolyte, the valence of the element whose activity is being measured, the absence of significant side reactions, and the verification of the alloy compositions and that the alloys were single-phase liquids.

-152Several methods are available for the verification that these assumptions were actually realized in the experimental processo Probably most important is the behavior of the galvanic cells themselveso Another direct procedure is chemical analysis of the electrodes. It is also possible to perform a direct experiment to verify the conductance of the electrolyte, the question of the metal valence, and the reversibility of the cells. Indirect verification is also possible by comparing the experimental results with those of independent investigations. Ideally, such comparisons should be made with data obtained by different experimental techniqueso This was not completely possible for the present results since the only activity studies reported in the literature for the zinc-bismuth system in the range 450-650~C had also been conducted by the galvanic cell method for higher concentrations of zinc, However, it was possible to compare the independent use of the galvanic cell method at these temperatures. In addition, the present results could be extrapolated to a temperature where data had been taken by a dynamic vapor pressure method,(36) The behavior of the cells has been previously discussed in detailo In general, all the cells were quite stable. The relations between electrode potential and temperature were linear over the range studied, except where discontinuities were noted because of transition to a single phase liquido For those systems where published phase diagrams indicated existence of a single phase liquid the slopes of the emf-temperature relations for binary and ternary alloys were almost

-153identical. Where transition to a single phase liquid did occur, the change in slope was quite marked (see Figures 40 and 42). Hence, the fact that the slopes for the binary and multi-component alloys for which no phase diagrams were available were almost identical and had no discontinuities could be taken as evidence that a given system was composed of a single-phase liquid at the temperature under considerationo The activity data obtained in the present studies of the binary alloys were shown to be logical extensions to the dilute region of the previous galvanic cell results of Kleppa(45) and Lantratov and Tsarenko(47) obtained at higher levels of zinc, Where the present experimental compositions overlapped those of Kleppa's investigation, the activity versus concentration relations were in good agreemento The present results were also extrapolated to 352~C by use of the temperature dependence relation for lny~ (see p, 90 )o This was the temperature at which Yokokawa et alo(16) had used a dynamic vapor pressure method to determine the activity of zinco The value of 7~ calculated from Yokokawa's data was 5o4, while the extrapolated value from these studies was 5~60 Kleppa(45) reported that chemical analysis of his samples showed no change in composition from the weighed-in values within the accuracy of the method of analysis (which was unstated)o He also noted that the electrode weights before and after the runs differed by only a few milligrams (the total weights were not reported). As part of the present investigation, chemical analyses were made on the electrodes from several binary and ternary alloy runs, In

-154addition, an attempt was made to determine the weight change of the electrodes, The results of these studies are reported separately, but in general they showed that negligible compositional and weight changes occurred from the weighed-in valueso In addition to the chemical analyses, weight-change studies, and the inferences drawn from cell behavior, a special experiment was conducted to verify the reversibility of the cell, the electrolyte conductance, and the valence of zinc, This procedure was termed the Faraday Yield experiment and is described below. 1. Faraday Yield Experiment This experiment was conducted after the emf-temperature relations had been established for Run 53 (Bi-Zn-Ga system, xZn - o025). The intent of the experiment was to change the composition of the binary reference electrode by deliberate transference of zinc from the standard electrodeo When the new composition had been established and its potential determined, the procedure would be reversed and the zinc transferred back to the standard. If the cell was reversible, the potential should return to the original valueo The transfer would be by passing a constant current between the standard and the binary electrode for a measured period of time - thus a specified number of coulombs of electricity would have been passed. By Faraday's Law, the electrochemical equivalent of zinc is given by MVioL. Wt. X 338 8g./c ollob -A —lee 3- 3.338 (96 ooO) V Iece., 2c(coo- (9Gsoo)

-155This expression gives the weight of zinc that will be transferred per coulombo The mole fraction of zinc in the binary electrode was.025 and the electrode contained.1404 grams of zinc plus 17.5037 grams of bismuth. To raise the content of zinc to.0375 mole fraction, it was necessary to increase the weight of zinc to o2132 grams. Thus it was necessary to transfer.0728 grams of zinc. Using the Faraday equivalent, and assuming a valence of two for the zinc, the amount of electricity required is 216 coulombso At a current of 150 milli-amps, 24 minutes are required, At the start of the run, the temperature of the cell was equilibrated at 500~C (and maintained at this value throughout the experiment) and the potentials of all the electrodes were measured, (The values will be summarized later,) The power source was a small electrolytic etching unit containing several fresh dry cells wired in series, and incorporating a DC voltmeter and a milli-ammeter. The positive terminal from the power source was connected to the lead from the pure zinc standard ih the cell and the negative terminal to the alloy leado The current within the cell passed from the standard electrode to the alloy and was maintained at 150 milli-amps by means of a rheostat. As current flowed and some polarization occurred, it was necessary to raise the voltage slightly. The initial voltage was about.5 volts and the final voltage reached.7 voltso Following the passage of 216 coulombs, the leads were disconnected and the cell allowed to equilibrateo The behavior of the electrode potentials is shown schematically in Figure 47.

a06 > 160- 1^ CO co E co) TERNARY $<0 -BINARY ELECTRODE LL) ELECTRODES co 0~~~~~~~~ o~~~~~~~~~~~~ 0 w 0~ /0BINARY 0 w ELECTRODE F w (U L 0 80...... W. 0 a S~ ^~l' C TERNARY ELECTRODES 0 a(0 /o I) P/l w ao > ~ ~ D(l) 82 ~ -80 14E 46 Hrs. 96 Hrs. RECOVERY _ RECOVERY PERIOD PERIOD TIME Figure 47. Schematic Diagram of Electrode Potential Behavior During the Faraday Yield Experiment.

-157About 46 hours were required for the electrode potentials to recover from polarization to a constant value, The point of complete recovery was taken as the time when the emf's became stable and the emf of the ternary alloy electrodes had returned to the values noted immediately before the passage of the currento At this point, the potential of the binary electrode had setteled out at a value quite close to that expected for an alloy containing ~0375 mole fraction zinc at 500~Co The external power source was then re-connected, this time in the reverse manner, and 216 coulombs were passed through the cello The current was monitored manually to remain at 150 milli-amps, This time the initial voltage was about 06 volts. It was necessary to raise the voltage to about lol volts at one point, but then it was reduced to about o7 volts in the latter stages of the experiment. Following the passage of the current, approximately 96 hours were required for recovery of the binary electrode to a constant potentialO The indicated emf's of the ternary electrodes were unaffected by the re-transfer operation. The run was discontinued when its was noted that no further change in the binary potential had occurred over a 16 hour period, At this point the potential of the binary electrode had returned to a value characteristic of a.025 mole fraction zinc alloy at 500~C, while the alloy electrodes over the entire time of the experiment had changed by only ol to o2 millivolts.

=158= A condensation of the experimental observations followsElectrode Potentials-mv Time Temperature Binary Alloy Alloy2 Alloy3 Before current transfer 500~C 80o55 79~88 80o04 80o41 12 hours after current transfer 499~6 45o38 57.46 57063 58o00 46 hours after current transfer (upon comr pletion of recovery) 500,0 67~86 79o89 80o03 80o31 13 hours after current re-transfer 499.4 101o97 80,05 80o20 80,47 96 hours after current re-transfer 500,0 8129 80o00 80o09 80o64 Nominal Expected Emf Binary Electrode (xZn -= 025 weighed-in) Composition (Stdo Value) Initial Emf 80o55mv xZn - 025 81,75mv After Current Passage and Recovery 67~86 XZn. 0375 68o24 Change in Emf 12069 Expected Change 13o51 After Second Current Passage and Recovery 81o29 Change from Intermediate Value 13 043 Average change in EMF - 13o06mv or 9605 per cent of expected changeo Change in Alloy Electrodes Over Alloy loy Alloy3 144 hours of the Experiment +0 12mv +0 05 +0 23 The behavior of the observed electrode potentials during the recovery periods is consistent with the assumption that the polarization took place at the surface of the electrode where the reaction Zn -: Zn++ + 2e took place. When zinc was transferred out of the standard

-159and into the binary electrode, all the potentials of the ternary alloys were affected with respect to the standard electrode, However, when full recovery from the polarization had taken place and the current was reversed to transfer zinc out of the binary electrode and back to the standard, only the potential of the binary electrode was affected relative to the standardo The average change in potential of the binary electrode for the two steps of the Faraday Yield experiment was 9605 per cent of the expected change for the planned alteration of zinc from.025 mole fraction to O0375 mole fraction and backo The theoretical or expected change was based on the assumptions of cell reversibility, ionic conductance of the electrolyte and a valence of two for zinc. It was concluded that the experimental results were well within the expected accuracy of the experiment and that the validity of the experimental assumptions was adequately verifiedo 2. Electrode Weight'&nranges An attempt was made to remove the electrodes from several cells in as quantitative a manner as possible to determine their weight change, if anyo The usual method of concluding the run was to water quench the cell to avoid the danger of the Vycor tube bottom cracking on contact with any solidifying electrolyte that might have condensed outside the cell crucible. The water quench usually caused some splashing of the liquid alloys and thus weight change determinations from normally concluded runs were not attempted.

-16oThe procedure for "quantitative" electrode removal was to withdraw the tantalum leads as carefully as possible, remove the Vycor cell tube from the furnace, and allow the entire assembly to cool in air with a rod extending into the salt melt. The salt solidified at about 350~C while the alloys, being almost pure bismuth, did not freeze completely until about 270~Co As soon as the salt had solidified sufficiently to hold the rod in place, the cell crucible was withdrawn by the rod and the Vycor tube filled with water. Of the two attempts to use this procedure, one of them ended with a cracked tube bottom. When it was certain that the electrodes were solid under their blanket of salt, the salt was dissolved in a stream of hot water. The electrode crucibles were then dried and weighed. Slight weight losses, ranging from o2 to 1.5 per cent were noted from the alloy electrodes (17-18 grams total weight). Of the two zinc standard electrodes, one apparently gained about,5 per cent and the other lost about o5 per cent in weight, The results are considered good within the expected accuracy but inconclusive since there was the possibility of losing material as the electrode leads were withdrawn and in washing out some other material with the salt. 3o Chemical Analysis Chemical analysis was conducted on a number of the electrodes for runs concluded by the normal "quenching" procedure. Both wet chemical procedures and emission spectroscopy were employed. The spectrographic

-161analysis was conducted on an ARL Spectrograph in the Department of Chemical and Metallurgical Engineering, University of Michigan, while the wet chemical analysis was done by an outside laboratory (Ledoux and Co,, Teaneck, New Jersey)o Several samples were analyzed by both means, The primary purpose of the analyses was to verify that no significant transfer of zinc had taken place during the operation of the cello Since the effect of the displacement reactions (if they occurred) would be expected to be greatest for alloys low in zinc and high in the amount of third element, the emphasis in the analysis was on these types of alloyso In addition, displacement due to the third element might be expected most likely for cadmium, tin, or lead-conta.ining alloyso The results obtained by emission spectroscopy are summarized in Table XIIo In addition to the alloys reported, attempts were also made to analyze for zinc in the presence of copper or tino In these alloys it was very evident that the presence of the third element interferred with the zinc analysiso The results for the cadmium-containing alloys appeared to be free of interferences, while there was some indication of a very slight interference by leado However, for the binary alloys the spectrographic determinations were not subject to third element interferences and the results appeared to be fairly goodo The standards for the spectrographic analysis were a series of binary zinc-bismuth alloys from 0075 to.050 mole fraction zinc that had been sealed in evacuated glass tubes, held molten in a furnace at 450~C for several hours, air cooled, and then removed from

TABLE XII RESULTS OF CHEMICAL ANAL YSES OF ALLOY rLECTRaDES (Analysis Made After Completion of Run) Nominal Composition Analyzed Composition (Weighed-In Value) by Emission Spectroscopy by Wet Chemistry y^p o. XZn % Zn XJ xn xtn % Zn* Xj Alloy Run No. a4ole Frac. Weight Mole Frac. Mole Frac. Mole Frac. Weight Mole Fr&c. Speotrseopi Stamndrds (not run as G-l.015.47 -.0152.48 eleetrodes) G-4.050 1.62 - - 0547 1.78 Binary Alloys 47.015.47 -.014/.016.0155.49 64.015 ~47 -.013 - -- 65.015.47 -.016.0152.48 68.015.47 -.012.0127.40 71.015.47 -.0145 - - 77.015.47 - 016 - - - 67.025.79 -.0255.0239.76 - 70.0375 1.20 -.032 - - 81.050 1.62 -.065.0502 1.63 - Ternary Alloy 64.015.47.015 Pb(1.50 wt.%).017.0127.40.0152(1.52 wt. 6).015.47.025 Pb.014 - - -.015.47.0575 Pb.0155 - - -.015.47.050 Pb.018 - - - 65.015.47.0575 Cd.015 - - -.015.47.050 Cd.0145/.015 - - 47.015.47.050 Sn(2.93 wt. %) -..018.45.0489(2.88 wt. %) 70.0375 1.20.015 Cu(.47 wt. %) -.0298.96.0118(.37 wt. %) Hexadie Alloy 83.015.47.015 each; In, Pb, Cd, Sn -.35 * Stated to be average of two determinations; reported aa weight percent and then converted to mole fraction.

-163the tubeso The samples analyzed from the completed electrodes had been water quenched from approximately 450~C, washed free of salt, broken out of their small crucibles, and then ground flat on one face. The spectrographic calibration curves prepared from the standards were generally good at the lower concentrations of zinc, but some scatter was evident about Zn =.025 o Since the solid solubility of zinc in bismuth is very small(55) the samples with larger amounts of zinc may have experienced some segregationo The spectrographic results for nominal zinc contents of o0375 and 050 may have thus been influenced by nonuniform zinc distribution in the solid standard, the unknown sample, or botho The results obtained on the sample containing o015 mole fraction nominal zinc content appeared to be randomly distributed about o015o The accuracy for commercial spectrographic analysis is quoted as from five to seven per cent of the element present(19) At xZn = 015, this would imply that +o001 mole fraction accuracy is the best that could be expectedo The present results appear to be in this range and it can be concluded, at least from the spectrographic data, that alteration of the zinc content was slight and that if it did occur, it was within the range of accuracy of analysiso The results of the wet chemical analyses are also summarized in Table XIIo In general these results agreed very well with the spectrographic analyses Two of the spectrographic standards were included with the specimens sent for wet chemical analysis in order to confirm

-164their composition and the consistency of the two types of analysiso The wet analyses were the average of two determinations. Their accuracy was not specified, however, the wet results obtained on the spectrographic standards suggest that it was probably at least +10 per cento There was no evidence from any of the chemical analyses that a significant increase in the zinc content had occurred in any of the alloy electrodes as a result of the galvanic cell runs, Consequently, the absence of significant displacement reactions, diffusion, or transference of zinc by current flow was confirmed, since all of these processes would have led to an increase in the zinc content, In several of the alloy electrodes the analyses indicated a slight decrease in the zinc content from the weighed-in valueo It is possible that the wet analyses may have been biased towards low values, particularly if complete gravimetric separation of zinc was difficult, However, it is also possible that some zinc could have been lost by volatilization or by preferential solution into the molten-salt electrolyteo Studies by Yamagishi and Kamemoto(61) had indicated the latter possibility, however, if either effect had been significant it would have been evidenced by a noticeable increase in the electrode potentials (see Figure 9 ) Since the electrode potentials were generally quite steady over the running time of the cell, the loss of zinc by these or any other process during the runs was smallo Analysis was also made for the third element content in three of the ternary electrodeso The results generally confirmed that there

-165was no substantial alteration of xj from the weighed-in values. The copper and zinc analyses in the Bi-Zn-Cu ternary sample were both lower than expected, however, they were low in the same proportiono This may have been due to segregation in the portion of the electrode analyzed since this system tended towards limited single-phase solubility at the low temperature from which the electrode had been quenched at the conclusion of the run,

V. DISCUSSION The discussion of the experimental results is divided into four main categories. The first considers the rationale for the observed interactions in the ternary alloys. The general approach is to consider explanations based on the structure and physical properties of the elements involved. The second category covers the applicability of simple solution models for the prediction of interactions. The approach of such theories is partly phenomenological and partly based on statistical thermodynamics. The third topic discussed is the validity of using ternary data for the prediction of activity in higher-order solutions. This stems from Wagner's suggestion of the Taylor series expansion for in activity coefficient and is thus a phenomenological approach. The last topic considered includes the limitations of the present results and suggestions for further work. A. Rationale for the Observed Interactions in Ternary Alloys One of the major premises on which this investigation was based is that substantial evidence existed from studies conducted in molten iron that the interactions of various third elements with a given primary solute were a periodic function of the atomic number of the added solute. The experimental results were fairly conclusive for non-metallic primary solutes such as carbon, hydrogen, nitrogen, or oxygen, as affected by additional metallic elements. However, the data were limited and inconclusive in the case where the primary solute was also metallic (such as chromium or nickel). -166

-167This investigation was designed to study the premise of periodicity for the case where both solutes were metals. Third-element additions from the 4th, 5th, and 6th Periods (Group B) were made to the basic system of zinc dissolved in molten bismuth. The results of these studies will first be discussed on the basis of periodicity, however, since this explanation did not completely account for the experimental observations a number of other possibilities are considered. Among these are included the electronegativities of the elements, thermodynamic interactions between the solutes alone, and atomic size factors. 1. Evidence for Periodic Behavior of Interaction Parameters Attempts were made to correlate the first —and second-order interaction parameters determined at 5500C with the position of the added solute elements in the Periodic Table. The attempted correlations were both by Period (Figures 48 and 50) and by Sub-Group (Figure 49). With the notable exceptions of tin and antimony, the first-order parameters almost obeyed a periodic rule. Within each period considered, the trend was for the first-order parameter to become less negative with increasing atomic number of the solute. However, in the 5th Period (the most complete sequence studied), the sequence of increasing parameter values showed an abrupt reversal in the middle. Considering the results by sub-groups (Figure 49); in three cases (Groups Ib, IIb, and IIIb) there was some tendency for the parameter values to decrease as the atomic number increased. However, the results for Group IVb (tin and lead) were in the opposite direction.

-168+ 10 -' I I i i i - +5 - Ga In Pb 0 Cd Sn Zn Cd Sn "~00 Hg Cu Ag t *5 Sb "-5 N -10 -15 - Au -20 -20 I- ----— i I - I I — 25 30 35 40 45 50 55 60 65 70 75 80 85 ATOMIC NUMBER OF THIRD ELEMENT (j) Figure 48. First-Order Zinc-j Interaction Parameters versus Atomic Number of j - dorrelation by period.

-169+10 I ii I I I Ga +5 In'rIn b Pb Zn - o —-- Sn 0! Ul) Cu _I Tr b -~Atomic Nmb of - correlatn by Ag C Sb X' b -10 - 15 Au -20 25 30 35 40 45 50 55 60 65 70 75 80 85 ATOMIC NUMBER OF THIRD ELEMENT (j) Figure 49. First-Order Zinc-J Interaction Parameters versaus Atomic Number-of j - correlation by sub-group.

-17020 I I i,., i I Zn Cd Sb 0 - U "O~~~~~~~~~I' I/ Hg Sn -20 C Ag 440 0 -0 In Au'-. N I' -60 -80-100 25 30 35 40 45 50 55 60 65 70 75 80 85 ATOMIC NUMBER OF THIRD ELEMENT (j) Figure 50. Second-Order Zinc-j Interaction Parameters versus Atomic Number of j - correlation by period.

-17150 I Sb Cd Pb o0 0 00 o OSn 0" x x AgCU N X X X | - O In -50 0Au -100 i a.) -20 -15 -10 -5 0 5 (cLnzn )X:o0 i XXZXn=O Figure 51. First-Order Zinc-j Interaction Parameters versus Second-Order Zinc-j, Interaction Parameters.

1'172 There appeared to be no periodicity in the values of the secondorder parameters (Figure 50) nor was there any discernible correlation between the first and second-order parameters involving a given solute element, (Figure 51) The primary basis for assuming periodic behavior is that many of the physical and chemical properties of elements can be explained on (74) the basis of electronic configuration.(74) As the atomic number is increased, the total number of electrons increase and the number of outer shell electrons vary cyclicallyo Schenck, Frohberg, and Steinmetz(l3) noted that periodicity of interactions leads to the supposition that the activity of the dissolved element is related to the number of valence electrons of the ternary elements or to the number of their electrons in unfilled shellso The electronic arrangement of the solute elements and the ternary interaction parameters are summarized below in periodic form, (Page 173) On examining this table it is noted that the elements copper, silver, and gold, all with an unfilled (s) shell, behave similarly and the interaction becomes stronger (more negative) as the total number of electrons (or Atomic Number) increases. The sequence zinc, cadmium, and mercury contains elements with a filled outer (s) shell. These elements also obey a similar sequence of decreasing interaction parameter as the atomic number increases. Gallium and indium each contain one electron in their outermost (unfilled) shell (this time a p-shell) and similarly show a relative decrease in the interaction parameter with atomic number.

Subgroup Tb TIb IIITb Vb Vb Total Outer Shell Electrons (s + p) Period 1 2 3 4 5 Atomic No., Outermost Shell, Interaction Parameter A.N. s1 A. A.N. N. 2 A.N. 3 e 4 29 Cu -2.4 30 Zn 0 31 Ga +5 +.7 5 47 Ag -2.5 48 Cd -.3 49 In +2.2 50 Sn -.5 51 Sb -4.5 6 79 Au -18.0 80 Hg -l,-2 - 81 Pb +1.3 82 (Bi) * Zn at 5500C; A.N. -Atomic Number; s1, s2, pl, p p -outer shell configuration Z n

-174Thus far, the "rule" would appear to be the following: a "deficiency" of (s) electrons relative to zinc results in a negative interaction, a "surplus" of (p) electrons causes a positive interaction, while the absolute value of the effect is moderated by the total number of electrons of the solute. However, when the interaction parameters and electron configurations of tin, lead, and antimony are considered, the "rule" breaks down. There is no way to account either for the negative interaction parameters of antimony or tin, or the relative increase in the parameter on going from tin to lead. Furthermore, there is no plausible way from this tabulation to explain the "rule" of why a "surplus" or "deficiency" of electrons in the outermost shell should account for a given direction of interaction. Wagner(23) suggested that the chemical potential of a solute could be separated into a portion pi due to its ions of valence zi and a portion which is the product of the valence and the chemical potential of the electrons, Zie. In discussing the marked periodicity for ternary metallic interactions with carbon or various gases dissolved in liquid iron, cobalt, or manganese (all transition metals with unfilled inner shells), Schenck, Frohberg, and Steinmetz(13) employed Wagner's concept and argued that the primary solute was dissolved as a positive, electron-donor ion. The electron liberated by such ionization then contributed partly to the chemical potential of free electrons, zipe, and partly to fill the unfilled inner shell of the solvent. A metallic third element addition can either provide electrons to the solution or complete its own shells. When ionized, the

-175third element might also affect the chemical potential of the free electrons. It might be expected that those additional solute elements with almost empty inner shells would be able to capture more electrons than the solvent, and thus the sign of the interaction would be negative since the chemical potential (or concentration) of free electrons would be decreased. A linear relation between interaction parameter and atomic number could then be explained by the fact that, inside a given period, the number of electrons given up or the degree of shell-filling would increase with the atomic number. The interaction would become more positive as the filling increases. The effect of the ions on the free electrons would either remain constant or also increase with the atomic number. The interaction thus would result from the competition between the solvent and the added solute for the electrons given up by the carbon, etc. Presumably (although not discussed by Schenck, et al.), the argument might be capable of extension to the case where the primary solute is a metallic transition element dissolved under the above conditions. However, virtually no experimental data are available for such systems involving transition metal solvents. The extension of the above argument to the case of zinc dissolved in bismuth does not appear possible. Although the zinc could be considered as dissolved as positive ions of charge two, bismuth has no unfilled inner shells and thus the electrons given up by the zinc would go only to increase the chemical potential of free electrons since there would be no competition for electrons between bismuth and the added solute. On the basis of

-176the relative shell-filling tendencies for the solutes considered in this investigation, a reversal of the interaction effect within a period (such as noted for the 5th Period in these studies) also could not be accounted for. The explanation would have to be constructed in terms of the outermost or valence electrons. Such a correlation by sub-group might provide a basis for assigning an "individuality" of behavior to antimony, however, the available data provide no basis for explaining why the subgroups behave as they do. It would appear that periodicity or electronic configuration per se does not provide a tenable explanation for the observed interactions. The partial periodicity shown in Figure 48 probably reflects only the fact that some other mechanism controls the interactions and that one of the factors in such a mechanism may have a periodic basis. 2. Alloying Considerations If purely periodic considerations are ruled out as the factors governing the solute interactions in the ternary systems, the explanation must be sought in some other property less directly related to atomic structure. We thus turn to the general field of alloying theory, solution thermodynamics and bonding between unlike atoms. A convenient approach to solution thermodynamics is in terms of the excess properties, that is, the deviations by which the thermodynamic properties of a real solution differ from those of an ideal solution of the same relative concentration. Indeed, the excess free energy of solution is given by the expression

-177AGA = RT A ti Hence, the interaction parameters - ) etc. merely express the relative change in the excess free energy due to solute additions. In a recent discussion of the thermodynamics of metalic solutions, Kleppa(25) recalled from the work on the theory of alloying and binary-phase diagram studies by Hume-Rothery et al., that such systems reflect the interplay on the excess properties by at least three important factors: the electrochemical or electronegativity factor, the size factor, and the valence factor. Usually the size factor causes a positive contribution to the excess properties, a difference in electronegativities causes a negative contribution, while a difference in valence can often produce an asymmetry in the excess functions over the range of solution of i in j or j in i a. Electronegativities The role of electronegativities in solution interactions as applied to Wagner's electron model has recently been discussed in detail (77,89). (19) by Sponsellor (7789) and Obenchain.9) The general basis of these discussions was to consider that electronegativities can be related to changes in the electron/atom ratio that take place in the alloying process. Wagner(5'23) had proposed that the activity of a solute will be increased by addition of a second solute if both solutes change the electron/atom

-178ratio in the same direction, or decreased if the solutes change the electron/atom ratio in opposite directions. The electronegativity itself represents the relative attraction of an atom for valence electrons in a covalent bond.(75'76) A number of investigators(75'7'79) have derived electronegativity values for the elements based on the relative heats of formation and dissociation for various compounds that could be formed. The electronegativity is regarded as a fundamental quantity of arbitrary and approximately known absolute value, but whose relative values express in a general way the electron affinity of an element. To some extent, the electronegativity increases with increasing valence. Thus it can be termed a measure of the effective valence. Furthermore, a change in electronegativity can be interpreted as a difference in effective valence depending on the environment the element finds itself in. In its application to the solution of a particular element in a liquid metal solvent, the electronegativity difference can be used as a measure of the relative transfer of electrons between solute and solvent. Elements which are more metallic than the solute will give up valence electrons, while those less metallic than the solvent will gain electrons. The free electron concentration in a particular solution may thus be increased or decreased by a solute addition. Thus far, the discussion is similar to the previously cited one (13) of Schenck et al.3) regarding the electron capture competition between solutes and solvent. However, Sponsellor(77'89) and later Obenchain(l9) PoeeSoslo

-179suggested the use of the electronegativity concept, thus allowing an "effective" valence or electron affinity to be employed rather than the more limited concept of the outermost shell electrons. Considering the chemical potential of a component in solution, Obenchain(l9) used the arguments of Wagner(23) and Himmler. (24) If the chemical potential is separated into that part due to the ions and that part due to the electrons, the activity of a component i in solution may be written as L[' rT (et (-)'^ 4rt - )d] (29) where p. = constant at temperature T If [i(+ ion) is taken as essentially constant, then changes in pe govern the activity of the constituent i, and it is assumed that e increases as the ratio of free electrons to atoms is increased or decreases as the ratio is decreased. Finally, if the electronegativities are taken as a measure of the electron affinity or "effective electrons per atom", then it is possible to predict the effect of a given solute addition on the activity of constituent i. The basis for the prediction is what effect the addition of solute j has on the electron/atom ratio already established by the solution of constituent i in the solvent. By considering the various cases to be encountered in Equation (29) for a third element addition to a given binary solution, Obenchain systematized Wagner's proposal and summarized a set of rules for predicting the sign of the interaction parameter. Without reproducing the

-18oreasoning in detail, they are: 1. If the added solute j affects the electron/atom ratio in the same way as did solute i in the binary, then the activity is increased and the interaction effect will be positive. 2. If the added solute affects the electron/atom ratio opposite to the case of the binary, then-the activity is decreased and the interaction will be negative. The employment of electronegativities in conjunction with these rules may be clarified somewhat by referring to the diagram below: U)@^ ~solvent base line + ~. interaction if forbidden region c ^Xj < solvent X Xi 1Q U) I Q) U z * forbidden region interaction if )w H Xj Xsolvent ______ X (electronegativity) Thus, for a solvent of given electronegativity, the position of the primary solute i fixes the manner in which the interaction parameter should behave as another solute is added. Those added solutes whose electronegativities fall to the same side of the base line as i should

-181(according to these rules) increase the activity of i and thus the interaction parameter should lie in the upper left (positive) quadrant of the diagram. Conversely, when the electronegativity of solute j falls on the opposite side of the base line from that of i, then the interaction parameter occupies the lower right (negative) quadrant. The other two quadrants are thus "forbidden" regions for the interaction parameters with solute i, In the case where the electronegativity of i is to the opposite side of the base line in the diagram, the roles of the quadrants are reversed. The previously "forbidden" regions then become the permissible regions and vice versa. Obenchain qualitatively tested these concepts for 80 systems for which interaction parameter data were available from the literature. In slightly over 80 percent of the cases considered, the correct sign of the interaction could be predicted. Considering the assumptions involved in establishing the model and the fact that the electronegativities are known only approximately and may vary depending on the environment, the results of this approach were remarkably good. Sponsellor(77 89 had used this method to predict the sign of third element interactions with calcium dissolved in liquid iron and was successful for three out of the four added solutes for which he had experimental data. Using these concepts, the first-order interaction parameter results of the present investigation are presented in Figure 52 as a function of the electronegativity of the elements involved. The test was made using the electronegativity values of several authors (which

-182TABLE XIII ELECTRONEGATIVITY VALUES FOR ELEMENTS INTERACTING WITH ZINC IN BISMUTH Electronegativity - (electron-volts/bond) /2 Element Pauling(75) Gordy(76) Furukawa(78) Allred (79) Bi 1.9 1.8 1.8 2.02 Zn 1.6 1.5 1.6 1.65 Cu 1.9, 2.0 1.8, 2.0 1.9 1.90 Ga 1.6 1.5 1.6 1.81 Ag 1.9 1.8 1.9 1.93 Cd 1.7 1.5 1.6 1.68 In 1.7 1.5 1.5 1.78 Sn 1.8, 1.9 1.7, 1.8 1.7 1.96 Sb 1.9 1.8, 2.1 1.8 2.05 Au 2.4 2.3 2.0 2.54 Hg 1.9 1.8 1.8 2.0o Pb 1.8 1.6, 1.8 1.7 2.33

-183-,, i r 1 [ ^S |- IQC Pauling's 1960 Values Furukawa (1959) In In On Zn I Zn Pb n 0 z n Hg 0 C C S ------- --- -- --- Cd n Cd Sn HgCd Ag,Cu Hg Ag,Cu In \) \ l ln ~ I _:-5-~Sb oSb -10 - -15 Bi i Au c/Au -20 I 1 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 (a) ELECTRONEGATIVITY (b) ELECTRONEGATIVITY Gordy (1958) Allred (1961) 5 - Ga I - Ga In' In 0 Z- Pb i Pb Zn -^'pZn opb 0 —------------ -0-0 —----- - ---------- o Cd Hg N Cd I Hg o 0 —---- Ag,Cu Cu Oo 5to I Ag -5-,b ----— t —'b oSb I \ -15 Bi I(~~~~ I.~~~~~ \. ~~Au Au 0 -20 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 (c) ELECTRONEGATIVITY (d) ELECTRONEGATIVITY Figure 52. First-Order Interaction Parameters for Zinc-j versus Electronegativit-es of the Added Solute j.

-184are summarized in Table XIII). Since the electronegativity of zinc is less than that of the solvent bismuth, it would be expected that positive interactions would be found with elements whose electronegativities were also less than that of bismuth and vice versa for those solutes with larger electronegativities than bismuth. If the values given by Pauling(75) (Figure 52a), Furukawa(78) (Figure 52b), and Gordy(76) (Figure 52c) are used, the results are generally similar. The interaction parameters for most elements fell in the permissible regions, the only clear exceptions being the cases of tin and cadmium. However, the "penetration" of the "forbidden" region was only slight. The electronegativities of mercury and antimony were close enough to that of bismuth so that some ambiguity existed as to what the prediction should have been. When Allred's values were used(79) (Figure 52d), the interaction parameters for lead, copper, silver, and mercury fell in the "forbidden" regions, but the parameter for antimony was clearly in a permissible region. The success of the electronegativity model for interaction prediction is summarized below: Electronegativity Scale Used Pauling Furukawa Gordy Allred Total Third Elements 10 10 10 10 Number of Unambiguous Predictions Possible 7 8 8 10 Number of Correct Predictions 5 6 6 4 Percentage of Success 71% 75% 75% 40%

-185Disregarding the results obtained using Allred's electronegativities, the percentage of successful predictions compares favorably with that of Obenchain who used Pauling's and Furukawa's values primarily. Figures 52a, b, and c also indicate that there is some quantitative basis to the electronegativity prediction model. The interaction parameters for those elements where a successful prediction can be made as to the sign of the interaction also tend to lie on a single line which passes through a value of zero interaction at the electronegativity of the bismuth solvent. The apparent quantitative nature of the result is quite interesting, however, additional data would be desirable on other systems before it could be termed conclusive. However, it does appear that a considerable number of the observed interactions can be accounted for on the basis of electronegativity difference, and furthermore, that the greater the difference, the greater the numerical value of the interaction parameter. Before an unequivocal interpretation could be placed on the "correlation", some theoretical explanation would have to be found for the slope or shape of the line, and for the penetration, even though slight, of the interaction parameters for tin and cadmium into the "forbidden" regions. b. Size Factors If electronegativities do not completely account for the observed interactions it would be logical, continuing the analysis summarized by (25) Kleppa,(25) to consider size factors - either alone or as moderating the electronegativities.

-186It was noted by Kleppa that differences in atomic size should give rise to positive contributions to excess properties. Accordingly, the experimental values for first-order interaction parameters with zinc in bismuth were studied with respect to a number of size-related factors. The quantities used were the Goldschmidt atomic radius, the atomic volume, and the ionic radii of Pauling and Goldschmidt.(76) The attempted correlations are shown in Figures 53 through 55. The results were completely negative. There appeared to be no regularity to the interaction parameters when size factors alone were considered. Attempts to group the sizefactor plots by Periodic Table considerations were also unsuccessful. In a discussion of the influence of chemical factors on the extent of primary solid solubility, Darken and Gurry(76) used a plot of the atomic radius versus the electronegativity. In general, elements which had appreciable solid solubility in a given solvent tended to be found in an ellipse of maximum width +15 percent of the value of the atomic radius and a height of +.4 electronegativity units. Waber et al. (8) later used this method to study solid solubilities in 1455 binary metallic combinations with about 77 percent reliability. Since a plot of this type appeared to be useful for simultaneous consideration of size and electronegativity effects in alloying, the elements used in the present interaction investigation were plotted in this manner. It was noted (Figure 56) that if a line were drawn between the coordinate points for zinc and bismuth, the elements lying to the right of the line were positive interactors and those lying to the left

-l87l v o I I I 5 - Go OIn OPb o - CdO OSn o Cu IHg O O0 in Ag - 5 oSb W -5-10 -15 - OAu -20 I i 5 10 15 20 25 ATOMIC VOLUME -CM3 Figure 53. First-Order Zine-j Interaction Parameters versus Atomic Volume of Added Solute J.

-18810,,,,,,,.,, a 5 _Ga In 0 OPb 0 ZnO Cd OOSn o cue o IHg )0 CuG O LS Ag L.O(Sb -, - -0 -10 -15 Zn OAu BI I I -20,', I. i, 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 GOLDSCHMIDT ATOMIC RADIUS - A~ Figure 54. First-Order Zinc-j Interaction Parameters versus Atomic Radius of Added Solute j.

-i8910,.,,,,. 5 3Ga OIn 0- Sno ozn OCd o JHg 0 OCu O O'0 Ag 1- _ -5 Sb c -10 - -15 - O Au -20 I 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 GOLDSCHMIDT IONIC RADIUS (CN6)-A~ Figure 55. First-Order Zinc-j Interaction Parameters versus Ionic Radius of Added Solute j.

3.0 w 2.5 ~$ g2.0 Cu Ag Hg / Cn) 49/ v^70 ~o 1 2. Cu Ag Hg 1. 0 1.5 2.0 2.5.0 0 w I 1.02) (76 1.0 1.5 2.0 2.5 3.0 ATOMIC RADIUS (CN 12) (76) Figure 56. Darken-Gurry Plot of Atomic Radius and Electronegativity for Zinc, Bismuth, and Added Solutes.

-191of the line were negative interactors. In addition, the distance of each solute element from such a base line was roughly proportional to the numerical value of its interaction parameter with zinc in bismuth. In general, therefore, negative interactions were associated with large values of electronegativities and small atomic radii, while positive interactors were associated with an increase in the atomic radius. These observations are consistent with Kleppa's comments (p. 177). The "correlation" shown in Figure 56 depends on the values used for the electronegativities - best results were obtained with Pauling's values. The results are interesting but hardly conclusive. A variation of this approach was then attempted on a basis that would combine both the electronegativity and the size factor. The method used was to divide the electronegativity for each element by a size factor. When atomic volume is used, the quotient is an "electronegativity per molar unit volume", while when the ionic radii or atomic radii are used, the quotient is a "hybrid" term that could be also considered as a measure of charge density or electron affinity density. Plots using these factors are shown in Figure 57. The only attempt that showed any promise was the plot of interaction parameter versus the quotient: electronegativity/atomic radius. The trend of results is similar to the plot versus electronegativity alone (Figure 52). However, an attempt to apply the prediction rules of page 180 to this plot did not succeed, Since zinc lies to the right of bismuth on the plot, the permissible regions would be expected to be the upper right and lower left quadrants. In Figure 57 only indium thus lies in the proper quadrant.

-192_! I, I I. Go 5- 0 In 0 PbO Cd Sn.0 - O O 0 ~ I0 Ag Cu 8Hg 0 0 o5 ~1 Sb C'U -10- I I -15 Bi Zn I I I I 0Au -20 I I I, I, I - 0.09 0.12 0.16 0.20 0.24 0.28 X/V - ELECTRONEGATIVITY/ATOMIC VOLUME BASE LINES FOR to. ^ APPLYING ELECTRON \ MODEL 5- AGO I I Go - In In e. PbO iSn Cd OPb O 0. --. 0 oSn&,0 1 0C 0 O Hg OCu k-5- I Ag Cu Ag OSb I -N vu -K)Bi Zn Bi Zn -15 I - I I AuI Au Au ~~~AI AG I I -20 i 0.5 I.0 1.5 1.5 2.0 2.5 I/A.R-ELECTRONEGATIVMTY/ATOMIC RADIUS X/I.R-ELECTRONEGATITY/IONIC RADIUS Figure 57. Electrogegativity-Size Factor Correlation Attempts for First-Order Zinc-j Interaction Parameters.

-193However, if the plot is considered on the basis that the effect of a given third element rests in the difference in the "hybrid" or density factor from that of zinc, then three of the four elements lying to the left of the value for zinc (lower values of the hybrid factor) produce positive interactions) while all of the elements to the right of zinc produce negative interactions, Thus, the deviation can be removed for tin that occurred where the rules were applied to the plots of eZn versus electronegativity alone. However, cadmium still remains anomalous. Furthermore, there is no theoretical justification for this correlation or why the atomic radius is the proper size factor to use. While an effective volume could be calculated from the atomic radius, the result would only be to spread the abcissa of the plot and would not change the relative position of the parameters. In effect, this interpretation calls for a reversal of the previous rule; implying that those elements with "hybrid" factor to the same side of the solute zinc as the solvent bismuth produce the positive interactions, while elements to the other side produce the negative interactions. The apparent correlation shown in Figure 57 must thus be considered primarily empirical, of questionable theoretical justification, and perhaps only coincidental, However, the value of the attempts at correlation by direct or semi-direct application of size factor is to show that if size effects enter into the interactions, their influence is secondary to the electronegativity effects, Finally, in considering "hybrid" or energy density effects on interactions, it was noted that the solubility parameter of Hildebrand(33)

_194also expresses such a quantitity. The solubility parameter is the square root of the energy of vaporization or sublimation per unit volume, and is described by Hildebrand as a measure of "internal pressure." While this concept will be used later in considering the applicability of regular solution theory to the prediction of interaction parameters (p.220), it should be mentioned at this point in conjunction with the correlation attempts. A plot of the first-order interaction parameters versus the solubility parameter is presented in Figure 58. There is no correlation in this plot and furthermore, as will be shown later, only half of the interaction parameters have the correct algebraic sign that would be predicted by regular solution theory. It would appear that the direct moderation of energy factors by size-factors does not lead to a reasonable explanation of the interactions for the systems studied, however, size effects undoubtedly play a part. 3. Thermodynamic Factors With the inability of usual alloying concepts of electronegativity and size difference to completely account for the observed interactions, attention was directed to qualitative and possibly quantitative thermodynamic considerations. The approach used was primarily phenomenological and began with consideration of the zinc-bismuth binary system itself. Strong positive deviations from ideal solution behavior characterize liquid alloys of zinc plus bismuth. The activity coefficient of zinc in the dilute region is about three and there is a region of liquid

-195+10 +5 - y In Pb 0 0 -5 C 20 Permissible Regions for Parameter, AccordSoing to Regular Solution Theory. -Is Bi Zn IL~~~ OAu -!0 0 10 20 30 40 50 60 70 80 90 100 110 SOLUBILITY PARAMETER (AEE /VYZ Figure 58. First-Order Zinc-j Interaction Parameters versus Solubility Parameter of Added Solute j.

-196immiscibility at higher zinc concentrations. Zinc and bismuth are only slightly, if at all, soluble in each other in the solid state. Kubaschewski and Catterall(53) attributed positive excess entropies of mixing in the zinc-bismuth system to the destruction of covalent bonds. Kleppa(45) showed that the partial molal heat of mixing of zinc —' 2 - could be represented by an equation of the type AH = Bi AHm, where Zn Bi Zn AHZn is the relative partial molal heat content of zinc at high dilution and 0 is the volume fraction of bismuth. This showed that the atomic sizes of zinc and bismuth, which differ by a factor of 2.1 (volume ratio) have a bearing on the number of bonds formed between them. All of these observations are consistent with a liquid state where the mixing of zinc and bismuth is non-random. Therefore, it might be expected that when a solute is added that would tend to alloy with or form a miscible solution with zinc, the activity of the zinc would tend to decrease, since effectively the bismuth would "see" less zinc. Conversely, a solute that by itself would tend towards immiscibility with zinc might effectively increase the activity of zinc relative to bismuth. Accordingly, an attempt was made to classify the experimental results by the type of solution formed by zinc plus the solute j. Table XIV qualitatively summarizes the solution behavior and also presents activity coefficient and excess free energy of mixing for zinc dissolved in an equi-molar solution of zinc with j. The data used were taken from Hultgren.5

TABLE XIV FACTORS FOR TRERMODYNAMIC EVALUATION OF ZINC-j BINARY SYSTEMS Type of Solution Calculated Activity Coefficient** Formed in Binary Alloy and Excess Free Energy at 5500C (deviation from ideality) in Zn + J in Zn + j / Solute-j Zinc + j j + Bi at xZ.5 at xZn. /g atom) Cu neg* PQS.57 -1610 Zn - pos Ga pos*? 1.14 +215 Ag neg* pos.88 -550 Cd poe neg 1.40 +550 In pos* neg 1.46 +625 Sn pos poe 1.3355 +460 Sb neg*?.54 -1000 Au neg* neg.04 -5500 Hg pos neg 1.20 +500 Pb pos* neg 2.53 +1550 * Direction of deviation from ideality consistent with sign of interaction parameter between Zn and j in Bi. ** Data from Hultgren et al. )

-198Considering only the type of solution formed by zinc in j, the expectation is that those elements forming a binary negativedeviating solution with zinc would also tend to produce negative interaction parameters with zinc in ternary solution with bismuth. Thus, the activity of zinc in solutions with copper, silver, antimony, or gold has a negative deviation from Raoult's Law (and there are many compounds formed in the solid alloys of these binaries). Negative interactions with zinc occurred in the ternary solutions formed by these metals in bismuth. Conversely, gallium, indium, and lead are positive interactors with zinc in bismuth, and the zinc activity in binary alloys with these metals exhibits a positive deviation from ideality. There is little tendency for solid solubility or compound formation in these binary systems. However, three of the solute elements, cadmium, tin, and mercury were slight negative interactors with zinc in bismuth yet the activity coefficient of zinc in binary alloys with these metals is greater than one, a positive deviation from ideality. The simple qualitative explanation based on the character of the solute binary systems is thus valid for seven of the ten cases considered. An attempt was made to extend the qualitative analysis to include the relative ideality of the solute j in bismuth (or the converse) but examination of the j-Bi solution behavior (Table XIV) failed to reveal any regularity that could be correlated with the interaction parameters. The success of this explanation was about as good as obtained using Wagner's electron model in conjunction with electronegativities. Where the interactions were large, the effect was successfully predicted.

-199The unexplained elements were the same for both models. The slight cadmium and tin effects could not be explained with either model, while mercury's electronegativity was the same as bismuth, rendering that prediction ambiguous. Therefore, the same factors involved in the binary zinc-j interactions probably entered into the electronegativity model. Considering the 70 percent qualitative success of the explanation, calculations were made to see if any quantitative basis existed. The premise was that the interaction between zinc and the solute j, in a ternary solution were both approach infinite dilution, might be governed by the excess free energy of mixing in a binary alloy of 50 mole percent zinc in j. The data in Hultgren(54) for AGn were extrapolated to 5500~C under the assumption that the heat of mixing remained essentially constant. The results are summarized in Table XIV and plotted in Figure 59 versus the interaction parameter. The results show that there is a definite quantitative basis to the effect. Furthermore, and perhaps coincidentally, the results can be grouped by the position in the Periodic Table of the elements involved. -xS In general, the less negative the AGzn in the binary zinc-j, the more positive the interaction in a ternary solution with bismuth. However, it will be noted that the exceptions to the qualitative rule, the slight negative interactions for cadmium, tin, and mercury, appear in the wrong quadrant of the quantitative diagram, and furthermore, while the 4th and 6th period elements appeared in order of atomic number going from left to right on the plot, the 5th period elements were not in order,

+ 10 I II, I lI Note: The elements F do not appear dn +5: Gay / periodic order. In 0 On -5- I Per.,/ LO -10 I -20 I | I I, i, } -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 - xs AGZn AT Xz=0.5, T=823~K, FOR BINARY SOLN. with j- cal/g-atom Figure jt9. First-Order Zinc-j Interaction Parameters versus Excess Partial Molal Free Energy of Mixing of Zinc for Equimolar Mixture of Zinc and j.

+5 - In 0 Pb o cu_^ —^.lo~~~f / 1n 0 Au~~~~~Cu 0 Sb Figur 60. irst- rder Interaction Para Per. c' Z %, 3Z1 Per.^ i - 10 -15 ) Au - 20 - 5000 -4000 -3000 - 2000 -1000 0 +1000 + 2000 + 3000 AH- AT XZn 0.5 at/g-atom Zn Zn Figure 60. First-Order Zinc-j Interaction Parameters versus Partial Molal Heat of Mixing of Zinc in Equimolar Mixture of Zinc and j.

-202Since the assumption of constant heat of mixing was involved -xs in the extrapolations of AGzn used in Figure 59, a similar plot of interaction parameter versus ZHZn was made. Figure 60 shows that the general - — Mxs form of the plots of en versus AlH or AG was the same however Zn Zn Zn there is a bit more scatter in the plot involving EHZn. Therefore the -xs entropy term involved in AGzn contributes to the interaction effect. Since the entropy can be divided into the configurational and vibrational contributions, it is probable that, despite the undetermined nature of the size effect, it is the configurational entropy term that accounts for the smoother curves of Figure 59 as compared to Figure 60. The comments regarding size difference effects on bonding energy made by Kleppa(45) and Kubaschewski and Catterall(53) (cited on page 196) support this conclusion. An attempt was made to relate the data of Figure 59 by subgroup in the Periodic Table but no regularity of behavior could be discerned. The results of these correlation attempts showed that most of the observed first-order interaction in the ternary alloys could be accounted for on the basis of binary interactions between the solutes. However, the slight interactions due to tin, cadmium, or mercury additions did not fit any of the simple explanations. 4. Other Physical or Chemical Factors In addition to the properties previously discussed, there are other physical or chemical properties that might be used to correlate the interaction effects. Where these properties may be within themselves

-203 periodic in behavior, the result of the correlation will be only as good as the attempt shown in Figure 48 for a direct relationship with atomic number. For instance, Batsanov(8) recently derived a system of electronegativities based on geometric considerations. Covalent, metallic, and geometric electronegativities were given, however, the values are all periodic with atomic number and no meaningful correlation could be made with the observed interaction parameters. As another example of a possible correlating factor, Figure 61 shows a plot of the first-order interaction parameters versus either the first or second ionization potentials of the solute elements (81) Turkdogan() had previously shown that third element effects on the-solubility of carbon in iron (which can be related to the interaction parameter) exhibited some linearity with the second ionization potential. The present results show no correlation with the second ionization potential and only a dubious relation with the first ionization potential. Sponsellor(77,89) attempted to use the first ionization potential in the same way that electronegativities were used in the application of Wagnerts electron model (p.180 ). His prediction of the sign of the interaction parameter for third element additions to calcium dissolved in liquid iron was correct in only two of four cases. Another physical quantity giving a rough measure of how readily an element gives up electrons is the work function. Sponsellor also applied the photo-electric work function to his data mentioned above and was correct in predicting the algebraic sign of the interaction for three

-2045 (-Ga Ga In 0 Pb OIn \ OPb 0 Ofb Zn Zn ~- Sn0 0~ 0 0 Cd HgI Sn Cd Ig 0o Cu C00Ag 0 AgO 0 to 0 0 LO -5- Sb Sb -10 Bi -I5 Au 0 I O Au -2 0, 5 6 7 8 9 10 15 16 17 18 19 20 21 Ist IONIZATION POTENTIAL-VOLTS 2nd IONIZATION POTENTIAL-VOLTS Figure 61. First-Order Zinc-j Interaction Parameters versus First and Second Ionizatinn Potentials of Bismuth, Zinc, and Added Solutes j.

-205out of the four cases considered. The present results were tested against the work function, but the range of values given for the photoelectric work function made meaningful predictions impossible. However, using average values of work function determined by a contact potential method, (81) the algebraic sign of the interaction was predicted correctly for eight out of nine solutes for which data were available,assuming that that the value for zinc is less than for bismuth. A plot of the results of this correlation attempt is shown in Figure 62. The only element clearly falling in a "forbidden" region is antimony but the degree of penetration is slight. In addition, there is a fair quantitative relation between work function and the interaction parameter. The correlation thus accounts for the interaction parameters of tin and cadmium, but at the expense of losing agreement for antimony. As another example, one might directly consider the relative chemical reactivity of the various solute elements. A correlation could be attempted using an electromotive force series or the free energy of formation for a series of similar compounds. Figure 63 shows such a relation in terms of the free energy of formation for chlorides.(62) The results appear to fit a two-branch curve which has its discontinuity at the free energy of formation of zinc chlorideo However, Figure 63 is purely empirical since there is no basis for drawing a two-branch curve other than the fact that the three positive interactors and the seven negative interactors may be classified in this manner. The extent of the interaction, as might be expected, depends on the relative difference in

-206+ 10. I I I " + 10 + 5 Go I Pb O 0 - Cd "'-' ~ Sn 00I iHg o ~Ag0 Cu 10 Ag L l 5Sb 01 -15 Bi ~~~~~~-20~0 LAu 3.0 3.5 4.0 4.5 5.0 5.5 WORK FUNCTION( BY CONTACT POTENTIAL METHOD) (81) Figure 62. First-Order Zinc-j Interaction Parameters versus Work Function for Bismuth, Zinc, and Added Solutes j.

.. 0 I I I Go +5 1 \In ~~~~~~0 0~~~ Zn o o o ACu / c OSb' N 5 V -5 ho o / l ~~~~~~~~~~/~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ / -10 / / / / -15 0 Au -20 0 10 20 30 40 50 60 70 80 -AG' (CHLORIDE) - KCal/g-atom Figure 63. First-Order Zinc-j Interaction Parameters versus Standard Free Enrrgy of Formation of Chlorides at 550~C.

-208o AG much as the previous explanations based on electronegativities or -xs AGZ for the binary systems formed by the solutes, however, the shape of the relation drawn in Figure 63 has no readily apparent theoretical basis. Weinstein and Elliott(87) attempted a correlation between the effective number of free electrons of the added solute and the interactions with hydrogen in liquid iron. Their correlation was only fair but there was some indication that the relation was a two-branch curve with a discontinuity at four effective free electrons. Due to a lack of data, is was not possible to apply this approach to the present results. Furthermore, there is no direct analogy between Weinstein and Elliott's curve and Figure 63. In summary, it would appear that a good chance for a structurerelated explanation of the interaction effects would lie in a model, such as Wagner's approach, which considers the chemical potential of the free electrons but which also incorporates some recognition of size-dependency. In a later section (pp. 217-239) the prediction of interaction parameters is considered on the basis of some presently accepted general solution models having their primary basis in statistical thermodynamics. 5. Temperature Dependence of Interactions The experimentally determined first and second-order interaction parameters were found to either be constant over the 450 to 650~C temperature range of this study, or else to exhibit a linear variation with the reciprocal absolute temperature. The data could be expressed in equations

-209of slope-intercept form, e.g., B' = A,..... Dealy and Pehlke(l6) stated that by using thermodynamic relationships the variation of the first-order interaction parameter with temperature is given by the expression: I4 _ I ( a H \ d(Tf) - 7;3i7.)7 — o (30) (The derivation of this equation, which has not been previously published, is given in Appendix B. together with an alternate version). Dealy and Pehlke further suggested that a useful extrapolation of interaction parameters could be accomplished by a linear plot, providing that the derivative of enthalpy was constant or else not a strong function of temperature. (22) Chipman and Corrigan(22) have also recently considered the question of the temperature dependence of interaction parameters and, under certain simplifying assumptions, derived an equation similar to that of Dealy and Pehlke. The assumption made was that the interaction paraXln7i meter xj = 0,remains constant over a finite range of concentration(at first glance this appears reasonable since it will be recalled that linear relations between ln7zn and xj were found for many of the solutes studied in this investigation). If the second-order effects

-210were neglected and the infinitely dilute solution was adopted as the reference state, then Wagner's Taylor series expansion reduces to: R, "/ =. (31) (note that by the change of reference state, lnyi = 0) Since the excess free energy of mixing is normally given by the expression AS* = R/T' (32) a new free energy term was introduced as G-J RT = T c (33) with the meaning that this expresses the "extra" free energy of component i due to the addition of component j and "within the limits of presentday experimental accuracy", is a linear function of concentration within a finite range. By defining "extra" terms for the entropy and enthalpy, Chipman and Corrigan then could write the following relations culminating in an expression for the temperature dependence of the interaction coefficient. G R/ T j~ -' -7S'Jy x - T C rj'(34)

-211where /7 = Xj. XJ J.eXJ I +9 = i(,T9) ='~ Ht~ (~35) The interpretation placed on this result by Chipman and Corrigan if the heat of solution of component i in a ternary alloy was the same as i ts binary alloy with the same solvent then H = and i as in its binary alloy with the same solvent, then HA = 0 and E. is independent of temperature. It was further stated that if the entropy change is the same for both cases, then _ f _ -_ (i( SK.O) (36) allT Thus a direct proportionality should exist between c and 1/T. This approach to temperature dependence is interesting since it places a more readily grasped interpretation to the situation than Dealy and Pehlke's less familiar quantity ( aH ) However, it is evident \xixj j that the two approaches are really similar. By restricting their treatment to first-order effects only, Chipman and Corrigan limited the generality of their results. Their approach may be extended to a more general consideration of the temperature dependence of dilute solute interactions by the following reasoning:

-212Writing the Taylor series expansion through the second-order terms and assuming the case of zinc-bismuth alloys where E. = c. = E= 0 1.1 i 0 0 0I 2 A < a 6. t /t + \j L C O JY, X/, xi.x.. (22)' Y t *1 +X>, XJ2 X' (37) L L J L 4 J J e -X S Since for a binary alloy, AG. = RTlnyi Then for a ternary or multicomponent alloy; multiplying (37) by RT?RTf(C4,y Gz> (38) ~eX i(j) is the "extra" excess free energy due to the addition of j (the same as defined by Chipman and Corrigan). This quantity may be separated into enthalpy and entropy terms Tj,','~ /'So (39),cJ) L(j ) ~' Rewriting Equation (37) G JJ 7T(- -t-~ WLTL. # X, J L (40) #' ^ L r - C (41)

-213If E' is zero, or else merely neglected, then Chipman and Corrigan's expression (Equation 34) is obtained in slightly different ex ^ex H S form by using the definitions H n and - a / /< (42) From this expression it can be seen directly that if a =, then e~ is directly proportional to 1/T if T = 0, then jE is a constant independent of T and equal to -c/R However, this expression is more general since it also shows that a plot of e versus 1/T can have a finite slope and intercept, or if such a plot is a straight line, then [ and a are constant. The slope is given by r/R and the intercept is - /R slope - R[ 1i ^ — - intercept = - - 1/T

221462H Furthermore, Dealy and Pehlke's term H is [ and is a axiaxj partial molar quantity. Its nature may be grasped more readily by the following definitions:'A --' = H AhYL 2,-j Qy S/ce Y;L C n T and thus it is the change in the partial molar enthalpy of i due to the addition of j i.e., the "extra" quantity defined by Chipman and Corrigan. This analysis also shows that it is not possible to have a constant first-order interaction parameter and a constant entropy of mix-eex ing (Si = O) unless the interaction parameter is zero. For the case where c? appears to be constant (within experimental error) for a limited range of temperature, T is probably very small, and hence E. is also quite small. This is consistent with the observed behavior for the ternary additions of cadmium, tin, and indium. The expected slight temperature dependence was detectable only for the case of lead. In the more general case, such as with the present experimental results where the second-order interactions are large or are not neglected, they may be included in this analysis by the following process: The "extra" enthalpy and entropy are divided into the firstorder and second-order components. H rj, = ~ -,H (43) ~s~~~~~,ifj = Sl ~Sv ~ ~(44)

-215Then using Equation (41) (6'G =eJ 0 T - (S.' So,] (45) The equation can be separated as follows.) [1. i (46),~x.. StR T, - - " and can be written as two parts "-,^ - 7,\ -- r (47) I I FH~ _ _ _ 1 Ge; -x, - T. T1 x'- -T ( C8) A linear relation, such as indicated in Table X, between C. and 1/T is predicted if the second-order "extra" terms are constant, Furthermore, additional equations could be obtained if the process were carried out through the third-order terms, etc. Experimentally in a galvanic cell at a given xi and xj it may be observed that'JS.j C) L (49).'=j $LKj' ) 6L6 I Syj (5~) (J' J (J-T

-21. - By using Equations (43) tirough (48), tne observed values can be expressed as the sc-ms of the first and second-order contributions to the "extra" excess quantities: a- a,~ I (-J/ I K t XJ 7j (52) In Chipman and Corrigan's second conclusion (Equation 36), a — ex direct proportionality was expected between ~ and 1/T if Si(j = 0 however, Equations (47) and (48) showed that proportionality may also be obtained when a and a2 are not zero. -e x Equation (49) was used to obtain values of Si(j) for several of the alloys studied in this investigation. The results which are tabulated below indicate that it is probably different from zero. while Table X had shown the proportionality between the interaction parameters and 1/T Si Si(3) (Xj =.015) Si(j) 3i-Si(j ) Zn-Bi Xn Binary j = Cu Ag Au Cu Ag Au.015 9.95e.u. 9.85e.u. 10.0 e.u. 10.80e.u. -0. 10eu. +0.10e.u. +0.85e.u..025 9.05 8.90 8.85 8.35 -0.15 -0.20 -0.70.0375 8.30 7.85 8.35 - -0.45 +0.05 -.050 7.60 7.25 7.75 6.85 -0.35 +0.15 -0.75 e.u. = entropy units, cal/~K/g. atom

-217= -ex The magnitude of Si(j) in the above tabulation is at the borderline of statistical significance. The estimated standard deviation of the slopes of the emf-versus-temperature curves had ranged from about.001 to.005 millivolts per ~C. Applying 90 percent confidence — ex limits to these values suggested that Si(j) as calculated from Equation (49) could range from +.10 to +.40 entropy units and still be interpreted as within the range of experimental uncertainty. Hence, it is believed -ex that the tabulation of Si(j) on page 216 has qualitative, but not necessarily quantitative validity - as exemplified by the differences in algebraic sign for a few values. Equation (51), in conjunction with the temperature dependence coefficients from Table X was also used to estimate Sx for several of i(j) these alloys. The results were of the same order of magnitude as the experimental values but the previously mentioned uncertainties precluded a quantitative comparison. The consideration of the temperature dependence of the interaction parameters has thus led to a potentially useful extension of the "extra" terms introduced by Chipman and Corrigan, The present studies suggest that the possibility of second-order effects must be considered in any attempt to interpret solute interactions in terms of thermodynamic quantities such as enthalpy and entropy. B. Prediction of Ternary Interactions From Simple Solution Models Thus far the discussion has been concerned with attempts to account for the observed interactions on the basis of structure-related

-218= physical or chemical criteria. The possible explanations were only semi-quantitative at best. On the other hand, the general development of the field of thermodynamics has led to the formulation of several theories which attempt to account for solution behavior. Insofar as these theories lead to expressions for the excess free energy of mixing, it is possible to use them for calculations of interaction effects. Oriani and Alcock(31) recently discussed some simple models of solutions and pointed out that they are primarily statistical and based on assumptions regarding the energy of an assembly of atoms, molecules, or ions without regard to the physical basis of the interaction. The models considered were the regular solution, the subregular solution, and the quasi-chemical model. The principal assumption involved is that the total energy of a collection of molecules is the sum of the interaction energies between the individual molecules taken two at a time, In the simple solution models only nearest-neighbor interactions are considered. The results of the present investigation were examined from the standpoint of the implications of these models and their utility for the prediction of interactions in ternary solutions. 1. Regular Solution Model The regular solution concept, which was introduced by Hildebrand,(33) is that the entropy of solution is ideal, i.e., the mixing is random, while the enthalpy of solution has some finite value. Using the convenient

-219concepts of the excess theymodynamic quantities, the following relations apply:(41) zGxs a A. -n' 4 RTR, - (53) L, S S, -'- S 0 -)<5 —,ejL - aH' A, -AX H. (55) Xs X- - /G\ ^ - / t;64 6, ZS,. S5 (56) then. T = ( A(o (57) The activity coefficients for regular solutions are thus directly related to the partial molal heats of mixing. For the case of a regular ternary solution, Hildebrand and Scott(33) showed that the activity coefficient of component i is given by the expression

-22012^~~ = V -' = - - )-;^ -~kt4(58) J is the volume fraction, V is the molar volume and 6 is the solubility parameter Z\vap The solubility parameter is defined as, the square root of the energy of vaporization per cubic centimeter, The meaning attached to the solubility parameter is that it represents the "cohesive energy density" of an element.(33) Equation (58) may be extended to the calculation of interaction parameters by differentiating with respect to the mole fraction of component j When the limit is taken at infinite dilution of the solutes, the expression for the interaction parameter ej is given by: i (~ /% s k Vko j c2()k} i2) (59) A similar expression may be derived for the self-interaction parameter C L __ __ t 0 /RT Vi (s -S (60) The interaction parameters thus become only functions of the molar volume and the solubility parameters of the elements involvedo The subscripts for i and j may be interchanged to show that Ci and hence Wagnerls reciprocity relation (Equation (3)) is j i

-221obtained directly. In addition, the form of the equation for CJ i predicts a linear relation with the reciprocal absolute temperature, Thus, depending on the validity of the assumption of regular solution behavior, the interaction parameters can be predicted from readily obtainable information. Hildebrand and Scott(33) tabulated values of V and 5 for a number of metals at 298, 500, and 1000~K, The change in solubility parameter with increasing temperature was quite small "thanks to small coefficients of thermal expansion and small ACp s, as well as small volume changes and heats of fusiono" Before attempting to apply these equations to the present results for ternary solutions based on zinc in bismuth, their utility was tested by calculations for several systems where substantially regular behavior was expected. Hildebrand and Scott examined galvanic cell data for several binary systems and noted that the agreement with regular solution behavior was "excellent for zinc in cadmium, good for cadmium in tin and lead, and not so good for zinc in tin and thallium in tino" Dealy and Pehlke's compilation of interaction parameters in non-ferrous systems(l6) gave the effect of several third elements on the activity of cadmium dissolved in tin or leado In addition, Boorstein and Pehlkels(2l) results for tin additions to cadmium dissolved in bismuth were useful, since examination of their cadmium-bismuth binary data revealed that regular solution behavior was obeyedo

-222The calculated and observed interaction parameters for those systems where the binary solutions were regular are compared below: Solvent Solute. Solutej T~C ce Predicted c~ Observed 1 j 1 1 Sn Cd Zn 700 1,l 1o6 Sn Cd Pb 500 -4.8 0 Pb Cd Sb 500 + 8 -1o6 Pb Cd Sn 500 +16 -1,4 Pb Cd Bi 500 - A4 - o9 Bi Cd Sn 500 +.8 +11o The algebraic sign of the interaction parameter was predicted correctly in only three of the six cases considered. Where the sign was predicted correctly, the numerical value of the interaction parameter was calculated with only fair success. The present data for the bismuth-zinc-j systems were then examined, however, it was immediately noted that the behavior of the zinc-bismuth binary was not regular. The observed values for the mixing entropy of zinc were appreciably greater than the ideal valueso Since the molar volumes of zinc and bismuth are widely different (9.2 cc/mole versus 21o3 cc/mole), an attempt was made to account for the excess entropy of zinc on the basis of the size effect. The calculation was made using an expression given by Hildebrand and Scott(39) for the ideal mixing entropy of unequal-size moleculeso L S - R -*w #- [ (61) (I L & kv/t J

-223A comparison of the observed and calculated mixing entropies for zinc in bismuth is given below: Si - S, Ideal Si i'2 XZn Equal-size Unequal-size Si - S. Molecules Molecules Observed o015 8~35 eu. 8.95 9~95 O025 7o35 7.95 9~05 o0375 6~55 7005 8.30 o050 5095 6o45 7060 (eu = entropy units, cal/~K/g-atom) These results show that even if the size disparity is taken into account, the zinc-bismuth binary system is not regularo (Oriani and Alcock(31) noted that an internal inconsistency in the definition of the regular solution can lead to an apparent ASi of a few-tenths of an entropy unit. The results given above exceed this limito) It was therefore not expected that the prediction of interaction parameters for bismuth-zinc-j systems using the regular solution model would be particularly successful. The calculated and observed parameters are compared in Table XVo The algebraic sign of the interaction was predicted correctly in only five of the ten cases consideredo None of the positive interactions were predicted correctlyo For the cases where the qualitative prediction was correct, the numerical values were only in fair to poor agreement with the observed values at 550"Co The regular solution model would thus seem to have limited usefulness in the prediction of interaction effects in metallic systems. The success of the model was only 50 per cent, regardless of whether the underlying binary solution conformed to regular solution behavioro

TABLE XV COMPARISON OF OBSERVED INTERACTION PARAMETERS WITH VALUES CALCULATED USING SIMPLE SOLUTION MODELS Regular Wada and Alcock and Richardson's Equations Solution Model Saito's Eqn. Observed ecn Predicted CZn at T, ~K Observed EL- Predicted EZn at 823~K Factors Used with Alcock and Richardson's Equations Interpolated -- YZn()a.aor Eqn. (67) Eqn. (71) Quasi-chemical (Bi)& ast x =.5 Zn(Bi)0 F - zn(j) Extrapolated Random Sol Quasi-chemical Solute EJ at 823~K - Model T, K T,~ K 7Zn T, K 7y F n(Bi)0(Bi) T,~K e e = l1F e = - 1] Zn Zn Zn Cu -2.4 -2.2 -5.2 to -6. 850 5.1 1300.6 1100 1.5.078 1100 -1.8 -2.5 -3.0 Ga +5.0 +.7 -1.6* -.3 to -.9 - 723 1.15 - -- - Ag -2.5 -1.8 -2.7 to -35.6 1000 2.6 1000.6 1000 2.1.11 1000 -1.6 -2.2 -2.6 Cd -.3 +.2* +5.1 to +7.4* 773 1.0 800 1.4 790 35.4.41 790 -.53 -.9 -.9 In +2.2 -1.0* +2.6 623 *3b 700 1.6 700 4.0 1.5533 700 +2.2 +0.3 +0.53 m Sn -.5 -2.53 -1.5 608 1.2 700 1.5 650 4.6.27 650 -.5 -1.3 -1.4 Sb -4.5 -1.1 -1.6 to -2.1 - 823.6 - - - - - Au -18.0 -2.4 -35.7 973.8 1048.09 1000 2.1.054 1000 -16.5 -2.9 -5.6 Hg -1 to -2 +1.5* +2.7 to +3.3* 594 1.46 608 1.2 600 5.0.16 600 -1 to -2 -1.8 -2.1 Pb +1.5 -. 3* +3.0 773.4 926 2.2 850 3.0 1.83 850 +1.3 +.6 +.6 No. of systems for correct prediction 5/10 7/10 8/8 8/8 of sign of interaction * Systems where algebraic sign of e was incorrectly predicted a Data from jultgren et al. (4) b Calculated from AHZn assuming regular solution c At average temperature for data of jBi) and d z = 7.5 J(Bi) ad7Zn(j)

-2252o Sub-regular Solution Model A variation of the regular solution model was proposed by Hardy(35) and termed the "sub-regular" model. In this formulation, the excess free energy of mixing is taken as a function of concentration. For a binary system, Yokokawa et alo(36) gave the sub-regular formulation for the excess free energy as x^ [/-\^ (i )Aj t~(62) A plot of Ge/xixj versus xi should thus be linear and have finite slope. The same treatment was extended to ternary systems by Yokokawa et al,, and the following expression was obtained C77 A X oA+y' - A a xXX, ~~ -x xL^ALkE + XL j A~i X; xL + x (63) tY XXXk A' + yk^ X< XJ- AJ where the A's are constants, characteristic of the appropriate binary system, This equation was further used to obtain a rather complex relation for the excess chemical potential of constituent i o As part of the present investigation the latter expression was differentiated with respect to the mole fraction of component j in order to derive

-226an expression for the interaction parameter in terms of the constants of Yokokawa's formulation of the sit-regular model. After considerable manipulation the final expression is given by c -n f4'A ~ Ak -AII/(A4J =A/K -AAk (64) It is assumed that the interactions A~ and A' are unaffected by the additional solutes and/or solvent. It will be noted that this relation predicts a linear relation between ~e and reciprocal absolute temperature Thus, providing that the constants are known for sub-regular behavior of the three binary pairs in the ternary alloy, the interaction parameter may be calculated provided that the binary alloys are either sub-regular in behavior or regular (in which case the constant A' = 0)o Yokokawa et al, gave constants for binaries involving tin, zinc, indium, and bismuth and it was possible to compare experimental results from two systems studied in this investigation with calculated values from the sub-regular model, The calculations are given below; System A AO A A A Ay ASn-Zn Zn-Bi BiSn Sn-Zn Zn-Bi BiSn Sn - Zn - Bi 1660 3100 =205 -430 1280 -125 A~ A~ A~ A' A l As In=Zn Zn-Bi Bi-In In-Zn ZnBi Bi In In - Zn - Bi 2320 3100 -700 -420 1280 0

-227(It was not possible to verify these constants; Yokokawa commented that the zinc systems could be considered sub-regular only in the zincdilute region,) Let Sn k = Bi, i = Zn, j = In Then using Equation (64) eSn _ 1 (1660 - 3100 + 275) + (-430 + 1280 + 125) Zn RT - (-190) RT at 550~C eSn -012 (Observed Sn = -0.5) Zn Zn'Zn -='- (2320 - 3100 + 700) + (-420 + 1280) 1 (780) RT at 550~C In = +0.50 (Observed eIn - +2,2) Zn Zn For both systems the proper sign of the interaction was predicted, however, the numerical values were low by a factor of about four. The results of this method are interesting and suggest that further consideration should be given to tests of the utility of Equation (64). However, the method has the disadvantage that the binary system

-228behavior must be known experimentally and that it conforms to the expression for a sub-regular solution (Equation 62). 3o Quasi-Chemical Solution Model The regular solution model proposed that the entropy of mixing was due only to perfectly random mixing of the constituentso This assumption was inconsistent with a finite energy of solution, however, the difference was stated by Oriani and Alcock(31) to be small and often negligibleo If the restriction of perfectly random mixing is removed, the resulting solution model becomes consistent with finite energies and gives rise to a solution model that had been termed the quasi-chemical theory. This model has been discussed by Guggenheim, (37) Oriani and Alcock(31) and applied to the problem of interaction effects by Alcock and Richardson(l17,27) and Wada and Saitoo(15) The principal assumptions of the quasi-chemical treatment are that bond energies between molecules are constant and the coordination numbers (number of nearest neighbors that each molecules "sees") are also constant. The implication is that the molten metal forms a quasicrystalline lattice and the resulting interchange energies between the like and unlike molecules can be treated by a statistical approach, Alcock and Richardson first used a chemical approach or a so-called random solution model to characterize the effect of an added dilute solute j on the partial heat of solution of a given dilute solute i o(17) By considering pairwise interactions, the following

-229expression was derived \Jt(9 ];) =(~xiJ.) — AH^)-/ kHj.( ) (65) Xj- o By assuming that at low concentrations of j, Si(j+k) = ASi(k) ieno "extra" excess entropy of solution (see pp. 210-212, the discussion of temperature dependence), then a ternary "regularity" of behavior occurs and the expression could be written as fT \ I Wx ~) - ^ ]i (k) (66) (Chipman and Corrigan(22) later felt this could only be true when EJ and cj were smallo) Alcock and Richardson then proposed the following set of assumptions 1. ASi(j) = Si(k) at xi = 2. Yi(j) and Yi(k) are taken relative to the same standard state for i 3o Aj(k) is ideal Under these conditions, Equation (66) reduced to the simple form tJ i< 4;~ 3~ ~ j "' -;~'(k -en- ~, (67) Ki losj ~ %k (

-230Richardson(l) stated that i should be present at concentrations substantially less than j. Wada and Saito(l5) pointed out that the same result is obtained if the three binary solutions are taken as regular solutions (see Equation 57). In a later publication, (27) Alcock and Richardson used a more rigorous quasi-chemical approach from a statistical viewpoint to obtain the following improved expression AEI(VIJ=/l- Ct' (gas'LXA. (68),'(]^J k(j J and Z is the coordination number taken constant in the solution. They differentiated this equation to produce an expression for the interaction effects "~~~'~,. i~ (69),l u((. = - ^-l! (69) For the case where both the solutes i and j are present at infinite dilution, Equation (69) was reduced to the following expression: 0((70)

-231and since yk(k + j) = 1 for k = 1 Z ( k -1 (71) It was pointed out by Alcock and Richardson that Equation (68) may be rewritten by taking advantage of the relation AH. i(jk) ny(jk) RT to yield the form + XJ ~-[4H ^ __"^ -] -ln L4Lk)] = - [=k RJ If the exponential terms in this equation were expanded and all except the first term are ignored, Alcock and Richardson stated that the resulting expression can be shown to be an integrated form of Equation (67)o Equation (67) was thus a first-order approximation to a quasi-chemical approacho Wada and Saito(15) based their calculations directly on the zeroth-approximation of Guggenheim's quasi-chemical method for regular solutions. By assuming that the coordination number in the liquid metal solution is about 10 to 12, and that the structure is very nearly close-packed, a face-centered cubic lattice can be taken as the model of the quasi-crystalline lattice. The dilute solutes are then assumed to be placed substitutionally in this lattice and the formal treatment

-232 of statistical thermodynamics can be appliedo Assuming random distribution and using Stirling's approximation to the factorial, the following equation was derived by Wada and Saito for the interaction parameter - R (j-W -^Wij (73) where W's are interchange energies between the subscripted componentso It was also shown that the expression for ci is the same, and thus J Wagner's reciprocity relation is satisfied. By comparing the various equations, it is seen that the random solution approach of Alcock and Richardson and the quasi-chemical approach of Wada and Saito produce identical results. Wada and Saito(l5) U " /T ( wj - -) (73) Alcock and Richardson(17) = ( - ) (66) However, the essential difference in these approaches lies in the means for evaluating the binary effects which are combined to predict the ternary interaction parametero Alcock and Richardson(l^) assumed regular solution behavior to obtain Equation (67) (or by the

-233quasi-chemical approach derived Equation (71) directly)o Both of these equations utilize experimental observations on binary alloys to predict the ternary effect. On the other hand, Wada and Saito proposed the use of an independent "analytical" equation for the determination of the interchange energies. By using an expression proposed by Mott(82) for the excess free energy of mixing in binary solutions (used in connection with studies of binary miscibility), and making use of an equation for Xs the excess free energy of mixing in a dilute solution, AG.. = Wij 0o.0 iiJ iJ j (this apparently comes from Hildebrand's development of regular solution theory), the equation for the estimation of the interchange energy is wj = V( s- s r- 2 3, 060 (x -x 1 (74) where Vm = molar volume, X = electronegativity, ~ = number of ij bonds, and S = solubility parameterso Since the binary molar volumes, solubility parameters, and electronegativities are known for nearly all elements, (33) the interaction parameters may be calculated directly, subject only to the knowledge of the value of n o Wada and Saito suggested that the smaller value of valency of the two components may be employed as the value of n and that Vm be taken as the arithmetic average of the atomic volumes of the two components in the solid stateo

-234The present results for the bismuth-zinc-j systems were studied using the solution models based on Equations (67) and (71) (Alcock and Richardson) and Equations (73) and (74) (Wada and Saito), The calculated and observed interaction parameters are compared in Table XV. The comparisons with Wada and Saito's model were made at 550~C and are presented as ranges of values depending on various assumptions regarding the value of n (81) Since the use of Alcock and Richardsonis equations requires knowledge of activity coefficients in the binaries, the comparisons were made at whatever temperature data were available in the literature. The binary data used were taken from Hultgren(54) and Dealy and Pehlke(86) and the present results for YZn(Bi) were extrapolated to the same temperature range, For Alcock and Richardson's quasi-chemical equation (Equation 71), the coordination number was taken as 7o5 since x-ray, electron, and neutron diffraction studies(83) have shown coordination numbers of molten bismuth in the range from 7-8The algebraic sign of the interaction parameter was predicted correctly in seven of the ten cases studied using Wada and SaitoYs method and in all of the eight cases for which data were available to test Alcock and Richardson's equationso The numerical agreement with experimental values obtained from Wada and Saito's equation (when the sign was predicted correctly) was excellent for silver or indium additions and in error by a factor of about two for all the other solute elements except goldo For gold, the predicted value was only 1/5-th of the experimental valueO

-235The numerical values obtained with Alcock and Richardson's equations were generally within a factor of two of the experimental results, The only exceptions were the predicted values for the indium or gold additions which were low by a factor of from five to seveno However, it was not expected that the numerical predictions for this equation would be very quantitative since the available data were not all at the same temperature and extrapolation outside the experimental temperature range was necessary for several of the values of y~ i Zn(Bi) In addition, the value of y7 for In in Bi was estimated under the assumption that the solution is regularo This estimate is only qualitative since the HHm values indicate that the In-Bi system is moderately sub-regular(48) (and even this may not be true since no values of AGxs were available to check sub-regularity)o Therefore, the only ternary addition for which the prediction can be termed significantly low is goldo Alcock and Richardson's quasi-chemical model gave slightly more negative parameters than the random solution model, however, the general agreement with the experimental values was about the sameo A variation in the assumed coordination number did not account for the disparity in the predicted values for goldo At a coordination number of 12 the predicted value was -3o39 at a coordination number of three or four the parameter was about -4o3, while a coordination number of two gave -6060 (A value of -17o5 was obtained by assuming a coordination number of one!)

-236Since random mixing on a quasi-crystalline lattice is an assumption in these models, their validity depends on how favorably the underlying binary systems behaveo Indeed, Alcock and Richardson's first equation directly assumes regular solution behavior in the binaries while Wada and Saito's method also has an implicit assumption of regularityo The regular solution model per se for the ternary interaction parameters (Equation 59) was successful for only 50 per cent of the cases consideredo This equation considered interchange effects (estimated as the difference in solubility parameters) only between the solutes and the solvento The quasi-chemical model admitted consideration of all three pairwise interactions, and in Wada and Saito's approach explicitly takes electronegativity differences into accounto This is consistent with the previous discussion of the results from the physical standpointo The quasi-chemical formulation of Alcock and Richardson appears to be advantageous since the coordination number appears in it explicitlyo However, it is still assumed that the coordination number is constant for all atoms in. solutiono There is very limited direct evidence available from diffraction studies that random mixing is not attained in some liquid binary systems(83) and indeed clustering is considered a major cause for the deviation of both solid and liquid alloys from idealityo (5385) Only when an absurd coordination number was assumed was it possible to obtain a "predicted" interaction parameter for the gold addition that agreed with the experimental valueso Thus, it is possible that the coordination number might be a function of the

-237alloying elements and may be variable within a given solutiono Possible corrections for such behavior were mentioned by Alcock and Richardson,(27) Wada and Saitots method has the disadvantage of a fixed coordination number taken as 10-12. However, this is probably good for most metallic elements since the effect in Equation (71) of changing the coordination number from 12 to 8 was only relatively slight. Thus, Wada and Saito's assumption of 10-12 probably does not introduce serious errorso Oriani noted(30) that the quasi-chemical theories are inadequate as exact statistical models because of the neglect of non-configurational factors and -the fact that pairwise interactions may vary with phase and concentrations. Such deviations of bond-energy models were also discussed in detail more recently by Oriani and Alcocko(31) Lumsden(38) considered the contributions of next-nearest neighbors to the excess free energy of mixing for a binary and first suggested a two-parameter expression that Oriani and Alcock(31) felt adequately expressed KleppaTs data for the zinc=bismulth system. However, this model breaks down in the Henry's Law regiono Lumsden proposed that three parameters are really needed to express the free energy of a metallic solution and discussed in detail why the regular solution model should not apply generally to metals. The three needed parameters were described as "adjustable o,, with no simple relation between themo' The possibility of extending Lumsden's equation to a ternary system and the derivation of an expression for the interaction parameter did not appear to be promisingo

-238From the standpoint of a readily available solution model for the estimation of interaction parameters for engineering purposes Alcock and Richardson's equations are suggested although it would appear that Wada and Saito's method is also capable of semi-quantitative resultso However, if the requisite binary data are available at the proper temperatures, Alcock and Richardson's method is preferred. The sign of the interaction effect was predictly correctly for all eight systems tested in this investigation, while Obenchain(l3) obtained similarly correct predictions for eleven other systems for which ternary interaction data were available from the literature. (Obenchain considered only the first or random solution equation of Alcock and Richardson,) Wada(9) recently reviewed the development of the Wada-Saito method and compared the predicted and experimental results for interactions occuring in molten iron. For the ferrous solvent the success of the method was somewhat better than attained in the present investigation. Obenchain found that Wada and Saito's method correctly predicted the sign of the interaction for eight of the twelve nonferrous systems for which he had taken experimental data. It would appear, therefore, that when suitable binary data are unavailable for applying Alcock and Richardson's method, Wada and Saito's equation can be used to estimate interaction effects in non-ferrous systems with about two to one odds that the results will be of proper algebraic sign and only a factor of two different from the proper numerical value, Unfortunately, neither method accounts for norpredicts a strong negative

-239interaction, however, this might be anticipated independently if the difference in electronegativities was large between the soluteso For the present data, the results of applying the quasichemical method are considered particularly promising since the zincbismuth system is definitely not a regular solution and can be considered only marginally sub-regularo The quasi-chemical model, while not exact, is thus shown to be a useful tool for estimation of dilute solute interactions in ternary liquid alloyso Co Validity of Wagner's Prediction Model for Multi-Component Solutions From a purely mathematical standpoint and without regard to a physical model for the solutionWagner's expression (Equation 1) for the activity coefficient in a multi-component solution is an application of the problem of representing a given function (in this case Inyi) by means of a sequence of polynomialso Since the activity coefficient at infinite dilution is the Henry's Law constant, both it and its logarithm have finite values (and finite derivatives) at the point xi = 0 o Hence a Taylor series may be used to expand lnyi about the origin,(72) The various interaction parameters are merely the coefficients of the Taylor series and are evaluated by the extrapolation of experimentally realizable quantities to the condition of infinite dilution with respect to the solutes, Mathematically speaking, we have a situation well exemplified by a statement attributed to Cauchy, "Give me five coefficients and I will plot an elephant; give me six coefficients and the elephant will wiggle its tail,"

-240The usual assumption here is that terms above the first order may be neglected, in which case the logarithm of the activity coefficient becomes a linear function of the mole fractions of the solutes and the resulting expression is presumed to be valid for any number of solutes so long as the solution may be regarded as "dilute". For ternary solutions it was previously shown (p. 19 ) that the second-order terms may be readily evaluated along with the first-order terms with no additional experimentation. Under the conditions that prevail in most of Bi-Zn-j systems, i.e., linearity of lnyZn with composition, the selfinteraction parameter and two of the three second-order parameters are zero. Thus, a second-order expression was shown to be rigorous for a ternary solution, and afforded the possibility of a better representation of the log activity coefficient in a higher-order solution. The utility of Wagner's prediction model or its extension, the second-order prediction model (pp.137-l4o0),rests in the ability of the truncated series to adequately represent the experimental facts in a system of higher order than ternary. The questions that might arise could include the following: What is the accuracy possible? How far away from dilute solution can these equations be extended; Does the inclusion of second-order terms significantly improve the calculations? Do the types and variety of interacting elements make a difference? What is the error introduced by truncation? etc. Several of these points were discussed in the course of presenting the experimental results on the multi-component solutions (pp.143-151). The agreement between the observed and predicted values

of the activity coefficient was generally quite good for the dilute solutions containing up to four solute elements in addition to zinc Near the limit of the Henry's Law region for zinc, (xZn ^ 05) the agreement was still good to excellent for the quaternary systems, and fair to good for the quinary and hexadic alloys studied~ If the zinc concentration was held well within the Henry's Law region, (XZn = 015) and the total solute concentration increased far beyond the "dilute" range, the observed activity coefficients were predicted in a solution with two negative interactions but were considerably lower than the predicted values for the two systems combining both a positive and negatively interacting solute with zinco Several quantitative studies were also made of the experimental results to aid in assessing the utility of the second-order prediction model. With regard to the accuracy of the prediction models, a study was made of the deviation between the experimental and predicted values of the activity coefficients. Only the absolute value of the deviation was considered - the sign of the deviation was left to a separate study, The results are shown in Table XVI as the average value of the deviation for each type of solution studied, both as a function of prediction model and the amount of zinc present, Two types of behavior were noted, Either the values deviated by about one per cent of the absolute level of the activity coefficient, (approximately three) or the deviation was about five per cento The results indicate that for higher-order solutions and for the higher content of zinc, (xZn = 005) the second-order model

-242= TABLE XVI QUANTITATIVE COMPARISONS OF MULTI-COMPONENT INTERACTION PREDICTION MODELS Average Value by Which the Predicted Activity Coefficient Differed from the Observed Activity Coefficient Average Arithmetic Value of (yobs - Ypred) Without Regard to Direction Type of Average-* XZna= O015 xZn = ~050 Solution Deviation: nModel I nModel II AModel I AModel II Quaternary.037.030.058.050 Quinary.045.042.14 025 Hexadic 024.030.12.025 Septenary.163.150 - Qualitative Frequency Distribution of (7observed - Ypredicted) Number of Observations for a Given Direction of Deviation XZn = o015 xZn = o050 rType of Model I Model II Model I Model II Solution (7obs - 7pred)~ - 0 - + -+ + Quaternary 3 1 7 6 0 5 2 2 4 0 Quinary 3 0 3 3 1 2 0 2 2 0 Hexadic 20 3 4 1 0 0 2 1 1 Septenary 2 0 1 2 0 1 - - Totals 10 1 14 15 2 8 2 6 7 1 n I7observed - 7predictedI * A = - - n a xZn mole fraction

-243did, as might be expected, produce noticeably closer predictions than the first-order modelo For all the solutions where xZn = o015, the deviations for either model were generally about the same, with some indication that the second-order model produced slightly better predictionso A separate compilation was made in Table XVI of the manner in which the deviations were distributed about the expected valueso For the dilute quaternary solutions, the first-order model tended to produce more positive deviations, that is, the observed activity coefficient was greater than predicted. The second-order model produced an almost equal number of positive and negative deviations, Random distributions of deviations were obtained for the quinary, hexadic, and septenary solutions with the first-order model, while the second-order model tended towards negative values for the hexadic and septenary solutionso On an over-all basis, it appears that the first-order model did not account for enough of the interaction, while the second-order model tended to account for too much. This behavior was also noted for the comparisons made at xzn = o050o A limited study was also made of the effect of temperature on the deviations. In virtually all of the cases examined, the deviation of the predicted value from the observed value was greater at 450OC and less at 650~C than the deviation noted at 550~Co It should be realized that such comparisons test not only the prediction models, but also the experimental values of the interaction parameters. Within experimental error, the inclusion of second-order

terms improved the predictions, however, the degree of improvement was slight and only really significant for the systems where xZn = 05 The implication of this is that in the solutions where xZn = o015 the remainder term of Talylor's Theorem for a truncated series is quite small and the neglect of the higher order terms is generally valido Since the predicted values account for virtually all the observed changes in the activity coefficients, the assumption that only the pairwise interactions are significant in the dilute ternary and higher-order solutions is probably a good one, As a practical limit for the "dilute" range, a bismuth content of approximately,9 mole fraction was a dividing line between "good" and "bad" behavioro However, in the one case of the quaternary system with silver and antimony additions, the "good" behavior extended to xBi =.785 o This probably resulted from the fact that the relations between lny and x. for Zn J the underlying ternary systems were linear far beyond the original experimental range. In the other cases, indium plus antimony and lead plus silver, the ternary behavior was probably not linear or else there was considerable interaction between the extra solutes that cannot be accounted for by the Taylor Series modelo For example, the intermetallic compound InSb is well known in the field of semi-conductors and, in particular, Ptak(84) has recently discussed the presumed formation (clustering?) of InSb in liquid solutions. However, there is no comparable compound in the lead-silver system. The inclusion of the secondorder term thus accounted for some but not all of the deviation from the first-order modelo

-245In the hexadic solutions, there was limited evidence that the agreement of the prediction models with the experimental results became poorer as xZn was increased while maintaining a constant level of the other solutes, There was also a very slight indication from the "dilute" quaternary results that the deviations from the predicted values for the combination of a positive and negative interactor was slightly greater than for the other combinations of two added interactors. Aside from this, and the previously noted behavior of the non-dilute quaternaries, there was no reason to believe that the types and varieties of the interacting elements made a difference in the predictionso In general, the first-order prediction model of Wagner adequately describes the experimental behavior where the zinc content is within the Henry's Law region and up to four additional solutes are considered. The inclusion of the second-order term improves the predictions, but the improvement is not significant unless both the zinc content and the total solute content are largeo As the temperature was increased, the interactions became less and the accuracy of the prediction models was greatero This may reflect the fact that only the simple pairwise interactions are significant at the higher temperatures, while at the lower temperatures the interactions between added solutes and the alloying behavior of the underlying solid system tend to play a greater part, The results of these studies of the additivity hypothesis were considerably better than those obtained by Okajima and Pehlke(73) for cadmium in leado The

-246difference is attributed to the fact that the interaction parameters for the Bi-Zn-j solution were accurately known in the dilute solution regiono The over-all success of the prediction models shows that the truncated Taylor series expansion proposed by Wagner is a useful engineering tool for the prediction of complex solution behavior, The applicability of the present interaction parameters for predictions far away from infinitely dilute solutions is of particular advantageo Do Limitations of Experimental Results The limitations of the experimental results primarily concern the specific alloy systems studied. Only a limited number of solutes were studied in a particular solvent over a particular range of temperatureo Bismuth was chosen as the solvent primarily for experimental convenience and to allow a maximum number of ternary systems to be studied by the galvanic cell method. However, bismuth cannot be considered as a "good" metallic solvent since its behavior is not typical and, indeed, bismuth is sometimes characterized as a semi-metalo In particular, the coordination number of liquid bismuth is less than that of the more common metalso However, the fact that simple solution models could account for most of the interactions is encouraging for their applicability to more "typical" solventso In addition, the liquid binary system of zinc in bismuth is a strong positive deviator from ideality reflecting the underlying immiscibilityo As a result most of the observed interactions were negativeo Additional corroboration would be desirable for a primary solute-solvent system exhibiting ideal behavior or a negative deviation from idealityo

-247Another possible limitation of these results is that the observed linearity of lny7n with x may not be general. Because J of the linearity, it was possible to utilize the experimental coefficients to account for the interactions over a wide range of solution concentration, The results for the gallium or gold additions showed that linearity is not necessarily obtained and imply that serious errors might result if the interaction parameters were used without realization that they might vary with composition, Finally, the studies of the multi-component systems were confined to representative combinations of interactorso Only two levels of zinc were considered and the mole fractions of the added solutes were made equalo The results obtained are believed to be an adequate test of the prediction models, however, the studies of the highly complex solutions were restricted. E Suggested Further Research The results of this investigation have opened several lines along which additional research might be basedo These would include not only additional theoretical or experimental studies of ternary interactions, but further exploration of the predictability of the activity in multi-component solutionso The ternary studies could be extended in several wayso First, consideration should be given to basic systems exhibiting different deviations from ideality than the zinc-bismuth system, In particular, cadmium in bismuth is almost an ideal solution, while lead in bismuth is

-248a fairly strong negative deviatoro By using some of the same third element additions as in the present investigation, information could thus be obtained on identical additions to three different bismuth-base solutions. Alternatively, the solvent might be varied while the primary solute and interacting elements are retained. For instance, zinc in lead is another strongly positive deviating system. The systems zinc in tin or cadmium are positive deviators but much closer to ideality, while zinc in antimony or silver are negative deviators from Raoult's Lawo The use of the other solvents might be limited from the standpoint of melting temperature and displacement reactions, however, enough additional data might be obtained to provide some general indications of the zinc-j interactions. Finally, consideration might be given to measuring some of the interactions with zinc in bismuth that could not be obtained by the present liquid-electrolyte galvanic cell studiesO This could involve the use of solid electrolytes or a dynamic vapor pressure methodo The elements that might be considered include thallium, germanium, tellurium, or selenium, and perhaps some of the Group-A elements From the theoretical standpoint, further tests of Alcock and Richardson's equations should be made as additional ternary interaction data are obtainedo Obenchain suggested that Wada and Saito's method might be improved by attempting to extend it to Guggenheim's first approximation of the regular solution(37) rather than the zeroth

-249approximation that was used, and also that the expression for the interchange energy (Equation 74) might be improved. In addition, the partial heats of solution might be used as the interchange energy, although such data are presently sparse. Yokokawa's sub-regular solution model (Equation 64) showed some promise and might warrant an intensive effort to search the literature for data from which the constants could be obtained for additional binary systemso The studies of multi-component solutions might be extended in several ways. Additional experiments could be conducted on septenary solutions and also extended to octetic or nonetic alloyso These could be made at two levels of zinc concentration. The quinary and hexadic alloys could be studied with increased added solute concentrations as far as and beyond the levels studied in the quaternary alloys. The relative amounts of the added solutes might be varied - this might be particularly interesting if the molar volumes or atomic size factors were quite different. The effect on a multi-component solution of a non-linear ternary interactor, such as gallium, might be considered. Finally, as ternary interaction parameters become available for different solute-solvent combinations, they should be extended to studies of corresponding higher-order solutions.

VIo SUMMARY AND CONCLUSIONS An investigation was made of the effect of additional dilute ternary solute elements on the activity of zinc in dilute solution with molten bismuth in the temperature range from 450~ to 6500Co The investigation was designed to test the hypothesis that the interaction effects in ternary solutions are periodic with atomic number of the added solute and that the ternary interactions may be used to predict the activity in higher-order solutionso The activities were measured in a multi-electrode galvanic cell apparatus utilizing a fused molten alkali-chloride electrolyteo The initial measurements defined the activity of zinc in dilute binary solution with bismutho The interaction effects in the ternary alloys were determined as a first-order interaction parameter, 6lny i 621nyi7 t - — J/, - |and a second-order parameter, c. where the 6xk^ 01 6 / parameters are the coefficients of a truncated Taylor Series for lnyi, as proposed by Wagnero The ten solute elements investigated were taken from the Group-B elements of the 4th, 5th, and 6th periods in the Periodic Table. Following this, the activity of zinc was determined in higher-order solutions through septenary, made by alloying combinations of these solute elements to the basic solution of zinc plus bismutho The ternary interactions were discussed on the basis of periodicity, alloying considerations, and thermodynamic factorso The applicability of several simple solution theories for predicting the interaction parameters was testedo The validity of the truncated Taylor Series for prediction of multi-component interactions was consideredo -250

-251The following conclusions are drawn from these studies: 1) The zinc-bismuth system obeys Henry's Law out to at least ~05 mole fraction zinc. Thus, the self-interaction of znc ilnyzn )\ zinc, cZn (-.n, was essentially zero. Zn X Zn= 0 Zn The Henry's Law constant may be represented by the following equation over the range 450-650~C: en - -.79 + 2) The additions of lead, gallium, or indium increased the activity of zinc in molten bismuth, The additions of cadmium or tin slightly decreased the activity, while additions of copper, mercury, silver, or antimony decreased the activity by a moderate amount. The effect of gold was to strongly decrease the activity of zinc, The interaction parameters determined at 550~C were as follows: /6lny-\ 62 ln7Zn Added Solute-j E1n = (=n j o Zn O n cx0Zn = ( x Zn Xi, ____~~__________ ____ __1__ Xzn,/j Cu -2,4 -20 Ga + 50 +,7 -100 20 Ag -2o5 -20 Cd -.3 0 In +202 -40 Sn - o5 -12 Sb -4.5 0 Au -18 0 =50 Hg -1 to -2? Pb +1.3 0 The estimated accuracy is + 7 to 10 per cento

-2523) Appreciable temperature dependence of the interaction parameters was found for the additions of copper, silver, antimony, and goldo There was a slight temperature dependence for the zinc-lead interaction parameter, and the parameters for the other solute elements were essentially constant over the experimental range of temperatureO The temperature dependence can be represented by equations of the formn = Aj + Bj/T, where BE/R is the "extra" excess enthalpy of solution of i and Aj/R the "extra" excess entropy of solution of i resulting from the addition of component j o 4) By observation of the point of discontinuity in the emfversus-temperature relations, the position of the singlephase liquid boundaries was determined in the bismuth-rich corners of the ternary systems Bi-Zn-Cu and Bi-Zn-Auo 5) The hypothesis of periodic variation in interaction parameter with atomic number was found to break dow n n the case of the tin and antimony additions, 6) The first-order interaction effects may be explained in a semi-quantitative manner through Wagner's electron model of a liquid alloy, where differences in electronegativities are used to express changes in the electron/atom ratio caused by the various solute additionso In addition, the excess free energy of mixing of zinc in the binary system

-253formed with the solute element gives a semi-quantitative explanation of most of the ternary interactions. No simple basis could be found to account for the effect of the size differences of the solute elements on the interaction parameters 7) Several simple solution models were found to give semiquantitative estimates of the interaction parameters despite the fact that the underlying binary zinc-bismuth system and most of the zinc-j systems are not regular and of uncertain coordination numbero Predictions based on quasi-chemical assumptions were better than those based on regular or subregular solution modelso Alcock and Richardson's equations which are based on the experimental behavior of the three binary systems correctly predicted the algebraic sign of all the ternary interactions considered and gave numerical results generally within a factor of two of the experimental values. Wada and Saitols equations which incorporate solubility parameters and electronegativities to estimate pairwise interchange energy gave comparable predictions for seven of the ten systems. The regular solution model was qualitatively successful in only five of the ten systemso The instances of failure of the latter two models were their inability to predict the proper direction of the interactiono

_2548) The validity of the truncated Taylor series proposed by Wagner to represent activities in multi-component solutions was found to be excellent for solutions containing up to seven components in the "dilute" range (Xn = o015, xBi 90) The general linearity of the relations between ln77n and x, permitted the multi-component solution predictions to be carried as far as XBi - 785 with fair success. The inclusion of a second-order term was found to significantly improve the prediction model for quinary and hexadic solutions where XZn o050o The forms of the prediction models areFirst-order model' SrA 1I > on Second-order modelo where m is the number of added solutes besides zinc. 9) A special Faraday yield experiment confirmed the valence of two for zinc and the assumptions on which the galvanic cell measurements were made, The cell potentials were generally reproducible to within one to two per cento

APPENDIX A RESULTS OF MULTI-COMPONENT ALLOY STUDIES Composition AT 550~C Comparison of Observed and Mole Fraction Calculated Interactions Experimental Results Calculated Activity Coefficients x x Type of Added Solute ZeeJ oIny Iny Run No. Cell *~ EMF of dE a Iny 7 7 7calc Zn Zn -Zn calc calc obs. calc cl n Interaction Elements n Z Model I Model IIFactor Alloys** T Corrected Model I Model II _ _ _ _ _ _ _ _ _ _ _______ M odel I M odel II _ _ _ _ _ __ _ _ _ __ _ _ _ __ _ _ _ Binary Alloys Basis for Comparison 3.04 Quaternary Alloys.015.015 + + Pb In +3.5 -38 1.162 1.154 71 -0.47 108.11.215.33 1.165 3.21 73 +1.24 107.17.24 1.143 3.14 30 3.17 Cd Sn -0.8 -12 1.098 1.096 71 -0.47 109.75.215.19 1.118 3.06 73 +1.24 108.50.216.17 1.105 3.02 3.00 2.99 + - Pb Sb -3.2 +8 1.062 1.064 72 +1.47 110.67.231.38 1.038 282 75 +0.19 111.75.226.33 1.043 2.84 2.9C 2.90 + Pb Sn +0.8 -10 1.122 1.118 73 +1.24 107.86.217.12 1.123 3.08 75 +0.19 109.91.230.46 1.095 2.99 3.07 3.05 Sn Ag -3.0 -32 1.065 1.058 73 +1.24 109.89.215.28 1.066 2.90 75 +0.19 111.34.227.37 1.055 2.88 2.90 2.88 Cu Ag -5.0 -34 1.035 1.027 71 -0.47 112.85.210.21 1.031 2.80 2.82 2.79 - = Sb Au -22.5 -40.772.764 72 +1.47 120.02.211.22.774 2.17 2.17 2.15 Ag Au -20.5 -66.802.788 72 +1.47 119.64.207.17.785 2.19 2.23 2.20 p.015.015 - - Ag Sb -7.0 -14 1.005 1.002 71 -0.47 113.26.212.26 1.020 2.77 2.73 2.72.025.935.930 76 -0.14 115.56.218.26.945 2.58 2.55 2.53.0375'.848.840 76 -0.14 119.81.220.41.825 2.28 2.33 2.31.050.760.750 76 -0.14 121.19.216.41.786 2.20 2.14 2.11.010 ".410.389 76 -0.14 134.45.210.32.412 1.51 1.51 1.48.050.015 " 1.005.994 81 +1.12 69.43.155.41 1.00oo6 2.74 2.73 2.70.050.760.725 81 +1.12 78.21.149.46.759 2.13 2.14 2.06.015.015 +- Pb Ag -1.2 -18 1.096 1.088 72 +1.47 109.64.227.30 1.067 2.9C 2.98 2.96 75 +0.19 111.12.229.45 1.061.025 " " 1.080 1.073 78 -0.09 112.16.230.32 1.o4o0 2.83 2.94 2.92.0375 " " 1.0o65 1.055 78 -0.09 113.31.230.34 1.007 2.72 2.90 2.87.050 " " 1.050 1.036 78 -0.09 112.53.227.35 1.029 2.80 2.86 2.82.100.990.963 78 -0.09 115.80.220.34.937 2.53 2.69 2.62.015.015 + - In Sb -2.3 -34 1.076 1.068 80 +1.78 109.73.214.18 1.056 2.87 2.93 2.90.025 " " 1.052 1.040 80 +1.78 111.60.214.16 1.003 2.73 2.86 2.82.050 " ".995.970 80 +1.78 115.45.207.06.895 2.45 2.70 2.64.100'.880.829 80 +.178 120.83.194.23.743 2.10 2.41 2.29.050.015' " " 1.076 1.050 81 +1.12 67.81.156.37 1.080 2.94 2.93 2.86.050 " ".995.910 81 +1.12 72.63.154.21.916 2.50 2.70 2.49

~~~~~~~~~~~~~~~~~~~~~~~~Composition ~Comparison of Observed and Mole Fraction Calculated Interactions Experimental Results Calculated Activity Coefficients x x Type of Added Solute SE Ze 1ny Tny Run No. Cell * EMF of oE b Iny alc'calc Zn Zn Zn calc cal obs. calc calc n Interaction Elements Zn'Zn ca ca Factor Alloys ** otf Corrected Model I Model II Interaction E lements Mo e o el I _ _ _ _ _ _ _ __ __ _ _ __ _ _ __ _ _ _ _ _ Model I Model II Quinary Alloys.015.01 5 --- Cu Ag Sb -9.5 -28.968.961 77 +0.69 114.55.212.08.950 2.59 2.63 2.62 i Cd Sn Sb -5.3 -6 1.030 1.029 77 +0.69 111.13.217.07 1.047 2.85 79 +0.96 109.15.207.10 1.067 2.90 2.82 2.8 +f Pb Cd Sn +0.5 -10 1.118 1.115 77 +0.69 109.03.219.08 1.106 3.02 79 +0.96 108.56.210.10 1.112 3.04 3.06 3.05.015.015 ++- In Pb Ag +1.0 -58 1.125 1.112 77 +0.69 108.76.214.15 1.114 3.04 3.08 3.04.030 i " 1.140 1.114 82 - (o108.57)c - - 1.138 3.11 3.12 3.05.050 " 1.160 1.116 82 - (107.17)c - - 1.178 3.24 3.19 3.06.050.030 " 1.140 1.053 84 +0.26 68.10.158.15 1.068 2.91 3.12 2.87.015.01 51 - Cd Sn Ag -3.3 -32 1.060 1.053 79 +0.96 109.52.208.09 1.085 2.96 2.89 2.87.030 " "1T 1" " 1.011.997 82 - (113.27)c - -.00oo6 2.73 2.75 2.72.015.015 - Pb Cd Ag -1.5 -18 1.087 1.084 79 +0.96 108.96.208.10 1.100 3.00 2.96 2.96.030 1.065 1.057 82 - (112.47)c - - 1.028 2.79 2.90 2.88.050.030 1.065 1.038 84 +0.26 69.09.160.13 1.o40 2.83 2.90 2.82 P0 Rexadic Alloys L.015.015 + — In Cd Ag Sb -5.1 -54 1.034 1.021 83 -0.14 112.03.222.18 1.045 2.84 2.81 2.78.050 I " I I I " 1.034.993 84 +0.26 70.52.159.14 1.000 2.72 2.8l 2.70.015.015 ++e- In Pb Sn Sb -1.5 -44 1.088 1.078 83 -0.14 110.70.227.16 1.082 2.96 2.97 2.96.015.015 ++'e In Pb Cd Sn +2.7 -50 1.155 1.144 83 -0.14 108.38.226.17 1.148 3.15 3.17 3.14.050 I"? I I I I " 1.155 1.118 84 +0.26 66.79.162.15 1.105 3.02 3.17 3.05.015.015' — Cd Sn Ag Sb -T.8 -26.993.987 83 -0.14 113.32.228.14 1.008 2.74 2.70 2.68.015.015 ++ — In Pb Ag Sb -3.5 -52 1.058 1.046 85 -1.04 112.f4.222.06 1.050 2.86 2.88 2.84 Septenary Alloys.015.015 +- In Cd Sn -5.6 -66 1.026 1.011 85 -1.04 110.81.217.04 1.105 3.02 2.79 2.74 - - Sb Ag.015.015 i - In Pb Sb -6.0 -66 1.020 1.005 85 -1.04 111.60.213.07 1.082 2.95 2.78 2.73 - - Ag Cu.015.0.5 ++a In Pb Cd -1.8 -44 1.083 1.073 85 -1.04 112.60.222.08 1.054 2.87 2.96 2.92 - Sn Sb * Cell factor is difference between standard EMF and observed EMF for binary electrode in indicated run. * Observed EMF of alloy electrode at 5500C; not yet corrected by cell factor. c Corrected EMF obtained by difference from standard EMF for binary.

APPENDIX B ALTERNATE DERIVATIONS OF THE TEMPERATURE DEPENDENCE OF INTERACTION PARAMETERS Dealy and Pehlke's unpublished derivation of the temperature dependence of the first-order interaction parameter (the result of which was quoted in Reference 16) was obtained by the following method: In a ternary solution, xi and x are independent. We can write; 4. Em _- a A fat. (B-l) where, ~' -/ -._ ___ X, and - fi- f f~ is the standard state fugacity, a function of temperature only. The partial molal free energy is defined rSw. < )nnvnSr (-L mIo/es O so t (B-2) Defining fugacity as d T Since TX -25-257

-258and as X > O T - s Hence, at constant temperature = g- (. ) I^,)' T (B-3) Differentiating with respect to xo and substituting J / A _ _ _ _ _ _ - J'/x~ -U-/- -o -'^ C -(4 j, =O o (B-4) <J X 0 We now define two new quantities Thus9 R T B Differentiating with respect to 1/T ~c~ c J A. T _ ((3 -- (,)r) 15 R J,),(B-7) K#) 3Gr) /<1

-259But since it can be shown that (T J - H (see Darken and Gurry)(76) _ _ ) (B-8) The analogous result is _ - /, - il_( )X^- j (B-9) TJ' i)'< SAL XJ X y J u This completed Dealy and Pehlke's derivation, The same result may be obtained by an alternate line of reasoning: For the solution of i and j as dilute solutes in k, we can write the change in free energy as aG- HS = (r + )CJ/ (B-10) By defining partial molal quantities, Gi and Gj "(r iGj d.\lt;J'j (B-ll) Forming Maxwell Relations, it is implied that _E 5.. _=_ -.. _. (B-l2) e)Qn'1j' <5^'<o; 3 <' a i^

-260For one mole of solution')-r7 -I J n — -- Therefore sX = ><^' 1' = n J; j(B-13) Hence, a — LJ ____ - _ _ _ (B-14) X c a xJ 3 x;. where G. is now a molar quantity By usual thermodynamic reasoning, from the definition C- = cr - T S (B-15) and since T —-, S (B-16) T? Re-writing and dividing both sides by and re-arranging (&- c) = T ) ((B-17) TG- - T H T (B-18) This is equivalent to the differential a / ^^I T I T: (B-19)

-261a2 Now operating on both sides of this equation with and re-arranging axiaxj - _<__ _... - - (B-20) c)/,,;' -(r) b J&1J Since the order of differentiation is immaterial,^J T,y^,,(J G)( - -^ (B-21) For one mole of dilute solution (a c ) = =, ^ RT f. (B-22) Substituting 3 j R i o /-. J )x L -: (B-23) XT' o Since the interaction parameter is defined as Then -CL __y'__ _._ V - Y,I JijL at) y xJ; ~ < ) ^ iJ O(B-24) (B-24)

-262Similarly, the second order parameter is obtained by differentiating once more with respect to x. and its temperature dependence is as_ ^ ^.^_ cJ\i/SH) I (B-25) (y-l,i aT' 3(-x =0 ) X 5o T J The thermodynamic quantifies involved are molar quantities and solutes i and j must be present at high dilutiono Note also that the pressure is constanto

APPENDIX C INTERACTION PARAMETER DETERMINATION BY LINEAR REGRESSION TECHNIQUE When the relations between lnyzn and xj appear to be linear it is possible to use statistical techniques of linear regression analysis to determine the interaction parameter and also to estimate the confidence limits, The data obtained on the Bi-Zn-Sb system were sufficiently linear to perform such an analysis, In addition, this afforded an opportunity to make a quantitative comparison of Methods I and II for the parameter calculationso The experimental data were taken from Table VIII and were originally obtained by runs made at constant mole fraction zinc. An IBM 7090 computer was used to make the initial calculations of the interaction parameter and to obtain the statistical quantities used in the calculations of confidence limitso The treatment which follows is based closely on the discussion of Bennett and Franklin(68) For convenience, the variable lnyz is Zn denoted y and the variable x. is denoted as x, (note xzn xZn) Thus the interaction parameter ( Yn ) is ( d, and the second-order 21 n \ n n \dxy \dx/ parameter aln7zn is d2y \ xj 6Xzn k dxVdxZn) The statistical analysis is based on the model that each value of yi is an observation of a random variable y which is normally distributed with constant variance a2 and mean value a + ~xi o It is assumed that the variance of each distribution is the same, and that average values for a given xi satisfy the "true" relation -263

- 264y + o< (C-1) The purpose of the statistical analysis is to test the model and to estimate a and 3 o The estimated value of a is a and the estimated value of D is b Thus, the relation y = C z At (C-2) is the best estimate of the average value of the random variables y. associated with the given x. o For convenience in calculations, the following quantities are first computedo -X - _ _ - _v _ _= ( c - 3 ),y.-. y (c-.)(z s (y- Z (\ -y - )Z y,: -, (o,) - = -, (c- ) w S Cn is the n ber - of ob (C6)s where n is the number of observations.

-265 - Then hb Sxy) (C-7) and..'-= y-^ l X (c-8) An estimate of the variance a2 of the observations of yi from the average value is given by the expression s = (Y _- =j - RX at y -bx y (c-9) 2 The square root of S is termed the standard error of y x estimate for y. The confidence limits for the slope b and the intercept a are obtained by considering the hypotheses that the difference (b - P) and (a - a) are significant with respect to the variance a (which is estimated as S x)o The tests of the hypotheses are made with the "Student" tdistribution using n - 2 degrees of freedom. The statistic for (b- I) is given by _(b/)JZ (x - = b-?) v/ 5 O() (C-10) n-i. ~ 5 y~ tSy'X

-266(b- S ) = ds-2, 2 (C-1L) h-2 s x) The statistic for a - is given by —. (A) 2 ^-d (C-12) o Y \ c Z + -x + X 5 By consulting a t-table at the appropriate degrees of freedom and by taking a desired confidence level, it is possible to calculate the values (a - a) and (b - 3) corresponding to that confidence levelo The computer print-out for the Bi-Zn-Sb data at 550C is reproduced on page 267~ Method I consisted of first determining the slopes of relationships between (lnyn - y) and (x - x) The values ( X^ cS (\dy \ X were then extrapolated versus Xzn to obtain yd at x -0 In Zn dx / 1 Zn the print-out, "ERROR" refers to the standard error of estimate S "SLOPE" is the second order parameter. Method II consisted of first extrapolating to XZn - 0 the values for y, assuming a linear relation. The slope of the relation between y(Xzn O) and x is the interaction parameter (d ) The results showed that identical values of the interaction parameters were obtained by either Method I or Method IIo The "error" terms for the individual intermediate extrapolations were in fair agreement but were assumed to be equal for the next step of the computationo

-267THE INPUT DATA ARE MOL FRAC ZN LOG GAMMA ZN AT INDICATED MOL FRAC J.005u.00000.01500.02500.03750.05000.01500 1.11000 1.06700.99200.93400.88400.02500 1.11000 1.04600.98600.93300.87500.03750 1.11000 1.03000.99100.94000.88700.05000 1.10900 1.04500.99400.96200.86800 INTERACTION PARAMETER FOR BI-ZN-SB AT 550 C MOL FRAC ZN SLOPE INTERCEPT ERROR METHOD I.01500 -4.74318 1.11835.01381.02500 -4.74917 1.11110.00521 (Standard error of.03750 -4.38804 1.10349.00629 estimate from least squares line).05000 -4.61063 1.11317.01549 AT ZERO MOL FRACTION ZINC, THE PARAMETER IS -4.82728 THE SLOPE IS 6.41645 THE ERROR IS.16915 (Intercept) BY EXTRAPOLATION AT CONSTANT J MOL FRAC J SLOPE LOG GAM ZERO ERROR METHOD II.C000OO -.02618 1.11058.00037.01500 -.66817 1.06830.01387.02500.10384.98744.00369.C3750.79183.91701.00757.050CO -.30519.88823.00896 AT ZERO MOL FRACTION J, THE PARAMETER IS -4.82728 THE INTERCEPT IS 1.11741 THE ERROR IS.02000 (Slope) THE FOLLOWING VALUES ARE PREDICTED dY THE EQUATION LN GAM=LN GAMZERO + X(J)*EPS(IJ) + X(I)*X(J)*EPS2(JII) MOL FRAC ZN PREDICTED LN;AM AT MOL FRAC J.00550.00000.01500.0?500.03750.05000.01500 1.11000 1.03903.99172.93259.87345.02500 1.11000 1.04C00.99333.93499.87666.03750 1.11000 1.04120.99533.93800.88067.05C00 1.109CO 1.04140.99634.94001.88368 PARAMETER BY METHOD 1 X - Y.015CO -4.74320.02500 -4.74920.03750 -4.38800.05000 -4.61060 N= 4 SUM OF X=.1275 SUM OF Y= -18.4910 SUM OF X*Y= -.585 SUM OF XSU=.005 SUM OF YSQ= 85.565 THE SLOPE= 6.4182 THE INTERCEPT= -4.8273 THE CORRELATICN COEFF=.5766 THE STD. ERROR OF EST.=.1692 STD. DEV. OF SLOPE= 6.4301 FOR SLOPE TEST, T=.9981 (For hypothesis that slope is significantly different from zero.) PARAMETER BY METHOD 2 Y.OCOO 1.11060.01500 1.06830.02500.98740.03750.91700.05000.88820 N= 5 SUM OF X=.1275 SUM OF Y= 4.9715 SUM OF X*Y=.120 SUM OF XSQ=.005 SUM OF YSQ= 4.979 THE SLOPE= -4.8281 THE INTERCEPT= 1.1174 THE CORRELATION COEFF= -.9833 THE STD. ERROR OF EST.=.0200 STD. DEV. OF SLOPE=.5155 FOR SLOPE TEST, T= -9.3658 (For gypothesis that slope is significantly different from zero.) *~** ALL INPUT DATA HAVE BEEN PROCESSED. AT LOCATION 10771

-268The interaction parameters were then redetermined with a separate linear regression program in order to obtain the quantities for calculation of the confidence limitso The print-outs of these results are reproduced on page 267 The confidence limits were obtained as follows: Method I: n = 4 Method II: n = 5 Confidence Level From t-tables: n - 2 90% 95% 2 t = 2o92 4~30 3 2o35 3o18 Method I The interaction parameter is the intercept of the relation between x and (d o The confidence limit for an intercept is given zn dx by Equation (C-12) (E = _'',i-.. (,'27S.

-269Method II The interaction parameter is the slope of the relation between Y(Xzn = O) and x o The confidence limit for a slope is given by Equation (C-ll) \2. (aL-)= o-o" - 22 o(b-^) -= -z (,&/,) Summarizing the results, Confidence Level 90% 95% 63 9 Method I Method of eSb - 48 + _ _ _3 Obtaining Z~n 3 10.21 1L64 Method II CS Zn These calculations show that either Method I or Method II can be used to obtain the interaction parameter, however, Method I allowed closer confidence limits to be fixed, The reason for this difference lies in the manner of calculation the statistic "t" for a slope as opposed to "t" for an intercept, The value of the first-order parameter, 4 83, obtained in the above calculations is slightly different from the result reported in Table IX, -4~45 [ The reason for this difference is that the linear regression equations used did not permit weighting the datao The data reported for this system in Table VIII included duplicate determinations

-270for several alloys containing o015 and.050 mole fraction zinc. The graphical determination of the interaction parameters in Figures 29 and 30 allowed visual weighting of these data to aid in calculating a "best" slope, whereas the computer program permitted only one value of lnyzn to be used for a given composition,

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-27663 Okajima, K, and Pehlke, Ro D,, "Thermodynamic Interactions and Liquidus Phase Boundaries in the Lead Corner of the Pb-Zn-Ag System," presented at Annual Meeting, AIME, Chicago, February 1965o 64~ Wilder, To Co and Elliott, Jo Fo, "Thermodynamic Properties of the Aluminum-Silver System," J Electrochem, Soc,, v, 107, no 7, pp. 628-35 (July 1960). 65 Weinstein, Mo and Elliott, J, F,, "Thermodynamic Properties of the Manganese-Lead-Bismuth System," Jo Electrochemo Soco, vo 110, no. 7, ppo 792-798 (July 1963)o 66~ Acton, Fo S,, Analysis of Straight Line Data, John Wiley and Sons, New York, 19590 67 Laitinen, H, Ao, Ferguson, W, S,, and Osteryoung, Ro A,, "Preparation of Pure Fused Lithium Chloride-Potassium Chloride Eutectic Solvent," Jo Electrochem, Soco, v, 104, no. 8, p 516 (August 1957), 68~ Bennett, C, Ao and Franklin, N, L,, Statistical Analysis in Chemistry and the Chemical Industry, John Wiley and Sons, New York and London, 1954, ppo 222=35, 69, Dunkerly, Fo J, and Mills, Go J,, "Application of Electromotive Force Measurements to Phase Equilibria," in Thermodynamics in Physical Metallurgy, ASM, Cleveland, 1950, ppo 47-84~ 70, Johnson, Io and Feder, Ho Mo, "Thermodynamics of the UraniumCadmium System," Trans. AIME, Vo 224, no, 3, p 468 (June 1962), 71o Chiotti, P and Stevens, E R,, "Thermodynamic Properties of Magnesium-Zinc Alloys," Trans, AIME, v. 233, no, 1, ppo 198-203 (January 1965)o 72, Thomas, G. B,, Calculus and Analytic Geometry, Addison-Wesley, Reading, Mass, 2nd ed., 1953o 73~ Okajima, Ko and Pehlke, Ro Do, "A Comparison Between Measured and Calculated Activity Coefficients in Multi-Component Lead-Base Liquid Alloys Containing Cadmium," Transo AIME, v, 230, p. 1731 (December 1964)o 74, Zwikker, C., Physical Properties of Solid Materials, Interscience Publishers, New York, 1954o 75~ Pauling, L,, The Nature of the Chemical Bond, Cornell Univ, Press, Ithaca, New York, 1960,

-27776~ Darken, Lo S. and Gurry, R, Wo, Physical Chemistry of Metals, McGraw-Hill, New York, 1953. 77. Sponsellor, Do L. and Flinn, R. A., "The Solubility of Calcium in Liquid Iron and Third-Element Effects," Trans. AIME, v. 230, no, 4, pp. 876-888 (June 1964). 78, Furukawa, K, J Japan Inst. Met,, V 23, p. A-322 (1959) cited in (9) 79. Allred, A, Lo, J Inorg. Nucl, Chem,, v, 17, p 215 (1961) cited in (9)o 80, Batsanov, S. S., "Geometric System of Electronegativities," Zh, Strukt, Khim., Vo 5, no. 2, p 293-301 (1964) (In Russian), - 81. Hodgeman, Eo D. (ed), Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., Cleveland, Ohio, 1958, 82, Mott, B, Wo, "Liquid Immiscibility in Metal Systems," Phil. Mag., Series 2, p 259-283 (1952)o 83 Kruh, R. F.,"Diffraction Studies of the Structure of Liquids," Chemical Reviews, v. 62, pp. 319-46 (1962). 84. Ptak, W, "The Problem of the Appearance of the Intermetallic Compound InSb in Liquid InSb Solutions in the Light of the Thermodynamics of Solutions," Archo Hutnictwa, v 8, pp. 21-36 (1963) (Chem. Abstracts 59:58487e)o 85 Swalin, Ro, Thermodynamics of Solids, John Wiley and Sons, New York, 1962. 86. Dealy, J. Mo and Pehlke, R. D., "Activity Coefficients in Binary Liquid Metallic Solutions at Infinite Dilution," Transo AIME, v. 227, no. 4, ppo 1030-32 (August 1963), 87. Waber, Jo T., Gscheider, K., Larson, Ao Co and Prince, Mo Yo, "Prediction of Solid Solubility in Metallic Alloys," Trans, AIME, v. 227, p 717 (June 1963)0 88o Weinstein, Mo and Elliott, J,, "Solubility of Hydrogen in Liquid Iron Alloys," Trans. AIME, v, 227, ppo 382-393 (April 1963), 89~ Sponsellor, Do Lo, "Third Element Interactions with the System Liquid Iron-Liquid Calcium," PhoDo Thesis, The University of Michigan, 1962,

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