THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE.OF ENGINEERING THE CORRELATION'OF BINARY.AD TERNARY LIQUID-LIQUID E.QUILIBRIA Thomas C., Boberg A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1959 December, 1959 IP-406

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ACKNOWLEDGEJENTS The author wishes to express his appreciation to Professor R. R. White, for his inspiration and invaluable guidance during this work, and to the members of the Doctoral Committee for their helpful suggestions. The assistance given by Mr. Glen Graves of the Statistical Research Laboratory, and Professor R. L. Bartels and the University of Michigan Computing Center staff is also greatly appreciated. The interest and suggestions offered by personnel of the Dow Chemical Company Computing Section, the DuPont Company Engineering Service Department, and the Humble Oil Company were helpful in this works and to those people the author wishes to express his thanks. To the National Science Foundation which provided fellowship support the past three years, the author extends his deepest gratitude. The author wishes to thank the staff of the College of Engineering Industry Program for their preparation of the manuscript, and to express his appreciation to his wife, Carol, for her assistance in preparing the rough draft. ii

TABLE OF CONTENTS Page AC K OWE I E G e e oo e e........o............... o.............. ii LIST OF TABLES....... o..........I o............... * Vi LIST OF FIGURES..,.... 0 0...,..,. o...o...............v iii NOMENCLATURE, o. o o, o. o o o. o o. b o.. o o. o o....... o o... *..o. e X I. INTRODUCTION. o, a... o. o o. o. o o o.. o Q........... 1 Ao Purpose of the Investigation, o b. i,.e. o o o.. o.o * o 1 B, Formulation of the Problem..o o o o o o...o......... 2 Co Summary o.o...o....o., o.oo. oo... o...o * 4 II. REVIEW OF THE LITERATURE RELATED TO LIQUID-LIQUID EQUILIBRIA o o o o o o....o e oo oo *......... 6... A. Review of Binary and Ternary Liquid-Liquid Phase Diagrams...o.~...o.o.* oooaoo......e.o. * o 9. *<* o. 6 B. The Representation of Equilibrium Data With Equations.. 14 1. Thermodynamic Foundations......oo,,..oo,.o o o o o., 14 2, Representations of the Molar Excess Free Energy... 22 The Wohl Representation of AGt.., oo..* 22 The Redlich-Kister Equations.., o.o.oo....o.. *... 24 Co Composition-Free Energy Diagrams........., o.o,.,,,.. 25 D, Critical Phases and Phase Stability....o...o..oo.... 32 E0 Temperature Corrections to Activity Coefficients...... 41 III APPLICATION OF THE REDLICH-KISTER EQUATIONSo..o......0..o.. 44 Ao Derivation of the Equations for the Liquid-Liquid Case. 44 lo Binary Case...... o o...........o o............ 45 2. Ternary Case...................... 46 B. Utilization and Flexibility of the Equations.....,... 48 1. Solution of the Equations..o*.oe..o Q 48 2. Demonstration of the Flexibility of the Equations.. 51 C. Critical Values of Redlich-Kister Coefficients......... 54 1o Binary Critical Point o P,,.l ooo...,...o o.,... 655 2. Ternary Critical Point or Plait Point......o..,...* 60 iii

TABLE OF CONTENTS (CONT'D) Page IV. DETERMINATION OF COEFFICIENTS FROM EXPERIMENTAL DATA....... 62 A. Binary Case.................... 64 B. Ternary Case............................ 67 1. Hand Methods..................... 68 2. Least Squares Using Approximate Linearizations..... 71 3. Least Squares Without Approximate Linearizations... 76 The Method of Steepest Descent......... 77 The Truncated Taylor Series Method................ 79 V. TESTING OF THE RESEARCH HYPOTHESES......................... 83 A. Binary Coefficients from Binary versus Ternary Data.... 84 B. Homologous Series of Systems................. 89 1. Water - Ethyl Acetate - Alcohols........... 91 2. Water - Benzene - Alcohols....................... 97 3. N-Paraffins - Sulfur Dioxide - Benzene........... 102 4. N-Paraffins - Water - Methanol e.................. 106 C. Correlation of Coefficients............................ 109 VI. CONCLUSIONS AND RECOMMENDATIONS............ 118 A. Conclusions...................... 118 B. Recommendations.................. 119 VII. APPENDICES...................... o 120 APPENDIX A: SUPPORTING DERIVATIONS AND MATHEMATICAL DETAILS...... ^ o.................... o 121 1. Expressions for AG e.ee...........o..... 121 2. Stability Criteria for the Binary Case.... 122 3. The Mathematics of Quadratic Forms........ 123 4. Supporting Details of the Derivation of the Binary Redlich-Kister Critical Relations................................o 124 5. Determination of aL where A is given by (4.29)....... A................... 126 ~~~~00 8~~0~~880~~~~~~~~126 iv

TABLE OF CONTENTS (CONT'D) Page APPENDIX B: DESCRIPI PROGRAM PROGRAM PROGRAM PROGRAM PROGRAM PROGRAM ZION OF COMPUTER PROGRAMS............ 130 1: Binary Vapor-Liquid Least Squares Program, Constant Temperature..... 131 2: Binary Vapor-Liquid Back Calculation Program, Constant Pressure.....,,, 134 3; Binary Vapor-Liquid Least Squares Program, Constant Pressure......o 136 4: Binary Vapor-Liquid Back Calculation Program, Constant Pressure....... 138 5: Program for Computing B/, and C/ from Binary Solubility Data....... 140 6: Solution Program for the Three Component Redlich-Kister LiquidLiquid Equations.....o..a........ 142 7: Linearized Ternary Liquid-Liquid Least Squares Program............. 148 8: Non-linear Ternary Liquid-Liquid Least Squares Program......... o. 156 PROGRAM PROGRAM APPENDIX C: TABLES OF CALCULATED VALUES APPEARING IN FIGURESo o O o o o o. o. o o o o. o o o. o. o..... o. o.. BIBLIOGRAPoHY. o o.o...oo.. o o..... o...............,............. o...O 170 185 V

LIST OF TABLES Table Page I Comparison of Third Order Expression for log ~ for a Binary System............................................. 24 II Comparison of Coefficients Determined from Binary versus TernaryData... 85 Ternary Data........................................ 85 III Homologous Series of Systemsg..............................90 IV Redlich-Kister Coefficients, H20(l)-Ethyl Acetate(2)-ROH(3), 20 C...,.*.. *......... *.*............. *, * 96 V Redlich-Kister Coefficients, (H20) (l)-Benzene(2)-ROH(3), 25~0C.*.** OO.O*OOO *..* 0*..... 96 VI Redlich-Kister Coefficients, N-Paraffins(l)-(S02):8(2)Benzene(3), 2OF..... 105 Benzene (,3) -20 ~F..,,.,..,................,...,.'............ 105 VII Redlich-Kister Coefficients, N-Paraffins(l)-HiO(2)Methanol(3), 15~C................1 0................5 105 VIII Lines of Constant _-M for the System BM = 1.3, B23 = 0, 4,O3 IRT B31 = 0.5, C12 = 0, Cs = 0, C1 = 0............. 171 IX Fictitious Systems B/ = 1.3, B13 = C, = C23 = C =...... 172 X Fictitious Systems B,2 = 1.3^ B31 = 1.0, C/2 = C3 = C1 = 0. 173 XI Fictitious Systems B,, = 1.3, B2, = B = 0.5, C/, = 0.2..... 173 XII Critical Values of the By, and CG,j Coefficients........... 174 XIII Examples of Fits Obtained Using Approximate Linearizations... 175 XIV B.j - C^. Fit of Binary x-y Data............................. 176 XV Comparison of Fits Obtained Using Coefficients Determined from Binary versus Ternary Data...................... 177 XVI H20-Ethyl Acetate-Alcohol Systems.................. 178 vi

LIST OF TiBLES (CO0T' D ) Table. Page XVII (H20)8-Benzene-Alcohol Systemso o..o.. o..........e e o 180 XVIII JN-Paraffins-(S02 )8 -Benzene Systems. *....,,o.,, *,. o o o..... 182 XVIX N-Paraffins-Water-Methanol Systems o,......1................... 183 XX Comparison of Predicted and Experimental Equilibrium Curves... 184 vii

LIST OF FIGURES Figure Page 1 Partially Miscible Binary Systems............................ 6 2 T-xA-x- Diagram, Type I......... e..e................. 7 3 T-X -X, Diagram, Type II................ **. *..*. 7 4 Isothermal Cuts of Figure 2...................... 8 5 Isothermal Cuts of Figure 3..... ~.................... 8 6 Examples of Ternary Isotherms - Two or More Partially Miscible Binaries.*....,... ~..,*.................. o 10 7 T-xA -XB Extended Temperature Range..........,......... 11 8-9 Isothermal Cuts of Figure 7..,... *.~ ~ ~............... 12 10 Vertical Section Taken Along the Linre pq in Figure 7..-.. 13 11 Ideal Binary at T.......... w......................... 26 12 Real Homogeneous Binary................................... 26 13 Binary Having Two Liquid Phases......................... 27 14 AG^ -x -xz, Partially Miscible Ternary Isotherms.......... 30 15 Hypothetical AG Surface as Represented by Equation (2.72).. 31 16-21 Plots of Equations (3,10), (3.11) and (3.12) for Various Values of B^ (B, = 1.3, B2 = C = C C = 0)....... 50 22-25 Plots of Equations (3.10), (3.11), and (3.12) for More Complicated Cases........................................ 52 26 Critical Values of B. and C.. versus x.................... 57 27 BC versus C.......................................... 58 28 Examples of Fits Obtained Using Linearizing Approximations.. 75 29 B. - C.. Fit of Binary x-y Data............................ 87 30 Comparison of Fits Obtained Using Coefficients Determined from Binary versus Ternary Data........................... 88 viii

LIST OF FIGURES (CONT'D) Figure Page 31-38 H20(1) - Ethyl Acetate(2) - Alcohols (3),. o... o,.,.. 92-95 39-44 (H20)8(1) - Benzene(2) - Alcohols(3)...................... 98-100 45-48 N-Paraffins(l) - (S02)8(2) - Benzene(3)................. 103-104 49-52 N-Paraffins(l) - H20(2) - Methanol(3)oo.,...................107-108 53 Binary B and C Coefficients versus Alcohol Boiling Point for the Homologous Series of Systems H20(1) - Ethyl Acetate(2) - Alcohols(3) 20 ~Cooo o o o oo. o. o ooo o o o. o o o o 110 54 Binary B and C Coefficients versus Alcbhol Boiling Point for the Homologous Series of Systems (H20)8(1) - Benzene(2) - Alcohols(3), 25~Cooo o o.... o o... O o o o o...o o o....o...o. 111 55 Binary B and C Coefficients versus Paraffin Boiling Point for the Homologous Series of Systems n-Paraffins(l) - (S02)8(2) - Benzene(3), -20.~F.oo.oooooo o o.. oo oe oooooooo.o 112 56 Binary B and C Coefficients versus Paraffin Boiling Point for the Homologous Series 6f Systems n-Paraffins(l) - H20(2) - Methanol(2), 15 o. o....o o o o o o o o o o o o...o... o... 113 57 Comparison of Predicted and Experimental Equilibrium Curves for Two Systems Using Coefficients Taken from Homologous Series Correlation o o o o. o.... o..o..o.... 115 ix

NOMENCI.ATRE a. bMa t C/a ' d/e Bj., C., D, C Cv D, D' D,, Dg FD F. f. fL, g,' hL G AG AGM,AHM H activity of component i coefficients in Equation (2.97) Redlich-Kister binary coefficients equation coefficients or parameters. When the RedlichKister equations are referred to, c,, ct, --- c6 is B,^ B, B 31 C,a Ca, Ci respectively number of components or first Redlich-Kister ternary interaction coefficients constant volume heat capacity critical phase determinants defined in Equations (2.88a), (2.95a) second Redlich-Kister ternary interaction coefficients meterized steepest descent correction factor coefficients in Equation (4.23) the ternary Redlich-Kister expressions for log, retaining only BJ, and CJ the function defined in Equation (3.10) fugacity of component i (L = 1, 2, -- 6), terms in (Eqtn.(3.10)), (eqtn.(3.11)), (eqtn.(3.12)) multiplying ct... C6 Gibbs' free energy equal H - TS the function defined in Equation (3.11) free energy, enthalpy of mixing excess free energy of mixing ideal free energy of mixing enthalpy x

NOMENCLATURE (CONT' D) L/, LA n P p 6 Q (1, a, a l q j a 'j k a *'~ R S T t U V V6 C the function defined in Equation (3.12) log^A.a,i log respectively number of moles number of phases or total pressure vapor pressure of the pure component i partial pressure of component i the Q function equal ^ _~r coefficients in Wohl's equations, see Equation (2.50) any extensive property or the gas constant entropy any arbritary function or system of functions absolute temperature, ~K temperature ~o internal energy variance or volume coefficients appearing in Scatchard's equations (see Table I) a weight factor, wee Equation (4.13) mol fraction of component i in liquid phase one, two, etc. terms in Equations (3.8) and (3~9) mol fraction of component i in the vapor phase experimentally determined values for the dependent variables y/, y, -—, and the independent variables z/, z --- ith data point; see Equation (4.12) w X^ I Xi 7 * Q X, X X3 X4. Y6 YI.= Y2 -- z@6 l",z -- xi

NOMENCLATURE (CONt D) the effective volumetric fractions in Wohl's equations defined in Equation (2.51) Z. ~. Greek: Letters: g, 3 e Subscripts*: i,, --- A, B, C, --- 1, 2, 3, --- magnitude of the steepest descent correction in the direction given by Di. population parameters in equations to be fitted by least squares activity coefficient of component i in solution operators indicating the first, second, third, etc. order terms in Taylor series expansion used alone: represents the sum of the squares of deviations to be minimized by least squares; used with another variable: means change from state one to state two arbitrary values to determine the degree of accuracy of iterative processes (see Appendix B) t/! equal FZ - F see Equations (3.10a)j (3.11b), (3.12c) the angle between the meterized steepest descent directions at the (n-l)th and nth trials the ith, jth, kth --- components component A, B, C component 1, 2, 3 * Subscripts having other meanings will be obvious from context. Sub Bars: R, -_ equal.: the,,:val.tae of any extensive property R per mole of solution xii

NOMENCLATURE (CONT'D) Superscripts * C Ir 0 refers to mathetically predicted values of dependent variables, no * refers to actual experimentally measured value of variables means critical value pertaining to a critical phase means excess, as excess free energy means ideal, as ideal free energy of mixing refers to the Nth iterxation, in a trial and error solution refers to the standard state, here taken to be the pure component in the liquid state at T Superbars: RZ equal b R L it where R is any extensive property

I 4 INTRODUCTION The prediction of equilibria in liquid phases is of greatest importance in the chemical and metallurgical industries. Although there has been a long continued effort in this field, accurate prediction of the phase equilibria in liquid phases is impossible, except in cases of mixtures which follow or deviate only slightly from Raoult's Law. This problem will ultimately be solved only through a thorough understanding of the interactions of molecules and molecular forces in solutions Such understanding unfortunately may be many years away, and the demand of the process industries for accurate prediction methods continues to grow as more complex separation problems are undertaken and higher product purities are desired, Because of the increased importance of solvent extraction processes, this thesis is devoted to the prediction of equilibrium in two phase liquid systems No attempt has been made to solve the long range problem of understanding solution behavior from a molecular view point. Rather, a means of extending our existing experimental data on liquid systems is investigated in this thesis which has the possibility of providing an immediate answer to the demand for accurate phase equilibrium prediction4 A. Purpose of the Investigation This research is an attempt to develope a correlation which will permit the prediction of parameters in equations suitable for the representation of phase equilibria4 Attention is focused on the case -1 -

-2 - of two phase liquid equilibria in three component systems, but the ideas evolved hopefully may be extrapolated to systems having any number of components, B. Formulation of the Problem Assume that a function or system of functions exist J(C, i C * ^ Z n t D * ',,)= 0 where the cts are parameters, the ' s are the compositions of phase one, the x"'s are the compositions of phase two, and n is the number of pairs of components. The questions that may be raised are: 1*. Do Equations(s) (1.1) satisfactorily represent the desired equilibrium? or are more-coefficients or better equations required? 2, Can the parameters in (1.1) be determined from systemscontaining fewer components than the case for which (11l) is written? 3- Consider a series of systems containing components A, B. C D...; A, B, C, D...; A, B, C3, D,.. etc, where C/, C,. C.. are homologs, Are the parameters in (1,1) simple functions of some monotonic property of component C/ Ca, C3.., such as molecular weight? Question two suggests a correlation in the form of tabulated parameters determined from simple systems which can be used in Equation (1.1);to predict the equilibria for more complex systems. Question three suggests a correlation in the form of graphs of the parameters in (l*l) plotted against some property of the variable component, C

-3 - Such a correlation has been detnonstrated by White (53) who used the Redlich-Kister equations to represent the binary vapor-liquid equilibria of hydrocarbon-mercaptan systems and found for mercaptan homologs, that equation parameters were nearly straight line functions of the number of carbon atoms in the mercaptan, The system of functions (l11) which have been proposed for the present work are the Redlich-Kister equations () which have been successfully used to represent vapor-liquid. equilibrium data. The questions stated above will be restated more specifically for the.case of the Redlich-Kister equations to represent ternary liquid-liquid equilibria, Many equations of the Redlich-Kister type such as Wohl's, Margulest, and Van Laarts equations and others(54 20 4739) have been used to represent phase equilibriao However because of their complexity they have not been used extensively to represent ternary liquid-liquid equilibria. Scheibel has recently discussed the use of the Redlich-Kister equations in representing ternary liquid-liquid equilibriao ) Other authors have used the Van-Laar equations(l8) and modified versions of the Van-Laar(4) to represent ternary liquid-liquid equilibria. Because of the complexity of the equations the methods used were approximate and usually graphical, By using high-speed digital computers in the present research more rigorous solution of the equations and more exact determination of the equation parameters from experimental data was possible. The subject of the effect of temperature on equilibria was!- not studied in this research,

-4 - C. Summary Section II of this dissertation is a review of liquid-liquid equilibria from a thermodynamic.viewpoint. In part A is a review of equilibrium diagrams. Part B gives the thermodynamic foundations of the equations for representing phase equilibria, and gives examples of the equations. The interrelation of the molar free energy of mixing and equilibrium diagrams is discussed in Section C. Those interested in the prediction of critical mixing points (ie,.plait points) will find Section II,, D, dspecially useful. Part E, is a brief review of methods to predict the variation of activity coefficients with temperature. The application of the Redlich-Kister equations to the cases of binary and ternary two phase liquid equilibria is taken up in Section III. The equations are presented and their ability to represent ternary liquid equilibria is demonstrated. Part Co presents relations enabling one to tell if a given set of Redlich-Kister equation parameters will represent a system having one or more than one liquid-phase-, Equations enabling prediction of critical points from equation parameters are given. Section IV deals with the problems of determining equation parameters from experimental data, Methods of determination of the parameters for the cases of binary liquid-liquid and vapor-liquid equilibria are discussed in Section A. The ternary liquid-liquid case is taken up in Section B, Methods of visual urve.fitting are discussed, The problems of the use of least squares in the ease of the Redlich-Kister equations are surveyed; approximate and exact methods of least squares curve fitting are applied. Those interested in fitting equations that are non-linear in equation parameters will find this section useful.

-5 - The feasibility of the Redlich-Kister equations to represent and correlate ternary liquid-liquid equilibria is determined in Section VO The questions of the preceeding section have been set down as research hypotheses for the specific case of the Redlich-Kister equations. Part A tests the hypothesis that equation parameters determined from binary data can be used to predict ternary liquid-liquid equilibria, Parts B and C attempt to answer question 3 of the preceeding section pertaining to the usefulness of homologous series. of systems in predicting liquidliquid equilibria.The conclusions as to the feasibility of the RedlichKister equations in representing phase equilibria areresented in Section VI as well as recommendations for future work,

II. 'REVIEW OF THE LITERATURE RELATED TO LIQUID-LIQUID EQUILIBRIA A. Review of Binary and Ternary Liquid-Liquid Phase Diagrams:(713122) Figure 1 is a plot of composition versus temperature for a two component system in which there is limited liquid phase miscibility. Below the temperature Tc, components A and B are only partially miscible in the range shown. Specification of a temperature T1, requires for this system that when there are two phases they must have a composition of xBl and xB2 respectively. This fact is given by the phase rule: 1l. B2 V=C +2 - P =2 + 2 - 2 =2 P1 T / I I I I I A XB1 Figure 1. XB XB2 Partially Miscible Binary System. B Specification of pressure and temperature, Pi and T1 therefore defines the system as seen from the diagram, Actually T-x diagrams for systems -6 -

-7 - consisting only of condensed phases are little affected by pressure and little error is introduced by not counting pressure as a phase rule variable for such systems..Addition of a third component, C, to the binary AB now causes the temperature composition diagram to become three dimensional. For the case where there are two condensed phases in equilibrium in a ternary the variance becomess V 3 + 1- 2 2 c This means that in addition to specifying the temperature we must specify one composition variable, say xA, the mole fraction of A in phase one, to specify the system. Figures 2 and 3 are two possibilities of the temperaturecomposition diagram for a system ABC where A and B are partially miscible and C is completely miscible with both A and B, The familiar trangular composition diagram has been used. A B A B Tc K / I _A T2 ' /B_/ I B T1 T __L 1 ' T1 Figure 2. T-XA-XB Diagram Type Io Figure 3o T-xA-XB Diagram Type II.

-8 -In Figure 2 there is no ternary critical point. The point K lies in the T-A-B plane and is the point (To, a) in Figure 1 the critical point of the binary solution. Figure 3 shows the case where there is a ternary critical point K, in addition to the binary critical point k. Isothermal cuts of the solid figures shown in Figure 2 and 3 are shown below: A B A Figure 4. Isothermal Cuts of Figure 2, C C B Figure 5. Isothermal Cuts of Figure 3.

-9 - Each of these isothermal cuts has at least one critical point or plait point, P where the two liquid phases become identicall At T2 in Figure 5 the two phase region lies entirely within the ternary region and there are two plait points P and Pt The curves in Figures 4 and 5 are known as binodal curves and the straight lines connecting points on the binodal curves are known as tie lines. The tie lines connect the compositions of liquid phase one (L1) in equilibrium with liquid phase two (L2), In Figures 2 and 3 the lines We: and k'Kk" are the loci of the plait points shown in the isothermal cuts. Since the tie lines in the ternary case are usually not parallel to the AB edge of trangular diagram P does not have to occur at the maximum point of the binodal curve. In the binary case the tie lines in the T-x figure parallel to the AB edge and the critical point does occur at the maximum of the temperature compos it ion curve, The line Pq in Figure 4 (T1) is known as a convolute line and is useful in correlating the tie lines, It is simply the locus of points which are the intersections of the dashed lines drawn from the end points of the tie lines parallel to the A-C and A.-B edges of the triangular diagram'Figure 6 shows isothermal cuts of some of the cases of systems consisting of liquid phases e theres more than one pair of parially miscible constituents a

-10 - A B C D Figure 6. Examples of Ternary Isotherms - Two or More Partially MisciE Binaries In case A there are two regions of limited miscibility. In case B these two regions have fused together, case C shows three two phase regions. In D the two phase regions of C have fused together to give a three phase region. Given T1,. the compositions of the three phases in equilibrium are fixed by nature since the phase rule states that: V = C + 1 - P =3 + 1 - 3 1 Therefore specification of only one phase rule variable (i.e. T) specifies'-the system. Note that in Figure 1l 2, and 3 the temperature.range has been limited to that range where only liquid phases exist -At lower temperatures for most systems solid phases appear and the resulting

-11 - phase diagrams become more complex, Figure 7 is an example of a ternary system of the type shown in Figure 4- The binary T-x diagrams are shown at the edges of the ternary diaagram. The solid phases.consist only of the pure components A, B, and C, The systems AC and BC are of the simple eutectic type. The system AB has two eutectics and 'a region of immiscibility in the liquid state similar to Figure 1, The triangular diagram shows the projection of lines of three phase equilibrium4 T1 T2 L22 T3 A T T L L Figure 7o T-xA-xB,.Extended Temperature. Range.

-12 - At T1 the isothermal section would look similar to. Figure 4, Figure 8 shows the isothermal section at T2. In addition to the two phase region LI-LI there now appears a second two phase region I-A. The dotted curve is the locus of points giving liquid composition in the three phase equilibrium L1-A-IA shown in Figure 7 -A C\/ C C Figure 8. Isothermal Cuts of ure 9. Isothermal Cuts of Figure 7. Figure 7. The cut at T3 is given in Figure 9. Since this temperature is below the AL1 eutectic the dotted line is now intersected by the saturation curves and a three phase region occurs in which A-L1 and L2 are in equilibrium. Note also the appearance of the two phase region I2B at this temperature. The L1-I2 region is now entirely within the ternary region. At the temperature corresponding to point a in Figure 7 the liquidliquid region will disappear. If one makes a vertical slice in the ternary temperature-composition figure perpendicular to the plane ABC at the line pq, Figure 10 is obtained,

-13 - T '1 -- L7z LI( /\A) (,A~c(,..)(c) A cc Figure 10, Vertical Section Taken Along the Line pq in Figure 7o The notation of LA(C) means a mixture of 12 and A ip equilibrium with Co Numerous graphical tie line correlations have been used in the past (14,41) A good summary of these may be fQund in Reference (46), These have.utility in cases where systems-hare been partially determened experimentally in predicting those portions of the equilibrium curve where there are no.data. They have little value in predicting systems for which there are no data at all. The work this paper discusses deals with the prediction of the cases of two phase liquid equilibria shown in Figures 4 and 6B as these cases are 'the important ones industrially However the methods of representation to be discussed in the next section are generally applicable to solutions of non-electrolytes,

-14 - B. The Representation of Equilibrium Data With Equations There are two general classes of equations which have been used to correlate phase equilibria. One class consists of empirical equations relating phase rule variables directly such as the relation y = ax + bx2 + cx3 (2.1) which has been used to correlate binary vapor-liquid equilibrium data, y being vapor phase composition, x being liquid phase composition (13) The second and more important class are those equations which express the dependence of the thermodynamic functions the activity coefficient, on liquid phase composition. Because of the marked success of this latter type of representation.of equilibrium data it has been,chosen as the basis for the correlation of ternary liquid-liquid equilibria being developed. 1. Thermodynamic Foundationsl (13) In the case of a multicomponent system consisting of nl, n2 -— nk moles of components 1, 2, -— k respectively the following is true of the free energy of the system: G = G(T, P, nl,n2 -— nk) or mT= ( (+ L X i (2.2) where it is understood that in the partial differentiation all variables are constant except that appearing in the denominator of the partial derivative

-15 - If we consider an infinitesi- ireversible process in which neither the amount nor the composition of the system changes, i e.o '1 = dY1 = ' ' ' o. =S = O Then Eat ion (2;,2) becomeso 6gG~ =(2 )a4T + (2 )P (2-3) For this case from the first and second laws of thermodynamics: = - iST+ VP. (2.4) Comparison of (2,3) and (2,4) gives ( )=-5 G ( p* (2.5) aLt= partial derivative 3- is called the chemical potential of the ith component and is denoted by/4 It is also frequently-denoted as Gi^ the partial molal frlo energy of the ith constituent. This notation is used interchangeably, On combination of (242) and (2 5) there resultss G=-ST + VP + t G n 0 (2Q6) For-the case of a system at constant temperature and pressure: CG= Gn [ T, P (2.7) Integrating (2,7) at constant composition: ~~G^Z^^t)~~~ =-~ i.(2.8) g=!

-16 - If we make infinitesimal changes in P, T and the number of moles, we obtain: G -1G5 = ( t+ 9XG)(G) (2.9) Subtracting (2.8) frt:n (2.9) and neglecting higher order differentials: 61G = ^ (G*6 + rp P 6) (2.10) z I On comparison of (2.10) with (2.6) there follows: S)T- VP+Z G = (2.11),'= I which is the Gibbs-Duh- emequation, ' At constant temperature and pressure, and dividing (2,11) by the total number of moles one obtains: X s7G =O LrT,P] (2.12) The fugacity fi, is defined by: t SG, = iRT 61 ^ (2.13) Combining (2.12) and (2.13) The atiit=y [,P1 (2.14) The activity, ai is defined: = 5/7i^O (2.15)

-17 - 0 where fi is the fugacity of component i at the same,temperature as fi, but in some standard state. In the case being considered the standard state would normally be the pure liquid i, at some specified temperature, Taking the-logarithm of both sides of (2.15) and differentiating we obtain since fi is constant: Thus Equation (2.14) may be written (2.16) u 1=/ Defining the activity coefficient as: [T, PJ (2.17) (2.18) we obtain upon Substitution into (2.17) C-/ Since Then since xid ln xi =-dxio Z=t (2.19) (2.20) (2.21)

-18 - It follows from (2.21) and (2.19) that: _, A, =- [=T,P] (2.22) i6S or A P1d =l = LT, P L This may also be arranged to the form: X m> = lo ITT (2.22a) Mixing and the Excess Free Energy'(13) Consider the change in free enaer'gy, AG^ in mixing the pure liquid components B, --- to formi a solution consisting of b moles of BJ C moles of C ---- (2.23) =b (Gn)- G-S) +- c-C, (G - A' 2 where X, G denote the partial molal free energies of the pure liquids B, C —, taken to be the standard states of B, C --- respectively. Note that combination of (2.16) with (2,13) and subsequent integration gives. r-5 = RT& Rg ~.(2.24) Substitution of (2.24) into (2.23) gives: k. AG z =/2 KFT ^S '(2.25) c /

-19 - From (2.18) and (2425): jy- L6 k C=, = (2.26) For an ideal solution the activity coefficients are equal to unity and (2-26) becomest AL^ = 51A (2.27) For real systems the activity coeffidients are usually not equal to unity. The difference /'G - G^ is called the excess free energy,^ G. aG'- 75 = ~R T' 4 6t=1 The partial molal excess free energy is obtained by tiating partially with respect to ni: (2.28) differen ~; —= a G = -4-n~ k RT, + RT>7, V Y j / ^ ET, P7.(2.29) Since by Equation (2,22a), - >I-' The Q functionis defined as The Q funetion fs defined -as; (2.30) Q-,o3 T Z v, ~ _iT O, 303 l T (2.31) where AG- represents the molar excess free energy,

-20 - The partial molar value of Q with respect to component i is: QZ = -<< 2.303R T '(2'32) From (2.32) and (2,30) it follows: Qia = oMo (2.33) If either AGC or Q are known as a function of composition, by partial differentiation with respect to ni it is possible to obtain a relation giving the dependence of the activity coefficient of this constituent on the composition of the solution, Since the mole fractions of the various components are not all independent AG.- is not equal to A6< and QZ, is not equal to. A formula may be derived which gives any partial molal quantity as a (5) function of partial derivatives with respect to composition. For any extensive property, R, as was shown in the case of Equation (2,2): gp= ZR62z6 (2.34) Integrating (2*34) at constant composition: R =;J R~ (2.35) is/ Differentiating (2,35): k - /e - ad1 Z/R - + Z ^ (2.36) Comparing (2.34) and (2,36) it follows that: ^_L~~~n~~l, =^0 ~(2.37) Z -

-21 - Considering one mole of solution it follows from (2.34), (2,35) and (2,37) that: ^R =; d - tR6Xi (2.38) ~R = L X R '(2.39) c=/ ''I & d_ i =O * (2.40) R means R per mole of solution in (2,38), (2,39) and (2.40), k From the fact that -Z =/ we obtain: -li > =r -Zd * (E.(2.41) Eliminating xk from (2o38) by means of (2,41); 'k-,. = ( t' k) (2.42) d,/ Since x/ -- x,_ are independent variables: ( 1) =R -Ri (2.43) Noting that 2j=/, (2..39) becomes (2o44) 5 = Z) (Z, a re R+ (2.45) Substituting (2^43) into (2.45-) and rearranging: k-l(2.46) R=_R- Z^;E (2.46) R,1 C%

-22 - This equation gives the partial molal value of R for the component k whose composition variable was eliminated. F*om (2.46) and (2.43) it follows that: R R+ - _- -R (2.47) Equation (2.47) gives RK for all components except the kth component. Note that for the case of the Q function, Q is an intensive property. Therefore replacing R with Q ' -=a /1 Z (2.48) k-l __zo \k = Q - %6 * '(2.49) Similar expressions may be obtained for G.G Expressions for the activity coefficients derived using,Equations (2,48) and (2.49) are automatically consistent with the Gibbs-Duhem equation at constant [T,P] 3) 2. Representations of the Molar Excess Free Energy: Ic,.(54) The Wohl Representation of AG: Wohl expressed the dependence of A Gt the molar excess free energy, on the composition of the liquid phase by the expansions 2,303 RT ZI ' +z zZt +z + Z6 ZZk Z,,jk... (2.50) L./~.

-23 - where q. qy qk,. --- are constants Wohl called the effective molar volumes of the constituents i, j, kJ 1, - and zz zj, z zk --- are the effective volumetric fractions of these constituents, z. is defined by: z. = (2.51) Equation (2.50) is written as an equation of the fourth ordprt In practice equations of order higher than this are rarely used.o Using Wohl's equation of the third order it can be shown that by making certain simplifying assumptions the third order equations of Van Laar(47) Margules (20) and Scatchard and Hamner(39) are obtained~ After -introducing the new constants A and B~ A =,(, + 3s,), B= (2, 2 + 3 2) the third order equation for the binary case becomes: 23O3 T - (/+ _2i ZZ,, BZ +4 zA (2.52) It:may then be shown that: &= z2[LA+ 2 z,(Bj -A)J (2,53) which is the Scatehard-Hamer representation of log, Table I gives the simplifications necessary to convert (2,53) to the forms derived by earlier authorso

TABLE I COMPARISON OF THIRD ORDER EXPRESSIONS FOR LOG 1 FOR A BINARY Type Simplification log Ref. Scatchard q/l = V/ (1- z)2[A+2z,(BV/V2-A) 39 Van Laar 2l" l =X B/A 47 Margules 2/q1 = 1 ( - xl) [A+2x,(B-A)] 20 Much more aomplete tables similar to Table I may be found in Reference (13) for the binary and ternary cases, The Redlich-Kister Equations: In 1948 Redlich and Kister introduced a simple, highly successful empirical representation of the Q function,(33'34) This representation can also be derived by suitable rearrangement of the Wohl equations.(13) For the binary case: Q =,[a2lB, + C,,-~)+ D,2 (z,-/) ' ' (2.54) Retaining only the B-. and Cc. terms, application of Equations (2.48) and (2.49) gives: 4 o =: L/B, + C,2, (-,) 7 (2.55) A,0 = [L B, + C2 (,z - ' 3 ) (2.56) For the case of a ternary Q is represented:,,. 131 (2a-x,) +, (X-I,) Z'" + + z24 023 + C23(X - ) +D3 (fz- C2 * (* + + 3zE 5 3 C (XC3-%z ) +03 ( - )f. *'. + + /,Zz% ZsC + Di, + D,2 ''-J (2-57)

-25 -where Ci, D,, DA are ternary coefficients and Bj, C$, are binary co-e.ficidnts'; Againrwhere-.on ly -th~(::'bitary B.iond0 C cOefficients a.,ae rretained applying.(2c49) g-ives cor -a ternaryf 4.._ = -/, z (-,)-$/ S3 z; +,3/ (i- x,). c,( (b(, (-I,+ ~ /- ) Q)), c (' 2 (-z3 X3)) +C,1 (3 (, (- - /a ) )) (2.58) A cyclic permutation of the subscripts gives the expressions for log, and log 3, Hence forth in this dissertationthe right hand sides of (2o58) and the expressions for log ~ and logy will be denoted as F, F and F3 respectivelyo The success of the Redlich-Kister equations in correlating vapor-liquid equilibria led to their selection as a basis for the correlation of ternary liquid-liquid equilibriao In practice it has been found that retaining only the binary BE, and Cj, coefficients is suf(53) ficient to correlate most cases of ternary vapor liquid equilibria,) and only these coefficients.have.been.retained in the equations to be used as the basis for the liquid-liquid.correlation, C0 Composition-Free Energy Diagrams 5 It can be seen from Equation (2~26) that at constant temperature, ALG, the molar free energy of mixing is a function only of composition: G,,,, f RT2, + ze-r-T^ (2.59) a~ ]} LM* eM;' i/ /

-26 - For the binary liquid case ~G~ may be plottedagainst x,, the mole fraction of component 1 in the liquid~ If the activity coefficients of 1 and 2 are unity then a curve of the function will be obtained as is, shown in Figure 11l L. Fo B Figure 11. Ideal Binary at TV. In real systems the function for the molar excess free energy aG is superimposed on the curve shown in Figure 11o Depending on the nature of the function AG, only one or more than one liquid phase will be encounteyed as x/ increases from 0 to l1 Figure 12 shows a case where the system is homogeneous throughout the liquid range, AG2M ^MA"^ ^!^ anG I 1 A- S SA N~~~~~~~~~~~~~~ A os ~ Figure 12. Real Homogeneous Binary.

-27 - It may now be seen graphically that for a solution or composition x8s the partial molar free energy of mixing of component A is given by the intercept with the A axis of the tangent to the /A, curve at X. This follows from Equation (2.46). (See Appendix A). /G,,, = ^-,a m MA rA AGy appearing in (2.61) is the value of the AG versus composition curve at the point x, shown0 Figure 13 shows the case of a binary system having limited miscibility in liquid phase. A Xb T 4-.I II Figure 130 BLnary Having Two Liquid Phaseso In this case there are two inflection points in the curveo Mixtures having a total compositio.- between xi and xs will break into phases of composition x, and x4, which are the abscissi of the points of. ommon tangency to the curve. Note that there is only -one set of these points

-28 - at a given temperature, a consequence of the.phase rule which.states that when there are two phases, given pressure and temperature a binary system is completely specified, In the general case equations similar to (2,6) may be written for any number of phases of which the total system may be composed: GG'- - S T + V P + G.' dv/ G" — - S '>T+ \V/;P + t; cl J- (2.62) Since G represents the total free energy of the system: G = G G' G " + ' G (2.63) Note also /G,5 = G- n A nB r) *~ (2.64) AG G"- 1'G: -nG * * (2.65) At equilibrium Gibbs(9) proved that dp = 0; therefore clG = JG'+ ~ G J ~dG" * =* 0 o(2.66) From (2,66) and (2,62): /. ZX (-,Ti +Z /o Ij-/ i * =0 (2.67) = / < =/

-29 - Since the total mass of each component remains constant' In order klA + /l^ + (A! o. c * =I t; ' to satisfy (2,67) and. (2-.68) it is necessary that: GA = - A - - A = = = * * =G ( o o.68):o69) From (2,65) it follows by partial differentiation with respect to.nI A-6' -G/- G Ay (2.70) It follows from (2 69) and (2.70) that-: I AS = LA = AL - = " a (2.71) The result given in (2,71) together with that of (2.61) is the mathematical statement of what is illustrated graphically in Figure 13. The common tangent at points b and c satisfies Equation (2o71), Another way of looking at this is that at equilibrium a system is in a state of minimum free energy, Rather than remaining a single phase of composition xa having a free energy corrqsponding to point a in Figure 13, the system can minimize its free energy by breaking:into two phases each having a molar free energy given at b and c, the tangent points,

Addition of a third component to the hypothetical system of Figure 13 results in a three dimensional representation of the dependence of a G on liquid composition shown in Figure 14. The case shown is one in which component C is miscible with both A and B, A and B being partially miscible. The locus of points of common tangency of planes to the free energy surface projects in the plane ABC as the familiar binodal curve for a ternary liquid-liquid system. The lines connecting common points of tangency of a given tangent plane project in the ABC plane as the tie lines, As the plait point is approached the reverse cleavage in the free energy surface becomes less and less finally damping out; the surface becomes convex downward, and the solution is homogeneous. Figure 14. AGZ - - X2, Partially Miscible Ternary Isotherm.

-31 - The Redlich-Kister representation for the ternary AG. surface(34) employing only the first two binary coefficients is: 2,3 r ET A/ Ri / P1 + _ X Z. t.3 x/ + /,%p (,+C,(,- ) 3r, ^ (fos C2 (4- X)) + X,(1>,+ C, (a-a,)) (2.72) Figure 15 gives the contours of hypothetical surface calculated by assuming B 3 B3 = 1.3, 0, = C 2 = CB/,3 0. The numbers on the graph are values of AOGM,3 03 RT MlA/ Figure 15. Hypothetical AG,^ Surface as Represented by Equation (2.72), B12 = 1.3, B31 = 0.5, B23 = B31 = C12 = C23 = C31 = 0.

-32 - D. Phase Stability and Critical Phases Of considerable importance in this research are those relations which enable one to predict phase stability and critical mixing points, A knowledge of the location of a plasit point in a partially miscible ternary is especially desirable in engineering considerations. The relations (9) which follow were originally derived by Gibbs. The development given (12) by Haase(12 has been abstracted and presented here, Consider a closed isolated system having a total entropy S. The second law of thermodynamics states that for any infinitesimal change: dS > 0 (isolated system) ~ (2.73) For an adiabatic change from state I to state II there are three situations: AS = S - S = 0 (reversible change) (2.74a) A =S = - Sz > 0 (irreversible change) (2.74b) -AS = S, - e < 0 (impossible change) (2.74c) If we propose a reaction "old system new system" then it follows from (2*74): AS > (old system unstable) AS = 0 (neutral equilibrium-a critical state) AS < 0 (old system stable) Since the system under consideration is isolated (that is there is energy transfer between system and surroundings, Q 0, W = -0) the additional relations hold: A = 0, AV = 0, An. =0 (i = 1, 2, N). where N is the number of components. (2.75a) (2.75b) (2.75c) no following (2.76)

-33 - The simplest case of stability is thermal and mechanical stability. Consider an isolated system consisting of a single phase of unchanging composition (UV), It is possible to form from n moles of the original phase n' moles of one phase (U + ALJ, V + AV) as well as n" moles of a second phase (U + U,, V + _V*) I(UV) -Y(u+u, v+ \V)+ (U+An i AV + ) (isolated system) In order that phase I be stable according to (2,75c)' /\S= A\_S + m"AS* < O (2.77) AUA = niJ + k l'IU = O (2.78) AV= )'AV + vl/NV* = (2.79) rl = ) V1/ V (2.80) Expanding AS as a function of U and V by Taylor series expansion: Similarly for AS* after utilizing (2.79), (2.80); -\ - — ' u - 5 '*' 2 (z s CV2

Combining these last two equations with (2.77) and (2o80): o l V L " ~ It follows from a property of the Taylor series that &(y,v)< O or ZS(UX ) is negative definite (2.81) where ( S(UiV) represents the entity in the brackets of the preeeding equation Similar consideration of additional terms in the series does not give additional independent relations. It follows from a rule of algebra (see Appendix A) that the quadratic form 6S(,,V) will be negative - definite if: F <0,~ ' balk ^As i > 0 * (2.81a) Phase I will have thermal and mechanical stability if the requirements of (2.81) are met. By reasoning analogous to the above, other relations for phase stability may be derived. In the case of the Gibbs free energy function, GI it is easily shown that for the reaction "old system to new system" at constant T and P: G <,0 (old phase unstable) (2.82) AG = 0 (neutral equilibrium-a critical state) (2.83) AG > 0 (old phase stable) (2.84)

-35 - For the case of a binary where Cv is positive, by introduction of suitable definitions it can be shown from (2.81) that for a stable phase SG (TP) > 0 (2.85) By an analogous procedure (see Appendix A) to that used in deriving (2,81) it may be shown that.in the cas.e of a binary for a stable phase ( ) >0 o r C G )],.p> ~ (2.86) (Note: the notation,^, ~ ~ refers to first order, second order terms, etc in the Taylor series expansion)4 Generalization of the development used to obtain (2,86) yields forf the case of N components the following stability criterion: G (%,,, * *, > [T P Cstb p co], (2.87) In words this states that,G(x, x, - ) must be positive definite if the phase under consideration is stable, An algebraic rule states that a quadratic form is.positive definite if the determinqant of its coefficients with all principal minors is positive..From (2,87) it follows then that for stable phase.s D>0; (2.88)

-36 - where a* T 2 Note that if bi(x,, --- x,.1 ) or D is negative the phase under consideration is unstable. Therefore the space curve D 0 (2.8a separates the region in which G(x,,.- x~., is negative >Ig Xa% an )iis (unstable phases) from the region in which (xr D is negative the phase D > O (2.82 positive (stabile. or metastable phases). Equation (2,89) is then the equation of the stability boundary. Graphically (2.89) is the curve b'Kc' in Figure 14 for a ternary system and is sometimes referred to as the spinodal curve. It is the loci of inflection points in the AG^M curve found in planes perpendicular to the x/, x2, plane and situated so that the tie lines lie in these planes. That region on the G(x, x) surface between bKc and b'Kct is known as the metastable region4.At a dritical point, an infinitesimal variation in state properties causes appearance of disappearance of a phase. It must lie therefore on the stability boundary and Equation (2.89) is known as the first critical relation.

-37 - The second critical relation may be derived as follows. Consider a two phase system such as the ternary system in Figure 14 Let us formulate a path in the G(xl x~ ) surface (at constant T and P) which passes through the unstable and metastable region, connecting two coexisting phases and which has zero length in the case of a critical phase, The equilibrium between two phases with N components at constant T and P is described by the N relations which follow from (2.69) together with b/_ ac' bG' G" G_ aG F _ ', ^ ^ ^,.-, -. -- 2i, daY ' E 2 >a-1/ }ui.1 11-1 N-/ (2.90) G'-JT'^ = G" -Zr/'2 If of the N quantities* 0 0 I &, ',a~a X........,, G -,& a, (2.91) N-2 are held constant, then a path is determined which behaves as that described above. -Since a two phase equilibrium with N components has a variance of N, the constancy of N intensive variables (T, P, and N-2 of the quantities in (2,91) which have the same value for the two coexisting phases).corresponds to a traversing of G surface between the two phases.

-38 - Consider now a point on the stability boundary (D - 0), For an arbitrarily small variation of the state of a phase, considering only variations on the prescribed path where: are constant: c O z + S= _D 3 - s;, (2.92) -, +.., ~ 3x ~rl a~2 3x, ~ ~N- (2.93) Proceeding in the direction prescribed by (2~93) one passes from metastable phases on one side of the boundary curve (D O) to unstable phases on the other side. Only in the limiting case of a critical phase (which must lie on the stability curve) does one succeed in the direction prescribed by (2,93) in going continuously from stable to stable phases. At the arbitrary location on the boundary curve the expression 6$,D can be positive as well as negative. If the point on the boundary curve coincides

-39 -with a critical point 6D must vanish4 If &D were negative one would succeed in arriving at unstable phases (D < ) on the path described by (2,93) which is impossible at the critical pointS If 6D were positive then it could be made negative by a sign change of the variations 6x 6x,,,:,x-/ o Therefore for a critical phase under the constraints (2.93): ED =0 (2.94) Equations (2,92), (2493), and (2,94) form a system of N-1 homogeneous linear equations consisting of N-= variables. In order that these equations can be satisfied, the determinant of the coefficients must vanish. ' Therefore / 0 -= (2.95) where D = | 2G t~G a~2G |(2095a) ~..O.... For a two component system at a critial solution point, thereFor a two component system at a critical solution points therefore, it follows from (2,89) and (2o95)~ 0==0, D~ L [T, Ep F (2.96a,b) of-i o, Di

Similarily for a three component system at a critical or plait point: a G - X12 er 2 2~; 3~ f7 [T, P(2.97a) [T, P]~(2.97b) 0- > ar 8,I 2 _ ^,w^ D- 37/ - For a quartenary, the locus of plait points will be given by the relations: D (=2~ n-^a1 2 zj a }it3 + ~aQ2 G 2. _ ~: &A2; K ~32<, G 2 3G aa - a a +C)| +, )2 G, 3 D A&,/ %,aZ W%/)X axl a EL P (2.98a) D. =_ uo ~W a 30 $ ~r 32 aG ~D \\ ~2 CaV < <^Xa7 <^J ( ' 2. rz~3 32G a ZG xi/~33/~ L[; Pj(2.98b)

The critical relations are of considerable utility in using algebraic representations of GG A further discussion of the critical *-M relations as applied to Redlich-Kister equations will be found in Part III. E, Temperature Corrections to Activity(3}34) Expressions for Q or AG involve a set of coefficients which are functions of temperature but not of composition, A given set of values for the coefficients pertain only to a single temperature, In order to predict equilibria at higher and lower temperatures a knowledge of B.and C,. as functions of temperature is required, It follows from (2#74) and the definition of activity that: (2.99) Differentiating partially with respect to temperature we obtain: aa = ^ (A- -)=_ - 7 H (2.100) AT R 3T RT7 where H. is the partial molal free enthalpy of i in solution, H~ is that of pure i. Since a; x-'b it follows that _ A___=__ U /G (2.101) T RT If (H - H ) is known as a function of eT, ln as a function of T may be found by integration of (2<93)e

-42 - From the definitions of L G, its enthalpy analog andAG given in (2 23), (2,26) and (2.38) it follows: AHN ^( \ 7) a T) = AH (2.102) and Gm T~ I( F &rt ARI AT) T AG T (2.103) On a basis of one mole of solution from (2.102), (2.103), and (2,31) we obtain: - Q L\z 3 C3 /'7 (2.104) A representation of LA\x has been proposed by Redlich and Kister similar to that for Q, For the binary case: A/i = atil (-if)Cni + (L*- ln+ o' (2.105) Partial differentiation of (2,54) gives: Q - T a,; (X 1C) +,-I —DII./-..yf^D).!.~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + Q '.'(2.106) From (2.104), (2.105) and (2.106) it follows that Z,^ _.- ^.aK b = Co. ^) Do3oR 2,$ ^, 3R (2.107)

If b,. and c/2 are known as fuhctions of temperature, B,/ and C/ can be found as. functions of T by integration of (24107) In the absence of additional data, if B,3 and C/.are known at two temperatures, b,, and ce can be assumed to be constant, and B1, -versus 1/T, and Cla versus 1/T plotted as straight lines Hala (13) found that B, CI/a, Taversus in P~plot as. straight lines. where P~ is the vapor pressure of some reference substance at the same temperature as B,2 and C,, Additional work in this area is needed.

III. APPLICATION OF TBE REDLICH-KISTER EQUATIONS A. Derivation of Equations for the Liquid-Liquid Case For the case of two phases in equilibrium it was seen in Equation (2.69) that G = G, G = G ' =G, GH [T,.P] Integration of (2.13) gives for either phase: ^G~- = /T4 t4o (3.1) If the standard states are chosen to be the same for component i in each of the phases then G = GSince a, is a definition fe/fo it follows from (2.69) and (3.1) that: / // / // / = ^-/, %.^ - ^ ^ ^ = *w ' (3.2) From the definition of the activity coefficient (2.18) it follows: \6; I I^-~~~ ~ ^ < >, / - (3-3) i =,1.~2. 2.) / / K; The Redlich-Kister equations for log,, retaining only the binary B,, C. coefficients may be written: -3 ' =-, (.', C. ',.-i) (3.4) 3~ re = i (T3, Cf * > Y, A -,-) (3.4) 13~~~~~~2 ~' 35 -44 -

-45 - From (3.3), (3.4), (3.5) it followsO ~." l) C " C-/,:( -4 2 " )~ (3.6) It is understood that these relations hold rigorously only at constant T and P (although small variations in P have negligible influence because we are dealing with condensed phases). Equations (3.6) and N equations in 2(N-l) unknowns if values for the parameters B'. - C, G.... are given. This means it is necessary to specify N-2 variables in addition to T and P to solve the system of of Equations (3.6). This result is obtained from the phase rule for in a system of N components and 2 phases at [TP]: V, =C C - P = /- (327) 1. Binary Case: For a two component system the Redlich-Kister representation of two phase equilibria retaining only the binary B^y and Cac coefficients follows from (2,55)^ (3o5), and (3.6): ~~/4 C,. ') (- -,'- 3.8) liA4 2// ZXA L3/2, =~,-g )tCj4,2 A /LA l_ -h]_x L /_ }( 9) where,= /-,/ and I ~ / - Xi, //

-46 - Given values for B/ and C/:Equations (3.8) and (3.9) may be solved for x, x. Nothing in addition to the coefficients must be specified (specification of the coefficients corresponds to specification of T and P) since N = 2 and from Equation (3.7), Vp,r = 0. Likewise specification of x,, x (binary solubility data) determines B.,, C,, in (3.8) and (3.9), 2. Ternary Case: By a procedure analogous to that used in the binary case, Equations (3.10), (3.11), and (3,12) may be written which represent ternary liquid-liquid equilbria and retain only bpnary coefficients, Bzj, CQ Because of the length of these equations, they will be written henceforth as (3.10a), (3,lla), and (3.12a), _9- 4 ir^ - B, f //- (/-Y/ + 2//](-zr/) + C, '4"(,</'( 4>(-, lX)-4+ i>-jj) - (2g'(-Z,'t2]/) C<3 L,( -2X8 Z- /i )-, (2zj (,/- 0 (3.10) +C," "z(2X2(- -/),-") / ^^/(/- /' - tC23 [ j(2/;'(-z + " "/)- ') z- ( (-z/) +C,[2(x& '(,"- 4- ~, '(z,'- o 0(3.11)

C/2L ('-,")- (-, +C/,2( %2 F4 C<-" -(' 4 / <j(r-^-/ -^ ^ '-> ) -1C, " 1;zI 1 e,1 IZ-31 + / -, _zi,(ZZ,_- /) 90- O'(3.12) where % = / -,Z, /- -3 ~ /- '"- = 0 (3.10a) = 0 = 0 a (3.11a) (312.3a)

Equations (3,10 - 12) are three equations in four independent variables (having specified the coefficients). Therefore by specifying one mole fraction we can solve the equations for the other three mole fractions, This coincides with the result from (3.7) for a ternary, B, Utilization and Flexibility of the Equations 1, Solution of the Equations Certainly if Equations (3,10), (3,11), and (3.12) can be used as a basis for a correlation, it must be possible.to solve them, given numerical values for the coefficients. Because of the nonlinearity of the equations it is apparent that successive approximation or iterative solution methods must be used,~ It was at once recognized.that;findin g an iterative technique,:that would converge to solutions having physical meaning would be a major problem in this research. (37) Newton-'s iterations(37) was the first method tried, It was found to be unsuccessful because it converges to the trivial solution where log, =, O (i = 1, 2, 3) Another method based on half interval convergence(8) also failed, The method that was found to be successful was far simpler, computationally than the Newton's or Half Interval methods, Having

-49 - specified the coefficients in Equations (3o10), (311l), (3.12) and one composition, say x,,, the steps in the procedure are as follows: 1l Guess X, A,. 2. Compute t. A. 3 ~ 3. Print, 2 3 / " if L ~// 4, - - / & where F is specified depending on the accuracy of the desired solution, 4. Compute aC = /O - /O 5. Compute = ( j 3//Q3+ 3 )/ where n denotes the number of the trial being performed, 6, Compute C =/-'/, = // " 3 / / 7- Compute.I - >-C (Note that x - x = c, // // -. / /1l / // o - C. x/ -- x- = d4. Since x, = ax and x = then = —. XI. x = bX3.hend bx = -'8 Compute = - = / 9. Using the N + 1 varlues of the mole fractions, go to step 2 and repeat the procedure until the condition of step 3 is satisfied.

-50 - Figure 17 Figure 16 831 - 1.0 Figure 18 Figure 19 83, = 0 B3-1= 0.5 Figure 21 B3 = 0.87 B3, =1.0 Plots of Equations (3.10), (3.11) and (3.12) B12 = 1.3, B23 = C12 = C23 = C51 = 0.

-51 - For a detailed description of the computer program based on this procedure, the reader is referred to Appendix B, Program 64 The procedure converges rapidly near a partially miscible binary, more slowly near a plait point4 The initial guess must be quite accurate near a plait point, to insure convergence, 24 Demonstration of the Flexibility of the Equations One of the first questions that can be raised is: do Equations (3410), (3,11), and (3,12) represent ternary liquid-liquid systems and if so can they represent a wide variety of cases? In order to answer this a number of fictitious curves were calculated, It was soon learned what the range of values of coefficients was that gave curves which looked like ternary liquid-liquid phase diagrams. Figures 16 - 21 were computed for one of the simplest of cases, that where B C 0 2 = C C/ 0 in (3,10). (3411), and (3.12), B,2 was set equal to 143 which is a value great enough to cause partial miscibility in the 1-2 binary4 B, -was varied from -2,0 to 140, It is noted that the slope of the tie lines is negative for negative values of Bl zero where B3/ = 0 and positive for positive values of B>3, For B$1 = 0;87 the solubility curve becomes tangent to the 3-1 edge at the plait point, and x/ = 0454 For B3 >.0087 the 3-1 binary is partially miscible Equations (3410), (3411), and (3412) are capable of more complicated representations by setting more than two coefficients not equal to zero4 In Figure 22 B3 0 o87k so that partial miscibility is just on

B12 1.3 B23 0.87 B31-1.0 B2 1.3 B23'B31-.0 C12z-C2s C31- 0 C12- Ca C31 0 82- 1.3 B23- B31 0.5 B12 -.3 B23 B31, -0.5 C12-.0 CZ3$ C3-C 0.3 C12 -0.2 C23- C31 0.5

-53 - the verge of appearing at the 2-3 edge, In Figure 23, B,, B,,, and:, are all greater than o087 therefore partial miscibility occurs for all three binary pairs, It must be realized that in the three phase region of Figure 23..Euations (3 10), (3a11), and (3412) are not valid in the physical senpe although they may be solved mathematically. One might imagine the system of Figure 23 to be the result of two superimposed systems of the type shown in Figure 22, one with partially miscible binaries 1-2 and 3-1, the other having partially miscible binaries 1-2 and 2-3, The calculation of Figure 23 was achieved by starting near the three binary regions and working in toward the center of the ternary region, Starting from the 1-2 binary and successively increasing x, a srstem identi.al to that in Figure.18 was computed. In general it has been found that systems of the Figure 18 type are obtained when Bjv = B,, and computation of the equilibrium curve starts at the ij binary, B,. being great enough to give partial miscibility. In computing those portions of the equilibrium curve of Figure 23 which terminate in the 3-1 and 2-3 edges of the diagram, convergence is very slow near the points where contact is made with the portion of the curve terminating -in the 1-2 edge. Figures24 and 25 show that setting the Cj coefficients not equal to zero produces more complicated systems than those illustrated in Figures 16-23- Setting C/2 0 produces asymmetry in the 1-2 partially miscible binary, Setting CG = C3l produces solutropic systems, that is systems in which the slope of the tie lines changes sign. Note that in Figures 24 and 25 the systems ar re soutropic and have asymmetric partially miscible binaries.

It was concluded from the above study that Equations (3.10), (3.11), and (3.12) are satisfactory representations of ternary liquidliquid equillibria and that they can represent a wide variety of systems. Co Critical Relations Equations which enable prediction of critical points given numerical values for Redlich-Kister doefficients are of considerable importance. From these relations one can determine whetiher a set of Redlich-Kister coefficients will yield a ternary system having one, two, three or no partially miscible binaries, whether or not there will be a plait point, and if there is a plait point, what its location will be, It was shown in Section II-D that at a critical mixing point the determinants D and D' vanish (Equations (2,89), (2.95)). These determinants involve-' the total free energy per mole of the system, G. Since the Redlich-Kister equations are representations of the molar free energy of mixing A Git is necessary to rewrite D and D' in terms of A\GM. The free energy of mixing is related to G, the total free energy of the system by AGM = G - n,Gl - GG KQ.GN (3.13) Dividing by VI, one obtains AG =G - G -0-G2 " " Z.. (3.14) where the xts are mole fractions of the total system. Taking the second partial derivative of G with respect to x; ALM E 2-Lr = v-^~2 {(3-15)

-55 - Noting that the determinants D and D' contain only derivatives of order two or higher, Equations (2,89) and (2,95) become: / / a6'a I 2^ _^ Ia ^^Ii a' I I.AG f (.A "L&M '^1 a1~c I CI ~Z!~......... — ~,,// = 0 (3.16) = 0(3.17) ~zi 14 Binary Critical Point: From Equations (3.16) and (3.17), Equations (3,18ab) may be written for the binary case: 2. b A, aGA = O 3 9 (3 G N~a-b) p (3.18a,b)

-56 - Note that it follows from (2.26), (2.28) and (2 31): /A\G,zj 4izo, ant + lA t 033 a (3.19) Substituting Q as given by Equation (2,54) and retaining terms multiplied by B,2 and C,2 it follows from (3.18ab) that 2. 303/3 s,= + ( )J (3.20) 3Z.~03~C,=,f-^ t(~/2.I r )2-((3.21) where xC is the composition of the system at a critical solution point (the point (T,,xi )in Figure 1), Equations (3,20) and (3,21) may be obtained by a different route without consideration of GM, For simplicity let us write (3.8) and (3,9) as L, = Be X, C-, X, (3.8a) Sl = n CX8+ CX4) (3.9a) Solvirig (3.8a) and (3,9a) for B' and C,, one obtains:,S i^,X4 - La^z, Ca x A'/\ (3.22) Xi 3 /Z Y3

-57 - I0 8 6 2 U b. 0 d0 0 -2 -4 -6 -8 -10 -12 0 0.20 0.40 0.60 0.80 XI Figure 26. Critical Values of B, C vs x1.

-58 - 2.0 1.0 0 -I.0 -2.0 - 3.0 -4.0 -5.0 -3.0 -2.0 -1.0 0 1.0 2.0 CC 3.0 Figure 27. Bc vs. Cc.

-59 - The critical values of the coefficients, BiP, C/, corresponding to the critical point (Tac x4 ) in Figure 1 may be expressed as 8v = A. B, C, =SOC, (3.23) where Be and C/, are given by (3,0) The limits in (3,23) may be evaluated by postulating that I4 / 3/. 2^~^~ =AL~ s ^~/hv^.-M A~ —,^/,3tg? and then showing for the ease in question < 3 2 => ^^ 3 /2 Taking the limits in (3,23) and applying ltHospitalts rule(56) one obtair* Equations (3,20) and (3,21).The details of the two derivations of Equations (3420) and (3,21) are given in Appendix A, Equations (3,20) and (3,21) are presented graphically in C Figure 26, as a plot of B/, and Ca versus x, and in Figure 27, as a plot of Bl versus C/2. Note that when C/,= 0, B/ goes through a maximum of 0,87 (2'0 if the Redlich-Kister equations are written with the logarithm to the base e rather than the base 10). Thus for the case of C/ =:0 values of B/ > 0,87 give a partially miscible binary, 1-2, and values of B <,0.87 give a homogeneous binary 1-2, This result

-60 - explains what was observed in Figures 16-21, that partial miscibility is on the verge of occurring in the i-j edge of a ternary when B.;j - 087, C; =-O, In general, points in Figure 27, that lie above the curve, B versus CC have as their coordinates values of B'j and CZ that give partially miscible binaries, those below the curve, homogeneous binaries. Ternary Critical Point or Plait Point As in the case of a binary mixture, the two constraints D = 0 and D' = 0 also hold for a ternary at a critical mixing point; thus it follows from (3l16) and (3,17): L)^ =A GM a'LM _(X0=o (3__3) D'_ a/G^ AD _ ~aLM aD -=o (3.24) For the special case where CG = C23 C31 =-0 substituting Equation (2,72) into Equations (3,2'3) and (3,24), carrying out the indicated differentiation and simplifying: ~ +A.. _/3 -, / 2f,,, 223( + -3X - 2 2 - — 2 +m+ 2-j(7" 3). Q(2..a (23+t 3 a,)_ 33 / 23 \3t # /l;I 3 3. *-iL~,ef't- ^H^2.X A- 4',-6 —^=a3A

-61 - Inclusion of terms multiplied by C/a, C,,, and C31 lead to considerably more complicated equations than (3,23a) and (3424.'a) Giiven numerical values for the coefficients, Equations (3c23) and (324) can be solved for x1 and xz, the composition at a plait point, A plait point.cannot be determined by direct experimental methods but must be estimated graphically from neighboring experimental tie lines, This graphical estimation is open to error, however, If one determines values for the Redlich-Kister coefficients which.enable Equations (3410), (3*11) and (3a12) to represent accurately a given set of experimental equilibrium data for a ternary liquid-liquid system, one may use these values of the coefficients in Equations (3,12) and (3,13) to solve for the plait point composition directly# Equations (3,16) and (3,1)7) are especially valuable in determining the locus of critical mixing points in systems of more than three components where it is not possible to.locate the critical graphically as in the case of a ternary systems. A knowledge of the experimentally determined coefficients in the coefficients,in the Redlich-Kister equations, or in any other representation of AG, is all that is needed with.Equations (3l,6) and (3,17) to solve for a critical locus, regardless of the number of components.

IV. DETERMINATION.OF COEFFICIENTS FROM EXPERIMENTAL DATA The determination of coefficients in the Redlich-Kister equations from experimental data is of prime importance, In the case of ternary liquid-liquid equilibria determination of the coefficients which enables the equations to best fit a given set of experimental data was by far the most difficult problem encountered during this research. In fitting a set of data to a mathematical function, or system of functions such as the Redlich-Kister equations one attempts to determine those values of a set of parameters that enable the function(s) to minimize the deviations of the experimental data from the mathematical curve. One seects a dependent variable or variables and minimizes the error in this variable, that is the difference of the experimental value from the mathematically predicted value for all of the data points. This presents the first major problem: what variable(s) should be chosen as the dependent variable(s)? The choice is arbitraryy There will be two kinds of deviations- first, those which are due to random scatter of the data, ie, experimental error, and second, those which are due to the fact that the mathematical functions are not capable of perfect representation of the data. In this research, the number of parameters in the equations has been limited to twice the number of pairs of components~ The number of data points for a given system is far more than twice the number of pairs of components4 Where this is so, deviations of the second type must be expected. -62 -

-63 - The second major problem occurs because of the nature of the Redlich-Kister equations, or for that matter any algebraic representation of the logarithm of the activity coefficient, These equations are long, complicated expressions which are implicit in the experimentally measured.variables such as compositions If one designates a directly measured quantity to be the dependent variable(s) the computational problem becomes quite considerable Let us consider first the simple case of an algebraic expression which relates the dependent variable, yl to independent variables ZI I ZA. 1 ZrA Z1 > ZI. Zmo Z., Y =f(z,, '* *Zo, /3, ' ' '/^) (4.1) where,g, *0 A *. are parameters, The experimental data points are denoted YC 0 6" 26' ' ' ' 'Z4. ' (2 ' K j4.2), ' = Statistical techniques are based upon the concept of a particular set of data called the sample, which is drawn from an infinitely large population of values, From the sample one obtains estimates (c/, d ~,. ) of the population Iparameters (,/<3,, ~A /Q3 ), Under quite general conditions, the appropriate statistical criterion for adjusting the values of the parameters in order-to best fit the data is the least squares criteriono(21) The assumption in chosing this criterion is that the deviations of the experimental data from the predicted values for the

(38) data should follow a Gaussian distribution. Let YZ denote the predicted value for y when- for the ith data point, the values of the independent variables are substituted into Equation (41l): X f= ^(f,Z*G v a ZM/6 s C/ C C X C ) (4, ) The least squares criterion states that the sample estimates of the population parameters shall have values such that n~tA =L -S (44') be a minimum, \ is a minimum when the partial derivatives of A with respect to each of the parameters vanish: = =,,,.,= =0 0 (4.5) ac, bc, ac0 Equations (4,5) are.k simultaneous equations in k unknown parameters, If Equations (4,5) are linear in the parameters, they may be solved directLy for c/, c2 ~,e ct If the equations are non-linear, problems in methodology present themselves as it is practically impossibLe to solve a system of non-linear equations of the type (45)o Methods of obtaining estimates of c/,. f,,obck for non-linear cases are discussed in detail in Section III* B, 1l Ao Binary Case There is considerable interest in the possibility of using binary Redlich-Kister coefficients determined from binary vapor-liquid,

-65 - or liquid-liquid (solubility) data to predict equilibrium in multicomponent systems The determination of the binary Bg and C,. coefficients from binary solubility data is trivial, Substitution of x/ and x/ into Equations (3.8) and (3,9) give two simultaneous linear equations, which may be easily solved for Bt~ and C2 X A computer program (Program 5) was written for this purpose. The determination of B., and C,. from binary vapor-liquid equilibrium data is less straight forward. A number of different variables may be chosen as the dependent variables for example: x/ y/y P 4 t 7nGM logg, Q....a etc. Suppose it were decided to minimize the deviation of the experimental value of Q from that value predicted by the Redlich-Kister equations, a _ ( - QY) (4.6) =/ where d is to be minimized, For the binary vapor liquid cases,*= ' o (o X +.2 4, ) (4.7) Q, = V,2(Q,, + (z,- )C.) (4.8)

-66 - (Equation (4,7) assumes that the total pressure, P, is low enough that fugacity corrections need not be included.) Equations (4,5) then become Z %a8,j (Q &) = o Z t 4Z ( - ) - 4) 0O (4.9) Note that the choice of Q as the dependent variable is advantageous because Equations (4,9) are linear in B/2 and C/2 and solution is straightforward.Another choice of dependent variable that leads to linear Equations (4.5) is logY, or logyz. Since it is normally desirable to minimize the error in log Y, and logY. simultaneously we may define 6 as: A =9~)2 (ZlfCA -^^t /^^ -c^ ^ 4j 2 (4.10) where = r ( R = J ^ (It is important to note here that B,. B,, but %, - CJ^ The reader is referred to Equations (2,55), and (2,56).) Two computer programs have been written to solve for the values of B/ and C/, which minimize the A defined in Equation (4,10), and these are described in Appendix B,. Program 1 handles data at constant temperature. Program 3 handles data at constant pressure. In the constant pressure (variable temperature) case B2 a C2, rigorously cannot be assumed constant since B/ and CG, are functions of temperature. If the temperature variation

is small, say 10~C, the error in assuming the coefficients to be constant is small. Figure 29 in Section Va A, gives x-y plots of four systems for which B/ and C/ were determined by minimization of a defined in (4.10). In each case the f it is excellent. Although the selection of Q or log/~, as the dependent variable leads to linear Equations (4,5), these variables may not be of direct engineering importance, In addition they are not quantities that are determined by direct physical measurement such as total pressure, but require introduction of the independent variable x/ or xX in their determination which under certain conditions may be undesirable, In a situation in which the total pressure, P, is especially important, one might wish to minimize A defined as A-i (p *F)2 ^(4.11) where -= a 7(, (2i+ (3o/-r Zz)C,) I(s (a/e- (3,z -je)C e) (, is assumed low enough here that fugacity correationsare not needed). For this case however Equations(4.5) are not linear in B. and C/ and it is somewhat more difficult to obtain the values of Br and C/1 that minimize A (4.11), M-arquardt(21) describes procedures for minimization of (4,11) for the case where P, is represented by the Van Ijar Equations (See Table I)o B, Ternary Case The ternary liquid-liquid case has curve fitting problems which differ somewhat from those of the case of binary vapor-liquid equilibria.

-68 - Use of the molar free energy of mixing or the Q function as the dependent variable is usually impossible because the partial pressures of the individual components are usually not measured in determining two phase liquid equilibria. Without a knowledge of the partial pressures of the individual components, the values of the activity coefficients, and thus of Q and LGM are not known, If the Equations (4,5) are to be linear in the Redlech-Kister coefficients for the ternary liquid-liquid case, some linearizing approximations must be introduced, Linearizing approximations as it will be seen often do not give good fits of the experimental data, The failure of linearizing approximations led to the development of "hand methods" for guessing successively better estimates of the coefficients. Scheibel(40) has described an approximate mnthod of determining Redlich-Kister coefficients. The basin of the method is that, theoretically, equilibrium data for two tie lines should determine the six coefficients in Equations(3*10), (311) and (3.12). This amounts to determining values of equation parameters that force the mathematical functions to pass through two specific polirts, This method was tried in the present work without success, It was found that although the equations are forced to pass through two selected points there is no assurance that they will pass through or near the remaining points. 1, HaRnd Methods: In many cases, good estimates of the coefficients in Equations (3.10), (3ll), and (3,12) can be obtained by visual curve fitting. Good initial estimates of the coefficients are desirable to insure convergence

-69 - of the iterative non-linear methods described in a later section for determination of highly accurate least squares estimates of the coefficients. A hand fitting procedure which usually gives good curve fits in a few trials is outlined below. For cases of ternary systems having one partially miscible binary, components 1 and 2: 1) Determine Big and C f.from binary solubility data for the 1-2 binary by solving Equations (3.8) and (3~9) after substitution of x/ and x / Extrapolation of the ternary data to the 1-2 binary may be necessary if the binary data are not available. 2) Note whether the experimental tie lines tend to parallel the 2-3 binary or the 3-1 binary on a triangular diagram as the plait point is approached. Assume that the coefficients corresponding to the binary edge which the tie lines do not tend to parallel are zero, If the tie lines tend to parallel the 3-1 binary; then set B^3, C, - 0. 3) Determine the values f C which cause the calculated plait point to coincide with the with the experimentally estimated plait point by solving Equations (3423) and (3.24) after substitution of the values of B/ and C/2 determined in part one and the composition of the plait point as established graphically from the experimental data. Alternatively. B;i and C t may be adjusted by trial and error to give an.approximate fit of the experimental equilibrium binodal and convolute curves. In order that the tie lines tend to parallel the 3-1 binary B33 must be greater than B3 if C23 - C/ 0.

-70 - 4) If after B^i and C3/ have been determined in part 3, additional accuracy of the fit is desired, B32 and CG3 may be adjusted by trial and error, In case it was found to be impossible to solve Equations (3,23) and (3,24) for real values of BI and Ci3 with B2 = Cz3 = O, B,3 and C -$ might be set not equal to zero and solution of' (3.23) and (3,2-) for B31 and C,1 tried again, Often the fit of points in the ternary region may be improved by modifying the values of Bl, and CI i.e. tolerating some error at the 1-2 binary, Increasing the values of the coefficients especially Bi, increases the area under the binodal curve, In guessing values of coefficients corresponding to binary systems that are homogeneous throughout it is advisable to use Figure 27, the plot of B.versus C, to make sure the assumed coefficients lie in the homogeneous region. If partial miscibility occurs in two binaries, say the 1-2 and the 3-1 binary: 1) Determine B/ C,1 B31 and C1 by solution of Equations (3*8) and (3,9) after substitution of the solubility data for the 1-2 and the 3-1 binaries respectively, 2) Calculate the ternary equilibrium curve by solution of Equations (3.10), (3,11) and (3.12) with B,2, B1 C,,, and Ci1 equal to the values determined in step 1, and B., = C2 = 0. If greater accuracy of fit is desired one may adjust the values of B23 and C23 by trial and error,.using Figure 27 to insure that assumed values of B.i and CG3 do not make the 2-3 binary heterogeneous,

-71-.It is often helpful to make use of any available binary vaporliquid equilibrium data to get initial estimates of B f CG for homogeneous binaries. Also it is advisable to study previously calculated ternary equilbrium curves before guessing coefficients.Note that the fictitious curves Figures 24 and. 25 illustrate that setting B, = B:1 and C, = C3/ gives equilibrium curves that are solutropic (i.e, the slope of the tie lines changes signs) Usually after a little experience is attained, less than four trials are needed to give a satisfactory curve fit if the above procedures are followed, Visual curve fitting:however suffers from the disadvantages that it is a trial and error pro cess, and does not necessarily minimize some specific deviation defined in a least squares sense, 2, ILast Squares Using Approximate Linearizations: When,one is concerned with a system of n equations in m+n variables and k parameters for estimation by least squares, n of the m+n variables may be designated as dependent variables and the remaining m., the dependent variables. Using the notation of Equation (4,2) where Y's are dependent variables predicted by the equations and Zts are the independent variables, the system of equations, may be written: ^ ^ J 2 10 I Z@Z Z2L'.% C,, C '^ C) (412) The least squares criterion states that the parameters will have such values that A f~= /jfz iu-y. ( ) Jz 7(4.13) -= el j J,,

-72 - be a minimum where Yo -is the experimental value for the jth dependent variable, ith data point, and w. is the corresponding weight factor. If all data points and dependent variables can be determined with equal precision and are of equal importance wj is set equal to unity, It is not necessary that the equations f be explicit in the dependent variables. All that is necessary is that given values for the parameters and the independent variables the equations/ may be solved for the dependent variables Yj * In the case under consideration in this research the Equations (3,10), (3.11), and (3,12) which comprise the system are implicit functions of four of the mole fractions x', X2 ^ x3 x x x, selected such that there are no more than two X2, I,~X:3 I X/I per phase (eg, x/ x x/ x3, not x/, xi x x ) If one arbitrarily selects x/, x. x, x3 as the variables in Equations (310), (3.11), and (3,12) and designates x2 as the independent variable then A becomes: I-A I\ = l(, -, + (ZJ )X2 + ( /' ) (4.14) Unfortunately the Equations (4,5) for A(4.l4) are not linear in the RedlichKister coefficients, and their solution is difficult. In order that the Equations (4.5) be made linear in the coefficients we modify A(4.4) by minimizing the deviations of the logarithms of the dependent variables. Assuming:w = w2 = w3 = 1 (4.14) then becomes: A( (,'- 44/2 A^/)a+( ) ''-.zJ, (4.15)

-73 - From Eauations (3,10), (3,11), and (3,12) it follows ^,^,,// = Al +,e<4,= - /^, AA-nIe M% 7y ^ -e", // pi j ~,R/ooz 4 11 ^.O 3 e3 *N l r (4.16) Substitution of the approximations (4,16), A (4,15) becomes: /ia=M L(' / )( 2. (X: g / L -,;~~ /, X ir or M A=Z: / I j x / 61 I/ (4.17) For the purposes of simplification let us write Equations (3*10), (3 11), and (3,12) as sI/U t/ I J t I — -/ j=. _l i a = I ~~~"3 P/~~~~( z - 0

where ca/.. c are the Redlich-Kister coefficients B/,B B^, C/ B CG, Col, The Equations (4,5) for (4,17) may then be written: ti i(+ ihA)] = t I +iX'. ^ 3wl=/ jsf = (k = 1, 2..,6) * (4.18) The six linear Equations (4.18) are linear in the coefficients co and may be solved directly for c/ *,. c., A computer program (Program 7, Appendix B) has been written which computes the values of c,/. c. that minimize (4,17) by solution of Equations (4,18). The success of the linearized approximation of a (4.15) depends on the accuracy of the approximation in Equations (4.16). This approximation is good when the experimentally determined compositions, the x's, are nearly equal to the xIs, the mole fractions obtained by solution of Equations (3,10), (3,11), and (3,12) using the parameters c,.. ce which minimize A (4,15). If the precision of the experimental data is not very good, or if Equations (3,10), (3,11), and (3,12) are not capable of highly accurate representation of the experimental data, i e, a fair amount of non-random error must be tolerated, then the approximations (4*16) will not be good and minimization of A(4.17) will not yield an estimate of the coefficients that will give an accurate fit of the experimental data, In particular it has been found that for systems where the tie lines are nearly parallel the data must be extremely precise if minimization of A (4.17) is to give satisfactory results,

-75 - - PROPANOL 0 --- — EXPERIMENTAL DATA ------ CALCULATED DATA 20 H20 20 C ETHYL 2 ACETATE ACETONE HO 31~C ETHYL.2 ACETATE Figure 28. Examples of Fits Obtained Using Approximate Linearization.

Figure 28 shows that the estimate of the coefficients by minimization of (4*17) was excellent in the case of i-propanbl-water-ethyl acetate 20C (2) where the equations were capable of highly accurate representation of the data, and the precision of the data is good, In the case of. the system ethyl acetate-water-acetone 300C(49) where the precision of the data is only fair and the tie lines are nearly parallel, the curve calculated using the coefficients obtained by solution of the Equations (4.18) is completely unsatisfactory, Although the solution of Equations (4*18) frequently does not give satisfactory values for the coefficients it is advisable to use the linearized approximation to see if it does give satisfactory values as it is much faster to solve Equations (4,16) than it is to use the hand methods described in section one or the non-linear methods described in the next section. If using the original experimental data in Equations (4,18) fails, and it is only desired to obtain an -initial estimate of the coefficients for use in the non-linear least squares method.described in the next section then data obtained from a smoothed curve through the original data should be used..3. 2least Squares Without Approximate Linearizations: It was seen in the..ast section that the appropriate A that should be minimized to obtain best estimates of the coefficients is of the type: f = f(Xj-4j + ( - ^ (Z - y43J (4.19) (abbe)

-77 - where j, k, and 1 can have any values from one to three. Solution of the Equations (4.5) for (4.19) is practically impossible. However, determination of the least squares estimates of the.coefficients is possible by direct, iterative, minimization of A (4.19). Two methods are available and these have been described in Reference (21). The Method of Steepest Descent: This method seeks at each trial to calculate corrections to the coefficients such that the value of A will decrease "most rapidly". The steepest descent direction is defined by the partial derivatives (^..... __ calculated at the nth trial values of the parameters c, o,. c6. Experience has shown that the minimum - is reached in.fewer trtals if the corrections to the parameters are not made exactly proportional to the partial derivatives of A but proportional rather to the normalized value of -~ after multiplication by (1 + ca ) Q =/ -( /f ( 1U^ ) g(4.20) The.n + 1 trial values of the coefficient c is calculated ^C =: C ~ + K a+ (4.21) where 0( is the step size which determines the absolute magnitude of the corrections

The calculation procedure that was found to be best suited to the Redlich-Kister equations may now be described, Assume that the calculation has proceeded to the nth iteration and it is desired to advance to the n + 1 iteration..a, D, (I = 1, 2....6) are calculated. aC. A^"'3,D^ b, If proce.d to step c., if not, divide CL used in calculating c2t by 4 and repeat step a, c, Calculate the cosine of the angle 6, between the meterized steepest-descent directions at the (n - l)th and nth trials cos =Z iD ~. (4.22) d. Calculate a tentative value for OC based upon cos G Note that if cose is negative (i,e. > 9.0~), it is desirable to reduce C0 tio prevent undesirable oscillation between successive tirial values of c/.,. c6, If cose is near one the angle between successive trial c 4 directions is too small and it is desirable to increase the value of ato speed the convergence. To achieve this' ehad the following formula has been devised: o'= O<-/e+, + e, Cos 3 (4.23) where e, and e2 are always positive. The combination e/ = 0.5 and eA = 1.0 has been found to be satisfactory.

-79 -e, Calculate the new trial values of the coefficients using (4,21)..and return to step a. This procedure converges from any reasonable initial guess of the parameters and it adjusts the step size automatically to the local topography of the A-c -c2 —.c. surface. Convergence is slow in the neighborhood of the minimum A however and it is generally desirable to supplement the steepest descent method with a second method described b-elow which converges only when the initial guesses are quite good but does converge rapidly. 2,1, 37) The Truncated Taylor Series Method: (2137) If values of the predicted mole fractions xj,^ xk x, are expanded about current trial values of the coefficients c,., c., and the series is truncated after terms that are linear in Ac, A c~ Xi (cac,, t c,t c) R(c,, c, +^z (4.24) and similarly for xh and xL Substitution of (4.24) into (4.19) gives: ' =~ (%i' -^jr ) ) + (Z -(~'.Z^i OcZ)) (/-a ( // o (4 25) The minimum A is reached, when for the nth trial (g_\....= -.^ \ 0 (4.26) I (ac, 3( [a) k { wc

-80 - which are six simultaneous equations linear in /\c/,, Az ~ The Equations (4.26) may be written after substitution of (4.25) and differentiation: f LL/\CA(S 8 # A a + AR1A-)S = -LjWT @ + A\^+4 # -t L\ z /-I (v-= 1, 2,.6) (4.27) where Ax. = X. - x (c,, ca ear c) and similarly for Axk and Ax L The six Equations (4.27) may be easily solved for Lc,... Ac, and the n + lth trial of the coefficients calculated: " =- C + AC (,= /, 2 - 6) (4.28) If the approximation made by truncating the Taylor series after linear terms is good enough then the method converges and aL C <I A C A computer program (Program 8 - Appendix B) for the IBM 704 has been written which uses both methods described here to obtain the estimates of the parameters which minimize XA WI [-, +, Ca ) -.( ^"- /)S, (4.29) eol L

-81 - The steepest descent method is used initially, then.when in the proximity of the minimum 6(4 29), the truncated Taylor series method is used, If the equations are not capable of fitting the experimental data with high precision the latter method diverges and the estimates of the coefficients obtained by the steepest descent must be used, The most difficult part of the two procedures is the computation of the A and the partial derivatives of A with respect to the coefficients, Note that solution of Equations (3,10), (3,11), and (3.12) is required for every experimental tie line where x3 = x (the chosen independent variable in the ease of Program 8). Thus the solution procedure for Equations (3410), (3.11) and (3.12) described in part III is required, U nortunately this solution procedure will not converge unless the initial guesses of the compositions are quite good and frequently, especially in the neighborhood of a plait point, use of the experimental data (the x:s) as an initial guess is not good enough that convergence to the values of the predicted mQolractions (the x s) is obtained. This problem was solved by a procedure of solving Equations (3,10), (3,11), and (3012) in very small steps of increasing xX from experimental tie line to experimental tie line, using as the initial guesses of the xts for each step those values obtained by linear extrapolation from the last two steps, This method has been found to be highly satisfactory, One must be especially careful that the.initial guesses of the, pa:rameters give an e ~qu~libri'um curve for which at every X3 = x/ a the parameters give an equilibrium curve for which at every x3 - x3 a

-82 - non-trivial solution of Equations (3,10), (311), and (3,12) exists, An equilibrium curve should always be calculated with the initial guess of the coefficients at the outset to make sure this is so before proceeding. The development of the partial derivatives A- is given in Appendix A. and follows the procedures outlined in text books on advanced calculus pertaining to the differentiation of implicit functions. All of the calculated curves for specific systems given in part V were determined using coefficients which minimize A (4,29-) w/ = w w =3. Initial guesses of the coefficients were determined by the hand methods described in Section 1L except for the systems ethylacetate-water with ethanol, iso-propanol, n-propanol and t-butanol, where the initial estimates were obtained using the linearized approximation, A (4.17)

V, TESTING OF THE RESEARCH HYPOTHESES The problem now remains to determine the feasibility of using the Redlich-Kister Equations (310), (3.11), and (3.12) to correlate and predict equilibria for liquid-liquid systems. Restating the questions raised earlier in the affirmative we may write the following research hypotheses for the specific case of the Redlich-Kister equations applied to ternary liquid-liquid equilibria: (1) The vast majority of ternary liquid-liquid equilibria can be represented by Equations (3410), (3,11), and (3,12) which retain only the binary Bj and C.. coefficients. (2) B. and C,, are independent of component k. That is BZ. and Cb. have the same values in the binary system i-j as in the ternary system i-j-k~ If this is true then coefficients determined from binary data can be used to predict ternary data, (3) B., and, C, are simple functions of some property of k. If this is true then coefficients for a homologous series of ternary systems i-j-kl ^ i-j-k2, D i-j —kpl where k, - k, k, are homologs could be used to predict the coefficients for a system i-j-kj for which no data are available, Hypothesis (1) has already been supported by the computation of the several fictitious equilibrium curves (Figures 16-25), Over twenty actual systems have been fitted by Equations (3,10), (311) and (3 12) during the course of this investigation as a further test of Hypothesis (1). These systems are illustrated later in this section. -83 -

A, Binary Coefficients from Binary Versus Ternary Data The possibility of prediction of multicomponent data from binary data has long interested workers in the field of phase equilibrium, Probably the most recent effort in this area was that of Kenny(18) who determined coefficients in the two suffix Van-Laar equations (see Table I) from total pressure measurement of binary systems to predict equilibria in the ternary system iso-octane-furfural-benzene 20~C. The predicted binodal curve deviated considerably from the experimental curve near the plait point and agreed fairly well near the partially miscible binary iso-octane-furfural, The predicted tie lines did not agree well at all with the experimental tie lines, It was found that including some ternary data in the determination of equation coefficients, improved the fit somewhat, Two ternary systems were found for which binary data are available at or near the same temperature as the ternary data, These systems are ethyl acetate-water-ethanol (70~C)(ll) with binary data: ethyl acetateethanol (1 atm,)(l1) ethyl acetate-water (700C),(11) ethanol-water (74,790C) (56) and benzene-water-l,4 dioxane (250C)(3) with binary data: benzene-water (250C),(43) benzene-dioxane (250C) (45) and dioxane-water (250c),(49) The binary B and C coefficients were determined in the cases of the partially miscible binary systems using Program 5, Appendix B, Program 1 determined least squares values for Be. and (C. for binary vapor-liquid systems at constant temperature, Program 3 was used for systems at constant pressure,

TABLE II COMPARISON OF COEFFICIENTS DETERMINED FROM BINARY VERSUS TERNARY DATA 120(1) - Ethyl Acetate(2) - Ethanol(3) 700C B12 B23 B31 C12 C23 C31 From Binary Data 10130 0,3630 005400 0o6910 -0.0092 -0.1730 From Ternary Data 1.172 -5.339 -7~733 0.6450 o 517 -3338 H20(1) - Benzene(2) - 1.4 Dioxane(3) 25~C oo B12 B23 B31 C12 C23 C31 From Binary Data 3.060 0o1080 0.7900 -0.2400 -0.0320 0.05730 From Ternary Data 2.704 0.1135 0.7659 -0,2407 -0.03483 -6.1283

-86 - All of the binary systems of ternary HO0-benzene-dioxane and the ternary itself were determined at 25~C, In the case of the ethyl acetateHI0-ethanol ternary although only the binary solubility data are at exactly the same temperature as the ternary, the correction of the coefficients for the ethyl acetate-ethanol system (1 atm,, 77.2-78.4~C) and the ethanolwater system (74,790C) to 70~C is negligible, Figure 2~ gives the Redlich-Kister Bz Cy Q least squares fits of the x-y data for the four vapor-liquid binary systems, In all cases the fits are good indicating that the coefficients have been accurately determined and that the data are thermodynamically consistent (otherwise the Redlich-Kister equations would have been unable to fit the data). The binary B and C coefficients were redetermined using only ternary data by minimization of A (4,29), utilizing Program 8, Table II compares the values for the coefficients determined from the binary and ternary data, Note that for the system ethyl acetate-water-ethanol at 70~C, except in the case of the coefficients that correspond to the partially miscible binary, the coefficients from binary data do not agree with those from ternary data, Figure 30 compares the fits of the experimental data using binary data to determine the coefficients (solid curve), The convolute lines have been drawn to enable location of the.tie lines, Note that all equilibrium curves reach the partially miscible binary regions at about the same location and that the accuracy of the curve predicted from binary data is best near the partially miscible binary, poorest near the plait point which agrees with Kennyts observation.

WATER (1) - ETHANOL (2) 74.79' BENZENE (I) - DIOXANE (2) 250 ~C 1.00 0.75 >- 0.50 0.25 0 0.25 0.50 0.75 1.00 0 0.2 0.4 X, Xi I.00 0.75 > 0.50 0.25 0 ETHANOL (1) - ETHYL ACETATE (2) 760 mm ~^ /~ / B, ~ 0.363 C,-0.0092 L__ __ 1 ___ H.O (I) 0 0.25 0.50 Xl 0.75 1.00 0 0.2 0.4 0.6 0.8 1.0 XI Figure 29. Bij - Cij Fit of Binary X-Y Data.

-88 - ETHANOL ---— EXPERIMENTAL CALCULATED FROM BINARY DATA CALCULATED FROM TERNARY DATA H20 70~ C ETHYL ACETATE DIOXANE BENZENE/25 ~C\VH BENZENE 25 0C HO2 Figure 30. Comparison of Fits Obtained Using Coefficients Determined from Binary Versus Ternary Data.

-89 - It must be concluded that for partially miscible systems there is enough ternary interaction that ternary equilibria cannot be accurately predicted from binary equilibrium data, except near a partially miscible binary. Hypothesis(2) must be rejected. Ternary interaction coefficients need not be introduced in Equations (3.10), (3.11), and (3.12) to enable correlation of the ternary liquid-liquid equilibria of the ethyl acetate-water-ethanol system at 700C, where a nearly perfect fit was obtained using values of B', and Cgj determined from ternary data. (dashed curve). The binary coefficients take on new values for the ternary liquid-liquid case reflecting the ternary interaction which could not be predicted from data on the individual binaries. The fit of the data using coefficients determined from ternary data for the system benzene-water-dioxane (25~C), though not perfect, is an improvement over the fit predicted from the binary data. Apparently more coefficients are required to correlate this system accurately. B. Homologous Series of Systems It remains to test Hypothesis (3) to determine whether the RedlichKister equations have utility as a basis for a correlation of ternary liquid-liquid equilibria. If Hypothesis (3) is valid then experimental data for systems i-j-k,, i-j-k,.... i-j-k. where k,, kz,.... k, are homologs can be used to predict the equilibrium in the system i-j-kL which has not been determined experimentally. A fairly large number of homologous series of systems have been determined experimentally. Table III gives some of these having at least three members per series.

TABLE III* HOMOLOGOUS SERIES OF SYSTEMS l' TOT-/ 'TTT\T'N T-T'"'AP-Pl~TQ.^vCqM~hA C TTn T TTATTrsT UV7n R A RR nM.R bi — bl'~l- I-VUVLVII U AIDILSuDlfuri i - j 1) Benzene 2) Benzene 3) Benzene Benzene 4) n-Heptane 5) n-Heptane 6) Methanol OTHER SYSTEMS 1) Acetic acid 2) Acetone Acetone 3) Acetone Sulfur Dioxide n-Heptane Water Water Water ' -thio* kl n-Butane t '-imino-* Ethanol n-Propanol Methanol Benzene k2 k3 n-Hexane n-Heptane ' -oxy-* p' -thio-* i-Propanol i-Butanol Ethanol Toluene t-Butanol n-Propanol p-Xylene Water n-Hexane n-Heptane- n-Octane Propyl** Butyl** Ethyl** n-Propyl** k4 n-Decane cf '-oxy-* n-Butanol Ethyl benzene n-Nonane n-Butyl** n-Propanol n-Butanol Butyric acid Temp. -20~F 25~C 25~C 38~C, 25~C 25~,30~,38~C 25~C 10~ & 15~c 30~C 30~C 30~C 30~C 20~ & 40~C 0~ & 20~C 0~ & 20~C 30~C 31~C 31~C Water Water Water Water 4) n-Butanol Water 5)Ethyl** Water Ethyl** Water 6) Ethyl benzoate Water 7) Ethylene Glycol Acetone 8) Ethylene Glycol Acetone 9) Methanol Water 10) M.I.B.C. Water 11) Di-Phenyl Ether Water 12) Di-Phenyl Ethyl Water 13) i. Propanol Water 14) Propionic acid Water Methyl** Methyl** Amyl** Ethyl Acetate Ethylene*** Methanol t-Butanol Formic acid Ethyl** Ethyl** Ethyl** Formic acid Methanol Acetic acid CC14 Ethyl** Ethyl Propionate Diethylenexx* Ethanol s-Butanol Acetic acid Butyl** Ethyl propionate n-Butyl** Acetic acid Ethanol Propionic acid CHC13 Ethyl** Ethyl Butyrate Triethylene*** i-Propanol i-Butanol Propionic acid Amyl** Ethyl butyrate Pentyl** Propionic acid n-Propanol Butyric acid CH2C12 Ethyl** Ref. (35) (36) (48), (26), (44), (51) (24),(1) (19), (42), (24) (36) (19) (25) (49) (49) (23) (2) (31) (30) (30) (32) (28) (27) (27) (17) (29) I kO 0 I 30~C Butyric acid 30~C n-Butanol 25~C 25~C 25~C 30~C *Dipropionitrile **Acetate ***Glycol M.I.B.C. = methyl isobutyl carbinol

-91 - One question that might be raised is: for the purpose of the correlation must one consider a homologous series to be that defined in its strictest sense (e.g. methanol, ethanol, n-propanol, n-butanol) or may isomers be included (e.g. methanol, ethanol, i- and n-propanol, t-, s-, i-, and n-butanol)? In compilation of Table III isomers have been included. In order to test Hypothesis(3), four homologous series of systems taken from Table III have been used, Three situations have been studied: (a) k, the variable component in the series, is miscible with both components i and j. Two series of systems having this characteristic were studied: water-ethyl acetate-alcohols (20~C) and water-benzene-alcohols (25~C). (b) k is partially miscible with j, completely miscible with i. The series studied having this characteristic is n-paraffins-sulfur dioxidebenzene (-20~F). (c) k is partially miscible with both i and j. The series studied was n-paraffins-water-methanol (150C). 1. Water-Ethyl Acetate-Alcohols: Counting both normal alcohols and iso-alcohols the data of Beech and Glasstone(2) provide the longest homologous series of experimentally determined systems found in the literature. Figures 31-38 illustrate the least squares fits of the data obtained using Program 8 which minimizes a (4.29). Convolute lines have been used to permit location of the tie lines. Calculated points (+) appearing in figures are calculated at x3 = x for each experimentally determined tie line, (') referring to the water phase, (") the ester phase.

-92 - METHANOL ------ EXPERIMENTAL — X — CALCULATED H20 20 ~C ETHYL ACETATE Figure 31. H20 (1) - Ethyl Acetate (2) - Methanol (3). ETHANOL H20/V V V \/\ \V\ H.O 20~C ETHYL ACETATE Figure 32. H20 (1) - Ethyl Acetate (2) -Ethanol (3).

-93 - i- PROPANOL ------- EXPERIMENTAL -— X — CALCULATED A7\/\\/\\ -BUN H 0 20 ~C ETHYL ACETATE Figure 33. H20 (1) - Ethyl Acetate (2) - i-Propanol (3). t - BUTANOL H20 20 OC ETHYL ACETATE Figure 34. H20 (1) - Ethyl Acetate (2) - t-Butanel 35).

-94 -n- PROPANOL ---- EXPERIMENTAL -- ---- CALCULATED H20 20 ~C ETHYL ACETATE Figure 35. H20 (1) - Ethyl Acetate (2) - n-Propanol (3). s- BUTANOL _y/\ A A A A /A T x T A A AA A GdH20 20 C ETH YL H20 20 ~C ETHYL ACETATE Figure 36. H20 (1) - Ethyl Acetate (2) - s-Butanol (3).

-95 - i -BUTANOL - 0- -- " LEXPERIMENTAL X- CALCULATED Water Phase Points are Given in AA Table XVI AAV v \// /X79 V /V\//\ \A/\/\/ \ H20 20 ~C ETHYL ACETATE Figure 37. H20 (1) - Ethyl Acetate (2) - i-Butanol (3). n - BUTANOL H20 20 ~C ETHYL ACETATE Figure 38. H20 (l) - Ethyl Acetate (2)- n-Butanol (5).

-96 - TABLE IV* REDLICH-KISTER COEFFICIENTS H20(1) - ETHYL ACETATE(2) ~ ROH(3), 20~C ROH Boiling B 23 B31 C12 C23 C31 / Method Pt. ~C Methanol Ethanol i-Propanol t-Butanol n-Propanol s -Butanol - i-Butanol n-Butanol 64.7 78.4 82.5 82.9 97.8 99.5 107.5 117 1.426 1.403 1.427 1.460 1.488 1.452 1.248 1.265 -2.612 -4.329 -3.6 7 -4.121 -3.666 -2.912 -1.458 -1.411 -2.838 -4.602 -5.244 -5.925 -4.910 -3-503 -1.724 -1.734 0.4677 0.4345 0.4017 o.4610 0.2967 0. 3601 0.4336 0.4296 1.005 1.642 0.8167 0.8599 0.8540 0.9161 1.201 1.219 -1.074 -1.758 -2.794 -3.282 -2.782 -2.527 -2.175 -2.050 0.395 1.605 0.0916 0.179 0.1005 1.685 7.161 1.621 S.D. S.D. Trunc. Trunc. Trunc. S.D. S.D. S.D. TABLE V* REDLICH-KISTER COEFFICIENTS (H20)8(1) - BENZENE(2) - ROH(3), 25~C ROH Boiling w B12 B23 B31 C12 C23 C312 A/ /5 Method Pt.~C A/l/ 5Mto Ethanol 78.4 2.782 0.8246 -0.9976 -0.5004 0.0150 0.0008 6.35 S.D. i-Propanol 82.5 2.823 0.2890 -0.2538 -0.5035 0.3741 0.2460 0.724 S.D. t-Butanol 82.9 2.773 0.0814 0.1848 -0.4747 0.5981 0.5246 8.60 S.D. n-Propanol 97.8 2.802 0.0000 0.0954 -0.4870 0.4553 0.3562 0.585 S.D. i-Butanol 107.5 2.778 0.4717 1.140 -0.5096 0.2163 0.3165 15.21 S.D. n-Butanol 117 2.754 0.4317 1.148 -0.5113 0.1438 0.2297 15.48 S.D. *A(4.29) divided by N, the number of experimental tie lines.

-97 - In those systems where there is only one partially miscible binary the fit of the data is excellent. The fits obtained for the systems having two partially miscible binary systems is good near the H.O- ethyl acetate binary but rather poor near the alcohol-water binary. This suggest the possibility that more than the B and C binary coefficients are required to correlate an entire ternary having two partially miscible binaries. Table VI gives the least-squares coefficients used to determine the calculated curves in Figures 31-38. It also gives the alcohol boiling point, A divided by N, N being the number of experimental tie lines, and the method by which A was minimized (steepest descent or truncated Taylor series). Note that in those cases where A was minimized by the truncated Taylor series method, one is fairly sure that A is a minimum. Where convergence using the truncated Taylor series was not obtained and results using the method of steepest descent must be relied upon, it is difficult to be sure that the minimum has been reached or if a region of exceptionally slow convergence is being traversed, such as a trough or saddle point. It is possible that this could explain the fact that the deviations from the experimental data are greater in the cases of the systems having two immiscible pairs as what appears to be A, was reached by steepest descent for these cases. 2. Water-Benzene-Alcohols: The water-benzene-alcohol series constitutes the second longest series of experimentally determined systems, six systems in all.- With the exception of water-benzene-n-propanol (38~C) all were determined at 25~C.

-98 - ETHANOL ----- EXPERIMENTAL -X CALCULATED */\AA /V\A/\AA7 (H20)8 25 ~C BENZENE Figure 40. (H20)8 (1) - Benzene (2) - i-Propanol (3).,AAX/ \ A (HO),25 OA 7VAAA,/v^7\A/v08 1)-Bezee(2 -iPrpv o ()

-99 - t - BUTANOL --— 0 --- —- EXPERIMENTAL -=( --- — CALCULATED (H20)8 25 ~C BENZENE Figure 41. (H20)8 (1) - Benzene (2) -t- Butanol (3). /\\,/\A \A N — 7/A (H20)8 Figure 42. (H20)8 - Benzene (2) - n- Propanol (3).

-100 - i- BUTANOL ------ EXPERIMENTAL /-'- X CALCULATED (H20)8 Phase Points are Found in Table XVII /\ (H20)8 25~C BENZENE Figure 43. (H20)8(1) - Benzene (2) i-Butanol (3). n - BUTANOL x X,7, (H20)8 250C BENZENE 28 ~ ///\//\A Figure 44. (H20)8(1) - Benzene (2) n-Butanol (5).

-101 - Because of the extent of the miscibility gap in the benzene-water binary these systems may be considered to be a severe test of the ability of Equations (3.10), (3.11), and (3.12) to fit ternary liquid-liquid equilibrium data. It was found that using 18 as the molecular weight of water (i.e. assuming a monomolecular specie) gave equilibrium curves for the benzenewater systems that could not be fitted accurately by the equations. Assuming water to exist as an octa-molecular specie, equilibrium curves were obtained that were readily fitted (see Figures 39-44). Excellent fits of the experimental data were obtained in the cases of i-propanol (Figure 40) and n-propanol (Figure 41). The fits were not as good for the other systems in the series, but certainly good enough for engineering purposes. In the cases of ethanol and t-butanol (Figures 39 and 41) the error of the calculated curve is greatest near the plait point. Note that in the cases of i-butanol (Figure 43) and n-butanol (Figure 44), the error in predicted values of x4t and x (the points denoted +) is fairly large, A (4.29) having been minimized and (") denoting the benzene phase. If (") is taken to denote the water phase and L (4.29) minimized the error in x3 is less than four mole per cent. Thus, although the quotient r given in Tables IV-VII is a measure of the relative goodness of fit, it is often misleading due to the arbitrariness of the definition of t. Either way it must be concluded that the error of fit is greater for the systems having two partially binaries than for the systems having only one partially miscible binary.

-102 - Table V gives the values of the least squares coefficients determined by minimization of A (4.29), the method used, and the alcohol boiling point. 3. N-Paraffins-Sulfur Dioxide-Benzene: (35) The data of Satterfield et. al. on hydrocarbon-sulfur dioxide (liquid) systems provide a series of systems in which the variable component, an n-paraffin, is partially miscible with liquid sulfur dioxide and completely miscible with benzene. It was found that assuming sulfur dioxide to exist as a polymolecular specie, (SO0), gave equilibrium curves upon conversion to mole fraction that were readily fitted by Equations (3.10), (3.11), and (3.12). Equilibrium curves based on the molecular weight of monomolecular SOZ could not be fitted by the equations. Multiplying the molecular weight of a specie by some factor has the effect of increasing the area under the binodal curve which appears to be helpful from a curve fitting standpoint as very flat binodal curves which subtend a small area have not been found to be well fitted by Equations (3.10), (3.11), and (3.12). Often as in the cases of (HO'0 and (SO.) there is some theoretical justification of the use of polymolecular weights. Because of the polarity of these molecules, the existance of associated species is more plausible than the existence of monomolecular H20 and SO, in the liquid state. The fit of the data for the system: n-Butane-(SO)8 -Benzene (Figure 44) is particularly good. As the molecular weight of the paraffin and the immiscibility in the paraffin-(SO^) binary increase, the error of the calculated curve in the neighborhood of the plait point increases, although not seriously.

-103 - BENZENE ----- EXPERIMENTAL ------ CALCULATED X/ n - BUTANE -20 OF (S02)8 Figure 45. n-Butane (1) - (S02)8 (2) - Benzene (3). (S02)8 Figure 46. n-Hexane (1) - (S02)8 (2) - Benzene (3).

BENZENE --- _ --- EXPERIMENTAL s -( ---- CALCULATED X \/V / /\0~~V n-HEPTANE -20OF (SO2)8 Figure 47. n-Heptane (1) - (S02)8 (2) - Benzene (3). BENZENE n-DECANE -20~F (SO2)l Figure 48. n-Decane (1) - (S02)8 (2) - Benzene (5).

TABLE VI* REDLICH-KISTER COEFFICIENTS N-PARAFFINS(1) - (S02)g(2) - BENZENE(3), -20~F N-Paraffin Boiling B1 B2 B C C C03 Method 12 23 31 12 C23 C31 Pt. o C n-Butane -0.6 0.9402 -0.4397 0.1217 0.4986 -0.2316 -0.4750 0.509 Trunc. n-Hexane 69.0 1.289 -0.0013 0.6041 0.2572 0.0002 -0.1044 1.089 S.D. n-Heptane 98.4 1.389 -0.0001- 0.6503 0.2190 0.0000 -0.0002 1.540 S.D. n-Decane 174. 2.095 -0.0203 0.7046 -0.1409 0.0048 0.0343 2.810 S.D. TABLE VII* REDLICH-KISTER COEFFICIENTS N-PARAFFINS(1) - H20(2) - METHANOL(3), 15~C N-Paraffin Boiling B12 B23 B31 C1 C2 C31 X/o Method Pt. OC n-Hexane 69.0 4.532 0.1756 1.027 0.0000 0.0501 -0.0037 242. S.D. n-Heptane 98.4 5.046 0.1674 1.160 0.0000 0.0508 0.1806 300. S.D. n-Octane 125.7 5.588 0.1745 1.182 0.0000 0.0495 0.1976 555. S.D. n-Nonane 150.5 6.100 0.1701 1.337 0.0000 0.0511 0.1514 236. S.D. * A(4.29) divided by N, the number of experimental tie lines. I 0 \J

-106 - Table VI gives the least squares values of the coefficients used in computing the calculated curves found in Figures 45-48. 4. N-Paraffins-HO-Methanol: In this series the variable component, an n-paraffin, is partially miscible with both of the other two components. These systems, provide probably the most severe test of the ability of Equations (3.10), (3.11), and (3.12) retaining only the binary B and C coefficients to represent equilibrium data, first because of the very high degree of immiscibility of n-paraffins with water, and second because there are two binaries with miscibility gaps. Table VII gives the values of the coefficients calculated by Program 8 and used to compute the calculated curves in Figures 49-52. The fit of the data is considerably better than one would conclude on the basis of A/N given in Table VII. The calculated binodal curves for these systems are in close agreement with those determined experimentally. There is some error in the predicted tie lines. The situation is that encountered in the cases of i-butanol-(H20)8 -benzene and n-butanol-(H 0) -benzene where if one minimized A (4.29), calling (') the hydrocarbon phase, as was actually done, the predicted values of xi, and x3 indicated as + in Figures 49-52 are in considerable error for every xX = C except near the binary regions. However, if (') designates the water phase the error in x!/ and x1 (now mole fractions of HO0, methanol in the hydrocarbon phase) is very much less. The cause of this error is the fact that the tie lines are pinched low in the hydrocarbon corner, a situation that apparently Equations (3.10), (3.11) and (3.12) cannot handle. It is recommended that the use of additional coefficients in the representation of systems having two partially miscible binary systems be studied.

-107 - METHANOL EXPERIMENTAL - -*t -- CALCULATED \/V\/\/ VB\/V N-H EXANE 152 C H0 Figure 49. N-Hexane (1) - H20 (2) - Methanol (3). METHANOL /\/\/\\/V\A7/\A/ N-HEPTANE 15~C H20 Figure 50. N-Heptane (1) - H20 (2) - Methanol (3).

METHANOL ----- EXPERIMENTAL -— X — CALCULATED N - OCTANE 15 C HO20 Figure 51. N-Octane (1) - H20 (2) - Methanol (3). METHANOL V\AAA/ 7vA20 FN-NONANE 15) C H20 Figure 52. N-Nonane (1) - H20 (2) - Methanol(3).

-109 - C. Correlation of Coefficients It now remains to be shown whether or not for a homologous series of systems the coefficients in Equations (3.10), (3.11), and (3.12) can be correlated with some property of the variable component, k. If molecular weight or number of carbon atoms is chosen as the correlating property, then isomers must be excluded from our definition of a homologous series. In order that isomers be included, the boiling point (at 1 atm.) has been chosen as the correlating property. Figures 53-56 are plots of B.. and C.. versus variable component boiling point for the four homologous series of systems discussed in the preceeding section. The least squares values of the coefficients given in Tables IV-VII have been plotted. For the series of systems H.0 (l)-Ethyl Acetate(2)-Alcohols(3) at 20~C (Figure 53) B& and C/a are nearly constant, as might be expected. The coefficients for the ethyl acetate-alcohol binary (B3, C.. ) and the alcohol-water binary (B3, C,, ) show some scatter. In the case of B B, Bs and C. lines were drawn only through the points for normal alcohols. It appears that some property other than boiling point as the correlating property is required if iso-alcohols are included in the homologous series. It is recommended that additional work be carried out on this subject. The scatter of the points may be attributed in part to inaccuracies of the data for it is true that equilibrium data are difficult to obtain. Since the vapor phase is usually not analyzed in liquid-liquid equilibrium determinations, it is generally not possible to check the data

-110 - 2 lG-0 - 0 - CI C2 i-C3 t-C4 n-C3 S-C4 i-C4 n-C4 -I / /x -2 - -3 - %N 0 X --- /.dP 0 o / / / -4 - -51 - I I I I 6 -60 70 80 90 100 110 120 ALCOHOL BOILING POINT, ~C 2. & - -0 C ~0 --- —-— (o O0 Cij -2 - x — x< X -..0 - 00010. X10 00-1190 X THE DASHED CURVES GO THRU POINTS FOR NORMAL ALCOHOL ONLY. -4k B,2 OR C12 0 B23 OR C23 * B3, R C31 I -61 60 I I 70 80 ALCOHOL 90 BOILING 100 POINT, ~C 110 120 Figure 53. Binary B and C Coefficients Versus Alcohol Boiling Point for the Homologous Series of Systems H20 (1) - Ethyl Acetate (2) - Alcohols (3) 20~C.

-111 - 3 2 Bij I 0 -I 70 80 90 100 110 120 ALCOHOL BOILING POINT, ~C 130 I 0 x Cij 0 F r\ B,2 OR C12 0 B23OR C23 0 I I i B3110R C31, X -I 70 80 90 ALCOHOL 100 BOILING 110 POINT, ~C 120 130 Figure 54. Binary B and C Coefficients Versus Alcohol Boiling Point for the Homologous Series of Systems (H20)8(1) - Benzene (2) - Alcohols (3) 25~C

-112 - 2.0 I.0 Bij 0 -I.0 -20.,..x...........X W 00 000000 I n-C4 I n-C6 n-C7 I I I I I n-Cio I I I I 0 20 40 60 80 100 120 140 160 180 200 PARAFFI N BOILING POINT,~C B12 OR C12 0 B23 OR C23 0 B31 OR C31 X 0.5 - Cij 0 X11 x y I I I I -0.5 L -20 I I I I I 0 20 40 60 80 PARAFFIN BOIL 100 120 140 160 180 200.ING POINT, ~C Fig-ore 55. Binary B and C Coefficients Versus Paraffin Boiling Point for the Homologous Series of Systems N-Paraffins (1) - (S02)8 (2) - Benzene (3).

-113 - 6 4 2 0 61 B12 OR C12 0 - B23 OR C23 ~ B31 OR C31 X n-C6 n-C7 n-C8 n-C9 -x- - x -- x - X - L0 80 100 PARAFFIN 120 BOILING 140 POINT, ~C 160 0.2 - w w w Cij 0 / '0 — Iqlp — -0.2 L 60 I I I I I 80 100 PARAFFIN 120 BOILING 140 POINT, ~C 160 Figure 56. Binary B and C Coefficients Versus Paraffin Boiling Point for the Homologous Series of Systems N-Paraffins (1) - H20 (2) - Methanol (3).

for thermodynamic consistency as it is in vapor-liquid equilibria determinations. However, liquid-liquid determinations appear to be less prone to experimental error than vapor-liquid determinations. In the only case found where two independent workers determined the same liquid-liquid systems at exactly the same temperature (benzene-water-ethanol at 25~C) '51) the data were found to be in close agreement. Another factor which may have contributed to the scatter of the points is that for some systems as few as six experimental tie lines are available for determination of sample estimates of the coefficients. The raw experimental data were used in all cases, and no points were read from smoothed curves. In all four series of systems discussed here the precision of the data is good, and the tie lines are located so as to satisfactorily describe the equilibrium curves through out their entire ranges. In view of this it is doubtful that additional experimental data would render the sample estimates of the coefficients to be significantly different from the values already obtained. Figure 54 for the series (H O\ (l)-Benzene(2)-Alcohols(3) shows that, as in the case of HzO-Ethyl Acetate-Alcohols, the coefficients for the (HO) -Benzene pair (Bp and C/ ) are virtually constant and have values close to those predicted from binary solubility data. The points for the 2-3 pair (benzene-alcohol) and the 3-1 pair (alcohol-(HO) ) are scattered but have a definite trend. The shortcoming of the use of boiling point as the correlating variable is noted in the cases of two alcohols, iso-propanol and t-butanol with benzene-(H O). Their boiling points being R 9

-115 - t-BUTANOL ----- EXPERI MENTAL 85 -- -- -- CALCULATED (H20)8 -20~F BENZENE BENZENE n-HEPTANE -20~F (SO2)8 Figure 57. Comparison of Predicted and Experimental Equilibrium Curves for Two Systems Using Coefficients Taken From Homologous Series Correlation.

-116 - close together (82.5~C and 82.9~C) would cause one to predict that they have nearly identical equilibrium curves which is not the case. Their equilibrium curves are similar but by no means identical. Far less scatter of the points is observed in Figure 55 for n-paraffins(l)-(SO, ) (2)-benzene(3) and in Figure 56 for n-paraffins(l)-HIO(2)methanol(3). In the case of the (SO2) -benzene systems coefficients for the (S2)g -benzene pair (B2 and C23) are not constant as might be expected. On the other hand, for the methanol-water systems the coefficients for the water-methanol pair (B3 and Cu ) are very nearly constant. No reason for the fact that B23 and C. are not constant in the (SO, ) -benzene case can be given. The use of Figures 53-56 to predict equilibrium is illustrated in Figure 57. The coefficients for the two systems given were taken from the smooth curves in Figure 54 and Figure 55. For (H20) (l)-Benzene(2)t-Butanol(3) the predicted coefficients are: Bi = 2.80, B3 =0.130, B3 = -0.180 C, = -0.50 C3 =0.38, C = 0.250 Except near the benzene-(HO)8 binary the calculated equilibrium curve is not in good agreement with the experimental curve as can be seen by comparison of the convolute lines. This is further evidence of the inadequacy of boiling point as the correlating property where isomers are involved. The coefficients read from the smoothed curves through the points in Figure 55 and used to calculate the equilibria for n-heptane(l)-(SO,) 6

-117 -(2)-benzene(3) are: Be = 1.500, B23 = 0.000, B3 = 0.690 C/a = 0.17, CZ 0. = 0.000, C = The fit of the experimental data is good. It may be concluded that where homologous series of systems include only homologs defined in their strictest sense, boiling point is a satisfactory correlating variable to enable prediction of liquidliquid equilibria.

VI. CONCLUSIONS AND RECOMMENDATIONS A. Conclusions 1. The Redlich-Kister Equations (3.10), (3.11), and (3.12) which retain only the B and C binary coefficients are capable of representing a large majority of cases of ternary liquid-liquid equilibria. Systems having one binary with a miscibility gap can be fitted more accurately than systems having two partially miscible binary systems. Systems having small miscibility gaps are better fitted than systems having large regions of immiscibility. 2. Equations of the Redlich-Kister type which are based on representation. of the molar excess free energy are especially useful in predicting the location of critical solution points. 3. Binary coefficients which have been determined from binary data cannot be used to predict accurately ternary liquid-liquid equilibria except near partially miscible binary regions. If the binary B and C coefficients are determined from ternary data, Equations (3.10), (3.11), and (3.12) are capable of highly accurate representation of the data for systems where the miscibility gap is not extremely large. 4. Experimental data for a homologous series of systems i-j-k1, i-j-ka,... i-j-k,, where k1, k2... k, are homologs (not including isomers) can be used to predict the equilibria for a system i-j-k, which has not been -118 -

-119 - determined experimentally. Redlich-Kister coefficients obtained for an homologous series of systems are simple functions of the boiling point or some other property of the variable component k. Boiling point is not a suitable correlating property to use when it is desired to'predict the equilibrium of series of systems including isomers, B. Recommendations 1. Research directed toward the prediction of solution behavior from theoretical considerations should be emphasized. Particular attention should be given to the prediction of the variation of activity coefficients with temperature. 2. The effect of additional coefficients in Equations (3,10), (311)* and (3*12) should be studied..3 The use of the Redlich-Kister equations to predict equilibria for systems having more than three components should be studied, 4* A study of correlating properties other than boiling point should be undertaken to.permit prediction of equilibria for homologous series of systems where isomers are included, 5. A study should be undertaken to compare the effectiveness of several different representations of the molar excess free energy in representing liquid-liquid equilibria.

VIib APPENDICES -120 -

APPENDIX A SUPPORTING DERIVATIONS AND MATHEMATICAL DETAILS 1. Expressions forAGS g (see 2.61) Consider a system composed of r, n2,.-.nk moles of the various components 1, 2, -- k having molar free energies in the pure liquid states 22, G, G0 The total free energy of mixing AG, is: Ie AG, = G -j n; (A.I) where G is the total free energy of the system after mixing. Takin the partial derivativer of AG with respect to n,: A, = G (A.2) From (2,47) it follows* G.= G + _b -(A-3) From (A*l): k-, G= LGM + G + Ak^ (A.4) Thus aG -^7 + G- (A.5) where x, is the mole fraction eliminated by the relation: = kX _ I_ E X,~~~~ -121 -

-122 - Substituting (A,4) and (A,.5) into (A,3) and simplifying~ - A, -: - e;+~ ^,+ (A.6) From (A,2) and (A,6):,IA.. I A5M -/ (Ao7) Similarly for the kth component from (2,46)o neAS G =A^-Z^ G~n _ G(A.8) For a two component system (k =2); LA~~ = A G. -, ^ (A.9) 2, Stability Criteria for the Binary Case: (12) The following derivation is analogous to that found in Section II, D,.where-S-was the quantity involvedo The following phase reaction at constant T P is proposed: I(^,%, )=' (t', IZ^+ 8X, 5 +^5)m*^ ( +A1n G+AG*) TF.PJ

-123 - It follows from (2.82)~ (2.83), (2.84) that for I to be stable: Lnr = e) I 1, Z\ G 7) (A.10) rn '= /+ VI / (A.11) ra z + r"' = 0 (A.12) By Taylor-series expansion and introduction of (A.10), (A.ll), (A,12): LAGG = tA-1+ 7 + 0 2 + / = - / n l x ) /.I Therefore A <? M Ax;^+ * * > 0 o x^ (A.13) Since if AG>,O each term of a Taylor series expansion must be greater than zero: 2 G t,x J > 0 (2.86) 3, The Mathematic of Quadratic Forms: (12) Consider a quadratic form = _ H + 2bhkk + ck 4-h z3 )+ (a c.- 2 a2. z)7

-124 - The table gives the correspondence of the sign of quantities in q to the sign of q. QUADRATIC FORMS q ac-b2 a c* positive-definite + + + Negative-definite' + - positive-semidefinite 0 + + negative-semidefinite 0 indefinite 4-.Supporting Details of the DeriVation of the Binary RedlichKister Critical Relations: (3.20), (32)1). a. Taking the second partial derivative with respect to x.,^ given in Equation (3419) and applying (3-18) one obtains: F of r R~' RT I x ^ [^A^,. I+ X; + 2.3s sl %, Xi I Xi X2.;r31 - 8,L + 6 (I-,)C, (A.4) * c is dependent on a and ac-b2

-125 - Taking the third partial derivative with respect to xl of -^ Rf and applying (3,18) a__- _-_, 2 -s,~ + 6 (1-,)c ] = i 3/3 J /2 C = O 0 (A.15) From (A,15) it follows after rearranging:.303o C,h - )7 -7 ~ y- Z3 ' (3.21) Substituting C,- from Equation (3.21) into Equation (A,14) it follows after rearrangement that: 2. 303B =0[ 1 (3.20).b, The alternate derivation of Equations (3ao20) and (3,21) consists of determining the following limits (see Equations (3.22)): - [L= xx - X.C [ (A.16) 1 %5 Xs A2,A 5 /2;gefX1 X, 2^,-2I A study of Equations (348) and (3-9) will show that the limits in (A,16) lead to the 0/0 case which must be evaluated by ltHospital's rule56)

-126 - // To evaluate these limits x# was first allowed to equal,2 the composition -at the critical point^ then: co~~ C;=A B, -1 X,I';:C a LXi- Lx3) 6 z(X, - L,X,) > X.'i (z1 = '-) (A. 17) (X,": Xz,) After taking the derivative with respect to xl four times of both the numerator and denominator in (A*16), that is n = 4 in (A.17), when xl - xl both numerator and denominator are finite and rnon-zero$ and 'Equations (3,20) and (3 21) are obtained, This result is also obtained by allowing xl — x, then solving for the limit as x -> x by ltHospitalts rule which indicates that the limits in (A1l6) are independent of path. 5, Determination of i where ' is given by (4,29)(55)o _ ______ac__ __ __________ The partial derivatives of A with respect to the RedlichKister coefficients in Equations (3410)o (3oll) and (3,12) denoted here

-127 - as eA, are of special importance in the determination of the coefficients that minimize D. Differentiating. (4.29) with respect to c: aCa L [2 )( % tK(4-//j ' (A.18) The problem now is determination of =, 0 and _ Let us. write Equations (3,10), (3.11) and (3.12) as x 6 Cs_ = 0 = -0 ~~= ~~e~gl. t CR gi2~= where x= 1 - x -x and / =1 - x where xl = 1 -':~. x3 and x/t/ By total differentiation of 7 (2 X, 4,, = I,~ (Al9) Cl 5 C2 ) CI C2 )... C)= 0 C.. ) I~~~~~~~~~~~/ a3 3~~~~~~~~~~~~~~~~~~~~~ C. 4 ' C2 =

-128 - we obtain? all:^~~ 3^j W55~ ac~ ^(A.20) and similar expressions for/andr/d/, which form three equations from which the differentials of x an may be eiminated Noting x2= dX may-be eliminated* Noting.that.now 4 (,', o,, ce.. * C,) z, = 2'(/j, C b c) x # - 7- /3/ ) c, I C;., v I CFO (A.21) and tbus0,3 x 3;%3 + Cz;=I C" L + bx/,/ + C l 7T z (Ao22)

-129 - The partial derivatives n, d and X are then obtained by comparison of the terms in (A.22) with those in the expressions obtained by solving the system of Equations (A,20) for. dx, dx, x dxj. These partial derivatives may be obtained from the matrix equation I } ~ \ 1b \ I-i \ IR b X e - l| (A.23) \~ / \3c/ \ ac The partial derivatives _ _/ are simply - -g -t and -h in Equation (Al19). The partial derivatives ofj,,, and ~ with respect to xi, x.,. and x$ are somewhat lengthy algebraically. They are written out in Fortran compiler notation in the program listing of Prohr gram 8, See the Correspondence of Variables section for the compiler statements It is understood that to obtain O_ ^ a_, A the values of xz x and x (predicted from the equations, not experimental data) are used in the algebraic expression obtained from (A,23)4

APPENDIX B DESCRIPTION OF COMPUTER PROGRAMS In the course of this research a number of computer programs have been written for the IBM 650 and. the IBM 704, Programs for the 650 were written in the Generalized Algebraic Translator Language(0) by B. Arden and R4 Graham of the University of Michigan computer research staffS.The language adheres to the rules of algebra in arithmetic statements. The reader is referred to Reference (10) for details of the language and of input- output format. Programs for the 704 were written in the Fortran language which is described in detail in References (15) and (16). Subroutines used were those available at the University of Michigan Computer Center, September 19594 Input and output format is controlled by the FORMAT statements found in the Fortran program listings 'These statements are described in detail in References (15) and (16)o The eight programs described here were *nt the only ones written during this research4 A number of other programs were written but were found to be less satisfactory than those presented herea -130 -

-131 - PROGRAM 1: Title: Binary Vapor-Liquid Least Squares Program, Constant Temperature, Language and Computer: GAT - IBM 650 Function: Accepts x, y, P data and determines B12 -and C,:in Equations (2,53), (2,59) by mi.nimiz.ationofA where A is: A/\~~~~~~\: =1. L(43zo';,1+A —;a^[^a'X rl.(4.10) =1,,/ +,,o~ ~ where log/o using the B/, Flow Sheet:?, is the binary Redlich-Kister expression for log, -and C/, binary coeff icients cc!, AAPLJ TF)4 - -.~ai 4L e, M=z(2+6), n= (b,c, + c) - - = TI(JC, +,2 C.,) o =, (C," C 2), Fp = f (), LI + A;L h,) O=Z(C (^/^a ^z2- =a u, * b, cI. ba, c., are the multiplyers of B,z&,.in log'i, logG~ respectively,

-132 - Correspondence of Variables; xt = Y1 l, = Y18 x. = Y2 14 = Y19 Y = Y3 m - Y13 Y? -- Yl4 n = Y.14 b! = Y1 o Y16 b =Y12,p Y15 I = Y9 q = Y17 c =oY10 k = 0 P = Y8 Ba = C1 P = Y5 CG, C2 P = Y6 Input: Y5, T Y6 then Yl, Y3 Y8 with O0 1 on last data point. Subroutines: - READ PUNCH, TFIX, FLOATq LOGIO.E

-133 - GAT Compiler Statements: 2 IS HIGHEST STATEMENT NUMBER 300 USED IN SUBROUTINES DIMENSION Y(20) C(2) ~1 READ Y2=1.-Y1 Y4=1.-Y3 Yll=Y2*Y2 Y12=Yl*Y1 Y9=Yll*(-Y2+3*Y1 ) Y10=Y12( Y1-3.*Y2) Y18=LOG10.((Y8*Y3)/(Y5*Y1)) Y19=LOG10. ((Y8*Y4)/ (Y6*Y2)) Y13=Y13+(Y11*Y11)+Y12*Y12 Y14=Y14+(Y9*Y11 )+Y10*Y12 Y15=Y15+(( YY1 Y8)+Y12*Y19 Y16=Y16+( Y9*Y9) +Y10*Y10 Y17=Y17+(Y9*Y18 )-Y10*Y19 GO TO 1 IF IOUO Y20=(Y13*Y16)-Y14*Y14 C1=( (Y15*Y16)-Y4*Y17)/Y20 C2=( (Y13*Y17)-Y15*Y14)/Y20 2 TC1 TC2 Y13=0. Y14=Y13 Y15=Y13 Y16=Y13 Y17=Y13 IO=0 GO TO 1 END

-134 - PROGRAM 2: Title: Binary Vapor-Liquid Back Calculation Program, Constant Temperature, Language and Computer: GAT - IBM 650 Function: Given B/, C,/a P, Pz, this program computes x, 1 y,, P in steps of any specified. /i by solving the Redlich-Kister equations for the binary vapor - liquid case, Flow Sheet: /O L2(B, (-LYB S zi,) c,l ) Y, =/ =2 /o

-135 - Correspondence of Variables: (same as previous program except as follows) = Y9 Y= Y10 P, = Yll P- = Y12 A, = YO Input: Cl, C2, Y5, Y6, YO Subroutines: READ, PUNCH, FIX, FLOAT, GENERAL EXPONENTATION GAT Compiler Statements: 3 IS HIGHEST STATEMENT NUMBER 400 USED IN SUBROUTINES DIMENSION Y(12) C(2) 1 READ 3,Y1IO.,YO,1.0, 2 Y2=3.-Y1 Y9=10.P(Y2*Y2*(C1+C2*(-Y2 N +3.*Y1))) Y10=10.P( Y1*Yl*(C+C2*(Y1- N 3.*Y2))) Y11=Y9*Y5*Y1 Y12=Y1O*Y6*Y2 Y8=Y11+Y12 Y3=Y11/Y8 Y4=Y12/Y8 3 TY1 TY3 TY8 GO TO 1 END

-136 - PROGRAM 3: Title: Binary Vapor-Liquid Least Squares Program, Constant Pressure, Lanuage and. Computer: GAT- IBM 650 Function: Accepts x, yY y T data and determines B, and C,:in Equations (2,55), (2,56) by minimization of A given in Equation (410)o Flow Sheet:,, P,COMP,TE I,, o, ) (see Pso, /) H=l 10 ~ ~ ~ ~ ~ ~ P...,.,,O ) j see. POG. I) )/ ~ Correspondence of Varriablesg (same atprogram 1 except as follows) a I D1 a = D4 bI = D2 ba = D5 c, - D3 a -- D6 t = X1 Input: Dl - D6 Y1, Y3, Xlj 10 = 1 (last data card) Subroutines: READ, PUNCH. FIX., FLQAT, GEN, EXPONENTIATION, LOGIO0 * a, b 7, c are the coefficients in the Antoine equation,

-137 - GAT Compiler Statements: 2 IS HIGHEST STATEMENT NUMBER 645 USED IN SUBROUTINES DIMENSION C(2)X(1)Y(20)D(6) N I(1) 1 READ Y2=1.-Y1 Y4=1.-Y3 Y5=10.P(D1l-D2/(X1+D3)) Y6=10.P(D4-D5/(X1+D6)) Yll=Y2*Y2 Y12=Y1lY1 Y9=Y11-(-Y2+3.*Y1 ) Y10=Y12( Y1-3.*Y2 ) Y18=LOG10.(Y8Y3/(Y5*Y1) ) Y19=LOG10 (Y8*Y4/(Y6Y2) ) Y13=Y13+Yll*Yll+Y12*Y12 Y14=Y14+Y9*Y1 l+YlO*Y12 Y15=Y15+Y11*Y18+Y12*Y19 Y16=Y16+Y9*Y9+Y10*Y10 Y17=Y17+Y9*Y18+Y10*Y19 GO TO 1 IF IOUO Y20=Y13*Y16-Y 14*Y14 C1= (Y15*Y16-Y14*Y17)/Y2 C2=(Y13*Y17-Y15*Y14) /Y2 2 TC1TC2TY8 Y13=0. Y14=Y13 Y15=Y13 Y16=Y13 Y17=Y13 I0=0 GO TO 1 END

-138 - PROGRAM 4: Title: Binary Vapor-Liquid Back Calculation Program, Constant Pressure. Function: Given Bt, C0^ P, x1 and vapor pressure data in the form of Antoine equation coefficients, Program 4 solves the Redlich-Kister equations in B & C binary coefficients for y, and t in steps of any specified x,, from x = 0 to 1l Flow Sheet: READ,A l,, c, 6,, CO PUTE Y, 1 s7#r\ I _ _I /_ I ----- = 2 <_y^^. b. c, # t c 3^ < ).........COMPU....=.. Yc - WI=^;~ 9 n= ' - bB ^, P" J x PXl P0 ovv~ =X4 ol Z=YPR90 S As-^ i p =XOo~ A~ ~o;'c z =3 t=Y 5 *Soli the Antoine equatins or each component for t and ting the a y erage brresponden e:of Variables: xI = X1 P; = X4 Z = Y.9 Ax, = xo a, = D3 t = Y5 xz = X2.b, = -D1 P = Y10 Y3 c, = D2 * Solving the Antotne equati-ojns for each component for t and taking the average

-139 - BM = C1 a. = Z3 C = = C2 bf = -Z1 Y. = Y4 c= =Z2 m = Y6 d = YO n = Y7 y, = Y1 P = X3 ye = Y2 Input: Dl, D2, D3, Zl, Z2, Z3, C1, C2, XO, Y5 Subroutines: READ, PUNCH, FIX, FLOAT, lOEXP., LOGIO. GAT Compiler Statements: 349 USED IN SUBROUTINES 3 IS HIGHEST STATEMENT NUMBER DIMENSION Y(10)X(4)D(3)Z(3) N C(2) 1 READ 3,X10,O.,XO,1.0, X2=1.-X1 Y3=1OEXP.(X2*X2*(Cl+C2*(-X2+ N 3.*X1) )) Y4=1OEXP.(X1*X1*(C1+C2*(X1-3.N *X2))) Y6=Y3*X1/Y1 Y7=Y4*X2/Y1 2 X3=1OEXP.(D1/( Y5+D2)+D3) X4=10EXP.(Z1/(Y5+Z2)+Z3) Y1=Y6*X3 Y2=Y7*X4 Y9=Y1+Y2 GO TO 3 IF YOV(A(Y9-1.)) X3=X3/Y9 X4=X4/Y9 Y5=(D1/(LOG1O.(X3)-D3)-D2+ N Z1/(LOG10. (X4)-Z3)-Z2)/2. GO TO 2 3 TX1TTX1TY1TY5 GO TO 1 END

-140 - PROGRAM 5: Title: Program for Computing B,/ and C,, from Binary Solubility Data. Function: Given x, x/ Program 5 computes B,, and CI, in the RedlichKister equations for a partially miscible binary by solving: Equations (3.8), (3.9), written as L,= X + I C,. Lp X3B,1 + XC, Flow Sheet~ / —\ | RAD l | COM/\PUTE START I x L, 311 LI - Y x 1 xI -Y2 L Y10 x;/X - XCX PNC22 ZX =-Y3 - XY4 X, Y8 X =Y9 Xt = Y3 X = Y9 'X 8 =Y

-141 - Xe = Y4 Y& = Y9 Input: Yl, Y5 Subroutines: READ, PUNCH, FIX, FLOAT, LOG10. GAT Compiler Statements: 2 IS HIGHEST STATEMENT NUMBER 645 USED IN SUBROUTINES DIMENSION Y(14)C(22) 1 READ Y2=1.-Y1 Y6=1.-Y5 Y3=(Y6*Y6)-Y2*Y2 Y8=(Y5*Y5)-Y1*Y1 Y4=(Y6*Y6*(-1.+4.*Y5))-Y2*Y2*N (-1.+4.*Yl) Y9=(Y5**5*(-3.+4.*Y5))-Y1*Y1*N (-3.+4.*Y1) Y10=LOG10.(Y1/Y5) Y11=LOG10.(Y2/Y6) Y12=(Y3*Y9)-Y4*Y8 Y13=(Y10*Y9)-Y4*Y11 Y14=( Y3*Y11 )-Y10Y8 C21=Y13/Y12 C22=Y14/Y12 2 TC21TC22 GO TO 1 END

PROGRAM 6: Title: Solution Program for the Three Component Redlich-Kister LiquidLiquid Equations. Language and Computer: GAT - IBM 650 or Fortran II - IBM 704. Function: Program 6 solves Equations (3,10), (3.11), (3.12) for values of x in steps of any specified Ax up to a specified final value of x / Program 6 begins by solving a point low on the binodal curve where x is very small, An initial guess of x1, x /, xJ is read in initially for -this first points The iteration then begins and solution is assumed when 6 > ~ / ~ ' / 6C being arbitrarily chosen to give the desired accuracya For the second points,[Z] /LQ / /C La D3 / = // y / = C: 7: (where [n: refers to the point number) are taken as initial guesses and the iteration starts again. In order to insure convergence Ax:.should be OoOl or smaller in almost all caseso For the third point and all successive points,initial guesses.are made by linear extrapolation from the last two points: r^17 E-] G Lz-lJ ( = /' 2,) =, Program 6 converges rapidly near a partially miscible binary region,,more slowly near a plait point.For a given C the accuracy of

-143 -solution is lowest near a plait point. For an average ease less than one minute is required to compute a complete binodal curve on the IBM 704. If divergence occurs,or if the logarithm of a negative number is taken during computation, a new set of Redlich-Kister coefficients is read in and computation proceeds. Flow Sheet: / -_.. |IAD 3,2, t32, B c,2 COM1PUTI / \ PT | 5TAf7T^C,3 ^ I L\ ^,, ^\ t~y C 'z^) F --- —- —: --- (7 T Co= ' I MPT / /oil 11 Q X//:/ 9 P/ R/N COLUMN ( t I _____ ^ ____ IC, I L('=/J2,3S) 7 r~~~~- - - `- --- 9~~~~~~~~~~~~~3/ 0 /V\ L) 7, 1z~~~~~~~~~~~ ~)-tb-l ii =~9PIICX ~ ~,4) ~c r~1 I —~tX $ I _ __1___.___Ii- I I Note: Dashed lines enclose solution routine used in Program 8

144..- = %6j (a': / 2) e 6X - 43- & /;b = /-x/, - -or X/ =,6 - / / Z3 4- /1 4= /-%I /xt = /:_," _Z; X 4^^+^~~~~~~~/ /9

Correspondence of Variables: B/ C1 / =Y8 x FY3 2 B = C2 Z = Z17 A'x = X21 B2 = C3 = Z8 X X22 C1z C4 = Z19 Ax, X23 %C = C.5 L, = D8 Ax,^ = X24 C, = C6 L = D9 x = X17 x, = Yl L = D xO X18 x' =Y2 fj = F(J' x, =X9 x' = Y3 g, = G(J) x' = X20 x/ =Y5 hj H(J) A = A x = Y6 = = Y8 x, = Y7 Ax =X Input: See READ INPUT TAPE Statement in program below, Format is in accordance with FORMAT statement 2f A is a divergence test constant normally set greater than one Subroutines Used: LOG (natural log), general exponentation plus the usual Fortran system routines,

-146 - Fortran Compiler Statements: DIMENSION C(6),F(6),G(6),H(6) 1 READ INPUT TAPE 7,2,(C(I),I=1,6),Y,Y3,XFY3,Y1,Y5,Y6,A 2 FORMAT (6E12.4/4E10.2/4E12.4) E=0.43429448 Y2=1.-Y1-Y3 Y7=1.-Y5-Y6 K=l WRITE OUTPUT TAPE 6,11,(C(I),I=1,6) 11 FORMAT (4HlC1=E12.44H C2=E12.4,4H C3=E12.4,4H C4=E12.4,4H C5=E12 X.4,4H C6=E12.4) WRITE OUTPUT TAPE 6,12 12 FORMAT (68H Y1 Y2 Y3 Y5 Y X6 Y7) 19 Y8=1. C26=1.-Y3 C27=1.-Y7 3 C16=1.-Y1 C17=1.-Y2 C18=1.-Y5 C19=1.-Y6 C20=2.*Y1 C21=2.*Y2 C22=2.*Y3 C23=2.*Y5 C24=2.*Y6 C25=2.*Y7 Y1O=Y2-Y1 X10=Y6-Y5 Z10=Y5-Y7 F(1)=Y6*C18-Y2*C16 F(2)=Y2*Y3-Y6*Y7 F(3)=Y7*C18-Y3*C16 F(4)=Y6*(C23*(X10+1.)-Y6)-Y2*(C20*(Y2+C16)-Y2) F(5)=C24*Y7*(Y7-Y6)-C21*Y3*(Y3-Y2) F(6)=Y7*(C23*(Z10-1.)+Y7)-Y3*(C20*(-Y3-C16)+Y3) G(1)=Y5*C19-Y1*C17 G(2)=Y7*C19-Y3*C17 G(3)=Y1*Y3-Y5*Y7 G(4)=Y5*(C24*(X10-1.)+Y5)-Y1*(C21*(Y10-1.)+Y1) G(5)=Y7*(C24*(C19+Y7)-Y7)-Y3*(C21*(C17+Y3)-Y3) G(6)=C23*Y7*Z10-C20*Y3*(Y1-Y3) H(1)=Y1*Y2-Y5*Y6 H(2)=Y6*C27-Y2*C26 H(3)=Y5*C27-Y1*C26 H(4)=C23*Y6*X10-C20*Y2*Y10 H(5)=Y6*(C25*(-C27-Y6)+Y6)+Y2*(C22*(C26+Y2)-Y2) H(6)=Y5*(C25*(Z10+1.)-Y5)-Y1*(C22*(C26+Y1)-Y1) 31 Z17=C( 1)*F(1) C(2)*F(2)+C(3)*F(3) +C(4)*F(4)+C(5)*F(5)+C(6)*F(6) Z18=C(1)*G(1)+C(2)*G(2)+C(3)*G(3)+C(4)*G(4)+C(5)*G(5)+C(6)*G(6) Z19=C(1)*H(1)+C(2)*H(2)+C(3)*H((3)+C(4)*H(4)+C(5)*H(5)+C(6)*H(6) D8=E*LOGF(Y1/Y5) D9=E*LOGF(Y2/Y6) D10=E*LOGF(Y3/Y7) Y7=((Y3/(10.**Z19))+Y7)/2. X8=10.**Z17 X9=10.**Z18 C26=1.-Y3 C27=1.-Y7 Y6=(C26-C27*X8)//X9-X8) Y5=1.-Y6-Y7 Y1=X8*Y5

Fortran Compiler Statements: Y2=X9*Y6 Y8=ABSF(D8-Z17)+ABSF(D9-Z18)+ABSF(D10-Z19) IF(Y8-Y)4,4,32 32 IF(Y8- A )10,1,1 10 IF(Y1/Y5) 1,1,14 14 IF (Y2/Y6)1,1,3 4 WRITE OUTPUT TAPE6,8,Y1Y2,Y3,Y5SY6qY7 8 FORMAT (1H 6E12.4) IF (FY3-Y3)1,13,13 13 IF (K-1)33,25,33 33 X21=Y1-X17 X22=Y2-X18 X23=Y5-X19 X24=Y6-X20 X17=Y1 X18=Y2 X19=Y5 X20=Y6 Y1=X17+X21 Y3=X+Y3 Y2=1.-Y1-Y3 Y5=X19+X23 Y6=X20+X24 Y7=1.-Y5-Y6 GO TO 19 25 X17=Y1 X18=Y2 X19=Y5 X20=Y6 Y3=X+Y3 Y2=1.-Y1-Y3 K=0 GO TO 19

-148 - PROGRAM 7: Title: Linearized Ternary Liquid-Liquid Least Squares Program, Language and Computer: GAT - IBM 650 Function: Program 7 determines the coefficients.in Equations (3.10), (3l11), (3,12) from ternary liquid-liquid equilibrium tie line data for three cases: 1. Where Equations (3,10), (3,11), (3.12) coefficients. 2, Where Equations (3.10), (311), (3,12) B/2, Bt, B 1 and C 3- Where Equations (310), (311), (3.12) C/2. G CR3, and C31" In order that Equations (4,5) be linear in the following A is minimized: retain all six retain only retain only B,,, the coefficients IA - L [/~,-g) (- 7/-. <'=/ The method used in solving the linear simultaneous equations is that of Jordan Elimination and the routine used here was written by Professor B, A. Galler and is described in Reference (8). For reasons discussed earlier, Program 7 does not always give satisfactory estimates of the coefficients. Because of its speed however it is always advisable to try Program 7 and see if it gives good estimates of the coefficients for a given set of data,

-149 -In comparing the flow sheet to the compile~ statements it will be recognized that GAT stores: C = C(1,1), C2 = C(1,2) ---- Ch = C(ln) C(n+l) = 0(21), C(n+2) C(2,2) ---- C(2n) = C(2,n) etco when in the DIMENSION-statement C'(n, n, n3), n3 1. Flow Sheet:

--- q ' L ' UOT.'e OOT U=T a ' ' % ' % ' '0 'v A % S' 'J.UToJJoG0o aqtq qTxa uo fautclnoa uoo^tuT.-Ira u'p1op aqqO pasoaoua SautT paIs'ep at1, ~ao I I-L I i~ ZiL - ~; - -- I1 L - - - - - t - -- '

I),7

-152 - Correspondence o'f Variables: (same as Program 6 except as follows) f, to f, = X1 to X6 b. = C(Il, i1) g, to g6 - Y1 to Y16 - (Il, Jl) h toh to = to Z6 L, to L3 = Dl to D3,n = IO Input: Yl, Y2, Y5^ Y6 with I0 = 1 on last data pdint., Subroutines: READ, PUNCGH. FIX, FLOAT LOGIO.C MATRIX SUBSCRIPTION

-153 - GAT Compiler Statements: 15 IS HIGHEST STATEMENTNUMBER 645 USED IN SUBROUTINES DIMENSION C(42,I2,1)Y(42)I(4)N J(1)K(1)Z(6)X(6)D(3) l ~ READ Y3= 1.-Y1-Y2 Y7=1.-Y5-Y6 Y8=1.-Y5 Y9=1.-Y1 X1=Y6*Y8-Y2*Y9 X2=Y2*Y3-Y6*Y7 X3=Y7*Y8-Y3*Y9 X4=Y6* ( 2*Y5* ( Y6+Y8)-Y6 )- N Y2*( 2.*Y1*( Y2+Y9)-Y2) X5=2.*(Y6*Y7*(Y7-Y6)-Y2*Y3 N *(Y3-Y2)) X6=Y7*(Y7-2.*Y5*(Y7+Y8))- N Y3*(Y3-2.*Y1* (Y3+Y9)) Y8=1.-Y6 Y9=1.-Y2 Y11=Y5*Y8-Y1*Y9 Y12=Y7*Y8-Y3*Y9 Y 13=Y1*Y3-Y5*Y7 Y14=Y5*(Y5-2.*Y6*(Y5+Y8) )- N Y1*(Y1-2.*Y2*(Y1+Y9 ) Y15=Y7*( 2.*Y6*(Y7+Y8 )-Y7)- N Y3*( 2.*Y2*( Y3+Y9)-Y3) Y16=2,*(Y5*Y7*(Y5-Y7)- N Y1*Y3*(Y1-Y3) ) Y8=1.-Y7 Y9=1.-Y3 Z 1=Y 1*Y2-Y5*Y6 Z2=Y6*Y8-Y2*Y9 Z3=Y5*Y8-Y1*Y9 Z4=2.*(Y5*Y6*(Y6-Y5)- N Y1*Y2*(Y2-Y1)) Z5=Y6*(Y6-2.*Y7*(Y6+Y8))-Y2 N *(Y2-2.*Y3*(Y2+Y9)) Z6=Y5*(2.*Y7*(Y5+Y8)-Y5)-Y1 N * ( 2.*Y3 ( Y1+Y9 )-Y1) D1=LOG10.(Y1/Y5) D2=LOG10 (Y2/Y6) D3=LOG10 (Y3/Y7) I1=1 I4=2 12,K1,1,1,6, I2=K1 I3=K1 11 CI1=CI1+XI2*XI3+Y(I2+10)*Y(I3N +10 )+ZI2*ZI3 I3=I3+1 I1=I1+1 GO TO 11IF 6WI3 I1=11+I4 12 14=14+1 I1=7 13=,I2,11,16,2+ Z CI1=C I+XI2*D1+Y( I2+10)*D2+ZIN 2*D3

-154 - GAT Compiler Statements, (Cont' d) TC1...C42 GO TO 1 IF IOUO C8=C2 C15=C3 C22=C4 C29=C5 C36=C6 C16=C10 C23=C11 C30=C12 C24=C18 C37=C13 C38=C20 C39=C27 C40=C34 C31=C19 C32=C26 14,I1,1,1,42, 14 YI1=CI1 I0=6 7 I2=IO+1 4,K 1, 1 1,IO 2J1II2,-1,K1, 2 C(K1,J1 ) =C(K1,J1)/C( K,K1) 4,Il,1,1,IO, GO 10 4IF IlUK1 3,J,I2,-1,K1, 3 C(C(I1,J1)=C(I1,J1)-C(Il,K1)* N C(K1,J1) 4 IO=I0 GO TO 8 IF KOU1 GO TO 5 IF IOU4 C1=C7 C2=C14 C3=C21 C4=C28 C5=C35 C6=C42 6 TC1TC2TC3TC4TC5TC6 C1=Y1 C2=Y2 C3=Y3 C4=Y4 C5=Y7 C6=Y8 C7=Y9 C8=Y1O C9=Yll C10=Y14 Cl=Y15 C12=Y16 C13=Y17 C14=Y18 C15=Y21 C16=Y22 C17=Y23 C18=Y24 C19=Y25 C20=Y28

-155 - GAT Compiler Statements (Cont'd) I0=4 GO TO 7 5 C1=C5 C2=C10 C3=C15 C4=C20 C5=0. C6=0. 9 TC1TC2TC3TC4TC5TC6 C1=Y1 C2=Y4 C3=Y5 C4=Y6 C5=Y7 C6=Y22 C7=Y25 C8=Y26 C9=Y27 C10=Y28 C11=Y29 C12=Y32 C13=Y33 C14=Y34 C15=Y35 C16=Y36 C17=Y39 C18=Y40 C19=Y41 C20=Y42 KO=4 GO TO 7 8 C1=C5 C2=0. C3=0. C4=C10 C5=C15 C6=C20 10 TC1TC23T3TC4TC5TC6 15, I1,1,1,42, 15 CI1=O. I0=0 KO=O GO TO 1 END

-156 - PROGRAM 8: Title: Non-Linear Ternary Liquid-Liquid Least Squares Program Language and Computer: Fortran II.- IBM 704 Function: Program 8 determines the six coefficients in Equations (3.10), (3.11), (3,12) from ternary liquid-liquid equilibrium tie line data by minimization of A defined as: A i/ [ (t Z f / (/- +n (f /- 3) The method of steepest descent described by Marquardt (1) is used to approach the minimum until a good.approximation is obtained after which the truncated Taylor series method is used which near the minimum A coniverges more rapidly than the steepest descent method. Very generally the following steps take place: 1. The experimental data and the initial guess of the coefficients are read in. 2. Equations (310), (3,11), (3,12) are solved for x3 equal to the xj of each experimental point starting with the lowest value of xl by the method of Program 6. 3. \ and the partial derivatives, and are computed. 4. Corrections to the coefficients are computed either by steepest descent or by the truncated Taylor series method. 5, The process is repeated starting with step 2 until > 1 ( -L c CI being arbitrarily small, '- /

-157 -Flow Sheet:

-158 - = /As- A *,, *

-159 -

-160 -

-161 - Jovdapi E/llwmiow o- Lputfie (ekltev dsOhed I/rhe resIon PROG,7)

-162 - Correspondence of Variables (same as Program 6 for B,/ through Ci3 c -=Y,. =Z 2 -= EPS Ax.= X ( = Z20 -- Z17 =- Z18 /= zi7 4: = Zl9 x = X21 AI= X22 Lx.- X23 A'[= X24 Z= Z13 = F(J) x1 x * / h-il Xg xI / x/ xl X/ / 3 x// x -/ 't a A~^ z 1)IE2 = x17 = Xl8 = x19 = X20 = X28 XI 'X2 X3 = X5 = x6 = X7 - X26 = Zll, / */ I x) } XA l /' = X27 = c7 C8O = 01 =C12 * = C13 J7= -10),d.G12 CP;= CL(L) - H(J) X 3 X3., X/ x,, X3 = 014 %* = C15 3' = YlO a = Z4 b = Z5 c = Z6 A = DEL A = DELL; ~h = P2(J) = PB2(J) = PB3(J) (CJ C1e 0= ON(L) D-13 = DL(L) ft; = G(J) Input: 1. C(1), C(2), C(3)t C(4), C(5), C(6) equal to initial guesses of B,, B, B 31 C, 0 CI 0 C3 3 2, Y =: which is greater than | % ' when Equations (3.10), (3411), (3,12) are solved *Those cases where correspondence is obvious are not included.

-163 - 3. Z = 6 which is greater than j (C - Cf ) when the program leaves the steepest descent routine to begin the truncated Taylor series routine.4 4, EPS — 6- which is greater than (Y - c'C when the minimum A has been reached by the truncated Taylor series method, 5. ElM E2 = e, e2 in the expression eA + e (cos) 6,.X = Ax 3 should be less than 001 and may be as low as 0,0005 in some cases to insure convergence, 7, 220= OC should always be negative, -0.2 say, 8, M; the number of tie lines for a given system. 9, NSDP if set equal to 1, causes the steepest descent method to be skipped - uses the truncated Taylor series method only. 10. I2; if set equal to 1, causes printing of Yl Y2, Y3, Y5, Y6, I */ t / '/ // ~ // // t Y7 equal x x2 -3, x/. x 3 x corresponding to every experimental tie line, for every trial of C(1) through C(6). 11. I33 if set equal to 1, causes printing of C7 -; 153 F(J), G(J) H(J) P2(J),PB2(J) P B3(J) D(J), (J = 1 through 6); DEL. 12, I4 if set equal to 1, causes printing of C(1) through C(6)9 DEL, Z20 for every trial of the coefficients. 13,3 I53 if set equal to 1 causes program to read in new data immediately after printing information caused by setting I3 - 1. 14- A; solution routine divergence control constant. Set equal one or greater. 15, x/, x, x x/, x w, we are read in as Yl, Y2, Y5, Y6, WI1 W2, W'3, in order of increasing xv, SubrQutines: LOGF (natural log), SQRTF. (square root), along with the normal FORTRAN, MICHIGAN EXECUTIVE SYSTEM routines

-164 - Fortran II Compiler Statements: DIMENSION CN(6), C(6), CL(6))F(6)- G(6)9 H(6)9 DL(6), R(120),D(6) X, WT(120), T(120),P2(6), PB2(6), PB3(6), B(6,7) 28 READ INPUT TAPE 7,47,(C(I)9 I=1,6)YZ9YEPE1,E29X,Z20,M,NSDI 2,I3, X I4,I5,A 47 FORMAT (6E12.4/7E10.2/614,E10.2) 27 FORMAT (1H1 6E12.4/7E10.2/6I4,E10.2) WRITE OUTPUT TAPE 6,27,(C(I),I=1,6),Y,Z,EP,E1,E2,X,Z20,M,NSDI2,I3 X,I4,I5,A 17 L=1 I=O 1 READ INPUT TAPE 7,29,Y1,Y2,Y5,Y6,W1,W2,W3 WRITE OUTPUT TAPE 6,69,Y1,Y2,Y5,Y6,W1,W2,W3 29 FORMAT( 4E12.4,3F6.0 ) 69 FORMAT (1H 4E12.4,3F6.0) R(L)=Y1 R(L+1)=Y2 R(L+2)=Y5 R(L+3 )=Y6 WT( L)=W1 WT(L+1:') =W2 WT(L+2)=W3 L=L+4 IF(I-M) L,30,30 30 N=1 NN=O 9 L=1 I=l I 1= K=1 D( 1) =. D(2) =0. D( 3) =0. D(4) =0. D(5 )=0. D(6)=0. DEL=O. DO 66 JI=1,7 DO 65 KI=1,6 65 B(KI,JI)=O. 66 CONTINUE Y1=R( 1 ) Y2=R(2 ) Y5=R(3) Y6=R(4) Y3=1.-Y2-Y1 Y7=1.-Y6-Y5 23 X1=R(L) X2=R (L+1) X5=R(L+2) X6=R(L+3) X7=1.-X5-X6 X3=1.-X1-X2 W1=WT(L) W2=WT(.L+1) W3=WT(L+2) J=O E=0.43429448 L=L+4 19 Y8=1.

-165 - Fortran II Compiler Statements: (Cont'd) C27=1 -Y7 3 C16=1.-Y1 C17=1.-Y2 C18= 1.-Y5 C19=1.-Y6 C20=2.*Y1 C21=2.*-Y2 C22=2.*Y3 C23=2.*Y5 C24=2.*Y6 C25=2.*Y C 2 5 = 2. *Y 7 Y10=Y2-Y1 X10=Y6-Y5 Z10=Y5-Y7 F( 1) =Y6*C18-Y2*C16 F ( 2 ) =Y2*Y3-Y6*Y7 F(3)=Y7*C18-Y3*C16 F(4)=Y6*(C234*(XlO+1.)-Y6)-Y2*(C20*((Y+C16)-Y2) F(5)=C24*Y7-*( Y7-Y6)-C21*Y3* ( Y3-Y2) F(6)=Y7*(C23*(ZlO-1.)+Y7)-Y3*(C20*(-Y3-C16)+Y3) G(1) =Y5*C19-Y1'C17 G(2 )=Y7*C19-Y3*C17 G(3)=Y1*Y3-Y5*Y7 G(4)=Y5*(C24*(X10-1. )+Y5)-Y1*(C21*(Y10-1.)+Y1) G(5)=Y7*(C24'( C19+Y7)-Y7)-Y3'*(C21'( C17+Y3)-Y3)?G( 6)=C23*Y7*Z10-C20*Y3 ( Y1-Y3) H(1)=Y1*Y2-Y5*Y6 H(2)=Y6*C27-Y2*C26 H(3)=Y5*C27-Y1-C26 H(4)=C23*Y6*X10-C20*Y2*YlO H(5)=Y6*(C25*(-C27-Y6)+Y6 )+Y2 ( C22 ( C26+Y2 )-Y2) H(6)=Y5*(C25*(Z1lO+l.)-)-5)-Yl*(C22(C26+Y1)-Yl) IF(Y8-Y)4,4,31 31 Z17=C(1)*F(1)+C(2)*F(2)+C(3)*F(3)+C(4), F(4)+C(5)*F(5)+C(6)*F(6) Z18=C( 1)*G(1)+C( 2)*-G(2)+C(3)*G(3)+C(4)*G(4)+C(5)*G(5)+C(6)*G(6) Z19=C(1)*H(1)+C(2)*H(2)+C((3)*H(3)+C(4)*H(4)+C(5)*H(5)+C(6)*H(6) D8=E*LOGF(Y1/Y5) D9=E*LOGF(Y2/Y6) D10=E*LOGF(Y3/Y7) Y7=((Y3/(10.**Z19))+Y7)/2. X8=10.**Z17 X9=10.**Z18 C26=1.-Y3 C27=1.-Y7 Y6=(C26-C27X8 ) / ( X9-X8) Y5=1.-Y6-Y7 Y1=X8*Y5 Y2=X9*Y6 Y8=ABSF(D8-Z17)+ABSF(D9-Z18)+ABSF(D10-Z19) IF (Y8-A ) 32,16,16 16 IF (N-l) 28,28,11 32 GO TO 3 4 IF (K-1)33i21,33 33 X21=Y1-X17 X22=Y2-X18 X23=Y5-X19 X24=Y6-X20 X.17=Y1 X18=Y2

-166 - Fortran II Compiler Statements: (Cont'd) X19=Y5 X20=Y6 X28=X+Y3 IF(X28-X3)26922,22 26 Y1=X17+X21 Y3=X28 Y2=1.-Y1-Y3 Y5=X19+X23 Y6=X20+X24 Y7=1.-Y5-Y6 IF ( J ) 23,19,23 22 X26=X3-Y3 X27=X26/X Y1=Y1+X27*X21 Y2=Y2+X27 *X22 Y5=Y5+X27*X23 Y6=Y6+X27*X24 Y3=X3 Y7=1.-Y5-Y6 K=1 GO TO 19 21 C16=C(1)-C(2) C18=C(3)-C(2) X9=0.43429448 Y4=X9/Y5 C7=(-X9/Y1)+C(1)*(YlO+l. )+Y3*C18+2.*(C(4)*((-YlO)*(Y10+1.) X+(C20-1.)*Y2)+Y3*-(C(5)-*(-C21+Y3)+C(6)*(-C20+Y3+1.))) C8=Y4-C(1)*(X1l0+1.)-Y7*C18-2.*(C(4)*((-XlO)*f(XlO+l.)+(C23-1.)*Y6) X+Y7*(C(5)*(-C24+Y7)+C(6)*(Y7-C23+1.))) C9=Y4-Y6*-C16-C(3)*(1.-Z10)-2.*(Y6*(C(4) *( C23-Y6-1.)+C(5)* (C25 X-Y6))+C(6)*(Z(Zl(Zlu-1.)+Y7*( 1.-C23))) ClO=X9/Y2+C( 1)*(Y1-1. )+Y3*C18+2.*(C(4)*( (-YlO)*(Y10-1,)+Y1 X*(C21-1.))+Y3*(C(5)*(Y3-C21+1.)+C(6)*(Y3-C20)) ) Cll=(-X9/Y6)-C(1)*(X10-1.)-Y7*C18-2.*(C(4)*((-XlO)*(X10-1.)+Y5*( XC24-1.))+Y7*(C(5)*(Y7-C24+1.)+C(6)*(Y7-C23))) C12=C16*C19+C(3)*Z10-2.-(C(4)*(Y5*(-C19)-Y6*'(X1O-1.))+C(5)*(Y7 X*(-C.19)+Y6*-(Y7+C19))+C(6)*( ZlO*Z 1-C23Y7 ) ) C13=C((1)*Y10-C18*C26+2.*(C(4)*(C2(0*Y2-Y10*YlO)+C(5)*( Y2 X*C26+Y3*(-Y2-C26))+C(6)-( Yl1C26-Y3*(Y1+C26))) C14=C( 1) (-XlO)+C18*C27-2* (C(4)*(C23*Y6-XlO*XlO )+C(5)*(Y6*C27 X-Y7*(C27+Y6))+C(6)* (Y5*C27-Y7*(Y5+C27))) C15=(-X9/Y7)-Y6*C16+C(3)*(Zl1+1.)-2.* (Y6S(C(4)* (C23-Y6)+C(5)* X(C25-Y6-1.))+C(6)*(ZlO*(Z1+l1.)+Y5;*(1.-C25))) C16=C9*C11-C12-'C8 C17=C12*C14-Cll*C15 C18=C15*C8-C14*C9 C19=C12*C7-C10*C9 C20=C10*C15-C13*C12 C21=C9*C13-C7*C15 C22=C10C8-C7*C 11 C23=C13*C11-C1O*C14 C24=C14*C7-C13*C8 Y10= (C15 —C22 ) +C9*C23+C12*C24 IF (ABSF(Y1O)-0.000001) 39,39972 72 Z4=Y2-X2 Z5=Y6-X6 Z6=Y7-X7 DEL=DEL+Z44*Z4*W1+Z5*Z5* W2+Z6*Z6*W3 DO 5 J=1,6

-167 - Fortran II Compiler Statements: (Cont'd) P2(J)=(H(J)*C16+F(J)-C1C7+G(J )*C18 )/YlO PB2 ( J ) = ( ( J) *C19+F ( J ) C2.+G ( J ).C21 )/Y 10 PB3 ( J ) = ( J) C22+F ( J )*C23+G ( J ) C24)/Y10 5 D(J)=D(J)+2.* (Z4*P2 (J)- *W1+Z5'PB2 J )*W2+Z6*PB3 ( J) *W3) T(I1)=Y1 T (I1+1 )=Y2 T(I1+2)=Y5 T (I1+3)=Y6 I 1=I 1+4 IF(I2-1) 41,39,41 39 WRITE OUTPUT TAPE 6,40,Y1,Y2,Y3,Y5,Y6,Y7,N,,NN,Y10 40 FORMAT (4H Y 1=E 11.4,4H Y2'=E11.4,4H Y3=E1.14,44H Y5=E11.4, X 4H Y6=E11.4,4H Y7=E11.4,4H N=I3,4H NN=13,5H Y10=E11.4 41 IF (I3-1) 57,42 57 42 WRITE OUTPUT TAPE 6,43,C7,C8,C9,C1lU,C1,C12,C13,C14, X C15, (F(IA),IA=l,6),(G(IA) IA=1, 6),(H( IA) IA=1,6) X, (P2( I A), IA=1 6 (P 2 IA) IA=1 96), ( PB3 I I )A=) 1(IA)6 ) X,(D(IA), IA-1.)6) tDEL 43 FORMAT (1H 9E12.5) IF (I5-1) 57,2'8,57 57 IF (NSD-1)18,48,18 18 IF (I-M)46,24,46 46 K=0 J=l 'f IF (I-1) 34,25,34 34 I=I+1 GO TO 26 25 X17=Y1 X18=Y2 X19=Y5 X20=Y6 Y3=X+Y3 Y2=1.-Y1-Y3 I=2 GO TO 23 24Z 13=0. DO 6 L=1 6 Z14=(1.+C(L)*C(L) )*D(L) Z13=Z13+Z14*Z14 6 D(L)=Z14 Z13=SQRTF(Z13) DO 7 L=1,6 7 D(L)=D(L) /Z13 IF (N-l) 35,8,35 35 IF (DEL-DELL) 36,11,11 11 Z20=Z20/4. DO 10 L=1,6 10 C(L)=CL(L) +Z20*DL(L) GO TO 9 36 COS=D( 1)DL(1)+D(2)*DL(2)+D(3)*DL(3)+D(4)*DL(4)+D( 5)DL(5)+D(6) X *DL(6) 12 Z20=Z20*ABSF(E2+E1*COS*COS*COS) 8 Zll=0. DO 13 L=1,6 CN(L)=C(L)+Z20*D(L) Z10=CN(L)-C(L) 13 Z11=Z11+Z10*Z10 IF (Z-Z11) 37,14,14 37 IF(I4-1) 80,68,80

-168 - Fortran II Compiler Statements: (Cont'd) 80 DO 15L=1,6 CL (L)=C(L) C (L)=CN (L) 15 DL(L)=D(L) DELL=DEL N=N+1 GO TO 9 14 WRITE OUTPUT TAPE 6,38,(C(IA),IA=1,6),DEL,N,NN 38 FORMAT (4H C1=E11.4,4H C2=E11.4,4H C3=E11.4,4H C4=E11.4, X 4H C5=E11.4,4H C6=E11.4,5H DEL=E11.4,3H N=I3,4H NN=I3) IB=4* (M-)+1 DO 70 11=1,IB,4 Y1=T(I1) Y2=T(I1+l) Y5=T(I1+2) Y6=T(I1+3) Y3=1.-Y1-Y2 Y7=1.-Y5-Y6 45 FORMAT(4H Y1=E12.4,4H Y2=E12.4,4H Y3=E12.4,4H YS=E12.4, X 4H Y6=E12.4,4H Y7=E12.4) 70 WRITE OUTPUT TAPE 6,45,Y1,Y2,Y3,Y5,Y6,Y7 IF (NSD-1) 56,28,28 56 NSD=1 GO TO 9 48 DO 61 KI=1,6 J=KI 60 B(KI J)=B(KI,J)+P2(KI)*P2(J)*W1+PB2(KI)*PB2(J)-*W2+PB3(KI)*PB3(J)* X W3 J=J+1 IF (J-6) 60,60,61 61 CONTINUE DO 62 KI=1,6 62 B(KI,7)=B(KI,7)-Z4*P2(KI )*W1-Z5*PB2(KI )*W2-Z6-*PB3(KI )*W3 IF (I-M) 46,49,46 49 DO 63 KI=1,5 J=KI+1 44 B(J,KI)=B(KI,J) J=J+1 IF (J-6) 44,44,63 63 CONTINUE DO 50 KI=1,6 JI=7 51 B(KI,JI)=B(KI,JI)/B(KI,KI) JI=JI-1 IF (JI-KI) 64,51,51 64 DO 50 II=1,6 IF (II-KI) 53,50,53 53 JI=7 52 B(II, JI)=B(I I JI )-B( I I,KI )B(KI JI) JI=JI-1 IF (JI-KI) 50,52,52 50 CONTINUE Zll=0. DO 54 L=1,6 CN(L)=C(L)+B(L, 7) Z10=CN ( L )-C(L) 54 Z11=Z11+Z10 *Z10 NN=NN+1 IF (EP-Z11) 55,14,14

-169 -Fortran II Compiler Statements: (Cont'd) 55 IF(I4-1) 81,68,81 81 DO 71 L=1,6 71 C(L)=CN(L) IF (NN-1) 76,9,76 76 IF (DEL-DELL) 78,28i28 78 DELL=DEL GO TO 9 68 WRITE OUTPUT TAPE 6i 75, (C(IZ) IZ=1,6) DELZ20 75 FORMAT (4H C1=E11.494H C2=E11.4+4H C3=F11.4+4H C4=E11.4, X 4H C5=E11.4,4H C6=E11.4,5H DEL=E11.4,5H Z20=E9.2) IF(NSD-1) 8C081,89

APPENDIX C TABLES OF CALCULATED VALUES APPEARING IN FIGURES For readers desiring to reproduce calculated plots appearing in this thesis with high accurracy, the following tables of calculated values follow. Concentrations (x's) are in mol fraction. -170 -

TABLE VIII* LINES OF CONSTANT FOR THE SYSTEM 2.5305RT B12= 13, B23 =, B31 = 5, C12 = 0, C2 = 0, C = 0 - -0.133 -0.20 -0.24 -0.28 -0.32 2.50 3RT xl X2 Xl X2 X2 X x X X X X2 0.100 0.822 0.100 0.750 0.100 0.710 0.100 0.636 0.100 0.520 0.200 0.720 0.200 0.647 0.200 o.590 0.200 0.518 0.200 0.360 o.300 o.61o 0.300 o. 54 0.300 0.476 0.300 0.373 0.200 0.290 0.400 0.493 0.400 0.416 0.400 0.340 0.300 0.150 0.100 0.300 0.500 o.386 0.500 0.311 0.500 0.210 0.200 0.155 o.6oo 0.290 0.600 0.185 0.500 0.020 0.100 0. 85 0.700 0.195 0.650 0.070 0.300 0.057 0.800 0.100 0.200 0.084 0.100 0.121 *Reference Figure 15. i H I

-172 - TABLE IX* FICTITIOUS SYSTEMS B12 = 1.3 B23 = C12 = C23 = C31 = B3 = -2 B31 = -1 X1 31 X1 ' X2 ' X 2" X1 X2 ' X Xi X2" 0.920 0.779 0.595 0.422 0.290 0.218 0.150 o.o69 0.059 0.071 0.107 0.167 0.223 0.309 0.070 o.o49 0.033 0.033 o.o49 0.070 0.100 0.929 0.941 0.907 0.787 0.626 0.515 0.410 0.873 0.806 0.712 0.553 0.491 0.416 0.076 00o83 o.096 0.135 0.157 0.193 o.o69 0.067 o.o66 0.076 0.082 0.097 0.922 0.909 0.876 0.783 0.731 0.657 B31 = 0 Xi1 X2" 1 211 0.865 0.083 0.083 0.865 0.761 0.107 0.107 0.761 0.676 0.133 0.133 0.676 0.579 0,170 0.170 0.579 0.449 0.239 0.239 0.449 0.378 0.290 0.290 0.378 B3 = 0.87 x x X2 x1 X 1 2 1 2 B31 = 0.5 y t x IY tt? lt X1 ' Xi" X2" 0o871 0.078 oo096 0.781 0.777 0.092 0.146 0.5(o 0.703.o106 0.192 0.473 0.624 0.125 0.250 0.373 0.535 0.154 0.329 0,282 0.487 0.171 0.403 0.221 B3 = 1.0 X Xt X x X1 2 1 2 0.885 0.777 0.728 0.679 0o628 0.604 o.o63 0.065 0,041 0.051 0.020 0.011 0.005 0.101 0.202 0.257 0.318 0.582 0.421 o.656 0.229 0.126 0.059 0.022.oo009 0.925 o.895 0.862 0.836 0.817 0.805 0.075 0.054 0.037 0.022 0.011.o004 0.075 0.099 0.130 0.158 0.180 0.195 0.925 0.584 0.326 0.170 0.077 0.025 *Reference Figures 16-21.

-173 - TABLE X' FICTITIOUS SYSTEMS B12 = 1.3, B31 = 1.0, C12 = C23 = C31 = - B2 = 0.87 B23 = 1.0 X-i 1 X X1 Xi tt x1 x2, Xl xX2" X1 X2 xl X21 o.905 0.075 0.075 0.895 0.905 0.075 0.075 0.905 0.776 0.102 0.121 0.700 0.775 0.105 0.105 0.775 0.686 0.120 0.310 0.295 0.596 0.160 0.160 0.596 o.666 0.104 0.320 0.204 0.596 o.16o 0.325 0.325 0.689 0.076 0.285 0.140 0.585 0.140 0.415 0.190 0.732 0.045 0.250 0.080 o.640 o.096 0.340 0.152 0.780 0.010 0.214 0.018 0.734 0.040 0.242 0.062 0.780 0.010 0.210 0.016 *Reference Figures 22-23. TABLE XI* FICTITIOUS SYSTEMS B12 = 1.3, B 3 B31 = 0.5, C12 = 0.2 c23 = c31 = 0.3 c25 = c31 = 0.5 Xlt X2 X1 x2 XIxl x2 x 0.935 0.042 0.110 0.882 0.935 o.o45 0.110 0.885 0.816 0.061 0.120 0.815 0.815 o.o60 0.105 0.858 0.725 0.085 0.145 0.710 0.655 0.110 0.183 0.599 0.634 0.116 0.224 0.494 0.550 0.133 0.285 0.314 0.543 0.145 0.325 0.301 0.494 0.130 0.257 0.258 0.492 0.165 0.406 0.229 0.480 0.085 0.180 0.191 0.470 0.035 0.136 0.092 0.420 0.000 0.132 0.000 *Reference Figures 24-25

TABLE XII* CRITICAL VALUES OF THE B AND C COEFFICIENTS X1 BC CC Bc C x1 Bc Cc 0.02 - 249.40 - 90.44 0.34 0.747 - 0.229 0.66 0.747 0.229 0.04 - 56,67 - 22.58 0.36 0.782 - 0.181 0.68 0.701 0.276 0.06 - 22.58 - 10.12 0.38 0.809 - 0.156 0.70 0.640 0.328 0.08 - 11.19 - 5.61 0.40 0.829 - 0.125 0.72 0.560 0.392 0.10 - 6.17 - 357 0.42 0.845 - 0.097 0.74 0.453 0.469 0.12 - 3.58 2.46 0.14 - 2.08 - 1.80 0.44 0.856 - 0.071 0.76 0.308 0.565 0.16 - 1.16 - 1.36 0.46 0.863 - 0.047 0.78 0.109 0.688 0.18 - 0.570 - 1.062 0.48 0.867 - 0.022.80 - 0.170 0.848 0.20 - 0.170 - 0.848 0.50 0.869 0.000 0.82 - 0.570 1.062 0.22 0.109 - 0.688 0.52 0.867 0.022 0.84 - l.16 1.36 0.24 0.308 - 0.565 0.54 0.863 0.047 0.86 2.08 180 0.26 0.453 0.469 0.56 0.856 0.071 0.88 - 358 2.46 0.28 0.560 0.392 0.58 o.845 0.097 0.90 - 6.17 3.57 0.30 0.640 0.328 0.60 0.829 0.125 0.92 - 11.19 5.61 0.32 0.701 - 0.276 0.62 0.809 0.156 0.94 - 22.58 10.12 0.64 0.782 0.181 0.96 - 56.67 22.58 0.98 - 249.40 90.44 *Reference Figures 26-27. I I

-175 - TABLE XIII* EXAMPLES OF FITS OBTAINED USING APPROXIMATE LINEARIZATIONS** H20 (1) - ETHYL ACETATE (2) - i-Propanol (3) 20~C xl1 0.9829 0.9639 0.9342 0.9027 0.8683 0.8502 x21 0.0161 0.0191 0.0248 0.0323 0.0427 0.0487 xltt 0.1147 0.1958 0.3639 0.5412 0.6850 0.7594 X2T 0.8807 0.7335 0.4880 0.2785 0.1470 0.0948 H20 (1) - ETHYL ACETATE (2) - Acetone (3) 31~C x ' x1 0.8083.6669 0.5792 0.5052 0.4541 0. 3630 x2t 0.1044 0.1557 0.1835 0.1975 0.1886 0.1697 x It x1 X2 x2 0.0o68 0.0591 0.0591 0.0620 0.0647 0.0677 0.8727 0.8955 0.9055 0.9059 0.9043 0.9018 * Reference Figure 28. * Calculated points only, mole fraction.

TABLE XIV* Bij - Cij FIT OF BINARY X-Y DATA* Ethanol (1) - H20 (2) 74.79~C Ethanol (1) - Ethyl Acetate (2) 1 atm H20 (1) - Dioxane (2) Benzene (1) - Dioxane (2) 25~C 25~C O.050 0.100 0.150 0.200 0.250 0.300 o.400 0.500 o.6oo 0.700 o.800 0.900 o. ooRefer *Reference 0.329 0.451 0.513 0.549 0.573 0.591 0.620 0.651 0.691 0.744 0.814 0.901 0.050 0.100 0.150 0.200 0.250 0.300 0.400 0.500 o. 6oo o.600 0.700 0.800 0.900 o.... 0.0895 0.160 0.217 0.266 0.308 0.345 0.409 o. 468 0.528 0.595 o.680 0.794 0.050 0.100 0.150 0.200 0.250 0.300 o.400 0.500 o.6oo 0.700 0.800 0.900 o.166 0.253 0.303 0.333 0.352 0.364 0.377 0.385 0. 397 0.421 0.474 0.599 0.050 0.100 0.150 0.200 0.250 0.300 o.400 0.500 o.6oo 0.700 0.800 o.900 o.9oo 0.140 0.255 0.350 0.430 0.498 0.557 o.654 0.730 0.794 0.849 0.899 0.949 I 0\ I Figure 29.

-177 - TABLE XV* COMPARISON OF FITS OBTAINED USING COEFFICIENTS DETERMINED FROM BINARY VERSUS TERNARY DATA Experimental x' XI x1 X 1 2 1 2 0.9760 0.0150 0.2520 0.7330 0.9640 0.0170 0.2980 0.6580 0.9400 0.0220 0.3940 0.5070 0.9170 0.0300 0.4650 0.4150 0.8990 0.0380 0.5230 0.3440 0.8840 o.o460 0.5550 0.3100 H20 (1) - Ethyl Acetate (2) - Ethanol (3) 70~C Calc. from Binary xI Xt x X1 x 1 2 1 2 1 0.9734 o.0146 0.2296 0.7447 0.9644 0.9234 0.0186 0.2409 0.6369 0.9448 0.8701 0.0239 0.2612 0.5232 0.9169 0.8151 0.0309 0.2928 0.4118 0.8861 0.7701 0.0379 0.3278 0.3289 0.8600 0.7282 0.0458 0.3674 0.2615 0.8301 0.6788 0.0572 0.4212 0.1961 0.8070 o.6199 0.0741 0.4999 0.1315 0.7797 Calc. from Ternary 2 1 0.0186 0.2805 0.0232 0.3559 0.0311 o.4501 O.0419 0.5241 0.0530 0.5715 0.0679 o.6168 0.0810 0.6502 0.0983 o.6903 x 2 0.6793 0.5667 0.4365 0.3431 0. 2880 0.2391 0.2056 0.1685 xI 0.9500 0. 7890 0.6860 0.6170 0.4600. 3690 0.3220 0.2900 0.2382 0.1388 0.1054 Experimental X2 1 0.0043 0.0002 0.0088 0.0003 0.0265 0.0003 0.0312 0.0006 0.0658 0.0010 0.0990 0.0022 0.1190 0.0027 0.1420 0.0031 0.1821 0.0064 0.3084 0.0130 0.3930 o.0209 x2 0.9900 0. 9550 0.9360 0.9200 0. 8750 0.8450 0.8270 0.8220 0.7820 0.7190 0.6720 Benzene (1) - Water (2) - Dioxane (3) 25~C Calc. from Binary x' x x' x" 1 2 X2 x2 0.9602 0.00065 0o.0016 0.9884 o0 0.7235 0.00280 0.0019 0.9281 0, 0.5179 0.00922 0.0023 0.8677 0, 0.3583 0.02287 0.0026 0.8074 0, 0.2380 0.04655 0.0030 0.7470 0. 0.1491 0.08312 0.0035 o..6865 0. o.o865 0.1348 0.0041 o.6259 0. 0.0640 0.1663 0.0045 0.5955 0. 0.0468 0.2007 0.0049 o.5651 0. 0.0340 0.2371 0.0053 0.5347 0. 0.0209 0.2946 0.0062 0.4888 0. xt.9386.7568.6747.6137.4660.3857.3409.3298.2433 1059 0439 Calc. from Ternary X2 X1 x 0.00161 0.00358 0.9867 0.00445 0.00374 0.9515 0.00703 0.00385 0.9324 0.00989 0.00397 0.9166 0.02274 0.00437 0.8717 0.03605 0.00472 0.8425 0.04682 0.00496 0.8247 0.04999ggg 0.00503 0.8201 0.08422 0.00565 0.7828 0.2102 0.00681 0.7252 0.3701 0.00752 0.6854 *Reference Figure 30.

-178 - TABLE XVI* H20 (1) - ETHYL ACETATE (2) - ALCOHOLS (3) Experimental x' 1 X2 2 xl 1 2 Methanol xl 1 Calculated x x 2t 1A x2 0.9564 0.0178 o.1446 0.8315 0.9268 0.0192 0.1721 0.7581 0.9030 0.0217 0.2151 0.6744 0.8763 0.0252 0.2273 0.6449 o.8192 0.0373 0.3173 0.5077 0.7579 0.0602 0.4022 0.3911 XI XI X11 Xf 0.9652 0.0177 0.1520 0.8150 0.9471 o.0194 0.2062 0.7100 0.9311 0.0218 0.2547 0.6342 0.9092 0.0263 0.2945 0.5688 0.8855 0.0334 0.3962 o.4360 0.8472 o.0496 0.5219 0.3027 xl X XI? X7 1 2 1 2 0.9711 0.0177 0.1736 0.7755 0.9577 o.o196 0.2249 0.6867 0.9477 0.0218 0.2972 0.5775 0.9333 0.0244 0.3867 0.4586 0.9168 0.0289 0.5015 0.3241 0.8986 0.0362 o.6044 0.2245 XI "X X" X1 I 0.9569 0.9232 0.8968 0.8667 0.8028 0.7386 0.6538 0.5795 0.0174 0.1433 0.8225 0.0228 0.1759 0.7491 0.0279 0.2045 0.6899 0.0348 0.2380 0.6254 0.0537 0.3069 0.5065 0.0795 0.3730 0.4076 0.1262 o.4581 0.2997 0.1785 0.5492 0.2049 Ethanol i-Propanol i-Butanol x1 X2 x1 x2 0.9635 o.0194 0.1585 o.8081 0.9428 0.0237 0.2164 0.7075 0.9252 0.0277 0.2728 o.6168. 9019o. 0336 o0.3341 0.5248 0.8785 o.o404 0.3823 0.4572 0.8448 0.0520 o.4386 0.3839 0.8131 0.0649 0.4827 0.3308 0.6988 0.1292 0.6317 0.1814 1 2 1 2 0.9705 0.0183 0.1672 0.7828 0.9566 0.0207 0.2365 0.6708 0.9468 0.0227 0.2927 0.5883 0.9319 0.0258 0.3921 0.4555 0.9162 0.0295 0.5050 0.3219 0.9016 0.0332 0.5969 0.2267 0.8820 0.0571 0.6791 0.1502 0.8500 o.0495 0.7741 0.0872 1 2 1 2 x'xjx x~~x~x~:% 0.9736 0.9674 0.9598 0.9510 0.9454 0.9307 0.0170 0.2027 0.0171 0.2844 0.0172 0.3760 0.0184 o.4916 0.0189 0.5698 0.0222 0.7117 0.7372 0.5998 0.4587 0.3132 0.2346 0.1244 0.9770 0.9702 0.9619 0.9535 0.9481 0.9360 0.9299 0.9195 0.0137 0.1988 0.7295 0.0144 0.2748 0.6117 0.0151 0.3881 0.4546 0.0158 0.5078 0.3077 0.0162 0.5765 0.2326 0.0168 0.6882 0.1279 0.0171 0.7284 o.0966 0.0175 0.7790 o.0634

-179 - TABLE XVI* (CONT'D) H20 (1) - ETHYL ACETATE (2) - ALCOHOLS (3) Experimental Calculated n-Propanol x XI x" XI 1; ";;2 12 0.9710 0.0175 0.1728 0.7606 0.9612 0.0183 0.2749 0.5949 0.9545 0.0193 0.3635 0.4705 0.9448 0.0214 0.4844 0.3232 0.9363 0.0255 0.5662 0.2424 0.9114 0.0313 o.6830 0.1469 1 2 1 2 0.9840 0.0150 0.1350 0.8600 0.9761 o.oi6o 0.2075 0.6750 0.9733 0.0152 0.3163 0.4974 0.9720 o.o146 0.4085 0.3572 o.9656 0.0121 o.4911 0.2457 0.9618 0.0097 0.5746 0.1434 0.9559 o.oo61 o.6503 0.0571 o.9500 0.0001 0.7800 0.0020 Xl x2 1 x2 0.9840 0.0150 0.1350 0.8600 0.9789 0.0161 0.2484 0.6206 0.9782 0.0152 0.3279 0.4631 0.9760 o.0l4o 0.3708 0.3792 0.9756 o.0106 0.4249 0.2614 0.9760 0.0080 0.4574 0.1738 0.9776 o.oo44 0.4766 0.0826 0.9780 0.0001 0.4500 0.0010 x1 X2x1 x2 s-Butanol i-Butanol n-Butanol x1 2 1 2 0.9679 0.0205 0.1620 0.7640 0.9574 0.0221 0.2756 0.5966 0.9508 0.0230 0.3743 o.4635 o.9423 0.0239 0.4895 0.5267 0.9353 0.0246 0.5598 0.2510 0.9164 0.0263 0.6882 0.1536 0.9050 0.0280 0.7395 0.0958 0.8912 0.0298 0.7748 0.0738 xt ~ x x" 1 x2 1 2 0.9819 0.0171 o.1094 0.8749 0.9760 o.ol6 0.2466 o.6150 0.9733 0.0155 0.3319 0.4803 0.9718 o.0148 0.3728 0.4205 o.9655 0.0122 0.5190 0.2310 0.9611 o.o0104 0.5956 0.1486 0.9545 0.0075 o.6830 0.0707 0.9468 0.0042 0.7494 0.0252 XI xt x X 1 2 1 2 0.9761 0.0229 0.1770 0.8143 0.9724 0.0226 0.2624 o.6443 0.9720 0.0214. 3343 0.4903 0.9756 o.0164 0.4173 0.5062 0.9756 0.0107 0.4840 0.1845 0.9764 0.0076 0.5230 0.1262 0.9768 0.0052 o0;5566 0.0823 0.9772 0.0018 o.6064 0.0262 xl xI xi2 1 2 1 2 0.9840 0.9818 0.9816 0.9817 0.9815 0.9817 0.9827 0.9830 0.0150 0.0158 0.0127 o.o116 0.0091 0.0074 0.0059 0.0005 o.1350 0.2634 0.5570 0.4118 0.4632 0.4937 o. 5077 0.5100. 8600 o.6049 0.4527 0. 420 0.2427 0.1578 0.0758 0.0010 0.9768 0.9740 0.9745 0.9755 0.9787. 9804 0.9824 o.9840 0.0222 0.0216 o.0198 0.0177 0.0119 0.0087 0. 0042 0.0000 0.1719 0.2815 0.5555 0.3855 0.4501 0.4857 0.5455 0.6147 0.8181 0.5987 0.4358 0.5601 0.2230 0. 1602 0.0748 0.0000 *Reference Figures 51-38.

-18o TABLE XVII * (E20)8 (1) - BENZENE (2) - ALCOHOLS (3) Experimental Calculated Ethanol X ' 1 x1 1 x2 0.6311 0.0026 0.0007 o.9681 0.4214 0.0073 0.0017 0.9346 0.3259 0.0172 0.0024 0.8964 0.2708 0.0271 0.0043 0.8681 0.1934 0.0774 o.oo60 0.8198 0.1450 0.1252 0.0090 0.7634 0.1114 o.1829 0.0128 0.7101 o.0906 0.2526 0.0176 0.6502 0.0835 0.2545 0.0190 0.6520 0.0598 0.3461 0.0327 0.5247 Xl X2 X x2 0.9294 o.oo40.oo0004 0.9892 o.8664 0.0051 0.0004 0.9802 o.8o64 o.oo0061 o.ooo4 0.9699 0.7649 0.0070 o.0004 0.9571 0.7106 0.0078 0.0004 0.9568 0.6491 o.oo89 0.0003 0.8870 0.6040 0.0097 0.0018 0.8112 0.5770 0.0102 o.oo46 0.7515 0.5498 0.0105 0.0101 0.6725 0.5391 0.0108 0.0151 0.6214 0.5241 0.0120 0.0200 0.5635 0.5082 0.0132 0.0253 0.5154 0.4999 0.0144 0.0321 0.4708 0.4836 0.0168 o.o406 0.4337 0.4738 0.0218 0.0445 0.4175 xI XI x2l X 1~ ~ 2 1 i-Propanol t-Butanol XI XI X xI 0.5757 0.0097 0.0003 0.9685 0.4345 0.0155 0.0003 0.9561 0.3329 0.0239 0.0002 0.8986 0.2794 0.0311 0.0003 0.8722 0.2062 0.0471 0.0003 0.8255 0.1437 0.0714 0.0003 0.7721 0.0995 0.1011 o.ooo4 0.7225 0.0614 0.1457 o.ooo6 0.6672 0.0518 0.1625 0.0007 0.6503 0.0144 0.2941 0.0013 0.5562 *,, X1 X2 X1 X2 0.9277 0.0057 0.0005 0.9933 o.8646 o.oo68 0.0005 0.9836 0.8041 o.0084 o.0005 0.9696 0.7620 0.0099 0.0005 0.9560 0.7059 0.0125 0.0007 0.9308 0.6412 0.0168 0.0009 0.8855 0.5924 0.0213.00oo6 o.8164 0.5626 0.0246 0.0031 0.7419 0.5320 0.0283 0.0067 0.6399 0.5201 0.0298 0.0089 0.6020 0.5042 0.0320 0.0120 0.5581 0.4870 0.0344 0.0159 0.5165 0.4787 0.0356 0.0178 0.4983 o.4624 0.0581 0.0222 o.4645 0.4567 0.0590 0.0257 0.4559 0.4012 o.0488 o.o0413 0.3628 0.2783.0817 0.1001 0.2175 0.2159 0.1091 0.1500 0.1569 Xi X1 X;1 XX 1 2 1 2 *X... -X-, 0.9659 0.9052 0.8278 0.7673 0.7378 0.7187 0.7028 0.6862 0.6482 0.5902 0.5423 0.0031 0.0007 0.9924 0.0039 o.ooi6 0.9758 o.oo46 0.0019 0.9276 0.0054 o.oo66 o.8046 0.0061 0.0180 0.6605 0.0064 0.0333 0.5345 0.0068 0.0536 0.4245 0.0074 0.0836 0.3138 0.0092 0.1352 0.2050 o.o0160 0.2028 0.155333 0.0236 0.2514 0.1048 0.9654 0.9024 0.8233 0.7607 0.7309 0.7117 0.6960 0.6798 0.6437 0.5928 0.5529 0.0056 0.00057 0.0067 o.ooo60 0.0091 0.00078 0.0120 0.00233 0.0131.0oog80 0.0154 0.01368 0.0136 o.01664 0.0137 0.01956 0.0137 0.02610 0.0134 0.05579 0.0130 0.04433 0.9958 0.9832 0.9458 0.8089 0.5762 0.5046 0.4590 0.4191 0.3453 0.2643 0.2127

TABLE XVII*(CONT'D) (H20)8 (1) - BENZENE (2) - ALCOHOLS (3) Experimental Calculated n-Propanol x' 1 X ' 2 xl 1 x"2 2 XI 1 X' 2 xt 1 *11 x2 2X, 0.8624 0.0051 0.0016 0.9482 0.8201 0.0099 o.oo48 0.8796 0.7978 0.0114 0.0042 0.8938 0.7720 0.0112 0.0073 0.8049 0.7057 0.0153 0.0273 0.5822 0.6656 0.0149 0.0439 0.4521 1 _ x2 X2 0.9784 0.00365 0.00038 0.9899 0.9507 0.00505 0.00103 o.9610 0.9272 0.00481 0.00173 0.9054 0.9152 o.oo460 0.00571 0.7861 0.9069 0.00353 0.01109 0.6792 0.9000 0.00316 0.01798 0.5674 0.8919 0.00245 0.02641 0.4445 0.8867 0.00227 0.03535 0.3391 0.8802 0.00191 0.04819 0.2355 0.8661 0.00069 0.06590 0.1155 0.8600 0.00010 0.10000 0.0020 XI Xt X1 X1 1 2 1 2 i-Butanol n-Butanol 0.8596 0.0079 0.0008 o.9440 0.8208 0.0092 0.0010 o0.9048 0.7992 0.0101 o.oo0014 0.8706 0.7720 0.0112 0.0023 0.8040 0.7071 0.0139.oo0096 0.5835 0.6650 0.0155 o.0168 o.4809 0.5192 0.0208 0.0506 0.2620 0.3528 0.0322 0.1174 0.1232 0.2707 0.0443 0.1798 0.0756 XI *X Xl x2 0.9762 0.00585 o.oo00063 0.9724 0.9495 0.00629 0.00095 0.9152 0.9254 o.00663 0.00181 0.8175 0.9131 0.00672 0.00330 0.7134 0.9037 o.oo668 0.00586 0.5976 o.8966 o.00654 0.00844 0.5132 0.8881 0.00628 0.01158 0.4307 0.8829 o.oo606 0.01346 0.3880 0.8764 0.00575 0.01581 0.3401 0.8619 o.oo488 0.02091 0.2505 0.8565 0.00450 0.02283 0.2212 1 2 1 2 1. I o.9601 o. 9490 o. 9416 0.9289 0.9235 0.9181 0.9092 0.9038 o. goo9004 0.8968 0.8933 0.8880 0.8846 0.8782 0.8741 0.8670 0.0036 0.0036 0.0036 0.0053 0.0053 0.0053 0.0071 0.0053 0.0070 0.0053 0.0053 0.0052 0.0052 0.0052 0.0035 0.0001 0.0003 o.ooo0004. 0005 0.0032 0.0086 0.0135 0.0183 0.0237 0.0281 0.0530 0.0385 0. 440 0.0485 0.0747 0.0913 0.1100 0. 9687 0.9489 0.9332 0.8478 0.7295 0.6471 0.5657 0.5029 0.4466 0.3822 0.3311 0.2899 0.2553 0.1155 0.0599 0.0020 0.9573 0.9461 0.9386 0.9276 0.9222 0.9169 0.9099 0.9029 0.9013 0.8961 0.8928 0.8877 0.8845 0.8784 0.8730 0.8641 0.oo64.00 oo65. oo66 o.0066 o. oo66 o.oo65 0. oo64 0.0062 o.oo6i 0.0059 0.0058 0.0055 o.o0053 0.0050 o.oo46 0.0039 0.0011 o.oo16 0.0021 0.0033 o.0044 0.0058 0.0083 0.0112 0.0120 0.0144 o.0160 0.0185 0.0202 0.0233 0.0262 0.0312 0.8948 0.8438 0.8000 0.7148 0.6624 0.6056 0.5292 0.4590 0.4439 0.3976 0.3695 0.3303 0. 067 0.2657 0.3215 0.1807 *Reference Figures 39-44.

-182 - TABLE XVIII* n-PARAFFINS (1) - (SO2)8 (2) - BENZENE (3) Experimental Calculated n-Butane xI 1 x, X1 x2 2 * x' 1 *I xl *I, x2 0.9496 0.0245. 3084 o. 5452 0.9332 0.0267 0.3089 0.5226 0.8958 0.0321 0.3400 o.4040 0.8571 0.0410 0.3719 0.3296 0.8267 0.0479 0.4178 0.2691 0.8011 0.0555 0.4642 0.2318 0.7415 0.0747 0.5893 0.1480 0.7071 o.0891 0.6248 0.1274 x! " " X' XI xl X2 0.9567 0.0203 o.1418 0.7540 0.9120 o.0260 o.1463 o.6009 0.8675 0.0323 0.1670 0.5068 0.8590 0.0329 o.1650 0.4712 0.7940 o.0433 0.1966 0.3731 0.7418 0.0520 0.2469 0.2907 0.7052 o.o6oo 0.2892 0.2458 0.6647 0.0715 0.3737 0.1931 0.6250 o.o840 o.4416 0.1580 0.5740 o.1014 0.5090 0.1261 XI Xt xyl Xi 1 2 1 2 0.9150 0.0249 0.1062 0.6589 o.8314 0.0336 0.1262 0.4602 0.7741 0.0426 o.1604 0.3401 o.6040 0.0737 0.2880 0.1938 0.5176 0.0952 0.3775 o.1463 0.4659 0.1124 0.4173 0.1284 xl "x x X 2 1.2 21 n-Hexane n-Heptane n-De cane 0.9394 0.0356 0.2905 0.5735 0.9246 0.0354 0.3021 0.5149 0.8928 0.0372 0.3370 0.4185 0.8590 0.0409 0.3809 0.3388 0.8294 0.0456 0.4227 0.2808 o.8046 0.0504 0.4595 0.2389 0.7587 o.0613 0.5306 0.1745 0.7234 0.0716 0.5873 0.1349 x1 2x x 2 0.9408 0.0362 0.1366 0.7701 0.8998 0.0382 o.1603 0.6211 0.8589 o.o408 o.1891 0.5025 0.8505 o.o0414 0.1957 o.4811 0.7906 0.0467 0.2474 0.3537 0.7415 0.0523 0.2953 0.-2755 0.7083 0.0569 0.3308 0.2324 0.6735 0.0627 0.3693 0.1951 0.6400 o.0690go 0.4079 0.1645 0.5971 0.0783 0.4825 0.1214 X1.2 1 x2 X 0.9069 0.0330 0o.1149 0.6807 0.8274 0.0376 0.1561 0.4720 0.7754 o.0413 0.1888 0.3713 0.6193 0.0584 0.3104 0.1830 0.5397 0.0731 0.3837 0.1301 o.488o 0.0903 0.3825 0.1313 *it *,, X1 x2 x12 0.9144 0.8378 0.7859 0.7230 O. 6454 0.5581 0.4455 0.0235 0.0278 0.0307 0.0364 0.0441 0.0545 0.0742 0.0267 0.0296 0.0386 0.0639 o.0904 0.1298 0.2287 0.6910 0.5082 0. 3891 0.2973 0.2418 0.1872 0.1364 0.9249 0.8513 o. 8012 0.7425 0.6701 0.5892 0.4888 0.0130 o.o143 0.0154 0.0169 0.0194 0.0233 0.0309 0.0159 0.0358 0.0554 0.0852 0.1320 0.1969 0.2984 0.7388 0.5202 0.4068 0.3011 0.2043 0.1300 0.0716 *Reference Figures 45-48.

-183 - TABLE XIX* n-PARAFFINS (1) - H20 (2) - METHANOL (3) -- -- - - - - - Experimental** Calculated n-Hexane *I *I X1 X, xit X1 X2 x" 1 xt 2 *,, xl - >, xI> 0.9998 0.00003 0.00003 0.9998 0.9813 (0.00014) (0.00061) 0.4164.968 (o.ooo00014) 0.0032 0.3487 0.9530 (0.00009) 0.0080 0.2649 0.9480 (0.00007) 0.0278 0.1544 0.8189.ooooo o.84 0.0000oooo 1 2 x1 2 o.9999 0.000009 0.000009 o 0.9999 o.9664 (0.000027) o.009640 0.1808 0.9661 (o.oooo49) 0.000137 0.3328 o.9400 (o.oooo16) 0.032200 0.0785 0.8189 00000oooooo0 0.069000. 0000 1 x2 1 2 0.9999 0.000003 0.000003 0.9999 0.9824 (o.oooo16) 0.00217 0.3332 0.9789 (.ooo000015) 0. 00276 o. 2604 0.9859 (.0000oooo) 0.00203 0.2271 0.9859- (0.000012) 0.00792 0.1823 0.9754 (0.000009) 0.01142 0.1284 0.9069 (o.ooooo6) 0.02755 0.0527 0.8422 0.000000 0.05800 0.0000 XI X2 x;I XI 1 2 1 2 n-Heptane n-Octane n-Nonane 0.9996 0.000029 0.00003 0.9989 o.9815 0.000033 0.00009 0.8710 0.9684 0.000035 0.00020 0.7711 0.9531 0.000037 0.00053 o.6510 o.9480 0.000037 0.00072 0.6128 0.8189 0.000000 0.1837 0.0000 0.8300 0.000020 O.08100. 0600 Xti Xt X X 1 2 1 2 0.9991 0.000009 0.000009 0.9995 0.9663 0.000011 0.000073 0.7701 0.9661 0.000011 0.000074 0.7679 o.9400 0.000012 0.000442 0.5615.8189 0.000000 ooooo.085230 0.0000oooo 0.8999 o.o00oo0 0.017230 0.1599 * * X,, *,, xl Xt XT1 XI 0.9997 0.000003 0.000003 0.9994 o.9816 o.ooo003 0.000008 0.8881 0.9789 0.000003 0.000011 oooo 0.8652 0.9850 0.000003 o.ooooo6 0.9122 0.9851 0.000003 o.ooooo6 0.9123 0.9755 0.000003 o.0000oooo4 0.8375 0.9069 o.ooooo4 0.002130 0.5179 0.8422 0.000000 0.06647 0.0000 0.8701 0.000003 0.01982 0.1012 1 2 1 2 3X- X-* 3CX-,. 9999 0.9842 0.9619 0.9139 0.8967 0.0000009 (0.0000070) (0.0000055) (0.0000013) 0.0000000 0.0000009. 000904 0.013790 0.022010 o.o45000 0.9999 0.2265 0.0624 0.0410 0.0010 0.9979 0.9829 0.9619 0.9139 0.8967 0.0000009 0.0000008 0.0000009 0.0000002 0.0000000 0.0000008 0.0000050 0.000121 0.025470 0.054230 0.9991 0. 8310 0.5409 0.3786 0.0000 *Reference Figures 49-52. **Parenthesised concentrations are estimated.

-184 - TABLE XX* COMPARISON OF PREDICTED AND EXPERIMENTAL EQUILIBRIUM CURVES (H20)8 (1) - BENZENE (2) - t-BUTANOL (3) X' 1 x' 2 X! 1 X2 2 0.9848 0.9332 0.8230 0.7297 0.6536 0.5366 0.4357 0.3280 0.2542 0.1149 0.0052 0.0058 0.0080 0.0113 0.0154 0.0244 0.0353 0.0530 0.0728 0.1821 0.00050 0.00051 o.000ooo63 0.00108 0.00258 0.01407 0.03851 0.08301 0.1302 0.1118 0.9984 0.9905 0.9527 0.8796 0.7638 0.5128 0.3426 0.2142 0.1479 0.1850 n-HEPTANE (1) - (S02)8 (2) - BENZENE (3) X' 1 X' 2 X" 1 X" 2. 9376 o.8440 0.7490 0.6515 0.5839 0.5315 0.0274 0.0310 0.0360 o.o435 0.0511 0.0585 Fingure 57 0.0725 0.1188 0.1822 0.2626 0.3277 0.4023 0.7885 0.4968 0.3082 0.1889 0.1345 o. 0960 *"Re ference - calculated curves only.

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UNIVERSITY OF MICHIGAN 3 9015 02229 1283 THE UNIVERSITY OF MICHIGAN DATE DUE 1l/22 t ill? 4/1 4 94?t),