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Prediction of non-ideal equilibria for separation operations Groves, William Douglas 1970

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PREDICTION OF NON-IDEAL EQUILIBRIA FOR SEPARATION OPERATIONS by W i l l i a m Douglas Groves B.A.Sc. U n i v e r s i t y of B r i t i s h Columbia, I960 B . S c , U n i v e r s i t y of A l b e r t a , 1962 A Thesis Submitted In P a r t i a l F u l f i l m e n t Of The Requirements For The Degree Of Doctor of Philosphy In The Department Of Chemical Engineering We accept t h i s t h e s i s as conforming to the re q u i r e d standard. THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In present ing t h i s thes is in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I fu r ther agree that permission for ex tens ive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h is r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be al lowed without my w r i t t e n permiss ion . Department of Chemical Engineering The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date December 20th, 1970 i ABSTRACT A r a t h e r general expression f o r the excess Gibbs f r e e energy of a nonionic l i q u i d mixture has been developed i n t h i s t h e s i s j and the expression has been t e s t e d on nine binary mix-tures e x h i b i t i n g a considerable range of nonideal behavior i n both v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a . The b a s i s f o r the f o r m u l a t i o n i s that of the constant c o o r d i n a t i o n number quasichemical mixture theory of Guggenheim 3 s u i t a b l y modified to account f o r the major e f f e c t s a r i s i n g from the nonconstant c o o r d i n a t i o n number s i t u a t i o n inherent i n a mixture of unequally s i z e d c o n s t i t u e n t s . The development of the geometric aspects of the theory i s c o n s i d e r a b l y based on the ideas of Hogendijk f o r mixtures of random-packed spheres. The g e n e r a l i z a t i o n of the theory to multicomponent systems i s based on the work of Barker on a s s o c i a t e d mixtures. Because the expression handles a nonconstant c o o r d i n -a t i o n number system, i t has f o r b r e v i t y been termed the NCZ equation. Although i n many cases the proposed equation performed no b e t t e r than those already a v a i l a b l e f o r c e r t a i n systems, no e x i s t i n g theory was able to deal w i t h a l l of the nine b i n a r i e s on which the NCZ expression has been so f a r t e s t e d . The proposed' method of f o r m u l a t i n g the excess Gibbs f u n c t i o n has brought to the f o r e the need f o r p h y s i c a l chemical measurement of data of a kind not p r e s e n t l y a v a i l a b l e on a large s c a l e , d e s p i t e the f a c t that the r e q u i r e d parameters are i n i i themselves standard thermodynamic q u a n t i t i e s , such as the pure-component fre e energy of a l i q u i d . The present extension of quasichemical theory i s seen as an i n d i c a t i o n of the basic v i a b i l i t y of the q u a s i -chemical type of approach f o r the p r e d i c t i o n of the excess p r o p e r t i e s of l i q u i d mixtures. Professor James S. Forsyth i i i TABLE OP CONTENTS FOREWARD MAIN TEXT CH. 1 CH.2 CH.3 CH. 4 CH .5 CH.6 CE.7 CH.8 CH.9 GENERAL ORIENTATION OF THE RESEARCH INTRODUCTION TO THE THERMODYNAMIC PROBLEM REVIEW OP CONSTANT COORDINATION NUMBER QUASICHEMICAL AND RELATED THEORIES REVIEW OF THE PHYSICAL BASIS OP CELL THEORIES RELATIONSHIPS INVOLVING THE PARTITION FUNCTION NONCONSTANT COORDINATION NUMBER MECHANICS CONFIGURATIONAL ENTROPY EFFECT CALCULATION OF EXCESS FREE ENERGY FOR THE SIMPLEST NCZ CASE EXTENSION OF THE NCZ EQUATION TO THE GENERAL CASE OF MULTICOMPONENT MIXTURES CR.1Q RELATION OF 2-LIQUID THEORIES TO QUASI-CHEMICAL THEORIES C H . l l CH.12 NOMENCLATURE REFERENCES APPENDICES ILLUSTRATION OF THE PREDICTIVE EQUATION CONCLUDING REMARKS Page 1 3 3 19 36 47 58 82 116 124 172 207 277 314 325 334 3 4 l i v APPENDICES Page 1. I o n i c S o l u t i o n Behavior 341 2. Methods of Te s t i n g G Functions Against E x p e r i -mental Data. 343 3. E f f e c t of NCZ Assumption on F l o r y ' s Expression 350 4 . Programming: (a) Logic Diagram f o r NCZ GE/RT C a l c u l a t i o n Program (b) 353 (b) FORTRAN-4 L i s t i n g f o r GE/RT Program (Monofunctional Monomer Binary Case) 356 (c) Table of Correspondences Between FORTRAN and A l g e b r a i c V a r i a b l e s 3 g 2 (d) Logic Diagram f o r NCZ GE/RT C a l c u l a t i o n Program (e) 3 6 6 (e) FORTRAN-4 L i s t i n g f o r NCZ GE/RT versus Composition. (General Case) 3 7 0 ( f ) A d d i t i o n a l Table of Correspondence Between FORTRAN and Al g e b r a i c V a r i a b l e s f o r Program (e) 3 7 8 V LIST OP ILLUSTRATIONS Page 1. Reader Routing Guide . . . 2 2. Thermodynamic R e l a t i o n s h i p s 22 3. The P a i r s - R e o r g a n i z a t i o n Reaction 2 5 4. Coordination Number Change due to Mixing of Unequal Sized P a r t i c l e s • 3 2 5 . Representative Packing Arrangements: (a) Dimer 1 0 1 (b) Short Rodlike Molecule 102 (c) i Short F l e x i b l e C h a i n l i k e Molecule 1 0 4 (c) i i Small n-mer Surrounded by Large Neighbors 1 0 5 (d) P a r t l y Rolled-up C e n t r a l Molecule ±07 6. Bundle-Species 1 3 4 LIST OP FIGURES Lo c a t i o n of Consolute Compositions Perfluoro-n-heptane Carbon t e t r a c h l o r i d e (a) NCZ Component Curves at 40°C. (b) ,(c) Component and Sum Curves f o r Free Energy at 40°C. (d) As above, at 40, 50, 60 and 70°C. (e) Consolute Envelope - C a l c u l a t e d and Experimental P e n t a e r y t h r i t o l t e t r a p e r f l u o r o b u t y r a t e Chloroform (a),(b) Component and Sum Curves f o r Free Energy at 2Q°C. (c) Consolute Envelope - C a l c u l a t e d and Experimental T y p i c a l R e l a t i o n s h i p Between the Terms of the NCZ p r e s s i o n and the Mole F r a c t i o n of the Larger Species Response of I n d i v i d u a l NCZ t-terms to x , _ ^ l a r g e r (a) Term t 1 (b) Term t 2 (c) Term t ^ (d) Term t ^ ' (e) Term t ^ " C f ) Term n-Butane Hydrogen s u l f i d e (a) NCZ Component Curves at 50°C (b) Excess Free Energy - C a l c u l a t e d and Experimental 2-Propanol Water Ca) NCZ Component Curves at 80°C. Caj* NCZ Component Curves at 80°C. (h) Excess Free Energy - C a l c u l a t e d and Experimental 8. Methanol Carbon t e t r a c h l o r i d e vii Page (a) NCZ Component Curves at 3 0 ° C 2 9 8 (b) Excess Free Energy - C a l c u l a t e d and Experimental 2 9 9 9. N i c o t i n e Water ( a ) j ( h ) Component and Sum Curves at a 3 0 2 S e r i e s of Temperatures (c) Consolute Envelope - C a l c u l a t e d and Experimental 3 0 4 10. Benzene S u l f u r , Toluene S u l f u r , m-Xylene S u l f u r (a) C a l c u l a t e d Consolute Envelopes 307 (b) Experimental Consolute Envelopes 3 0 8 (c) C a l c u l a t e d Consolute Envelopes w i t h Radius Ratios Interchanged 3 1 1 (d) Energy Parameters P l o t t e d against Temperature 3 1 2 v i i i LIST OF TABLES Page 1. S t r u c t u r a l Assumptions of Quasichemical and Related Theories 2 4 8 2. Expansion of Table 1 with I n t e r n a l Text References 2A9 3. Values of Some B e s t - F i t Thermodynamic Constants from Wilson and NCZ Expressions 2 5 2 4. C l a s s i f i c a t i o n of S o l u t i o n Types i n R e l a t i o n to the NCZ Equation 255 5- Comparisons Made Between Theories and E x p e r i -mental R e s u l t s 2 7 9 6. NCZ Parameter S e t t i n g s f o r Systems Tested 2 8 2 7. References to Data f o r Systems Tested 2 8 3 8. Parameter S e t t i n g s f o r Related Theories f o r Systems Tested: (a) NRTL Parameters 2 8 4 (b) Wilson Parameters 2 8^ (c) ASOG Parameters 2 8 5 (d) S o l u b i l i t y Parameters 2 8 7 (e) Barker Parameters 2 8 7 9. Summary of Main G e n e r a l i z a t i o n of CZ Binary Quasichemical Theory 10. R e l a t i o n s Between y. and G /RT and Experimental Comparison Methods 3 4 9 ix ACKNOWLEDGMENT I would l i k e to thank Professor J.S. Forsyth for his u n f a i l i n g help and encouragement i n a l l stages of t h i s work, not the least of which was the aid i n transforming the work into f i n i s h e d form. Thanks are also due to Dr. J.R. Sams of the Chemistry Department for his a i d i n s t a t i s t i c a l l i q u i d mixture-theory, and for discussions about the nature of intermolecular a t t r a c t -i ons; and to Mr. J . Lielmezs of the Chemical Engineering Department for enlightening discussions on various thermo-dynamic aspects of the work. Numerous others, including the other members of my Committee, namely Drs. S.D. Cavers and N . . Epstein, have been very h e l p f u l i n bringing to my attention key a r t i c l e s i n the l i t e r a t u r e i n areas of i n t e r e s t to the work, and also for providing h e l p f u l ideas and c r i t i c i s m s i n the course of the numerous seminars given on the work as i t pro-gressed. Thanks are also due to Miss B.FE. Ford, who, i n the course of summer work on the project, c a r r i e d out an exhaustive l i t e r a t u r e search which proved most h e l p f u l . The writer i s also indebted to the following organi-zations for f i n a n c i a l assistance: the National. Research Council of Canada, Finning Tractor Corporation, the Pan American O i l Company, and the University of B r i t i s h Columbia. X Dedication to:-rolus a r v i d barb bev dale donna f reda Judy johann f r i n d & f r i e n d and to peter pat david bob robin david ken v a l e r i e and don who contributed so f r e e l y l i n e s symbols help counsel expertise enthusiasm i n s i g h t example i n s p i r a t i o n books thoughts and i n some cases new molecules for the molecule c o l l e c t i o n . FOREWORD 1 A Ph.D. t h e s i s is. read 'by i n d i v i d u a l s of very d i f f e r e n t background I n t e r e s t s and d e t a i l e d e x p e r t i s e i n the f i e l d of the t h e s i s . To avoid tedious and needless r e a d i n g , three c h a r t s have been provided showing p o s s i b l e reading routes to meet the p o t e n t i a l needs of three types of readers. 1} The expert who i s f a m i l i a r w i t h the background and needs only to consider what i s new i n the t h e s i s , both i n the theory and i n the r e s u l t s . 2 ) The reader who i s only g e n e r a l l y knowledgable i n the f i e l d and who might t h e r e f o r e wish to be reminded of r e l e v a n t background. 3) The reader who i s only concerned i n the r e s u l t s obtained and who i s prepared to take d e r i v a t i o n s on t r u s t . I l l u s t r a t i o n l provides a r o u t i n g guide f o r readers having one of the above purposes i n mind. G E N E R A L L Y R E S U L T S -E X P E R T K N O W L E D G A B L E O R I E N T E D R E A D E R R E A D E R R E A D E R I* I I I 7 I 8 8 10 10 10 11 II I I 12 \ ALL APPENDICES 12  1"^  APPENDIX 4 12 * CHAPTER NUMBER I l l u s t r a t i o n 1. Reader Routing Guide. 3 • CHAPTER 1  GENERAL' ORIENTATION OF THE RESEARCH 1 . THE EFFECT OF NON-IDEALITY ON MULTICOMPONENT DISTILLATION 2. FIVE POINT RESEARCH PROGRAM A. General B. Search f o r P r e s e n t l y A v a i l a b l e Data C. Measurement of Vapor-Liquid E q u i l i b r i u m D. Measurement of Heat of V a p o r i z a t i o n E. C a l c u l a t i o n Program F. Development of More Powerful P r e d i c t i v e Expressions 3. THE DESIGN IMPLICATIONS OF THERMAL AND EQUILIBRIUM DATA 4. THERMODYNAMIC CONSISTENCY TESTS 5. CONSISTENCY BETWEEN EQUILIBRIUM AND THERMAL DATA 6. THE NEED FOR VERSATILE EXPRESSIONS FOR PROPERTY PREDICTION 7. THE NEED FOR A GENERAL CLASSIFICATORY SYSTEM FOR NON-IDEALITY 8. USEFUL ATTRIBUTES OF SUCH A CLASSIFICATION 9. RESULTS OF THE PRESENT STUDY CHAPTER 1  GENERAL ORIENTATION OF THE RESEARCH 1. THE EFFECT OF NON-IDEALITY ON MULTICOMPONENT DISTILLATION At the outset of the work reported herein, the author embarked on a study of the ef f e c t of non-ideality i n the phases present i n f r a c t i o n a t i n g columns on such important parameters as the number of stages required, the minimum r e f l u x r a t i o and the d i s t r i b u t i o n of the various components between two or more outlet streams. The intended mode of attack was to consider 3-component systems for which extensive vapor l i q u i d ternary equilibrium data as well as a l l the necessary thermal values were a v a i l a b l e . Knowing these data for several ternary systems, i t would have been possible to examine the behavior of systems to see which features of t h e i r non-ideality were most important and, i f these features affected the f i n a l per-formance of the column i n d i f f e r e n t ways, to enumerate these e f f e c t s for the guidance of future users. It was hoped that among other things, the r e s u l t s might be amenable-to summary i n some way which would give a clear general understanding of the behavior of a column hand-l i n g non-ideal mixtures i n the way that the equations of Underwood^ ' and t h e i r extension and graphical i n t e r p r e t a t i o n by Forsyth and F r a n k l i n ( 2 ) has done for the case of i d e a l * wherever p o s s i b l e , format of reference i s according to reference.(97). 5 mixtures. Even i f t h i s were not p o s s i b l e i t was hoped that by doing a number of column c a l c u l a t i o n s i t would be p o s s i b l e to o f f e r some g e n e r a l i z a t i o n s of use to a designer. I t would not be p o s s i b l e to produce any g r a p h i c a l g e n e r a l i z a t i o n s f o r systems w i t h more than four components, but the i l l u s t r a t i o n s r e f e r r i n g to three and p o s s i b l y four components together w i t h an extension of the arguments by analogy might give some general guidance to the behavior of multicomponent systems. 2. FIVE POINT RESEARCH PROGRAM A. General A research program was set up comprising f i v e con-s t i t u e n t p a r t s , a l l of which would be needed to achieve the f i n a l g o a l , y e t , each being s u f f i c i e n t l y independent of the others that some progress could h o p e f u l l y be made i n a given p a r t , independent of d i f f i c u l t i e s which may have a r i s e n i n the meantime i n the p a r a l l e l s t u d i e s . Some of these programs were c a r r i e d out e n t i r e l y by the author, but some formed work c a r r i e d out by undergraduates as part of t h e i r graduating r e s e a r c h p r o j e c t s and so, while being supervised by the author, were only to a l i m i t e d extent c a r r i e d out by him. In a d d i t i o n , part of the considerable l i t e r a t u r e search f o r experimental data,which w i l l be mentioned later,was c a r r i e d out under guidance by an undergraduate student working during the summer v a c a t i o n . For c l a r i t y , i t . i s convenient to l i s t the f i v e sub-6 sections at t h i s point. B. Search for Presently Available Data A considerable search of the l i t e r a t u r e was made i n order to f i n d as many systems as possible containing three or more components for which a 'complete' set of physical data was a v a i l a b l e . These data had to include binary vapor-liquid equilibrium for a l l possible b i n a r i e s , and some multicomponent vapor - l i q u i d equilibrium data. In addition was required the enthalpy of a s u f f i c i e n t l y large number of multicomponent l i q u i d and vapor mixtures, and l i q u i d and vapor enthalpy of the pure components together with r e l i a b l e heats-of-mixing. C. Measurement of Vapor-Liquid Equilibrium Before the search f o r data had been underway for very long i t became obvious that i t was u n l i k e l y that even a very l i m i t e d amount of data meeting the s p e c i f i c a t i o n s given would become avail a b l e through a l i t e r a t u r e search, and, accordingly, a l i m i t e d program was u n d e r t a k e n w / for deter-mining three component vapor-liquid equilibrium data for the system Benzene-Toluene-Xylene. This system had already been examined as f a r as the Benzene-Toluene- and Toluene-Xylene b i n a r i e s , but the t h i r d binary had not been studied and no ternary data were a v a i l a b l e . Experimental and a n a l y t i c a l technique ava i l a b l e to hand proved inadequate,.and the work must be regarded as unsatisfactory. 7 D. Measurement of Heat of V a p o r i z a t i o n P a r a l l e l w i t h above undertakings, an attempt was made to measure the d i f f e r e n c e between the a c t u a l heat of v a p o r i z a t i o n of Benzene-Toluene-Xylene mixtures, and that which would have been p r e d i c t e d by i d e a l mixing r u l e s ^ * ^ . E. C a l c u l a t i o n Program I f the above three programs had come to f r u i t i o n , or even i f only the f i r s t had been s u c c e s s f u l , then data would have been a v a i l a b l e f o r use i n f r a c t i o n a t i n g column c a l -c u l a t i o n s , e i t h e r as t a b u l a r data, or f i t t e d to standard da t a -( 6 7 ) r e d u c t i o n expressions 5 . To t h i s end computer programs were devised and t e s t e d to use the data when i t became a v a i l -a b l e . C e r t a i n assumptions had to be made about the form the data would take when i t became a v a i l a b l e , but w i t h i n these r e s -t r i c t i o n s programs were w r i t t e n and t e s t e d ^ \ I t was hoped that these would be f l e x i b l e enough to deal w i t h a wide range of d i f f e r e n t s i t u a t i o n s . One of the main concerns of the o v e r a l l work was to examine the e f f e c t of n o n - i d e a l i t y on the minimum r e f l u x r a t i o i n f r a c t i o n a t i n g columns. P a r t i c u l a r a t t e n t i o n was given to t h i s f e a t u r e . R e a l i z i n g that no p l a t e - t o - p l a t e c a l c u l a t i o n could ever be done e x a c t l y at the minimum r e f l u x r a t i o , p r o-v i s i o n was made f o r c a l c u l a t i n g at a r e f l u x r a t i o s l i g h t l y above and a l s o s l i g h t l y below the minimum value. The l a t t e r c a l c u l a t i o n gives r e s u l t s l e a d i n g to compositions negative i n one or more components, but these r e s u l t s , although u n r e a l , can serve to evaluate the minimum r e f l u x r a t i o more c l o s e l y 8 than would be possible i f only r a t i o s greater than the minimum (9) could be used .. F. Development of More.Powerful Predictive Expressions * The f i f t h prong of attack on the o v e r a l l problem, and the one which, i n the event, has been given the most attention and„which forms the bulk of the matter i n t h i s t h e s i s , was an attempt to pre d i c t , on t h e o r e t i c a l grounds, consistent vapor-liquid e q u i l i b r i a and thermal data. I n i t i a l l y , t h i s -attempt(3) was made at what i s now r e a l i z e d was an i n -s u f f i c i e n t l y profound l e v e l , and made use of the Clausius-Clapeyron equation to r e l a t e r e l a t i v e v o l a t i l i t i e s of pairs of components d i r e c t l y to the difference i n t h e i r heats of vaporization, rather than deriving both thermal and v o l a t i -l i t y data from a free energy expression,<as i s more usual(lO)-This work i n i t i a t e d a t r a i n of thoughtwhich l a t e r , led to a generalization of pre - e x i s t i n g theories to include, i n a new way, the e f f e c t s of unequally sized molecules. However, i n due course, the use of the Clausius-Clapeyron approach i t s e l f had to be given up as lacking s u f f i c i e n t f l e x i b i l i t y of a p p l i c a t i o n * , and a considerable study of solution theory was * The u n e n f o r c i b i l i t y of constant T and P over a range of compositions does not in v a l i d a t e the Clausius-Clapeyron equation as a point transformation at a given composition, though i t may make i t inconvenient to apply i n pr a c t i c e . 9 undertaken. It now appears that t h i s f i f t h avenue of attack has produced useful r e s u l t s and that the author now has a v a i l -able to himself, consistent ( i f somewhat incomplete, i n the case of the thermal data) estimates of thermal and equilibrium data which can be used i n the programs already developed. This l a t e r step has not yet been taken. 3. THE DESIGN IMPLICATIONS OF THERMAL AND EQUILIBRIUM DATA The r e l a t i v e s e n s i t i v i t y of column calculations to equilibrium data on the one hand, and to thermal data on the other, depends upon the intended operating conditions of the column. At high r e f l u x r a t i o s , the thermal properties of the l i q u i d and vapor have very l i t t l e e f f e c t on the column per-formance. Except inasmuch as the t o t a l throughput of the apparatus may be dependent on those thermal properties, the separation performance i s minimally affected. However, near the minimum re f l u x r a t i o , the quantity of the l i q u i d and vapor phases flowing at any one point i n the column may have a de-c i s i v e e f f e c t on the separation, and so under these conditions (and these are the conditions near which commercial columns operate) the thermal properties become important as well. Not only must such relevant data be availa b l e for c a l c u l a t i o n , butthey must possess an adequate degree of r e l i a b i l i t y to avoid excessive contingency allowances i n physical column design. 4. THERMODYNAMIC CONSISTENCY TESTS Experimental data may have i n i t sources of errors 10 which might well be divided into two kinds. F i r s t l y , there i s the random scatter about the true l i n e which a l l experimental r e s u l t s display, and secondly, the systematic error i n y-versus-x a r i s i n g from experimental technique, or some other source of constant error. It i s e n t i r e l y possible that a set of data containing a systemmatic error w i l l , when the random errors are removed by c u r v e - f i t t i n g , produce a curve which shows consistency with the Gibbs-Duhem equations method of t e s t i n g ^ 1 1 ) . It i s therefore pointed out that while meeting t h i s standard i s a necessary condition i t i s by no means s u f f i c i e n t , and i t i s e n t i r e l y possible that some data Is being accepted as r e l i a b l e because i t has met t h i s consistency test whereas i n fact i t contains unrevealed discrepancies. I f , however, i t were possible to evaluate the Gibbs free energies of the mixtures involved, then equilibrium data derived' from t h i s would, of n e c e s s i t y ^ s a t i s f y the Gibbs Duhem consistency t e s t s , obviating them, and t r a n s f e r r i n g the question of uncertainty of data to a more fundamental l e v e l than that which can be inspected by consistency t e s t s . In t h i s t h e s i s , p r e d i c t i v e expressions are formulated i n the excess free energy form. These are conformal with the p r i n c i p l e of thermodynamic consistency as long as the transformations r e-s u l t i n g i n the expressions derived from the mixture free energy are c o r r e c t l y performed. . ... 11 5. CONSISTENCY BETWEEN EQUILIBRIUM AND THERMAL DATA Consistency between vapor-liquid e q u i l i b r i a and corresponding thermal data must also be considered. Of course i n p r i n c i p l e , i f the vapor-liquid e q u i l i b r i a prove to be consistent, and i f i t can then be shown they conform with the heat of mixing or heat of vaporization by any v a l i d thermo-dynamic r e l a t i o n s h i p , under the pertinent conditions of tem-perature and pressure for a given mixture, then the given thermal data w i l l perforce be consistent with the equilibrium data. Such a v a l i d thermodynamic r e l a t i o n s h i p i s exemplified by a modified form of the Clausius-Clapeyron equation r e l a t i n g the vapor pressure r a t i o of a p a i r of com-ponents to the difference i n t h e i r p a r t i a l molal heats of vaporization i n a given mixture. This method of producing equilibrium data from thermal data was i n fact used for some time i n the present work. Unfortunately, the apparent straightforwardness of such a means of securing consistency between thermal and equilibrium data encountered the following d i f f i c u l t y . There i s an extreme difference i n degree of s e n s i t i v i t y of vapor-pressure data and heat of vaporization data to volume changes i n the l i q u i d phase upon mixing: the former are most i n s e n s i t i v e to volume change e f f e c t s , and the l a t t e r extremely s e n s i t i v e . Since the measurements useful for column design are the i s o b a r i c ones ( i n which l i q u i d ex-pansion e f f e c t s are free to occur), the fact that no very r e -l i a b l e quantitative means exists for assessing the e f f e c t of l i q u i d volume changes upon thermal properties e f f e c t i v e l y 12 undermines the usefulness of applying consistency tests between the two sets of properties - no matter how formally correct are the algebraic thermodynamic descriptions which are pre-sently a v a i l a b l e v ' , nor how d i l i g e n t l y these tests are applied. 6. THE NEED FOR VERSATILE EXPRESSIONS FOR PROPERTY PREDICTION Even i f one were to ignore the operational d i f f i c u l -t i e s i n transforming one set of data into another, and even allowing that such transformations are e n t i r e l y v a l i d , i t i s obvious that no demonstration of the interna'l consistency of an experimentally determined set of data i s going to a s s i s t with the problem of p r e d i c t i n g the behavior, even of the same system away from conditions under which i t was tested. The above i s true even with the same system to be used under d i f f e r e n t conditions of T and P, and of course becomes even more obviously true i f the composition range i s switched from one region to another. Indeed what i s required more than a means of t e s t i n g the consistency of measured data (and the worthwhileness of such t e s t i n g has already been questioned by authors such as Van Ness \x3) and Chang and Lu ),is rather the means to predict data away from the regions i n which meas-urements have been made, even i f these predictions are some-what less than completely r e l i a b l e . S p e c i f i c a l l y i t would appear that from an engineering point of view, even rather poor data on a 5-component system which has to be processed i s more valuable than extremely r e l i a b l e data on any of the bi n a r i e s . 13 In f a c t , i n such a case the designer would probably be prepared to accept l e s s c e r t a i n t y i n h i s data, i f , i n r e -t u r n , he were able to increase the comprehensiveness of p r e -d i c t i o n s about what might be expected to happen i n the mixture he wishes to employ under various design c o n d i t i o n s , even i f t h i s guidance were only r a t h e r q u a l i t a t i v e . I t i s obvious that a designer must have guidance as to what e f f e c t s on the performance of h i s column w i l l come from the p a r t i c u l a r k i n d of n o n - i d e a l i t y which h i s system i s l i a b l e to e x h i b i t . For example, i f a system had a n o n - i d e a l i t y such that the r a t i o of l i q u i d to vapor molal r a t e s i n c r e a s e d down the column from the top p l a t e , then a d i f f e r e n t s i t u a t i o n might a r i s e w i t h respect to the onset of a pinch than i f the reverse were t r u e . E q u a l l y , i f the K-value of one component i n a multicomponent system were reduced to a value near to u n i t y because of a high c o n c e n t r a t i o n of some other component, then the normal e x p e c t a t i o n that the former component would d i m i n i s h r a p i d l y e i t h e r as one proceeded down or up the column would not be met, and would q u i t e s e r i o u s l y i n t e r f e r e w i t h the column's s e p a r a t i o n performance. Such an example i s i n c l u d e d f o r purposes of p e r s p e c t i v e , to show that there can be r e a l problems, and that these problems can a r i s e e i t h e r from the e q u i l i b r i a n o n - i d e a l i t y or from the flow n o n - i d e a l i t y . 7. THE NEED FOR A GENERAL CLASSIFICATORY SYSTEM FOR NON-IDEALITY In the f i e l d of p r e d i c t i v e t h e o r i e s of n o n - i d e a l i t y , much recent work i s concerned w i t h the study of v a p o r - l i q u i d 14 n o n - i d e a l i t i e s . Various broad c l a s s i f i c a t i o n s of non-i d e a l i t i e s have become recognized, investigated, and named. As long as a system i s showing only one kind of non-ideality, the s i t u a t i o n i s not confusing. But i t i s i n the nature of things that experiment and c l a s s i f i c a t i o n of t h i s kind can usually only t e l l of the r e s u l t i n g o v e r a l l e f f e c t , and cannot often d i s t i n g u i s h whether or not there i s only one kind of non-i d e a l i t y present or whether there i s more than one kind of non-i d e a l i t y at work, with diverse e f f e c t s r e i n f o r c i n g or can-c e l l i n g to an unknown extent. In fact one might be thinking i n terms of one type of non-ideality when the system a c t u a l l y exhibits another (or others), so that any attempt to predict parameters for a system under such circumstances would r e -present e f f o r t s made i n the wrong context. Hence workers i n t h i s f i e l d have been s t r i v i n g for many years to produce mathe-matical expressions of the kind which attempt to incorporate some s p e c i f i c idea of the physical behavior of l i q u i d s into theirmathematical expressions, and which i t i s hoped w i l l describe q u a n t i t a t i v e l y a l l the features of a system dis p l a y i n g more than one s p e c i f i c pattern of nonideality. 8. USEFUL ATTRIBUTES OF SUCH A CLASSIFICATION So f a r r e s u l t s have been f a r from complete, and the s i t u a t i o n i s by no means resolved. It would be useful however to consider the form that such an expression might take. Some desired v a r i a b l e , and for reasons which w i l l appear l a t e r l e t E us choose the excess Gibbs free energy G 3 would be equated to a number of terms, each of which would i n c o r p o r a t e , together w i t h the necessary mathematical operators, only 'recognizable' thermodynamic and p h y s i c a l parameters which would, h o p e f u l l y , a l s o be e a s i l y measurable. Each parameter could be d i s -t r i b u t e d i n various f u n c t i o n a l r e l a t i o n s h i p s through the thermodynamic c l a s s e s of component terms present i n a general expression. The parameter might appear i n s e v e r a l of the com-ponent terms of the o v e r a l l expression and be mis s i n g from s e v e r a l . I t could appear i n one guise i n some terms and i n another i n other terms. Terms would be formulated so that the numerical extent of any k i n d of n o n - i d e a l i t y would be r e f l e c t e d i n the numerical magnitude of component terms of the o v e r a l l expression. In t h i s way, a statement of n o n - i d e a l i t y would be unambiguous, inasmuch as i t would take numerical cognizance of each k i n d of p o s s i b l e n o n - i d e a l i t y , and wh i l e r e c o g n i z i n g the c o n t r i b u t i o n of these various kinds of n o n - i d e a l i t y would, i n t o t a l , give the o v e r a l l n o n - i d e a l i t y of the system however c o n t r i b u t e d . I m p l i c i t i n the foregoing i s the assumption that a l l the r e l e v a n t p h y s i c a l parameters must be i n c l u d e d i n the e x p r e s s i o n , even i f some of t h e i r e f f e c t s were n e g l i g i b l e i n any one p a r t i c u l a r system. These parameters would con-s t i t u t e what i s known mathematically as a spanning set of i n -dependent v a r i a b l e s f o r the system. I t would be understand-able that a number of p o s s i b l e spanning sets could be s p e c i f i e d but the exi s t e n c e of a minimum set i s mandatory i n the argument. I t might be that the numerical magnitude of r e c o g n i z a b l e members of the spanning set f o r a given system would i n d i c a t e 16 i n some way the type of n o n - i d e a l i t y which the system would e x h i b i t , but i t i s much more probable that the same p h y s i c a l parameter w i l l be i n v o l v e d i n a number of d i f f e r e n t ways and the author f e e l s i t more probable that the terms of the over-a l l e x p r e s s i o n , r a t h e r than the parameters which are used to construct them, would be the ba s i c u n i t s f o r d e s c r i p t i o n of n o n - i d e a l i t y . The above observations have been w r i t t e n w i t h some h i n d s i g h t , because such an expression has indeed been developed and seems to have most of the p r o p e r t i e s d e s c r i b e d . In de-v e l o p i n g such an expr e s s i o n , which a f t e r a l l i s r a t h e r an am-b i t i o u s t a s k , a reasonable place to s t a r t would be to see t o what extent e x i s t i n g t h e o r i e s which, although fragmentary, are s u c c e s s f u l i n the more l i m i t e d contexts f o r which they were designed, might be brought together. I t d i d seem most pro-bable that the t h e o r i e s which can most e a s i l y be brought t o -gether would be those which had the common b a s i s of a p h y s i c a l imagined p i c t u r e of molecular behavior. This c o n c l u s i o n i s strengthened by the f a c t that the seemingly most e f f e c t i v e t h e o r i e s at the moment are those which do_ i n c l u d e such a p h y s i c a l p i c t u r e i n t h e i r f o r m u l a t i o n s . The way i n which the present work i s based upon such e x i s t i n g t h e o r i e s w i l l be i l l u s t r a t e d i n the chapters d e a l i n g w i t h the d e r i v a t i o n of formal r e s u l t s . A l s o , an expression developed from w e l l - e s t a b l i s h e d f o r m u l a t i o n s has the double advantage of reducing the amount of new f o r m u l a t i o n necessary while at the same time p r e s e r v i n g 17 structures i n formulation which are at least p a r t l y f a m i l i a r to -the reader. It ,is of course well recognized that other approaches are possible, including those of pure data reduction nature (7). Such procedures are not r e a l l y thermodynamic i n nature and therefore are r a r e l y p r e d i c t i v e - at least beyond a very l i m i t e d range. Although they are useful manipulative devices, they provide a rather l i m i t e d insight into the nature of the problem of recognizing s p e c i f i c n o n - i d e a l i t i e s , and while un-doubtedly of very considerable value i n engineering design, are i n e v i t a b l y suspect when the predictions are extrapolated to any noticeable extent. It i s also r e a l i z e d by those who have had to make use of them that empirical polynomial f i t s are notoriously s e n s i t i v e to the data from which they have been derived, and i t i s t h i s very s e n s i t i v i t y which makes the user cautious about using extrapolations. The author therefore proposes not to devote more than cursory attention to highly empirical approaches except by way of i l l u s t r a t i o n , and for comparison purposes l a t e r on i n the discussion of r e s u l t s . While recognizing the power of highly empirical approaches to deal with known experimentally determined r e s u l t s , because of t h e i r lack of u t i l i t y i n pre-d i c t i n g r e s u l t s , and above a l l i n t h e i r incapacity to elucidate the problem of the existence of various kinds of non-ideality, such attention should s u f f i c e for the p a r t i c u l a r purposes of t h i s t h e s i s . 18 9. RESULTS OF THE PRESENT STUDY' The reader w i l l f i n d i n due course that the ex-pression developed i s complex, a complexity which r e f l e c t s the generality of the concepts embodied i n i t . Inherently the complexity of an expression i s , nowadays, no b a r r i e r to i t s use provided i t s predictions are superior to those of alte r n a t i v e s (52) ava i l a b l e . I f , as the author hopes, the expression can, by the examination of a lim i t e d number of systems for which data i s more or less available, prove to be a p o t e n t i a l l y useful and powerful one, then he believes that more data w i l l be generated, t h i s data being of a form most conveniently usable i n the ex-pression . 19 CHAPTER 2 INTRODUCTION TO THE THERMODYNAMIC PROBLEM 1. GENERAL 2. FRAMEWORK OF STANDARD THERMODYNAMIC AND STATISTICAL RELATIONSHIPS 3. KEY INDEPENDENT VARIABLES 4. THE QUASICHEMICAL THEORY OF SOLUTIONS 5. THE PRINCIPLE OF MINIMUM FREE ENERGY IN EQUILIBRIUM 6. THE PARTITION FUNCTION 7. THE SCOPE OF THE QUASICHEMICAL THEORY 8. PRESENT EXTENSION OF THE THEORY A. General B. Basic Nature of the Extension 9. GEOMETRIC INTERPRETATION OF THE NCZ CASE 10. ENERGETIC RAMIFICATIONS OF NCZ PACKING GEOMETRY 11. ENTROPY EFFECTS OF NCZ PACKING GEOMETRY A. The Primary C o n f i g u r a t i o n a l Entropy E f f e c t B. Secondary Entropy E f f e c t s 12. CONCLUDING REMARKS 20 CHAPTER 2 INTRODUCTION TO THE THERMODYNAMIC PROBLEM 1. GENERAL I t has now been shown how the t h e o r e t i c a l thermo-dynamics problem which became the t o p i c of the t h e s i s arose out of the o r i g i n a l context of an engineering study of the e f f e c t s of n o n - i d e a l i t y i n multicomponent d i s t i l l a t i o n . I t i s now time to i n t r o d u c e , i n the f o l l o w i n g s e c t i o n , the out-l i n e s of the thermodynamic research i n t o mixture n o n - i d e a l i t y which became the main work c a r r i e d out by the author. 2. FRAMEWORK OF STANDARD THERMODYNAMIC AND STATISTICAL RELATIONSHIPS A review of standard thermodynamic r e l a t i o n s ^ 1->) w i l l now be given. These r e l a t e the mixture p r o p e r t i e s being sought to other, more fundamental though more e a s i l y c a c u l a b l e q u a n t i t i e s . The more d e t a i l e d observations to be made i n the present work w i l l be then i n s e r t e d i n t o the standard r e l a t i o n -s h i p s . The standard r e l a t i o n s h i p s used are those between the a c t i v i t y c o e f f i e n t CY) of a component present i n the mixture, molar heat of mixing (H ) (two p r o p e r t i e s sought f o r d i s t i l l a -t i o n purposes) and the excess Gibbs f r e e energy of mixing (G ), and a fundamental q u a n t i t y from s t a t i s t i c a l mechanics known as ± j » n± , n ± » R , A , T , n) f Parameter Estimation Molecular Thermodynamics, •Statistical Mechanics. Relations between Statistical Mechanics and Thermodynamics. Classical Thermodynamics , HE Heat Balance Equilibrium Compositions. Engineering Applications Illustration 2. Thermodynamic Relationships. 22 the p a r t i t i o n function (designated i n t h i s work as Q.). For brevity, these are f i r s t introduced i n elementary notation to indicate the general nature of the relationships and to i n t r o -duce the set of independent variables chosen to be incorporated. Y» H E = f 1 ( G E ) G E = f 2(Q) £ = Q.(xi,Ri,ni,r1jlwij,Ai,n,T) The organization scheme to be followed i s shown i n I l l u s t r a t i o n 2. By consolidation of the above r e l a t i o n s , the o v e r a l l E E plan was to e s t a b l i s h a set of r e l a t i o n s between G and H and the independent variable set e n t i t i e s , as summarized by Y , H E = f 0OK. ,R. ,n ,u. , 10 . .,A. ,II,T) (4) E These r e l a t i o n s indicate that y and H are obtainable by thermodynamic methods from the Gibbs excess free energy of mix-F F ing G , while i n turn G i s obtainable through a standard formalism from the p a r t i t i o n function Q. for the mixture. Other thermal properties needed for column c a l c u l a t i o n s , namely en-thalpy of the pure l i q u i d and pure vapor and non-ideality of the vapor are taken for present purposes as being available from tables of data and are not given any further detailed con-ID ( 2 ) ( 3 ) 23 s i d e r a t i o n i n t h i s part of the t h e s i s . Customarily, the g thermodynamic v a r i a b l e G i s that f o r which e x p l i c i t ex-pre s s i o n s are w r i t t e n , and t h i s p r a c t i c e w i l l be maintained i n the present work. 3. KEY INDEPENDENT VARIABLES One set of independent v a r i a b l e s which define Q g and hence G , i s that bracketed i n Equation 3. Note th a t the set e x p l i c i t l y contains the molecular r a d i u s v a r i a b l e R • This I n c l u s i o n i m p l i e s that a l l molecules i n the mixture (more e x a c t l y monomer u n i t s ) need not be considered as being of the same s i z e . S p e c i f i c a l l y , the i n c l u s i o n of the v a r i a b l e R i s to permit of f o r m u l a t i n g equations not n e c e s s a r i l y pre-assuming that a l l molecules are of the same s i z e . This i s one d i f f e r e n c e between the work now to be proposed and that described i n e x i s t i n g t h e o r i e s , and w i l l be the subject of very considerable a t t e n t i o n l a t e r i n the present work. 4. THE QUASICHEMICAL THEORY OF SOLUTIONS One mainstream of e x i s t i n g theory, and the one which i s a l s o f o l l o w e d i n t h i s work may be c a l l e d broadly the " q u a s i -chemical theory of s o l u t i o n s " . The name 'quasichemical' des-ignates one of the primary features of the theory, namely that the process of mixing two l i q u i d s which r e s u l t s i n u n l i k e mole-cules being juxtaposed ( i n c o n t r a s t to the J u x t a p o s i t i o n of molecules of the same species i n the pure component s t a t e ) i s conceived of i n the same terms as those d e s c r i b i n g a r e v e r s i b l e 24 chemical r e a c t i o n consistent' w i t h such a r e o r g a n i z a t i o n . I f a t t e n t i o n i s focused on the i n d i v i d u a l p o i n t s of contact of molecules i n nearest-neighbor r e l a t i o n s , then the 'forward' r e a c t i o n i n v o l v e s the r e o r g a n i z a t i o n of two ' l i k e ' p a i r s of such c o n t a c t s , one p a i r f o r each of the two types of molecules being considered, and w i t h the formation of two 'un-l i k e ' p a i r s r e p r e s e n t i n g the mixed s i t u a t i o n . The 'reverse' r e a c t i o n simply r e t u r n s the two u n l i k e p a i r s to the l i k e - p a i r s found i n r e s p e c t i v e pure component s t a t e s . I l l u s t r a t i o n 3 i n d i c a t e s the ideas of c o n t a c t s , ' p a i r s ' , and the r e o r g a n i z a t i o n r e a c t i o n j u s t d e s c r i b e d . This r e o r g a n i z a t i o n i s a s s o c i a t e d w i t h the r e l e a s e or a b s o r p t i o n of energy which i s the heat of mixing. For example, the forward ' r e a c t i o n ' i s exothermic i f the cohesion of the u n l i k e p a i r bond formed i s stronger than the a r i t h m e t i c average of the two o r i g i n a l l i k e - p a i r bonds. The e q u i l i b r i u m mixture of l i k e and u n l i k e p a i r a s s o c i a t i o n s , where a l a r g e number of l i k e p a i r s of each substance was i n i t i a l l y p r esent, i s the one f o r which the o v e r a l l system has assumed i t s lowest f r e e energy, subject t o the a d d i t i o n a l system r e s t r a i n t s which are the p h y s i c a l p ro-p e r t i e s of the component substances and t h e i r r e l a t i v e abundance. The conception of the l i q u i d as an assemblage of p a i r s of contacts i s the b a s i s of what was c a l l e d the 'inde-pendent p a i r s l i q u i d ' by Guggenheim^ ? I t w i l l be seen th a t t h i s r e p r e s e n t a t i o n of the l i q u i d i s w e l l s u i t e d f o r formal a n a l y s i s of mixing phenomena, i n that i t leads to expressions f o r m a l l y i d e n t i c a l w i t h the mass law expressions i n r e a c t i o n L I K E P A I R S U N L I K E P A I R S I l l u s t r a t i o n 3. The Pairs Reorganization Reaction. chemistry used to r e l a t e e q u i l i b r i u m concentrations of products and rea c t a n t s to standard f r e e energy changes. In the independent p a i r s l i q u i d , l i k e p a i r s of the pure component s t a t e s are the 'reactants' and the u n l i k e p a i r s formed on mix-i n g are the 'products' of the mixing ' r e a c t i o n ' . Hence the name quasichemical s o l u t i o n theory. 5. THE PRINCIPLE OF MINIMUM FREE ENERGY IN EQUILIBRIUM O p e r a t i o n a l l y what has to be done to minimize the fr e e energy of the mixture can be summarized i n step 1) to 3) f o l l o w i n g : 1) Evaluate 0. f o r the mixture 2) A = - kT Jin Q. 3 A. 3) A -»- min. f o r values of X. such t h a t : (— ) = 0 (Then to f i n d X. c o n s i s t e n t w i t h A . ) i mm. -4) Solve (30 f o r x., the independent v a r i a b l e 'minimized', whose e q u i l i b r i u m value i s thus determined. A . i s c o n s i s t e n t w i t h stage 3 ) . Minimum f r e e mm ° energy values of the independent v a r i a b l e set (the x i ) are obtained v i a step 4 ) . The symbol X^, here used i n step 3) i n the general sense,is c o n s i s t e n t w i t h the n o t a t i o n of v a r i -a t i o n a l c a l c u l u s to de s c r i b e any s p a t i a l v a r i a b l e . In par-t i c u l a r , here, since A = A(T,V,N i), A T y = A(N ±) , so that i f g_ „• i s p r o p e r l y formulated, c o n d i t i o n 3) r e s u l t s i n a simple a l g e b r a i c e x p r e s s i o n , which i n the case of a binary mixture i s s o l u b l e f o r the e q u i l i b r i u m p o p u l a t i o n of u n l i k e p a i r s , 27 designated i n t h i s thesis as X , related to the unlike pairs concentration X by a simple normalization. (Inc i d e n t a l l y , the d i f f i c u l t y of maintaining a notation scheme which i s both f a i t h f u l to conventions i n the f i e l d of o r i g i n , suggestive of the physical s i t u a t i o n and also compatible with notation of other areas of t h i s work becomes apparent. To resolve i t , unique symbols are employed whenever possible - where i t was not, s p e c i a l meanings are always given i n close conjunction with the expression, i n which they appear, as i s the case here for x ± ). 6. THE PARTITION FUNCTION It i s seen that Q. , the p a r t i t i o n function, (more pr e c i s e l y the Canonical Ensemble P a r t i t i o n Function)is the basic s t a r t i n g quantity upon which operations are performed. 0. i s d i r e c t l y r e l a t e d to system parameters: a cursory de-f i n i t i o n of 0. i s that i t i s the sum of the unnormalized s t a t i s t i c a l weightings of a l l the distinguishable subclasses of events e x i s t i n g i n the system. The weightings take into account the r e l a t i v e populations of the subclasses , and energy and geometry e f f e c t s . The most important single thing to notice at t h i s point i s that the weightings are d i r e c t l y ex-pressable i n terms of the independent variable set i n con-junction with a set of s p e c i f i c rules governing the behavior of the l i q u i d being modelled, and the general s t a t i s t i c a l rules governing the type of p a r t i t i o n function being con-structed. 28 7. THE SCOPE OF THE QUASICHEMICAL THEORY The quasichemical theory presents a f o r m u l a t i o n which permits of the use of a very small number of b e h a v i o r a l ob-se r v a t i o n s to determine,through the appropriate mathematical s t r u c t u r e j t h e p a r t i t i o n f u n c t i o n s r e q u i r e d f o r c a l c u l a t i o n of thermodynamic p r o p e r t i e s . (See Chapter 4 f o r a d i s c u s s i o n of these b e h a v i o r a l observations ) • The quasichemical theory,as o r i g i n a l l y formulated, i m p l i e d t h a t a l l molecular species w i l l be of the same s i z e , and i t must be obvious to the reader that the component mole-c u l a r species of a r e a l l i q u i d mixture w i l l not n e c e s s a r i l y be of equal s i z e . I t was t h i s l i m i t a t i o n which r e s u l t e d i n quasichemical theory f a l l i n g i n t o d i s u s e ; and other t h e o r i e s , incompatible w i t h i t but which d i d i n c o r p o r a t e molecular s i z e parameters, have become prominent. In view of the r e s i l i e n c e and g e n e r a l i t y of the b a s i c p r i n c i p l e s of the quasichemical theory, t h i s s i t u a t i o n seemed unfortunate, and so a r e a s s e s s -ment of the theory was undertaken. A b r i e f i n v e s t i g a t i o n r e -vealed that i t might be u s e f u l l y considered that the q u a s i -chemical s o l u t i o n theory comprises two re c o g n i z a b l e p a r t s . The f i r s t of these i s the 'quasichemical r e a c t i o n ' p a r t , namely that governing the formation of u n l i k e p a i r s from l i k e p a i r s , and the second i s the group of concepts of l i q u i d  s t r u c t u r e i n t o which the quasichemical r e a c t i o n idea had to be i n c o r p o r a t e d to produce an o v e r a l l s o l u t i o n theory. I t was t h i s group of s t r u c t u r a l concepts which needed to be broadened beyond the case of equal s i z e d molecules. The b a s i c 29 a t t r a c t i v e n e s s of the theory nonetheless remained: i f i t i s l e g i t i m a t e to t r e a t a l l nonionic l i q u i d s as anassemblage of p a i r s , then the concept of the quasichemical f o r m u l a t i o n of p a i r s i s a powerful one, and capable of g e n e r a l i z a t i o n i n t o any type of molecular geometry which s u i t a b l y describes the c l a s s of l i q u i d s to be analysed. The bulk of t h i s work was to extend quasichemical theory to inco r p o r a t e a more f l e x i b l e s t r u c t u r a l b a s i s , and thus extend the usefulness of the q u a s i -chemical theory as a means of p r e d i c t i n g excess f r e e energies of mixing. 8. PRESENT EXTENSION OF THE THEORY A. General Any theory which a s p i r e s to comprehensiveness must in c o r p o r a t e a l l the v a r i a b l e s which i n f l u e n c e the behavior of the system being examined. The argument of equation ( 3 ) i s p r e c i s e l y such a s e t . In p a r t i c u l a r i t should be noted that the molecular r a d i u s parameter R has been i n c l u d e d . In p a r t i c u l a r t h i s extension and g e n e r a l i z a t i o n of quasichemical theory to the case of unequally s i z e d molecules i n v o l v e d e l u c i d a t i o n of geometric c o n s i d e r a t i o n s governing the nature of the e f f e c t s of c o n t a c t i n g unequally s i z e d molecules, a fea t u r e m i s s i n g from the o r i g i n a l theory. S t a r t i n g from a p i c t u r e of the model of a l i q u i d as being analogous to that of a bed of spheres of unequal s i z e , the r a m i f i c a t i o n s of t h i s concept have been pursued to produce new expressions p r e d i c t i n g the behavior of such l i q u i d s . The 30 t e s t of the expression developed has been i n i t s a b i l i t y to p r e d i c t the behavior of a number of experimental systems. B. Basic Nature of the Extension The b a s i c mathematical i m p l i c a t i o n s of the extension are f o l l o w e d through the whole body of the theory, r e s u l t i n g i n a q u i t e c o n s i d e r a b l y changed f o r m u l a t i o n . Only the most general aspects of the extension need to be noted at t h i s p o i n t . B a s i c a l l y , the conversion i n v o l v e s changing the geometric con-ce p t u a l b a s i s of the l i q u i d model from that of an assemblage of molecules arranged on the p o i n t s of i n t e r s e c t i o n of a 3-dimensional r e g u l a r l a t t i c e , to one which regards the assemblage as l a r g e l y equivalent i n o r g a n i z a t i o n to that of a close-random packed bed of unequally s i z e d spheres. The r e s u l t of t h i s change i n outlook i s that the number of nearest neighbors per molecule i s no longer regarded as a constant as i n the ' l a t t i c e ' case, but becomes a f u n c t i o n of the r e l a t i v e s i z e of the mole-c u l a r species i n v o l v e d , and the p r o p o r t i o n s of these s p e c i f i e d components, as w e l l as of secondary m o d i f i c a t i o n s i n l o c a l o r g a n i z a t i o n r e s u l t i n g from b i n d i n g energy e f f e c t s . The terminology to be used d i s t i n g u i s h i n g the two methodsof geometric assignment, which a l s o serves to designate the two types of quasichemical t h e o r i e s u s i n g the d i f f e r e n t geometric bases, i s that of constant c o o r d i n a t i o n number f o r the l a t t i c e case (or "CZ", f o r constant-z, where Z i s the symbol f o r c o o r d i n a t i o n number), and non-constant c o o r d i n a t i o n number ("NCZ") to c h a r a c t e r i z e the case of packing of unequally 31 sized molecules. It w i l l be r e a l i z e d even from t h i s b r i e f mention of the two s i t u a t i o n s , namely CZ and NCZ, that the CZ case e f f e c t i v e l y constitutes, at least i n the context of quasi-chemical property estimates, a s p e c i a l case of NCZ which may be achieved i n the l i m i t where the constituent molecular species are of equal s i z e . This statement i s l a t e r shown to be quantitative, i n that the expression for evaluating G i n the NCZ case reduces mathematically to that for the CZ case upon formally s e t t i n g size r a t i o parameters to unity i n the former. • 9. •GEOMETRIC INTERPRETATION OF THE NCZ CASE I l l u s t r a t i o n 4 indicates schematically the geometry of NCZ packing for a mixture of unequal sized spheres, an ef f e c t previously studied by Hogendij k^1"^) and co-workers for packed beds. This diagram i s only q u a l i t a t i v e , i n that the representation i s two-dimensional rather than . showing the three dimensional nature of the actual s i t u a t i o n , but NCZ ef f e c t s are nonetheless c l e a r l y manifested by the I l l u s t r a t i o n . Actual c a l c u l a t i o n of coordination number ( z ) i s made on the basis of t h i s bed of spheres model, with suitable modification for the energy e f f e c t s that exist i n the r e a l l i q u i d . 10. ENERGETIC RAMIFICATIONS OF NCZ PACKING GEOMETRY From the general thermodynamic formula F E E G = H - TS (5) Z s = 6 PURE COMPONENT ' CONDITION Z 2 = 5 MIXTURE CONDITION I l l u s t r a t i o n 4. Coordination Number Change due to Mixing of Unequal Sized P a r t i c l e s . 33 i t i s seen that the free energy of a mixture contains both F E heat (H ) and entropy (S ) of mixing e f f e c t s . Both of these i n v o l v e not only the nature but a l s o the number of contacts per molecule. In the NCZ case, the act of mixing i n e v i t a b l y -a l t e r s both the nature and number of p a i r contacts from t h e i r pure component values. The a d d i t i o n a l energetic r a m i f i -c a t i o n s of NCZ a r i s e out of the cohesion changes due to changes i n the p o p u l a t i o n of l i k e pairs,due to c o o r d i n a t i o n number  changes, as w e l l as the more f a m i l i a r heat of mixing e f f e c t s due to the p a i r s r e o r g a n i z a t i o n . r e a c t i o n accounted f o r i n the o r i g i n a l CZ theory. 11. ENTROPY EFFECTS OF NCZ PACKING GEOMETRY . A. The Primary C o n f i g u r a t i o n a l Entropy E f f e c t P a r t i c u l a r entropy e f f e c t s are more d i f f i c u l t to desc r i b e at the l e v e l of t h i s i n t r o d u c t o r y chapter, but s u f f i c e i t to say, the main e f f e c t a l s o a r i s e s out of a change i n the number as w e l l as i n the k i n d of contacts per molecule upon mixing. The s i t u a t i o n e x i s t s because of a change i n the number of ways i n which a new c o n f i g u r a t i o n of the system can be define d i n the NCZ (versus the CZ) case and hence i n the c o n f i g u r a t i o n a l term i n the p a r t i t i o n f u n c t i o n , hence u l t i -E E mately i n S , the c o n f i g u r a t i o n a l c o n t r i b u t i o n to G . B. Secondary Entropy E f f e c t s For c e r t a i n other excess entropy c o n t r i b u t i o n s to be E reckoned i n t o the complete G expression, i t i s e a s i e r to employ 34 an a l t e r n a t i v e f o r m u l a t i o n f o r entropy than that a f f o r d e d i n connection w i t h the p a r t i t i o n f u n c t i o n . This a l t e r n a t i v e approach i s that of the ' p r o b a b i l i t y d i s t r i b u t i o n ' (£ P±) i approach, coupled w i t h the use of the "Boltzman" d e f i n i t i o n of / -i o \ e n t r o p y v ' w r i t t e n as a f u n c t i o n of p r o b a b i l i t i e s . The basi c f o r m u l a t i o n i s given by equation ( 6 ) S = -k £ p In p ( 6 ) i Because two of the entropy terms i n the general expression are obtained i n t h i s way, the s e c t i o n of Chapter 9 i n which these are discussed i s set apart from the d e r i v a t i o n of terms i n which the us u a l p a r t i t i o n f u n c t i o n approach i s used. Because of the r e l a t i o n of p r o b a b i l i t i e s to c o n c e n t r a t i o n s , c e r t a i n c o n c e n t r a t i o n terms used i n the general e x p r e s s i o n are a l s o ob-t a i n e d through t h i s ' p r o b a b i l i t y ' approach. The development of t h i s approach i n i t s a p p l i c a t i o n to the quasichemical (19) theory of multicomponent s o l u t i o n s was due f i r s t to Bar k e r v ' j Bare mention of the existence of t h i s a l t e r n a t i v e approach has been made at t h i s p o i n t , simply f o r purposes of i n t r o d u c i n g the i d e a , without attempting to elaborate f u r t h e r on I t . 12. CONCLUDING REMARKS Although no attempt to do more than mention the ex-i s t e n c e of the s a l i e n t p o i n t s of the t h e s i s has been made so f a r , i t i s hoped that the reader i s now s u f f i c i e n t l y f a m i l i a r -i z e d w i t h the concepts i n c l u d e d i n i t to f a c i l i t a t e r eading the more d e t a i l e d treatment to f o l l o w . 35 Of the f o l l o w i n g Chapters, Chapter 6 introduces the idea of packing f r a c t i o n u n i t and e x p l a i n s the method of e v a l u a t i n g a p p r o p r i a t e c o o r d i n a t i o n numbers. In Chapter 7 the c o n f i g u a t i o n a l e f f e c t s of NCZ are introduced. In Chapter 8 an e x p o s i t i o n of the extension of the v a r i a b l e co-o r d i n a t i o n number concept to the simples t type of r e a l l i q u i d mixture i s given. In Chapter 9, an extension of the ex-p r e s s i o n i s undertaken, r e s u l t i n g i n the general form of the excess f r e e energy exp r e s s i o n . In Chapter 10 an extensive comparison of the theory to r e l a t e d ones i s made. Chapter 11 i l l u s t r a t e s the comparison of the theory to experimental r e -s u l t s r e p o r t e d i n the l i t e r a t u r e . Chapters 3 to 5 are of an i n t r o d u c t o r y nature, i n c l u d e d to make the Thesis r e l a t i v e l y more s e l f - c o n t a i n e d . 36 • CHAPTER 3 REVIEW OF CONSTANT COORDINATION NUMBER QUASICHEMICAL AND RELATED THEORIES 1. EXTENSIONS OF THE CZ CASE A. General B. T r i p l e t s of Contacts C. M u l t i f u n c t i o n a l S o l u t i o n s - P a i r s Concentration M a t r i x D. Quasichemical n-mer Theory 2. LESS CLOSELY RELATED THEORIES A. Regular S o l u t i o n s B. The 2 - L i q u i d Theory •• C. L o c a l Composition Theories 3. THE PROBLEM OF THE CZ LATTICE 4. SUMMARY OF THE SOURCES OF EARLIER WORK USED 37 CHAPTER 3 REVIEW OF CONSTANT COORDINATION NUMBER  QUASICHEMICAL AND RELATED THEORIES 1. EXTENSIONS OF THE CZ CASE A. General The independent p a i r s l i q u i d , the simple mechanical p r o p e r t i e s of which have been o u t l i n e d i n Chapter 2, i n v i t e s examination of the en e r g e t i c s of the s o l u t i o n as a whole u s i n g a n a l y s i s of changes i n the p a i r w i s e l i n k a g e e n e r g e t i c s as the (16) t o o l . The c o n t r i b u t i o n of Guggenheim i n t h i s area was, through operations on the 'independent p a i r s l i q u i d ' , t o i n c l u d e a c o r r e c t i o n f o r the e f f e c t of l o c a l c l u s t e r i n g i n the con-c e n t r a t i o n p o r t i o n of the heat of mixing term of the fre e energy. S p e c i f i c a l l y he c o r r e c t e d the heat of mixing term to in c l u d e the s t o i c h o i m e t r i c e f f e c t of e n e r g e t i c a l l y induced l o c a l c l u s t e r i n g . Guggenheim's approach, described i n simple terms was to minimize the f r e e energy of the system, subject to the r e s t r a i n t s of constant temperature, volume, and o v e r a l l composition, l e a v i n g as the s i n g l e disposable degree of freedom the c o n c e n t r a t i o n of u n l i k e p a i r s . He i l l u s t r a t e d the pro-cedure to be used f o r the case of a binary mixture of mono-f u n c t i o n a l * s p e c i e s . In such a system, the * one type of surface f u n c t i o n a l group per molecule. Carbon t e t r a c h l o r i d e and butane would be examples of monofunctional speci e s . 38 e f f e c t of the energy-induced c l u s t e r i n g upon the equilibrium concentration of unlike p a i r associations i s evaluated through the solution of a mass-law expression explicitly i nvolving the pure component concentrations, the coordination number, and the net reorganization energy change <o . The mass-law ex-pression i t s e l f i s obtained formally by the v a r i a t i o n a l method, i n which the free energy of the system i s mathematically mini-mized. The i n t e r n a l consistency o'f approach indicated by the above comments accounts for the elegance of t h i s part of his method. Schematically, the process i s set f o r t h by the oper-ations numbered 1) to 4) i n section 5 of Chapter 2. Guggenheim's main objective In formulating the con-cept of the pairs l i q u i d evidently was/to be able to provide a r a t i o n a l basis for c a l c u l a t i n g the e n e r g e t i c a l l y induced l o c a l c l u s t e r i n g i n mixtures. However, the v e r s a t i l i t y and basic soundness of the approach has made possible various extensions of his i n i t i a l theory, although there have also been cert a i n problems, inv o l v i n g c a l c u l a t i o n a l d i f f i c u l t i e s i n solving the algebra,which a r i s e i n cases less r e s t r i c t e d than that con-sidered by Guggenheim himself. This chapter w i l l serve to introduce f i v e l i n e s of i n v e s t i g a t i o n r e l a t e d to quasi-chemical theory. In t h i s chapter w i l l be indicated some of the d i r e c t i o n s of extensions made by Guggenhejm and other workers to the o r i g i n a l theory, and seme of the problems which have already been overcome by more or less r e l a t e d (but compatible) approaches,as well as by some seemingly incompatible ones. 39 This section should also serve as an introduction to c e r t a i n problems which remain, and which form the main burden of l a t e r chapters of t h i s work. The chapter concludes with a b r i e f resume of previous source work. The numbered steps of section 5, Chapter 2, while fundamentally correct and leading to useful answers, may prove to be very d i f f i c u l t to handle numerically. So much so, that r e s t r i c t i o n s on the generality of the concept have been introduced at various stages to simplify the mathematical manipulation. It w i l l now be considered how these r e s t r i c t i o n s have been removed or relaxed by other workers, and l a t e r i n the work to draw attention to s t i l l other relaxations properly the work of the author. The f i r s t r e s t r i c t i o n to be i n v e s t i -gated i s the v a l i d i t y of assigning the properties of the p a i r s -l i q u i d to pairs only, rather than considering t r i p l e t s of con-t a c t s , or quadruplets of contacts. Next, comes the question of removing the r e s t r i c t i o n of the method which confined the method to apply f i r s t only to binary and monofunctional cases; next, to monomer cases. Pin-a l l y , the r e s t r i c t i o n imposed by requirements of a l a t t i c e con-f i g u r a t i o n a l model concomitant with a formalization which im-p l i e s that the mixture i s made up of equally sized molecules i s investigated at length. B. T r i p l e t s of Contacts The o r i g i n a l theory applies to the case of constant-z monofunctional monomer bi n a r i e s . The f i r s t ex-tension, of the theory of Guggenheim, Chang et a l , as d i s -cussed by Guggenheim^ 1 6^, was for the same 40 c l a s s of mixtures i n which not only p a i r s of contacts were con s i d e r e d , but a l s o the case of t r i p l e t s of contacts ( i . e . i n t e r a c t i o n s between three c o n t a c t i n g molecules), and the case of quadruplets of c o n t a c t s . Theirs was p r i n c i p a l l y a study of the magnitude of p r e d i c t e d quantum-mechanical d e v i a t i o n s from the s t r i c t l y a d d i t i v e p a i r w i s e nature of d i s p e r s i o n type i n t e r a c t i o n s f o r groups of i n t e r a c t i n g molecules l a r g e r than two. However, the r e s u l t s of these s t u d i e s mainly confirmed the o r i g i n a l p r e d i c t i o n that estimates of excess f r e e energy ob-t a i n e d by accounting f o r p a i r w i s e i n t e r a c t i o n s could be only s l i g h t l y improved by these r a t h e r l a b o r i o u s , though i n t e r e s t i n c o n s i d e r a t i o n s of higher order e f f e c t s . While w e l l worth the t r o u b l e from a research p o i n t of view, such c o n s i d e r a t i o n s no longer a r e , from the pragmatic p o i n t of view of the r e s u l t s . Other, more se r i o u s problems f i r s t needed a t t e n t i o n . However such s t u d i e s are important i n another way. In the i n v e r s e d i r e c t i o n , they i n d i c a t e d that q u i t e accurate estimates of l o c a l ' s t r u c t u r e s ' i n l i q u i d s (the word ' s t r u c t u r e ' meaning simply s t a t i s t i c a l l y most probable a s s o c i a t i o n s of groups of c o n t a c t s , hence of l o c a l a s s o c i a t i o n s of molecules i n the s o l u t i o n ) could be obtained i f a means could be found to e s t a b l i s h j u s t the e q u i l i b r i u m p a i r w i s e a s s o c i a t i o n prob-a b i l i t i e s of c o n t a c t i n g s i t e s of various p o s s i b l e types i n a s o l u t i o n c o n t a i n i n g more than two chemical s i t e types. C. M u l t i f u n c t i o n a l ' S o l u t i o n s - P a i r s Concentration M a t r i x A s o l u t i o n becomes m u l t i f u n c t i o n a l e i t h e r when one component of a b i n a r y mixture contains more than one type of 41 f u n c t i o n a l g r o u p , o r w hen m o r e t h a n t w o c o m p o n e n t s a r e p r e s e n t i n t h e m i x t u r e . I n t h e g e n e r a l c a s e b o t h s i t u a t i o n s e x i s t . T h e p r o b l e m o f how t o c a l c u l a t e t h e p a i r w i s e p r o b a b i l i t i e s i n m u l t i f u n c t i o n a l s o l u t i o n s a n d t h u s t h e p r o b l e m o f t h e d e -d u c t i o n o f ' t h e s t a t i s t i c a l l y f a v o r e d s t r u c t u r e s ' o f t h e t y p e d e s c r i b e d a b o v e , was s o l v e d b y B a r k e r (19) > R e c o g n i z i n g t h a t t h e v a r i a t i o n a l a p p r o a c h l e d t o i n s o l u b l e c o u p l e d s e t s o f p s e u d o q u a d r a t i c e q u a t i o n s i n t h e u n k n o w n p a i r w i s e c o n c e n t r a t i o n s w h e n m o r e t h a n t w o s p e c i e s o f f u n c t i o n a l g r o u p s ( h e n c e m o r e t h a n o n e u n k n o w n ) w e r e i n v o l v e d i n t h e a l g e b r a o f s t e p 4 ) o f s e c t i o n 5 o f C h a p t e r 2, B a r k e r p r o p o s e d a n a p p r o x i m a t e s o l u t i o n f o r t h e m u l t i f u n c t i o n a l . c a s e b a s e d o n s y m m e t r y a r g u m e n t s . T h i s a p p r o a c h , w h i l e p e r h a p s o v e r s y m m e t r i z i n g t h e p r o b l e m s l i g h t l y , s t i l l r e -s u l t e d i n e q u i l i b r i u m p a i r w i s e c o n c e n t r a t i o n s f r o m w h i c h t h e r m o -d y n a m i c p r o p e r t i e s i n c l o s e a c c o r d w i t h e x p e r i m e n t a l o n e s c o u l d b e o b t a i n e d . S i n c e t h e f o r m u l a t i o n was n o t a v a r i a t i o n a l o n e b a s e d o n p e r t u r b a t i o n s a b o u t t h e maximum r a n d o m n e s s c a s e , b u t was r a t h e r b a s i c a l l y a ' p r o b a b i l i t y ' a p p r o a c h a s m e n t i o n e d i n C h a p t e r 2, e v e n v e r y l a r g e e n e r g y e f f e c t s c o u l d b e t a k e n i n t o a c c o u n t , r e n d e r i n g t h e m e t h o d u s e f u l i n d e s c r i b i n g e f f e c t s i n s o - c a l l e d ' a s s o c i a t e d s o l u t i o n s ' o f p o l a r m o l e c u l e s . B e c a u s e o f t h e g e n e r a l i t y o f B a r k e r ' s a p p r o a c h , n o p r o b l e m s a r e e n -c o u n t e d i n t h e c a s e o f m u l t i c o m p o n e n t s o l u t i o n s o f e i t h e r mono-f u n c t i o n a l o r m u l t i f u n c t i o n a l s p e c i e s , . D. Q u a s i c h e m i c a l n - m e r T h e o r y A n o t h e r e x t e n s i o n o f t h e o r i g i n a l t h e o r y o f G u g g e n h e i m , d u e t o Change , F l o r y v ; a n d H u g g i n s ^ ' was t o develop a quasichemical theory of constant-z n-mers. (A n-mer i s a chain of n monomer u n i t s strung together. A constant-z n-mer s o l u t i o n i s one i n which both the monomer species and the n-mer species are assumed to have e q u a l l y s i z e d monomer u n i t s . In t h i s context a monomer i s not to be considered as a t o t a l molecular species but r a t h e r as a s i n g l e u n i t of a molecule, t h i s u n i t being r e c o g n i z a b l e only by i t s con-f i g u r a t i o n a l p r o p e r t i e s . Considerably a m p l i f i e d d e s c r i p t i o n of the p h y s i c a l s i t u a t i o n represented by the contents of t h i s p a r e n t h e s i s w i l l be given i n Chapter 6 ). The formal r e l a t i o n of quasichemical theory to the polymer t h e o r i e s of P l o r y -( 2 3 ) Huggins , e t c . , i s thus i n d i c a t e d . A r e s u l t of the Flory-Huggins theory was that the co n c e n t r a t i o n u n i t r e l e v a n t t o the c a l c u l a t i o n of standard heat of mixing and c o n f i g u r a t i o n a l e f f e c t s of mixtures i n a form comparable t o th a t f o r monomers was considered t o be the volume f r a c t i o n , i f one of the components of the mixture was ai n-mer of asymptotic l e n g t h . 2 . LESS CLOSELY RELATED THEORIES A. Regular S o l u t i o n s The connection between polymer theory and q u a s i -chemical theory had important consequences i n another f i e l d of s o l u t i o n theory, namely that of 'regular s o l u t i o n s ' , pioneered by Hildebrand^- 2^ >^\nd l a t e r a l s o developed by S c o t t ^ ^ . Hildebrand s u b s t i t u t e d volume f r a c t i o n s f o r mole f r a c t i o n s i n the heat of mixing term i n the expression f o r the 43 excess free energy of a l i q u i d mixture. In fact t h i s use of the volume f r a c t i o n can be shown hot to be a formal extension of quasichemical theory. Problems associated with the use of volume f r a c t i o n concentration units are discussed i n Chapter 10. B. The 2-Liquid Theory In various recent semiempirical theories r e l a t e d to quasichemical theory, renewed emphasis has been placed on the primary importance of l o c a l molecular configurations (on the nearest neighbor scale) i n the determination of excess mixture properties. The evolution of t h i s type of thought began with (2 6) Scott's^ proposal of the two-liquid theory, wherein the l o c a l composition about one species was considered to be d i f f e r e n t both c o n f i g u r a t i o n a l l y and e n e r g e t i c a l l y from the l o c a l environment of the other species. The l i q u i d was thus v i s u a l i z e d as an interpentrating mixture of the two l i q u i d s , each of which had the composition of one of the two l o c a l environments, s t i l l subject of course, to the constraint of the 'overall material balance on the component species present. C. Local Composition Theories The underlying idea of ' l o c a l composition' was elaborated i n various ways by Wilson^ 2"^ , Prausnitz et a l . ^ 1 0 \ H e i l ^ ^ ) and Renon^^) m The notable success of these theories has lent fresh impetus to investigations of l o c a l com-p o s i t i o n i n l i q u i d s , begun by the analysis of energ e t i c a l l y induced l o c a l c l u s t e r i n g e f f e c t s i n the theory of Guggenheim. Some of the aspects of the theories of these workers 44 w i l l be discussed i n Chapter 10, i n comparison with the approach taken i n t h i s work. 3. THE PROBLEM OF THE CZ LATTICE In t h i s section the problem of the r e s t r i c t i o n of the o r i g i n a l quasichemical formulation to the CZ case i s i n t r o -duced. For a ' l i q u i d ' v i s u a l i z e d as an assemblage of d i s - embodied pai r s of contacts between molecules, mixing energies and l o c a l clusterings a r i s i n g therefrom have been obtained. Next, for purposes of c a l c u l a t i o n of configurational entropies, a change i n v i s u a l i z a t i o n i s r e q u i s i t e ; namely, to revert attention to the l i q u i d as an assemblage of molecules, and then to consider the interchange behavior of the assemblage so regarded. In the quasichemical theory of solutions as o r i g i n -a l l y stated, i t was at t h i s point assumed that each molecule was located at the i n t e r s e c t i o n point of a regular l a t t i c e , a -facet of the theory d i r e c t l y borrowed from the l a t t i c e theories of the s o l i d state. In t h i s model, the number of possible species assignments to the s i t e s of a given con-f i g u r a t i o n of the system i s given by a combinatorial term enumerating possible interchanges of the two species (for the binary case) on an N-point 3-dimensional z-connected l a t t i c e . The problem i s that t h i s l a t t i c e configurational model i s i n -capable of mathematically accommodating l o c a l packing e f f e c t s (energetic or geometric). 45 These geometric e f f e c t s a r i s e out of changes i n the number of nearest neighbors found about a molecule of a given species i n a mixture of species of unequal s i z e s , r e l a t i v e to the number of nearest neighbor contacts the given species makes i n the pure component s t a t e . The f a c t that the l a t t i c e model might a l s o seem to over-order the system, i n that both l o c a l and long range geometric o r d e r i n g i s i m p l i e d , i s not the prob-lem, since the l a t t i c e i s used only as a b a s i s f o r counting p o s s i b l e interchanges without regard to the r e g u l a r i t y , per se, of the large s c a l e a r r a y . The r e a l problem of the CZ l a t t i c e i s that l a r g e s c a l e consequences r e s u l t i n g from the l o c a l p e c u l i a r i t i e s induced by packing e f f e c t s are not acknowledged by the method. While f o r mixtures of equal s i z e d molecules, the CZ theory s t i l l produces the c o r r e c t 'ideal' entropy of mix-i n g term, f o r mixtures of unequal s i z e d molecules i t does not, and i n such instances various problems a r i s e i n the a p p l i c a t i o n of the s t r a i g h t f o r w a r d CZ approach. F o r t u n a t e l y , the other assumptions and b a s i c thermo-dynamic and s t a t i s t i c a l arguments of the quasichemical approach are i n no way i n h e r e n t l y dependent upon the above ' l a t t i c e ' assumption, though formulations are thereby g r e a t l y s i m p l i f i e d . Since l i q u i d s do_ have l o c a l conf i g u r a t i o n a l anomalies not ex-p l i c i t l y i n c l u d a b l e i n the constant-z l a t t i c e model, a major, and, i t seems, an unnecessary, a r t i f i c i a l i t y i s introduced i n t o the theory by r i g o r o u s adherence to the d i c t a t e s of a l a t t i c e c o n f i g u r a t i o n a l model. How t h i s r e s t r i c t i o n was r e -moved by the author i s the t o p i c of subsequent chapters. 46 4. SUMMARY OF THE SOURCES OF EARLIER WORK USED The foregoing I n t r o d u c t i o n to constant-z approaches i s s u f f i c i e n t to provide a context f o r the e f f o r t s of the. present work o u t l i n e d i n subsequent chapters. I t i s seen that the CZ basis from which the present work i s g e n e r a l i z e d i s that of Guggenheim ( S - r e g u l a r s b i n a r y quasichemical model of the ' p a i r s ' l i q u i d ) and that of B a r k e r * ^ ) (extensions of quasichemical theory to the multicomponent case). The present work has a l s o b e n e f i t t e d more i n d i r e c t l y from ideas of the f o l l o w i n g men: Langmuir(30) (surface i n t e r a c t i o n s ) ; Scatchard^ 31) (surface ( po \ • f r a c t i o n s ) ; Flory-Huggins J J (volume f r a c t i o n r e p r e s e n t a t i o n f o r polymers); H i l d e b r a n d ^ ^ (volume f r a c t i o n s f o r monomers); Scott ' ( t w o - l i q u i d t h e o r y ) ; Bethe ( e n e r g e t i c second neighbor c o r r e c t i o n f a c t o r s ) , - and from the recent work of Wilson*' 2 7'* , P r a u s n i t z et a l ( ' 1 0 ' ) . , Renon^ 2^ , . H e i l ^ 2 8 a n d Derr and Deal ( a l l concerned w i t h the idea of ' l o c a l ' com-p o s i t i o n , an idea r e s u l t i n g i n expressions such as that of Wilson, or that of H e i l , or the NRTL equation f o r G ). In (34) • (35) a d d i t i o n , recent work by Sams , Bondi , and Kreglewski (3 6) ^ , on i n t e r m o l e c u l a r f o r c e s , and that of Lielmezs and Bondi (•57) V J ' , on Van der Waals r a d i i , have been found to be u s e f u l . * Note: S-regular i s the term used by Guggenheim to des-c r i b e a s o l u t i o n i n which the c o n f i g u r a t i o n a l energy term was the same as i n a Raoult's law s o l u t i o n . Regular s o l u t i o n s are those defined by Hildebrand as e x h i b i t i n g a systematic de-parture from the i d e a l mixture due to s i z e induced e f f e c t s . 4 7 • CHAPTER 4 • REVIEW OF THE PHYSICAL BASIS OF CELL THEORIES 1. GENERAL 2. THE NATURE OF LOCAL ORDERING IN A LIQUID 3. CONSEQUENCES OF LOCAL ORDERING IN MODEL BUILDING 4 . LIMITATIONS IMPLICIT IN A SINGLE-OCCUPANCY CELL MODEL 5. STATISTICAL AND CONFIGURATIONAL ASPECTS OF SINGLE-OCCUPANCY CELL MODEL 6. ENERGETIC ASPECTS OF THE PHYSICAL MODEL A. Nearest-Neighbor A l l o c a t i o n of L i q u i d Cohesion B. The P a i r w i s e Nature of Inter m o l e c u l a r Cohesions 1. P a i r w i s e A d d i t i v i t y of Cohesions 2 . P a i r w i s e Independence of Cohesions. 3. Axiomatic A p p l i c a b i l i t y to the Multicomponent Case 7 . CONSTANT-VOLUME PROPERTY ESTIMATES FROM A QUASICHEMICAL MODEL A. E f f e c t on Estimate of G E B. E f f e c t on Estimate of H 8. A SPANNING SET OF INDEPENDENT VARIABLES 48 CHAPTER 4 REVIEW" OF THE PHYSICAL BASIS OF CELL THEORIES 1. GENERAL Before embarking on the d e t a i l s of the t h e o r i e s i n t h i s t h e s i s , a review of c e r t a i n p h y s i c a l p r i n c i p l e s and ob-servations i s provided f o r r e f e r e n c e . Two groups of phenomena are enumerated and discussed: one, c o n f i g u r a t i o n a l or geometric or s t r u c t u r a l , the other, energetic or cohesive. A d e s c r i p t i o n of the independent v a r i -able set employed i s a l s o provided, as w e l l as a mention of c e r t a i n important s t a t i s t i c a l and thermodynamic p r i n c i p l e s bearing on model c o n s t r u c t i o n . A b r i e f resume'' of p r o p e r t i e s of types of p a r t i t i o n f u n c t i o n encountered i s i n c l u d e d as an extension of the d i s c u s s i o n of u n d e r l y i n g s t a t i s t i c a l p r i n -c i p l e s . 2. THE NATURE OF LOCAL ORDERING IN A LIQUID -C o n f i g u r a t i o n a l l y the dense l i q u i d s t a t e e x h i b i t s long range and long time chaos yet l o c a l , short-time o r d e r i n g . I t should be r e a l i z e d that these two sets of c h a r a c t e r i s t i c s are not mutually c o n t r a d i c t o r y , and i n f a c t both c h a r a c t e r i s t i c s must be e x h i b i t e d by any r e a l i s t i c model of a l i q u i d . By • l o c a l ' i t i s s p e c i f i c a l l y meant t h a t , i f a given molecule i s chosen as. the c e n t r a l one, then the number, nature, and p o s i t i o n of the molecules i n i t s nearest neighbor s h e l l i s i n f l u e n c e d by c e r t a i n p r o p e r t i e s of the c e n t r a l molecule. For example the c o o r d i n a t i o n number, which i s defi n e d as the number of such nearest neighbors, i s i n f l u e n c e d by l o c a l geometric, and ener-g e t i c c o n s i d e r a t i o n s as w e l l as by general s t a t i s t i c a l con-s i d e r a t i o n s . The o r d e r i n g e f f e c t of a c e n t r a l molecule dimin-ishes so r a p i d l y beyond the f i r s t neighbors s h e l l , however, that f u r t h e r - n e i g h b o r o r d e r i n g w i t h respect to a given c e n t r a l molecule may be ignored f o r purposes of the present theory. On the other hand, the i n f l u e n c e of the c e n t r a l molecule l s _ so strong upon i t s nearest neighbors that the c o o r d i n a t i o n number can be thought of as a ' m a t e r i a l property' a t t r i b u t a b l e to each species of c e n t r a l molecule i n a mixture. As w i l l be seen i n Chapter 6, the magnitude of t h i s property can be c a l c u l a t e d q u a n t i t a t i v e l y . . 3 . CONSEQUENCES OF LOCAL ORDERING IN MODEL BUILDING I t w i l l be seen from a model-building p o i n t of view that the dense l i q u i d can o f t e n be viewed f a i r l y s u c c e s s f u l l y simply as a s u p e r p o s i t i o n of nearest neighbor s t r u c t u r e s . The f a c t that these s t r u c t u r e s i n t e r p e n e t r a t e needs to be accounted f o r only i n second order adjustment to estimates of the l i q u i d ' s (32) p r o p e r t i e s . Moreover, i t becomes obvious that average l o c a l composition of nearest neighbor c e l l s t r u c t u r e s need not n e c e s s a r i l y conform e x a c t l y (or even c l o s e l y ) to the o v e r a l l average composition of the mixture. L o c a l and o v e r a l l average 50 compositions only c o i n c i d e i f there are no l o c a l geometric or cohesive r e s t r a i n t s imposed on the maximum-randomness homo-geneity of the mixture. Whenever these l o c a l geometric and cohesive e f f e c t s are not the same f o r a l l species present, both of these e f f e c t s b i a s l o c a l compositions to some extent. E f f e c t i v e t h e o r i e s must account f o r such l o c a l e f f e c t s w i t h some accuracy, f o r , while the e f f e c t s are l o c a l w i t h respect to any one molecule, they are a l s o general i n the sense that they a f f e c t a l l the i n d i v i d u a l molecules of a given type i n the system, hence they b i a s the p a r t i a l molal p r o p e r t i e s of the o v e r a l l system. 4. LIMITATIONS IMPLICIT IN A SINGLE-OCCUPANCY CELL MODEL I t i s p o s s i b l e to make use of a conceptual " c e l l " framework i n the d e s c r i p t i o n of a p a r t i c l e aggregate under d e n s e - l i q u i d c o n d i t i o n s . Such a conceptual framework serves as a b a s i s f o r c a l c u l a t i n g the cohesion of a molecule w i t h i n a c e l l a r i s i n g from i t s i n t e r a c t i o n s w i t h molecules forming the c e l l (which are t h e r e f o r e those _in adjacent c e l l s ) . The same concept a l s o serves as a guide to i n d i c a t e which s t a t i s t i c a l manipulations are a p p l i c a b l e f o r the aggregate. The p r i n c i p a l l i m i t a t i o n s on r e a l behavior r e q u i r e d to make v a l i d the assumption of a single-occupancy c e l l model are t w o f o l d : i ) f i r s t l y , because the space occupied by the l i q u i d i s thought of as being d i v i d e d up i n t o i n d i v i d u a l c e l l s of a s i z e to hold one molecule each, the a c t u a l number of empty c e l l s or m u l t i p l y - o c c u p i e d c e l l s at any one moment must be small enough 51 to be i n s i g n i f i c a n t f o r present purposes. This s i t u a t i o n t y p i c a l l y e x i s t s , at l e a s t to a good approximation, i n the dense l i q u i d s t a t e . i i ) secondly, the frequency of interchange of p o s i t i o n s of molecules between neighboring c e l l s must be much l e s s than the frequency of random thermal b u f f e t i n g motions w i t h i n a given c e l l . This assumption, i m p l i c i t i n quasichemical t h e o r i e s , has been borne out by molecular dynamics s t u d i e s ^ 8 ) _ 5. STATISTICAL AND CONFIGURATTONAL ASPECTS OF SINGLE- OCCUPANCY CELL MODEL On the s c a l e of nearest neighbors s h e l l s , adherence to s i n g l e occupancy c e l l behavior means that a given c e n t r a l molecule i s c o n s t r a i n e d c o n f i g u r a t i o n a l l y to_ i t s c e l l or cage of nearest neighbors s u f f i c i e n t l y permanently that various caging arrangements can be formulated, then.averaged i n some appropriate way to o b t a i n i n f o r m a t i o n about the e f f e c t s of a given species of molecule's average environment of nearest neighbors. The averaging procedure used i n c o r p o r a t e s a formal r e c o g n i t i o n of the f a c t that the frequency of any p a r t i c u l a r nearest neighbor c o n f i g u r a t i o n i s a l s o governed by the s t a t i s t i c s of the mass as a whole. The r e s u l t of making the assumptions of s i n g l e c e l l occupancy, caging, and the meaningfulness of an average l o c a l environment f o r a given species of molecule i s that the mathematical s i m p l i f i c a t i o n of breaking up the 3-N space of an N - p a r t i c l e aggregate i n t o N f r e e l y permutable 3-spaces becomes p o s s i b l e . This s i m p l i f i c a t i o n r e s u l t s i n the. , -. avoidance of c e r t a i n i n t r a c t a b l e 3-N space c o n f i g u r a t i o n a l i n -t e g r a l s d e s c r i b i n g the p o t e n t i a l energy of the l i q u i d , and i n -stead a l g e b r a i c a l l y approximates these i n t e g r a l s by o r d i n a r y 3-space i n t e g r a l s f o r I n d i v i d u a l p a r t i c l e s , plus a purely s t a t i s t i c a l g e n e r a l i z a t i o n of the r e s u l t s f o r the i n d i v i d u a l p a r t i c l e s to form a d e s c r i p t i o n of the whole aggregate. The mathematical end-product of these s i m p l i f i c a t i o n s i s c a l l e d a p a r t i t i o n f u n c t i o n . P a r t i c l e s of a given species are r e -garded as I n d i s t i n g u i s h a b l e . As a matter of f a c t , the average cohesion per p a r t i c l e of a c l a s s of p a r t i c l e s i s used to c h a r a c t e r i z e I n d i v i d u a l p a r t i c l e energies i n the f o l l o w i n g C39) treatment . For geometric purposes a p a r t i c l e i s a monomer u n i t ; f o r e n e r g e t i c purposes, i t i s a ' p a i r 1 . 6. ENERGETIC ASPECTS OF THE PHYSICAL MODEL E n e r g e t i c a l l y , two p r o p e r t i e s of dense l i q u i d s proved to be very u s e f u l i n a r r i v i n g at mathematically t r a c t a b l e f o r m u l a t i o n s f o r G . They are the approximation of nearest neighbor a l l o c a t i o n of l i q u i d cohesion, and the approximation of p a i r w i s e independence (hence p a i r w i s e a d d i t i v i t y ) of the E E i n t e r m o l e c u l a r cohesion, and the approximation of G ^ = A ^ A. Nearest-Neighbor A l l o c a t i o n of L i q u i d Cohesion S t u d i e s ^ ^ of the cohesion energy b i n d i n g a g i v e n molecule i n t o the l i q u i d s t a t e i n d i c a t e that over two t h i r d s of the cohesion a r i s e s from I n t e r a c t i o n s w i t h nearest neighbors. * The term ' p a r t i t i o n f u n c t i o n ' i s thus used i n a r e s t r i c t e d sense i n t h i s t h e s i s to apply only to the p o t e n t i a l energy part of the Hamiltonian. 53 A major s i m p l i f y i n g assumption a l l o c a t i n g the e n t i r e cohesion to nearest neighbor i n t e r a c t i o n s i s thus found to be p r a c t i c -a b l e , at l e a s t f o r purposes of c a l c u l a t i n g net cohesion changes produced by mixing i n nonionic s o l u t i o n s . The e v a l u a t i o n of mixture cohesion changes f o r a nearest-neighbor l i q u i d model i s thus, to a f i r s t approximation, p h y s i c a l l y meaningful. B. The P a i r w i s e Nature of I n t e r m o l e c u l a r Cohesions 1. P a i r w i s e A d d i t i v i t y of Cohesions The second property of cohesion, t h a t of i t s approx-imate p a i r w i s e a d d i t i v i t y , was f i r s t r e v e a l e d by the quantum (41) ' mechanical s t u d i e s of London , who demonstrated that the p r i n c i p l e holds c l o s e l y f o r d i s p e r s i o n - f o r c e systems ( i n v o l v i n g a t t r a c t i o n s between nonpolar molecules). Adoption of the assumption i n g e n e r a l , w h i l e not s t r i c t l y c o r r e c t , proves to be a u s e f u l s i m p l i f y i n g assumption when used w i t h s u f f i c i e n t c a u t i o n . . The concept of p a i r w i s e a d d i t i v i t y of cohesion, r e -duced to mechanical terms, i s that the cohesion between any two molecules i s not a f f e c t e d by the presence c r absence of other molecules. Of p a r t i c u l a r i n t e r e s t i s the p a i r w i s e a d d i t i v i t y of cohesions of neighboring ("contacting") molecules. 2 . P a i r w i s e Independence of Cohesions In a d d i t i o n , the concept of p a i r w i s e a d d i t i v i t y of cohesions contains i n i t . t h e assumption of p a i r w i s e i n - dependence of cohesions. This assumption of independence i s important i n the present work, si n c e i t e n t a i l s the i d e a , that i n a nearest neighbors s h e l l of molecules about a ' c e n t r a l ' one, the t o t a l cohesion of the c e n t r a l molecule I s changed when the number of neighbors changes as w e l l as when the k i n d of neighbors changes. I f the process of mixing causes a modi-f i c a t i o n of packing e f f e c t s due to the s i z e d i f f e r e n c e of the vari o u s components i n the nearest-neighbors s h e l l , then the cohesion a t t r i b u t a b l e to the c e n t r a l molecule w i l l change because of both these e f f e c t s . A much f u l l e r development of t h i s theme of m o d i f i c a t i o n of packing e f f e c t s due to mixing of unequal s i z e d molecules Is given i n Chapter 6. 3. Axiomatic A p p l i c a b i l i t y to the Multicomponent Case Another f a c e t of the p a i r w i s e a d d i t i v i t y assumption i s t hat i f the nature of a given type of ' u n l i k e ' p a i r w i s e a t t r a c t i o n i s determined i n the simplest case of a b i n a r y mix-t u r e , then the same value w i l l c l o s e l y apply f o r that i n t e r -a c t i o n i n the multicomponent mixture case, under the assumption that a given p a i r i n t e r a c t i o n i s independent both of the number and nature of other p a i r contacts i n t o which the two molecules enter. In a c o r r e c t l y formulated e x p r e s s i o n , the g e n e r a l i z -a b i l i t y of binary energy parameters i n t o multicomponent s i t u -a t i o n s i s thus a t e s t of the degree t o which nearest neighbor cohesions are of a p a i r w i s e a d d i t i v e nature. 7. CONSTANT-VOLUME PROPERTY ESTIMATES FROM A QUASICHEMICAL  MODEL A u s e f u l a t t r i b u t e of l i q u i d mixtures from the p o i n t of view of f o r m u l a t i n g G i s the c l o s e approximation of the Gibbs to the Helmholtz f r e e energy f o r a dense l i q u i d . Since G - A + iiv- , and since nv for the l i q u i d state i s only a few percent of the magnitude of -A »G - A A. E f f e c t on Estimate of G E Since A = A (TjVjN^), then for an equilibrium mixture W N ,T = 0 ' Thus, at least to a f i r s t approximation, volume changes due to mixing do not enter into estimates of G ~ A , although various second-order e f f e c t s are of course non-zero. S t i l l , i n t h i s work, such terms w i l l not be included as being of small magnitude, and not easy to c a l c u l a t e . Hence i n t h i s E E work the standard approximation G - A w i l l be made. B. E f f e c t on Estimate of H E I f i n expressions for GE = H E - TSE ( 5 ) E the H terms are separated out and used as the basis for heat of mixing estimates, H (v,T) ^ s a n incomplete estimate of heat of mixing, and may be i n serious discrepancy with H (JJ,T) which i s the property relevant to d i s t i l l a t i o n c a l c u l a t i o n s . 3H If -jjy n T d i f f e r s s i g n i f i c a n t l y from zero, the volume change term may even i n some cases overshadow the constant volume term ( 1 2 ) i n equation ( 5 ) just stated . The heat of mixing behaviour of ethanol-water solutions i s a case i n point. 8. A SPANNING SET OF INDEPENDENT VARIABLES Now that the set of general geometric and energetic observations about the dense l i q u i d state relevant to model bu i l d i n g have been noted, the set of independent variables found to be s u f f i c i e n t to describe a general l i q u i d mixture needs to be introduced. These v a r i a b l e s have already been enumerated In the f u n c t i o n bracket of equation (3) of Chapter 2, namely the set (R±,n±,r\±,x±,in±.A^n.T) . They are now to be b r i e f l y d e s c r i b e d . R i i s the e f f e c t i v e dimension-l e s s * diameter of a molecule (or molecular segment, i n the case of a long molecule) occupying one u n i t of the packing arrangement. The chain l e n g t h parameter n^ i s the number of u n i t s of the packing arrangement occupied by species i when the species i s a long molecule. The n-j_ stands f o r the p r o p o r t i o n of a molecule of a given species devoted to the i type of f u n c t i o n a l group f o r a m u l t i f u n c t i o n a l molecule. I f only one type of f u n c t i o n a l group e x i s t s , the species i s monofunctional and n ± assumes the t r i v i a l value of 1. The r e l e v a n t energy v a r i a b l e i s 03 i j > namely the net p a i r w i s e r e -o r g a n i z a t i o n energy, r e p r e s e n t i n g the c o e f f i c i e n t of the heat of mixing e f f e c t . I t i s r e s p o n s i b l e f o r e n e r g e t i c a l l y caused l o c a l c l u s t e r i n g e f f e c t s i n the l i q u i d . A^ i s the pure component Helmholtz f r e e energy of the i s p e c i e s . The p o p u l a t i o n f r a c t i o n or mole f r a c t i o n of .a given species i s given by the u s u a l symbol x^ . T and n have t h e i r u s ual thermodynamic meanings of system temperature and pressure. For monofunctional mixtures, i r e f e r s to a mole-c u l a r s p e c i e s . For m u l t i f u n c t i o n a l o n e s , i r e f e r s t o a p a r t i c u l a r * w i t h respect to an a r b i t r a r i l y chosen reference r a d i u s to reduce R to a r a d i u s r a t i o , hence dimensionlessness. 57 type of chemical s i t e , and the molecular species index is. re p l a c e d by the Index "a". Retention of i and j f o r a mole-c u l a r species as w e l l as f u n c t i o n a l group index f o r mono-f u n c t i o n a l l i q u i d s i s to maintain consistency w i t h the no-t a t i o n of Guggenheim: subsequent adoption of 'a' as a mole-c u l a r species index, r e t a i n i n g I as the f u n c t i o n a l group i n -dex a r i s e s out of the n e c e s s i t y to d i s t i n g u i s h 'chemical' and 'geometric' aspects of the molecule i n the l a t e r treatment of m u l t i f u n c t i o n a l substances. A l l v a r i a b l e s are f u r t h e r de-f i n e d both i n nomenclature and o p e r a t i o n a l l y i n subsequent chapters. 58 CHAPTER 5 RELATIONSHIPS INVOLVING' THE PARTITION FUNCTION 1. THERMODYNAMIC BASIS A. The Nature of A and 0. B. A and Q. as Functions of Volume C. A and Q. as Functions of Concentration D. R e l a t i o n of A to E and S E. D i f f e r e n c e of the Temperature Response of A from that of E and S F. The Gibbs Helmholtz Equation f o r Free Energy H. Summary 2 . FORMS OF THE PARTITION FUNCTION A. The Canonical P a r t i t i o n Function B. The Grand Canonical P a r t i t i o n Function C. The G.C.P. i n D e s c r i p t i o n s of L o c a l (Nearest-Neighbor S h e l l ) Behavior 3. APPROXIMATION OF THE G.C.P. BY A MULTINOMIAL SERIES 4 . PROBLEMS IN THE DIRECT USE OF THE G.C.P. 5 . USE OF THE G.C.P. TO ILLUSTRATE LOCAL PHENOMENA IN SOLUTIONS 6. DEVIATION OF LOCAL FROM AVERAGE COMPOSITION 7. PARTIAL MOLAL PROPERTIES 8. MACROSCOPIC SYSTEM PROPERTIES DERIVED FROM MOLECULAR SCALE PARAMETERS 59 9. USE OP DIFFERENT SCALES IN THE ft AND" CANONICAL PARTITION FUNCTION £(N) FACTORS OF THE f. 1 60 CHAPTER 5 RELATIONSHIPS' INVOLVING' THE PARTITION FUNCTION 1 . THERMODYNAMIC BASTS A. The Nature of A and £ As indicated i n a very cursory way i n Chapter 2, the ( 2 3 ) chief i n t e r e s t i n 0. (the Canonical Ensemble P a r t i t i o n Function' or C.P.F.) for the purposes of t h i s thesis arises from the fact that the usual route followed to obtain thermodynamic property predictions (such as those for Y and H ) from a given molecular model of a l i q u i d Involves the incorporation of modelling assumptions into a formulation for the p a r t i t i o n function ( Q. -for one mole of the mixture i n t h i s case), and then subsequent-ly obtaining Y and H by carrying out cer t a i n standard mathe-matical operations on Q, as shown schematically i n I l l u s t r a t i o n 2. The r e l a t i o n A = -kT iniQ) ( 7 ) can be rearranged as follows. An i n i t i a l t r i v i a l t r ansposition y i e l d s : -A/kT = In Q. ( 8 ) In t h i s form, the equation i s dimensionless. I f the operational d e f i n i t i o n of 0. , that £ i s the sum 61 of Boltzman-factor weightings f o r the n p a r t i c l e s comprising the system i s w r i t t e n out f o r m a l l y , one o b t a i n s : * n -E./kT -A/kT = Hn [I e 1 ] (9) i -A/kT v -e./kT (10) e = I e x The dlmenslonless Boltzman f a c t o r argument, (-A/kT) i s thus d e f i n e d as being the equivalent to the sum of the argu-ments of the Boltzman f a c t o r s of the R.H.S. s e r i e s . This i s the b a s i c o p e r a t i o n a l d e f i n i t i o n of Helmholtz Free Energy A, or r a t h e r of the dimensionless q u a n t i t y (-A/kT). To convert the sum A t o a mean value of A per p a r t i c l e , the r e l a t i o n s n ( e ~ a / k T ) = 6" A/ k T = I e i / k T ( I D i may be solved f o r a. A i s thus the s i n g l e energy f a c t o r which c h a r a c t e r i z e s the energy exponents of the s e r i e s represented by the C.P.F. I t i s seen that the compact form of the L.H.S. of equation (10) permits of mathematical manipulations which would be extremely awkward w i t h a sum of exponentials as i s the R.H.S. The r e l a t i v e compactness of the L.H.S. of equation (10) compared to i t s R.H.S. i s most important, when not Just one but * i f Q., the molar s c a l e p a r t i t i o n f u n c t i o n i s being d i s c u s s e d , n = N . 62 a sequence of transformations must be made on 0. . Consider, for example the series of operations by means of which Y i (the a c t i v i t y c o e f f i c i e n t of component i) , Is derived from the expression for Q. • B. A and d as Functions of Volume In the. case where e j i s the average value of the point p o t e n t i a l over a small volume domain v^ (the domain volumes need not be of equal s i z e ) , and V i s the t o t a l volume ( i . e . v = I v. ), one can write j j e-A/kT = 1 ; V k T ( / v ) (12) j 3 where (u./v) i s a volumetric weighting factor or s p a t i a l prob-a b i l i t y of the j t h domain. . By transposing the common denominator V to the L.H.S. the expression i s rearranged to e - A / k i [ v i - 1 r ^ / k T „. ci3> , In the l a t t e r form, which i s most applicable to a c e l l model of a l i q u i d , i t i s self-evident that (-A/kT) i s the Mean-Value - Theorem Boltzman factor argument which, when applied to the whole volume V, make the L.H.S. of equation (13) equivalent to the sum of the R.H.S. series of l o c a l weighting factors and domain volumes. Note now that i n equation (13) A bears the same r e l a t i o n to the Individual e as E bears to j i the E within the j domain. Two points should be raised i n 63 t h i s regard. F i r s t l y , the v are f o r m a l l y volumes of i n t e g r a t i o n over which e i s the appropriate average energy. In the g e n e r a l case, these volumes of i n t e g r a t i o n can o v e r l a p , or i n t e r -p enetrate, or be as l a r g e as the t o t a l p h y s i c a l volume of the system. Thus, i m p l i c i t i n the r e l a t i o n £ v. = V i s the j J a d d i t i o n a l assumption that domain volumes correspond to r e a l volumes of the order of V/n ( f o r an n - p a r t i c l e system), so t h a t , i n consequence the v do not s e r i o u s l y overlap. This i s one of the s p e c i a l s u p p o s i t i o n s r e q u i r e d of a c e l l theory. Secondly, that the "appropriate form of average" i s the average exponent i n t h i s case. The l e v e l of s u b d i v i s i o n i n t o domains defines the l e v e l at which one begins t o c a l l the e x p o n e n t i a l average energy of a domain (which i s r e a l l y thus a f r e e energy) simply "the energy". This degree of s u b d i v i s i o n corresponds to the degree of " f i n e - g r a i n i n g " adopted i n r e -garding the system. A i s seen to vary not only w i t h the e.. , but a l s o w i t h t h e i r d i s p o s t i o n i n space, and w i t h the r e l a t i v e abundance of the p a r t i c u l a r species present, i f these species have d i f f e r e n t c h a r a c t e r i s t i c values of e j and v j C. A and 0. as Functions of Concentration* I f the k^*1 species has c h a r a c t e r i s t i c values of energy and domain s i z e ek and \ r e s p e c t i v e l y , and i s present i n con-c e n t r a t i o n \ then f o r an N - p a r t i c l e assemblage * r e a l l y , most d i r e c t l y ^ of N.. not of x. 64 e-A/kT [ v ] = N ^ -e, /kT (14) D. Relation of A to U and S The difference i n nature between A (the Free Energy) and U (the Total Energy), and t h e i r r e l a t i o n to S (the system's Entropy) are c l e a r l y borne out through d e f i n i t i o n s of each as relat e d to 0.. In p a r t i c u l a r , i t i s of in t e r e s t to r e l a t e A to U, because u , as a t o t a l energy, i s an inherently more under-standable quantity than A, hence serves as a point of reference for the understanding of A. The difference betwen the natures of A and U i s shown by expanding the axiomatic d e f i n i t i o n of E, namely s V = I e. p. -e /kT P t - e /Q. U5) (16) Equation (15) i s simply a formal expression of the fact that the o v e r a l l energy i s the average of the i n d i v i d u a l component energies of the system. ( I t i s an average, not an average exponent, regardless of the fact that Boltziman factors may have been used to weight t h i s average). 65 By simple operations on Q. i t can be shown that equation (15) i s equivalent to. U = k T 2 H n ( ) • /(17) (as compared to the d e f i n i t i o n of A In terms of 0., namely A=-kT In Q. ) The r e l a t i o n between U and A, invo l v i n g S, i s ob-tained straightforwardly by expanding the " p r o b a b i l i t y " de-f i n i t i o n 3 ^ S,and,after i n s e r t i n g the e x p l i c i t form of P± given by equation (16) -e./kT -e. S = -k I (e 1 /Q) [—| - an QJ (18) e. S - ~k I j± p. + k Sin a (19) S = | + k An a (20) whence i s obtained, by a t r i v i a l rearrangement, -kT £n Q. = U - TS (21) F i n a l l y , by s u b s t i t u t i n g the d e f i n i t i o n of A for the L.H.S., * leaving aside, for the moment, the necessity for the P± to be the p r o b a b i l i t i e s of independent events (See Chapter 9) 66 i s obtained •A - IT- TS (22) The reason for the term 'free' energy for A i s that A i s the difference between the t o t a l energy u> and the pro-duct of temperature times the system Entropy S. E. Difference of the Temperature Response of A from that of U and S A and U, apart from having quite d i f f e r e n t d e f i n i t i o n s i n terms of 0. (and hence responding d i f f e r e n t l y to e k > a?k» are shown by equation (22) to have b a s i c a l l y d i f f e r e n t temper-a t u r e - c h a r a c t e r i s t i c s . A i s always an e x p l i c i t function of T (because of the TS product i n equation (22)), whereas S and U are only i m p l i c i t functions of T i n t h i s regard. Also, because i n equation (16), P k for a species k involves i t s mole f r a c t i o n #k , equation (22) implies that A i s a much more active function of concentration than U, again because of A's dependence on S, hence on« , through the entropy of mixing e f f e c t . F. The Gibbs Helmholtz Equation for Free Energy S can be eliminated from equation (22) by making the sub s t i t u t i o n H A ) V S = " ~ 9 T ~ ~ C*3') 67 (which can a l s o be a r r i v e d at from the above d e f i n i t i o n s of A and S) . The s u b s t i t u t i o n gives r i s e to the Gibbs-Helmholtz equation WT) = ( 2 h ) 8(1/T) U U ^ or 9 (A/T) = _ U_ ( ) The Gibbs-Helmholtz r e l a t i o n holds e q u a l l y f o r p a r t i a l molal excess q u a n t i t i e s A.^  andU.^ i n place of A and u i n equation ( 2 5 ) . Since SL n Y ± = A^/RT (26) the p a r t i a l molal v e r s i o n of equation (25) serves to p r e d i c t the primary temperature dependence of a c t i v i t y c o e f f i c i e n t s as an i n t e g r a b l e d i f f e r e n t i a l equation i n T: = ~ ^ 2 ( 2 ? ) F F i n a l l y , because A - G i n systems considered,H. can be used i n place of U ± i n equation (27.). 68 H. Summary Though some of the foregoing d e f i n i t i o n s are un-avoidably being drawn on out of order In terms of the l o g i c a l u development of the arguments of the t h e s i s , the compactness of the framework of thermodynamic and s t a t i s t i c a l mechanical re-l a t i o n s among the primary quantities Q., A, U, S, T, and also among V, x and y, alluded to In Chapter 2, i s revealed, and the •operational' nature of r e l a t i o n s between A and Q. i n p a r t i c u l a r i s also s u i t a b l y i l l u s t r a t e d for use-in l a t e r discussion. 2. FORMS OF THE PARTITION FUNCTION At t h i s point It w i l l be useful to make some general comments about the technique of applying p a r t i t i o n functions to the present study, about the realm of a p p l i c a b i l i t y of c e r t a i n types of p a r t i t i o n functions', and about r e l a t i o n s between types of p a r t i t i o n functions and the scales of assemblage which must be considered i n order to develop various aspects of the theory. A. The Canonical P a r t i t i o n Function One convenient,type of p a r t i t i o n function used i s the Canonical Ensemble P a r t i t i o n Function (C.P.F.) designated by the symbol Q.. This obtains for a fixed population imposed by the assumption of a closed system, for example for one mole of a mixture of a s p e c i f i e d composition. A mathematical feature of the C.P.F. important to t h i s work i s that i t can be r e -presented as a mathematical product of two types of factors. This i s shown by the formalism; 69 i a = (n) >< (TT f±) (28) « where the Individual p a r t i c l e weight f ^ i s given by: -E./kT f ± = g. e 1 (29) where e i s a generalized energy of a molecule, and g^ i s a generalized geometry factor per molecule depending on i t s domain volume. i The portion (II f ) i s the weight of a given •per- mutation' of the p a r t i c l e s In a given configuration of the system, hence i t s e l f i s a product of the weights (the f i ) of the i n -d i v i d u a l p a r t i c l e s of the assemblage, and fi i s a combinatorial term, enumerating the number of distinguishable species assign-ments possible for the chosen system configuration. I f the CP.P. for an i d e a l i z e d binary n - p a r t i c l e system, consisting of n^ p a r t i c l e s of species 1 and p a r t i c l e s of species 2 i s designated Q.(nn ,n ?), then a(n 1,n 2) = [n!/( n ; L!n 2!)] x g l 6 -£./kT l n. g 2 e -e2/kT <30) Factors g^ and are again generalized geometry fa c t o r s , and E 1 , and E 2 are energy factors per molecule, but now s p e c i f i c a l l y applying to the average cohesion per molecule of 1 and 2, r e s -p e c t i v e l y , i n the mixture. 70 B. The Grand Canonical P a r t i t i o n Function A second type of p a r t i t i o n function used i n arguments i n t h i s work i s the Grand Canonical Ensemble P a r t i t i o n Function (G.C.P.)*, designated by the symbol /. The G.C.P. applies to an open system, i n which the molecules are free to enter and leave, although i t i s usually written for the case where the net balance of entrance and egress rates maintains a constant t o t a l population within the system. While the choice be-tween regarding the system as being "open" or "closed" i s often a matter of convenience, the thermodynamic quantities eventually extracted from the r e s u l t i n g p a r t i t i o n functions w i l l depend on the choice of basis which was made. Whereas / the " c h a r a c t e r i s t i c " function of A = _ k T ^ n £, the character-i s t i c function corresponding to £ for an open system i s n V = RT £n 7 ( 3 1 ) -From the-point-of view of estimating Free Energies, t h i s c h a r a c t e r i s t i c function for i. means that A can only be ob-tained from f by a rather roundabout thermodynamic route, which i n fact proves to be rather unsatisfactory. The con-cept of the G.C.P. i s most useful however, i n i l l u s t r a t i n g the role of l o c a l - s c a l e behavior i n solutions ^ and for t h i s i t i s introduced here. * These abbreviations; namely G.C.P. and C.P.F. are those employed by H i l l ^ 3 ) 71 The G.C.P. f o r n p a r t i c l e s i n a 2-species system i s given by the general formalism f ( n ) = I 4 ( n l f n 2 ) X^ X^ (32) a l l n 1 » n 2 3'(n 1+n 2=n) where X^ i s the independent p r o b a b i l i t y of the presence of a s i n g l e p a r t i c l e of species 1, w i t h i n the bounds of the system and * 2 Is that f o r species 2. The symbol X ±s a l s o c a l l e d the absolute a c t i v i t y of a single p a r t i c l e ^ ^ . Thus the G.C.P. f o r n p a r t i c l e s i s seen to be the sum (over a l l p o s s i b l e n.^  + n 2=n) of C.P.F. s f o r each given value of n 1 and ngj times the i n d i v i d u a l species p r o b a b i l i t i e s to powers n^ and . In form - 6./kT X± = x± e g± (33) where8 ±/kTand g ± are energy and geometry f a c t o r s r e s p e c t i v e l y , depending on the system, not e x a c t l y i d e n t i c a l to the energy and geometry f a c t o r s of the C.P.F. C. The G.C.P. i n D e s c r i p t i o n s of L o c a l (Nearest-Neighbor  S h e l l ) Behavior In an open system, the system boundary can u s u a l l y be designated i n any way that proves to be convenient. In one p a r t i c u l a r a p p l i c a t i o n of the general formalism f o r ~£ (equation (32)), n i s set equal to z, the nearest neighbor co-o r d i n a t i o n number. So doing enables one to i n v e s t i g a t e the form of the G.C.P. applicable to a set of z nearest neighbors of some designated central molecule. For the case of a binary species mixture, a p a r t i c u l a r s h e l l might contain n^ molecules of 1, and of 2, so that ( n 1 + n 2 = z). Hence n l ,n2 Hz) = I £ ( V n 2 ) \2* ( 3 4 ) n l ' n 2 3 (n 1+n 2=z) where the summation i s over a l l possible numbers n-^  and n 2 from 0 and z, to z and 0, respe c t i v e l y . In order to evaluate equation (34 ) i n the presence of l o c a l e f f e c t s ( i . e . those produced by the cen t r a l molecule species about which the s h e l l i s gathered), one would have to st a r t by speci f y i n g about which species of ce n t r a l molecule the z nearest neighbors were being considered. A l t e r n a t i v e l y , and more generally, one could write the G.C.P. representative of a l l the arrangements of (z+1) molecules, for a l l p o s s i b i l i t i e s of z neighbors plus one central molecule, namely the extra 1 shown i n the (z+1) bracket, for a l l central species. I f now z-^ i s s p e c i f i e d as being the coordination number around the^D) centra'l molecule species, (with c e n t r a l i t y indicated here by an i n d e x - c i r c l i n g convention) the G.C.P. would be m f ( z + l ) A ^ f C z ^ ) ( 3 5 ) 73 C i r l i n g of the s u b s c r i p t T) of each of the G.C.P. s on the R.H.S. of equation (35) designates that each of the m ( f o r m species present i n the mixture G.C.P. s ( i . e . each £ (z-^)) p e r t a i n s to the l o c a l environment of the i ^ species of c e n t r a l molecule. By t h i s means i t i s p o s s i b l e f o r m a l l y to i n c o r p o r a t e l o c a l e f f e c t s i n t o each of the /.(z^O which are otherwise, i n form, as i n d i c a t e d by equation (32:). 3. APPROXIMATION OF THE G.C.P. BY A MULTINOMIAL SERIES I f the independent p r o b a b i l i t i e s *1 and *2 are a p p r o x i -mated by x and x^*, the mole f r a c t i o n s of species 1 and 2, r e s p e c t i v e l y , i n the o v e r a l l mixture, equation (32) may be approximated by n l n2 f(z) = I Q.(nvn2) x 1 i x£ (36) a l l n 1 , n 2 3T (n 1+n 2=n) Thus _ -e /kT n -e 9/kT n = I W n v n 2 ) (g e x±) 1 * ( g e x ) Z ] (37) a l l n l 9 n 2 The terms of equation (37) are recognizable as being those of a bi n o m i a l expansion, whose sum i s t h e r e f o r e given by -<S. /kT * A = a e 1 x • While f o r an i d e a l s o l u t i o n of equal s i z e d molecules gi-> 1 and <$ -> 0, f o r r e a l s o l u t i o n s the appro-ximation o f ;\ by x i s more or l e s s i n a c c u r a t e . 74 Hz) = [§1 e -e,/kT z (38) In the case of more than two components, the terms of the corresponding G.C.P. correspond to those of a m u l t i n o m i a l d i s t r i b u t i o n . 4. PROBLEMS IN THE DIRECT USE OF THE G.C.P. Of course, i t i s p o s s i b l e to make thermodynamic pro-perty estimates i n ways other than u s i n g the c h a r a c t e r i s t i c r e l a t i o n s A=-kt l n Q, or PV=kTln £ i n the G.C.P. case. In one e a r l y attempt to evaluate s o l u t i o n p r o p e r t i e s , weights (as given by i n d i v i d u a l terms of equation G7 ) ) , normalized by t h e i r sum (as given by equation ( 3 8 ) ) , were used simply as weighting f a c t o r s i n a computation of the p a r t i a l molal energy (3) of v a p o r i z a t i o n of a given s p e c i e s . o f c e n t r a l molecule • The method c o n s i s t e d of summing the product of each weight times the cohesion due to the corresponding nearest neighbor assem-blage about the given c e n t r a l molecule. The r e s u l t i n g energy was then r e l a t e d to the p a r t i a l molal Helmholtz Free Energy through a s u i t a b l e form of the Gibbs-Helmholtz equation. By t h i s means, q u i t e good estimates of Yj_ (as r e l a t e d to the p a r t i a l molal excess f r e e energy of mixing, obtained by s u b t r a c t i n g the pure component f r e e energy from the above) would r e s u l t from even a very crude form of the G.C.P. f o r each c e n t r a l molecule's spectrum of nearest-neighbor arrangements. However, apart from 75 i t s inherent lack of formal elegance, t h i s route proved to involve other d i f f i c u l t i e s . These w i l l be b r i e f l y mentioned to i l l u s t r a t e how e a s i l y one can become trapped by the use of an inappropriate formulation, even i f the formulation i s sound i n p r i n c i p l e . Because the Gibbs-Helmholtz r e l a t i o n s (or Clausius Clapeyron type integrations of the Gibbs-Helmholtz equation), do not e x p l i c i t l y contain S (see equation 25)> expressions for A developed through the above route are not amenable to i n -elusion of S terms (see Chapter 9, for such a general ex-E ~ E pression for A ~ G ). Also, the method does not provide a mechanism for c a l c u l a t i n g l o c a l concentrations or net mixing energies, which are consistent with the p r i n c i p l e of minimum free energy, as does the quasichemical method. Eventually then, the above-described 'd i r e c t ' method of estimating free energy was abandoned i n favor of the quasichemical approach. 5. USE OF THE G.C.P. TO ILLUSTRATE LOCAL PHENOMENA IN SOLUTIONS Semi-quantitative arguments involving the G.C.P. are most usefu l i n i l l u s t r a t i n g l o c a l e f f e c t s i n solutions. One begins by considering a l o c a l assemblage, consisting of an open system of the z nearest neighbor positions about a chosen cen-t r a l molecule. This open system exists i n equilibrium with "the environment". The environment consists of an i n f i n i t e res-e r v o i r of the mixture, whose o v e r a l l species mix i s characterized by the mole fractions of the various species. It i s possible for a z - p a r t i c l e assemblage to be con-76 s t i t u t e d i n any proportions from the species present i n the environment, as long as the t o t a l number present i n the system remain equal to z. In f a c t , the r e l a t i v e frequencies of occurrence of each of the variously composed nearest neighbor s h e l l s i s given by an i n d i v i d u a l term of7>^(z-i0 (the G.C.P. about the i cen t r a l molecule). The energetic and geometric variations between i n d i v i d u a l terms of / ^ ( z ^ ) serve to skew the d i s t r i b u t i o n s , so that each £r>^ w i l l depart more or less from £. It must also be remembered that a d i f f e r e n t average coordination number w i l l exist for each central species; hence each of the w i l l depart more or less from z. Differences i n size of the various molecular species present w i l l exert the largest influence on z^. En e r g e t i c a l l y induced c l u s t e r i n g tendencies r e s u l t i n enrichment of nearest neighbor assemblages i n the species which intera c t most strongly with the i t h c e n t r a l molecule. This also contributes to skews in/.^(z^>. The re-l a t i o n between the l o c a l behavior incorporated i n £( z) and that on the molar scale i s i l l u s t r a t e d by considering the o v e r a l l N- molecule assemblage as being comprised of N over-lapping assemblages, each consisting of z nearest neighbors and a c e n t r a l molecule (momentarily ignoring the secondary accounting problems a r i s i n g from variations i n the z-^). For each of the ^ species present,, the G.C.P. w i l l be given by the corresponding 6. DEVIATION OF LOCAL FROM AVERAGE COMPOSTION Because each of t h e / ^ z ^ ) contains l o c a l compositions, 77 energy e f f e c t s and geometry e f f e c t s , the modal (most pro-bable) compositions of such l o c a l d i s t r i b u t i o n s need no longer correspond to the o v e r a l l composition. Rather, each average l o c a l composition may depart from the o v e r a l l -sometimes markedly - but the departures of l o c a l compositions with respect to the various species present from the o v e r a l l composition must be i n compensating amounts and d i r e c t i o n s , so that the o v e r a l l material balance on each species present i n the mixture i s conserved. The magnitude of such l o c a l c l u s t e r i n g e f f e c t s i s also tempered by the tendency of the solution as a whole to remain as near to maximum-randomness (complete homogeneity) as i s possible under the r e s t r a i n t s im-posed by the l o c a l energy-induced c l u s t e r i n g e f f e c t s and l o c a l geometry e f f e c t s . The average configuration of the mixture thus represents the equilibrium balance between the l o c a l c l u s t e r i n g and o v e r a l l randomizing tendencies above described. 7. PARTIAL MOLAL PROPERTIES In p a r t i c u l a r , the modal compositions of each of the —/T)^ziS' w^-ll d i f f e r , i f d i f f e r e n t i a l energy and geometry ef f e c t s e x i s t . Thus, the properties of each species of c e n t r a l mole-cule which depend on the composition of the nearest neighbor s h e l l , such as cohesion,will also d i f f e r on account of l o c a l com-po s i t i o n differences. ' 78 On the macro-scale, the, p a r t i a l molal p r o p e r t i e s w i l l r e f l e c t the l o c a l p r o p e r t i e s corresponding to the most probable l o c a l compositions about each species of c e n t r a l mole-cule r a t h e r than upon the o v e r a l l average composition, where these are not the same. 8. MACROSCOPIC SYSTEM PROPERTIES DERIVED FROM MOLECULAR SCALE  PARAMETERS I t i s o b v i o u s l y always p o s s i b l e to t a k e , as an a p p r o x i -mation of the average value of a property of a z-scale assemblage, the value of that property c a l c u l a t e d f o r the composition c o r -responding to the modal term of i t s One might a l s o reasonably suppose th a t t h i s estimate could be improved by a l s o i n c l u d i n g i n i t a p p r o p r i a t e l y weighted c o n t r i b u t i o n s from com-p o s i t i o n s corresponding to off-modal terms of the d i s t r i b u t i o n . This might be so on the z - s c a l e , but one i s i n f a c t p r i m a r i l y concerned w i t h e s t i m a t i n g , i n s t e a d , the p a r t i a l molal p r o p e r t y , on the N-scale. I f the G.C.P. on the N-scale, namely £(N) can be thought of as being made up of a l a r g e number of r e p l i c a t e d subsystems on the z - s c a l e , then i t can be shown that due to a c o m b i n a t o r i a l e f f e c t i n v o l v e d i n the 'scale-up' from the z-scale to the N-scale In such a s i t u a t i o n , that there i s an enormous incr e a s e i n sharpness of the d i s t r i b u t i o n of a property i n the N-scale composite system wi t h respect to that f o r any one of the z-scale r e p l i c a t e s . I t can a l s o be shown that the compositions corresponding to the modal terms of N-scale and z-scale d i s -t r i b u t i o n s c o i n c i d e . Thus, while c h a r a c t e r i z a t i o n of the z-scale system's p r o p e r t i e s by the z-scale modal term estimate may be a rather poor approximation, the approximation of the 79 average p r o p e r t i e s of the N-scale system by the modal pro-p e r t i e s of any one of the z-scale systems i s very good indeed. The consequences are 3 - f o l d : 1) Modal z-scale p r o p e r t i e s -the most obvious example being z i t s e l f - are compatible w i t h N-scale p a r t i t i o n f u n c t i o n s . 2) Modal z-scale estimates form extremely good estimates of N-scale p r o p e r t i e s . 3) Because of the above-mentioned modal-term dominance, the z-scale c a n o n i c a l p a r t i t i o n f u n c t i o n 2(z) can be used i n place of £ ( z ) , and Q.(N) can be used i n place of _/(N) even f o r an open system. The combined r e s u l t of 1 ) , 2 ) , and 3 ) , i s that modal z-scale pro-p e r t i e s , p a r t i c u l a r l y z^, can be i n c l u d e d i n the d e t a i l e d f o r m u l a t i o n of Q_(N), which i s the p a r t i t i o n f u n c t i o n most d i r e c t l y r e l a t e d to A f o r a mole of the l i q u i d mixture being 'modelled' by Q[N) . The above arguments on the c o m p a t i b i l i t y of modal pro-p e r t i e s of r e p l i c a t e d small ( c l o s e d or open) ( " l o c a l " ) en-sembles used i n the l a r g e s c a l e ensemble composed of a l l the r e p l i c a t e s , a l s o extends q u i t e g e n e r a l l y to such other instances as the use of most-probable p a i r s p r o p e r t i e s i n N-scale thermo-dynamic estimates. As a f i r s t p o i n t i n summary of the above, i t i s hoped that the above comments should help the reader understand the p e r m i s s i b i l i t y of working at various s c a l e s to o b t a i n v a r i o u s terms i n the o v e r a l l e x p r e s s i o n , such as the ex-p r e s s i o n f o r excess f r e e energy of mixing obtained as the end product of the formulations undertaken i n the t h e o r e t i c a l s e c t i o n of t h i s t h e s i s . As a second p o i n t , the r e l a t i o n of p a r t i a l molal pro-perty estimates to l o c a l compositions should have been established. ' , . Having used the G.C.P. to make the above points, the 'discussion may now be"redirected to further consideration of applications of the C.P.F. to the present work. i TT 9. USE OF DIFFERENT SCALES IN THE » AND ( f^FACTORS OF THE OjN)CANONICAL PARTITION FUNCTION It i s now apparent that for a monofunctional monomer binary system d i s p l a y i n g only moderate energe t i c a l l y induced l o c a l c l u s t e r i n g , the free energy of mixing can be obtained d i r e c t l y from Q.(N), the C.P.F. for the entire N-particle system. However, for systems which are multifunctional, or contain more than two components, or contain very strong energy e f f e c t s , the properties of z - p a r t i c l e systems must again be considered as well. This i s because i n c e r t a i n cases, the method of p a r t i t i o n functions becomes unmanageable, and an-other method altogether (the p r o b a b i l i t y d i s t r i b u t i o n method) must be resorted to. The p r o b a b i l i t y d i s t r i b u t i o n method of property formulation i s discussed i n Chapter 9. The configuration normally chosen i n which to evaluate 1 the 1 [f^ term of Q.(N) of equation (28 ) i s that corresponding to the most probable single configuration of the system. The fa c t o r ft i s seen to be a combinatorial expression specifying the rumber of distinguishable assignments by species of the p a r t i c l e s of the system that can be made i n the chosen system configuration. Since computing A from Q.(N)(via the formula A =-kTln Q.(N)) e n t a i l s t a k i n g the log a r i t h m of £(N), the pro-duct of the two p o r t i o n s of Q.(N) i s transformed Into a sum of the logarithms of the two f a c t o r s In the r e s u l t i n g f r e e energy expression. Such s e p a r a t i o n of terms w i l l s h o r t l y be seen to be a powerful technique and i s made use of i n t h i s t h e s i s , and i n a n t i c i p a t i o n of such use the reader might note t h a t : as long as care Is taken that the bases of f o r m u l a t i o n i of (ft) and (] [fi^portions of the p a r t i t i o n f u n c t i o n are not inco m p a t i b l e , the b a s i s of f o r m u l a t i o n f o r each p o r t i o n need not be i d e n t i c a l . Indeed, the form of 0 (N) most convenient f o r t h i s work i s one i n which the two p o r t i o n s of Q. are f o r -mulated by c o n s i d e r a t i o n of assemblages of d i f f e r e n t s c a l e . F a c t o r ft i s formulated on a molecular s c a l e , i n which the i n d i v i d u a l " p a r t i c l e " i s regarded to be a molecule, and the 1 (] [f±) f a c t o r i s formulated on a " p a i r s " s c a l e , where the i n d i v i d u a l " p a r t i c l e " i s a " p a i r " , an e n t i t y c o n s i s t i n g of a c o n t a c t i n g p a i r of s i t e s on neighboring molecules. (See Chapter 8 ) . The (^ ) term gives r i s e to the conf i g u r a t i o n a l entropy of 1 mixing e f f e c t s ,while the ("] [f ) product gives r i s e , i n simple cases, to heat of mixing e f f e c t s . In the f o l l o w i n g d i c u s s i o n of the quasichemical exr-p r e s s i o n of Guggenheim f o r monofunctional monomer bi n a r y mix-t u r e s , and the extension of t h i s theory to more complex types of s o l u t i o n s , by h i m s e l f and other workers, s p e c i f i c note should be taken by the reader of the forms of the p, and i (] ff ) p o r t i o n s of the C.P.F. employed. 82 CHAPTER 6 NONCONSTANT-COORDINATION NUMBER MECHANICS 1. GENERAL 2. THE ANALOGY OF A PACKED BED OF SPHERES 3. EXPECTED RANGE OF APPLICABILITY OF THE GEOMETRIC MODEL 4. THE COORDINATION (OR CONTACT) FRACTION 5. CONSIDERATION OF z ± AS A PARTIAL MOLAL PROPERTY 6. ROLE OF K IN LOCAL CONTACT FRACTIONS IN A REAL LIQUID 7. LOCAL CLUSTERING EFFECT 8. THE CONCENTRATION MEASURES OF SCATCHARD AND HILDEBRAND 9. CALCULATION OF COORDINATION NUMBERS z AND z a ave 10. CLOSED FORMULA FOR COORDINATION NUMBER IN A SIMPLE BINARY 11. TERMINOLOGY FOR A SOLUTION CONTAINING n-MERS 12. PACKINGS INVOLVING ELONGATE MOLECULES 13. DIFFICULTIES IN THE DEFINITION OF A MONOMER UNIT 14. IDEALIZATION OF POLYMER GEOMETRY IN CLASSICAL TREATMENTS 15. CONTRAST BETWEEN IDEALIZED AND ACTUAL POLYMER GEOMETRY 16. NOMENCLATURE FOR CHAINS OF NONUNIFORM MONOMER UNITS 17. CALCULATION OF PACKING NUMBERS WITH n-MERS PRESENT 18. THE EFFECT OF ROLLED-UP CONFIGURATIONS ON THE EFFECTIVE PACKING NUMBER 19. S FOR AN n-MER CONTAINING SOLTUION 20. EFFECTIVE MONOMER RADIUS 83 CHAPTER 6 NONCONSTANT-COORDINATION NUMBER MECHANICS 1. GENERAL In t h i s chapter a q u a n t i t y c a l l e d the contact f r a c t i o n ( 5 i ) i s i ntroduced as a measure of the number of c o n t a c t i n g s i t e s a s s o c i a t e d w i t h the i t h molecular species i n the mixture. The usefulness of the contact f r a c t i o n a r i s e s from i t s a b i l i t y to c h a r a c t e r i z e the p a i r - f o r m i n g c a p a c i t y of d i f f e r e n t con-s t i t u e n t species of molecule i n the case of mixtures c o n t a i n i n g d i f f e r e n t s i z e s . A c a l c u l a t i o n scheme r e l a t i n g ^ , and x±,the contact f r a c t i o n , c o o r d i n a t i o n number, and mole f r a c t i o n of the i species present i n the mixture r e s p e c t i v e l y , i s then i l l u s t r a t e d . The treatment i s r e s t r i c t e d i n i t i a l l y to monofunctional binary mixtures, which permits c l o s e d mathematical s o l u t i o n s to c e r t a i n aspects of the theory which are not a v a i l a b l e i n the multicomponent case. A d d i t i o n a l methods which must be i n v o l v e d i n the m u l t i f u n c t i o n a l and multicomponent cases are t r e a t e d i n Chapter 9, although t o -ward the end of t h i s chapter c e r t a i n c o n f i g u r a t i o n a l aspects of n-mer s o l u t i o n s are discussed (without regard as to whether they are monofunctional or m u l t i f u n c t i o n a l ) , i n order to g e n e r a l i z e the c a l c u l a t i o n scheme f o r contact f r a c t i o n i n t o the n-mer case. 84 The question of the m o d i f i c a t i o n of nearest neighbor c o o r d i n a t i o n number and of the composition of the average nearest neighbors s h e l l by geometry and energy e f f e c t s might be approached e i t h e r i n terms of the e f f e c t s produced i n "most-probable" nearest neighbor s t r u c t u r e s , or more g e n e r a l l y i n the context of the p a r t i t i o n f u n c t i o n s discussed i n the previous chapter. In f a c t - i t i s convenient to make use of both l e v e l s of a n a l y s i s , and t h i s procedure i s used i n subsequent formulations i n t h i s work. The e f f e c t s of geometry and energy on the average nearest neighbor s h e l l composition are easy to understand q u a l i t a t i v e l y . F i r s t , consider the energy e f f e c t . About a given c e n t r a l s p e c i e s , the neighbor species whose net heat of mixing w i t h the c e n t r a l molecule i s most exothermic i s r e -l a t i v e l y favored, as most lowering the o v e r a l l mixture f r e e energy. Secondly, consider the geometry e f f e c t . A l a r g e r c e n t r a l molecule tends to be able to accommodate a l a r g e r number of nearest neighbors of the average s i z e than would a smaller one. Thus such a l a r g e r species i s exposed to more nearest neighbors and hence tends to p a r t i c i p a t e as a nearest neighbor i n more arrangements, than would a s m a l l e r s p e c i e s . A l a r g e species would thus tend to exert an i n f l u e n c e i n nearest neighbor arrangements d i s p r o p o r t i o n a t e l y above i t s simple c o n c e n t r a t i o n e f f e c t . In t h i n k i n g about nearest neighbor arrangements an important symmetry axiom must be observed, concerning 'neighbor' and ' c e n t r a l ' p o s i t i o n s , namely: 85 h^)= Pj-T) ( 3 9 ) In r e l a t i o n ( 3 9 ) , p i s the p r o b a b i l i t y of a given type of pairwise i n t e r a c t i o n . Subscripts i and j stand for two types of molecules which can form contacts In the so l u t i o n , and the c i r c l e d index designates the a r b i t r a r i l y chosen central mole-cule of a nearest neighbors arrangement. The symmetry axiom, together with the statement preceding i t , forms the basis of quantitative c a l c u l a t i o n of the species coordination numbers i n a mixture of unequal sized molecules by a method originated (17) by Hogendijk. In the present work, the method has been generalized to include, i n addition, a correction for energetic-a l l y induced c l u s t e r i n g and for the e f f e c t s of long molecules. 2. THE ANALOGY OF A PACKED BED OF SPHERES Hogendijk's method was o r i g i n a l l y developed for c a l -c u l a t i n g the average number of contacts per sphere of a given species i n a close-random-packed bed of spheres of discrete s i z e s , each 'species' being i d e n t i f i e d by i t s discrete s i z e . In order to adapt the method for coordination number c a l -c u l a t i o n to a r e a l l i q u i d , the analogy of a r e a l l i q u i d to a packed bed of spheres i s drawn. The analogy between a packed bed of spheres and the organization of a dense l i q u i d does not imply a hard sphere l i q u i d model i n the usual f u l l sense of the word^ 2^ ,since the 86 molecules do not need to be 'hard' to permit such a method of c a l c u l a t i o n of c o o r d i n a t i o n number, but need only be a s s o c i a b l e w i t h a c h a r a c t e r i s t i c domain s i z e , from which they s t a t i s t i c a l l y tend to ' d i s p l a c e ' or 'exclude' t h e i r nearest neighbors. Thus, the p e r t i n e n t s i z e of a molecule i s measurable by means of the e f f e c t i v e diameter which can be a s s o c i a t e d w i t h the sum of i t s equivalent hard sphere volume surrounded by a s h e l l of s i z e e quivalent to i t s average f r e e volume. The assignment of 'packing-based' c o o r d i n a t i o n numbers to molecular species i n l i q u i d s , e s p e c i a l l y i n mixtures w i t h l a r g e and a n i s o t r o p i c cohesion e f f e c t s , may seem at f i r s t to be a crude p r o p o s i t i o n , s i n c e at some p o i n t or other the 'packing' mechanism of determining of the number of neighbors i s overwhelmed by other f a c t o r s . I t i s thus important t o know what some of these f a c t o r s are, and how and at what p o i n t they impose e f f e c t i v e l i m i t a t i o n s upon the a p p l i c a b i l i t y of the f r e e energy expression. 3. EXPECTED RANGE OF APPLICABILITY OF THE GEOMETRIC MODEL One bound on the r e g i o n of a p p l i c a b i l i t y of the analogy i s imposed by the ' s t e r e o - s p e c i f i c ' ' f i x i n g ' e f f e c t of very st r o n g s p e c i f i c i n t e r a c t i o n energies. At some p o i n t these energies s e r i o u s l y d i s r u p t the 'packing-dominated' mode, of l i q u i d o r g a n i z a t i o n . In s t i l l other cases, the onset of l o c k -and key ' f i t ' e f f e c t s c h a r a c t e r i z i n g c h e l a t e s , or concerted nearest-neighbor a c t i o n s , as i n the case of c o o r d i n a t i o n com-p l e x e s , should by d e f i n i t i o n cause any p a i r w i s e theory to break down. 87 The very rapid expansion of the l i q u i d phase as con-d i t i o n s approach the mixture's vapor-liquid c r i t i c a l point sets another l i m i t on the range of a p p l i c a b i l i t y of a l i q u i d model incorporating a packing method of assigning coordination num-bers. F a i l u r e of a packing configurational model i s bound to occur with the breakdown of the effectiveness of the 'caging' phenomenon when the l i q u i d becomes s u f f i c i e n t l y expanded. Nevertheless, one experimental system (system 7 ) * was found to be amenable to the expression developed,under pressures up to 0.4 of the c r i t i c a l pressure. In f a c t , one of the main points i l l u s t r a t e d by the ex-amples of r e a l systems chosen for the experimental part of the thesis i s that the quasichemical model works to a useful degree i n nonionic solutions as long as interactions between geometry and energy e f f e c t s are properly accounted for, even i n the case of some highly polar mixtures and r e l a t i v e l y high pressure. 4. THE COORDINATION (OR CONTACT) FRACTION It i s convenient to r e f e r to a molecule or sphere which contacts z nearest neighbors as being z-coordinated. S i m i l a r l y a l a t t i c e , or packing arrangement i n which each molecule or sphere i s z-coordinated i s c a l l e d a z-coordinated ' l a t t i c e ' or packing. (It i s remembered that a system i n which z i s constant has been termed a CZ system and one i n which coordination numbers of species vary with composition a nonconstant-z system, or NCZ system). In a bed of random-packed spheres of unequal s i z e , the Using the numbering system of Table 4, Chapter 10. 88 numerical values of the c o o r d i n a t i o n numbers of the species are not the same as would be found i n a bed of spheres a l l of equal s i z e . Mutual d i s t o r t i o n of c o o r d i n a t i o n number occurs, augmenting the c o o r d i n a t i o n number of the l a r g e r species and decreasing that of the smaller from the s e l f - p a c k i n g c o o r d i n -a t i o n number. To t h i s extent, In a mixture, the c o o r d i n a t i o n number could be regarded as being changed or d i s t o r t e d by the mere act of mixing. I t i s s t i l l q u i t e p o s s i b l e however, though not n e c e s s a r i l y so, f o r the t o t a l number of contacts i n the mix-tur e to remain the same as that of the sum of the s u b t o t a l s of the two species populations as pure components, i n s p i t e of the d i s t o r t i o n s j u s t d e s c r i b e d . I f species i i s z^ coordinated and species j i s " z. co-o r d i n a t e d , the t o t a l number of contacts i n the assemblage i s z i z i ~2~ N i + 2 N j ' * "^or a m i x t u r e of I\L spheres of species i and spheres of species j , the f r a c t i o n of the t o t a l number of con-t a c t s due to species i , namely £^ i s given by „ '1/2 " i w E l z l / 2 H l + 'ill N3 ( ' Upon d i v i d i n g numerator and denominator by (N. + N.) and a l s o 1 J by the o v e r a l l average c o o r d i n a t i o n number z a v e » a n d- n o t i n g that N./(N. + N.) = x. (41) and that z.. ~ z , the f i n a l form of 5. obtained i s i j ave 5 I 89 h - z i / z i j x V ( z i / z i j x * i + z j / z i j X x j ) C42)-5. CONSIDERATION OF z ± AS A PARTIAL MOLAL PROPERTY While z^ i s a function of the mixture composition and species s i z e , i t Is more strongly a property of the size and con-f i g u r a t i o n of the chosen central molecule than of the environ-ment, though, obviously i t depends on both. Thus i t is_ meaning-f u l to associate a c h a r a c t e r i s t i c z^ with each component of the mixture. To a f i r s t approximation, z^ can thus be regarded as a material property of the central species. It should more p r e c i s e l y be regarded as a p a r t i a l molal quantity, dependent both on the species present, and on t h e i r concentrations. 6. ROLE OF K IN LOCAL CONTACT FRACTIONS IN A REAL LIQUID For a given mixture of composition a» and x. i n which energy e f f e c t s are n e g l i g i b l e , the proportion of i - i and j - i •pairs' contacts around an i-type sphere due to i t s nearest neighbors i s given by 5 . and not by x. and x. . The over-i- 0 l J a l l p r o b a b i l i t y of a given type of 'pair' contact i n a bed of spheres, say of type i - j , i s the product of the proportion of the t o t a l population coordinated by j , (K^ ) times the nearest-neighbor p r o b a b i l i t y of the i ^ * 1 species (namely £ ) so that 90 ( 4 3 ) where P(i- j ) i s the normalized p r o b a b i l i t y of the given p a i r event. This p r o b a b i l i t y Is more simply w r i t t e n - a s p S i m i l a r expressions f o r p ( i - i ) and p(j-j) can be obtained by s e t t i n g i=j 7. LOCAL CLUSTERING EFFECT In a binary mixture of h i g h l y s p h e r i c a l molecules, as f o r example Argon-Krpyton w e l l below i t s c r i t i c a l p o i n t , one would expect to encounter ' l o c a l o r g a n i z a t i o n s ' analogous to those which would occur i n a bed of unequally s i z e d spheres which had been randomly packed, namely an o r g a n i z a t i o n h e a v i l y dominated by the packing c o n s i d e r a t i o n s brought about by the d i f f e r e n c e i n the s i z e of the spheres. In the usual molecule-packings, however, these simple ideas of c o o r d i n a t i o n number der i v e d from the n o n - i n t e r a c t i n g spheres model w i l l be modified by the f a c t that the molecules a t t r a c t each other i n a way that the s o l i d spheres do not, and th a t these a t t r a c t i o n s w i l l be species-dependent. A b e t t e r r e p r e s e n t a t i o n of the p r o p o r t i o n s of I and j contacts around a molecule of i i s given by -w../kT ( 4 4 ) i i 91 and -u ,/kT q - e ^ 1 * <*»5> respectivelyjwhere u.. i s the net reorganization energy of the quasichemical pairs reaction mentioned i n Chapter 3. Since by d e f i n i t i o n to.. = 0 , the Boltzman factor multi-p l y i n g the f i r s t term Is unity, so that the f i r s t term degener-ates simply to£^. It i s obvious that any p r e f e r e n t i a l c l u s t e r i n g of one species around another must produce an oppositely compensating ef f e c t somewhere else i n the o v e r a l l mixture and that such se- condary r e d i s t r i b u t i v e e f f e c t s cannot be zero as long as any value o f u i s nonzero. This secondary r e d i s t r i b u t i v e e f f e c t i s i l l u s t r a t e d by the statement that the more i n t e r a c t i v e species tends to exclude the less i n t e r a c t i v e from the nearest neighbor s h e l l of any chosen species of c e n t r a l molecule, which thus tends to r e s i d u a l l y concentrate the less i n t e r a c t i v e species about some other 'central' species. The species about which r e s i d u a l concentration c h i e f l y occurs i s the most weekly i n t e r -active species i n the mixture. Ultimately, of course, the re-jected material may form a second l i q u i d phase, i f d i f f e r e n t i a l a t t r a c t i o n s are large enough, and indeed, l i q u i d - l i q u i d e q u i l i -b r i a have been predicted on p r e c i s e l y t h i s basis. The i n d i r e c t e f f e c t s of en e r g e t i c a l l y induced c l u s t e r i n g tendencies are more f u l l y accounted for by the quasichemical model of Chapter 8, so that further discussion i s unnecessary at t h i s point. 92 8. THE CONCENTRATION MEASURES OF SCATCHARD AND HILDEBRAND The contact f r a c t i o n £^ (as d i s t i n c t from the mole f r a c t i o n x^) i s the appropriate c o n c e n t r a t i o n measure f o r a l i q u i d model i n which the behavior of the contact p o i n t s be-tween molecules i s to be emphasized. The contact f r a c t i o n has connotations more s i m i l a r to Scatchard's surface f r a c t i o n * as given by the symbol S : 2/3 x. p . .. s = 1 '2/3 ^ X ± p i j + x. than to Hildebrand's volume f r a c t i o n <)>. = v.x./J v . x . ,.,„. i 1 i h i i ( 4 7 ) where v^ i s the molar volume of component i . 9. CALCULATION OF COORDINATION NUMBER z AND z a a v e The scheme used i n c a l c u l a t i o n s of c o o r d i n a t i o n numbers, whence contact f r a c t i o n s , i s now presented. S u b s c r i p t 'a' i s now used i n place of ' i ' to denote s p e c i e s . S u b s c r i p t 'a' has the s p e c i f i c connotation of molecular species and i t i s now i n -tended that ' i * w i l l subsequently be reserved f o r use i n r e -rip") * The n o t a t i o n of equation ( 4 6 / i s adopted from Scatchard v ' and no attempt has been made to t r a n s l a t e the n o t a t i o n i n t o t hat of t h i s t h e s i s s i n c e the equation w i l l not subsequently be used and i s i n c l u d e d here f o r comparison purposes onl y , v. p_,.=— = volume r a t i o of two components. See p. 19 of r e f . ( 1 2 ) 1 J V J f e r r i n g to a p a r t i c u l a r type of contact. Since a molecule may have a number of d i f f e r e n t chemical f u n c t i o n groups s e v e r a l e n e r g e t i c a l l y d i s t i n g u i s h a b l e kinds of p a i r s may be formed from the same molecule 'a'. The value given to z i s the number of molecules con-° a t a c t i n g a molecule of species 'a' f o r the average l o c a l com-p o s i t i o n of the nearest neighbor s h e l l about that s p e c i e s . Also i t i s u s u a l l y assumed that a l l s e l f - p a c k e d s p e c i e s , i . e . , those i n pure component form, have the same pure component monomer packing number, z . In the pure component s t a t e , a l l ' c e n t r a l ' molecules are surrounded by z i d e n t i c a l molecules. By simple geometry s (and r e f e r r i n g by way of example to species 'a') the view f a c t o r of a nearest neighbor (and here 'hard sphere' geometry i s being employed) as seen from the centre of the c e n t r a l molecule i s a 0 a = R a / ( 4 X R a ) 2 ) ( W To c o r r e c t t h i s estimate f o r the n o n - r i g i d p r o p e r t i e s of mole-c u l e s , i n order to conform w i t h the r e a l experimental s i t u a t i o n , the p r o p o r t i o n a l i t y constant k i s i n t r o d u c e d , thus 1/z = a ^ / k s a a) Because the s e l f - c o o r d i n a t i o n number of a l l molecules i s taken to have the same numerical value i n t h i s work, k i s assumed 94 constant and i d e n t i c a l f o r a l l s p e c i e s , and i s a l s o assumed to apply to the mixed as w e l l as pure component s t a t e s , I t i s now convenient f o r purposes of n o t a t i o n , to i n t r o -duce index 'c' t o denote the nearest-neighbors p o s i t i o n . The 'blanking e f f e c t ' Inherent i n the 'pairwise' contact between a 'c' species nearest neighbor and a central'a) i s w r i t t e n a q ^ / k Also w i t h a mixture of species present, the ra d i u s of the r i n g of centres of neighbors i s no longer simply 2 R-^but.is des-I ignated as R Thus, a f t e r e s t a b l i s h i n g the i n i t i a l con-& ave a, ' ° r v d i t i o n s 5 = x , 5, - a ( 5 0 ) a a b b ave -a) ra 'a) s i t i s p o s s i b l e to f o l l o w the i t e r a t i o n : R R „ + IK R /J g ave.1i) -li) u Q c 6 c -.v c " e c „y a (51) < (52) W R c / ( 4 x « a v e ^ ) ) 2 ) (53) 95 (a) a (55) and so, by using a l l the species present i n turn, obtain new values of ? a , KH , RAVE3>» z a * a n d zb » a n d t o r e i n s e r t these i n the next i t e r a t i o n beginning at the f i r s t step (equation (52.)). Convergence i s rapid. Of course, i t i s r e a l i z e d that a l l molecules are equally •c e n t r a l ' , and therefore that the device of assuming the cen-t r a l i t y o f 1 ) has been only a c a l c u l a t i o n a r t i f i c e . Thus use of the subscript 'c' to denote generic neighbor species i s now superfluous and, over the solution as a whole, z ( i . e . , with a no connotation of c e n t r a l i t y ) can be regarded as being a mater-i a l property associated with species 'a' whose value i s equal to z^ > z i s defined by aj ave J = ^ Ka ZA 5a (56) z ave a a The e f f e c t of energy-induced l o c a l c l u s t e r i n g i s i n -cluded i n the i t e r a t i v e scheme for c a l c u l a t i n g z , as outlined by equation (52) to (55),through'the use of l o c a l species con-centrations of c around c e n t r a l a. S p e c i f i c a l l y , Barker's l o c a l concentration functions* which are functions of £ and * Barker functions X. are defined i n Chapter 9 under notes J on the general equation - see program l i s t i n g for exact algorithm In Appendix 4 ( c ) . 96 reduce to £ i n the monofunctional zero-energy i n t e r a c t i o n case., are used as weighting f a c t o r s i n place of f o r mixtures con-t a i n i n g m u l t i f u n c t i o n a l molecules. The computation s t i l l r e -s u l t s In convergent values of z f o r a l l systems so f a r s t u d i e d . a In the case of la r g e net energies of p a i r s r e o r g a n i z a t i o n , z becomes a f a i r l y strong f u n c t i o n of energy as w e l l as of r e -l a t i v e molecular s i z e . I f u n l i k e species a t t r a c t , z /z, w i l l be l a r g e r than f o r the case of u.. = 0. I f the s e l f i n t e r -a c t i o n of a species Is very s t r o n g , m o l e c u l a r l y u n l i k e p a i r s are discouraged, hence z /z, r a t i o s w i l l be c l o s e r to u n i t y than a D f o r the nonenergetic case, because z & and z^ both s e p a r a t e l y tend toward z g as the molecular species tend to segregate. This crude geometry can be j u s t i f i e d only i n t h a t , i n the expressions developed to c a l c u l a t e G , r e l a t i v e changes i n c o o r d i n a t i o n numbers are o f t e n more important than the absolute value of these numbers. 10. CLOSED FORMULA FOR COORDINATION NUMBER IN A SIMPLE BINARY For a binary mixture, and when the w f a c t o r s are not too l a r g e , i t can be considered that . Zave z s " Za^ Zb ^a^b ^ab ( 5 7 ) so that the approximations z, - z x (R b s ab a (58) and z = R ab b (59) a are obtained. Here 'a' s p e c i f i c a l l y a p p l i e s to the l a r g e r molecule. The just-mentioned approximations are very u s e f u l simply b i n a r y mixtures. 11. TERMINOLOGY FOR A SOLUTION CONTAINING n-MERS I f at l e a s t one of the molecular species i n a mixture i s s u f f i c i e n t l y elongated to behave as a rod or concatenated s t r u c t u r e r a t h e r than as a s i n g l e compact e n t i t y i n the packing arrangement, t h i s a d d i t i o n a l complexity must be taken i n t o account i n packing c a l c u l a t i o n s . Before s p e c i f y i n g the method of doing t h i s , a b r i e f note i s needed as to how the degree of concatenation of a molecule i s to be de f i n e d . From the organic chemist's viewpoint, the word polymer i m p l i e s a species of long chain molecule made up of r e p e a t i n g 98 chemical u n i t s , descrihed In notation as R, - CRw) - R-,J where 1 dm 3 R 1 and R^ are terminal groups between which are m repeating units of the group R 2 . The word 'n-mer' was coined by physical (16) chemists to characterize a molecule of n units linked to-gether .into a physical chain structure without any d i s t i n c t i o n being made between end and i n t e r n a l u n i t s . The analogy be-tween chemical and physical chemical terminology for concatenated species i s evident, but It i s i n another sense unfortunate that polymer chemists and physical chemists chose such s i m i l a r term-inology for uses which were b a s i c a l l y d i f f e r e n t : a succinct d e s c r i p t i o n of chemical composition on the one hand and an i n -d i c a t i o n of a p h y s i c a l c h a i n - l i k e structure on the other. Nevertheless since both usages are current, the most that can be done i s to point out that i n t h i s thesis the physical chemist's d e f i n i t i o n emphasizing morphology rather than chemistry i s used. In t h i s work the s t r u c t u r a l chain connotations of the word n-mer are retained without including the r e s t r i c t i o n that the n units be i d e n t i c a l or even s i m i l a r chemically to each other, nor need they be chemically s i m i l a r to other species pre-sent i n the mixture. In the context of t h i s t h e s i s , an n-mer i s any molecular species which occupies n unit c e l l s i n the packing arrangement. I f n assumes the t r i v i a l value of one, the species i s a monomer species. It follows that a monomer unit i s operationally defined as that portion of a molecule which occupies one unit of the packing arrangement. The pack-99 i n g arrangement i s determined i n concert by a l l the- species present. When these species are of d i f f e r e n t monomer s i z e s , there Is no exact one-to-one r e l a t i o n s h i p n e c e s s a r i l y to be expected between the chain l e n g t h of a molecule i n terms of the number of chemical u n i t s , and the molecule's e f f e c t i v e con-f i g u r a t i o n a l n-mer chain length i n a p a r t i c u l a r type of mixture. 12. PACKINGS INVOLVING ELONGATE MOLECULES The d i f f i c u l t i e s , which have j u s t been o u t l i n e d , a r i s i n g l a r g e l y because of nomenclature problems, may to some extent be re s o l v e d by c o n s i d e r i n g the i l l u s t r a t i o n s e r i e s 5, showing v a r i o u s examples of p o s s i b l e r e l a t i o n s h i p s between e f f e c t i v e c e l l s i z e i n the packing arrangement and the e f f e c t i v e n-mer chain l e n g t h of concatenated species present i n the mixture. In each i l l u s -t r a t i o n , the u n i t c e l l boundaries about 'monomer' segments of the c e n t r a l molecule are shown i n heavy dashed l i n e s . C e l l centres are shown by cross e s , and nearest neighbor r i n g s are i n d i c a t e d by a l i g h t dashed l i n e through the centres of the neighbors. The a c t u a l form of the c e n t r a l molecule Is shaded. Values or ranges of values f o r n, z ., and V i n d i c a t e d , c o r r e s -& ' e x t s ' ponding to e f f e c t i v e chain l e n g t h , e x t e r i o r * c o o r d i n a t i o n number and the p r o p o r t i o n of nearest neighbors which are p r o x i -mal chain u n i t s . Because the packing arrangements shown are * The s i g n i f i c a n c e of the word " e x t e r i o r " i n a s s o c i a t i o n w i t h c o o r d i n a t i o n number I n d i c a t e s the number of e x t e r n a l neigh-bors (as d i s t i n c t from proximal chain u n i t s ) a given monomer u n i t can c o o r d i n a t e . 100 two-dimensional r a t h e r than corresponding to the a c t u a l t h r e e -dimensional s i t u a t i o n i n the l i q u i d , parameter values are not p h y s i c a l l y meaningful, but r a t h e r serve only to i n d i c a t e the concepts being used. Let us f i r s t consider I l l u s t r a t i o n 5(a). The c e n t r a l molecule (shaded) i s , f o r the present I l l u s t r a t i o n regarded as comprising two p a r t s , A and B, each of which can be considered as occupying a s i n g l e c e l l . However, note that u n i t B r e -presents one of the nearest neighbor r i n g p r o p e r l y belonging to u n i t A and s i m i l a r l y u n i t A i s part of the nearest neighbor r i n g of B. I t w i l l be seen that r e g a r d l e s s of how f l e x i b l e the bond between the p o r t i o n s A and B may be, there are two adjacent c e l l s of the packing arrangement occupied by segments of the c e n t r a l molecule. The number f o r z , , which represents the e x"C o v e r a l l e x t e r i o r c o o r d i n a t i o n number per monomer u n i t , must take i n t o c o n s i d e r a t i o n that one nearest neighbor f o r both A and B i s a proximal chain u n i t . Since the c e n t r a l molecule i s being t r e a t e d as f i l l i n g two adjacent c e l l s , n equals 2. These numbers are attached to I l l u s t r a t i o n 5(a) although they have been explained i n t h i s t e x t so that the corresponding I l l u s t r a -t i o n s i n s l i g h t l y more complex cases can be compared. I l l u s t r a t i o n 5(b) d e p i c t s a s i m i l a r but more complicated case. In t h i s s i t u a t i o n the c e n t r a l molecule, although of roughly the same s i z e as i n I l l u s t r a t i o n 5 ( a ) , i s considered to be of such a chemical s t r u c t u r e that no f r e e r o t a t i o n around A's chemical bond i s p o s s i b l e , and so c o n f i g u r a t i o n a l l y the . c e n t r a l molecule behaves as a r i g i d blob as shown again by 101 Illustration 5(a). Representative Packing Arrangements. Dimer. 102 I l l u s t r a t i o n 5 (b). Representative Packing Arrangements. Short Rod l i k e Molecule. 1 0 3 shading. Nevertheless i t can be seen that i t i s s t i l l l e g i t i m a t e to t r e a t the A and B f r a c t i o n s as occupying adjacent c e l l s , and once again A f r a c t i o n has B as i t s proximal chain u n i t s neighbor. A has 7 e x t e r n a l neighbors, g i v i n g z = 8 . The B p o r t i o n , on the other hand, has a proximal chain nearest neighbor and 5 e x t e r n a l neighbors, g i v i n g z = 6 . This drawing has been d e l i b e r a t e l y adjusted so that the A end i s r a t h e r l a r g e r than the B end to show that one need not have the same c o o r d i n a t i o n number around each part of the dimer. The s t a -t i s t i c a l average of the number of e x t e r n a l nearest neighbors around the complete molecule, d i v i d e d by the number of monomer u n i t s w i l l represent the average c o o r d i n a t i o n number per monomer u n i t of that c e n t r a l molecule. In t h i s case, the e x t e r n a l co-o r d i n a t i o n number of the average monomer u n i t w i l l l i e between 5 and 7 . Now consider the s i t u a t i o n represented by the p a i r of I l l u s t r a t i o n s 5 ( c ) i and 5 ( c ) i i . These have been drawn to i l l u s t r a t e a poin t about the e f f e c t of the s i z e of surrounding molecules a c t i n g as neighbors to n-mer u n i t s . I l l u s t r a t i o n 5(c) i has been drawn so that each monomer u n i t of a 4-mer molecule has been considered to be occupying one c e l l and the neighbor molecules have been drawn to surround i t as comfortably as p o s s i b l e . However, i n I l l u s t r a t i o n 5 ( c ) i i , where an n-mer of small monomer u n i t s i z e i s surrounded by l a r g e neighbors, o p e r a t i o n a l l y , the c e n t r a l molecule behaves as a 3-mer. In t h i s case the c e n t r a l 2 monomer u n i t s of the c e n t r a l molecule are a l l o c a t e d to one c e l l . I t w i l l be seen that t h i s l a s t arrange-Illustration 5 ( c ) i . Representative Packing Arrangement Short Flexible Chain lik e Molecule CENTRAL MOLECULE BEHAVES AS 3 - M E R I l l u s t r a t i o n 5 ( c ) i i . Representative Packing Arrangements. Small n-mer Surrounded by Large Neighbors. 106 ment i s a more comfortable one than an attempt to f i n d a pack-i n g i n which the molecule could act as a 4-mer under the given circumstances. In I l l u s t r a t i o n 5 ( c ) i a four monomer u n i t c e n t r a l mole-cul e has been i l l u s t r a t e d as being surrounded by nearest neigh-bors of s i z e s approximately equal to the monomer u n i t , and no great d i f f i c u l t y i s encountered i n r e c o g n i z i n g the n e s t i n g pro-cess, i n i d e n t i f y i n g the i n t e r i o r neighbor and e x t e r n a l neigh-bors. I t can be seen from I l l u s t r a t i o n 5 ( c ) i no d i f f i c u l t y a r i s e s i n d e f i n i n g the order of the polymer i n such a case. Case 5 ( c ) i i s handled by c l a s s i c a l methods i n which the s i z e of the molecules, or r a t h e r , here, the surrounding molecules and monomer u n i t , are approximately the same. D i f f i c u l t i e s only begin to occur when r e a l s i z e d i f f e r e n c e s e x i s t between nearest neighbors and monomer u n i t s as seen i n I l l u s t r a t i o n s 5(b) and 5 ( c ) i i . F urther complications can e x i s t . In I l l u s t r a t i o n 5(d) i s shown a c e n t r a l molecule of 5 f l e x i b l y connected u n i t s . I t probably can most comfortably be represented as a dimer, the c o o r d i n a t i o n number of the two u n i t s A and B of the dimer being unequal. Appropriate data i s attached to the I l l u s t r a t i o n . The r a m i f i c a t i o n s of case 5(d) are more f u l l y d iscussed l a t e r i n the chapter. 1 3 . DIFFICULTIES IN THE DEFINITION OF A MONOMER UNIT The d e f i n i t i o n of a monomer u n i t as the amount of a mole-cul e which occupies one u n i t of the packing arrangement, w i t h the 107 I l l u s t r a t i o n 5 Cd). Representative Packing Arrangements. Partl y Rolled-up Central Molecule. 108 'packing arrangement' i n t u r n being c o o p e r a t i v e l y determined by a l l the species present, may seem a r a t h e r noncommittal set of statements i n view of the obvious interdependence of monomer un i t and c e l l s i z e i n the packing arrangement. However, one must r e a l l y not ask whether the above d e f i n i t i o n s are c i r c u l a r (indeed, they a r e ) , but r a t h e r whether they can form the b a s i s of a set of mathematical statements by means of which e f f e c t i v e l y convergent values of chain l e n g t h could be obtained by an a p p r o p r i a t e I t e r a t i v e procedure ( t h i s i n f a c t a l s o i s . s o ) . Hence, i n the operative sense, the above d e f i n i t i o n s are ad-equate. 14. IDEALIZATION OF POLYMER GEOMETRY IN CLASSICAL TREATMENTS The p r e c i s i o n of the d e f i n i t i o n of monomer u n i t t o be found i n c l a s s i c a l treatments of the p r o p e r t i e s of n-mer s o l u -t i o n s (-23) i s achieved by i d e a l i z i n g the p r o p e r t i e s of n-mers to the p o i n t where they become compatible w i t h a r e g u l a r l a t t i c e s t r u c t u r e , so that thermodynamic estimates of n-mer mixtures may be obtained through the use of l a t t i c e s t a t i s t i c s . In such i d e a l i z e d s o l u t i o n s , a l l monomer u n i t s are assumed to be i d e n t i c a l i n s i z e , and a l s o i d e n t i c a l i n s i z e to the monomer u n i t s of other species present i n the mixture (see I l l u s t r a t i o n 5 ( c ) i . I n t r a - and i n t e r - m o l e c u l a r distances are considered to be i d e n t i c a l . The n-mers are considered to be of a s p e c i f i e d degree- of r i g i d i t y - u s u a l l y s p e c i f i e d as being p e r f e c t l y f l e x -i b l e , so that any proximal l i n k on a chain i s able t o occupy any nearest neighbor s i t e i n the l a t t i c e not already occupied. 1 0 9 Packing i n the l a t t i c e model, hence the nearest-neighbor pack-i n g number, i s a s t r i c t l y predetermined and i n v a r i a n t pheno-menon d i c t a t e d by the assumed c o o r d i n a t i o n number of the l a t t i c e . The problem of interdependency of assigned c e l l s i z e and e f f e c t -i v e chain l e n g t h Is circumvented under such s t r i n g e n t I d e a l -i z a t i o n . By d e f i n i t i o n , ( i n the l a t t i c e model), an i n t e r i o r monomer u n i t has 2 chain connections, hence ( z - 2 ) e x t e r n a l neighbors, a t e r m i n a l has one chain connection, hence ( z - 1 ) ex-t e r n a l neighbors, and a monomer species has no chain connections, and z e x t e r i o r nearest neighbors, w i t h z being i n a l l cases the same constant. 1 5 - ' CONTRAST BETWEEN IDEALIZED AND ACTUAL POLYMER GEOMETRY I f one ' l i b e r a l i z e s ' the above regime to the case of a r e a l s o l u t i o n of molecules of d i f f e r e n t s i z e s , shapes, l e n g t h s , and e n e r g e t i c a l l y induced c l u s t e r i n g tendencies, and where some of the molecules may be In p a r t l y r o l l e d up c o n f i g u r a t i o n s , one has to r e t u r n to the k i n d of o p e r a t i v e d e f i n i t i o n s o f chain l e n g t h estimate based on the amount of e f f e c t i v e c e l l occupancy as given In the present work. In f a c t , one can adopt e i t h e r of two viewpoints on the matter. The f i r s t i s that chain l e n g t h i n the circumstances j u s t described i s indeterminate. Such a s i t u a t i o n i s c o ntrary to the p h y s i c a l i n t u i t i o n that s o l u t i o n s almost a l -ways s e t t l e to an unique lowest f r e e energy c o n f i g u r a t i o n . For such a minimum to e x i s t , the mixture must respond to a unique set of p h y s i c a l parameters, one of which o b v i o u s l y Is chain l e n g t h . One i s thusJxeartened to take the more p o s i t i v e viewpoint, that there is_ an e f f e c t i v e chain l e n g t h , even i f i t i s not r e a d i l y measurable. One can then e i t h e r t r y to c a l c u l a t e an e f f e c t i v e 110 chain length, or f a i l i n g that, simply to include e f f e c t i v e chain length as an emperical constant i n the equations. A completely I n t e r n a l l y consistent treatment of the general case above would obtain the e f f e c t i v e value of n from a d d i t i o n a l funamental i n -formation about a l l the species present i n the mixture through an i t e r a t i v e c a l c u l a t i o n ; but i n t h i s t h e s i s , n i s treated as a parameter to be varied, or set according to the best estimate avai l a b l e from external physical observations, or assigned the value which gives the best thermodynamic r e s u l t s . This thesis was not p r i m a r i l y Intended to be a study of chain e f f e c t s i n mixtures, though n had to be included for completeness i n a l l the formulations. This accounts for the lack of r i g o r i n evaluating n, despite i t s i n c l u s i o n in.the algebra. The reader might wish to r e f e r back to I l l u s t r a t i o n 5(d) at t h i s point. 16. NOMENCLATURE FOR CHAINS OF NONUNIFORM MONOMER UNITS Consistent with the above l e v e l of r i g o r with regard to n, the case of a molecule composed of monomer units of d i f f e r e n t sizes has not been included i n the formulation. I f the para-meter n i s a species e f f e c t i v e chain length, then the parameter R becomes i t s e f f e c t i v e monomer radius, which i n the case of a heterogeneous n-mer i s an average of i t s constituent monomer-unit r a d i i . Furthermore, i t i s hoped that defining n as a chain length for packing considerations, while employing n i to d i f f e r e n t i a t e f r a c t i o n s of the surface of a monomer unit on chemical grounds, i s not f e l t to be a problem. Excessive con-cern over such second order l e v e l refinements i n d e f i n i t i o n s of what Is, operatlvely, a f i r s t order theory, would tend to imply I l l a non-existent corresponding degree of refinement to be ex-pected i n the thermodynamic estimates a r i s i n g from the c a l c u l a -t i o n s . 17. CALCULATION OF PACKING NUMBERS WITH n-MERS PRESENT The c a l c u l a t i o n of packing number per monomer u n i t has been extended i n t h i s work t o the case where n-mer species are present, without m o d i f i c a t i o n to the basic c a l c u l a t i o n method employed (equations (52) to (55)) i n the 'all-monomers' case. The f a c t of concatenation was accounted f o r by i n c o r p o r a t i n g a c o r r e c t i o n f a c t o r , E a , such t h a t : z ( e x t e r i o r ) = E z ( i s o l a t e d monomer) C60) a a a when the u n i t 'a' belongs to an n-mer molecule. For the a t h species of n-mer length n ,( see I l l u s t r a t i o n 5 ( c ) i ) E = [1 - (1 - 2 V )(n - l)]/n ( 6 l) a a a a where Y > 1/z a — (62) For long chain species Ea~- < * 8 - 2 ) / . a (63) For a dimer E * (z - l)/z (.64) a s s and for monomer molecules, E = 1 (65) a The symbol z appearing i n equations (52) to (55) i s i n the general case taken to be the exterior coordination number per monomer unit of species 'a*. It i s obtained from the value of the coordination number obtained from purely packing-geometry considerations, namely z (i s o l a t e d monomer) by equation (60) and ( 6 l ) . Because coordination numbers per monomer unit i n mix-tures are understood to be exterior ones, E^ only e x p l i c i t l y appears as applied to z i n the formulations of t h i s work. 1 s Since packing number calculations include only nearest neighbors, the presence or absence of remote units of long chains i n the nearest neighbor s h e l l i s immaterial to the c a l -c u l a t i o n of packing numbers*. Whether a nearest neighbor i s a remote unit of the same chain, or whether i t belongs to a di f f e r e n t chain, does not change the considerations which deter-mine the average packing number for a species of monomer unit -namely the sizes and o v e r a l l proportions of various kinds of monomer units present i n the mixture.- The permanence of p r o x i -mal chain units i n the nearest neighbor s h e l l of the m^1monomer unit of concatenated species simply requires that i n the c a l -c u l a t i o n of packing numbers such proximal units be accounted for i n the weighted average of nearest neighbors. However, i n * This i s . o f course the case for ' l a t t i c e ' c a lculations of configurational entropy, where dispositions of whole n-mer chains d i s t i n g u i s h system permutations of the N-particle system . 113 the c a l c u l a t i o n of the exterior packing number (the number of nearest neighbors which are constantly being interchanged by the random thermal motion of the liquid,hence are available for p a r t i c i p a t i o n i n mixing effect s ) the number of proximal units for the average monomer unit of a given species must be sub-tracted from the t o t a l packing number to obtain the average exterior packing number per monomer unit of the given n-mer species f o r a p a r t i c u l a r mixture. 18. THE EFFECT OF ROLLED-UP CONFIGURATIONS ON THE EFFECTIVE  PACKING NUMBER In connection with packing number calculations involving concatenated species, i t should be borne i n mind that while the cji apo.si tjon of remote monomer units poses no problem, a further complication can ar i s e - due to l o c a l molecular configurations. For example the sigma bonds of proximal ^methylene groups of •'-p a r a f f i n hydrocarbons permit the skeleton of the molecule to rotate i n such a way that the molecule can assume a much more compact form i n some configurations that i n those of maximum ex-tension. (See again I l l u s t r a t i o n 5 ( d ) ) . Normal butane, for (79) example, may tend to exist i n a compact form which acts, as f a r as i t s packing number goes, l i k e a large monomer species with a large packing number, rather than as-an n-mer (with n> 1 ) , whose packing number per monomer unit i s lower. The e f f e c t i v e chain length of a r e l a t i v e l y long f l e x i b l e molecule, namely n , a i s not immediately deducible from i t s chemical formula, though the r e l a t i o n 114 v = K n x R' ,2 a a (66) must hold, where K i s a p r o p o r t i o n a l i t y constant. Since the roles of R and n are not interchangeable i n t h e i r e f f e c t on G , departure of n_ from the value of n for the molecule's con-a a. f i g u r a t i o n of maximum extension a l t e r s the e f f e c t i v e value of R. Nevertheless, the above complication due to the existence of p a r t l y " r o l l e d up" configurations for c e r t a i n f l e x i b l e n-mers s t i l l does not change the mechanics of the c a l c u l a t i o n of z, given n and R. Thus the problem was not given extensive atten-t i o n ; even though i t does bear on the a p r i o r i values assigned n and R for a given species of long molecule, and thereby a f f e c t s the value of z which i s e f f e c t i v e i n the mixture. 19. gaFOR AN n-MER CONTAINING SOLUTION The only modification of the packing cal c u l a t i o n s scheme necessitated to accommodate the presence of an n-mer species i s the appropriate i n c l u s i o n of factor E into the algebra to modify the value of the packing number per monomer uni t , remember-ing that for monomer species, E a assumes the t r i v i a l value of 1.0. Thus when n-mer species are present, the contact f r a c t i o n of species a i s generalized to z x n x a / z n x L a a , a a ( 6 7 ) . -a 115 where z Is understood to he the exterior coordination number a per monomer unit as discussed above. 20. EFFECTIVE MONOMER RADIUS A note on the c a l c u l a t i o n of ^ a , the monomer-unit radius, i s appropriate at t h i s point. For a binary mixture, 1 R a / R b = R a b = ( V V 3 ( 6 8 ) where v a and v^ are the pure component molal l i q u i d volumes. This estimate Is a very approximate one, p a r t i c u l a r l y i n the case of d i l u t e solutions, but, i t can be somewhat improved i f the p a r t i a l molal volumes are used. In such a case the better estimate i s provided by 1 Ra / Rb = ( V V 3 ( 6 9 ) " The more sophisticated methods described by Reid and Sherwood and also that of Lielmezs and Bondi^Vere used i n the actual c a l c u l a t i o n s , the r e s u l t s of which appear i n Chapter 11, 116 CHAPTER: 7  C O N F I G U R A T I O N A L E N T R O P Y E F F E C T 1. T H E I R R E G U L A R M U L T I P A R T I C L E ARRAY 2. I N C O R P O R A T I O N OF z I N N - S C A L E ENTROPY FORMULATIONS 3. C O M P A T I B I L I T Y WITH L A T T I C E S T A T I S T I C S 4. P A I R S - P E R M U T A T I O N S I N T H E NCZ C A S E 5 . ENTROPY OF M I X I N G E X P R E S S I O N I N TERMS OF CONTACT F R A C T I O N S 6. SUMMARY 117 C H A P T E R 7  C O N F T G U R A T T O N A L E N T R O P Y E F F E C T . 1. T H E I R R E G U L A R M U L T I P A R T I C L E ARRAY C l o s e e x a m i n a t i o n o f G u g g e n h e i m ' s t h e o r y o f CZ b i n a r y m i x t u r e s s u g g e s t e d t o t h e a u t h o r ( f o l l o w i n g t h e e x a m p l e o f (44) c e l l t h e o r i e s , o f w h i c h l a t t i c e t h e o r i e s a r e o n e l i m i t i n g c a s e ) t h a t t h e i d e a o f a t h r e e - d i m e n s i o n a l n e t w o r k o f i n s t a n -t a n e o u s m o l e c u l a r l o c a t i o n s w o u l d b e e q u a l l y w e l l s e r v e d b y c o n s i d e r i n g a ' s n a p s h o t ' o f t h e l i q u i d a s a n a d e q u a t e b a s i s o n w h i c h t o d e f i n e N i n s t a n t a n e o u s m o l e c u l a r p o s i t i o n s . P e r -m u t a t i o n i n t e r c h a n g e s c o u l d e q u a l l y w e l l b e c o u n t e d i n s u c h a f r a m e w o r k , w h e t h e r o r n o t t h e l a t t i c e o b t a i n e d b y c o n n e c t i n g c e n t r e s h a d a n y l o n g r a n g e r e g u l a r i t y . T h e u s e f u l i d e a o f p e r m u t a t i o n i n t e r c h a n g e s c o u l d t h u s b e s e p a r a t e d f r o m t h e c o m -p l e t e l a t t i c e s t r u c t u r a l I d e a a n d w o u l d p e r m i t t h e i n c o r p o r a t i o n o f t h e e f f e c t o f d i f f e r e n t c o o r d i n a t i o n n umbers f o r t h e v a r i o u s s p e c i e s p r e s e n t i n t h e m i x t u r e i n t o t h e p e r m u t a t i o n i n t e r c h a n g e f o r m u l a s f o r t h e s y s t e m . A s t u d y o f t h e c o n s e q u e n c e s o f a b a n d o n i n g t h e r e s t r i c t i o n o f a c o n s t a n t c o o r d i n a t i o n n u m b e r , a s r e q u i r e d i n c l a s s i c a l q u a s i c h e m i c a l e x p r e s s i o n s , b e c a m e o n e o f t h e m a i n a r e a s o f f o r m a l i n v e s t i g a t i o n o f t h e p r e s e n t w o r k , a n d r e s u l t e d i n a g e n e r a l i z a t i o n o f q u a s i c h e m i c a l t h e o r y t o t h e n o n - c o n s t a n t c o o r d i n a t i o n n u m b e r c a s e . I n d e v e l o p i n g t h i s 118 i d e a i n d e t a i l i t w i l l b e r e v e a l e d t h a t t h e l a t t i c e c o n c e p t i s s i m p l y a c o n c e p t u a l d e v i c e p e r m i t t i n g t h e r e d u c t i o n o f t h e 3-N s p a c e o f t h e c o n f i g u r a t i o n a l i n t e g r a l f o r t h e s y s t e m o f N p a r t i c l e s i n t o N i n d e p e n d e n t i n t e g r a l s , e a c h i n i t s own 3 - s p a c e o f p o i n t d i m e n s i o n s . T h e I n t e r c h a n g e o f a n y t w o m o l e c u l e s o n t h e i r r e s p e c t i v e l a t t i c e s i t e s d e f i n e s a new c o n f i g u r a t i o n f o r t h e w h o l e s y s t e m . T h e b a s i c s i m p l i f i c a t i o n i n v o l v e d i n t h e a p p r o x i -m a t i o n o f 3-N s p a c e b y N I n d e p e n d e n t 3 - s p a c e s i s a l s o p o s s i b l e f o r a n y ' c e l l ' m o d e l a s l o n g a s t h e a p p r o x i m a t i o n o f s i n g l e -o c c u p a n c y o f t h e c e l l s b y t h e m o l e c u l e s i s i m p o s e d . F o r p e r -m u t a t i o n p u r p o s e s , a ' l o c a t i o n ' i s now d e s i g n a t e d a s " a n y w h e r e w i t h i n a s p e c i f i e d c e l l " ( t h a t i s , r a t h e r t h a n i n some o t h e r • c e l l ) . I n a d e n s e l i q u i d f a r f r o m i t s c r i t i c a l r a n g e , t h e m o l e c u l a r d y n a m i c s s t u d i e s o f W a i n w r i g h t a n d A l d e r ( 3 8 ) s n o w t h a t a m o l e c u l e t e n d s t o r e b o u n d w i t h i n t h e c o n f i n e s o f t h e ' c a g e ' o r ' c e l l ' f o r m e d b y i t s n e a r e s t n e i g h b o r s , a n d o n l y r a t h e r i n f r e q u e n t l y b r e a k s o u t o f t h e c a g e t o p e n e t r a t e a n a d -j a c e n t c e l l . On a t i m e s c a l e e s t a b l i s h e d b y t h e p e r i o d o f t h e r m a l r a n d o m n e s s m o t i o n s , t h e n , N d i s t i n c t s p a t i a l d o m a i n s a r e d e f i n e d b y t h e N c a g e s c o n f i n i n g t h e N m o l e c u l e s o f t h e s y s t e m . A t l e a s t i n t h e o r y , a g i v e n c o n f i g u r a t i o n o f t h e w h o l e s y s t e m c o u l d b e d e f i n e d b y m e a n s o f a 3 - d i m e n s i o n a l p h o t o g r a p h o f t h e s y s t e m t o ' s t o p t h e a c t i o n ' h e n c e d e f i n e t h e p o s i t i o n o f t h e N c a g e v o l u m e s i n t h e g i v e n s y s t e m c o n f i g u r a t i o n . T h e n u m b e r o f d i s t i n c t p e r m u t a t i o n s o f m o l e c u l e s w i t h r e s p e c t t o t h e N c a g e s d e f i n e d b y s u c h a p h o t o g r a p h I s g i v e n b y : 119 = N!/(N±!N !) ( 7 0 ) . f o r N. m o l e c u l e s o f i a n d N. o f j , w h e r e N. + N. = N. T h i s i s t h e same r e s u l t w h i c h w o u l d h a v e b e e n o b t a i n e d f o r t h e n u m b e r o f d i s t i n c t p e r m u t a t i o n s o f m o l e c u l e s a b o u t t h e p o s i t i o n p o i n t s o f a l a t t i c e . I f a m o l e c u l e b r e a k s o u t o f i t s c a g e , i t i n e f f e c t i n t e r c h a n g e s p o s i t i o n s w i t h a n e a r e s t n e i g h b o r . T h i s d e f i n e s a new p e r m u t a t i o n o f t h e w h o l e s y s t e m . 2 . I N C O R P O R A T I O N OF z I N N - S C A L E E N T R O P Y F O R M U L A T I O N S I n o r d e r t o d e r i v e t h e t h e r m o d y n a m i c p r o p e r t i e s o f t h e N - s c a l e s y s t e m ( a n d i t i s o f c o u r s e a t t h i s s c a l e t h a t t h e f i g u r e s a r e u l t i m a t e l y r e q u i r e d ) i t i s m o s t c o n v e n i e n t t o a c c e p t t h e c o n c l u s i o n s o f C h a p t e r 6, a n d f o r m u l a t e t h e N - s c a l e p r o p e r t i e s i n c o r p o r a t i n g n e c e s s a r y z - s c a l e v a r i a b l e s . I n o t h e r w o r d s a n i n v e r t e d v i e w p o i n t i s b e i n g u s e d : now t a k i n g t h e c o m -p a t i b i l i t y o f l o c a l p r o p e r t i e s w i t h t h o s e o f t h e N - s c a l e f o r g r a n t e d , t h e w a y s i n w h i c h N - s c a l e c o n f i g u r a t i o n a l f o r m u l a s a r e d e p e n d e n t o n t h e z ^ i n t h e NCZ c a s e a r e d e s c r i b e d , a n d t h e m o d i f i e d N - s c a l e c o n f i g u r a t i o n e x p r e s s i o n s a r e t h e n w r i t t e n . 3. C O M P A T I B I L I T Y WITH L A T T I C E S T A T I S T I C S W h i l e t h e c o n c e p t o f t h e s n a p s h o t h a s b e e n u s e d t o i l l u s t r a t e a n i d e a ( t h a t o f t h e g e o m e t r i c b a s i s o f t h e p r e s e n t 120 m o d e l ) a n d w h i l e i n p r i n c i p l e a c o u n t i n g p r o c e s s o n t h e s n a p s h o t w o u l d e n a b l e t h e e v a l u a t i o n o f z ^ t o b e c a r r i e d o u t , i n r e a l i t y s u c h u n s o p h i s t i c a t e d p r o c e d u r e s w o u l d h a v e t o b e r e p l a c e d b y a m o r e f o r m a l c a l c u l a t i o n m e t h o d . T h i s i s a c c o m p l i s h e d b y r e a l i z i n g t h a t t h e i d e a o f N C Z , b o t h o n t h e z - s c a l e a n d o n t h e N - s c a l e , d o e s n o t r e q u i r e a n i r r e v o c a b l e m a t h e m a t i c a l d e p a r t u r e f r o m t h e N - s c a l e CZ l a t t i c e v i e w p o i n t o f t h e o r i g i n a l q u a s i c h e m i c a l t h e o r y . What was d o n e was t o r e f r a m e t h e m a t h e m a t i c s o f p a r t i c l e i n t e r c h a n g e c o m p u t a t i o n s b y c o m b i n a t o r i a l m e t h o d s i n a m o r e f l e x i b l e c o n c e p t u a l c o n -t e x t . I n s u c h a c o n t e x t , t h e i n t e r c h a n g e c o m p u t a t i o n s b e c o m e a m e n a b l e t o t h e i n c o r p o r a t i o n o f t h e e f f e c t s o f l o c a l p r o -p e r t i e s o f t h e l i q u i d ( s p e c i f i c a l l y t h e v a r i a b l e z ) , a s w e l l a s t o p a i r w i s e e n e r g y a t t r a c t i o n s . 4. P A I R S - P E R M U T A T I O N S I N T H E NCZ C A S E T h e c o n s t a n c y o f c o o r d i n a t i o n n u m b e r o f a r e g u l a r z - c o o r d i n a t e d p a c k i n g c o n v e n i e n t l y e n s u r e s t h a t t h e p e r m u t a t i o n o p e r a t i o n i n v o l v i n g t h e i n t e r c h a n g e o f t w o m o l e c u l e s a f f e c t s o n l y z s i t e s o r c o n t a c t s i n e a c h c a s e . T h e n u m b e r o f p h y s i c -a l l y p e r m i t t e d i n t e r c h a n g e s o f p a i r s o f c o n t a c t s o r ' p a i r s ' i s c o n s t r a i n e d b y t h e f a c t t h a t i n a n y g i v e n p e r m u t a t i o n o p e r a t i o n , e a c h molecule m u s t b e m o v e d a s a w h o l e , t h u s z / 2 p a i r s m u s t b e i n t e r c h a n g e d a s a g r o u p . T h e f a c t o r o f 2 a r i s e s f r o m t h e f a c t t h a t e a c h p a i r i s a s s o c i a t e d w i t h 2 m o l e c u l e s r a t h e r t h a n 1 , a n d s o t h e n u m b e r o f p a i r s a t t r i b u t e d t o a n y s p e c i f i c m o l e c u l e i s ( z / 2 ) , t o a v o i d d u p l i c a t i n g t h e c o u n t . 121 W h i l e a p e r m u t a t i o n o p e r a t i o n i n t e r c h a n g i n g t w o m o l e c u l e s a f f e c t s j u s t z/2 p a i r s i n e a c h c a s e , i f t h e t w o m o l e -c u l e s a r e z - c o o r d i n a t e d , i n c r e a t i n g n ew m o l e c u l a r c o n f i g u r a t i o n s o n e i s n o t r e s t r i c t e d t o p e r m u t i n g o n l y t w o m o l e c u l e s a t a t i m e . I f s p e c i e s i I s z ^ c o o r d i n a t e d a n d s p e c i e s j i s z_. c o o r d i n a t e d a n d z a v e i s t h e o v e r a l l a v e r a g e c o o r d i n a t i o n n u m b e r , a n i n t e r -c h a n g e i n v o l v i n g A ^ m o l e c u l e s o f I a n d B. m o l e c u l e s o f j a f f e c t s t h e zi x p a i r s a n c h o r e d b y t h e A.^  a n d t h e z^ * B_. p a i r s a n c h o r e d b y t h e B., w i t h o n l y s e c o n d o r d e r a c c o m m o d a t i n g m o v e -m e n t s r e q u i r e d o n t h e p a r t o f t h e o t h e r ( u n p e r m u t e d ) m o l e c u l e s o f t b e s y s t e m . A s l i g h t n o t a t i o n a l s i m p l i f i c a t i o n e x i s t s f o r a b i n a r y w h e r e we c a n r e f e r t o z a v e s i m p l y a s z . T h u s i t may b e s e e n t h a t i f 1 i s t h e l a r g e r m o l e c u l e , t h e n t h e r e l a t i v e n u m b e r o f p a i r s p e r m o l e c u l e d i r e c t l y a f f e c t e d b y i t s m o v e m e n t i s ( z . / z . . ) , a n d f o r t h e s m a l l e r j o n l y ( z . / z . . ) . T h e s e t w o 1 I J j i j r a t i o s r e p r e s e n t t h e r e l a t i v e i m p o r t a n c e p e r i n d i v i d u a l m o l e -c u l e o f t h e t w o s p e c i e s f o r p a i r s p e r m u t a t i o n p u r p o s e s . I f a l a r g e r n u m b e r , N, o f m o l e c u l e s (1\L o f c o o r d i n a t i o n n u m b e r z^ a n d N j o f c o o r d i n a t i o n n u m b e r z j ) a r e p e r m u t e d , t h e t o t a l n u m b e r o f d i s t i n g u i s h a b l e a n d p e r m i t t e d p a i r s p e r m u t a t i o n s r e p r e s e n t e d b y t h e s y m b o l , n ' i s g i v e n n o t b y , <Ni + V ! (71) (N.!N.!) 1 J b u t m o r e c l o s e l y b y 122 (z./z., . N. + z./z., . N.) ! , _ 1 i.i l l i.i .1 (z./z.. N,)!(z./z.. N.)! l xj i j i j 2 (72) Where z^ and z^ are not too d i f f e r e n t , equation (72) w i l l be 1 s u b s t a n t i a l l y correct. The form of ft i s consistant with a transformation - from a r e a l population of N^ groupings of pairs of group size and N.. groupings of pairs of group size Z j / 2 to pseudopopulations of z±/z^ja^ groupings of type i and Z j ^ Z i J N j g r o u P i n g s °f type j of the same grouping s i z e , namely of the o v e r a l l average size z ± ^ 2 - r r n e term ft' i s then i n the form which equalizes the siz e of pairs bundles for interchange purpose, hence conserves the number of pairs interchanged i n a given permutation of spheres. This same transformation w i l l be found useful i n Chapter 8. 5. ENTROPY OF MIXING EXPRESSION IN TERMS OF CONTACT FRACTIONS It should be noted i n passing that the expansion of the negative logarithm of ft' by Ste r l i n g ' s approximation for the logarithms of f a c t o r i a l s , a f t e r normalizing, y i e l d s an ex-pression i n £. and E. namely, t ~ 7 7 ST^I 7 = £• £ n + £• l n • (73) (z./z. .N. + z,/z..N.) 1 i 2 i ' 1 i j 1 J i j 3 In the l i m i t of equal sized spheres (the constant- z case)ft' re-duces to ft and we obtain simply 123 - r r - — , „ v = x. In x. + x. Hn x . (N + N ) 1 1 j j ( 7 4 ) 6. SUMMARY I t h a s b e e n s h o w n t h a t c o u n t i n g t h e n e a r e s t n e i g h b o r c o n t a c t s o f a g i v e n c e n t r a l m o l e c u l e e s t a b l i s h e s i t s n e a r e s t n e i g h b o r c o o r d i n a t i o n n u m b e r . I n a m i x t u r e o f u n e q u a l s i z e d s p h e r e s , e a c h s p e c i e s a s s u m e s a d i f f e r e n t v a l u e o f z , a n d t h u s t h e z ^ may b e c o n s i d e r e d a s a m i x t u r e - d e p e n d e n t m a t e r i a l p r o p e r t y o f t h e g i v e n s p e c i e s . C o n s i d e r a t i o n o f t h e p o s s i b l e i n t e r c h a n g e s o f c o n t a c t s f o r t h e NCZ c a s e h a s p r o v i d e d t h e f o r m o f t h e c o n f i g u r a t i o n a l m i x i n g t e r m , t o b e i n c o r p o r a t e d i n t o t h e f o r m u l a t i o n o f t h e e x p r e s s i o n f o r t h e e x c e s s f r e e e n e r g y o f a r e a l l i q u i d i n t h e n e x t c h a p t e r . 124 CHAPTER 8 CALCULATION OF EXCESS FREE ENERGY FOR THE  SIMPLEST NCZ CASE 1. OBJECTIVE 2. REFORMULATION OF CONSTANT COORDINATION NUMBER THEORY FOR THE NCZ CASE A. Review of the P a i r s L i q u i d B. Review of the Mixing Process Seen as a Reaction 3. STOICHIOMETRIC ADAPTION TO THE NCZ CASE 4. ENERGETIC INTERPRETATION OF THE INDEPENDENT PAIRS ASSUMPTION 5. BUNDLE-SPECIES MOLECULAR SCALE QUANTITIES 6. FORMULATION OF THE PARTITION FUNCTION 7. REVIEW OF FORMULATION OF THE G E EXPRESSION FOR THE CZ CASE 8. THE G E EXPRESSION FOR THE CORRESPONDING NCZ CASE A. General B. P r i m i t i v e Form of the G Expression C. The Problem of the Free Energy Constnat . 1. Q u a n t i t i v e D i f f i c u l t y 2. Q u a l i t a t i v e J u s t i f i c a t i o n D. Other Considerations Involved i n the C o e f f i c i e n t a E E. G i n Terms of P a i r s Concentrations F. The F i n a l NCZ Expression f o r a Monofunctional Monomer Binary 9. MASS LAW C O N S I D E R A T I O N S A. T h e S i m p l e s t NCZ F o r m B. A p p r o x i m a t i o n s I n v o l v e d i n U s i n g M a s s - L a w - A v e r a g e . C o n c e n t r a t i o n s C. M o r e F o r m a l l y C o r r e c t F o r m o f t h e M a s s L a w E x p r e s s i o n 10. E F F E C T OF VOLUME CHANGES UPON E S T I M A T E S OF G E AND H E 11. E X P E R I M E N T A L E V I D E N C E FOR NCZ E F F E C T S I N R E A L S O L U T I O N S A. G e n e r a l B. P e r f l u o r o - n - H e p t a n e C a r b o n T e t r a c h l o r i d e C. P e n t a e r y t h r i t o l - T e t r a p e r f l u o r o b u t y r a t e C h l o r o f o r m 12., SUMMARY 126 C H A P T E R 8 C A L C U L A T I O N OF E X C E S S F R E E E N E R G Y FOR  T H E S I M P L E S T NCZ C A S E 1. O B J E C T I V E p I n t h i s c h a p t e r , t h e q u a s i c h e m i c a l e x p r e s s i o n f o r G f o r a m o n o f u n c t i o n a l m onomer b i n a r y m i x t u r e o f u n e q u a l l y s i z e d m o l e -c u l e s i s d e v e l o p e d . F i r s t t h e b a s i s o f t h e CZ q u a s i c h e m i c a l t h e o r y i s s e t o u t I n a f o r m s u i t a b l e f o r e x t e n s i o n i n t o t h e NCZ c a s e , a n d t h e n t h e f o r m a l e x t e n s i o n s t o t h e p r e s e n t w o r k a r e s h o w n , a n d t h e i r c o n s e q u e n c e s d i s c u s s e d f o r t h i s s i m p l e s t p o s s i b l e c a s e i n F w h i c h NCZ e f f e c t s c o n t r i b u t e t o t h e e x p r e s s i o n f o r G . 2. R E F O R M U L A T I O N OF CONSTANT C O O R D I N A T I O N NUMBER THEORY FOR T H E NCZ C A S E A. R e v i e w o f P a i r s - L i q u i d T h e t r e a t m e n t b e g i n s w i t h a d e t a i l e d i n s p e c t i o n o f t h e e n e r g e t i c a s p e c t o f t h e l i q u i d m o d e l , i n w h i c h t h e l i q u i d i s r e g a r d e d a s a n a s s e m b l a g e o f i n d e p e n d e n t ' p a i r s ' . A p a i r i s a n e n t i t y w i t h b o t h a g e o m e t r i c a n d e n e r g e t i c i d e n t i t y . B. R e v i e w o f t h e M i x i n g P r o c e s s S e e n a s a R e a c t i o n E n e r g e t i c a l l y , t h e m i x i n g p r o c e s s i s r e g a r d e d i n t h e l i g h t o f a ' q u a s i c h e m i c a l ' r e o r g a n i z a t i o n r e a c t i o n o f p a i r s , w h e r e i n 127 t h e l i k e p a i r s p e c i e s ( i - i ) a n d ( j - j ) f o u n d i n t h e p u r e c o m -p o n e n t s t a t e r e o r g a n i z e t o f o r m u n l i k e p a i r s ( i - j ) . T h e r e -a c t i o n i s w r i t t e n ( i - i ) + ( j - j ) ^ ( i - j ) + ( j - i ) (75) a n d i s a s s o c i a t e d w i t h t h e p a i r s c o h e s i o n e n e r g i e s e. . + £. . ^ 2zA . (76) i i J J i J T e r m s e i i a n d e „ a r e t h e l i k e - p a i r s c o h e s i o n e n e r g i e s ^ a n d e i j i s t h a t o f t h e u n l i k e p a i r s . L i k e - p a i r c o h e s i o n s a r e o b -t a i n e d f r o m t h e p u r e c o m p o n e n t e n e r g y o f v a p o r i z a t i o n E v a p ± a t t h e g i v e n t e m p e r a t u r e , b y t h e r e l a t i o n e.. = E / ( Z /2) (77) 1 1 v aP.£ s e - v a l u e s c a r r y a n e g a t i v e s i g n , s i n c e t h e y a r e c o h e s i v e ( b i n d i n g ) e n e r g i e s . T h e n e t r e o r g a n i z a t i o n e n e r g y , w h i c h , b e c a u s e o f i t s o r i g i n i s u s u a l l y a n i n s e n s i t i v e f u n c t i o n o f t e m p e r a t u r e , i s d e f i n e d a s * a. . = [e. . - % ( e . . + e. .)] (78) i J i J i i J J * I t s h o u l d b e e m p h a s i z e d t h a t t h e a r i t h m e t i c a v e r a g e r e -f e r e n c e u n l i k e p a i r c o h e s i o n e n e r g y , g i v e n b y % ( £ ± ± + i n e q u a t i o n (78) > s i m p l y r e p r e s e n t s t h e i d e a l m i x i n g c a s e f o r t h e p r o c e s s d e s c r i b e d b y r e l a t i o n (75) - P h y s i c a l l y i t i s r a r e t o e n c o u n t e r a v a l u e o f e i j s u c h t h a t = ^ ^ e ± ± + e j j ) s o t h a t = 0 . x n d i s p e r s i o n - f o r c e t y p e s y s t e m s ( n o n -p o l a r i n t e r a c t i o n s ) a p p r o p r i a t e v a l u e s f o r a r e o f t h e t y p e C o n t i n u e d 128 I f u n l i k e p a i r c o h e s i o n i s m o r e b i n d i n g t h a n t h e a r i t h m e t i c a v e r a g e o f l i k e - p a i r c o h e s i o n s , u i s n e g a t i v e a n d u n l i k e p a i r i n g i s t h u s e n e r g e t i c a l l y f a v o r e d . N o t e t h a t a^j i s i n d e p e n d e n t o f t h e d i f f e r e n c e i n t h e p a i r c o h e s i o n s p e r s e . S i n c e n o g e n e r a l m e t h o d o f e v a l u a t i n g e . . , h e n c e o f w . ., b y d i r e c t c a l c u l a t i o n h a s y e t b e e n d e v e l o p e d , u^.. m u s t u s u a l l y b e e x p e r i m e n t a l l y d e t e r m i n e d b y a t l e a s t o n e m e a s u r e m e n t p e r b i n a r y i n v o l v e d i n t h e m i x t u r e . T h e a s s u m p t i o n o f t h e p r e s e n t t h e o r y i s t h a t i s i n d e p e n d e n t o f c o m p o s i t i o n , a n d a l s o t h a t i t i s t h e same I n a m u l t i c o m p o n e n t m i x t u r e a s i t i s i n t h e b i n a r y m i x t u r e T h e c a l c u l a t i o n f o r w^ .. d o e s n o t c o n t a i n a n y a s s u m p -t i o n a s t o w h e r e o r how t h e r e a r r a n g e m e n t o f 2 l i k e p a i r s t a k e s p l a c e . H e n c e i t s i m p l y a n d c o m p l e t e l y a c c o u n t s f o r t h e C o n t i n u e d 1 1 J j o r i j - i i J J w h i c h may t e n d t o w a r d e . . = E . . , w h e r e i i s t h e l e s s c o h e s i v e J i j i i s p e c i e s , s o t h a t i s n o r m a l l y p o s i t i v e ^ - 5 ; . . When s p e c i f i c i n t e r a c t i o n s o f a p o l a r n a t u r e a r e i n v o l v e d , t h e m a g n i t u d e o f e^j i s u s u a l l y m u c h l a r g e r t h a n t h e a r i t h m e t i c - m e an o f t h e u n -l i k e p a i r c o h e s i o n s , s o t h a t io b e c o m e s l a r g e a n d n e g a t i v e , a n d i s a s s o c i a t e d w i t h a n ' e x o t h e r m i c ' h e a t o f m i x i n g e f f e c t . S e e f o r e x a m p l e r e f e r e n c e ( 3 4 ) . 129 l o w e r i n g o f t h e o v e r a l l a s s e m b l a g e ' s e n e r g y a s a r e s u l t o f o n e s u c h a d d i t i o n a l 4 - m o l e c u l e r e a r r a n g e m e n t r e s u l t i n g i n t h e j u x t a -p o s i t i o n o f u n l i k e s p e c i e s . T h u s , w h i l e n o t i n c o m p a t i b l e w i t h c o n s i d e r a t i o n o f t h e b i n d i n g e n e r g y o f a c e n t r a l m o l e c u l e b y i t s n e a r e s t n e i g h b o r s , t h e q u a s i c h e m i c a l r e a r r a n g e m e n t e n e r g y ("^j) f o r m u l a t i o n i n n o way ' i s o l a t e s ' o r ' d e c o u p l e s ' t h e n e a r e s t n e i g h b o r s a s s e m b l a g e f r o m t h e l i q u i d a s a w h o l e . T h i s i s t h e k e y c o n c e p t u a l d i f f e r e n c e b e t w e e n q u a s i c h e m i c a l e n e r g y c o n s i d e r a t i o n s a n d t h o s e i n v o l v e d i n t h e 2 - l i q u i d t h e o r i e s d i s c u s s e d i n C h a p t e r 10, w h i c h do r e g a r d e a c h n e a r e s t n e i g h b o r a s s e m b l a g e a s b e i n g ( e n e r g e t i c a l l y ) i s o l a t e d f r o m t h e r e s t o f p a r t i c l e a s s e m b l a g e . 3. S T O I C H I O M E T R I C A D A P T I O N TO T H E NCZ C A S E T h e s t o i c h i o m e t r y o f e q u a t i o n (75) ( i . e . t h e n u m b e r s o f p a i r s o f ' r e a c t a n t s ' a n d ' p r o d u c t s ' i n v o l v e d i n p a i r s r e -o r g a n i z a t i o n ) i s o n l y s t r i c t l y c o r r e c t f o r t h e c o n s t a n t - z c a s e . I n t h e NCZ c a s e , w h e r e c o o r d i n a t i o n n u m b e r s b e c o m e d e p e n d e n t o n c o m p o s i t i o n , i n o r d e r t o e n s u r e c o n s i s t e n c y w i t h a n o v e r a l l s t o i c h i o m e t r i c b a l a n c e , o n e w o u l d f i r s t w r i t e t h e o v e r a l l r e -o r g a n i z a t i o n r e a c t i o n , n a m e l y t h a t s t a r t i n g w i t h l i k e - p a i r s i n t h e p u r e c o m p o n e n t s t a t e , f o r t h e n u m b e r o f p a i r s c o -o r d i n a t e d p e r m o l e c u l e . f ( i - i ) +f- ( j - j ) ^ ^ f i ( i - j ) + *-f - ( j - D (81) ( T h e r e a d e r may f i n d i t h e l p f u l t o l o o k a g a i n a t I l l u s t r a t i o n 4, 130 page 32, i n connection with t h i s process). Dividing by z r e s u l t s i n the desired p a i r s - s c a l e reaction: ( i - i ) + a-j) ^ H r 1 ) (i-j) (82) s Hence the e f f e c t i v e reorganization energy per unlike p a i r formed, u>'. . Is given by I f (z /z..) s e r i o u s l y departed fron unity, the s i j stoichiometry of the pairs reorganization reaction would depart correspondingly from the integer form and, for one thing would become appreciably, dependent on z, hence on composition, i n the NCZ case. However z i j B z s (84) for solutions encountered, so that u may be used as an adequate approximation to u>.!.. without appreciable loss of accur-acy. The fact that w^j characterizes the o v e r a l l mixing pro-cess, s t a r t i n g from the pure component state, prevents u>^  • from being a highly concentration-dependent r e l a t i o n depending on z. and z. applied to s e l f - p a i r cohesions, i n a form such as " i j = [ £ ± j - * ( [ ( 2 i / z . . ) x E i i + x e ^ U ] ( 8 5 ) 131 as w e l l as z. . being a p p l i e d to the u n l i k e p a i r cohesion. 4. ENERGETIC INTERPRETATION OF THE INDEPENDENT' PAIRS  ASSUMPTION I m p l i c i t l y i n c l u d e d i n equation ( 8 l ) i s another major ene r g e t i c assumption of quasichemical theory, which i s even more important i n the NCZ case than f o r constant-z s o l u t i o n s : namely that the p a i r w i s e cohesion i s regarded as independent of c o o r d i n a t i o n number. The assumption i s merely an emphasis on one aspect of the meaning of independent p a i r s ' i n t e r a c t i o n s . S o l u t i o n s i n which cohesion i s dominated by London-force or d i s p e r s i o n - f o r c e i n t e r a c t i o n s conform w i t h t h i s assumption, because the d i s p e r s i o n a t t r a c t i o n between any two molecules i s a quantum mechanical e f f e c t , shown by London not to be s e r i o u s l y perturbed by the presence or absence of other mole-(41) cules .The p a i r w i s e independence of i n t e r a t i o n of any a r b i t r a -r i l y chosen " c e n t r a l " molecule and a 'nearest neighbor', would be a case i n p o i n t . Hence, i n d i s p e r s i o n - f o r c e systems the cohesion of the c e n t r a l molecule to one nearest neighbor i s not s e r i o u s l y perturbed by the presence of a second, t h i r d . .z^ nearest neighbor, where packing r e l a t i o n s are such t h a t the cen-t r a l one i s z i coordinated. The assumption of p a i r w i s e i n -dependence has been used as an approximation f o r a l l systems In t h i s work, even those i n which i n t e r a t i o n s are f a r from p u r e l y ' d i s p e r s i o n ' i n nature. A dire c t c o r o l l a r y •  of the assumption of p a i r w i s e i n -dependence of energy i n t e r a c t i o n s i s that packing-induced 132 c h a n g e s o f a s p e c i e s ' c o o r d i n a t i o n n u m b e r , a s d e m o n s t r a t e d i n C h a p t e r 6 f o r a p a c k e d b e d , a n d h e r e e x t e n d e d b y a n a l o g y t o t h e c a s e o f a r e a l l i q u i d , c a u s e d i r e c t l y c a 1 c u 1 a h 1 e c h a n g e s i n t h e t o t a l c o h e s i o n p e r m o l e c u l e , a n d h e n c e i n t h e s o l u t i o n a s a w h o l e . w h e r e a s t h e c o n s t a n t - z t r e a t m e n t f o c u s e s o n t h e e f f e c t s o f c o h e s i o n c h a n g e s i n a m i x t u r e d u e t o n o n - z e r o ^ . . , t h e NCZ t r e a t m e n t a l s o a c c o u n t s f o r t h e c o h e s i o n c h a n g e s w h i c h c o u l d a r i s e d i r e c t l y f r o m p a c k i n g - i n d u c e d z c h a n g e s * • P a c k i n g i n -d u c e d c h a n g e s t h u s r e s u l t i n t e r m s i n t h e NCZ f o r m o f t h e e x c e s s f r e e e n e r g y o f m i x i n g e x p r e s s i o n w h i c h a r e n o t p r e s e n t i n c o n s t a n t - z f o r m u l a t i o n s . T h e s e e f f e c t s c a n o c c u r e v e n i n t h e c a s e w h e r e t h e OJ . . t e r m s a r e a l l z e r o . I n f a c t , b e c a u s e o f t h e p h e n o m e n o n o f t h e n e a r e q u a l i t y o f z . . a n d z i n e q u a t i o n ( 8 3 ) , i t may b e s e e n t h a t t h e 1 j s a d d i t i o n a l c o h e s i o n e f f e c t s o f t h e NCZ c a s e a r e n e a r l y i n - d e p e n d e n t o f t h e c l a s s i c a l o n e s o f c o n s t a n t - z q u a s i c h e m i c a l t h e o r y . F o r m u l a t i o n a n d s u b s e q u e n t a s s e s s m e n t o f t h e 'new' t e r m s t h u s b e c a m e a m a t t e r o f c o n s i d e r a b l e p r a c t i c a l i n t e r e s t . 5. B U N D L E - S P E C I E S M O L E C U L A R - S C A L E Q U A N T I T I E S I n C h a p t e r 7 was d e r i v e d a n e x p r e s s i o n f o r n' , t h e * T h e u s e o f w . . ( a s a n r a t h e r t h a n t h a t o f i n p o r t a n t c h o i c e . H e n c e t h i s a d e q u a t e a p p r o x i m a t i o n t o u ^ ) t h e NCZ c a s e r e p r e s e n t s a n i m -n o t e . 133 a p p r o p r i a t e c o m b i n a t o r i a l t e r m d e s c r i b i n g t h e n u m b e r o f p o s s i b l e i n t e r c h a n g e s o f c o n t a c t s i n t h e NCZ c a s e , w h i c h was o b t a i n e d f r o m a t r a n s f o r m a t i o n o f ft , t h e n u m b e r o f p o s s i b l e a n d d i s t i n g u i s h a b l e i n t e r c h a n g e s o f m o l e c u l e s i n a b i n a r y m i x -t u r e . T h e m a i n c o n f i g u r a t i o n a l r e s t r a i n t u p o n a r e a l l i q u i d r e g a r d e d a s a n a s s e m b l a g e o f p a i r s o f c o n t a c t s i s t h e s a m e : n a m e l y t h a t f o r c o n f i g u r a t i o n a l p u r p o s e s , p a i r s m u s t b e r e -g a r d e d a s r e a s s e m b l e d i n t o m o l e c u l a r - s c a l e e n t i t i e s . T h e m o l e c u l a r - s c a l e e n t i t y f o r m e d f r o m t h e p a i r s a s s e m b l y i s h e r e a f t e r d e s i g n a t e d a ' p a i r s - b u n d l e ' , o r g r o u p o f z/2 p a i r s . U n l i k e a n a c t u a l m o l e c u l e , w h i c h i n a m i x e d e n -v i r o n m e n t f o r m s v a r i o u s s p e c i e s o f p a i r s , a p a i r s - b u n d l e i s c o m p o s e d e x c l u s i v e l y o f o n l y o n e p a i r s - s p e c i e s . I n a s i m p l e b i n a r y m i x t u r e o f i a n d j t y p e s i t e s , t h e a c t u a l p a i r s p o p u l a -t i o n s c a n b e r e f o r m e d b y a n i m a g i n a r y s e g r e g a t i o n p r o c e s s i n t o f o u r p o s s i b l e t y p e s o f b u n d l e s w i t h b u n d l e p o p u l a t i o n s made c o n s i s t e n t w i t h a n o v e r a l l b a l a n c e o f t y p e s o f p a i r s . T h e f o u r b u n d l e - s p e c i e s w o u l d t h u s b e t h o s e o f i s u r r o u n d e d b y i , j s u r r o u n d e d b y j , i s u r r o u n d e d b y j a n d j s u r r o u n d e d b y i . T h e r e f o r m a t i o n o f a m i x t u r e i n t o f o u r k i n d s o f b u n d l e s p e c i e s i s s h o w n i n I l l u s t r a t i o n 6 . T h e p o i n t o f d o i n g t h i s i s s o t h a t e a c h t y p e o f p a i r c a n b e a s s o c i a t e d w i t h a c o o r d i n a t i o n n u m b e r . I t i s c o n v e n i e n t t o a m a l g a m a t e t h e t w o u n l i k e - p a i r b u n d l e - s p e c i e s I n t o o n e u n d i f f e r e n t i a t e d u n l i k e - p a i r b u n d l e s p e c i e s . F o r c a l c u l a t i o n p u r p o s e s , t h u s , t h r e e b u n d l e - s p e c i e s e x i s t : i - i t y p e b u n d l e s o f b u n d l e s i z e z./2, j - j t y p e b u n d l e s o f s i z e z./2 , a n d I - j t y p e b u n d l e s o f s i z e z . .12 . D i v i d i n g ACTUAL CONFIGURATION 2-®PAIRS j -©PAIRS 2-®PAIRS I - © PAIRS BUNDLE BUNDLE BUNDLE BUNDLE IDEALIZATION AS BUNDLE-SPECIES MOLECULAR SCALE GROUPINGS (NOTE: IN OVERALL MIXTURE, I - © BUNDLE POPULATION = Zj/2X ( I - ® ) ACTUAL PAIRS POPULATION ) I l l u s t r a t i o n 6. Bundle-Species. 135 b y 2 r e d u c e s t h e n u m b e r o f c o n t a c t s p e r b u n d l e t o t h e n u m b e r o f c o n t a c t s a s s o c i a t e d w i t h , o n e m o l e c u l e . S i n c e z . , z . a n d 1 J z . . a l l h a v e d i f f e r e n t v a l u e s i n t h e NCZ c a s e , r e f o r m u l a t i o n o f t h e c o n f i g u r a t i o n a l b e h a v i o u r o f t h e p a i r s a s s e m b l a g e i n t o t h a t o f ' b u n d l e s ' d i f f e r s f r o m d i r e c t c o n s i d e r a t i o n o f r e a l -m o l e c u l e b e h a v i o r I n a n o n - t r i v i a l w ay. T h e c o n c e p t o f ' b u n d l i n g ' t o g e t h e r w i t h a m e t h o d f o r e v a l u a t i n g t h e e q u i l i -b r i u m p o p u l a t i o n s o f t h e b u n d l e s p e c i e s * i n a m i x t u r e f o r m s t h e b a s i s f o r d e f i n i n g m i x t u r e b e h a v i o u r i n t h e q u a s i c h e m i c a l m o d e l . 6. F O R M U L A T I O N OF T H E P A R T I T I O N F U N C T I O N T h e b a s i c m e c h a n i c a l a n d e n e r g e t i c p o s t u l a t e s o f t h e q u a s i c h e m i c a l m o d e l a r e now d e f i n e d i n a way c o m p a t i b l e w i t h a n NCZ m i x t u r e . I t i s now n e c e s s a r y t o c o n s t r u c t a m o l a r s c a l e p a r t i t i o n f u n c t i o n f o r t h e m i x t u r e a n d p u r e c o m p o n e n t i , n a m e l y , Q l n i x a n d r e s p e c t i v e l y , f r o m w h i c h t o o b t a i n t h e e x c e s s f r e e e n e r g y o f m i x i n g f r o m t h e s t a n d a r d r e l a t i o n : A . . = -kTUn 0 . - T £n Q°] ( 8 6 ) mixing TIIIX ~ S i n c e t h e m e c h a n i c a l m o d e l i s f o r m u l a t e d o n t h e ' p a i r s ' * T h i s m u s t b e d o n e c o n s i s t e n t w i t h e n e r g e t i c a l l y i n d u c e d l o c a l c l u s t e r i n g t e n d e n c i e s , s t o i c h i o m e t r i c r e s t r i c t i o n s , a n d c o n f o r m i t y w i t h t h e p r i n c i p l e o f maximum r a n d o m n e s s . 136 a n d ' b u n d l e ' l e v e l s , i t i s f i r s t n e c e s s a r y to. d e f i n e p a r t i t i o n f u n c t i o n s a t t h e s e l e v e l s , a n d r e l a t e t h e s e t o t h e m o l a r s c a l e o n e s . T h e p a r t i t i o n f u n c t i o n f o r a n i - j . p a i r i s d e -f i n e d a s - e . ./kT q i i = ( e 1 3 Vf ) ( 8 7 ) i - j T e r m i s t h e p a i r w i s e c o h e s i o n o f t h e i - j p a i r a n d ^ i s a v o l u m e - o f - i n t e g r a t i o n f a c t o r d e f i n i n g t h e 3 - s p a c e r e g i o n > o v e r w h i c h t h e v a l u e o f i s t h e a v e r a g e v a l u e o f t h e c o h e s i o n , vf = (w/N)/(z 12) . ( 8 8 ) . w h e r e v = m o l a r v o i d v o l u m e o f t h e l i q u i d , N i s A v o g a d r o ' s n u m b e r a n d i s t h e c o o r d i n a t i o n n u m b e r a p p r o p r i a t e t o t h e i - j b u n d l e s p e c i e s . v f i s n o t a n e a s i l y e v a l u a b l e n u m b e r , i - j a n d w h e n e v e r p o s s i b l e f o r m u l a t i o n s r e q u i r i n g d i r e c t a s s i g n -m e n t o f a n u m b e r t o i t a r e a v o i d e d . T h e m o l e c u l a r s c a l e p a r t i t i o n f u n c t i o n ( o f t h e i - j b u n d l e s p e c i e s ) i s d e s i g n a t e d b y z . . / 2 Q . . - ( q ± 1 ) 1 J ( 8 9 ) . I J 1 J I f q . . i s t a k e n t o b e i n d e p e n d e n t o f c o n c e n t r a t i o n ( a s s u m e d i n i j t h i s f o r m u l a t i o n ) t h e n i s e x p l i c i t l y a f u n c t i o n o n l y o f r e -l a t i v e b u n d l e s i z e z . . / 2 s o t h a t i n p a r t i c u l a r i j ' 137 o r m o r e s p e c i f i c a l l y z.l 2 Q,, = C O 1 ( 9 1 ) X I ' X X F o r N m o l e c u l e s o f p u r e c o m p o n e n t i 4° = (Qj ± ) N ± (92). F o r a b i n a r y m i x t u r e * N.-X.. N.-X.. X.. X.. * -In e q u a t i o n (93) t r a n s f o r m e d b u n d l e p o p u l a t i o n s a r e u s e d i n ^ • T h e a c t u a l p o p u l a t i o n s , w h i c h a r e t h e o n l y p o p u -l a t i o n s c o n s i s t e n t w i t h s t o i c h i o m e t r y , a r e u s e d a s t h e i n d i c e s o f t h e q t e r m s . T h e r e a s o n i s t h a t ^ i s r e d u c i b l e i n t o t e r m s o f r a t i o s o f p o p u l a t i o n s , w h e r e a s t h e p o w e r s o f t h e q t e r m s r e -p r e s e n t p o p u l a t i o n s s o t h a t n u m b e r s o f b u n d l e p o p u l a t i o n s r a t h e r t h a n a b a l a n c e o n n u m b e r o f c o n t a c t s m u s t b e m a i n t a i n e d . p o r t h e m e a n i n g o f ' t r a n s f o r m e d ' , r e f e r b a c k t o C h a p e r 7, e q u a t i o n s (.71) a n d ( 7 2 ) . 138 w h e r e I\L a n d N. a r e t h e m o l e c u l a r p o p u l a t i o n s o f c o m p o n e n t s i a n d j a n d X „ I s t h e e q u i l i b r i u m u n l i k e - p a i r b u n d l e - p o p u -l a t i o n i n t h e m i x t u r e . 7. R E V I E W OF FOR M U L A T I O N OF T H E G E X P R E S S I O N FOR T H E CZ C A S E T h e H e l m h o l t z f r e e e n e r g y c h a n g e d u e t o m i x i n g i s o b -t a i n e d b y s u b s t i t u t i n g e q u a t i o n s (89) a n d (90) i n t o ( 9 2 ) , a n d (92) i n t o ( 8 6 ) , t h e n ( 9 1 ) , (89) a n d (93) i n t o ( 8 6 ) , t h e n w r i t i n g o u t t h e b u n d l e s p e c i e s p a r t i t i o n f u n c t i o n s i n t e r m s o f t h e i r p a i r s p a r t i t i o n f u n c t i o n s v i a ( 8 7 ) . T h i s r e s u l t s - i n : A . . k T * n (z./z..N. + z./z..N.)! (z./z..N.)!(z./z..N.)! o Z i / 2 hi (N.-X..) 1 13 o Z3/2 33 (N.-X^) Z i j / 2 £n o Zs/2 q i i N. + Jin 13 o Zs/2 q 3 i q i 3 5 I j / 2 13 N, (94.) I n t h e c o n s t a n t - z c a s e , z . = z . = z . . = z = z 1 3 13 s (95) A s ummary o f t h e r e d u c t i o n o f e q u a t i o n ( 9.4) i n t h e c o n s t a n t - z c a s e I s s e t o u t b e l o w f o r p u r p o s e s o f c o m p a r i s i o n t o c o r r e s p o n d i n g o p e r a t i o n s f o r t h e NCZ c a s e n e x t t o b e c o n -139 s i d e r e d . _ mixing kT J t n + £n (N.+N.)! N.!N .! i J + £n q. .q.. q. .q.. > J J . I J r o -r q i i l l z/ 2 N i + Jin r o i J J . z/2 N j v C96) The l a s t two terms are zero through the assumptions of equation (91). In the second term of equation (96) q i i * q i i o o r =£n -e /kT _ e i 1 / k T U f . v f . e x e J w - i ^ i i - . i -e../kT i i e x e -e../kT v, „, x - i J - J (97) -2 x [e.. - %(e. . + e . .) ] = An f. . f, . i - ^ i ^ i l - i J- 8! (9$) -20). /kT. i j (99) The extensive c a n c e l l a t i o n s r e s u l t from the assumption that the r a t i o of volume f a c t o r s i s e s s e n t i a l l y u n i t y , and from the i d e n t i f i c a t i o n of the term i n square brackets as the net re-^ o r g a n i z a t i o n energy, u±y By s u b s t i t u t i n g the r e s u l t of (99) back i n t o ( 9 6 ) , and e v a l u a t i n g the ft term i n ( 9 6 ) , the f i n a l e xpression i s obtained f o r one mole of mixture: 1 4 0 A co. = (x. inx. + x. Inx.) + X. . z -±1 (100) RT i i j j i j RT w h e r e X. . = X. ./(N. + N.) ( 1 0 1 ) i s t h e e q u i l i b r i u m u n l i k e - p a i r s c o n c e n t r a t i o n , e v a l u a t e d f r o m m a s s l a w c o n s i d e r a t i o n s a s X. 2 / ( 3 . . + 1) ( 1 0 2 ) w h e r e 3 ^ i s a c o r r e c t i o n f a c t o r f o r e n e r g e t i c a l l y i n d u c e d c l u s t e r i n g , g i v e n b y 2co../kT p • • 3.. = (1 + hx±x. x [e ^ - 1 ] ) 2 e l 0 3 ) S u b t r a c t i n g f r o m e q u a t i o n ( 1 0 0 ) t h e i d e a l m i x t u r e c o n -f i g u r a t i o n a l e x t r o p y m i x i n g t e r m , n a m e l y xjinx^ + x^lnx^ r e s u l t s i n t h e e x t r e m e l y s i m p l e G u g g e n h e i m f o r m f o r t h e e x c e s s E E f r e e e n e r g y o f m i x i n g , A ~ G , GE/RT = X.. z - | i ( t„") ( 1 0 4 ) I n t h e p r e s e n t w o r k , t h e n a m e t ^ " i s a s s i g n e d t h i s e x p r e s s i o n , a s I t w i l l p r o v e t o b e t e r m t ^ " i n t h e g e n e r a l f o r m o f t h e NCZ e q u a t i o n o b t a i n e d b y t h e e n d o f C h a p t e r 9, a l t h o u g h t h e 141 f o r m u l a f o r X.. w i l l b e m o d i f i e d b y NCZ e f f e c t s i n t h a t c a s e , S i x o t h e r " t " t e r m s w i l l b e s i m i l i a r l y I d e n t i f i e d a s t h e y a r i s e i n t h e t w o c h a p t e r s t o f o l l o w . 8. T H E G E X P R E S S I O N FOR T H E CORRESPONDING NCZ C A S E A. G e n e r a l L e a v i n g a s i d e f o r t h e moment t h e q u e s t i o n o f m a s s -l a w e x p r e s s i o n s * f o r X.., t h e r e d u c t i o n o f e q u a t i o n (94) i n t h e NCZ c a s e I s now u n d e r t a k e n . I n t h i s c a s e t h e n e a t c a n -c e l l a t i o n s w h i c h w e r e p o s s i b l e I n t h e c o n s t a n t - z c a s e a b o v e c a n n o l o n g e r b e made. T h e s p i r i t o f t h e i n v e s t i g a t i o n a t t h i s p o i n t was o n e o f d e v e l o p i n g t h e a l g e b r a i c c o n s e q u e n c e s o f t h e NCZ i d e a , i n -s o f a r a s p o s s i b l e , a s a n e x e r c i s e i n d e a d - r e c k o n i n g , a n d n o t t o d i s c a r d t e r m s f o r t h e s a k e o f c o n f o r m i t y w i t h a n y p r e -c o n c e i v e d i d e a s a s t o w h a t t h e r e s u l t s s h o u l d l o o k l i k e . N e x t , t e r m s w e r e r e d u c e d t o a s e l e g a n t a f o r m a s p o s s i b l e t o a i d i n p h y s i c a l a n d t h e r m o d y n a m i c i d e n t i f i c a t i o n ; i n some c a s e s c e r -t a i n f o r m a l s i m p l i f i c a t i o n s h a d t o b e f o r c e d t h r o u g h s i m p l i -f y i n g a p p r o x i m a t i o n s . C a r e was e x e r c i s e d n o t t o i n t r o d u c e t o o many o f t h e s e a p p r o x i m a t i o n s t o o e a r l y , a n d t h e r e b y l o s e t r a c k o f t h e u n d e r l y i n g f o r m a l m e t h o d , s i n c e t h e b a s i c a p p r o a c h was * A s e c t i o n o n m a s s l a w c o n s i d e r a t i o n s i n i n c l u d e d a t t h e e n d o f t h i s c h a p t e r . 1 4 2 f e l t to be both powerful enough and ,well enough, v e r i f i e d on r e a l 'bonstant-z"systems to bear the' pressure put on i t by the extension. The only point at which the extension appears to be somewhat unclear concerns the c o e f f i c i e n t of one of the new terms, which experimental r e s u l t s Indicate should d i f f e r from the derived value by a factor of z/2. B. Primitive Form of the G Expression By straightforward algebraic manipulation, the NCZ equivalent of equation d.Q.4 ) i s obtained, as indicated below. + »± - V (1 -. Sr. q " . < 1 0 5 > z £n q° S u b s t i t u t i n g i n * X - (N. - X. .)/N (io6) m i i 1 1 J * substitutions are leftward conformal with FORTRAN practice The subscript m indicates that the proportions are properties of the t o t a l molecular population. The s i g n i f i c a n c e of t h i s d e t a i l emerges i n equations (117) to (121) l a t e r i n the chapter. Note that equation (105) contains approximations.N'= N. 143 X = (N, - X . J / N m.. J iJ (107) *. = X. ./N m i j 1 J (108) and r e a l i z i n g that formally, i l n q ° . = *n(q°. Z s / 2) -"£n Q?. - -A°/N kT 2 H J J M J J J J J (109) where A = the molal Helmholtz free energy of species i n the pure component state, one would expect the NCZ equivalent of (l04) to be given by ^ifiS£= Z.MZ. + S.AiiS. + X z. . RT i i i J m. . x i RT " ^m./ 1 " z i / z a ) i f f i 1 ^ + Xm . . ( 1 " Z j / Z s } RT J J + X m (1 - z../z ) x (A° + A°)/RTV (110) C. The Problem of the Free Energy Constant 1. Quantitive D i f f i c u l t y However i t i s found that (110) overstates each of the terms included i n the braces \ [by a factor of very approximately 144 z/2 . The reason e v i d e n t l y l i e s i n the assumption that the l i k e - p a i r s hundle p a r t i t i o n f u n c t i o n s are f u n c t i o n s s t r i c t l y as simple as i s i m p l i e d by equation (.90). In f a c t , i t Is .o found that (110) gives b e t t e r r e s u l t s upon r e p l a c i n g A± by a i = \ k l \ n z s / 2 ) (111) a s u b s t i t u t i o n which however has apparent mass law i m p l i c a t i o n s not yet f u l l y understood by the author. 2. Q u a l i t a t i v e J u s t i f i c a t i o n However, a j u s t i f i c a t i o n of the b a s i c f r e e energy nature o of the a ± c o e f f i c i e n t I s given e x p e r i m e n t a l l y by the markedly d i f f e r e n t temperature dependence of the heat of mixing and NCZ f r e e energy terms. Whereas i s e i t h e r constant w i t h I n -c r e a s i n g temperature ( d i s p e r s i o n energy c a s e ) , or r a p i d l y a t t e n u a t i n g w i t h i n c r e a s i n g temperature ( d i p o l e energy c a s e ) , |a°| and |a°| i n c r e a s e at increased temperature, such that has a much s l i g h t e r r a t e of temperature a t t e n u a t i o n t h a n R T would a heat of mixing e f f e c t . D. Other Considerations Involved l n the C o e f f i c i e n t a As a b r i e f aside on the question of the "a" c o e f f i c i e n t at t h i s p o i n t , a thermodynamic c o n d i t i o n f o r e q u i l i b r i u m i s given by (—) - 0 (112) 145 T h a t t h e f o r m o f a 0 ^ a n d a ° ^ a l s o r e p r e s e n t s a n o v e r s i m p l i -f i c a t i o n w i l l be e v i d e n t i f a T a y l o r ' s s e r i e s i n A(z.. z.) i s r e l a t e d t o o n e i n A ( v . , v . ) t h r o u g h t h e a p p r o x i m a t e r e l a t i o n s b e t w e e n z i a n d v ^ . ( S e e e q u a t i o n s ( 5 8 ) , (59) a n d (69)) . E q u a t i o n (110) w o u l d be ' e x a c t ' i f t h e r e w e r e no t h e r m o d y n a m i c d i f f e r e n c e b e t w e e n t o t a l e n e r g i e s a n d f r e e e n e r g i e s . Remember-i n g t h a t t h e f o r m o f q i s g i v e n b y q 1 4 - (e 1 1 v, ) ( H 3 ) i - i I n e q u a t i o n (113) w h e r e u i s t h e v o l u m e - o f - i n t e g r a t i o n a p p r o p r i a t e t o a p a i r , t h e a s s u m p t i o n t h a t t h e r a t i o o f v o l u m e f r a c t i o n s i n t h e m i x i n g t e r m , i . e . , i n t h e r a t i o J - - 1 J-J a i (114) i - j i-J i s a g o o d a p p r o x i m a t i o n . H o w e v e r , t h e r e i s no s u c h a p p r o x i -m a t i o n t o h e l p d i s p o s e o f t h e v f a c t o r s p r e s e n t i n t h e t e r m s i - i i n b r a c e s i n ( 1 1 0 ) , s o t h a t , i n o r d e r t o a v o i d e v a l u a t i n g t h e v f f a c t o r s e x p l i c i t l y , t h e y a r e l e f t c o m b i n e d w i t h t h e i r i - i B o l t z m a n f a c t o r s , a n d t h e w h o l e t e r m c o n t a i n i n g t h e m i s e v a l u -a t e d a s a f r e e e n e r g y . T h e t o t a l c o h e s i o n p e r m o l e o f t h e s o l u t i o n i s c h a n g e d o n m i x i n g b y c h a n g e s i n t h e s t r e n g t h o f p a i r b o n d s , a s r e f l e c t e d b y 146 X_ (1 - zjzj x (-e ± 1 x z./2) + X m (1 - Z j / z s ) x (- £ j j x m i i 1 s JJ + \iil.-'±i/Zs) X ( " [ e i i + E J J ] X Z i / 2 ) (115) However, the change i n the t o t a l p a i r s p o p u l a t i o n s ' f r e e energy z. . a Is o depends on a s i m i l a r f u n c t i o n i n ~~ s- n u f , which p a r t l y i ~ j compensates the NCZ cohesion change shown. g E. G i n Terms of P a i r s Concentrations F i n a l l y , i t Is r e q u i r e d to express the molecular p a i r s c o n c e n t r a t i o n terms i n equation (110) by X.., X.. and X.., noting l j l i j J that x.. - 5 . - X,. i i * i i j (116) and X.. = 5. - X.. JJ J i J (117) by t h e i r r e l a t i o n to the po p u l a t i o n p a i r s f u n c t i o n s x m. . IJ X. . x z Iz. . i j ave i j (118) x m i i m. JJ X. . x z / z . x i ave l X. . x z Iz. JJ ave j (119) (120) 147 The X^. p a i r s c o n c e n t r a t i o n s are p r o p o r t i o n s of a p o p u l a t i o n of p a i r s t r e a t e d as being z - coordinated. The X.. form a v e i j i s more convenient to c a l c u l a t e In the multicomponent case, so i s introduced here i n the i n t e r e s t of c o n s i s t e n c y . The c o o r d i n a t i o n number r a t i o s r e l a t e the X.. to the X p a i r s x j m . . c o n c e n t r a t i o n s a c t u a l l y a r i s i n g i n the derivation. P. The' F i n a l ' NCZ Expression f o r a Monofunctional Monomer  Binary Making the approximation GE(n) = AE(V) ( 1 2 1 ) RT RT one obtains: E = U ^ n q + C ^ n ^ - (XjAna^ + x.lnx^)} ( t ^ ) z 0 ) . . + X i j RT l V o o +• (X.. ( 1 - z . / Z i . ) ( ^ i ) ^ + X. . ( l - z . / z . . ) ( f i l ) i } ( t 5 ) ( 1 2 2 ) Equation ( 1 2 2 ) i s the form of the NCZ equation f o r a mono-f u n c t i o n a l monomer binary mixture. In equation ( 1 2 2 ) , two more terms of the general ex-p r e s s i o n f o r G have a r i s e n , namely t ^ and t ^ , and the form of t h " has been a l t e r e d . Note that the approximation 148 Z = ZAA (123) ave i j has . been made, and that the i d e a l entropy of mixing term (x.lnx. + x.lnx.) has been subtracted i n term t , of equation 1 1 J J . , 1 (122) to a r r i v e at the usual d e f i n i t i o n of excess energy, G . Equation (122) i s the form usable f o r many mixtures of f a i r l y round monomers of unequal s i z e , and of r a t h e r low , i n which there i s only one k i n d of f u n c t i o n a l group on each molecular s p e c i e s . 9. MASS LAW CONSIDERATIONS A. The Simplest NCZ Form Although i n v e s t i g a t i o n s of the forms of the mass law, and the problems a s s o c i a t e d w i t h the constant-volume nature of the formulations of quasichemical theory, r e a l l y do not f i t very concordantly i n t o the development of the mainstream of de-velopment of thought at t h i s p a r t i c u l a r p o i n t ; s t i l l they con-s t i t u t e c o n s i d e r a t i o n s which must be borne i n mind (and w i l l have to be r e f e r r e d back to from the next chapter) before the NCZ equation can be g e n e r a l i z e d , as at l e a s t the former causes s e r i o u s problems when attempts are made to apply quasichemical theory to mixtures of more than two components. The l a t t e r i s another l i m i t a t i o n of quasichemical t h e o r i e s , but can be shown to be of a much l e s s s e r i o u s nature than the former. I t i s th e r e f o r e proposed to conclude the more formal d i s c u s s i o n of t h i s chapter w i t h these two somewhat apocryphal t o p i c s , hoping not to. t r y the reader's patience too f a r i n the process. I t was mentioned, apropos equation (122), that the formula f o r X.. r e s u l t e d from mass-law c o n s i d e r a t i o n s . Since these are modified i n the NCZ case, a b r i e f account of the r e -l a t i o n s should be i n c l u d e d f o r the case of a binary mixture. The mass law expression f o r X.. i s one c o n s i s t e n t w i t h the c o n s t r a i n t of lowest system f r e e energy, c h a r a c t e r i z e d as A . -> minimum. (12A) mix / Because A . - -kT An 0. . ( 1 2 5 ) mix mix and since can be formulated i n terms of X.., the unknown e q u i l i b r i u m p a i r s p o p u l a t i o n , i t i s p o s s i b l e to solve f o r X.. through s o l u t i o n of the extremum c o n d i t i o n 8An Q. . . mix _ n (126) op e r a t i n g upon a s u i t a b l e f o r m u l a t i o n f o r the molar s c a l e mixture p a r t i t i o n f u n c t i o n , then r e l a t i n g X.. to X.. through equation .(lOl) • The NCZ case mass-law expression i s computed from the perturbed mixture p a r t i t i o n f u n c t i o n shown by equation (127). 150 ( z . / z . . N . + z./z. .N.)! ^ W (z./z..N.)!(z./z..N.)! Hrkz./z. .N. - X . . ) ] ! [-^"(z./z. .N. [ ^ ( Z ^ . ^ - X y ) ] ! [^.j/Vj-V ^ V ^ ^ 1 ' i i 1 ( W o Z j / 2 (N.- X i.) q Z ± J / 2 Z i j / 2 X - JJ i J X - i j x i j This i s the same form as the mixture p a r t i t i o n f u n c t i o n of equation (93."), except f o r the i n c l u s i o n of the term i n braces above. The term i n braces i s the r a t i o of the c o m b i n a t o r i a l degeneracy of the p a i r s assemblage i n the maximum randomness case, i n which OK., i s zero, where the e q u i l i b r i u m u n l i k e p a i r s p o p u l a t i o n i s given by z±^' 13 3 to that of the p a i r s assem-blage i n the a c t u a l case i n which OK., i s "nonzero, r e s u l t i n g i n an e q u i l i b r i u m u n l i k e p a i r s p o p u l a t i o n z±^2*±j • Note that i n c a r r y i n g out the operations i n d i c a t e d by equation (126"), X^j i s not a f u n c t i o n of \j > n o r l s n' (the f i r s t term of Q."(N) ,as expanded i n equation (127) . S u b s t i t u t i n g equation (127) i n t o equation (126) , and computing the expression f o r the p a r t i a l d e r i v a t i v e w i t h respect t o X and r e a r r a n g i n g , y i e l d s the mass law expression. 151 _2co1j \ i X i J , . , R T (128) U./zt.h - X y l , [ y z . ^ - X..] whence [z./z. .N ] x [z /z. .N ] x = — 1 — T J — ^ V - x 2/(g!. + 1) n o c n *Lj [z./z. .N. + z./z. .N.] ^ / V P i j . (129) X.. ( i s the p r o p o r t i o n of the t o t a l p a i r s p o p u l a t i o n which are i - j ' s ) i s given by X i i = [z./z.,N. + z./z..N.] (130) = [(z./z..)x. + (z./z. )x ] x g F x 2/(0' + 1) C131) 1 1 J 1 J J J J J where now 2co../RT , 0j = (1 + 4 5 ± 5 j [« 1 3 " ID* (1 32) 152 I f a l l values of z are made equal, the above r e -l a t i o n s reduce to those r e s u l t i n g i n the constant-z mass law expression f o r a b i n a r y . B. Approximations Involved i n Using Mass-Law-Average Concentrations (45) Rushbrooke showed that the use of mass-law average p a i r s c o n c e n t r a t i o n i n a free energy expression i s not s t r i c t l y c o r r e c t , though i s w i t h i n a few percent f o r moderate values of w . This approximation i s inco r p o r a t e d i n the present work. C. More Formally Correct Form of the Mass Law Expression The more s t r i c t l y c o r r e c t form f o r the mass-law ex-p r e s s i o n than equation (128) i s obtained by l e a v i n g terms of the p e r t u r b a t i o n term of the expression i n [ z i / 2 ( N i "X^..)]! form, which r e s u l t s In X.. X.. -2u../RT Ciir>\ 12 12 E I J (133) z.lz. . z .1z. . (N. - X ) 1 1 J x (N. - X. .) J 1 3 which may be (only approximately) solved f o r X„ z / z z / z x =X../N = [ G O 1 i j ] x [ G O J i j ] x 2/(3" ) (134) z.1z. . where the c o n c e n t r a t i o n f u n c t i o n i n 3V. are a l s o [ G O 1 1 J ] z .1z. . and [ G O J " L" J]. Hence while the co n c e n t r a t i o n f u n c t i o n [ 5 a l 3 z / z i s more t r a c t a b l e that [ G O ], e s p e c i a l l y f o r l a t e r ge-n e r a l i z a t i o n i n t o the multicomponent case, i t somewhat understates the NCZ e f f e c t at higher r a t i o s , i . e . i s seen to provide a con-s e r v a t i v e estimate of NCZ e f f e c t s . Note a l s o i n both forms of 153 the mass law expression the value o f ^ a r i s i n g from s t r a i g h t -forward s u b s t i t u t i o n of equat i o n (127 ) i n t o equation (126 ) i s "V. = [ e. . - * ( ( z . / z . .)e. . + (z./z. . ) E . .)] ( 1 3 5 ) 1 3 1 3 1 1 3 1 1 3 1 3 3 3 while the form shown i n equation (128) Is the same form as i n CZ quasichemical theory, namely ui = re _ %(e + e )] (78) i j i j i i j j The reason f o r u s i n g i s that \^ corresponds c l o s e l y to , the net energy of the u n l i k e p a i r s ' r e a c t i o n * of the o v e r a l l mixing process ( i . e . , where the components s t a r t i n pure component form, z coordinated, and end up i n the mixture, s with u n l i k e - p a i r s pair-bundles being z.. c o o r d i n a t e d ) . While, where bundle s i z e i s ( z / 2 ) ¥ (z, ./2) d 3 6 ) i 1 3 3g o)" i s c o m p o s i t i o n a l l y dependent through a dependence of z on i j a x , to' i s very i n s e n s i t i v e to composition since i n equation, a 1 3 ( 3 2 ) 5 z-• - z • Since l i t t l e i s gained u s i n g w! i n place of i j s 1 3 to , i t was decided to use to. ., the constant-z case value, f o r i j ^ i J the sake of s i m p l i c i t y . * remember that z. and z are interchangeable f o r a mono-x a D f u n c t i o n a l species a c o n t a i n i n g only i - t y p e f u n c t i o n a l groups 154 10., EFFECT OF VOLUME CHANGES. UPON ESTIMATES OF G E AND H E The problem of the e f f e c t of volume of mixing changes E E upon estimates of G and H was introduced In Chapter 4, where i t was i n d i c a t e d that G E i s i n s e n s i t i v e to V^ (the excess E volume, or volume change due to m i x i n g ) , whereas H was sen-E s i t i v e to V . How these conclusions are reached i s I l l u s t r a t e d by c o n s i d e r i n g the d i f f e r e n c e i n the response of H and G to V i n an e q u i l i b r i u m mixture. At low pressure ( i n the a t -mospheric pressure range) f o r a l i q u i d , G(T,II) - A(T,V) (137) The c o n d i t i o n of e q u i l i b r i u m i n a mixture i s a c o n d i t i o n on the fr e e energy, and not on the t o t a l energy, of the system. Thus, to a cl o s e approximation, because of equation (137) the e q u i l i b r i u m c o n d i t i o n i s that <SA(T,V) = 0 (138) which ensures that ^3V ;T, eq u i l i b r i u m (139) Hence I t w i l l be r e a l i z e d that the v a r i a t i o n of the fr e e energy w i t h respect to volume changes due to mixing (namely due to V ) must be of no lower order than 2 mathematically i n the mixture at e q u i l i b r i u m . However, s i n c e the e q u i l i b r i u m con-3HS d i t i o n i s not a r e s t r i c t i o n on H, per se, ^ would not be expected to be zero i n the e q u i l i b r i u m mixture, nor i s i t i n p r a c t i c e . Thus the i n t e g r a l of t h i s q u a n t i t y with respect to V over l i m i t s zero to V , represe n t -E E i n g the e f f e c t of V upon H , could a l s o not be expected t o be n e g l i g i b l e . E E A c o r r e c t i o n f o r the second order V dependency of G - g can of course be'added to the constant volume G r e s u l t , a l -though the d i f f i c u l t i e s i n c a l c u l a t i n g the c o r r e c t i o n are such that i t may w e l l have to be ignored. This c o r r e c t i o n term cannot be c a l c u l a t e d d i r e c t l y by the methods of t h i s t h e s i s , (12). but Scatchard gives a d e t a i l e d a n a l y s i s of the e r r o r s l i k e l y to be i n c u r r e d by i t s omission. Problems a r i s i n g out of i g n o r i n g the e f f e c t s of volume changes only become s e r i o u s when heat of mixing estimates made E E from quasichemical G expressions are being considered. H expressions obtained by a b s t r a c t i n g the H terms out of the G E expression are not themselves complete estimates of H (nor are E they, i n consequence, a c c u r a t e ) , though such H expressions are compatible w i t h the G expression from which they came. E I f G i s expressed as a group of constant-volume terms E E (grouped as G ), and a second group of terms e x p l i c i t i n V p (the excess volume at constant pressure) i s appended, one ob-t a i n s the formalism GET = H E - TS E - Z x.SLnx. V V V . l 1 (140) But G E = { H E + T S J - I x±lnx±} + {H (Vp) - T S E ( V E ) } ( 1 4 D The f o r e g o i n g expressions employ Scatchard's scheme of nota-t i o n . The second group of terms i n braces i n equation ( l 4 l ) E E represents the e f f e c t s of V upon G under the i s o b a r i c con-d i t i o n s which e x i s t i n d i s t i l l a t i o n columns. I f H + 0 ( 1 4 2 ) as a c o n d i t i o n of e q u i l i b r i u m , then H E ( V p ) - T S E ( V E ) . ( 1 4 3 ) E E This would be true even though the H (V ) component of the heat of mixing terms may i n f a c t be l a r g e , and i n some cases E . of opposite s i g n to Hy E The value of H determined from the termperature de-E pendence of G v i a the Gibbs Helmholtz r e l a t i o n , namely H E . _ R T2 M&m. {M) a * i s not 157 (145) S i m i l a r l y , i f the temperature dependence of an a c t i v i t y c o e f f i c i e n t i s to be p r e d i c t e d from the p a r t i a l molal form of equation (.144), namely dSLn-y. (G^/RT) —r I — . - H J / R T 2 ( 1 4 6 ) 9 T 8 T i the p a r t i a l molal heat of mixing at constant volume H E v i s the q u a n t i t y i n v o l v e d . Thus, values of H obtained v i a the Gibbs Helmholtz equation s t a r t i n g from quasichemical E E estimates of G are estimates of H obtained by summing d i r e c t -E E l y the e x p l i c i t H terms of the G expression and are thus estimates of heat of mixing at constant volume. E I f i s o b a r i c H values are d e s i r e d , as i s the case f o r E d i s t i l l a t i o n column p l a t e heat balances, the p r e d i c t i o n of H by the quasichemical method i s incomplete. This problem i s E E common to a l l c e l l - t h e o r y approaches to G and H . I t should be noted that vapor pressure responds ex-E p o n e n t i a l l y to G changes, whereas the heat balance reponds only l i n e a r l y to heat of mixing changes (or to the degree of mis-estimate i n heat of mixing r e s u l t i n g from using Hy ). In p l a c i n g the above ob s e r v a t i o n i n t o the context of d i s t i l l a t i o n column c a l c u l a t i o n s , the c o n c l u s i o n i s that vapor pressures and a c t i v i t y c o e f f i c i e n t s are s e n s i t i v e f u n c t i o n s of G , but not of V , whereas though phase flows are not u s u a l l y E E E s e n s i t i v e to H , H is_ s e n s i t i v e to V . In p r a c t i c e , as was mentioned i n Chapter 1, one i s somewhat saved i n t h i s respect by the u s u a l l y greater importance of p a r t i a l vapor pressure changes (induced by y e f f e c t s ) i n column c a l c u l a t i o n s than of ope r a t i n g l i n e curvature e f f e c t s due to changes i n p a r t i a l m o l a l e n t h a l p i e s with change i n composition of the d i s t i l l i n g m ixture, except at c l o s e to minimum r e f l u x r a t i o s . 11. EXPERIMENTAL EVIDENCE FOR NCZ EFFECTS IN REAL SOLUTIONS A. General The chapter i s concluded w i t h a d i s c u s s i o n of two ex-amples of the p r e d i c t i v e use of equation (122) which have been chosen to provide evidence f o r the existence of NCZ e f f e c t s i n r e a l s o l u t i o n s . For the purpose of examining the equation (122), a search was made f o r experimental data f o r monfunctional 'monomer' b i n a r i e s which would pre-eminently d i s p l a y e f f e c t s of d i s p a r i t y of molecular s i z e . L a t er i n t h i s work a d d i t i o n a l ' terms w i l l be added to equation (122), and the s i g n i f i c a n c e of these w i l l be t e s t e d i n due course. Now i t i s the I n t e n t i o n to examine systems which can be described s a t i s f a c t o r i l y by the three terms, t-^, t ^ " , and t ^ , so f a r obtained. The search f o r monofunctional molecules of very un-equal s i z e i n binary mixtures l e d to systems which had been (24) used by Hildebrand ' as a t e s t f o r h i s S o l u b i l i t y Parameter theory, and were regarded by him as extreme examples of such 159 systems. The f i r s t system to be examined w i l l be perfluoro-n-heptane carbon t e t r a c h l o r i d e (a l i q u i d - l i q u i d equilibrium system). The choice of a l i q u i d - l i q u i d system arises because extreme molecular size d i s p a r i t y w i l l (normally) r e s u l t i n ex-treme l i q u i d non-ideality and, i n the l i m i t , l i q u i d immis-c i b i l i t y . Figure 1 shows the excess free energy of such a mixture (curve a) and the i d e a l free energy of mixing (k). Forming the algebraic sum of a and k gives the free energy of the mixture (curve n). Common tangent PQ drawn to curve n locates the com-p o s i t i o n of the coexisting l i q u i d phases. This i s true because of the requirement that the chemical p o t e n t i a l of a given species be the same i n two phases at equilibrium. Because the chemical p o t e n t i a l of a component i , i s defined* by u i = y° + RT£na ± (147) where = J£ (1^8) , i on.i . l|n^ = const * In equation (146), In notation consistent with standard texts, such as Hougen, Watson and R a g a t z ^ ^ , a^ stands for the a c t i v i t y of component i . Thus i n equation (147), a^ = a\ • Here also, n^ stands for the population of molecules of the i ^ species. 161 Then f o r a two component system, equation (.147) i s represented by the tangent to the G . versus composition curve at any J D mix p o i n t . I f the G - versus composition curve i s i n f l e c t e d ^ mix ^ such that i t admits of two tangents of the same slope (a "common tangent"), the two p o i n t s of tangency mark the composi-(47) t i o n s of the two l i q u i d phases i n e q u i l i b r i u m v 1 ' . Figure 1 i s drawn f o r a given constant value of tem-perature. For systems i n which the v a r i a t i o n of f r e e energy w i t h temperature i s to be examined, the c o n s t r u c t i o n s of Figure 1 must be repeated at the various temperatures r e -q u i r e d . In c o n s t r u c t i n g a consolute envelope from data such as Figure 1, a s e r i e s of curves such as n must be determined at d i f f e r e n t temperatures. From the curves, sets of p o i n t s PQ are obtained, and the phase compositions p l o t t e d against temperature. In using such a method i t should be noted that i n many cases, h i s s m a l l , and drawing tangents a c c u r a t e l y i s t h e r e f o r e d i f f i c u l t . In extreme cases, c a r e f u l p l o t t i n g and c l o s e l y spaced p o i n t s on the G . /RT versus x , curve are r e -mi x q u i r e d f o r accurate l o c a t i o n of the phase compositions. I f h i s small I t i s because the phases change l i t t l e i n w i t h Thus, f o r such a system, phases separ-ate w i t h d i f f i c u l t y . This i s c o n s i s t e n t w i t h the o b s e r v a t i o n of Hildebrand and co-workers that c e r t a i n b i n a r y mixtures c o n t a i n i n g l a r g e f l u o r o c a r b o n compounds took a very long time to come to phase e q u i l i b r i u m from an i n i t i a l l y t u r b i d s t a t e ( p a r t i c u l a r l y i n the r e g i o n j u s t below the upper c r i t i c a l s o l u t i o n temperature H In other words 162 when the "energy preference" of the system f o r two phases I n -stead of one was very s l i g h t , achievement of two phase e q u i l i b r i u m was correspondingly slow. B. Perfluoro-n-Heptane Carbon T e t r a c h l o r i d e * Figure 2(a) plots, the values of t-^, t r , and t ^ " against x at 40° C. Because i t i s f r e q u e n t l y d e s i r e d to separate the e f f e c t s of those terms i n the NCZ equation which a r i s e only i n the case of nonconstant c o o r d i n a t i o n number, from others (such as the simple heat of mixing e f f e c t given by t ^ " ) , the combined curve ( t n + t._) i s u s u a l l y drawn. This i s the case i n i o Figure 2 ( b ) . The values f o r t ^ " , (t-^ + t ^ ) , and k, the i d e a l f r e e energy of mixing curve are added to give curve n of Figure 2(c) upon which the c o n s t r u c t i o n s of Figure 1(b) can be performed. The operations used to produce Figure 2 (at 40° C) have been repeated at 50° C, 60° C, and 70° C, and are shown as the s e r i e s Figures 2(d). The r e s u l t i n g p a i r s of con-s o l u t e p o i n t s PQ are p l o t t e d against temperature i n Figure 2(e) Also shown i n Figure 2(e) i s the e x p e r i m e n t a l l y determined (46) curve * System (5) i n the c l a s s i f i c a t i o n system of Chapter 10. ** The c a l c u l a t i o n program used was f o r a general form of the NCZ equation (equation ( 2 1 2 ) , which reduces to equation (122) to w i t h i n terms of a t r i v i a l s i z e f o r the given system. These a d d i t i o n a l terms are i n c l u d e d i n n but are not d i s p l a y e d i n -d i v i d u a l l y i n the above F i g u r e s . 163 4 0 ° C 4 I I I 1 0-0 0-4 0-8 00, MOLE FRACTION PFH Figure 2(a). Perfluoro-n-heptane Carbon t e t r a c h l o r i d e . NCZ Component Curves at 40°C. 164 tr o E X, MOLE FRACTION.. PFH Figure 2(b) and 2(c). Perfluoro-n-heptane Carbon tetrachloride. I Component and Sum Curves for Free Energy at 40°C. 10 05 CE 00 o -05, -10 00 H "Of CC I -0 2 o -03 1 r 40°C H 1 -. r-50 °C H 1 1 (--l 1 1 r 60° C H 1 1—• H _l I L. 70°C H 1 1 1-j l I i -0 10 005 00 £ CD -005 -010 00 -01 H or •02 J H-0 3 OO 04 08 00 04 08 00 04 08 OO 0 4 08 10 MOLE FRACTION PFH Figure 2(d). Perfluoro-n-heptane. Carbon tetrachloride. As above, at 40, 50, 60 ana /0"C. 60 S» 50 LLI DC ZD !5 CC LU CL 40 IxJ 30-(NCZ SINGLE PHASE BY 65° C) - O d / / O-NCZ EQUATION EXPERIMENTAL SOLUBILITY PARAMETER EXPRESSION \ 0 0 0-2 0-4 0-6 Xt MOLE FRACTION PFH Figure 2(e). Perfluoro-n-heptane Carbon tetrachloride.. Consolute Envelope. Calculated and Experimental. 167 The b a s i c shapes of the curves produced by t ^ " , t ^ and tj - remain s u b s t a n t i a l l y unchanged over the temperature range considered, but t h e i r r e s p e c t i v e amplitudes do change (see Figure 2(d) . The o v e r a l l r e s u l t i s to f o r c e the c r i t i c a l s o l u t i o n composition of the system w e l l away from the e q i m o l e c u l a r , towards mixtures r i c h i n the s m a l l e r component. The p r e d i c t i o n of c r i t i c a l s o l u t i o n compositions i n t h i s r e g i o n , u s i n g a p h y s i c a l argument b a s i c a l l y d i f f e r e n t from that of Hildebrand i s one f e a t u r e making an equation based on NCZ of r e a l i n t e r e s t . For comparison, a l s o shown on Figure 2(e) i s a curve c a l c u l a t e d from the S o l u b i l i t y Parameter expression f o r excess f r e e energy. For t h i s e x p r e s s i o n , the d i f f e r e n c e between a c t u a l l i t e r a t u r e values of the s o l u b i l i t y parameter f o r (15) Perfluoro-n-Heptane (6^ = 6.0) and Carbon T e t r a c h l o r i d e (6 = 8.6) was not great enough to produce two l i q u i d phases. The s o l u b i l i t y parameter curve f o r comparison i n Fi g u r e 2(e) i n v o l v e s an assumed value of 6 = 5 . 6 . A somewhat b e t t e r f i t might have been obtained by a t t e n u a t i n g the d i f f e r e n c e 2 . (6^ - 6 2) w i t h i n c r e a s i n g temperature, as has been suggested •(24) by Hildebrand , but t h i s l i n e of refinement was not pursued. As i n the case of the NCZ equation, the fr e e energy of mixing curve r e s u l t i n g from the S o l u b i l i t y Parameter expression I n f l e c t s only j u s t s u f f i c i e n t l y t o permit the drawing of the common tangent i n d i c a t i v e of the existence., of two l i q u i d phases. This i s p a r t i c u l a r l y t r u e near the upper c r i t i c a l s o l u t i o n temperature. This i s the phenomenon a s s o c i a b l e w i t h v a n i s h i n g l y 168 small values of h i n Figure 1, discussed e a r l i e r . The f i t of the two curves (namely those obtained from NCZ and S o l u -b i l i t y Parameter e x p r e s s i o n s ) i n t h i s system should be regarded as e q u a l l y good, though the assumptions about the l i q u i d be-haviour u n d e r l y i n g the two models are s i g n i f i c a n t l y d i f f e r e n t . C. P e n t a e r y t h r i t o l T e t r a p e r f l u o r o b u t r a t e Chloroform* This system i s very s i m i l i a r i n general character to the Perfluoro-n-Heptane-Carbon T e t r a c h l o r i d e system. The d i f f e r e n c e i s q u a n t i t a t i v e r a t h e r than q u a l i t a t i v e inasmuch as the am-p l i t u d e of the curve generated by (t-^ + t^) i s much greater than i n the previous system. In t h i s l a t t e r case ( t ^ + t,_) dominates i n l o c a t i n g the two-phase re g i o n r a t h e r than o p e r a t i n g i n concert w i t h the t j j " term as was the case i n the previous system. The curves f o r the present system are shown i n Figure 3(a) a l l drawn on the same s c a l e ; whereas i n the previous system curve ( t ^ + t,-) was drawn on a g r e a t l y expanded s c a l e . For the pre-sent system, the sum (curve 3(b)) a has t h e r e f o r e the l e f t w a r d minimum at a much lower c o n c e n t r a t i o n of the l a r g e r molecule than i s the case i n the previous system, Repeating these c a l c u l a t i o n s at a s e r i e s of temperatures, and p l o t t i n g the appr o p r i a t e PQ p a i r s , r e s u l t s i n the p a i r s of c i r c l e s on Figure 3 ( c ) . I t i s seen that the envelope i s narrower, sharper, and c l o s e r i n t o the side represented by the s m a l l e r molecule * System (6) i n the general c l a s s i f i c a t i o n system of Chapter 10. 170 50 10 T T O-NCZ EQUATION EXPERIMENTAL SOLUBILITY PARAMETER EXPRESSION J_ 0 0 0-2 0-4 JF, MOLE FRACTION PFB Figure 3 ( c ) . P e n t a e r y t h r i t o l tetraperfluorobutyrate. Chloroform. Consolute Envelope. Calculated and"Experimenta'l. 171 than i s the case i n the previous system. Figure 3 ( c ) , a l s o i n c l u d e s f o r comparison purposes the S o l u b i l i t y Parameter pre-d i c t i o n f o r the system, as w e l l as the exp e r i m e n t a l l y d e t e r -(24) ' mined envelope. . The r e l a t i v e p e r s i s t e n c e of the envelope to higher temperatures i s due to the p o s i t i v e temperature d i -pendence of (a°/RT) and could not have been achieved without i t . The S o l u b i l i t y Parameter expression gives a f i t (see Figur e 3(c)) which i s as good as that p r e d i c t e d by the NCZ equation, although the envelope shown r e s u l t s from v a r i a t i o n o f the S o l u b i l i t y Parameter (6 ) values by the author to give a b e s t - f i t l i n e . 12. SUMMARY Having demonstrated i n these two examples very s t r o n g i n d i c a t i o n s t h a t NCZ e f f e c t s do_ e x i s t i n r e a l s o l u t i o n s , i t i s now reasonable t o proceed t o introduce t h i s concept i n t o a general e x p r e s s i o n f o r m u l t i f u n c t i o n a l multicomponent s o l u t i o n s . T h i s w i l l be the burden of the next chapter, a f t e r which more exte n s i v e comparisons w i t h experimental data can f r u i t f u l l y be made. 172 • CHAPTER 9 EXTENSION OF THE NCZ EQUATION TO THE  GENERAL CASE OF MULTICOMPONENT MIXTURES 1 . GENERAL 2. PAIRS CONCENTRATION FUNCTIONS FOR MORE THAN TWO TYPES OF FUNCTIONAL GROUPS A. I n s o l u b i l i t y of the Mass-Law Expression f o r a More Than B i f u n c t i o n a l Mixture B. The P a i r s Concentration M a t r i x C. Extension of the P a i r s - C o n c e n t r a t i o n M a t r i x to the NCZ Case D. G e n e r a l i z a t i o n of the Method to Include M u l t i f u n c t i o n a l Species E. C a l c u l a t i o n of the Energetic Component of P a i r s -Concentrations Non-Randomness F. Summary of Remarks on the P a i r s - C o n c e n t r a t i o n M a t r i x 3. MODIFICATION OF THE HEAT OF MIXING TERM FOR A MULTI-FUNCTIONAL SPECIES 4. HEAT OF MIXING TERMS FOR THE CASE OF n-MER SPECIES 5. ADAPTION OF TERM (THE PURE COMPONENT FREE ENERGY TERM) TO THE GENERAL CASE 6 . ORIGIN OF THE MINOR TERM t ' 5 7. ADDITIONAL ENTROPY TERMS 173 A. C o n f i g u r a t i o n a l Terms (Due to Non-Unit n and n._ -a l a D i s t r i b u t i o n Function Approach. 1) D i s t i n c t i o n between P a r t i t i o n Function and D i s t r i b u t i o n Function ( P r o b a b i l i t y ) Approaches 2) A p p l i c a t i o n of the D i s t r i b u t i o n Function Approach 3) A d d i t i o n a l C o n f i g u r a t i o n a l Terms - The Problem of the Correct Scale 4) A d d i t i o n a l Scale Considerations i n the Presence of n-mer Species 5) The Meaning of 5. i n an n-mer Containing System cl 6) N e c e s s i t y f o r Separate Methods of Approach f o r Con-f i g u r a t i o n a l and Hindered R o t a t i o n Entropy E f f e c t s 7) D e r i v a t i o n of A d d i t i o n a l C o n f i g u r a t i o n a l Entropy Terms B. E n e r g e t i c a l l y Hindered R o t a t i o n a l Entropy Term 1) General 2) D e s c r i p t i o n of the Concept of Hindered R o t a t i o n 3) Formulation of the Hindered R o t a t i o n E f f e c t . 4) Hindered R o t a t i o n Terms i n P a r t i a l M o l a l Form 5) D e r i v a t i o n of the Hindered R o t a t i o n Term 6) Conversion of R e s u l t s i n t o Desired 'Excess' Form 8. THE GENERAL NCZ EXPRESSION FOR GE/RT 174 CHAPTER 9 EXTENSION OF THE NCZ EQUATION TO THE  GENERAL CASE OF MULTICOMPONENT MIXTURES 1. GENERAL In t h i s chapter, the a d d i t i o n a l c o m p l e x i t i e s of c a l -c u l a t i n g e q u i l i b r i u m p a i r s concentrations i n m u l t i f u n c t i o n a l and/or multicomponent mixtures of monomers are f i r s t i n t r o d u c e d . Then the c o n f i g u r a t i o n a l e f f e c t s of n-mer components i n the NCZ s o l u t i o n are set f o r t h . Next the a d d i t i o n a l terms f o r excess f r e e energy a r i s i n g i n the case of s t r o n g l y a s s o c i a t e d m u l t i -f u n c t i o n a l systems are d e r i v e d . F i n a l l y , the terms which have been developed s e p a r a t e l y i n Chapters 6, 7 and 8 are gathered i n t o a f i n a l general NCZ equation i n which the various e f f e c t s d e scribed appear i n d i v i d u a l l y as terms t ^ through t ^ . I t w i l l then appear that the goal of d e r i v i n g an expression the terms of which i n d i c a t e d i n one way or another the n o n i d e a l i t y of the system, as was described i n Chapter 1, has now i n f a c t been achieved. 2. PAIRS CONCENTRATION FUNCTIONS FOR MORE THAN TWO TYPES OF FUNCTIONAL GROUPS A. I n s o l u b i l i t y of the Mass-Law Expression f o r a More Than  B i f u n c t i o n a l Mixture When three types of l i k e p a i r species are present i n the 175 mixture i n s t e a d of two, ^-"(JJ) ( L N e 1 u a t i o n (125) Chapter 8 ) , becomes a f u n c t i o n of 3 u n l i k e p a i r s e q u i l i b r i u m p o p u l a t i o n s , say X. . , X.. , and X , and i n place of a s i n g l e equation l i k e XJ J K IK equation (128) of Chapter 8, a set of 3 coupled pseudoquadratic equations (of order c l o s e to 2) a r i s e which are not d i r e c t l y s o l u b l e f o r the three unknowns. Consequently an a l t e r n a t e route must be employed to o b t a i n e q u i l i b r i u m p a i r s concentrations i n the m u l t i f u n c t i o n a l case r a t h e r than attempting to argue through mass-law type f o r m u l a t i o n s . B. The P a i r s Concentration M a t r i x A s u c c e s s f u l a l t e r n a t i v e approach to t h i s problem was (19) o r i g i n a t e d by Barker , and w i t h s l i g h t m o d i f i c a t i o n i s adapted h e r e i n t o the NCZ case. His approach w i l l be reviewed i n a form as b r i e f as p o s s i b l e to show the method of i n c o r p o r a t i n g NCZ e f f e c t s . Development of the method i s most e a s i l y shown by ex-ample. In a b i n a r y mixture of equal s i z e d molecules f o r which a). . = 0, c o n c e n t r a t i o n o f the i - j p a i r s i s given by x.. = x.x. (149)" This r e s u l t can be obtained e i t h e r f o r m a l l y , by o p e r a t i n g on a s u i t a b l e p a r t i t i o n f u n c t i o n as i n equation (127) i n Chapter 8, or i n t u i t i v e l y , c o n s i d e r i n g x . and x. to be the p r o b a b i l i t i e s X J of two kinds of independent events i and j r e s p e c t i v e l y . The c o n c e n t r a t i o n of the j - i p a i r s i s given by t h e i r product, again by equation ( l49_) . The t o t a l c o n c e n t r a t i o n of i - j p a i r s , when o r i e n t a t i o n i s not considered, i s • 2x^x • T h- e subject of or d e r i n g of the u n l i k e p a i r s of any given c o n f i g u r a t i o n of the assemblage d i d not enter e x p l i c i t l y Into d i s c u s s i o n of the de-r i v a t i o n of the excess f r e e energy expression ( i n i t s simplest form i n equation (104) f o r the constant-z case). No d i s -t i n c t i o n was needed between the two p o s s i b l e o r i e n t a t i o n s of u n l i k e p a i r s ( I - l e f t w a r d i - j p a i r s , and j - l e f t w a r d i - j p a i r s , t o demonstrate the method of d i s t i n c t i o n between the two) f o r ener g e t i c purposes. However, equation (104) i s equ i v a l e n t to G E / R T = ( X ± j + X i ; J ) ^ L ' ° ^ i (150) z.. where co^ i s now the r e o r g a n i z a t i o n energy per u n l i k e p a i r bundle. I f the p o s s i b l e p a i r s concentrations are w r i t t e n In a matrix a r r a y , the r o l e of o r i e n t a t i o n becomes both more u s e f u l and more apparent. Hence, X(T) X±  X(T)X3 X i i . X i J (151) The r e p r e s e n t a t i o n of (151) i s used to imply t h a t terms i n corresponding matrix l o c a t i o n s are equal. For example, x^x. - X ± j (152) In r e l a t i o n (151), v a r i o u s s e l f - e v i d e n t . r e g u l a r i t i e s may be f o r m a l i z e d which may then be extended i n t o cases of more i n t e r e s t . These are as f o l l o w s : 177 T h e c o n c e n t r a t i o n p a i r s m a t r i x i s s y m m e t r i c a c r o s s t h e m a i n d i a g o n a l . j B e c a u s e 1 X i ~ 1 (155) j S u m m i n g o v e r o n e i n d e x " r e m o v e s " o n e s e t o f c o n c e n t r a t i o n s C . E x t e n s i o n o f t h e P a i r s - C o n c e n t r a t i o n M a t r i x t o t h e NCZ  C a s e I n a b i n a r y m i x t u r e o f u n e q u a l - s i z e d m o l e c u l e s , r e l a t i o n (151") may b e r e w r i t t e n f o r r e l a t i o n s i n t e r m s o f c o n t a c t f r a c t i o n s Vl) X± KfT)**3 X l i X i j 6 O c i x j i x j j (156) 178 Symmetry r e l a t i o n ( 153) remains unchanged. The row normal-i z a t i o n r e l a t i o n (154 ) becomes (157) because by d e f i n i t i o n z £j - l (158) I f the complications of l o c a l energy-induced c l u s t e r i n g e f f e c t s are i ntroduced i n t o r e l a t i o n (158) one could w r i t e _o>. . /RT -9-.. e 1 J . X.. X., 1 ) 1 11 vi) J X J 1 1 ^-J (159) O J . . / R T C ^ C . e " l j X ^ X.. •j) 1 a 1 O J J J 1 A u x i l i a r y functions^-. . are introduced to account f o r secondary r e o r g a n i z a t i o n e f f e c t s of the primary (Boltzman f a c t o r ) energet a l l y induced c l u s t e r i n g f a c t o r s i n the i n t e r a c t i n g s o l u t i o n . Obviously, a l s o , since co. . = cu. . , the Boltzman f a c t o r s are equal and symmetric across the main d i a g o n a l . 179 Barker introduced the p o s t u l a t e that the . were symmetric product f u n c t i o n s such that •3 J (160) Array (159) becomes jo../RT (161) _ c o. . /RT The r e a l contact f r a c t i o n c o n c e n t r a t i o n v a r i a b l e s , %^ and Q± fu n c t i o n s are now replaced by a set of s i n g l e - i n d e x e d pseudo-c o n c e n t r a t i o n such that x i=^i 9 i (162) Thus the l e f t h a n d array becomes to . ./RT x 0 x. x (i)V~ 1 : 1 V XiJ j o../RT x G> x i e " J 1 xO>xJ V x^ (163) Because assignment of c e n t r a l i t y (designated by c i r c l i n g ) Is a r b i t r a r y x = x ' C164) (l) x i 180 The set of row n o r m a l i z a t i o n r e s t r i c t i o n s are e x e m p l i f i e d by t o ^ ./RT I X. e~ ^ J = 5, ^ L 6 5 ) J This may be transposed i n t o an equation set of the form _ u , ^ /RT X ( i ) = 6 ^ ( l 6 6 ) The equation set (166) i s s o l u b l e by i t e r a t i v e s u b s t i t u t i o n s of a set of x i n t o the r i g h t hand s i d e , r e c a l c u l a t i n g a set of X ^ t h e n r e s e t t i n g x / ^ = ^ • Convergence occurs w i t h i n a few i t e r a t i o n s . For an I n i t i a l approximation, the value of X^  i s set = . The c i r c l i n g convention denotes a r b i t r a r y ' c e n t r a l -i t y ' of the row indexed s p e c i e s . D. G e n e r a l i z a t i o n of the Method to Include M u l t i f u n c t i o n a l  Species R e l a t i o n s (163) are i n no way r e s t r i c t e d to a b i n a r y mixture i n order t o be s o l u b l e , as opposed to the method of e s t a b l i s h i n g the x p a i r s c o ncentrations by means of the mass-law . For m u l t i f u n c t i o n a l l i q u i d s , i f the contact f r a c t i o n of the i t h type of contact on the a ^ h molecular species i s de-signated by 181 and dropping the "a" index (each type of contact species i s considered to be a separate species i n the matrix) equation set (166) becomes X / 7 N = $^>./ E X. e -WRT where the summation on j Is now over a l l f u n c t i o n a l group species on a l l molecules. That the row-sum n o r m a l i z a t i o n con-d i t i o n i n c o r p o r a t e d i n equation set -C168-) i s v a l i d i s shown by the t r u e r e l a t i o n , E $ i a ' • 5 a . E n i a " Ka ( 1 6 9 ) l a 1 a on on E. C a l c u l a t i o n of the Ener g e t i c Component of P a i r s -Concentrations Noh-Randomness I f another symmetric matrix whose r e p r e s e n t a t i v e element Is given by U J ® " U l ® ( 1 7 0 ) i s d efined by u . ± = X . ^ / ( $ . x - X i ^ / $ r a ) $ i ( 1 7 1 ) 182 then the elements represent the p r o p o r t i o n a t e energy skew e f f e c t , or the energetic component of non-randomness p a i r s a s s o c i a t i o n of given j - j p a i r s i n t e r a c t i o n s . The are used i n o b t a i n i n g the 'hindrance of r o t a t i o n ' entropy term l a t e r i n the chapter. F. Summary of Remarks on the P a i r s - C o n c e n t r a t i o n M a t r i x In summary i t i s noted that NCZ e f f e c t s are i n c o r p o r a t e d i n the Barker r e l a t i o n s ' f o r the e q u i l i b r i u m p a i r s c o n c e n t r a t i o n i n the mixture, namely the X^ , simply by f o r c i n g row sums to equal € ,or i n the case of m u l t i f u n c t i o n a l systems $. Barker's p o s t u l a t i o n of the symmetric decomposition of the f a c t o r s i n t o 6.^  x 8^  i s unproven, and probably o v e r s t a t e s the symmetry of the s o l u t i o n . Nevertheless, the method i s again j u s t i f i e d on the b a s i s of i t s p r a c t i c a l success. 3. MODIFICATION OF THE HEAT OF MIXING TERM FOR A MULTI- FUNCTIONAL SPECIES While the heat of mixing term r e s u l t i n g from u n l i k e p a i r formation between molecular species f o r which n. = 1 r xa (monofunctional species) remains as shown i n (1Q4 ), namely X i j Z i ^ u i j / R T y a m o d i f i c a t i o n must be made i n the form of the heat of mixing terms i n the case of m u l t i f u n c t i o n a l mole-c u l a r species ( f o r which < 1 ). The c o r r e c t i o n i s again due to the exis t e n c e of u n l i k e f u n c t i o n a l group p a i r i n g i n m u l t i f u n c t i o n a l l i q u i d s i n the pure component s t a t e . The reference s t a t e f o r the p a i r s r e o r g a n i z a t i o n r e a c t i o n i s an assemblage i n which only l i k e - p a i r s are present. Hence, when 183 m u l t i f u n c t i o n a l components are present i n the mixture, the net change In the amount of a given type of u n l i k e p a i r i n g r e s u l t i n g from the mixing process i s not simply X j _ a j a but r a t h e r (X. . - x x x° . ) (172) iaia a z iaia J ave J X? . Is the e q u i l i b r i u m c o n c e n t r a t i o n of the i - j p a i r per xaja <J sr r mole of species "a" In the pure component s t a t e . The f a c t o r z E S Si ( ) c o r r e c t s f o r the f a c t that X . , i s a p r o p o r t i o n of a z i a i a ave J p a i r s p o p u l a t i o n whose o v e r a l l average e x t e r n a l c o o r d i n a t i o n number i s z a v g , whereas the c o o r d i n a t i o n number f o r the pure component assemblage i s z g * E^ . Th e f a c t o r x a accounts f o r the simple d i l u t i o n by other components which would be present i n an i d e a l mixture. In an i d e a l ( Raoult's Law) mixture the term given by (172") caused by d i l u t i o n by (1 - x ) moles of the 3, other i d e a l l i q u i d s would be zero f o r a l l values of #a . While X . - . „ by i t s e l f i s always p o s i t i v e , the net p a i r s c o n c e n t r a t i o n xaj a change term may be e i t h e r p o s i t i v e or n e g a t i v e , depending on the r e l a t i v e strengths of u ^ f a c t o r s f o r p o s s i b l e p a i r s a s s o c i -a t i o n s i n the mixed and pure component s t a t e s . For c e r t a i n h i g h l y a s s o c i a t e d mixtures, the value of (X. . - x -S-± x ° . ) laja a z iaia ave J f o r the most s t r o n g l y a s s o c i a t i n g p a i r species may be p o s i -t i v e f o r part of the c o n c e n t r a t i o n range and negative f o r o t h e r s . 184 This behavior c o n t r i b u t e s to the unusual heat of mixing pheno-mena of t e n observable In such mixtures. The f i n a l form f o r the heat of mixing c o n t r i b u t i o n to the excess f r e e energy of mixing f o r such a general mixture may be w r i t t e n as the sum of two sets of terms, one d e s c r i b i n g the simple heat of mixing, the other the net heat of mixing of un-l i k e p a i r s from m u l t i f u n c t i o n a l components j u s t d i scussed. This sum I s thus represented as z E z to.. r. r ,-0 s a, ave x i E £ [n X. . - n x X. . ] x (t') ave xaia a a xaia z RT 4 xa j a ave (173) + I E n X. z co. ./RT (t' 1) ., ave xaib ave xi H xa jb J 4. HEAT OF MIXING TERMS FOR THE CASE OF n-MER SPECIES In the f i n a l form of the NCZ equation f o r GE/RT the e f f e c t of concatenation I s a l s o taken i n t o c o n s i d e r a t i o n by the i n c l u s i o n of f a c t o r s n and n . These f a c t o r s account f o r a. V 0 cl the f a c t that the f i n a l NCZ expression i s w r i t t e n f o r one mole of molecules, which In the case of a mixture whose average chain l e n g t h i s n a v e » i s n a v e moles of monomer segments. The t o t a l p o p u l a t i o n of p a i r s ' i n t e r a c t i o n s , hence the t o t a l heat of mixing e f f e c t , must thus be modified by the appropriate c h a i n - l e n g t h c o r r e c t i o n f a c t o r s . These w i l l be shown i n c l u d e d i n the f i n a l f o r m u l a t i o n of the heat of mixing terms. 185 5. ADAPTION OP TERM t c (THE PURE COMPONENT FREE ENERGY TERM) 5 TO THE GENERAL CASE For the simplest b i n a r y case (equation (122)Chapter 8) o a. C 5 = X i i X ( 1 - Z J \ ? X ^ I j ' V lk (174) o a. + X x (1 - z./ z..) x (z../z.) ^ In the case of a m u l t i f u n c t i o n a l molecule, the t o t a l amount of s e l f - a s s o c i a t i o n of the molecular species i s found by adding a l l the p a i r s c o n c e n t r a t i o n terms f o r a l l the types of p a i r s p o s s i b l e between various f u n c t i o n a l groups of that type of mole-c u l e . This sum i s represented by the term Z E X. (175) i a j a i a J " a on on For example, i f the molecule were b i f u n c t i o n a l , of group types 1 and 2, the sum would then be X l a l a + X l a 2 a + X 2 a l a + X2a2a ( 1 ? 6 ) showing that f u n c t i o n a l l y u n l i k e p a i r types of terms of the Barker X.. matrix are counted twice (once f o r each ordering) i n the sum. The term (1 - z./z..) of the simple binary case (equation -L 1 J (174)) i s r e w r i t t e n f o r the general case as 1 - ( z /z E ) . The a s a term accounting f o r the t r a n s f o r m a t i o n from x to x m form i s no longer w r i t t e n as 186 ( z . . / z . ) , but as (z /z ), where molecular s u b s c r i p t i n g must i j i ave a ' • • replace the s i t e s u b s c r i p t i n g appropriate to the monofunctional binary mixture case. The approximation z - z . . i s a l s o made, J ave i j s where z i s the o v e r a l l mixture average e x t e r n a l monomer co-ave D o r d i n a t i o n number. F i n a l l y , because a° i s c a l c u l a t e d as a molecular s c a l e q u a n t i t y , the term t ^ i s m u l t i p l i e d by n , the mixture average j ave chain l e n g t h , to convert to the monomer p o p u l a t i o n of a mole of molecules, then d i v i d e d by n to reduce t h i s monomer p o p u l a t i o n to the number of n l e n g t h molecules. The g e n e r a l i z e d form of t,- i s thus 5 o t . - E x ( Z Z X. . ) x (1 - -^-) ( Z a v e n a v e ^ (177) 5 . . i a j a z E z n RT a l o n a j o n a s a a a 6. ORIGIN OF THE MINOR TERM t» In m u l t i f u n c t i o n a l and/or multicomponent mixtures, another term of the same nature as a r i s e s , f o r m a l l y due t o s l i g h t departures of the c o o r d i n a t i o n numbers z ^ (the co-o r d i n a t i o n number analogous to z.., but f o r multicomponent XJ m u l t i f u n c t i o n a l mixtures) from the s e l f - c o o r d i n a t i o n numbers, namely z ^ and z E . This term, c a l l e d t'. r was i n c l u d e d i n s a s u z) the general expression f o r the sake of completeness, mainly to see whether I t ever assumed s i g n i f i c a n t p r o p o r t i o n s compared w i t h other terms. I t d i d not; so, beyond showing i t s form, no 187 f u r t h e r mention w i l l be made of I t , z 5 II s a ave ab a o a E ( E E ab p a i r s i on a j on b on a^b X i a j b x [ ( 1 - ) ( z s a ( 1 7 8 ) + (1 - x (-o z , n a, s b ave. D z . n ' RT ab a 7 . ADDITIONAL ENTROPY TERMS A. Conf i g u r a t i o n a l Terms (Due to Non-Unit n and r\.n -— • a—— = xa."=— D i s t r i b u t i o n Function Approach 1) D i s t i n c t i o n between P a r t i t i o n Function and D i s t r i b u t i o n Function ( P r o b a b i l i t y ) Approaches The c o n f i g u r a t i o n a l c o n t r i b u t i o n t o the value of 9 . /RT (the reduced form of the f r e e energy of mixing) f o r round mono= mer systems, namely E £ £n £ Is obta i n a b l e by two methods, a a a The one already presented i n equation ( 7 3 ) , Chapter J, was t o formulate fi' which i s the c o m b i n a t o r i a l degeneracy term i n the c a n o n i c a l p a r t i t i o n f u n c t i o n f o r the mixture, then o b t a i n the fr e e energy terms through the operations on ft' i n d i c a t e d by - k T An(ft') . This method i s c a l l e d "the p a r t i t i o n f u n c t i o n " approach. The second method, c a l l e d "the d i s t r i b u t i o n f u n c t i o n " approach, which has c e r t a i n formal advantages f o r the purposes of t h i s s e c t i o n , i s now Introduced, The term £ £ &n § can be obtained d i r e c t l y by expanding the d e f i n i t i o n of the re p r e s e n t i n g the c l a s s e s of 'Independent* events present i n the system. a Boltzman entropy .(78) of the normalized p r o b a b i l i t y d i s t r i b u t i o n 188 2 ) . A p p l i c a t i o n of th.e D i s t r i b u t i o n Function Approach I f <% Is the p r o b a b i l i t y of a s i t e type on the a a molecular s p e c i e s , then f o r a mixture, the normalized p r o b a b i l i t y d i s t r i b u t i o n of events <% i s simply a Z K = 1 (179) a a and the negative of the Boltzman entropy i s given by - r = S 5 In ? (180) k a a a where k i s a p r o p o r t i o n a l i t y constant, i n t h i s case equal to R f o r a mole of mixture. In the l i m i t i n g case of constant-z, equation (180) reduces t o £ x £n x the i d e a l mixing term. 3 3 a The normalized p r o b a b i l i t y d i s t r i b u t i o n f o r a'general' l i q u i d can a l s o be formulated i n a form e x p l i c i t i n C as w e l l a as i n the a d d i t i o n a l c o n f i g u r a t i o n a l v a r i a b l e s n and n to c l 1 3 d e s c r i b e n-mer and m u l t i p l e f u n c t i o n a l group type c h a r a c t e r -i s t i c s of the system r e s p e c t i v e l y . The c o n f i g u r a t i o n a l mixing term obtained by expansion of the Boltzman entropy of i t s p r o b a b i l i t y d i s t r i b u t i o n can be w r i t t e n i n a form i n which the IE, In £ term already obtained, which i n c o r p o r a t e s the co-3 3 a o r d i n a t i o n number d i s t o r t i o n e f f e c t , i s a separate term from those i n which a d d i t i o n a l c o n f i g u r a t i o n a l v a r i a b l e s n and n^ 0 • a l a appear. Since such a s e p a r a t i o n i s d e s i r a b l e f o r s o l u t i o n -c a t e g o r i z a t i o n purposes, the p r o b a b i l i t y d i s t r i b u t i o n route i s 189 the one now employed. 3) A d d i t i o n a l C o n f i g u r a t i o n a l Terms - The Problem of the Correct Scale In the ' p a r t i t i o n • f u n c t i o n ' approach to the f o r m u l a t i o n of excess f r e e energy terms of Chapter 8, the m o l e c u l a r - s c a l e r e s t r a i n t on c o n f I g u r a t i o n a l behavior was b u i l t i n t o the p a r t i t i o n f u n c t i o n by means of bundling of p a i r s i n t o molecular s c a l e en-t i t i e s . By c o n t r a s t , In the present ' p r o b a b i l i t y - d i s t r i b u t i o n ' approach, i n s t e a d of ' b u i l d i n g - i n ' such c o n f i g u r a t i o n a l mole-c u l a r s c a l e c o n s t r a i n t s Into the p a i r s model a p r i o r i , the en-tropy of the assemblage i s f i r s t formulated on the " s i t e s " l e v e l , though the formalism i s w r i t t e n to i n c l u d e the molecular s c a l e p r o b a b i l i t i e s by expressing the s i t e s p r o b a b i l i t i e s as dependent p r o b a b i l i t i e s of the molecular s c a l e ones. Then the molecular s c a l e i s established,ex-post,by f o r c i n g the f i n a l r e s u l t i n t o a form i n which the f a m i l i a r molecular s c a l e term E 5 5 3. 2. a appears as a separate term. Since the l a t t e r i s the expression to which the general expression must reduce i n the s p e c i a l case where n and n. become u n i t y , and since £ £ ^ n 5 a has already a l a a been shown to be c o n s i s t e n t w i t h a 'molecular s c a l e ' l i q u i d , other terms present i n the general expression w i l l a l s o f o r c i b l y be c o n s i s t e n t w i t h the same c o r r e c t s c a l e , i n an expression which so reduces. k) A d d i t i o n a l Scale Considerations i n the Presence of n-mer Species• In the matter of s c a l e I t must f u r t h e r be e s t a b l i s h e d whether, i n the case of a species which behaves as an n-mer , 190 the proper c o n f i g u r a t i o n a l s c a l e i s that of the whole molecule, or whether I t i s that of a monomer segment. The l a t t e r choice i s c o n s i s t e n t w i t h the Flory-Hugglns theory of n-mer s o l u t i o n s . The i m p l i c a t i o n Involved I s that the entropy based on such an n-mer chain i s an upper l i m i t f o r r e a l (that i s l e s s than com-p l e t e l y f l e x i b l e ) polymers. However, i n the Flory-Huggins treatment, a r i g i d n-mer has the same entropy as a dimer, s i n c e t h i r d and higher numbered segments must l i e i n the l i n e d e f i n e d by the f i r s t two segments placed In the l a t t i c e . A l s o , no d i s t i n c t i o n e x i s t s between the cases of r i g i d or f l e x i b l e dimers. The i n d l s t i n g u i s h a b l e n e s s of r i g i d and f l e x i b l e dimer behavior i n such a treatment i n d i c a t e s the i n s e n s i t i v i t y of the entropy of mixing term to assumptions of degree of r i g i d i t y f o r s u f f i -c i e n t l y short molecules, which are the c l a s s of substances most commonly encountered i n systems of p a r t i c u l a r i n t e r e s t i n t h i s work, In the approach below, monomer s c a l e i s to be taken as the s c a l e corresponding to a c t u a l c o n f i g u r a t i o n a l behavior of the l i q u i d , and l i t t l e f u r t h e r formal a t t e n t i o n w i l l be de-voted to the question of the r i g i d i t y of the s h o r t - c h a i n s p e c i e s . 5) The Meaning of £ i n an n-mer Containing System a In the general d e f i n i t i o n of I f o r n-mer s o l u t i o n s as a used i n t h i s work K = ii E z x / I n E z x (181) a a a a a a a a a a 5 i s thus the p r o b a b i l i t y of encountering the e x t e r i o r p o r t i o n 191 of a monomer segment of the a ^ ' molecular s p e c i e s * . F i s a thus a monomer-scale q u a n t i t y In the n-mer s o l u t i o n case, j u s t as i t i s In the monomer case, o b t a i n a b l e from (.181 ) by reducing a l l n t o 1 , so that the E a l s o become 1 by d e f i n i t i o n of E a. ' . a a (See equation ( 6 5 ) ) . 6) N e c e s s i t y f o r Separate Methods of Approach f o r Con-f i g u r a t i o n a l and Hindered R o t a t i o n Entropy E f f e c t s In a l i q u i d where nonzero n>_^  e x i s t , e q u i l i b r i u m p a i r s c o n c e n t r a t i o n s are no longer e x a c t l y given by b i n a r y products of a p p r o p r i a t e contact f r a c t i o n s , but d e v i a t e more or l e s s from such b i n a r y products due to e n e r g e t i c a l l y induced l o c a l c l u s t e r -i n g . M a t hematically, w h i l e the p a i r s p r o b a b i l i t i e s are indepen- dent p r o b a b i l i t i e s , because they have been f o r c e d to be so by the p a i r s c o n c e n t r a t i o n maxtrix method as d e s c r i b e d , they d i f f e r from those of an i d e a l s o l u t i o n . S p e c i f i c a l l y , they are d i f f e r e n t l y d i s t r i b u t e d , and thus the e n t r o p i e s of e n e r g e t i c a l l y unhindered and e n e r g e t i c a l l y hindered d i s t r i b u t i o n s w i l l not be the same. In c o n t r a s t to the treatment of Barker, the net ex-cess f r e e energy component due to e n e r g e t i c a l l y induced c l u s t e r -i n g i s , i n t h i s work, separated from the p u r e l y c o n f i g u r a t i o n a l t e f f e c t s d e s c r i b e d above. To a f i r s t order, these two e f f e c t s should be separable: the c p n f i g u r a t i o n a l v a r i a b l e s n and R d e s c r i b e molecular p r o p e r t i e s , hence p r o b a b i l i t i e s , and w d e s c r i b e i n t r a -*At t h i s p o i n t the reader may wish to r e f e r back t o the d i s c u s s i o n of contact f r a c t i o n s i n Chapter 6. 192 molecular p r o p e r t i e s and p r o b a b i l i t i e s . The method of e s t i -mating the two types of effects, thus need not be the same, which co n s i d e r a b l y s i m p l i f i e s the method of f o r m u l a t i o n . In the next s e c t i o n I t i s proposed to deal w i t h the a d d i t i o n a l con-f i g u r a t i o n a l e f f e c t s due to non-unit n and n. , and to leave • . a i a ' f o r m u l a t i o n of the e f f e c t s of the to., to a l a t e r s e c t i o n . 7) D e r i v a t i o n of A d d i t i o n a l C o n f i g u r a t i o n a l Entropy Terms The most d e t a i l e d p o s s i b l e d e s c r i p t i o n of a p a r t i c u l a r s i t e would be to s p e c i f y i t as a p a r t i c u l a r type of f u n c t i o n a l group on a p a r t i c u l a r segment of a molecule of a p a r t i c u l a r molecular spec i e s . Only a s l i g h t l o s s of s p e c i f i c i t y ( i n -s i g n i f i c a n t f o r purposes of c a l c u l a t i o n of net mixtures pro-p e r t i e s ) would r e s u l t from counting the t o t a l number of con-t a c t s of each f u n c t i o n a l group type, f o r example those on the a^*1 molecular s p e c i e s , l e t t i n g n . be the f r a c t i o n of the t o t a l ^ ' • 6 i a due to the i ^ * " 1 type of f u n c t i o n a l group, and then t o apply n. ia e q u a l l y to each segment. The v arious segments of the n-mer could then be regarded as e n e r g e t i c a l l y i d e n t i c a l , d i s t i n g u i s h -able only by r e l a t i v e p o s i t i o n on the chain. th L e t t i n g n ^ a be the normalized p r o b a b i l i t y of the i f u n c t i o n a l group type on the given A segment of the a^*1 mole-c u l a r s p e c i e s , the p r o b a b i l i t y of i t s occurence i n the s o l u t i o n would be given by ^ i A a n i a n ^a a (182) 193 so that the d i s t r i b u t i o n of a l l such p r o b a b i l i t i e s , namely-E p. A a , 1 (183) a l l i,A,a Equation (182) i n v o l v e s the use of dependent p r o b a b i l i t i e s j n. i s mathematically dependent on ( i ) and ( i ) i s mathematics i a n n a a a l l y dependent on 5 The negative of the Boltzman entropy of c l - - - - - -the p r o b a b i l i t y d i s t r i b u t i o n would then be given by != J PiAa £ n PiAa ( 1 8 4 ) Purely mechanical expansion of t h i s d e f i n i t i o n y i e l d s the ex-p r e s s i o n I- = E E, In I + E % ( E W An W ) k a a -a . Aa Aa a " a A on a (185) + ? ? a x [ E ( E n . A a ^ n . A a ) ] a A on a l on A on a Here i s the p r o b a b i l i t y of the A segment on the a mole= c u l a r s p e c i e s . A l l o w i n g n•~ to be the s i t e f r a c t i o n of s i t e type i on an average segment of molecule 'a', and a l s o now dropping the d i s t i n c t i o n between segments on a given molecular-species so that r = w = i -Aa Ba n a (186) 194 equation 0-85) i s reduced t o - T- (configurational) = £ £ &n £ (187) + Z £ x [-in n + ( Z a a a I on a The a d d i t i o n a l c o n f i g u r a t i o n terms are grouped by the second summation of the expression. B. E n e r g e t i c a l l y Hindered R o t a t i o n a l Entropy Term 1) General In the s e c t i o n on c a l c u l a t i o n of p a i r s c oncentrations i n the general case ( see s e c t i o n 2D. ) the elements of the u ^ matrix were defined as the r a t i o of a c t u a l e q u i l i b r i u m p a i r s c oncentrations (the X..) t o those (the x $ products) i n a xj xa j b s o l u t i o n i n which a l l w. . were zero. When at l e a s t one of the energies of net p a i r s rearrangement (one of the w.. terms) i s nonzero i n the mixture, e n e r g e t i c a l l y induced c l u s t e r i n g e f f e c t s are r e f l e c t e d i n non-unit values of the U i • terms. E x p l i c i t y , f o r such a s o l u t i o n : u ¥ l (188) When such l o c a l c l u s t e r i n g e f f e c t s become s i g n i f i c a n t , they give r i s e to an excess f r e e energy of mixing e f f e c t c h a r a c t e r i z e d by Barker as the 'hindrance of r o t a t i o n entropy e f f e c t * . 195 2) D e s c r i p t i o n of th.e Concept of Hindered R o t a t i o n C o n s i d e r a t i o n of the r o t a t i o n a l hehavior of an a r b i t r a -r i l y chosen ' c e n t r a l ' molecule w i t h i n i t s nearest neighbor 'cage' serves to I l l u s t r a t e the aptness of the hindrance of r o t a t i o n d e s c r i p t i o n a t t r i b u t e d to the e f f e c t i n ques t i o n . I f s i t e s on the * c e n t r a l ' molecule can give r i s e to values of <*>.. of various magnitudes w i t h the contacts of va r i o u s members of i t s nearest neighbor s h e l l , the c e n t r a l molecule w i l l tend to spend a d i s -p r o p o r t i o n a t e l y l a r g e amount of i t s time i n o r i e n t a t i o n s f o r which co i s s t r o n g l y negative ("binding") and correspondingly l e s s In o r i e n t a t i o n s where co i s zero or p o s i t i v e ("unbinding"). This p a r t i a l allignment tendency w i l l be r e f l e c t e d as a p a r t i a l decrease i n the c e n t r a l molecule's f u l l 3-dimensional r o t a t i o n a l freedom w i t h i n the neighbor case above and beyond the r e s t r i c t i o n s Imposed by the geometry of the molecule. The c e n t r a l molecule's r o t a t i o n a l degrees of freedom w i l l thus be f r a c t i o n a l l y decreased hence the c e n t r a l molecule's c o n t r i b u t i o n to the r o t a t i o n a l en-tropy of the system w i l l he correspondingly decreased. The over-a l l r e s u l t of the hindrance of the r o t a t i o n of a species i s an increase i n the p a r t i a l molar f r e e energy of that s p e c i e s . 3) Formulation of the Hindered R o t a t i o n E f f e c t . Since any a r b i t r a r i l y chosen c e n t r a l monamer u n i t i s bound by z^/2 p a i r s cohesions simultaneously, i t i s necessary to formulate f i r s t the e n e r g e t i c a l l y induced nonrandomness component of the Boltzman entropy of the a c t u a l p a i r s d i s t r i b u -t i o n i n terms of the u.. component matrix and then to transform t h i s r e s u l t to apply to a d i s t r i b u t i o n of z/2 grouped s i m u l t a n -eous events. 196 4) Hindered R o t a t i o n Terms, i n P a r t i a l M o l a l Form When one r e a l i z e s , however, that each, molecular species i s a s s o c i a t e d with i t s own s p e c i f i c c o o r d i n a t i o n number, the a c t u a l t r a n s f o r m a t i o n to an expression i n v o l v i n g the appro-p r i a t e c o o r d i n a t i o n number e f f e c t s becomes somewhat more, i n t r i -cate than the general procedure j u s t o u t l i n e d would i n d i c a t e . In f a c t , the nonrandomness. component of the entropy of the p a i r s d i s t r i b u t i o n must be r e w r i t t e n i n p a r t i a l molal form so that each p a r t i a l molal entropy component may be m u l t i p l i e d s e p a r a t e l y by i t s own appropriate c o o r d i n a t i o n number e f f e c t , and the i n d i v i d u a l r e s u l t s combined i n t o a mole f r a c t i o n weighted average which c o n s t i t u t e s the f i n a l r e s u l t . 5) D e r i v a t i o n of the Hindered R o t a t i o n Term I f a (new) general p r o b a b i l i t y p of an i n d i v i d u a l event i , Is now considered and normalized so that then the entropy of such a d i s t r i b u t i o n of 'one-grouped' events i s The procedure may be made e x p l i c i t as f o l l o w s : E P. = 1 x (189) i (190) I f J ^ J I s the simultaneous p r o b a b i l i t y of a p a i r of such independent e v e n t s , then p 3 197 (191) and the entropy of the independent p a i r s p r o b a b i l i t y dis-t r i b u t i o n can be shown by simple expansion to be . S l 2 l = Z E p . . t a p . . = 2 x ( - ^ ) (192) In g e n e r a l , f o r grouping independent events: S(m-grouped) = m/n x S(n-grouped) (193) In an e n e r g e t i c a l l y random s o l u t i o n of monomers X. . - K. x (194) The p a i r p r o b a b i l i t y x. . i s 2-grouped. The p r o b a b i l i t y of a given set of z nearest neighbors i s that of a z-group ., so that by equation (19 3 ) , (monomer sc a l e _ z ^ x (_ S ( p a i r s ) ^ (195) k const z ) k From the d e f i n i t i o n of p a r t i a l molal entropy, S Is defined o p e r a t i o n a l l y by S = £ x S a a a (196) 198 Then a This would a l s o apply to p e r f e c t l y f l e x i b l e polymers. For p e r f e c t l y r i g i d polymers (or very short s t i f f groups of n-seg-ments) n'z S (pairs) _ mixture, n-mers, a a , _a . a g g . S (NCZ e f f e c t s ' E 2 Xa C k } ( 1 9 8 ; a In g e n e r a l , one c o u l d . t h i n k of n' as the equivalent ' s t i f f -c l polymer' chain l e n g t h , not n e c e s s a r i l y simply r e l a t e d to s t r u c t u r a l chain length.' This s t i f f polymer chain l e n g t h i s u s u a l l y In the order of 1 to 5. n* < n (199) a — a For an i n f i n i t e l y f l e x i b l e polymer n' i s always equal to 1. The p a i r s entropy of the unhindered s o l u t i o n can now be de-si g n a t e d * as S ( i j ) such that R - S = E X R. Jin X R = E E $,$". Jln($.$.) (200) k . . I J i j - j j i j i J i»J J i j J * S u p e r c r l p t R, stands f o r "random", s u p e r s c r i p t NR stands f o r "non-random", and s u p e r s c r i p t NR-R stands f o r the net non-randomness component (non-random minus random), as i n the ex-Continued 199 The a c t u a l (.energetically c l u s t e r e d ) p a i r s entropy i s given by S C i j ) NR - S = Z in X N R = Z Z $.$. u. . An($.$. u. .) (201) i , j 1 J 1 J i j 1 2 X 1 1 2 1 J where u i s d e f i n e d by equation ( 1 7 1 ) . I f we can expand the d i f f e r e n c e s y m b o l i c a l l y as NR-R NR-R. . .. ON/R-N C N / - - N § - (- AS R ( l-^) - (- *-f&> - (- ^ - ^ ) (202) Continued p r e s s i o n (- A S N R _ R ) . The b u l k i n e s s of the e x p l i c i t form f o r the term i n the bracket make i t d i f f i c u l t to w r i t e i n f u l l and i s not p a r t i c u l a r l y i n f o r m a t i v e i n d e t a i l . Thus, a f t e r having NR.—R shown the e x p l i c i t form of the term, the a b b r e v i a t i o n (AS ) i s r e t a i n e d i n the general expression i n the above symbolic form f o r s u c c i n c t n e s s . The e x p l i c i t expression by means of which the term i s computed i s given on p.200, and a l s o as a footnote under the general excess f r e e energy expression i n i t s f i n a l form on p.205. 2 0 0 into, terms of the form* x 9 , then each of the "9 " are a a a i d e n t i f i e d . T h i s i s done through the simple mechanism of i s o l a t i n g the c o e f f i c i e n t of x and I d e n t i f y i n g i t w i t h " e a " f o r each component. The 9 are then the r e q u i r e d p a r t i a l cl molal mixture non-randomness terms r e q u i r e d f o r use i n equation ( 1 9 6 ) - . By expanding the r i g h t hand s i d e of equation ( 2 0 1 ) and that of (2Q0").» and s u b t r a c t i n g , equation ( 2 0 3 ) i s obtaind. f E r\rr\r-\x T£ $. x (u. ^  - 1) /T\ r~\ i).a) " xa 3 1 ) (203) AS NR-R "R x £n $. + U . ^7N£II U.,-.-N]> Since x Z n ~ ^3) ave ave (204) Z yi , \ a a — The term i n braces of equation (203 ), times 2 ~ , i s the 9. ave ave sought. D e f i n i n g , f o r the present the term w i t h i n braces i n r N R - R i equation (203 ) as being represented by \ _ r » then z n a a ~ m u l t i p l i e d by ^ n I s equal to e a ave ave / N R - R I® / ' * 9 i s used here simply as a g e n e r a l i z e d unknown. 201 NR-R . n'z . z n CM-D^ -DI AS .mixture, n-mers, _ a a x„ a a x | N K - > K I ( 205) R NCZ e f f e c t s ~ & 2 a z a v e n a v e \Q J NR-R . n'z i-NR-R-v AS .mixture, n-mers. = a a I 1 ( 2 Q 6 ) R V NCZ e f f e c t s 1 2 ® I^ N / a (aj 6) Conversion of R e s u l t s i n t o Desired 'Excess' Form Equation (206) now Incorporates the new terms present i n the f r e e energy of the mixture due t o the e f f e c t s of non-unit n , n and u. . v a r i a b l e s , a' i a i j Because hindrance of r o t a t i o n can obv i o u s l y occur i n pure m u l t i f u n c t i o n a l l i q u i d s (such as ethanol or water, f o r ex-ample), the pure component e f f e c t must then be subtracted from that f o r the mixture i n order to o b t a i n the net change due t o mixing. For x moles of pure component 'a', the new terms i n a and n of the second l i n e of equation (187) become of the i a a form X {-In n + (' E n . i n n, )} (207) a a . i a i a i a so that the new mixture excess term due to n and n. e f f e c t s a i.a i s 202 £ (5 - x ) x [-An n + ( I n. An n. )3 a a a . l a i a a i a ( t 2 ) (208) The form of (206) f o r x moles of pure component "a" i s given a by X x n'z E cNR-R^o a a s a J I ( 2 Q 9 ) where fNR-RiO o o { ^} " <"3«®" \\ V { - J ® - 1 H " "da + ® " J ® 1 ( 2 1 0 > /NR-R-v which i s the form of | ^ I f o r pure component a. A term such as u 0 ^ s i s the value of f o r pure component "a". For m u l t i -f u n c t i o n a l molecular s p e c i e s , p a r t i c u l a r l y i n the case where f u n c t i o n a l l y u n l i k e groups tend to i n t e r a c t s t r o n g l y ( f o r ex-ample "0" and "H" groups i n a pure a l c o h o l ) elements of the matrix depart s t r o n g l y from u n i t y . The excess mixing e f f e c t corresponding to equation (206) i s thus 203 { -NR-R) o E x a a 3 ( t 3 ) (211) which. I s term t ^ In the NCZ expression. This term becomes im-portant when l i q u i d s of d i f f e r i n g p o l a r i t y are mixed. 8. THE GENERAL NCZ EXPRESSION FOR GE/RT Since each of the terms of the NCZ equation has been I n d i v i d u a l l y set f o r t h by t h i s p o i n t , i t i s now p o s s i b l e to w r i t e the NCZ equation f o r (G /RT) f o r the general case of a m u l t i f u n c t i o n a l and/or multicomponent n-mer s o l u t i o n . These terms are assembled In equation (212). (See a l s o reference (49)). + E(E - x ) x . [ - A n n a + ( E n ± a * n n ± a ) ] . i a a a 204 G E / R T = E 5 i n 5 " xa An X Q ^ a <2 a v e L 2 * a \ / 2 a n I a J 3 Z CO . . o , I - / \ i ave 11 . + E E [n X . . - n x X . . ( z E / z ) ] zp^- t , 1 ave i a i a a a l a j a s a ave RT * i a j a + E E n X . . z x co. . / R T tA a v e i a i a a v e i j • i a 3b z n ave ave.. + E x ( z E X ) x (1 - z a / z s E a ) (-^V^) >< a a / R T t,. a i o n a j o n b a a + 1 ( 1 \ X > x [ ( 1 - z a b / z s E a ) x a a ab p a i r s i on a j o n b ao a ab b (212) 205 1. Notes on General NCZ Equation: •p a) t ^ i s c a r r i e d i n G /RT i n c a l c u l a t i o n s but not elsewhere discussed /•NR-R-i ^ 1 0 / = J a : n ® a X [ j $ j X " l H n $ J + U ® J ^ U ® J ) ] c) = pure component form of b) j-NR-R-j o \ " f = I n i ^ [ £ n i a [ u i i " l U n n o o , + U^TN . i n u ^ . . ] l a d) For simple binary X. . = X , = (z n a: + z,n ar )/(n z ) i j ab a a a b b b ave ave 2u>. . /kT ^ ^ b = ( 1 " 4 ^ ( 1 " ^ » ' 206 2. Multifunctional and/or Multicomponent System u),,/kT XJJL = X^X. e~ 1J i j i J X^ = Barker type pseudocomposition to. ./RT X - $ ( E X. e~ 1 3 ) a l l j J obtained by i t e r a t i n g from i n i t i a l estimate on RHS of above equation. e) E = [1 - (1 - 2 ¥ )(n - l ) / n a a a a Y > 1/z a — s E = exterior f r a c t i o n of a monomer a E -> 1 as n -> 1 , otherwise E < 1 a a a n^= e f f e c t i v e stiff-polymer chain length <_ n. n = £ n /E a; ave a a a a a 207 CHAPTER 10 . RELATION OF 2-LIQUID THEORIES  TO QUASICHEMICAL THEORIES 1. GENERAL 2. INTRODUCTION 3. REDUCTION OF THE CONFIGURATIONAL INTEGRAL TO PARTITION FUNCTIONS 4 . THEORIES FOR 0^ 5. INDIVIDUAL APPROACHES BASED ON A. The S o l u b i l i t y Parameter Expression B. The Wilson Expression C. D i f f e r e n c e Between Quasichemical and 2 - L i q u i d Theory Energy Weighting F a c t o r s 6. LOCAL COMPOSITION IN MASS LAW TERMS 7. THE NRTL EQUATION 8. THE ASOG EQUATION 9. AN NCZ ANALOGUE TO THE WILSON EQUATION TP A. G /RT Expression B. A c t i v i t y C o e f f i c i e n t s 10. SUMMARY OF RELATIONS BETWEEN QUASICHEMICAL AND RELATED THEORIES 11. TABULAR COMPARISONS BETWEEN G E EXPRESSIONS 12. THERMODYNAMIC VALIDITY OF PARAMETERS 13. CLASSIFICATION OF G E EXPRESSIONS WITH RESPECT TO STANDARD TYPES OF SOLUTIONS 208 14. FUNCTIONAL RESPONSE BEHAVIOR OF THE NCZ EXPRESSION 15. INTERCOMPARISON OF FUNCTIONAL RESPONSES OF VARIOUS EXPRESSIONS 209 CHAPTER 10 RELATION OF 2-LIQUID THEORIES  TO QUASICHEMICAL THEORIES 1. GENERAL At t h i s p o i n t , two e q u a l l y important c o n s i d e r a t i o n s must be attended t o : 1) To show the r e l a t i o n of the NCZ equation to more or l e s s r e -l a t e d approaches, and 2) To t e s t equation (212) against a wide range of experimental data. Neither c o n s i d e r a t i o n has c l e a r precedence, but the r e s u l t s of one to some extent i n f l u e n c e the arguments of the other. The course taken has been r a t h e r a r b i t r a r i l y , t o f o l l o w through 1 ) , then 2) i n t h i s and the suceeding chapter. The d i s c u s s i o n of Related Theories i n t h i s chapter has been conducted on a r e -l a t i v e l y fundamental l e v e l i n v o l v i n g , i n p a r t i c u l a r , p o i n t by point comparison of major t h e o r e t i c a l precepts of quasichemical t h e o r i e s and 2 - l i q u i d t h e o r i e s ; whereas i n Chapter 11, the com-p a r i s o n w i t h Related Theories i s made on the pu r e l y p r a c t i c a l l e v e l . In Chapter 11 the r e l a t i v e performance of v a r i o u s ex-cess f r e e energy p r e d i c t i o n t h e o r i e s i s compared w i t h the excess f r e e energy versus composition curve d e r i v e d from experimental r e s u l t s . While each chapter provides i n s i g h t i n t o the other, 210 s t i l l the reader who i s more p r a c t i c a l l y o r i e n t e d i s urged to read Chapter 11 f i r s t (or o n l y ) : the reader who i s more t h e o r e t i c a l l y i n c l i n e d i s urged t o read both chapters i n the order presented. 2. INTRODUCTION The p r e d i c t i o n of l i q u i d mixture excess fr e e energies i s but one aspect of the i n v e s t i g a t i o n of the l i q u i d s t a t e and the p r e d i c t i o n s of mixture p r o p e r t i e s f o r dense l i q u i d s which are a l s o n o n i o n i c * i s but a small p a r t of t h i s . Notwithstand-i n g , the number of nonionic mixture excess f r e e energy pre-d i c t i v e expressions i s con s i d e r a b l e and t h e i r d e t a i l e d formu-l a t i o n s not only d i v e r s e , but a l s o not i n a l l cases e n t i r e l y compatible • A somewhat d e t a i l e d examination of the s i t u -a t i o n i s thus r e q u i r e d . The o r g a n i z i n g i d e a of the f i r s t s e c t i o n of t h i s chapter i s t o argue the formal development of S o l u b i l i t y Parameter, Wilson and NRTL (nonrandom 2 - l i q u i d ) expressions on the one hand, and quasichemical t h e o r i e s on the other from fundamentals, i n order t o show inherent d i f f e r e n c e s i n the p a r t i t i o n f u n c t i o n s from which the two groups of t h e o r i e s can be shown t o r e s u l t , and then to show p o i n t s of p a r a l l e l i s m and d i f f e r e n c e i n the subsequent steps of the d e r i v a t i o n s of the two sets of t h e o r i e s . * See Appendix 1 211 This i s followed by a d e s c r i p t i o n of the A.S.O.G. (Associa t e d S o l u t i o n of Groups) theory, which i s a g e n e r a l -i z a t i o n of the Wilson Equation to the groups or ' p a i r s ' l e v e l , thus which i s a l s o formulated on the energy s c a l e used i n quasichemical t h e o r i e s . To summarize the foregoing arguments i n graphic form, some of the c h i e f s t r u c t u r a l assumptions of quasichemical and ' r e l a t e d ' t h e o r i e s discussed are then d i s p l a y e d i n Table 1 on p. 248 . This i s fol l o w e d by Table 2 which footnotes the b r i e f headings of Table 1 and 'indexes' these w i t h references to s p e c i f i c l o c a t i o n s elsewhere i n the t h e s i s where the p o i n t s i n q u e stion have been d e a l t w i t h i n d e t a i l . Thus Table 2 i s a l s o u s e f u l as a 'point of access' to the t h e s i s f o r those who wish f i r s t to look at the r e s u l t s , and then r e f e r these back to e a r l i e r d i s c u s s i o n s . Next, the question of'thermodynamic' c r i t e r i a of a p p l i -c a b i l i t y of expressions (as d i s t i n c t from t h e i r wider engineer-i n g use as e m p i r i c a l c u r v e - f i t t i n g expressions) i s b r i e f l y d i s -cussed i n terms of degree of correspondence between 'best f i t ' and independently measured constants f o r a given system. Some of these comparisons f o r NCZ and Wilson t h e o r i e s are set out i n Table 3 on p . 252. For the f u r t h e r a s s i s t a n c e of the reader, s p e c i f i c ex-amples of r e a l systems, t h e i r commonly understood type names, and t h e o r i e s which are u s e f u l f o r the p r e d i c t i o n of t h e i r phase e q u i l i b r i a are given i n Table 4. Also i n Table 4 are set out the parameters which are prominent i n determining the behavior 212 of these systems, hence those which a p p l i c a b l e p r e d i c t i v e ex-pr e s s i o n s should i n c l u d e i n t h e i r f o r m u l a t i o n s . A l s o i n c l u d e d i s a s e c t i o n on f u n c t i o n a l response of component terms of the NCZ equation (shown g r a p h i c a l l y as G /RT vs a: curves) t o encourage the reader to see the excess f r e e energy behavior of systems chosen f o r a n a l y s i s i n the frame of re f e r e n c e of t h i s work. 3. REDUCTION OF THE CONFIGURATIONAL INTEGRAL TO PARTITION FUNCTIONS I t i s convenient to begin arguing from fundamentals at the stage o f c o n s i d e r i n g the form of the c o n f i g u r a t i o n i n t e g r a l a p p l i c a b l e to c e l l t h e o r i e s . The steps by which the r e d u c t i o n of the c o n f i g u r a t i o n i n t e g r a l t o p a r t i t i o n f u n c t i o n s takes place have f a r - r e a c h i n g consequences on a l l subsequent f o r m u l a t i o n s t e p s , although the a c t u a l assumptions themselves are q u i t e s t r a i g h t f o r w a r d . The f o r m a l l y c o r r e c t N - p a r t i c l e c o n f i g u r a t i o n i n t e g r a l i n 3-N space may be w r i t t e n : n f v t \ ( ( -U(all-all)/kT _ • £ 0 0 - j ... j e dar1 ... dx 3 N ( 2 1 3 ) 1=1 i=3N Upon making the standard c e l l theory s i m p l i f i c a t i o n that p a r t i -c l e movements are l o c a l i z e d t o w i t h i n d e f i n a b l e " c e l l s " , ( a n d assuming s i n g l e occupancy of each c e l l ) the equivalent p a r t i t i o n f u n c t i o n f o r the N - p a r t i c l e assemblage i s obtained. (214) - e ( 2 - a l l ) / k T N, 2 d * 2 d y 2 d z 2 Comparing equation (214) t o (213)1 i t i s seen that the p o t e n t i a l U ( a l l - a l l ) of a l l molecules-bound-by-all-molecules i n equation ( 2 1 3 ) 3 l s decoupled i n (214) i n t o c o n t r i b u t i o n s of e ( l - a l l ) (the p o t e n t i a l of one molecule of 1 bound by a l l ) and con-t r i b u t i o n s of e ( 2 - a l l ) (the p o t e n t i a l of one molecule of 2 bound by a l l ) ) s u c h that -The a c t u a l process of e v a l u a t i o n of one of the s i n g l e molecule p a r t i t i o n f u n c t i o n s (the terms i n the square brackets' i n equation ( 2 l 4 ) ) s u c h as i s v i s u a l i z e d as f o l l o w s . The whole N-molecule system r e -p r e s e n t i n g the mixture i s f i r s t imagined as being arranged i n t o i t s most probable c o n f i g u r a t i o n . (This c o n f i g u r a t i o n gives U ( a l l - a l l ) = K1 e ( l - a l l ) + N 2 e ( 2 - a l l ) (215) (216) 214 the best s i n g l e estimate of the p r o p e r t i e s of the system). Indeed, because of the l a r g e s i z e of the whole system, (a mole of molecules i s u s u a l l y chosen) the most probable c o n f i g u r a t i o n i s enormously dominant over l e s s probable c o n f i g u r a t i o n s , d u e to the sharpness of the p r o b a b i l i t y d i s t r i b u t i o n s f o r systems of such a s i z e about t h e i r l a r g e s t term. From here on t h e r e -fo r e ft1 w i l l be considered to be simply the degeneracy -of the most probable c o n f i g u r a t i o n . The l i m i t s of i n t e g r a t i o n are d e f i n e d f o r any i n d i v i d u a l molecule as the volume of permitted movement w i t h i n the cage o f i t s nearest neighbors. This domain of i n t e g r a t i o n i s v f f o r a i molecule of species i . For each poi n t i n t h i s s m a l l volume v , 1 the p o i n t value of E ( l - a l l ) i s evaluable (or would be i f the exact form of the i n t e r m o l e c u l a r p o t e n t i a l f u n c t i o n were known). The i n t e g r a l s i n equation (214) could normally be evaluated by a summation of these p o i n t values of e all^/'kT such that e " £ l / k T = I e - £ ( l - a l l ) / k T (217) all volume elements i n u 1 d T i where now (-—=) i s the volumetric weighting f a c t o r f o r each V dr. poi n t value of the exponent, (TJ- "^) i s thus a conf i g u r a t i o n a l ( s p a t i a l ) type of p r o b a b i l i t y . From the form of equation (217) i t i s r e a l i z e d that the appropriate average value of the exponent of e ( l - a l l ) over the volume, i . e . , e i s a f r e e energy average*, since equation (217 ) can be rearranged such that - ^  i s a s i n g l e exponent r e -pr e s e n t i n g the r i g h t hand side p a r t i t i o n f u n c t i o n . Thus t~ s /., E l / k T N v - e ( l - a l l ) k T , ( qP = ( f e ^ = i 1 1 (218) a l l volume elements i n u 1 I f the smooth i n t e g r a t i o n over the r e g i o n i s rep l a c e d by summation as shown, the number of volume increments summed (23) over represents the degree of " f i n e - g r a i n i n g " of the approximation of t h i s sum of the RHS of equation (218) t o t n e i n t e g r a l on the LHS. Now i f a l l the energy b i n d i n g 1 i s a t t r i b u t e d to i t s z^ nearest neighbors, as i s the formal assump-t i o n of the quasichemical method, the op e r a t i o n on the r i g h t hand side of equation (218 ) becomes, i n s t e a d of the v o l u m e t r i c -a l l y weighted sum over p o i n t energies, the product of the (assumed) independent p a r t i t i o n f u n c t i o n s f o r each of the z/2 p a i r s i n t e r a c t i o n s given as q 1_. * z/2 . (z/2) / k T .. N(z/2) (219) f. < q i ) = T T ~ q i - j = ( e 1 J v* } i - j pairs i = l * In f a c t , the free-energy average may be appromixated by a measured mass-law average q u a n t i t y i n most c a s e s ^ ^ . Thus the p a i r s - l i q u i d assumption corresponds to a coarseness of g r a i n i n g corresponding to d i v i d i n g up the t o t a l volume of i n t e g r a t i o n per molecule i n t o z/2 reg i o n s . In a f u r t h e r assumption, i t Is now; considered that each p a i r s i n t e r a c t i o n need he i n t e g r a t e d only over a domain of s i z e vf / ( z / 2 ) . I t • i i s seen from equation (219 ) t h a t the f r e e volume per p a i r i s of the form ' . v = (v ) ( 2 / z i } (220) i - i i I f i t i s argued from the p a i r s p a r t i t i o n f u n c t i o n p o i n t of view, whereas i t Is vf = V /(z/2) (221) f i - i f i 1 from the c o n s i d e r a t i o n of a d d i t i v e volumes. This discrepancy i s one of the drawbacks of the ' p a i r s - l i q u i d ' , and i n f a c t deters one from d i r e c t l y c o n s i d e r i n g f r e e volume per p a i r terms e x p l i c i t l y i f there i s any way of a v o i d i n g i t . N e vertheless, d i s r e g a r d i n g t h i s d i f f i c u l t y , the development i s pursued by i n s e r t i n g the a d d i t i o n a l assumption whereby the f i n a l form of the p a i r s l i q u i d used i n the quasichemical model i s obtained. Rather than a c t u a l l y e v a l u a t i n g equation (219 ), i t i s more p r o f i t a b l e to evaluate four 'molecular-scale bundle s p e c i e s ' , 1-surrounded by l ' s , 1-surrounded by 2's, 2-surrounded by l ' s and 2-surrounded by 2's.and t o regard q 1 and as composed of appro-p r i a t e averages of these f o u r . Thus: 217 - E n / R V i / 2 -( Z le u/2kT) [y f e . ] 11 -e 2 2/kT» /2 q 2 ® = [ V i e -(z e 1 9/2kT) [v, e ^ ^ ] 21 - E 1 2 / k T , Z 1 2 / 2 q i - 2 y [ \ _ 2 e i - ( z 1 2 £ ; L 2 / 2 k T ) [yf e ] 12 -e 2 2/kT z 2/2 q 2 ® = [ \ _ 2 6 ] -(z_e 9 9/2kT) [ V e ] 22 (220) The 4 bundle-species p a r t i t i o n f u n c t i o n s are the u l t i m a t e b u i l d -i n g b l o c k s from which f r e e energy of mixing estimates can be assembled. A l t e r n a t e l y , however, (as i s the case i n 2 - l i q u i d t h e o r i e s ) i t i s a l s o p o s s i b l e t o re-compose the s i n g l e molecule s c a l e p a r t i t i o n f u n c t i o n s f o r species 1 and 2 i n the average  environment of the mixture by making use of what are considered to be a p p r o p r i a t e l y weighted averages of the bundle species p a r t i t i o n f u n c t i o n . Thus q l = ?l(l) q10 + C2(T)q20 (221) 218 " C l ( 2 ) q l ( ~ 2 ) + ^2 2, ^2(2) (222) The weighting f a c t o r s , namely the may be volume f r a c t i o n s , l o c a l volume f r a c t i o n s , mole f r a c t i o n s , contact f r a c t i o n s , l o c a l contact f r a c t i o n s , e t c . , depending on the i n t e r p r e t a t i o n of the l o c a l c o n f i g u r a t i o n of the l i q u i d being used. Equation (214 ) then becomes = n'(N',N2,N')(q) 1 (q 2) 1 (223) This i s the form of the p a r t i t i o n f u n c t i o n from which amal-gamated (t-^ + t^") type excess f r e e energy expressions of the Wilson type are de r i v e d (hence the s u b s c r i p t w). The primes on N-j^  and i n d i c a t e adjustment of populations to the primed ones of equal c o o r d i n a t i o n number i f the treatment r e q u i r e s i t I f the four molecular s c a l e bundle species p a r t i t i o n f u n c t i o n s are used.instead,the r e s u l t i s obtained: n l l , " l 2 0^ = O/CN^N'.N'Kq^) " ( q ^ ) "21 , "22 x ( q 2 ^ ) ( q 2 ^ ) (224) This form of the c o n f i g u r a t i o n a l p a r t i t i o n f u n c t i o n may be used t o o b t a i n seperate t ^ athermal ( P l o r y type)and t ^ " thermal (Guggenheim type) terms i n the excess fr e e energy expression as 219 used by H e i l (hence the s u b s c r i p t H ) The symbols n ^ n ^ and represent the populations of the four molecular s c a l e bundle species.at e q u i l i b r i u m . Athermal and thermal terms must be expressed s e p a r a t e l y i n t h i s case, since the F l o r y terms i n v o l v e concentrations of the two a c t u a l molecular species and whereas the thermal terms i n v o l v e c o n c e n t r a t i o n f u n c t i o n s i n v o l v i n g the four bundle species p o p u l a t i o n s , which are quadratic f u n c t i o n s of and N through mass-law r e l a t i o n s . The c o m b i n a t o r i a l term fi,(N-^, N^, N') i s the same func-t i o n i n both (223) and (224). Whatever assumptions are made about the number or nature of bundle s p e c i e s , these c o m b i n a t o r i a l terms must be the same i n 0 as i n QTT, and a l s o must be, w i t h i n w H the l i m i t s of a l l o w i n g f o r e q u a l i z i n g the s i z e of the p a i r s bundles, c o n s i s t e n t w i t h the number of a c t u a l molecular con-f i g u r a t i o n s defined by Q. (N^, N 2, N) f o r molecules, and z z z z , — N 9 , - ± - N . + - 2 - N . ) (225) z12 1 Z12 " Z12 X Z12 L f o r permitted p a i r s permutations. In b r i e f summary up to t h i s p o i n t : (1) A p r e s e n t a t i o n has been made of the us u a l treatment of volume-of-Integration f a c t o r s , which i n t u r n l a y s the stage f o r the three main ways of d e a l i n g w i t h the v o l u m e - o f - i n t e g r a t i o n e f f e c t s more q u a n t i t a t i v e l y . The three ways are: (a) by s u b s t i t u t i o n , through the somewhat tenuous connection between v o l u m e - o f - i n t e g r a t i o n and the molecular displacement 220 volume per c e l l (V/N),or with the molal or p a r t i a l molal volume of the components i n s o l u t i o n , or wi t h the f r e e or v o i d volume (used i n e v a l u a t i o n of Q.^); (b) by w r i t i n g formulations from which the volume-of-i n t e g r a t i o n f a c t o r s tend to cancel (the method used i n e v a l u -a t i o n of Q.JJ) , or (c) by l e a v i n g n o n - c a n c e l l i n g p o r t i o n s i n u together w i t h i t h e i r energy exponents (thus i n fre e energy form) and thus t r y -i n g d i r e c t l y to evaluate the r e s u l t i n g f r e e energies (the method used i n e v a l u a t i n g t ^ terms i n the NCZ equation). C l a s s i c a l methods i n v o l v e (a) and (b) only. (2) Probably more important, the f i r s t b a s i c b i f u r c a t i o n i n approach t o G expressions has been r e v e a l e d , that of the choice between Q.^  or Q.^  as a s t a r t i n g p o i n t f o r d e t a i l e d f o r m u l a t i o n of mixture property p r e d i t i o n s . H i s t o r i c a l l y , Van Laar and Hildebrand began w i t h the Q. ^  form, and Guggenheim i n i t i a t e d use of Q.JJ. Since the b a s i c form of Q. expressions has been out-xl ri l i n e d i n the f i r s t part of Chapter 8, H e i l ' s equation which s t a r t s from 0.^ w i l l not be discussed i n d e t a i l , s i n c e i t i s of t h i s type. Instead, a b r i e f resume' of r e s u l t s l e a d -i n g up t o the S o l u b i l i t y Parameter and Wilson expressions w i l l now be given. 4. THEORIES FOR 0.^ N a t u r a l l y the o r i g i n a l p r e s e n t a t i o n s of the r e l a t e d t h e o r i e s were not made i n as u n i f i e d a form as i t i s now hoped to employ, and i n p a r t i c u l a r these were not n e c e s s a r i l y brought 221 about e x p l i c i t l y through the c o n s i d e r a t i o n of operations on the 0. ^  p a r t i t i o n f u n c t i o n per se. Nevertheless, i n the i n t e r e s t s of c o n s i s t e n c y , i n the present work steps l e a d i n g to f o r m u l a t i o n of expressions f o r the excess f r e e energy f o r non-ionic s o l u t i o n s proposed by p a r t i c u l a r workers i n the f i e l d of quasichemical and r e l a t e d s o l u t i o n t h e o r i e s w i l l be argued, from the p a r t i t i o n f u n c t i o n s from which t h e i r expressions can be* shown to r e s u l t . I f we w r i t e the mixture p a r t i t i o n f u n c t i o n i n the most general way p o s s i b l e , namely N' £ . , = n»(N' N' N*) n q, (226) mxxture 1 2 . a l l i where -e./RT q ± = Ce V u f ) (227) i I t i s seen that the p a r t i t i o n f u n c t i o n (226 ) responds to three q u i t e d i f f e r e n t types of independent v a r i a b l e s : (1) Numbers N^ of various molecular species present, which appear i n o v e r a l l c o n f i g u r a t i o n a l terms i n the mixture f r e e energy expression. When the l o g of the n ? c o m b i n a t o r i a l term i s expanded v i a S t e r l i n g ' s approximation these N^ generate terms of the form (2) T h e a p p r o p r i a t e c o h e s i v e e n e r g y p e r m o l e c u l e , e a n d (3) t h e v o l u m e o f i n t e g r a t i o n u f i P a r t s (2) a n d (3) make u p t h e c o n s t i t u e n t s o f t h e s i n g l e -m o l e c u l a r - s c a l e p a r t i t i o n f u n c t i o n s , <1 i ( e q u a t i o n (227 ) ) . I n a t t e m p t i n g t o e v a l u a t e f i x t u r e ( e ( l u a t l o n (226 )) t h e p a r t o f t h e o v e r a l l o p e r a t i o n r e p r e s e n t e d b y fil p r e s e n t s no p a r t i c u l a r f o r m a l d i f f i c u l t i e s a s was s e e n i n C h a p t e r 7. T h e e v a l u a t i o n o f E i n (227 ) a g a i n p r e s e n t s no f o r m a l d i f f i c u l t i e s i n a s m u c h a s i t c a n i n p r i n c i p l e be e v a l u a t e d f r o m t h e l a t e n t h e a t o f e v a p o r a t i o n o f t h e two p u r e c o m p o n e n t s , a n d t h e m e a s u r e -ment o f o n e h e a t o f m i x i n g . T h e r e a l d i f f i c u l t y a r i s e s i n t h e e v a l u a t i o n o f v f , o r more p r e c i s e l y , i n t h e r e l a t i o n s b e t w e e n i t h i s v a r i a b l e o n t h e p a i r s a n d m o l e c u l a r s c a l e s , l e a d i n g t o t h e p a r a d o x ( s e e a g a i n i n e q u a t i o n s (220) a n d (221)) v f = ( u f * 2 / z ) z / 2 (229) i 1 w h i c h o c c u r s when t h e o p e r a t i o n s w h i c h t r a n s f o r m p a i r s - s c a l e e n e r g i e s t o m o l e c u l a r s c a l e e n e r g i e s a r e a p p l i e d t o t h e c o r r e s -p o n d i n g f r e e v o l u m e f a c t o r s . R e l a t i o n (229 ) i n d i c a t e s t h a t t h e p r o b l e m o f m a n i p u l a t i n g a n d e v a l u a t i n g v o l u m e f a c t o r s m u s t be l e s s s t r a i g h t f o r w a r d t h a n t h a t o f t h e i n t e r a c t i o n e n e r g i e s , i n t h e p r o d u c t i o n o f e x c e s s f r e e e n e r g y e x p r e s s i o n s t h r o u g h o p e r a t i o n s o n p a r t i t i o n f u n c t i o n s . N e v e r t h e l e s s v o l u m e - o f - i n t e g r a t i o n f a c t o r s do_ a p p e a r i n t h e p a r t i t i o n f u n c t i o n , a n d t o some e x t e n t t h e f o r m u l a t i o n s made i n r e l a t e d t h e o r i e s h a v e b e e n d e p e n d e n t o n how t o g e t r i d of the d i f f i c u l t i e s introduced by t h i s f a c t o r . ( N o t e : the paradox of equations (220 ) and (221) i s avoided (at l e a s t o s t e n s i b l y ) i f molecular s c a l e formulations, not p a i r s s c a l e f o r m u l a t i o n s , are employed.) I t i s remembered that i n 2 - l i q u i d t h e o r i e s , the d i f f i c u l t y has been circumvented by i n c o r p o r a t i n g the v i i n some way i n the (fi' ) c o n f i g u r a t i o n a l aspect of the o v e r a l l e x p r e s s i o n . I t i s avoided i n Guggenheim's f o r m u l a t i o n , by making the assumption that ^ . i . i " 1 a i (230) i - i i-j an assumption which was l e g i t i m a t e i n CZ form u l a t i o n s and a l s o holds q u i t e c l o s e l y f o r the NCZ s o l u t i o n . In the present work the problem of e v a l u a t i o n or ope r a t i n g on v in the t term i-j 5 has been again circumvented by i n c l u d i n g t h i s term i n ex-pres s i o n s i n v o l v i n g "a" and thus being able to regard i t as a component of the f r e e energy. In the f o l l o w i n g e x p o s i t i o n of i n d i v i d u a l r e l a t e d t h e o r i e s p a r t i c u l a r care has been taken to d i f f e r e n t i a t e between various . approaches taken toward N. , e-• ,and v f e f f e c t s , w i t h s p e c i a l i i J J-^  a t t e n t i o n given to the l a t t e r . 5. INDIVIDUAL APPROACHES BASED ON Q, w A. The S o l u b i l i t y Parameter Expression The f i r s t approach of t h i s type was that due to Hildebrand. The mixture p a r t i t i o n f u n c t i o n was regarded as being: 224 l * t u r e " ^vV> * ^y1^2 ( 2 3 1 ) y i e l d i n g G . mixture „ , „ „ • ^ = x^ in x^ + # 2 ^ n x2 ~ xi ^ N - ^2 Q 2 ^^ ~>^ > Since the r e a l mixtures s e l e c t e d f o r t e s t i n g the theory happened to have u n l i k e p a i r cohesions c l o s e to the a r i t h m e t i c average of l i k e p a i r cohesions, energy e f f e c t s were disregarded, Average values of the molecular p a r t i t i o n f u n c t i o n were thus considered to be ql = Q 2 = xi ql + X2 Q 2 = Xl \ + X2 \ ( 2 3 3 ) ( S u p e r s c r i p t ° i n d i c a t i n g pure component s t a t e q u a n t i t i e s ) whence, by s u b s t i t i o n of ( 2 3 3 ) i n t o ( 2 3 2 ) G . . m l g n g = x± In Xl + x2 Sin x2 - (a^ + a ^ J l n C a ^ v° + x2 v° ) ( 2 3 4 ) + cc. Jin v ° + ar„ In v ° 1 1 2 The ensuing amalgamation of pure component and mixture e f f e c t s y i e l d s , a f t e r s u b t r a c t i o n of the i d e a l mixing term: | j = x1 an ( < f i 1 / a : 1 ) + x2 Jin ( + 2 / a : 2 ) (.235) 225 where the volume f r a c t i o n 4> , i s defined: x v° 1 f 1 1 x v° AX \J° (236) l f l l f 2 The form of the excess f r e e energy expression (equation (235 ) ) i s the same as P l o r y ' s f o r an athermal mixture, though Hildebrand's method of d e r i v a t i o n r e l i e s upon amalgamation of (what are u l t i m a t e l y ) l o c a l volume^of^.integration f a c t o r s i n t o the o v e r a l l c o n f i g u r a t i o n a l term to produce the volume f r a c t i o n e f f e c t (though simultaneously assuming a purely random form of fi)? r a t h e r than u s i n g the p u r e l y c o n f i g u r a t i o n a l arguments em-ployed by F l o r y i n the d e r i v a t i o n of the polymer theory. The r e s u l t I s a f u n c t i o n of volume f r a c t i o n i> , defined i n terms of molar volumes r a t h e r than chain lengths. T r i v i a l c a n c e l l a t i o n of x and d i v i s i o n of each term by i t s numerator v i n equation i (235 ) y i e l d s the expression: GE/RT = -x± ln(x± + (v° /v° )x2) - x2 In (v° /v° )x± + x£ (237) B. The Wilson Expression Wilson pursued the theme f u r t h e r by i n c l u d i n g the ~ E energy of the q terms i n the r e s u l t i n g r e l a t i o n f o r G , which would have the e f f e c t of f u r t h e r modifying equation ( 2 3 7 ) • The expression f o r the f r e e energy of mixing then becomes: -e'j/kT -e2/kT Gmixing/RT = Xl l n Xl + X2 *n X2 " (*1 + * 2 ) £ n ( a r l 6 Vf± + X2 6 -e°/kT -e°/kT + x^ In (e v ) + x in (e v ) 1 2 (.238) 226 Amalgamating,thendividing throughout by the numerator terms -e /kT (e v f ) i n each amalgamated term, and s u b t r a c t i n g the 1 i d e a l mixing terms, y i e l d s the r e s u l t F - ( e -e JkT G^ /RT = i n (x. + e 1 Z 1 1 (u° /v° )ar0) 1 1 f2 f l 2 - ( E - E )/kT -x 2 in (e Z 1 Z Z (v° /v° )a?1 +x2) (239) In.Wilson's " A " n o t a t i o n GE/RT = - x1 in (x1 + A/^ 2 x2)- x^ln (A./^ ± x1 + x^ (239A) A more s u c c i n c t r e p r e s e n t a t i o n of equation (239) i s GE/RT = xx in ((frj/a^) + * 2 in (^/a^) (240) The <j>^  are 'volume f r a c t i o n s ' (see the form of (235)), now a l s o containing energy weighting f a c t o r s of the form shown i n (239) • C. D i f f e r e n c e Between Quasichemical and 2 - L i q u i d Theory  Energy Weighting Factors Except i n the case where l*> 1 2 l m 1 £u - z22\ (241) (which conforms to Kreglewski' s (3.6), o b s e r v a t i o n t h a t , f o r l i g h t hydocarbon systems f o r example,,£12 = E 2 2 where e 2 2 i s the s e l f cohesion of the more weakly bound s p e c i e s ) , the a l g e b r a i c form of the energy exponents i n expression (239 ) v i o l a t e s the p r i n -c i p a l assumption of the quasichemical approach as o u t l i n e d i n 227 Chapter 8. This problem e x i s t s because r e l a t i v e p r o b a b i l i t i e s of c o n f i g u r a t i o n ( l o c a l c o n f i g u r a t i o n Included) e x h i b i t i n g l i k e versus u n l i k e p a i r i n g depend not on versus e^^ or versus but upon uT.2' n e ^ p a i r w i s e r e o r g a n i z a t i o n energy, which i s the a c t u a l measure of the amount by which the c o n f i g u r a t i o n r i c h e r i n u n l i k e p a i r s w i l l lower the energy (and i n t h i s case the f r e e energy) of the system as a whole. The use of a c t u a l values of (E.. - £..) type energy f a c t o r s i n s t e a d of to . . f a c t o r s I J i i J ^ & J i j can be expected sometimes to exaggerate, or i n some cases to reverse the estimates of the energy induced l o c a l c l u s t e r i n g e f f e c t s caused by the same species of molecule. A l s o , the unequal s p l i t of energy weightings, i n t o (^2 ~ e i l ^ a n <^ ^ e21~ E22^' r a t h e r than i n t o two i d e n t i c a l f a c t o r s (both w^^),further de-v i a t e s from the p r e s c r i p t i o n s of the mass law. However, the s i t -u a t i o n i n p r a c t i c e i s not as grave as i t seems, s i n c e d i f f e r -ences are a c t u a l l y assigned by e m p i r i c a l curve f i t t i n g ; and s i n c e e n e r g i e s , (hence w-|_2^ m a v w e i l b e somewhat c o n c e n t r a t i o n -dependent i n r e a l s o l u t i o n s . Because the regions of maximum s e n s i t i v i t y of the a c t i v i t y c o e f f i c i e n t s to t h e i r p a r t i c u l a r energy f a c t o r s are at the opposite ends of the b i n a r y r e g i o n , the p o s t u l a t i o n of two energy f i t - f a c t o r s unconstrained by mass law symmetry r e s t r i c t i o n s may be u s e f u l as i t i s then p o s s i b l e to f i t each a c t i v i t y c o e f f i c i e n t of a given binary i n d i v i d u a l l y i n i t s d i l u t e r e g i o n . In the f i n a l form of the Wilson ex-p r e s s i o n , s u b s t i t u t i o n of p a r t i a l molal volume r a t i o f o r volume of i n t e g r a t i o n f a c t o r r a t i o s i s made before the energies are f i t t e d . Thus i n the Wilson equation there are three disposable 228 parameters, C^/v^) and the two energy f a c t o r s , i.c12 ~ Ei 'i7 a n d is 21 ~ e22^' ^ S a n e m P i r i c a l equation, the Wilson expression has had notable success f o r the p r e d i c t i o n of vapor l i q u i d e q u i l i b r i a because of i t s remarkable b r e v i t y and f l e x i b i l i t y of form. I t has however the inherent s t r u c t u r a l problems which are seen to be i n e v i t a b l e consequences of the contracted type of p a r t i t i o n f u n c t i o n used as the s t a r t i n g p o i n t i n i t s d e r i v a t i o n . U n f o r t u n a t e l y , these same feat u r e s c o n t r i b u t e to i t s I n a b i l i t y to d e a l very s u c c e s s f u l l y w i t h 2 l i q u i d phases, r e s u l t i n g i n systems e x h i b i t i n g very l a r g e excess f r e e energies of mixing. 6. LOCAL COMPOSITION IN MASS LAW TERMS Designation of z r a t h e r than v as the e x p l i c i t geometry f a c t o r i n the NCZ equation i s a main p o i n t of d i s t i n c t i o n be-tween the viewpoint on geometric e f f e c t s upon l o c a l con-c e n t r a t i o n s taken i n the present work, and that taken by Hildebrand and Wilson. In a d d i t i o n t o the problems w i t h the p a i r w i s e energy i n t e r a c t i o n terms mentioned.above, amalga-mation of the t ^ " simple heat of mixing term i n t o the o v e r a l l con-f i g u r a t i o n a l term t-^, engenders the d i f f i c u l t y that the term t j j " i n v o l v e s i n h e r e n t l y quadratic c o n c e n t r a t i o n f u n c t i o n s where-as the c o n f i g u r a t i o n a l term i n v o l v e s l i n e a r f u n c t i o n s of con-(29) c e n t r a t i o n a p p l i e d to l o g a r i t h m i c terms. Renon r e c e n t l y s t u d i e d means of f o r m a l l y 'uncoupling' the mass law expression f o r the e q u i l i b r i u m u n l i k e p a i r s c o n c e n t r a t i o n i n an attempt to reduce the qua d r a t i c c o n c e n t r a t i o n f u n c t i o n s i n t o a product of f i r s t order f a c t o r s , the sum of whose logarithms would thus be more suitable for amalgamation with those of the t ^ term. It w i l l now be shown that Renon's method of in v e s t i g a t i o n admits of various answers, depending upon how one choses to make a key su b s t i t u t i o n . I f the mass law expression i s written i n a form includ i n g free volume factors per p a i r , i t s form can be repre-sented as X 2 1 X 1 2 2 w 1 2 / k T = e x X 1 1 X 2 2 1-2 2-1 1-1 2-2 (242) or X 2 1 X 1 2 - 2 " i 2 / k T Q X X 1 1 X 2 2 rv v • 1-1 2-2 2-1 1-2 = 1 (243) I f the product of any other four factors K 1 2 a r b i t r a r i l y chosen i s such that K 2 1 K 1 2 K K 11 22 = 1 (244) The equation (243) and (244) can be equated through t h e i r RHS s. Then by equating appropriate parts of K 1 2 r a t i o s to selected factors of the LHS of (243 ), Renon obtained two expressions of f i r s t order: K 2 1 X 2 1 w 1 2 / k T V l ~ e (245) K l l X l l v. 1-1 230 a n d K 12 _ X 1 2 <*L2 / K T F 2 - l K ~~ x e v ( 2 4 6 ) 22 A 2 2 E q u a t i o n s ( 2 4 5 ) a n d ( 2 4 6 ) c a n e a c h be' r e a r r a n g e d i n t o a f o r m , a s t y p i f i e d b y t h e r e a r r a n g e m e n t o f ( 2 4 5 ) , s u c h a s e q u a t i o n ( 2 4 7 ) . . X21 "21 -»12 / k T % - l T h e o n l y r e s t r i c t i o n t h a t h a s b e e n a r r i v e d a t i n o b t a i n i n g e q u a t i o n ( 2 4 7 ) i s t h e v e r y w e a k o n e ( a n o r m a l i z i n g a s s u m p t i o n ) i m p o s e d b y e q u a t i o n ( 2 4 4 ) . S i n c e t h e e x a c t n a t u r e o f x 2 l ^ X l l I s n o t k n o w n , a p r i o r i , n o new l i g h t I s s h e d o n t h e e x a c t n a t u r e o f K 2 i / K n by t h e d e v e l o p m e n t , a l t h o u g h I t s f o r m i s s u g g e s t i v e . I f t h e f a c t o r k 2T/ K;Q I s r e g a r d e d a s a l o c a l c o n c e n t r a t i o n r a t i o o f 2 a b o u t 1 o v e r 1 a b o u t 1 f o r . t h e e x a m p l e s h o w n , o n e may s u b -s t i t u t e f o r i t t h e l o c a l c o n c e n t r a t i o n r a t i o o f o n e ' s c h o i c e . T h e n a t u r e o f s u c h l o c a l c o m p o s i t i o n s i s t h u s l e f t t o b e d e c i d e d b y o t h e r p h y s i c a l a r g u m e n t s . F o r r e a s o n s o u t l i n e d i n C h a p t e r 7 , ( r a t h e r t h a n x ^ a s i s u s e d b y R e n o n ) s e e m s t o b e s u g g e s t e d i n t h i s work. as. t h e m o s t r e l e v a n t l o c a l c o n c e n t r a t i o n v a r i a b l e . T h e l o c a l s i t u a t i o n a r o u n d a m o l e c u l e o f 2 f o r a b i n a r y , may t h u s b e r e p r e s e n t e d b y t h e r a t i o 5, - c o . „ / k T [ V f . „ ] ( 2 / z 1 2 ) J L « e 1 2 -2=1 ( 2 4 8 ) h [ V f 2 _ 2 ] ( 2 / Z 2 > 2 3 1 Since h = z2x2 (240) then _12 = f l _1 ( f l - 2  X22 X2 Z2 U f 2 _ 2 2 / z12 -u> /kT } 2 / Z 2 e ( 2 5 0 ) Equation (250 ) accentuates the need f o r c a r e f u l e v a l u a t i o n of ' l o c a l geometry e f f e c t s ' which e v i d e n t l y combine packing (z) and displacement volume Cv) e f f e c t s . Note t h a t i n equation ( 2 5 0) the volume r a t i o i s present i n a l o c a l c o n c e n t r a t i o n r a t i o only to the 2/z p o w e r , whereas i t w i l l be seen that the volume r a t i o present i n the Wilson f o r m u l a t i o n of the same e f f e c t i s to the power one. On the other hand, the z r a t i o occurs i n equation (250 ) t o the f i r s t power. While c h a r a c t e r i z i n g l o c a l geometry f a c t o r s by the z r a t i o (which i n v o l v e s R^ 2 to the f i r s t power) alone perhaps somewhat understates them, use of the volume r a t i o (which v a r i e s as R^to the t h i r d power) alone i s bound to o v e r s t a t e them. In a d d i t i o n to t h i s , the z r a t i o , while not a very s e n s i t i v e f u n c t i o n of composition, c e r t a i n l y responds d i f f e r e n t l y to composition than does the volume r a t i o . I t Is the o p i n i o n of the author that i n equation' ( 2 5 0 ), the e f f e c t s of the z r a t i o overshadow those of the volume r a t i o , and consequently, NCZ f o r m u l a t i o n s r e t a i n the z r a t i o . As a matter of f a c t , i n some f u t u r e work i t would be i n t e r e s t i n g to see how w e l l the z r a t i o to an 232 immediate power, say two, would represent l o c a l geometry e f f e c t s 7. THE NRTL EQUATION The NRTL equation Is best a t t r i b u t a b l e to Q.R } though i t has elements of both approaches. Renon's i n v e s t i g a t i o n s Into 'decoupling' the mass law e v e n t u a l l y lead to the NRTL ex-p r e s s i o n . I f one were to begin w i t h the simplest form of the t ^ " term from equation (104') i n Chapter 8 f o r a CZ binary mix-t u r e , namely G E = xf2™12 (251) and were to expand the expression f o r w2.2» one would o b t a i n E 1 G = 2x^2 z / 2 [ e 1 2 - y ^ - Q + z 2 2 } ^ (252) = x1x2 z / 2 ( e 1 2 - e 1 ; L) + x ^ z / 2 ( e 1 2 - e 2 2) I f now, as i n d i c a t e d by the usual convention of c i r c l i n g the i n -dex on the ' c e n t r a l ' molecule, the connotation of l o c a l com-p o s i t i o n were given to the now-separated p a i r of terms of the RHS of equation (252), one could w r i t e I f the'Wilson' form o f ' e n e r g e t i c a l l y induced l o c a l c l u s t e r i n g weighting f a c t o r were used to r e l a t e x 2 ^ i ) t o x2> a n <^ xl'j£) to x, , one would make the s u b s t i t u t i o n s .233 - ( e 2 T T e = xn x-1 0 )/RT • ( e 0 ^ - E , ^ )/RT (254) ana x 1 ^ = a;-. e ~v ^ - < e l ^ _ £ 2 ^ ) ) / R T + x1 e ^ + Xr (255) S u b s t i t u t i n g ( .254 ) i n t o ( 253 ) f o r x ^ and (. 255 ) i n t 0 (.253 ) f o r would give r i s e to the NRTL form of expression f o r a binary mixture. I t should be noted however, that i n i t s g e n e r a l l y accepted form, the NRTL expression i s w r i t t e n c o n t a i n i n g energy parameter on a molecular s c a l e . I f i s s u b s t i t u t e d f o r z/2 * e 1 2 and the parameter f o r 2/z, and the c i r c l i n g convention i s not e x p l i c i t l y used, one a r r i v e s at the NRTL expression i n n o t a t i o n compatible w i t h that of Renon. The r e s u l t of these transformations of v a r i a b l e s i s given by E _ e G — "^2^2 - a ( e 2 1 - e i l ) / R T -a(e' - e' )/RT, X (fe21 € l i * x^ + a; , , e '21 '11' + %2X\ x - a ( e | 2 - £ 2 2)/RT : " a ( e l 2 " £ 2 2 ) / R T ) x1 e + x 2 (256) 1 12 22; F i n a l l y , the' molecular s c a l e energies are in v e s t e d w i t h the p r o p e r t i e s of f r e e energies by the formal replacement of 234 12 by G 12 (257) and the substitutions ^G±2 = G i 2 ~ G l l and A G 2 1 = G 1 2 - G 2 2 a r e made, leading to the f i n a l form of the expression for a binary mixture - a ( A G 2 1 ) / R T °E = X1X2 X " -a(AG21)/RT X A G 2 1 (x1 + x2 e ) - a ( A G 1 2 ) / R T + X2X1 X " - a ( A G ) / R T X A G 1 2 (,x1 e + x2) (258) This i s now c l e a r l y the form of the NRTL expression as o r i g i n -( 2 9 ) a l l y reported . The expression i s regarded as containing three f r e e l y variable f i t parameters, namely a , ^ G ± 2 and A G 2 i • Parameter a i s usually assigned a value close to 0.3- G i s then empirically represented as the sum of two skewed parabolic component curves, either or both of which can be p o s i t i v e or negative depending on the sign given A^ 1 2 and ^ G2i ° The form of the sum curve i s thus quite f l e x i b l e , ranging from unimodal symmetrical (either p o s i t i v e or negative) to increasingly as symmetrical unimodal,or even to righthand or lefthand p o s i t i v e sigmoid, i n the extreme case of energies of large magnitude and d i f f e r e n t signs. 235 O p e r a t i o n a l l y , the NRTL Expression could be defined as another I n t e r e s t i n g type of h y b r i d expression. Instead of amalgamating the heat of mixing term i n t o the c o n f i g u r a t i o n a l term, and o b t a i n i n g a l o g a r i t h m i c expression as was the case w i t h the Wilson equation; the NRTL expression i s an example of amalgamating Wilson-type l o c a l c l u s t e r i n g f a c t o r s Into an ex-panded form of the heat of mixing term. While the operations transforming equation (251) i n t o equation (252) are not in c o n -s i s t e n t w i t h the mass law, the use of the AG f a c t o r s i n the i j exponent terms i s s t i l l subject to the same problems experienced w i t h t h i s form i n the Wilson equation. Note a l s o that no ex-p l i c i t volume weighting f a c t o r s appear i n the f i n a l expression f o r G , although the a f a c t o r , i f given a d i f f e r e n t value i n each exponent, has the e f f e c t of a geometry f a c t o r . 8. THE ASOG EQUATION The ASOG (Associated S o l u t i o n of Groups) model of a. l i q u i d , i s composed of a Plory-Huggins c o n f i g u r a t i o n a l term r e g a r d i n g the s o l u t i o n on a molecular s c a l e f o r c o n f i g u r a t i o n purposes, and a Wilson type treatment of the p a i r w i s e i n t e r -a c t i o n e f f e c t s which are considered on a groups s c a l e . The co n c e n t r a t i o n v a r i a b l e employed i n the Plory-Huggins term i s the m o l e - f r a c t i o n , and that i n the Wilson term i s the "group f r a c t i o n " . The group f r a c t i o n i s the p r o p o r t i o n of the t o t a l p o p u l a t i o n of groups ( c o n s i s t i n g of a l l groups on a l l mole-cu l e s ) which are of a given f u n c t i o n a l group s p e c i e s . In the ( 3 3 ) treatment of Derr and Deal , f o r example, i n a s o l u t i o n , of 2 3 6 b u t a n o l , methanol and water, the group f r a c t i o n of OH would d e r i v e from a l l three molecular species: present. Only two group species are considered to be present; namely OH and methyl. The p r o p o r t i o n of molecular surface represented by a given f u n c t i o n a l group, and volumetric displacement e f f e c t s are represented by d i f f e r e n t (and independently evaluated) sets of parameters i n the ASOG theory. 'Group numbers' d e s i g n a t i n g number per molecule of a given type of group f o r p a i r w i s e con-t r i b u t i o n s to the heat of mixing are denoted by v ^ ( f o r group k on molecule I ) . The displacement volume and/or chain length, parameter used In the volume f r a c t i o n f a c t o r s i n the Flory-Hugglns term of the ASOG expression i s designated by FH v f o r species i . The e f f e c t s of e n e r g e t i c a l l y induced l o c a l c l u s t e r i n g are not in c l u d e d i n the ASOG f o r m u l a t i o n . Geometry FH . , determining parameters v k i and v ^ are not c o n c e n t r a t i o n de-pendent. Each group p a i r w i s e i n t e r a c t i o n i s c h a r a c t e r i z e d by two p a i r w i s e i n t e r a c t i o n Wilson type constants. Measurements are made of the a c t i v i t y c o e f f i c i e n t s of r e a l s o l u t i o n s con-t a i n i n g the two i n t e r a c t i n g groups. Then, by mathematical a n a l y s i s , 'group' e f f e c t s are a l l o c a t e d . The d e s i r e d f i n a l parameters c h a r a c t e r i z i n g the (sometimes h y p o t h e t i c a l ) s o l u t i o n s of 1 ) groups A i n i n f i n i t e d i l u t i o n i n a s o l u t i o n of group B, and 2 ) group B i n i n f i n i t e d i l u t i o n i n a s o l u t i o n of group A. Adjustment of these two energies per p a i r of groups thus adds u s e f u l degrees of freedom to the expression to help accommo-date not only c o m p o s i t i o n a l dependence of u n l i k e p a i r i n t e r a c t i o n 237 energy, but a l s o that of c o o r d i n a t i o n number and e n e r g e t i c a l l y induced, l o c a l c l u s t e r i n g tendency, which are not d i r e c t l y c a l -(33) c u l a t e d . In a recent paper, , the success of the ASOG Equation In f i t t i n g numerous systems has been demonstrated. One of the equation's f e a t u r e s i s t h a t , l i k e any sol u t l o n - o f - g r o u p s approach, an economy of parameters i s e f f e c t e d , r e l a t i v e to a m o l e c u l a r - l e v e l treatment, In that many molecular species can be formed from a given set of group types. The ASOG equation embodies the c h a r a c t e r i s t i c s of the Wilson equation on the p a i r s l e v e l , A 'group' would be the cl o s e equivalent of a ' s i t e ' of a given species i n the terminology of t h i s t h e s i s . The form of the excess f r e e energy expression according to the ASOG theory i s w r i t t e n : GE/RT = I x± in ^/x± + I - X^ in (£ X' a ^ ) (259) i k I where the f i r s t summation over a l l molecular species i s the Plory-Huggins term, and the second summation over a l l group species i s the Wilson type of expression f o r the energy i n t e r a c t i o n p a r t of G /RT w r i t t e n on a groups-scale. Concentration v a r i a b l e x i s the mole f r a c t i o n of molecular species i , and X^ i s the group f r a c t i o n of the k species of f u n c t i o n a l group. <j>" i s the form of the volume f r a c t i o n used i n the.ASOG equation. 9. AN NCZ ANALOGUE TO THE WILSON EQUATION p A. G /RT Expression Since the Wilson equation has probably been the most 238 widely s u c c e s s f u l of the (G /RT). expresssions d e r i v e d from Q^ , i t was f e l t to. he. u s e f u l to f o r c e a l g e b r a i c p a r a l l e l i s m be-tween the a c t i v i t y c o e f f i c i e n t expressions obtained from the Wilson equation and the expressions r e s u l t i n g from I n c o r p o r a t i n g the NCZ e f f e c t s of terms t± and t ^ i n t o the O^form of the p a r t i t i o n f u n c t i o n . The object was to demonstrate under what c o n d i t i o n s the two s e t s of a c t i v i t y c o e f f i c i e n t expressions become i d e n t i c a l , at l e a s t i n a l g e b r a i c form. One begins w i t h 0^ i n the u s u a l way ~ N i ~ N o / poo >, = n W N ^ X q ^ 1(q 2) 2 ^13) But now Ni = z i / z i 2 N i ( 2 6 o ) and N' = z 2 / Z l 2 N 2 (261) S u b s t i t u t i n g i n t o a general statement of the f r e e energy of mixing, namely equation (262) SsigS& = - i n 9^ _ + in a° Q° (262), mixture r e s u l t s i n z z z z • f i x i n g Q Z12 1 Z12 2 ~ Z12 1 ,- «12 2 ~~RT = * n " T Z X ( ql> X ( q 2 ) (263) Z12 1 Z12 Z Z l Z12 z 1 9 1 z 1 9 2 + Hn (q°) ^ + In (q°) i Z At t h i s p o i n t the q terms are on a molecular s c a l e , •with populations adjusted to e q u a l i z e the s i z e of the p a i r s bundles f o r each s p e c i e s . The reference populations of 1 and 2 have a l s o been correspondingly a d j u s t e d , so that only the mixing e f f e c t i s f e l t by the molecular p a r t i t i o n f u n c t i o n s , r a t h e r than a combination of p o p u l a t i o n and mixing e f f e c t s . (This amounts to Ignoring term t,_ i n the general expression (122 ) of Chapter 8. Expansion of equation (263) leads t o : ^ k f ^ i * n i ^ T N j + N 2 £ n i p i ? [ ( 264) + N| in (q^/q-^ + N 2 in (q^/q^ Since , f o r a monofunctional monomer binary when heat e f f e c t s are low, N' 2— = r (265) N | + N 2 ?1 and N 2 o o q. m i^ n g = N i *n h + N 2 *n ? 2 + Nl £ n + N 2 £ n ^ ^ 2 6 7 ) R e a l i z i n g that the 'molecular' p a r t i t i o n f u n c t i o n r a t i o s represent ' p a i r s ' bundleSjand f o l l o w i n g the example of Hildebrand, one may s u b s t i t u t e f o r the molecular p a r t i t i o n f u n c t i o n a p p r o p r i a t e l y weighted averages of these f o r the molecular 240 p a r t i t i o n f u n c t i o n s o b t a i n i n g i n the mixture. Thus s u b s t i t u t i n g one obtains q l = h q l + K2 q 2 < 2 6 8 ) q 2 = 5 X q° +. ? 2 q 2 ^ 2 6 9 ) A f t e r s u b s t i t u t i n g and m u l t i p l y i n g through, the r e s u l t i s ob-t a i n e d : o G m i ^ n S = N! i n E_ + Nl In E.+ N.' i n ( - 1 -) ( 2 ? 0 ) kT 1 l z z J- £ a + E a h q l ^2 q2 b ^2 + Nl i n ( — S -) £ 1 q 1 + 5 2 q 2 Although the next step i s not f o r m a l l y d e f e n s i b l e , one suspects from the mass law expression that the p e r t i n e n t l o c a l r a t i o s of q^ are r e a l l y those of the p a i r s p a r t i t i o n f u n c t i o n s designated ^ q2' / ^ ) a n d q 2 ^ w n e r e t n e second c i r c l e d index now i n d i c a t e s the c e n t r a l molecule. Amalgamation of c o n f i g u r a -t i o n a l and energy terms as u s u a l , r e s u l t s i n ; k T 1 5 l q l ^ l ) + ? 2 q 2 0 2 ? l q l 3 + ^2q2D ( 2 7 1 ) 24l G . . £ , q , 0 e 2 q 2 2^) mixing _ . , — L X i + x„ An (— 4 ^ ~)) RT 1 Sqll) + ^ V ® 2 £lql®+ £ 2 q 2 D (2"?2) The excess fr e e energy expression i s obtained by sub-t r a c t i n g the i d e a l mixing term x1 An x^ + x2 An x2 • Doing so, w i t h a s l i g h t rearrangement i n t o denominator form, the f o l l o w i n g r e s u l t i s obtained: GE ' . + 22 - a ) 1 2 / R T / Z l / z 1 2 > „ , , " l - a ) 1 2 / R \ ( z 2 / z 1 2 ) - x An (x. + — e ) + 1*1 (1 " zl/ z12> * ^ 3 + 1*2 ^  ( 1 " (273) B. A c t i v i t y C o e f f i c i e n t s Wilson shows that the a: - e x p l i c i t form f o r two com-ponents can be obtained by f o l l o w i n g through the operations p i n d i c a t e d by Equation (26.)*, whereby £ny± i s obtained from G , For the sake of p a r a l l e l i s m constants r a t i o s are shown i n Wilson's n o t a t i o n of a and b: * more e x p l i c i t forms f o r computation are: a GE(N.) X. a l l k ,E-, E (274) J * 1 J (275) 2 4 2 Here -u).0/RT a' = (z 2/ Z ; L) 1 2 (276) -u 1 2/RT b' = ( Z l/z 2) ( 2 7 7 ) whereas i n ¥ilson's expression *1 " ( c21 - € 1 1 > / R I ( 2 7 8 ) v - ( e 1 0 - £_ 9)/RT 2 By s t r a i g h t f o r w a r d o p e r a t i o n i n d i c a t e d by the f i r s t r e l a t i o n mentioned i n the footnote and r e - e x p r e s s i n g the r e s u l t s i n N i terms of x ^  = y N , the expr e s s i o n : i 1 in y1 = ( z ^ z ^ ) x [-£n (a^ + a'a^) ^ ( 1 - a') z. a ; ( l - b') +*2<- 4 + + ( b \ + * 2 > » ( 2 8 o ) i s obtained. Whence i n y± = ( z ^ z ^ ) [-in a' + ( z ^ z ^ d - b')] z s „ z l w e l l „ . , Z 1 ~ Z 2 , (281) + [ - f ( l - ^ ) C - ^ ) ] + (-^—^) s 12 and z_ x (1 - b') z x (1 - b') in y 2 = 0 ^ ) x [-in (b'x 1 + - ^ + ^ + (-) ^ + ^ >1 z z_ E z a: 9(z 1 - z„) 2 z 1 2 R l z 1 2 / z 1 2 (282) 2M3 * n * 2 ( . ) " ( ^ * [ " £ n b ' + ^ * ( 1 " a ' ) ] ar2->0 ' z2 z2 e22 ( z2 " z l * 2 z RT z ( 2 8 3 ) Equations (280) and (281) correspond t o the a c t i v i t y c o e f f i c i e n t s i n l o g a r i t h m i c form f o r a monofunctional binary l i q u i d mixture embodying the assumptions and approximations de-t a i l e d above. These reduce to the exact a l g e b r a i c form of Wilsons' l o g a r i t h m i c a c t i v i t y c o e f f e c i e n t s my^ 3 and U Y 2 (as (51) given i n h i s t h e s i s ) upon s e t t i n g and r e p l a c i n g a' and b' by a and b as defined above. Thus, though the a c t i v i t y c o e f f i c i e n t expressions of Wilson and that obtained by f o r c i n g the NCZ equation i n t o a 'Wilson' form are a l g e b r a i c a l l y s i m i l a r , the two types of a c t i v i t y c o e f f i c i e n t ex-p r e s s i o n w i l l not behave i d e n t i c a l l y i n p r a c t i c e , because of the d i f f e r e n t z - r a t i o exponents, the f a c t t h a t a' and b' are u s u a l l y somewhat d i f f e r e n t as w e l l as stronger f u n c t i o n s of composition than a and b, and f i n a l l y because of the e x t r a terms i n the NCZ case. While the experimental p a r t of t h i s t h e s i s i s r e s t r i c t e d t o i n v e s t i g a t i n g the main features of the NCZ equation's a b i l i t y t o match G /RT behavior w i t h e x p e r i m e n t a l l y d e r i v e d r e s u l t s , the c l e a r l y d i f f e r e n t response behavior of NCZ and Wilson a c t i v i t y 244 c o e f f i c i e n t p r e d i c t i o n s Is of some I n t e r e s t , p a r t i c u l a r l y i n the d i l u t e s o l u t i o n r e g i o n , i n that i t i s i n d i l u t e s o l u t i o n that the a c t i v i t y c o e f f i c i e n t of the d i l u t e species i s both l a r g e s t and E a l s o most s e n s i t i v e to. composition, whereas G /RT i n the same re g i o n tends toward zero. The study of d i l u t e s o l u t i o n a c t i v i t y c o e f f i c i e n t behavior i s a l s o Important In that the removal of minor components i s a most important aspect of commercial product separations by d i s t i l l a t i o n . 10.: SUMMARY OF RELATIONS BETWEEN QUASICHEMICAL AND RELATED  THEORIES E n e r g e t i c a l l y , the p r i n c i p a l d i f f e r e n c e that emerges be-tween the NRTL, Wilson and ASOG expressions on the one hand and those of Guggenheim, Barker, and the NCZ expressions on the other i s the number of energy i n t e r a c t i o n constants per u n l i k e p a i r . In the case of the former, there are two, namely A i j = ( g i j " 8 i i } and A . - - (g.. - g j j ) whereas In the l a t t e r there i s only one, namely "ij = I g i j ~ ( g i i + g j j ) ] (287) In equations (285) and ( 2 8 6 ) , the symbol g.. i s i n t r o -duced as a g e n e r a l i z e d f r e e energy per molecule without the de-t a i l e d f u r t h e r connotations discussed e a r l i e r . I t s form i s (285) (286) , 245 c o n s i s t e n t with, the n o t a t i o n of P r a u s n i t z and Cukor*.-. Also note "that i n the above equations, the same, value of cu.. i s used both f o r the ' i j and j l senses of i n t e r a c t i o n . The second group of t h e o r i e s are c o n s i s t e n t w i t h the mass law ••' , whereas the f i r s t do not appear to be. P r a u s n i t z * has i n d i c a t e d that the reason NRTL and Wilson expressions depart from apparent con-f o r m i t y w i t h the mass law Is explained by the f a c t t h a t they de-r i v e from the 2 - l i q u i d theory of S c o t t v ; . In the 2 - l i q u i d t h e o r y , two i n t e r p e n e t r a t i n g l i q u i d s c o n s i s t i n g of '2 around 1' and '1 around 2' are combined by l i n e a r s u p e r p o s i t i o n to make up the o v e r a l l mixture, w i t h no energy i n t e r a c t i o n s between the two ' l i q u i d s ' . I t seems to the author that e n e r g e t i c a l l y , such a model would most a c c u r a t e l y d e s c r i b e two mutually i s o l a t e d l i q u i d f i l m s (of t e r m o l e c u l a r t h i c k n e s s ) of molecules i n a vacuum. This Is i n view of the assumption of no competition f o r nearest neighbor species by c e n t r a l molecules s h a r i n g such neighbors, i m p l i c i t i n the 2 - l i q u i d theory. On the p u r e l y e m p i r i c a l l e v e l , the use of two disposable energy constants per p a i r i n t e r a c t i o n (which a r e , i n p r a c t i c e , f i t t e d independently) i s most convenient i n that compensation of these values to help account f o r the c o n c e n t r a t i o n dependence of l o c a l c l u s t e r -i n g e f f e c t s , e t c . , can be accommodated i n the two constants. This i s p a r t i c u l a r l y t r u e when the two constants are evaluated, as i n the case of the ASOG equation, at opposite i n f i n i t e d i l u t i o n * P r i v a t e communication. regions of the appropriate b i n a r y ; but on the other hand, the p r a c t i c e l e a d s to twice as many energy constants per hinary as do the Barker, Guggenheim and NCZ models. I t i s admitted that t h i s gain i s at the cost of more computing to o b t a i n l o c a l com-(52,53) p o s i t i o n s ' . When computing f a c i l i t i e s are a v a i l a b l e , the l e s s d a t a - r i c h model may be pr e f e r a b l e , . e s p e c i a l l y w i t h regard to the thermodynamic I n t e r p r e t a t i o n of the e m p i r i c a l l y determined energy constants found to produce a b e s t - f i t . C o n f i g u r a t i o n a l l y , the most pronounced d i f f e r e n c e be-tween the " 2 - l i q u I d " and "quasichemical" sets of t h e o r i e s i s that the vo l u m e - o f - i n t e g r a t i o n f a c t o r s of the single-molecule p a r t i t i o n f u n c t i o n (see equations ( 226 ) and ( 227 )•) a r e mani-f e s t e d as molar volume r a t i o geometry f a c t o r s i n the Wilson and ASOG t h e o r i e s , whereas, i n the NCZ theory, packing numbers c a l c u l a t e d from monomer u n i t diameters are considered to exert the primary p u r e l y c o n f i g u r a t i o n a l e f f e c t ( i n c o r p o r a t e d i n the term t-^) manifested as r a t i o s of c o m p o s i t i o n a l l y v a r i a b l e co-o r d i n a t i o n numbers. The vo l u m e - o f - i n t e g r a t i o n terms are seen to "cancel out" i n the heat of mixing term of quasichemical t h e o r i e s , though i n the NCZ equation v o l u m e - o f - i n t e g r a t i o n f a c t o r s are in c o r p o r a t e d i n the fre e energy "a" constants of the t c term. The fundamental d i f f e r e n c e of choice of con-5 f i g u r a t i o n a l v a r i a b l e Is that between a "v" e f f e c t from the "q" term i n 2 - l I q u i d t h e o r i e s , and an "fi " e f f e c t i n q u a s i -chemical t h e o r i e s ( r e f e r back to equation ( 2 2 6 - ) ) . While the numerical values assigned the two e f f e c t s may not, i n p r a c t i c e , be very d i f f e r e n t , the formal d i f f e r e n c e between the two types 247 of conf i g u r a t i o n a l v a r i a b l e , at l e a s t , i s . fundamental, and adoption of the approach i n v o l v i n g the l a t t e r gives r i s e to two a d d i t i o n a l and d i s t i n c t i v e terms i n the NCZ equation not a r i s i n g In 2 - l i q u i d t h e o r i e s . 11. TABULAR COMPARISONS BETWEEN G EXPRESSIONS As a means of summarizing the d i s c u s s i o n of the f o r e -going chapter, and r e l a t i n g i t to the account of the v a r i o u s quasichemical t h e o r i e s o u t l i n e d i n previous c h a p t e r s , some of the p r i n c i p a l assumptions and s t r u c t u r a l a t t r i b u t e s of v a r i o u s workers' expressions are set out In t a b u l a r form In Table 1. As was mentioned i n the i n t r o d u c t i o n to t h i s chapter, Table 2 immediately f o l l o w i n g Table 1 a m p l i f i e s and references e n t r i e s i n Table l to a p p r o p r i a t e passages In the t e x t of t h i s t h e s i s . For maximum u t i l i t y , the two Tables should be considered by the reader In concert. Table 1 shows In summary form the d i v e r s i t y of . approaches e x i s t i n g i n excess f r e e energy expressions f o r non-i o n i c l i q u i d mixtures, as w e l l as d i f f e r e n c e s i n extensiveness of treatment of the problem. A dashed entry i n Table 1 i n d i c a t e s that the given equation i s not d i r e c t l y concerned w i t h the given f a c e t of s o l u t i o n behavior. One would thus not normally expect an expression f o r which c e r t a i n columns of Table 1 were dashed to conform very c l o s e l y w i t h the e x p e r i -mentally d e r i v e d excess f r e e energy f o r the c l a s s of mixtures i n which the given e f f e c t s were prominent. E q u a l l y , one would not expect to f i n d a p a r t i c u l a r l y high c o r r e l a t i o n between be s t -.Hal a Wohl Hildebrand Wilson NRTL ASOG NCZ Plory-Huggins Barker Guggenheim Expression Column - | mol. 3 o H 3 o 3 o (—1 mol. mon. •uoui i 1 Scale of term. H mol. mol. mol. mol. mol. m >-s o •e pairs i pairs pairs S cale of H E term. ro 3 o 3 o i yes yes 3 O yes i yes yes C l u s t e r i n g i i i 3 o 3 o 3 O yes i 3 o 3 0 z=f(x) ? -p-i i < < none < N <! N N Prime geometry v a r i a b l e . i i 1 ' 1 (0 111 1 1 ' NCZ e f f e c t s . 0-1 i i 1 i i ; + 1 1 7-1 , S( energy cst./RT' <*>rp -] i i yes 3 o 3 O 3 O •yes 1 yes yes Obeys mass law. CD 1 v H i — 1 ' ro ro ro 1 M Energy constants VO 1 i 1 yes 1 yes Hindered rotation. H O 1 i yes yes I yes 3 o 3 O yes Large monomer = polymer? H H H > \-> w > CQ ro 1 Separate S E & H E teims. ro 1 1 yes yes I yes ! 3 o yes 1 • Plory S E only. H VjJ yes 1 i i I i i , i i 1 1 E l e c t r o l y t e s ? M 4=-w o c+ C P H > c r H* O 3 co O i-t> .O P 3" » 3 CL CD M P rt-CD CL H3 3* CD O H 8^2 TABLE 2 Expansion of T a b l e 1 w i t h I n t e r n a l Text R e f e r e n c e s -Cey to C o l . of T a b l e 1 Notes Ref. i n Text 1 1 S c a l e ' d e f i n e d by p a c k i n g u n i t em-p l o y e d : m o l e c u l e , o r monomer u n i t . CH9: 7A(3) and (4) 2- •Cohesion reckoned p e r : m o l e c u l e , monomer u n i t , o r p a i r o f con-t a c t s . CH8:, CH10:5 to 10:9 3 E n e r g e t i c a l l y - i n d u c e d l o c a l c l u s t e r i n g accounted f o r e x p l i c i t l y . Eqns. 102 and 103, CH8:9A, CH9:2B. it Is argument' o f z x - e x p l i c l t ? CH6:9 and 6:10 5 Dominant e f f e c t o f unequal m o l e c u l e s i z e : p a c k i n g ( z ^ e f f e c t s a r i s i n g i n ft , o r d i s p l a c e m e n t - volume (v) e f f e c t s a r i s i n g i n the A ( f : ) p a r t o f 2 . ~ 1 CH10:6, App . 3 , CH6:8 6 C o n s i d e r a t i o n o f c o n f i g u r a t i o n a l and/or e n e r g e t i c e f f e c t s o f NCZ p a c k i n g -CH7:5, term t^' eq/.122 7 Energy constants (normalised by RT) i n c r e a s e ( + ), d e c r e a s e (-1, w i t h T , or a r e of b o t h types ( + , - ) . CH8: C:2 8 L o c a l e n e r g y - c l u s t e r i n g f a c t o r i s same as i n mass-law e x p r e s s i o n . • p.241,21.5 g No. of energy c o n s t a n t s a s s i g n e d p e r u n l i k e s p e c i e s b i n a r y i n t e r -ac t i o n . p.235 ,236 and pp.244-216 10 E x p l i c i t SE term f o r r o t a t i o n a l h i n d r a n c e e f f e c t . CH9:7B 11 Same p r o p e r t i e s a t t r i b u t e d to ' l a r g e monomer as t o polymer c h a i n o f same d i s p l a c e m e n t volume. CH10:5A. C o n t r a s t t h i s toApp . 3 12 H E and T S E c o n t r i b u t i o n s to G E shown s e p a r a t e l y o r not? V a r i o u s p o s s i b i l i t i e s coded:. S = s e o a r a t e terms. A = amalgamated. 1 = H E o n l y . 2 = T S E o n l y . CH10: Eqns (223), (224) More gen-e r a l l y 10 : 3 to 10:8 i n c l u s i v e 13 C o n t a i n s o n l y F l o r y - H u g g i n s form o f S E term. App 3, p.350, CH10:5 A and B l't • E x p r e s s i o n a p p l i e s to electrolyte components in mixtures. App.l 250 f i t values, of constants of the expression i n question to the values o f the same constants obtained from independent thermo-dynamic property, estimates. Apart from the a b i l i t y of. a given e x p r e s s i o n t o produce a " f i t " , a t e s t of the expression's •thermo-dynamic' a p p l i c a b i l i t y i n the given s i t u a t i o n i s a f f o r d e d by the correspondence of ' b e s t - f i t ' constants w i t h those estimated by independent means. In a c t u a l i t y , most s i t u a t i o n s are f a r from c l e a r - c u t : l a c k of experimental data, b a s i c s i m i l a r i t y i n the f u n c t i o n a l form of component curves r e p r e s e n t i n g q u i t e d i s s i m i -l i a r thermodynamic e f f e c t s , formal o b s c u r i t y , the e x i s t e n c e of s e v e r a l compensating e f f e c t s , and the predominence of i n t e r -mediate cases a l l c o n t r i b u t e to the d i f f i c u l t y of d i f f e r e n t -i a t i n g e f f e c t s as c l e a r l y as might be d e s i r e d from the viewpoint of thermodynamic a n a l y s i s . 1 2 . THERMODYNAMIC VALIDITY OP PARAMETERS For example, i n the Wilson equation i f b e s t - f i t values of are obtained from f i t t i n g a system, then the pure component volume r a t i o of the two pure components of the given b i n a r y may be estimated by the expressions * However, c e r t a i n rudimentary t e s t s are o f t e n p o s s i b l e . E E vap_ - vap 1 2 RT ( 2 2 8 ) where g ±r5) = 8J0 ( 2 2 9 ) * Here one takes E as a magnitude (a p o s i t i v e q u a n t i t y ) 251 and g. <TN= E x 2/z 1 £ v a P ± ( 2 9 0 ) where these symbols have been defined e a r l i e r i n t h i s work. From the heats of v a p o r i z a t i o n of the two components, at the given temperature, measurements of Wilson constants A_ and an estimate of z, values of pure component volume r a t i o from the equation can be compared wi t h volume r a t i o r e s u l t i n g from d i r e c t property measurement. This comparison i s shown i n Table 3 along w i t h the (l / 3)and f i r s t power of pure. component r a d i u s r a t i o s from the NCZ equation f o r the correspond-i n g system. A general c o n c l u s i o n to be drawn from the above d i s -c u s sion i s that the r e g i o n of thermodynamic a p p l i c a b i l i t y ( i n d i c a t e d by a c l o s e c o r r e l a t i o n between best f i t and i n -dependently determined c o n s t a n t s ) , should probably be expected, to be somewhat more r e s t r i c t e d than the r e g i o n i n which a f i t i s o b t a i n a b l e by u n r e s t r i c t e d v a r i a t i o n of constants. Thus, two c r i t e r i a f o r the appropriateness of a given expression must be considered: goodness of f i t of the p r e d i c t e d to the ex-•p perimental G /RT curve, and a l s o the thermodynamic v a l i d i t y of the constants used to produce t h i s f i t . The l a t t e r c o n s i d e r -a t i o n is. of p a r t i c u l a r Importance when the data obtained f o r one component i n one systems i s used to p r e d i c t i t s behavior i n another. TABLE 3 Values of some Best-Fit Thermodynamic Constants from Wilson and NCZ Expressions. System No.*** System AT 2 A ^ 2 ( v ^ v ^ T (Wilson) (v 1/v 2^T (Exptl) z s Cals./am.mol. E 1 vap - j— R12 (NCZN T°C 7 n- C l )H 1 0 CP -H2S (2) (.525) (.525) .96 1.30 6 (4000.)** (3500.) 1.55 50°C 8 2-PROP ( l ) -Kg0 (2) (.25) (1.30) .750 1.58 8 9550. 9920. 1.5 80 *C 9 CH^OH (1) -CC14 (2) (.470) (.010) 1.67 1.59 8 8500. 6700. 1.3 30°C v ^ / V g c a l c u l a t e d from equation (288) i n text. Prom (Ref. 7M ( ) = c ontain estimated temperature c o r r e c t i o n s . Numbering i s consistent with Table 4: 1 = the l a r g e r component. * »* *** 253 13. CLASSIFICATION OF G E EXPRESSIONS: WITH RESPECT TO .STANDARD  TYPES OF SOLUTIONS In Table 2j , another method of i n d i c a t i n g regions of a p p l i c a b i l i t y of various expressions for excess free energy i s shown. These: expressions are the Wilson, ASOG, NRTL, Hala*, Barker, Hildebrand S o l u b i l i t y Parameter, Van Laar, Guggenheim, Flory-Hugglns and NCZ expressions. The purpose of Table 4 i s both c l a s s i f i c a t o r y and a n a l y t i c . The two sections of the Table are separated by a heavy black v e r t i c a l l i n e . The two halves of the Table have been set i n juxtaposition to i l l u s t r a t e i t s c l a s s i f i c a t o r y and a n a l y t i c aspects i n r e l a t i o n to the same set of physical systems. The c l a s s i f i c a t o r y section w i l l be dealt with f i r s t . To the l e f t of the heavy black l i n e i s indicated a number of systems, a generic p h y s i c a l and chemical de s c r i p t i o n of these systems,and a t y p i c a l example. Also indicated are those systems for which the NCZ equation i s useful or applicable, as well as other equations which have been found to be useful or applicable. It i s not attempted to define the bounds of the usefulness of the other solution theories mentioned, but simply to indicate those systems for which they are most popular or e a s i l y applied. Also indicated i n a separate column i s the maximum number of components which the theories can accommodate. It i s not unreasonable to comment that some of the solu t i o n theories quoted as> being able to handle only two components have i n the nature of t h e i r formalism no such limitation,,although the p r a c t i c a l d i f f i c u l t i e s of handling more than two com-ponents by the given expression have proved insurmountable. * See Appendix 1 254 While the NCZ expression i s d e l i b e r a t e l y formulated In a form not r e s t r i c t e d to binary mixtures, i t i s true that up to now,only two-component systems have been t e s t e d In d e t a i l against experimental r e s u l t s . The c a l c u l a t i o n program has been run on an n-mer c o n t a i n i n g t e r n a r y system comprising f i v e f u n c t i o n a l group types, and no o p e r a t i o n a l d i f f i c u l t i e s were ex-perienced. The c a l c u l a t e d r e s u l t s have not yet been t e s t e d i n d e t a i l against experiment and so are not quoted here. In Table 4 the procedure f o l l o w e d has been to show the k i n d of systems f o r which various s o l u t i o n s t h e o r i e s can apply but what does not appear, and what i s well-known to anyone versed i n the f i e l d , i s that f o r any one of these t h e o r i e s there i s a con s i d e r a b l e range of the systems quoted f o r which i t i s not a p p l i c a b l e . There may be very r e a l d i f f i c u l t y i n knowing where the boundaries e x i s t between the regions of a p p l i c a b i l i t y and n o n - a p p l i c a b i l i t y of any one system, and t h i s p e r f o r c e en-genders a f e e l i n g of d i s q u i e t when a theory i s t o be used i n an area i n which i t has not been t e s t e d . The s i t u a t i o n i s even more d i s t u r b i n g when working i n a r e g i o n where two t h e o r i e s give c o n f l i c t i n g r e s u l t s . Less s e r i o u s from the point of view of ob-t a i n i n g numbers, but at l e a s t as * d i s q u i e t i n g from a s a t i s f a c t o r y t h e o r e t i c a l p o i n t of view, i s the case when two theories which are fundamentally incompatible give e s s e n t i a l l y the same numerical answers from d i f f e r e n t input parameters. A s i t u a t i o n of t h i s kind i s d i s q u i e t i n g because, towards the boundary of a p p l i c -a b i l i t y of any one theory, i n the absence of any t h e o r e t i c a l background one i s unable t o judge whether unknown f a c t o r s are 255 c o • H •M rt 3 cr w N l C J 0 o •r-> c o OS C i - r — (D ft E -C o • H +J 3 r - l O C/J <+-t o c o rt o • H <-H • H 10 to rt r H U O J - J i 8 1^ o • J ' S a a t- S 5 Iss Hi- o o O Si O O O ; o o O S I o o o o S o O o o 3 « i ^ 1 z$ e a o o o S, o o o o o o o 0 3 * ' t * - -o o o o o S, s, s, S i - Si o ( s, o « fl o o o S. o o o O Si o o je c « 1 ? ? II C K i t o • * o e * • o B * 1 a o B • -o B • w w « o rH G 1 i 1 *> 3 « a c w i»- w p -3 k k ^ E H r l O « *> 3 a e " A •» ** M C M « tc p.(i rt u B * • 3 « i vll o • S B E H r l U « « • • B B _ E?E E H U M O c J " -* IT J » ^r-l y j j-> J~ j> i * * c ^ ti - o JC C « k. » e myf • E t c a & o fl fc tt U CD S, Si Si Si 1^ Si Si S I Si S, a r * . « u o f 3 S S 4J 2 IBs c * • ti m 5 > « 1 t o c g *• 1 1 S c •> fl X m c J o c " i l 1 S £ 3 >. o \l ^ 2 S . a « US s» i o e w r-• D.-0 « C C " • 5 | l o >,c •> Kfl>« « fc- k. v c m o c o • (7 n • « e K f l C « C % M * 3 -« 8 * K s >> * • >• >, O e o c ll a k * V G | -c fl I w e e-ss k ^ c I 3 0 w E a Sv C t) £g •i | 1 fl O • §s c ** St I ? 1 « £ to • ti v « C « 3 « 8 k T> S 3 a: •! i ** * k I S I c i l S " 8 . • i= v. o * « 3 * * ^ i >. k £ U V s a w 3 c M e c a t 65 . 3 C K X * « S £ c tz i.« . *J * c « X 6 k p. O.U 3 -c o c c « o K L 3 o e k o J « ^ 3 " J ""3 S. * " Cr3 3 « M fl 8" < a. i. fc « w « « 3*3 " • - O S t — O D O- k « • w *- ** k « E O « 3 • » ^ — * a " k E « W 3 k fl D £ CJ M 6 ? O C I 01 1 9 m I c t « £ 5 2 ISOI. " u * c o ,fi « a -k » V g - fl c fl O k N ** 2 ** c o o « • O B a. ^  i l l x c -• < E 3 is CM c o * k %! CSV o o U-2 3 C O k (X*1 B. £ *> O O k *i k o t § 2 sc" t t i fill 3 0 C c u * to 1 o it o 9 &^ t i 1 • (\* • k * iZi ait • e <J k o • m K t l c • Ii £5 £ S i K i l « o - Cy * CD O <H a H 256 beginning to obtrude and make the expression i n a p p l i c a b l e . I t i s p r e c i s e l y : t h i s l a c k of knowledge of the causes f o r equation f a i l u r e s which f o r c e s the cautious designer to use equations only w e l l w i t h i n the realms f o r which they have been proved to be u s e f u l . I t i s t h i s atmosphere of i n d e f i n a b l e doubt which p r e -vents s p e c i a l i z e d t h e o r i e s being used p r e d i c t i v e l y , even In those cases f o r which they would, i n f a c t , have given r e l i a b l e . / -I nformation. I t w i l l be w e l l r e a l i z e d that i f a theory has been s t a t e d i n a form which Im p l i e d the i n c l u s i o n of l e s s t h a n : a l l the m a t e r i a l contained i n terms t ^ through t^-'of the NCZ equation, yet the omitted terms were, In the event, zero f o r the system chosen, then a p r e d i c t i o n made u s i n g only terms which were, non.—v-.-zero would s t i l l be v a l i d . A change i n the system might., .„ however, render t h i s s i m p l i f i c a t i o n i n v a l i d because one of the terms which had been omitted might a l s o now prove to be non-zero. C l a s s i c a l theory arose w i t h i n a number of compartments, w i t h i n each of which reasonable success has r e s u l t e d In the study of systems w i t h p a r t i c u l a r p r o p e r t i e s . Conversely, the compartmentalization of c l a s s i c a l s o l u t i o n theory could be r e -garded as a r e s u l t of c o n s i d e r i n g only some of the terms t ^ through t,-' at a time. H i s t o r i c a l l y t h i s p r a c t i c e made; very good sense because of the complexity of the s u b j e c t , but i t now appears p o s s i b l e to decompartmentalize the subject and con-s i d e r a l l seven terms simultaneously, r e c o g n i z i n g of course, th a t f o r many cases some terms i n the NCZ equation may w e l l be n e g l i g i b l y s m a l l . 257 The f a c t t h a t some of the terms of the NCZ. equation can be. reduced i n s p e c i a l cases t o c l a s s i c a l equations i s i n -deed not s u r p r i s i n g s ince the present o v e r a l l equation has been developed as a g e n e r a l i z a t i o n of these s p e c i a l cases. I f , i n the NCZ equation, c o o r d i n a t i o n numbers are made constant some terms vanish and some do not. The non-vanish-i n g terms, a r e , f o r t h i s s p e c i a l case, r e c o g n i z a b l e as the c l a s s i c a l ones. The terms which do v a n i s h are those which, h i t h e r t o , have been concealed by a f o r m u l a t i o n which demands constant c o o r d i n a t i o n number. Even the non-vanishing terms, however, have been d i s t o r t e d i n t h e i r c l a s s i c a l forms by t h i s i m p o s i t i o n and d i f f i c u l t i e s encountered i n d e s c r i b i n g e x p e r i -mental r e s u l t s u n n e c e s s a r i l y engendered. Table 4 sets out, f o r a number of chemical systems, a statement of which terms of the NCZ equation are not n e g l i -g i b l e . The chemical systems l i s t e d i n c l u d e some which have proved amenable to c l a s s i c a l methods but a l s o systems i n t r a c t -able t o such a n a l y s i s are named. Observation of Table shows th a t over a range of system types much greater than t h a t f o r any a l t e r n a t i v e theory, the general NCZ expression h o l d s , and t h e r e f o r e i t i s w i t h some hope of success that the author sets about u s i n g the NCZ equation not now p r i m a r i l y as a device f o r o b t a i n i n g numerical values f o r (G /RT), but as a device f o r c l a s s i f y i n g systems w i t h respect to t h e i r behaviour, as r e -vealed by the f u n c t i o n a l response of the v a r i o u s terms of equation ( 212) t o t h e i r c o n s t i t u e n t independent v a r i a b l e s . 258 14. FUNCTIONAL RESPONSE BEHAVIOR OF THE NCZ: EXPRESSION . I t has now: been seen t h a t the NCZ equation has con-s i d e r a b l e g e n e r a l i t y , so o f f e r s at l e a s t the p o s s i b i l i t y of being c l a s s i f i c a t o r y . To see whether i t could c l a s s i f y e f f e c t i v e -l y , the equation must be shown to have: 1) S e l e c t i v i t y of response to s i n g l e independent v a r i a b l e s ( o r s m a l l groups of t h e s e ) , and 2) C h a r a c t e r i s t i c , or i d e n t i f i a b l e (hence " a n a l y t i c " ) responses of the i n d i v i d u a l terms t , to t._ t o these parameters. 1 5 I t i s now proposed to examine the a n a l y t i c proper-t i e s of the NCZ equation under considerations 1) and 2 ) . An example of a very severe and s p e c i f i c response to terms t ^ and tr- has already been given and d e s c r i b e d i n Chapter 8 • I t i s now proposed to d e l i n e a t e the f u n c t i o n a l response of a l l the terms i n a more comprehensive way, and t o t h i s end Figure 4 i s i n t r o -duced. In F i g u r e 4 are l i s t e d the s i x s i g n i f i c a n t terms of the NCZ equation ( i t w i l l be remembered that t ^ f was mentioned on page . l87, and discarded as n e g l i g i b l e ) . The f i r s t three of these terms correspond to the thermodynamic f u n c t i o n S, used i n the graphs presented as TS t o convert to energy u n i t s , the next two terms to H, and the l a s t to G. At t h i s p o i n t the reason f o r the numbering system of the terms t 1 through t,- appears, whereas l o o k i n g at them as they arose throughout the t e x t , the numbering system of the terms was not apparent. I t w i l l be seen t h a t the parameters to which the i n -d i v i d u a l terms respond s e n s i t i v e l y are few - never more than Term D e s c r i p t i o n Term Thermo- P r i n c i p a l Mode shown Number of Responds dynamic P o s s i b l e a l t e r n a t i v e model(s) NCZ S e n s i t i v e l y Nature Equation to: "5 V N.CZ. Con-f i g u r a t i o n a l term "Flory"term Hindered ro-t a t i o n term Net heat of mixing term Heat of mixing term N.CZ. Free energy term ab w a ' " l a ' " i J ; T 1 i a Rab' Aa ( i ) ( l i ) ( i l l ) . ( i v ) (v) ( v i ) RT ^3: X(LARGER)-^ . ^ H k  - | k r ^ t l ^ - H I ^ H Figure 4. Typical Relationship Between the Terms of the NCZ Expression and the Mole Fraction of the Larger Species. 260 two, and the t o t a l n'umher of parameters f o r a l l terms i s p r e -c i s e l y that of the spanning set on page 2 2 , remembering that x , n and T are the v a r i a b l e s subject to e x t e r n a l manipulation and which are not p r o p e r t i e s of the m a t e r i a l s being examined. The p o t e n t i a l i t y of the NCZ expression as an a n l y t i c a l t o o l stems to no s m a l l extent from the f a c t t h a t , as w i l l be seen i n Fi g u r e 4, terms t ^ , and t,_ respond to molecular geometry and i n the case of t<- to f r e e energy a l s o , but not to s i t e f r a c t i o n (n.) , heat of mixing (to..) or polymer le n g t h (n ) . Terms X X J X t^> t j j 1 and t ^ " on the other hand do not respond e x p l i c i t l y to molecular geometry ( i . e . not to R a b o r IK ) . While terms t^y t ^ ' and t ^ " a l l respond both to heat of mixing and s i t e f r a c t i o n (n 1) , the approximate magnitude of term t ^ i s f r e q u e n t l y estimable a p r i o r i through e x t e r n a l evidence (such as d i f f e r e n c e of p o l a r i t y between corresponding s p e c i e s ) . Terms tjj " and tjj ' are b a s i c a l l y s i m i l a r and simply appear as two separate terms to d i f f e r e n t i a t e between the simple heat of mixing e f f e c t between monofunctional s p e c i e s , and the more complex r e l a t i o n s h i p between m u l t i f u n c t i o n a l ones. Fur t h e r comments on the behavior of these i n d i v i d u a l terms i s probably more conveniently l e f t u n t i l a f t e r the response of terms t ^ to t ^ to composition, as c h a r a c t e r i z e d by the sketches on the r i g h t hand side of Figure 4,have been expanded i n Figure 5 ( a ) t o 5 ( f ) . The conventions of Figures 5(a) to 5(f) must f i r s t E be e x p l a i n e d . The coordinates'are normalized energy (G /RT) p l o t t e d against the mole f r a c t i o n of the l a r g e r molecule, and the R ab 0, * 02, large a b<bc ZT, < J 2 0J0~, 0C\ LARGER (00a) i.o Figure 5(a). Response of NCZ t-Terms to X L A R G E R - T e r r a t i-h- a i / c I T T -\ncnVia • ab < be both quite large 0.0 ^ L A R G E R ( - ^ a ) 1.0 Figure 5(b). Response of NCZ t-Terms to x T e r m t . LARGER 2 POSSIBLE ALTERNATE FORM SMALL NEGATIVE ( j j ] a i a , 7Jia x Via A ).o LARGE NEGATIVE Wjbjb LARGE NEGATIVE <J f aj 0 SMALL POSITIVE O J I a j f l , 7Jia X V]0 moderate ab < be 0 . 0 •^LARGER (3?Q) 1.0 Figure 5(d). Response of NCZ t-Terms to X L A R G E R - T e r m t4 POSITIVE « „ , V i a x V j b t Figure 5 (e). Response of NCZ t-Terms to x A R r F R • Term t. rv) Figure 5 ( f ) , Response of NCZ t-Terms to x T 4 R r F R - Term t 267 examples c i t e d are f o r binary mixtures. In the t r i v i a l case of a Raoult's law s o l u t i o n , a l l the curves are h o r i z o n t a l s t r a i g h t l i n e s ( i . e . of zero amplitude). The F i g u r e s drawn t h e r e f o r e a l l c h a r a c t e r i z e a t t r i b u t e s of mix-t u r e s which a r e , i n one way or another n o n i d e a l . Features of a t y p i c a l curve of the s e r i e s 5(a) t o 5 ( f ) are constructed as f o l l o w s : -•1) The amplitude (s) are marked as J (or and J^)-2) A l l the curves have zeroes f o r the pure component s t a t e s (designated as a. and c ), but some a l s o have one a d d i t i o n a l zero i n the mixture range, (designated as b ). Some of the curves ( t ^ ' f o r example) are of unlmodal c e n t r a l maximum type. The composition corresponding to the maximum i s a l s o l a b e l l e d b . 3) Also shown on a l l the curves of Figures 5(a) to 5 ( f ) , i s the general d i r e c t i o n of movement of the composition of some p o i n t . o f i n t e r e s t i n the mixture range (such as a zero value of G /RT f o r a sigmoid curve, as at b ), or an extremum i n a unlmodal curve, such as a curve maximum (again i n d i c a t e d by b ). Also shown i s the amplitude response t o input parameters. The convention used i s that the p o i n t of i n t e r e s t w i l l move i n the d i r e c t i o n of the arrow shown as the stated parameter i n c r e a s e s . 4) Also i n d i c a t e d are e 1 and e 2, the angles of approach of the (G /RT) curves to the abscissa i n the i n f i n i t e d i l u t i o n r e g i o n s . 5) In some cases, a l t e r n a t i v e forms of the curves being d i s -cussed a r i s e i n s p e c i a l cases, which are described i n con-268 j u n c t i o n with, s p e c i f i c systems, In Chapter 11. Figures 5(a) to 5 ( f ) have i l l u s t r a t e d the a n a l y t i c f e atures of the terms of equation ( 2 1 2 ) . Table 4 demonstrated the g e n e r a l i t y of t h i s equation, and the subsequent Figures demonstrate, i n c o n j u c t i o n w i t h equation (212) and the para -meter groupings on the r i g h t hand s i d e of Table 4, the s p e c i f i c i t y of response of the terms of equation (212) to the v a r i o u s subgroups of independent v a r i a b l e s of Figure 4. I t i s t h e r e f o r e apparent that a t-term c l a s s i f i c a t i o n corresponds t o , but ex-tends and u n i f i e s , the c l a s s i c a l c l a s s i f i c a t i o n scheme provided by standard c a t e g o r i e s of s o l u t i o n s , a s i n c l u d e d i n Table 4 . In the immediately f o l l o w i n g Chapter , a number of r e a l systems w i l l be examined w i t h r e s p e c t . t o c a p a c i t y of the NCZ equation to p r e d i c t r e s u l t s , and on the way opportunity w i l l be taken to di s c u s s and con t r a s t i t w i t h other s o l u t i o n t h e o r i e s . 15. INTERCOMPARISON OF FUNCTIONAL RESPONSES OF VARIOUS EXPRESSIONS I t i s apparent from Figure 4 that terms t^, t ^ , t ^ ' and t j j " are a l l capable of g i v i n g r i s e to what might be termed unlmodal c e n t r a l ( c e n t r a l maximum or minimum) excess f r e e energy component curves, of roughly the same shape, whereas t ^ and t,-component curves are i n d i v i d u a l l y d i s t i n g u i s h a b l e and not even approximately s i m i l a r i n shape to the above unimodal c e n t r a l group, being i n s t e a d ' l e f t w a r d minimum' and 'l e f t w a r d maximum' sigmoid i n shape (with respect to rightward increase In mole f r a c t i o n of the l a r g e r molecular s p e c i e s ) . The t ^ and t,-curves are shown by c a l c u l a t i o n to be not s i n u o s o i d a l , and are only p a r t l y compensating. 269 The a l g e h r a i c form of term t 2 has a modified l o g a r i t h m mic argument, that of t ^ , while: complex, u s u a l l y e x h i b i t s a. ' c e n t r a l maximum' response, and curve t ^ " i s roughly p a r a b o l i c . Nevertheless, the gross Interresemblance of these three types of t versus x curves would a l l o w the f i t t i n g of experi m e n t a l l y de r i v e d excess f r e e energy curves which are of a general ' c e n t r a l maximum' type w i t h q u i t e good success by means of an expression c o n t a i n i n g any one of the above-described f u n c t i o n s provided the excess f r e e energy of the system Is not too l a r g e . Thus, f o r example, the Wilson equation, whose form give r i s e to a modified, l o g a r i t h m i c ' c e n t r a l maximum' shape, achieves q u i t e good success i n f i t t i n g systems e x h i b i t i n g any or a l l of the three NCZ e f f e c t s mentioned, as does that of Hildebrand, Barker, the NRTL equation and the ASOG equation, although thermodynamic i n t e r s p r e t a t i o n of the b e s t - f i t parameters w i l l vary c o n s i d e r a b l y depending on which theory i s being a p p l i e d , A general comment which w i l l prove to be u s e f u l i n the present d i s c u s s i o n should, be introduced at t h i s p o i n t . I t w i l l be seen from the NCZ equation that a t t e n t i o n i s being focussed on what might be c a l l e d a n " 5" l i q u i d r a t h e r than on an "a:" l i q u i d . That i s , the emphasis i s being put on the p r o p o r t i o n of d i s t i n g u i s h a b l e p a i r - s p e c i e s of various kinds e x i s t i n g w i t h i n the o v e r a l l mixture, r a t h e r than simply on the p r o p o r t i o n of molecules of various kinds w i t h i n the same mixture, When entropy effects, are of prime concern the parameter % Is s i n g l y i n v o l v e d ; when i n t e r a c t i o n e f f e c t s are of concern the product 5 a * £ b i s i n e v i t a b l y i n v o l v e d . • • = • 270 Noting that z x n = 5 * 3 (219) 'a b % *b n b and adopting the convention that z >z, , then 5 /£. f o r monomers w i l l always be gr e a t e r than x /x-r. S p e c i f i c a l l y s i n c e the a D product £ x£, w i l l have a maximum where £ =5, t h i s w i l l occur a b a b> 1 where x < x I t w i l l be seen that terms t , ' and t , " c o n t a i n a b 4 4 the product £ x 5. e x p l i c i t l y (see equation (212)) and term t Q contains t h i s product i n a more obscure way, and t h e r e f o r e a l l three of these terms must show skewing when the c o o r d i n a t i o n numbers are not equal. By way of i l l u s t r a t i o n of the j u s t -mentioned skewing e f f e c t , the maximum i n the heat of mixing term ( t ^ " ) , which i n v o l v e s p a i r products (5 x K^)3 i s somewhat s h i f t e d toward compositions d i l u t e i n the l a r g e r component f o r mixtures of unequal s i z e d molecules i n the NCZ treatment. However, i t i s p o s s i b l e to skew central-maximum curves (notably the e q u i v a l e n t of t^") i n v a r i o u s other ways, and t h e r e -by use terms d e r i v e d to represent ' c e n t r a l maximum' e f f e c t s to produce single-maximum curves, the composition of the maximum being i n the r e g i o n where the t ^ curve of the NCZ equation has i t s p o s i t i v e maximum. For example, t h i s e f f e c t i s accomplished, i n the r e l a t i o n of Hildeb r a n d , by us i n g the volume f r a c t i o n product i n the p a r a b o l i c heat of mixing term. In the case of mixtures of s u f f i c i e n t l y l a r g e g l o b u l a r f l u o r c a r b o n s and small g l o b u l a r s o l -vents , maxima i n the heat of mixing f o r m u l a t i o n c h a r a c t e r i z e d i n the S o l u b i l i t y Parameter expression are thereby h i g h l y d i s p l a c e d i n t o the r e g i o n d i l u t e i n the l a r g e r s p e c i e s , where maxima i n the e x p e r i m e n t a l l y observed excess f r e e energies are seen to occur. 271 The problem w i t h using the heat of mixing term to ex-p l a i n excess f r e e energy maxima i n a r e g i o n so f a r removed from the equimolar, emerges i n systems i n which there i s evidence of a s u p e r p o s i t i o n of both a l e f t w a r d maximum, and a r e l a t i v e l y c e n t r a l l y p o s i t i o n e d e f f e c t , both of comparable Importance. While the NCZ a n a l y s i s of both systems 5 and 6 i s based on the ex i s t e n c e of both e f f e c t s , the s i t u a t i o n r e a l l y becomes a n a l y t i c f o r both when the heat of mixing s t a r t s to become l a r g e . Such Is the case i n the system P e n t a e r y t h r i t o l t e t r a p e r f l u o r o -b utyrate c-Octamethyl s i l o x a n e s t u d i e d by Shinoda and Hildebrand . The uncomfortable question then a r i s e s as to how to f i t both e f f e c t s w i t h a s i n g l e term i n such a system. Em-p i r i c a l l y , t h i s I s accomplished i n the case of the NRTL equation^ by c r e a t i n g two p a r t s to the heat of mixing term, the p a r t s of which can now be made to maximize at f a i r l y widely d i f f e r i n g compositions, provided s u f f i c i e n t l y d i f f e r e n t parameters are i n s e r t e d i n t o the two heat of mixing constants appearing i n the expression (see equation ( 2 5 8 ) ) . S o l u b i l i t y Parameter Theory, however, has no such p r o v i s i o n . Where both a ' c e n t r a l maximum' and a f a r l e f t w a r d and/or ri g h t w a r d e f f e c t are e x h i b i t e d i n the excess f r e e energy curve at the same time f o r a system, the p a r t i c u l a r u s e f u l n e s s of the NCZ equation at the p u r e l y e m p i r i c a l l e v e l i s shown. For a system of unequally s i z e d molecules, both ( t ^ + t^) and t ^ " curves e x h i b i t maxima, which are c o n s i d e r a b l y d i s p l a c e d from each other. An example of the l a t t e r behavior encountered i n the systems s t u d i e d i n t h i s work i s n-Butane-Hydrogen s u l f i d e (system 7 ) . 272 In the comparison of f u n c t i o n a l responses of various expressions, the temperature c h a r a c t e r i s t i c s of the va r i o u s types of terms employed need a l s o to be considered. A s i g n i f i c a n t property of the t,_ term of the NCZ ex-p r e s s i o n i s that I t s energy constants are fre e e n ergies, not heats of mixing ( i . e . not enthalpy d i f f e r e n c e s ) . While the f u n c t i o n (H E/RT) diminishes with'T, the f u n c t i o n (A E/RT) i s capable of e i t h e r decreasing or_ i n c r e a s i n g somewhat j and at l e a s t i n general 'attenuates' with increased temperature much more  s l o w l y , i f indeed i t attenuates at a l l . This accounts f o r the b a s i c a l l y d i f f e r e n t temperature c h a r a c t e r i s t i c of the t,_ curve compared w i t h the heat of mixing type of term. : The t,- e f f e c t p e r s i s t s to much higher temperatures than does a heat of mix-i n g e f f e c t . I t i s t h i s behavior which i s thought to e x p l a i n the extremely narrow (and le f t w a r d ) two-phase envelope of the more extremely unequal s i z e d f l u orocarbon-solvent mixture of Hildebrand and coworkers. In another system i n c l u d e d i n t h i s work, the sulfur-benzene system, a l s o f i t t e d by the NCZ ex-p r e s s i o n , i n c r e a s e of the f r e e energy constant of s u l f u r at high temperatures accounts i n part f o r the system's unusual high temperature phase behavior. This increase i s thought to e x p l a i n the reappearance at a high temperature of a two-phase r e g i o n , . with a lower c r i t i c a l s o l u t i o n temperature, i n the composition range d i l u t e i n the l a r g e r component ( i . e . d i l u t e i n Benzene). The advantages and disadvantages of d i s p l a c i n g c e n t r a l maximum curves by various means, the b e n e f i t s and drawbacks of the e x i s t e n c e of at l e a s t three i d e n t i f i a b l y d i f f e r e n t types of ' c e n t r a l maximum' e f f e c t i n s o l u t i o n s , and the v a r i e t i e s of temperature response: exhibited, i n various., f o r m u l a t i o n s f o r mixture e f f e c t s i n s o l u t i o n s are c o n s i d e r a t i o n s which have r e -levance t o most of the t h e o r i e s discussed i n t h i s Chapter. Some f u r t h e r e f f e c t s of l a r g e p a i r s energy i n t e r a c t i o n s on p a i r s -c o n c e n t r a t i o n s t h e o r i e s (that of Barker and the NCZ expression) need to be added to the d i s c u s s i o n before concluding the formal side of the explanations of the behavior of excess f r e e energy expressions'responses to input parameters. The p o s s i b i l i t y of a sigmoid response of t ^ ' (the net heat of mixing term of a m u l t i f u n c t i o n a l s o l u t i o n ) has been i n d i c a t e d i n F i g u r e 5(d) , as has that f o r the t ^ term i n Figure 5(c); Any v e r b a l o b s e r v a t i o n on the behavior of these two terms f o r s o l u t i o n s of groups of widely d i f f e r i n g a f f i n i t y (such as s o l u t i o n s c o n t a i n i n g OH and alkane groups) i s l i k e l y to be an o v e r s i m p l i f i c a t i o n . As i s evident from Chapter 9 , the ex-i s t e n c e of any one unusually l a r g e value of u>. . r e d i s t r i b u t e s the p a i r w i s e associations, of the e n t i r e x.. matrix. Hence e f f e c t s e i t h e r d i r e c t l y dependent on the behavior of i n d i v i d u a l p a i r w i s e c o n c e n t r a t i o n s w i t h composition (as i s the case w i t h t j j 1 ) , or upon a l l of them at once (as i s the case w i t h t ^ j which estimates the entropy decrease due t o the r e o r g a n i z a t i o n of the X . . matrix from the e n e r g e t i c a l l y random i d e a l value c o r r e s -ponding to maximum entropy), may respond i n very complex ways to such energy disturbances i n the s o l u t i o n . In the case of t ^ ' , i f (D.J.J f o r a given i n t e r a c t i o n i s more negative than about -4000 cals/mole of p a i r s at o r d i n a r y temperatures, and i s a l s o much l a r g e r than other a t t r a c t i o n s , a r a t h e r unexpected r e s u l t can 21k occur.. Very energetic p a i r s of i n t e r a c t i n g • g r o u p s present i n a m u l t i f u n c t i o n a l l i q u i d may c l u s t e r more h i g h l y when ' i n e r t d i l u e n t ' is. present than when the l i q u i d i n v o l v e d i s i n the pure component s t a t e . This occurs when the Increase i n mutual o r i e n t a t i o n made p o s s i b l e by the presence of 'Inert d i l u e n t ' molecules more than o f f s e t s the s t r a i g h t f o r w a r d d i l u e n t e f f e c t , so that the p o p u l a t i o n of p r e f e r r e d i n t e r a c t i o n s a c t u a l l y I n d i c a t e d increases r a t h e r t h a n decreases as a r e s u l t of d i l u t i o n . This behavior then gives r i s e to a negative r e g i o n i n that p a i r ' s c o n t r i b u t i o n to t ^ ' , contrary to what one would suppose from the si g n of u>. . alone. F i n a l l y , of course, the d i l u t i o n e f f e c t overcomes that of increased s p e c i f i c i t y of o r i e n t a t i o n , and the component of the t ^ ' term due to tha t p a i r then becomes p o s i t i v e . When var i o u s very s t r o n g (but not e q u a l l y strong) p a i r w i s e i n t e r -a c t i o n s e f f e c t s between m u l t i f u n c t i o n a l molecules e x i s t , the rearrangements of the X.. m a t r i x , hence the s i g n of the various p a i r w i s e c o n t r i b u t i o n s to t ^ ' become d i f f i c u l t to p r e d i c t short of doing c a l c u l a t i o n s . The term t ^ can a l s o become sigmoid, but u s u a l l y under r a t h e r d i f f e r e n t energy c o n d i t i o n s : i n a weakly e n e r g e t i c s o l -u t i o n , l a r g e geometry e f f e c t s and small energy e f f e c t s may domin-ate i n d i f f e r e n t composition ranges and cause a s m a l l amplitude sigmoid curve. In h i g h l y e n e r g e t i c s o l u t i o n s , p a r t i c u l a r l y those e x h i b i t i n g l a r g e d i f f e r e n c e s i n i n t e r a c t i o n energy,t^ i s almost always p o s i t i v e , r e l a t i v e l y l a r g e and unimodal. P a r t i c u l a r l y when the strong i n t e r a c t i o n s are p o l a r , t ^ tends to attenuate more r a p i d l y w i t h temperature than does t ^ ' or t ^ " . 275 A f i n a l comment on the e f f e c t of l a r g e p a i r w i s e energy i n t e r a c t i o n s on the behavior of various terms Involves the presence of very l a r g e p o s i t I've (.'unmixing') p a i r w i s e i n t e r -a c t i o n s . In the case of d i l u t e s o l u t i o n s of very l a r g e mole-cules In small ones between which a l a r g e p o s i t i v e i n t e r a c t i o n energy a l s o e x i s t s , NCZ e f f e c t s are f i r s t severe ( i n very d i l u t e s o l u t i o n ) , then r a p i d l y attenuate In more concentrated s o l u t i o n s , as the l a r g e p o s i t i v e i n t e r a c t i o n energy f o r c e s the species to segregate (even I f no phase break r e s u l t s ) , hence f o r c e s the co-o r d i n a t i o n number r a t i o toward u n i t y . This e f f e c t appears to operate i n the n i c o t i n e water system reported i n t h i s work, and a l s o i n c e r t a i n s o u r - g a s - l i g h t hydrocarbon systems c u r r e n t l y under I n v e s t i g a t i o n (though not in c l u d e d i n the d e t a i l e d r e s u l t s of t h i s work). In d i s c u s s i o n s of the next chapter, the f o r e g o i n g general c o n s i d e r a t i o n s and comparisons w i l l be d e t a i l e d w i t h s p e c i f i c systems, although i t . s h o u l d be by now r e a l i z e d t h a t programming the NCZ equation w i t h a p p r o p r i a t e parameter values does not r e l y on the observers judgement of what, a p r i o r i , should happen i n very complex cases, but simply e x h i b i t s the r e s u l t s i n response to the given i n p u t s . In simple cases, l i k e most of those described above, the consequences are pre-d i c t a b l e i n advance; i n very complex cases, some very unusual behavior may r e s u l t which i s f a r from being obvious. I f the equation Is a n a l y t i c i n simple cases, and f o r m a l l y a p p l i c a b l e i n the complex cases, I t s performance In the l a t t e r cases can be hoped to give i n s i g h t i n t o some of the more s o p h i s t i c a t e d e f f e c t s exhibited by such, solutions . 277 CHAPTER 11  ILLUSTRATION OF THE PREDICTIVE EQUATION NATURE AND METHODS OF COMPARISONS BASIS OF DETAILED GRAPHICS VAPOR-LIQUID SYSTEMS A. n-Butane Hydrogen S u l f i d e B. 2-Propanol Water C. Methanol Carbon T e t r a c h l o r i d e LIQUID-LIQUID SYSTEMS A. Systems w i t h an Upper C r i t i c a l S o l u t i o n s Temperature Only B. The System N i c o t i n e Water C. The SystemsBenzene S u l f u r , Toluene S u l f u r and m-Xylene S u l f u r 278 CHAPTER 11  ILLUSTRATION OF THE PREDICTIVE EQUATION 1. NATURE AND METHODS OF COMPARISONS In t h i s chapter, q u a n t i t a t i v e comparisons i n g r a p h i c a l and numerical form are made between the ex p e r i m e n t a l l y de-r i v e d excess f r e e energy of mixing versus composition curves f o r seven b i n a r y systems and those r e s u l t i n g from various p r e d i c t i v e e xpressions. The systems have been chosen as r e p r e s e n t a t i v e examples of the c l a s s e s of systems discussed i n Chapter 9* and are among those l i s t e d i n column 1 o f Table 4. Table 5 i n d i c a t e s which expressions have been con-s i d e r e d i n conduction w i t h which system. Emphasis i s placed on examining a broad range of r a t h e r h i g h l y nonideal system types r a t h e r than on e x h a u s t i v e l y t e s t i n g any one group. The t o t a l number of systems t e s t e d was not l a r g e , because each one r e -presented a new set of p a r t i c u l a r problems. Not a l l systems are compared by a l l methods, but the comparisons made are f e l t to be i l l u s t r a t i v e . As i s discussed i n Appendix 2, there are a number of ways i n which the values of CG /RT) versus composition express-ions can be compared w i t h the corresponding e x p e r i m e n t a l l y de-r i v e d values. However, f o r reasons given i n that appendix, i t was decided to produce l i q u i d mixture excess f r e e energy TABLE 5 Comparisons Made Between Theories and Experimental Results. Sys-tem no. System* Phass Expt. NCZ NRTL 8(a)** Wilson 8(b) ASOG 8(c) S o l . - Earke 1 8(e) Key 5 PFK-CCl^ L A • V i 1 Y// O O PFB-HCC1, L/L y 1 j 2 i 3 i \ % 1 K \ : ? Good f i t V \ 7 L A / V Poor f i t - - ' >y Not applicable-' p P-PROP-H ?0 L A / Pub % < , m 9 CIT-OH- L A / • • • l i s h e d i % £ 1 I parameters had to be modified to get ppad-fit 2 LD Good f i t expected j but not t r i e d out 3 CD . | Probably i n a p p l i c -able 10 NIC -H^ O L A • 9 11(a) 3Z-S L A • ^ i ! 1 S •A y/A 11(b) TOL-S L A ^ i 1 ! i I 1 1 11(c) ra-XY-S i L A 1 h m • N u m b e r i n g c o n s i s t e n t w i t h T a b l e U, * Keys t o T a b l e s 8(a) t o 8(e). f o r parameter settings. 280 (G /RT) versus composition curves, from vapor l i q u i d e q u i l i b r i u m data f o r comparison with. the. c a l c u l a t e d expressions. For those systems e x h i b i t i n g l i q u i d - l i q u i d e q u i l i b r i a , experimental versus t h e o r e t i c a l l y d e r i v e d consolute envelopes were compared. The numbering system was e s t a b l i s h e d by Table 4 i n Chapter 10. The systems to be discussed i n the chapter are i n -cluded In Table,5-, u s i n g the same numbering p l a n . Systems 1, 2 and 3 which, are not to be discussed i n d e t a i l , are r e p r e s e n t a t i v e of mixtures f o r which the c l a s s i c a l p r e d i c t i o n equations were de-r i v e d , although not a l l of them apply to a l l three c l a s s e s . System 4 i s one of intermediate d i f f i c u l t y , which has r e c e n t l y y i e l d e d t o the work of Wilson, P r a u s n i t z and Renon, Derr and Deal. Phase e q u i l i b r i u m p r o p e r t i e s of System 5 have been p r e d i c t e d by Hildehrand's method, as are those f o r System 6, regarded by Hildebrand as an extreme example of the c l a s s t y p i f i e d by System 5. Systems 7 to 9 have been found by v a r i o u s workers to pre-sent r e a l d i f f i c u l t i e s In p r e d i c t i o n , and no s i n g l e equation i n present use works f o r the whole group. The behavior of System 10 and Systems 11 has not, to the author's knowledge yet been pre-d i c t e d . A l l of the systems, w i t h the exception of Systems 12 and'13, which by d e f i n i t i o n are u l t r a v i r e s , were found to y i e l d q u i t e w e l l t o the equation proposed. (In regard to. the i n -a p p l i c a b i l i t y o f the NCZ equation to f i t systems 12 and 13, the word ' c l a t h r a t e ' i m p l i e s h i g h l y concerted i n t e r a c t i o n s be-tween nearest neighbors, thus completely i n v a l i d a t i n g the i n -dependent p a i r s assumption. The inherent inappropriateness of the NCZ Equation f o r type 13 Clonic) systems w i l l be apparent 281 from the comments made i n Appendix 1 ). In t h i s chapter a c l o s e r look w i l l now be taken at Systems 5 t o l l i n c l u s i v e . I t w i l l be convenient to begin by d i s c u s s i n g systems i n which v a p o r - l i q u i d e q u i l i b r i u m i s i n v e s t i g a t e d . In order t h a t the argument w i l l be as l i t t l e en-cumbered as p o s s i b l e , the numerical data from which the NCZ curves f o r these systems have been c a l c u l a t e d i s given i n Table 6 , r a t h e r than being l i s t e d here. Reference t o other workers' experimental data and p a r t i c u l a r methods of e s t i m a t i n g p h y s i c a l p r o p e r t i e s i s shown i n Table 7 • Tables 8' l i s t the parameters used i n the present work when an e x p r e s s i o n of some other worker had to be shown i n connection w i t h the experiment-a l l y d e r i v e d (G /RT) r e s u l t s . In some cases the parameters given by the e a r l i e r workers had been those most a p p r o p r i a t e t o the c o r r e l a t i o n of many systems and thus d i d not g i v e the best p o s s i b l e f i t f o r a s p e c i f i c system. In order that these a l t e r n a t e expressions might be seen i n t h e i r best l i g h t , such parameters were r e f i t t e d to s u i t the s p e c i f i c system, under ex-amination, and these values are the ones reported i n Table 8 . The same procedure was f o l l o w e d where values of c e r t a i n para-meters d i d not e x i s t , or at l e a s t could not be found i n the l i t e r a t u r e to hand. 2. BASIS OF DETAILED GRAPHICS The b a s i s f o r the d e t a i l e d graphics i s now shown i n the Figures of t h i s Chapter. The numerical c o n t r i b u t i o n of each of terms t ^ through t,- of the NCZ equation to the o v e r a l l excess f r e e energy of the 282 TABLE 6 NCZ PARAMETERS FOR SYSTEMS TESTED VALUES USED IN THE NCZ EQUATION SYS NO COMPONENTS Trc) «« 1. 1. *".) ".1 " i l " l l n , *. N N •MC. JETMOO -OH X,, 5 FOP FII)-CI(2)* CK2) G c c i . i a 40 1.2 1. 200. 1. 1 1 8 6000. 2625. ^4264. 50 1.2 1. 190. 1. 1 1 8 6355 2992 1.33 60 12 I- 180. 1. 1 1 8 6720. 3133. 6 IC,F700CH«)4- FID FIU-AI2) A(2) G CID-CHcijia 20 1.78 1. 100. 1. 1 1 8 3222. 1375. 15000" 4 BSOO* 4 35 1.78 1. 6 5 . 1. 1 1 8 3636. 1464. 1.88 50 1.78 1. 42. 1. 1 1 8 4071. 1559. 7 n-C4H^D- S(2) S-H HI2) H-I 1(1) 1-S B 50 1.65 0.5 -50 05 -40 1. .40 1 1 6 900. 750. 5500 1.30 8 c c y i ) - CIO) H(2)-CI(I) 0(2) 0-H H(2) H-I 1 0-1 112) I-CI O-CI B -CHjOH(2) 30 1.3 1. -900 0.3 -5000 .15 15 055 15 1 1 8 1234. 1783. 11000 4 450Q 4 1.59 9 2-C^,CH(l>- Oil) OID-HII) HID HID - 012) 0(1)- H(2) III) 1-0 0(2) OI21-H12) H(2) I-H ill co MjO BO 1.3 .15 -3500 .12 -3300 .73 20 .65 -2800 .35 20 1 1 8 2100. 200. IIPOO 4 800 4 1.58 ill co eo 1.35 .15 -2750 .12 -3250 .73 20 65 -2800 .35 20 1 1 8 1950. 200. 11000 4 800 4 1.58 ill co 10 C.oNjH^D-H20I2) - Nil) N(l)-H(2) « D + -N 1(1) N-0 *-0 -Q 0(2) 0(2)-H12) HQ N-1 B 25 1.75 0.4 -6000 0.3 -2500 0.3 20 0.7 -2200 0.3 -20 1 1 8 800 100 75|a' 4^a 2.03 60 1 75 0.4 -5300 03 -2200 0.3 20 0.7 -2100 0.3 -20 1 1 8 1200 ISO .100 1.75 0.4 -4100 0.3 -2000 0.3 20 0.7 -2050 0.3 -20 1 1 8 1500 500 140 1.75 0.4 -3700 0.3 -1800 0.3 20 0.7 -1800 0.3 -20 1 1 8 1750 600 175 1.75 0.4 -3200 0.3 -1600 0.3 20 0.7 -1400 0.3 -20 1 1 8 2000 800 208 1.75 0.4 -2800 0.3 - 1300 0.3 20 0.7 -MOO 0.3 -20 1 1 8 2500 BOO 225 1.75 0.4 -2700 0.3 - 1200 0.3 20 0.7 - 950 05 -20 1 1 8 25O0 800 3300 4 2.01 lllo) C,H,(I) -SO <p{\) «S-S SI2) 6 100 1.550 1. •2405 1. 1 r 6 1815 IS32 140 1.550 1. .234 B 1. 1 i 6 1910 1950 1.55 160 1.550 1. .231.6 1. 1 i 6 1965 2035 11.551" 180 1.550 1. •228.6 1. 1 i 6 2020 2132 200 1.550 1. .223.8 1. 1 i 6 2064 2250 220 1550 1. • 223. 1. 1 i 6 2120 2420 250 1.550 1. •218.4 1 . 1 i 6 2200 2880 nib) C7H,(I)-S(2) 4>{» <p-S SI2) 100 1.595 i . •225.4 1. 1 i 6 2110 1832 1.65 140 1595 i. •220.6 1. 1 i 6 2240 1950 (1.66)" 160 1595 i . • 218.2 1. 1 i 6 2300 2035 / * 180 1 595 t. • 216. 1 . '1 i 6 2365 2130 200 1595 i. •213.4 1. 1 i 6 2425 2245 220 1.595 i. •211 . 1. 1 . 6 2482 2395 250 1.595 i. .207.4 1. 1 i 6 2580 2685 lllc) m-C,iy D-SI2) <#II *-S S(2) 100 1 620 i . • 220. 1. 1 i 6 2440 1832 1.72 140 1.620 r. •214.6 1. 1 i 6 2585 1950 ll.76r 160 1.620 I. .212. 1 i 6 2625 2035 1 80 1 620 i . .209.4 1. 1 i 6 2730 2132 Q 200 1 620 1. .207. 1. 1 i 6 2800 2240 220 1.620 I. •204.3 1. 1 i 6 2875 2380 250 1.620 1. •200.4 '• 6 2980 2660 DIRECT ESTIMATE KEY TO TABLE 6' F • FLOURINE 0• OXYGEN Cl • CHLORINE A • UNDIFFERENTIATED TYPE N • NITROGEN f '. AROMATIC H • ACTIVE HYDROGEN I • ALKANE 1)„ • SITE FRACTION i ON SPECIES 0. Ul,, • PAIRWISE INTERACTION ENERGY IN CALS/MOLC OF PAIRS. BRACKETING CONVENTION'- EXAMPLE' SII) MEANS S ON COMPONENT I. UNBRACKETED SITE TYPES ARE EITHER UNAMBIGUOUS OR SAME ON EITHER COMPONENT. * MEASURED AT I2I*C (REF81) ' E MEANS ESTIMATED FROM RELATED COMPOUNDS ( )' ESTIMATED FROM SUMMING ATOMIC VOLUMES Ml) AND (2) IDENTIFY . THE MOLECULES. (I) IS ALWAYS 'THE LARGER TABLE 7 Data References for. Systems Tested System No. System 5 C 7 F l 6 ( l ) - C C l 2 j ( 2 ) 6 (C 3P 7COOCH 2) 4-HCCl 3(2) 7 n - C 1 | H 1 ( ) ( l ) - H 2 0 ( 2 ) 2 - C 3 H 7 0 H ( l ) - H 2 0 ( 2 ) 8 CCl i |(l)-CH 3OH(2) 10 C 1 0 N 2 H l 4 ( l ) - H 2 0 ( 2 ) 11a) C6Hg-S l i b ) Tol-S 11c) mXylene-S Quantity, i n NCZ Equation Exp. Data A CD A (2) 67 68 ( 6 7 ) ( 7 7 ) C 7 3 ) (68) 60 i j 80 55 59 65 61 ( 6 4 ) 63 68 62. (66) (71) 63 (88),82 (88),81,86 (82) 35 75 35 (83) (88),81,86 (82) (88),81,86 t I n d i c a t e s r e l a t e d compound m (1) v (1) n j ( 2 ) v (2) 80 ( 4 8 ) t — 80 - . . . . 55 - 55 - (58) 43 - 57 — 76 35 74 75 72 75 74 (35) 69 70 (35) _ 74 72 — 85 - 72 85 72 85 B.. Type Source J Ref — Consolute 80 — Consolute 55 78 79 56 76 75 — Consolute 69 - Consolute 92,87,' - Consolute 92 - Consolute 92 ro CO 284 TABLE 8(a) NRTL Parameters System No. System AG 2 l/RT* &G1 /RT Ref. n - C 4 H 1 0 ( l ) - ( 0 . 4 ) t ( 0 . 4 ) H 2S ( 2 ) T = 50 . °C * See Equation (258) of t h i s t h e s i s . TABLE 8(b) Wilson Parameters System No. System h^)1 Ref. 7 n - C 4 H 1 0 ( l ) - ( 0 . 6 2 5 ) t (0.625) -1 H 2S(2) 2-PR0P(l)- (0.25) (1.30) 8 H^ O (.2) 9 CH 30H(1)- CO.47Q) (0.010) -C C 1 4C2) See Equation (239A) of t h i s t h e s i s . t own f i t . 285 TABLE 8(c) i ASOG Parameters SYSTEM: 9 2-PR0PC1) - H20C2) ' D. £ D Parameters Assignment of Geom. Factors COMPOUND Group Counts fv, .) v k i Flory-Huggins Counts CH 2 OH (v .) 2-PROP (1) (3)* 1. 4. H 20(2) - 1.4 1. Pa i r - i n t e r a c t i o n Wilson Parameter Array Neighbor Group CH 2 OH 1 2 Central Group CH 2 1 1.0 * * 0.305 OH 2 0.0147 1.0 Bracketed values due to the author. Note that D. § D assign CH 2 group counts to alkane alcohols equal to the number of carbons-i . e . i n D. § D's notation, a 0H/CH2 = 0.0147 = a 2 when " c e n t r a l " group i s c i r c l e d : a 0H/CH2 = .0147 = a 2 a CH2/0H = 0.305 = a x These parameter values are from Figure i v , p. 41 of D. § D's paper (Ref. 33). They are the ' b e s t - f i t ' values for a set of alkane alcohol, ether and water systems. See Fig. 8(c). TABLE 8(c) i i ASOG Parameters 286 B e s t - f i t values, t h i s work, f o r the s p e c i f i c system 2-PR0P- H^ O only. SYSTEM:9 2-PR0P(l) H 20C2) Assignment of Geom. Factors COMPOUND Group Counts ( V i ^ ) Flory-Huggins Counts CH 2 OH Cv .) 2-PROPC1) C3.) 1. (4.5) H 20(2) - 1.4 1. P a i r - i n t e r a c t i o n Wilson Parameter Array Neighbor Groups CH 2 1 OH 2 Central Group CH 2 1.0 (0.001) 1 OH 2 (0.001)* 1.0 * Bracketed values due to the author. 287 TABLE 8(d) S o l u b i l i t y Parameters Syst. No. System (cals/cm 61 3,1/2 (cm 3  V l /mole) V 2 Ref. 5 PFH(l) CC1 4(2) (5.6)?* [ 6 ] * 8.6 226. 97. (80) (80) (15) 6 P P B ( l ) -HCC1 3(2) C7.D [7.7] (9.0) [9.3] 553. 82.6 (55) (24) 10 N I C ( l ) -H 20(2) (18.2) 23.5 151. 18. (46) ( 6 9 ) * [ ] denote l i t e r a t u r e values which d i d not work. values c a l c u l a t e d i n t h i s work TABLE 8(e) Barker Parameters See references (.19, 7 5 ) . 288 s y s t e m i s c a l c u l a t e d f o r a s e r i e s , o f c o m p o s i t i o n s . T h e s e i n d i v i d u a l v a l u e s , t o g e t h e r w i t h t h e i r sum a r e p l o t t e d a g a i n s t c o m p o s i t i o n . T h e r e s u l t i n g sum c u r v e i s t h a t o f t h e e x c e s s f r e e e n e r g y f o r t h e s y s t e m a n d c a n b e u s e d d i r e c t l y t o c o m p u t e E Y v a p o r - l i q u i d e q u i l i b r i a v i a r e l a t i o n s g i v e n b e t w e e n G a n d I . W h e r e t h e c o n t r i b u t i o n o f a n y t e r m o f t h e NCZ e q u a t i o n i s s m a l l a n d n o t o f I n t e r e s t , t h a t t e r m I s n o t p l o t t e d a l t h o u g h i t s c o n -t r i b u t i o n i s a l w a y s i n c l u d e d I n t h e sum c u r v e . I n t h e t h r e e e x a m p l e s r e l a t e d t o v a p o r - l i q u i d e q u i l i b r i u m a b o u t t o b e d i s c u s s e d , t h e d a t a i s p r e s e n t e d i n t h e f o r m o f a c o m p a r i s o n b e t w e e n t h e n e t e x c e s s f r e e e n e r g y a s c a l -c u l a t e d b y t h e NCZ e q u a t i o n a n d t h a t d e r i v e d f r o m e x p e r i m e n t a l v a p o r - l i q u i d e q u i l i b r i u m d a t a . T h e e x p e r i m e n t a l l y d e r i v e d E v a l u e o f G /RT i s b a c k - c a l c u l a t e d b y m e a n s o f t h e e q u a t i o n : GE/RT = [ x . £ n Y j L (292) i w h e r e Y^ ^ h a s f i r s t b e e n o b t a i n e d b y s o l v i n g t h e s i n g l e c o m p o n e n t p h a s e e q u i l i b r i u m r e l a t i o n (TT-P?) Yi *J ir = Y ± * ± P° exp (v. ~ ^ - ) ( 2 9 3 ) w h e r e t h e $± a n d $± a r e f u g a c i t y c o e f f i c i e n t s d e t e r m i n e d a s i n r e f e r e n c e ( 1 0 ) . T h e m e t h o d o f p r e s e n t a t i o n o f t h e s y s t e m s t o b e e x a m i n e d i s e s s e n t i a l l y t h a t p r e s e n t e d f o r t h e p e r f l o u r o - n h e p t a n e c a r -b o n t e t r a c h l o r i d e s y s t e m o n p a g e 162 . T h e i n t e r e s t i n g a n d 289 n u m e r i c a l l y s i g n i f i c a n t curves out of.the group of t ^ through w i l l be d i s p l a y e d . Some terms w i l l not be shown i n -d i v i d u a l l y , but In a l l cases a l l terms were i n c l u d e d i n the . F computation of the f i n a l sum of terms c o n s t i t u t i n g (G /RT). .1 3. VAPOR-LIQUID SYSTEMS •" ' , A. Butane Hydrogen S u l f i d e For t h i s system'these curves.are d i s p l a y e d i n Figures 6. Complicating f a c t o r s i n the a n a l y s i s of t h i s systems were: 1) The r e l a t i v e l y h i g h presssure (ranging from about 0.1 to Q.M of the c r i t i c a l pressure) and 2) The low l i q u i d cohesions, r e s u l t i n g in a rather expanded l i q u i d phase. I n reducing vapor pressure data f o r purposes of ob-F t a i n i n g the e x p e r i m e n t a l l y d e r i v e d (G /RT) curve, a p p r e c i a b l e vapor phase f u g a c i t y c o e f f i c i e n t s are i n v o l v e d . H 2S i s a weakly s e l f - i n t e r a c t i n g molecule which i s s l i g h t l y p o l a r . The s i z e d i f f e r e n c e between the two species i s moderately l a r g e . Under such circumstances s i g n i f i c a n t c o n t r i b u t i o n s to the over-a l l net f r e e energy a r i s e from t ^ , t 2 , ( t ^ ' + t^") and t r of the NCZ equation, (see Figure 6(a)). The value of term t 2 shown r e s u l t s from two assumptions. These are that the H 2S molecule has two types of s i t e s , and that the e f f e c t i v e chain l e n g t h of butane i n i t s most probable c o n f i g u r a t i o n can be con-s i d e r e d as u n i t y . The approximately symmetrical shape of t 2 taken a l s o w i t h the r a t h e r s i m i l a r shape of ( t ^ ' + t ^ " ) makes i t d i f f i c u l t , a n a l y t i c a l l y , t o a p p o r t i o n the c o n t r i b u t i o n of the i n d i v i d u a l curves. Nevertheless, i n t h i s case, t 2 has been i n c l u d e d , i n order to i l l u s t r a t e i t s form i n an a c t u a l 0-0 0-2 0-4 0-6 0-8 1-0 Xx MOLE FRACTION BUTANE Figure 6Ca). n-Butane Hydrogen sulfide. NCZ Component Curves, at 50°C. 291 system. In Figu r e 6(b) the sum curve f o r the'NCZ expression i s p l o t t e d , along w i t h the ( s u i t a b l y reduced) data of Hughes. I t w i l l be seen that the agreement i s good e s p e c i a l l y a l l o w i n g f o r the unusual shape of the curve, but i t must be admitted t h a t t h i s i s one of the. cases where a p r e d i c t i v e method f o r determining the u n l i k e p a i r vapor second v i r i a l co-e f f i c i e n t had to be r e l i e d upon. Because i n the system vapor n o n i d e a l i t y i s of the same order of magnitude as l i q u i d phase E n o n i d e a l i t y , the shape of the exp e r i m e n t a l l y d e r i v e d G /RT versus composition curve i s f a i r l y s e n s i t i v e t o the magnitude of the p r e d i c t e d value of the u n l i k e v i r i a l c o e f f i c e n t . F i g u r e 6(b) shows the c l o s e s t f i t t i n g of the NRTL and Wilson curves as w e l l as the already mentioned NCZ curve and t h a t e x p e r i m e n t a l l y d e r i v e d f o r excess free energy of mixing of the system. I t i s apparent from i n s p e c t i o n of Figure 6(b) that E both Wilson and NRTL expressions tend to produce (G /RT) curves which are too "rounded" i n t h i s case, so th a t a good f i t i n the d i l u t e regions produces a curve which i s too high i n the equimolar r e g i o n , and a good f i t i n the equimolar r e g i o n gives r i s e to a curve which i s too low i n the d i l u t e r e g i o n s . In the case of the Wilson equaton, these problems may have c o n t r i b u t e d to the r a t h e r l a r g e d e v i a t i o n s between the Wilson f i t and the e x p e r i -mental values f o r the Butane - E^S> System, report e d by Robinson and Saxena (^ 9) ^ I t i s evident that the good f i t produced by the NCZ expression a r i s e s from the exist e n c e of the CH \ U J CD 0-2 01 00 50° C NCZ EQUATION WILSON EQUATION NRTL EQUATION O-EXPERIMENTAL RESULTS - HUGHES _L 00 0-2 0-4 0-6 0-8 X, MOLE FRACTION BUTANE Figure 6(b). n-Butane Hydrogen Sulfide. Excess Free Energy - Calculated and Experimental. 29-3 t ^ , and components of the o v e r a l l excess f r e e energy. B. 2-Tropanol-Water This system represents an example of hydrogen bonding. The p a i r w i s e a t t r a c t i o n s are very l a r g e , but to some c o n s i d e r -able degree, are compensating because the a t t r a c t i o n between un-l i k e molecules i s comparable i n magnitude to that between l i k e molecules. There i s , however, a s i g n i f i c a n t r e s i d u a l heat of mixing. The hindered r o t a t i o n term i s l a r g e r than any of the others and r e l a t i v e l y symmetrical. Although the system has a f a i r l y l a r g e r a d i u s r a t i o , the magnitude of p a i r w i s e i n t e r a c t i o n s tends to reduce the s i g n i f i c a n c e of t h i s e f f e c t , sis d i s c u ssed i n Chapter 10. . ' For t h i s case two sets of input parameters have been used to i l l u s t r a t e the changes which can r e s u l t i n the excess f r e e energy from changing the i n t e r a c t i o n energies assumed. The r e s u l t s from both these sets are p l o t t e d i n Figure 7(a) and i n F i g u r e 7(a)' showing i n both cases the curves f o r t ^ , (t-j^ + t,-) and ( t ^ ' + t ^ " ) . A p o i n t of i n t e r e s t i s tha t the ( t ^ + tj_) curve i s i n t h i s case negative f o r a l l compositions, systems whereas, i n some other d i s c u s s e d i t shows a p o s i t i v e p o r t i o n i n mixtures r i c h i n the sm a l l e r molecule. F i g u r e 7(b) shows the experimental r e s u l t s f o r excess f r e e energy and p o i n t s c a l c u l a t e d from the data used i n producing Figure 7(a) and 7 ( a ) ' (T •25 •20 •15 •10 •05 0-0 8 0 ° C EXPERIMENTAL • - N C Z EQUATION A S O G EQUATION (2 DATA SETS) WILSON EQUATION 0 0 •2 -4 -6 -8 X, MOLE FRACTION 2-PROPANOL 10 Figure 7(b). 2-Propanol Water. Excess Free Energy - Calculated and Experimental. ro vo 296 Excess f r e e energy of mixing p r e d i c t i o n s given by the ASOG and Wilson expressions are i n c l u d e d f o r purposes of comparison i n t h i s system i n Figure 7 ( b ) . The best f i t was obtained f o r the ASOG e x p r e s s i o n . though the para-meter s e t t i n g s g i v i n g t h i s f i t were not* the "best f i t " f o r A l k y l p l u s OH group systems report e d by Derr and D e a l ^ 3 ) m The excess f r e e energy curve f o r the Derr and Deal parameters i s shown ("D & D"). An e x c e l l e n t f i t of the p r o p a n o l - r i c h p o r t i o n of the range i s af f o r d e d by the Wilson equation, which , however, r e s u l t s i n excessive values i n the w a t e r - r i c h r e g i o n . Para-meter values f o r the two ASOG curves and the Wilson curve are given i n Tables 8 . R e s u l t s f o r t h i s system i n d i c a t e t h a t the NCZ expression i s able t o p r e d i c t r e s u l t s f o r hydroxyl-containing systems, though perhaps not as w e l l as the ASOG equation, i n t h i s p a r t i c u l a r i n s t a n c e . While Barker's equation would have been of i n t e r e s t i n . t h i s case, f i t t i n g i t i s d i f f i c u l t because of the l a r g e number of disposable parameters. I t i s d i s -cussed w i t h reference t o the next system, which a l s o i s s t r o n g l y p o l a r . C. Methanol-Carbon T e t r a c h l o r i d e The p a r t i c u l a r f e a t u r e of s i g n i f i c a n c e i n t h i s system, and the reason why Barker s t u d i e d i t , i s the extreme e f f e c t of hindrance of r o t a t i o n r e p o r t e d . In t h i s case the primary term i s t0 (the term r e l a t e d to hindrance of r o t a t i o n ) . Terms * f o r reasons discussed at the beginning of t h i s Chapter. 297 Ct'ij' + t^") and Ctj +• t<-) are a l s o p l o t t e d i n Figure 8(a) • F i g u r e 8(b) shows the experimental curve together w i t h the c a l -c u l a t e d p o i n t s . Because of i t s e x c e p t i o n a l shape, the c a l -c u l a t e d p o i n t s f o r the heat of mixing (given by ( t ^ ' + t ^ " ) ) are a l s o given i n F i g u r e 8 ( b ) , compared w i t h experimental data. I t w i l l be noted that i n t h i s case the f i t i s not n o t i c e a b l y b e t t e r than that of Barker. Reference curves s e l e c t e d f o r t h i s system are those of Barker and Wilson, both of which are shown on Figure 8(b) along w i t h the experimental curve and that d e r i v e d from the NCZ e x p r e s s i o n . In t h i s case the Wilson equation i s regarded as g i v i n g marginally the best f i t . However, again, the f i t pro-duced by the NCZ equation i s comparable, i n d i c a t i n g that i t i s capable of use i n systems e x h i b i t i n g extreme hindrance of r o t a t i o n e f f e c t s . Again, parameter values f o r the Wilson and Barker equations are given i n Tables 8 . 4. LIQUID-LIQUID SYSTEM A. Systems w i t h an Upper C r i t i c a l S o l u t i o n Temperature Only R e s u l t s f o r the two systems e x h i b i t i n g t h i s behavior (systems 5 and 6 i n the current numbering scheme) have already been d i s p l a y e d i n Figures 2(e) and 3 fc)of Chapter 8, where t h e i r phase behavior was c i t e d as p r o v i d i n g e s p e c i a l l y s t r o n g evidence f o r the presence of the terms most a n a l y t i c of NCZ e f f e c t s , namely terms t^ and t ^ , along w i t h a s m a l l simple heat of mixing term t j j " . Since both systems y i e l d e d to treatment as nonpolar mono-f u n c t i o n a l monomer b i n a r i e s , the other terms of the general NCZ 1 1 1 1 1 1 1 I 7 MOLE FRACTION f CARBON TETRACHLORIDE Figure 8(a). Methanol Carbon tetrachloride. NCZ Component Curves at 30°Cs X, MOLE FRACTION CARBON TETRACHLORIDE Figure 8(b). Methanol Carbon tetrachloride. Excess Free Energy -Calculated and Experimental. ro vo vo 300 expression introduced subsequent to that p o i n t , namely terms t^> t^> a n c^ 'would e i t h e r be zero ( and t^') or i n -s i g n i f i c a n t Ct^) f o r the two systems. Rapid a t t e n u a t i o n of the simple heat of mixing e f f e c t was i n d i c a t i v e t h a t t h e i r un-l i k e p a i r i n t e r a c t i o n was b a s i c a l l y of " d i s p e r s i o n " type as expected f o r such r e l a t i v e l y c h e mically i n s e r t s p e c i e s . Prom the point of view of comparison of the e x p e r i -mentally measured consolute envelope w i t h that d e r i v e d from excess f r e e energy e x p r e s s i o n s , other than NCZ and S o l u b i l i t y Para-meter 1 the choice i s somewhat l i m i t e d . The Wilson equation i s not to be used i n cases where two l i q u i d phases are i n -volved ^90)^  Since the c o n s t i t u e n t s are both monofunctional molecules the group f r a c t i o n of a given type of group i s the same as i t s volume f r a c t i o n , so that the Wilson term of the ASOG equation reduces t o the Wilson equation. The separate m o l e c u l a r - s c a l e P l o r y term of the ASOG ex p r e s s i o n , i s , i n any case, smoothly unlmodal and approximately symmetrical, hence could not c o n t r i b u t e to the change of curvature of the f r e e energy of mixing curve r e q u i r e d to produce a two phase r e g i o n . The author was unable to f i n d a set of parameters f o r the NRTL equation which could produce such an a s y m e t r i c a l two phase r e g i o n . A l s o , i n view of the monofunctional nature of the sp e c i e s , Barker's e x p r e s s i o n , p a r t i c u l a r l y u s e f u l where a stro n g hindrance of r o t a t i o n e f f e c t r e s u l t s from u n l i k e p o l a r i n t e r -a c t i o n s , could not be expected to be p a r t i c u l a r l y h e l p f u l . In a d d i t i o n t o re a d i n g the for e g o i n g b r i e f comments, the reader may wish at t h i s p o i n t to review the Figures s e r i e s 2 and 3 to r e f r e s h h i s mind on the s a l i e n t aspects of the two 301 systems before p a s s i n g on to the next system. B. The System Nicotine-Water An example f o r system whose l i q u i d - l i q u i d e q u i l i -brium phase behavior i s r e l a t i v e l y complex i s ' t h e system N i c o t i n e -Water. This system has always been one of very c o n s i d e r a b l e i n t e r e s t to p h y s i c a l chemists because I t d i s p l a y s both an upper and a lower c r i t i c a l s o l u t i o n temperature. Prom the viewpoint of the present work, the f a c t o r s which are of importance and which lead to t h i s remarkable s o l u b i l i t y r e l a t i o n s h i p , are an e x t r a o r d i n a r i l y l a r g e r a d i u s r a t i o , the abnormal r e l a t i o n s h i p between f r e e energy and temperature f o r water, the p a r t i a l com-pensation of the l a r g e nitrogen-water b i n d i n g i n t e r a c t i o n by l a r g e H-OH b i n d i n g of the water, and the r a p i d decrease of these ( d i p o l e t y p e ) i n t e r a c t i o n s w i t h temperature. Curves generated by ( t ^ ' + t ^ " ) , t ^ and (t.^ + ^5) NCZ equation are shown f o r a s e r i e s of temperatures i n Figure 9(a) . The heat of mixing given by term ( t ^ ' + t^") i s at f i r s t l a r g e and negative. I t becomes p o s i t i v e and f i n a l l y attenuates as p a i r w i s e i n t e r a c t i o n s d i m i n i s h as temperature i s i n c r e a s e d . The change of s i g n i s p r i m a r i l y due to change i n . s i g n of the term t ^ ' , which i s negative f o r high magnitudes of O-H i n t e r a c t i o n , and p o s i t i v e f o r lower ones. The term t ^ " Is always a negative component and i s mainly due to the s t r o n g i n t e r a c t i o n between the n i t r o g e n atoms of the n i c o t i n e molecule and the hydrogen of water. I t i s t h i s change of s i g n i n the term t ^ 1 which p r i m a r i l y causes the onset of the two phases at about 60°C. Figure 9(a) and 9(b). Nicotine Water. Component and Sum Curves at a Series of Temperatures. 303 Term t ^ i s very l a r g e and p o s i t i v e at low temperature and i n d i c a t e s the extreme hindrance of r o t a t i o n caused by the a t t r a c t i o n between the nitrogens of the n i c o t i n e and the hydrogen of water. The e f f e c t of i n c r e a s i n g temperature i n reducing p o l a r I n t e r a c t i o n energies and i n In t r o d u c i n g randomi-z i n g e f f e c t s , causes t ^ to attenuate r a p i d l y . The maximum of the curve f o r t ^ i s s t r o n g l y towards the l e f t and t h i s determines the c o n s i d e r a b l e l e f t w a r d l o c a t i o n of the consolute envelope at the lower temperature. Examination of the s e r i e s 9(a) w i l l now r e v e a l that at the intermediate temperatures the de-crease of the magnitude of the t ^ curve Is more than o f f s e t by the Increase i n magnitude of the curve f o r ( t ^ ' + t^") (now p o s i t i v e ) , and the two phase r e g i o n i s maintained. At the highest temperatures the only curve of s i g n i f i c a n t magnitude i s that f o r the heat of mixing ( t ^ 1 + t ^ " ) . Were i t not f o r the somewhat sigmoid shape of the ( t ^ + t^) curve, the necessary i n f l e c t i o n i n the sum of a l l these curves, i n c l u d i n g of course that f o r the i d e a l heat of mixing, would e i t h e r not be able t o d i s p l a y the minimum f r e e energy which i s e s s e n t i a l , o r , a l t e r n -a t i v e l y , t h i s minimum would occur much more towards the equimolar composition. The s e r i e s 9(b) shows the o v e r a l l net f r e e energy f o r the s e r i e s of temperatures and i n d i c a t e s the two-phase r e g i o n s . Figure 9(c) d i s p l a y s t h i s data along w i t h the experimental con-s o l u t e curve f o r the system. The choice of reference curves f o r comparison f o r t h i s system i s a l s o somewhat l i m i t e d . The Wilson equation i s r u l e d out because of two l i q u i d phases being present. I t .is d o u b t f u l 30H Figure 9(c). Nicotine Water. Consolute Envelope - Calculated and Experimental. 305 i f a set of NRTL parameters could he found to produce the ob-served degree, of asymmetry or c l o s u r e of the two-phase r e g i o n . In any case, the equivalent of both, an energy-induced l o c a l c l u s t e r i n g e f f e c t i n order to e x h i b i t the equivalent of the sigmoid ( t ^ " ) curve, and the e q u i v a l e n t of a hindered r o t a t i o n ( t ^ ) curve would probably have to be invoked to generate the lower c r i t i c a l s o l u t i o n p o i n t of a c l o s e d two phase envelope. The ASOG equation which does not jnclude e i t h e r f e a t u r e was not t r i e d . The s o l u b i l i t y parameter expression of H i l d e b r a n d , u s i n g the given molecular volumes, the S o l u b i l i t y Parameter f o r water, and a b e s t - f i t value f o r that of n i c o t i n e , provides a curve as shown on F i g u r e 9(c) . (The parameter s e t t i n used are recorded i n Table 8(d)).The S o l u b i l i t y Parameter r e -l a t i o n provides no mechanism f o r a lower c r i t i c a l s o l u t i o n tem-pe r a t u r e ; so, d e s p i t e the a b i l i t y of the expression to be f i t t e d to the c o r r e c t c o n c e n t r a t i o n range f o r the upper c r i t i c a l s o l -u t i o n temperature, i t does not provide a very complete r e -p r e s e n t a t i o n of the a c t u a l behavior of the system i n t h i s case, i n c o n t r a s t to i t s usefulness i n Systems 5 and 6 . The e x p r e s s i o n which most c l o s e l y resembles the NCZ e x p r e s s i o n f o r t h i s system i s that of Barker. Due to i t s l a r g e number of d i s p o s a b l e parameters (when a l l i t s geometric f a c t o r s are v a r i e d as w e l l as i t s energy parameters), the author has not to t h i s p o i n t , attempted to f i t i t t o the system n i c o t i n e -water, but b e l i e v e s that i t might be capable of producing a c l o s e d two phase envelope. Whether i t would be able to p r e d i c t the c o r r e c t c o n c e n t r a t i o n range f o r the upper c r i t i c a l s o l u t i o n temperature i s f a r from c e r t a i n , i n view of the f a c t that t h i s 3 0 6 range is. fixed i n the NCZ f i t by the emergence of a a l i g h t p o s i t i v e l e f t w a r d maximum In the ( t ^ + t^.) curve of the NCZ equation, due t o the temperature i n c r e a s e of (ag/RT) f o r water at high temperature. Consequently t h i s question must be l e f t open f o r the moment, whi l e r e g r e t I s expressed f o r having not had the time so f a r to make an exhaustive attempt t o f i t Barker's expression to Nicotine-Water. Although Barker's expression does not take NCZ e f f e c t s i n t o account, these, i n the case of Nicotine-Water at low temperatures, are almost completely suppressed by the very h i g h p a i r w i s e I n t e r a c t i o n energies. (See values of these parameters In Table 6 ) . C. The Systems Benzene S u l f u r , Toluene S u l f u r and m-Xylene S u l f u r F igures 1 0 ( a ) and 1 0 ( b ) show NCZ equation p r e d i c t i o n and 'experimental' consolute behavior of three Aromatic S u l f u r systems. I n a l l r e s p e c t s but one, the systems behave o p e r a t i o n a l l y as simply as do systems 5 and 6 : a l l three systems are a l s o a n a l y s a b l e as monfunctional monomer b i n a r i e s , i n which the terms of equation ( 1 2 2 ) are the dominant ones. What accounts f o r the s t r i k i n g departure from systems 5 and 6 (namely, the appearance of an upper consolute region) i s the f a i r l y h igh p o s i t i v e temperature r a t e of inc r e a s e of the magnitude of the '' a^' constants i n the term t ^ . This tem-perature c h a r a c t e r i s t i c of the t,- terms i s p r i m a r i l y r e s p o n s i b l e f o r the appearance of the upper d i s j o i n t consolute envelope f o r Benzene-Sulfur shown on Figure 1 0 ( a ^ ) . Also shown are those c a l -c u l a t e d f o r Toluene S u l f u r and m-Xylene S u l f u r f o r comparison purposes. Note t h a t m-Xylene S u l f u r , whose NCZ e f f e c t s are r a t h e r more severe than those of Benzene S u l f u r , does not have a m l s c i b l e 307 BENZENE —TOLUENE" — XYLENE MOLE FRACTION OF HYDROCARBON Figure 10(a). Benzene Sulfur, Toluene Sulfur, m-Xylene Sulfur. Calculated Consolute Envelopes. 308 240 220 200 o o 180 u i CC ZD CC LU 0_ UJ r -160 140 120 BENZENE - TOLUENE - XYLENE 100-0.0 0.2 0.4 0.6 MOLE FRACTION OF HYDROCARBON Figure 10(b). Benzene S u l f u r , Toluene S u l f u r , m-Xylene Sul f u r . Experimental Consolute Envelopes. 309 r e g i o n i n the 160 - 190°C temperature range. (91) ' Other workers have c i t e d as the b a s i s f o r the appearance of the upper consolute r e g i o n i n Benzene S u l f u r the resurgence of a l a r g e endothermic heat of mixing e f f e c t due to the s o l v a t i o n - i n d u c e d opening of Sg s u l f u r r i n g s i n the mixture r e g i o n at high temperatures. The e x p l a n a t i o n o f f e r e d i n t h i s work does not r e q u i r e the p o s t u l a t e , although the oper a t i o n of such an e f f e c t would tend to widen (rightwards) the upper con-s o l u t e r e g i o n w i t h respect to the curve shown corresponding to the NCZ c a l c u l a t i o n . I t i s i n t e r e s t i n g i n t h i s connection that the r i g h t hand limb of the upper consolute r e g i o n (pre-s e n t l y appearing on p.308) appears to be somewhat too l e f t w a r d . N e v e r t h e l e s s , the b a s i c NCZ nature of the phase behaviour i s i n -d i c a t e d by the more l e f t w a r d l o c a t i o n of the l e f t hand side of the upper consolute curve than of the lower, and by the d i s -placement of both envelopes toward the l e f t hand side of the composition r e g i o n . When the Benzene-Sulfur system was s t u d i e d by H i l d e b r a n d , only the lower consolute r e g i o n was reported. However,the exis t e n c e of the upper one had been reported by other (92) workers Except f o r the high-temperature behavior of the Benzene-S u l f u r system, i t s other s i m i l a r i t i e s to systems 5 and 6 are s u f f i c i e n t l y great that comments on the a p p l i c a b i l i t y of ex-pre s s i o n s other than the S o l u b i l i t y Parameter and NCZ ones would l a r g e l y be a d u p l i c a t i o n of what has already been s a i d about systems 5 and 6 i n t h i s r e g a r d , so f u r t h e r comment on the subject 310 Is omitted here. I t Is p o s s i b l e that the phenomenon of an upper d i s j o i n t consolute r e g i o n might w e l l be revealed f o r systems other than Benzene-Sulfur and Toluene-Sulfur as phase measurements were extended to temperatures w e l l above the normal b o i l i n g p o i n t s of the given mixtures i n s u i t a b l e apparatus. The eventual r a p i d i n crease i n the 'a' constants f o r many substances would seem to i n d i c a t e that t h i s behavior might not be as r a r e as i s p r e s e n t l y r e p o r t e d . While f o r no member of the s e r i e s Benzene S u l f u r , Toluene S u l f u r or m-Xylene S u l f u r , p r e d i c t e d and expe r i m e n t a l l y d e r i v e d consolute behavior matches p e r f e c t l y ; s t i l l the a b i l i t y of the NCZ equation to d e p i c t a systemmatic p r o g r e s s i o n of phase r e l a t i o n s f o r members of a homologous s e r i e s Is c l e a r l y demonstrated, as reference t o the parameter values i n Table 6 w i l l i n d i c a t e . Figure 1 0 ( d ) , which d i s p l a y s the energy parameter values f o r the three systems as f u n c t i o n s of temper-ature f u r t h e r bears out the r e g u l a r i t y noted above. The dominant r o l e of r a d i u s r a t i o s i n determining the degree of m i s c i b i l i t y of a given aromatic - s u l f u r b i n ary i s borne out, i n F i g u r e 1 0 ( c ) , where the interchange of r a d i u s r a t i o s i n the m-Xylene S u l f u r and Benzene-Sulfur systems r e s p e c t i v e l y r e s u l t s i n c a l c u l a t e d e q u i l i b r i a more c l o s e l y resembling the system whose r a d i u s r a t i o i s em-ployed r a t h e r than r e t a i n i n g the aspect of the system whose energies are being used. At t h i s p o i n t , i t i s hoped that the reader w i l l r e a l i z e t hat the NCZ equation has been t e s t e d both against d a t a , and the expressions of f i v e other workers, i n d i c a t i n g i t s scope of a p p l i c a b i l i t y and a b i l i t y to f i t a l a r g e range, i f not so f a r a 311 240 220 A I i A 200 o o UJ CC 53 CC UJ Q. s U l r I i I 180 60 40 20 - I 100 A' / XYLENE * ENERGY DATA,-BENZENE RADIUS BENZENE ' ENERGY DATA, " XYLENE RADIUS \ \ \ \ \ \ 0.0 0.2 0.4 0.6 MOLE FRACTION OF - HYDROCARBON 0.8 F i g u r e 1 0 ( c ) . Benzene S u l f u r , m - X y l e n e S u l f u r . C a l c u l a t e d C o n s o l u t e E n v e l o p e s w i t h R a d i u s R a t i o s I n t e r -c h a n g e d . Figure 10(d). Benzene Su l f u r , Toluene Su l f u r , m-Xylene S u l f u r . Energy Parameter P l o t t e d against Temperature. 313 l a r g e number of p h y s i c a l systems. Whether the t e s t s shown have l e g i t i m i z e d the expression and i l l u s t r a t e d i t s usefulness s u f f i c i e n t l y i n the reader's mind, at t h i s p o i n t must be l e f t up to the reader. I t must be remembered th a t i n a t h e o r e t i c a l en-deavor, j u s t o b t a i n i n g the expression to be t e s t e d i n a usable form accounts f o r a l a r g e p o r t i o n of the e f f o r t s i n v e s t e d i n the p r o j e c t . Also the t e s t i n g of extreme systems i s a slow b u s i n e s s , but was f e l t by the author t o be a more u s e f u l route t o f o l l o w than to mass-produce Intermediate cases which are always r a t h e r easy to f i t but correspondingly l e s s c o n v i n c i n g , i f not only the u t i l i t y of the e x p r e s s i o n , but a l s o i t s v a l i d i t y i s i n q u e s t i o n , as i s bound to be the case when a new r e s u l t I s produced. In the way of u t i l i t y , though, i t i s hoped that what has been de-monstrated i s the a b i l i t y of the NCZ equation to operate e f f e c t i v e l y i n the extremely nonideal r e g i o n of l i q u i d - l i q u i d e q u i l i b r i a , an area becoming more important w i t h the present t r e n d towards renewed i n t e r e s t i n l i q u i d e x t r a c t i o n , as w e l l as i n the l e s s extreme r e g i o n of n o n i d e a l i t y c h a r a c t e r i z i n g l i q u i d -vapor e q u i l i b r i u m property c a l c u l a t i o n germane to i n d u s t r i a l d i s t i l a t i o n . 314 CHAPTER 1 2 CONCLUDING REMARKS 1 . JUSTIFICATION OF THE NCZ ASSUMPTION 2. NOTE ON NCZ POLYMER SOLUTIONS 3. JUSTIFICATION OF THE NCZ FREE ENERGY TERM 4. CONSISTENCY OF THE NCZ EXPRESSION WITH THE MASS LAW 5. MULTICOMPONENT SYSTEMS 6. DIRECTIONS OF FUTURE INTEREST 315. • CHAPTER 12 •CONCLUDING' REMARKS 1. JUSTIFICATION OP THE NCZ ASSUMPTION Thi s work has had s e v e r a l main preoccupations but a l l have heen f a c e t s of the major premise that the concept of the p a i r s - l i q u i d of quasichemical mixture theory could be con-s i d e r a b l y extended In I t s range of a p p l i c a b i l i t y by a s u i t a b l e formal i n c o r p o r a t i o n of the e f f e c t s of com p o s i t i o n a l v a r i a t i o n of nearest neighbor c o o r d i n a t i o n number f o r the u s u a l l y - en-countered case of mixtures of unequal s i z e d molecules. Problems inherent i n CZ bi n a r y quasichemical theory and a u x i l i a r y problems i n v o l v e d i n a r r i v i n g at an NCZ m u l t i -component f o r m u l a t i o n f o r G /RT, and means by which these have been s a t i s f a c t o r i l y overcome are summarized i n Table 9. There seemed no r e a l l y inherent reason to abandon the quasichemical approach i n favour of volume f r a c t i o n t h e o r i e s ( s o l u b i l i t y parameter, Wilson and NRTL t h e o r i e s f o r example) i f the formal r e s t r i c t i o n of a constant - z f o r m u l a t i o n could be l i f t e d . By d i n t of s e v e r a l assumptions and cons i d e r a b l e 316 TABLE 9 Summary of Main G e n e r a l i z a t i o n s of CZ Binary Quasichemical Theory-L i m i t a t i o n of CZ Theory Approach to Obtain NCZ M u l t i -Component Formulation 1} Geometric: CZ l a t t i c e Place l a t t i c e 'mechanics* i n 'bed of spheres' con-c e p t u a l framework. (This work). 2) Numerical r no means of c a l c u l a t i n g mixing co-ord Ina tlon-numb er changes Evaluate 'bed of spheres' c o o r d i n a t i o n number. (Adapted from Hogendijk). 3) Formal: l i m i t a t i o n to bi n a r y mixtures Adopt ' p r o b a b i l i t y ' approach ( B a r k e r ) , r a t h e r than 'par-t i t i o n f u n c t i o n ' approach (Guggenheim). a l g e b r a , the author f e e l s that he has managed to do j u s t t h i s , or at l e a s t make a s t a r t at i t , and f e e l s that the success of the NCZ expression i n coping w i t h the wide range of systems such as Butane - H 2S, Perfluoro-n-Heptane-Carbon T e t r a c h l o r i d e , Pen-t a e r y t h r i t o l - t e t r a p e r f l u o r o b u t y r a t e Chloroform, and Xylene-S u l f u r , a l l of which seem to e x h i b i t q u i t e n o t i c a b l e NCZ packing e f f e c t s i n t h e i r excess f r e e energy behavior, j u s t i f i e s the work so f a r undertaken In t h i s d i r e c t i o n . 317 Another necessary preoccupation of the work has been the attempt to demonstrate, i n s o f a r as p o s s i b l e , the formal c o m p a t i b i l i t y of the NCZ f o r m u l a t i o n w i t h that of other q u a s i -chemical t h e o r i e s , notably that of Barker, who l i f t e d the other major formal r e s t r i c t i o n of the o r i g i n a l quasichemical theory of Guggenheim, namely i t s r e s t r i c t i o n to monofunctional b i n a r y systems, a r e s t r i c t i o n imposed simply because a r i g o r o u s mass law approach to l o c a l compositions leads to i n s o l u b l e coupled sets of equations when more than one u n l i k e - p a i r species i s present. The comparable performance of the NCZ expression w i t h respect to that of Barker f o r the system Carbon Tetracho-ride-Methanol I n d i c a t e s that f o r t h i s m u l t i f u n c t i o n a l system (3 group s p e c i e s : C l , CH 2 and OH), which a l s o e x h i b i t s a severe h i n d r a n c e - o f - r o t a t i o n entropy e f f e c t , i s i n d i c a t i v e that the two expressions a r e , at l e a s t i n p r i n c i p l e compatible. The i n c l u s i o n of NCZ e f f e c t s i n Barker's m u l t i f u n c t i o n a l p a i r s c o n c e n t r a t i o n matrix i s the c o n t r i b u t i o n of the author i n t h i s area.. 2. NOTE ON NCZ POLYMER SOLUTIONS While the Flory-Huggins theory gives r i s e to an athermal excess f r e e energy term which i s always negative, the NCZ athermal term i s u s u a l l y negative i n mixtures d i l u t e i n the l a r g e r compound, and p o s i t i v e i n mixtures r i c h i n i t . Constant- z quasichemical t h e o r i e s are ( c o n f i g u r a t i o n a l l y ) compatible w i t h that of Plory-Huggins f o r s o l u t i o n s of polymer chains of e q u a l l y s i z e d monomer u n i t s mixed w i t h monomer solvent 318 molecules of the same monomer s i z e , i n that both use l a t t i c e permutation s t a t i s t i c s t o c a l c u l a t e t h e i r c o n f i g u r a t i o n a l entropy of mixing. The NCZ expression i s a l s o compatible w i t h the Flory-Huggins polymer theory i n t h i s way. In the c l a s s i c a l Flory-Huggins case ( f o r f l e x i b l y - l i n k e d constant-z n-mer c h a i n s ) , there i s a decrease i n the t o t a l permutations of the system due t o the f a c t that l o c a t i n g the t e r m i n a l u n i t of an n-mer molecule r e s t r i c t s the r a d i u s of placement of i t s successive u n i t s to w i t h i n a s p h e r i c a l domain of r a d i u s of n-l a t t i c e s i t e s * , r a t h e r than "anywhere on the l a t t i c e " . However, i n the Flory-Huggins case, t h i s i s always more than compensated f o r by the i n c r e a s e i n the t o t a l number of d i s t i n g u i s h a b l e methods of p o s i t i o n i n g the n u n i t s of the n-mer c h a i n , w i t h r e -gard t o an equal number of monomer u n i t s . Thus by t h i s theory the presence of n-mers i n a mixture always i n c r e a s e s the number of combinations f o r l a t t i c e s i t e occupancy, hence gives r i s e to a p o s i t i v e excess entropy of mixing term. This r e s u l t s i n a negative Flory-Huggins athermal f r e e energy term. I t has been shown that i n NCZ s o l u t i o n s , even those of un-equal s i z e d monomers, athermal f r e e energy of mixing terms may be of e i t h e r s i g n , depending on the composition of the mixture. Term t ^ , i n f a c t , assumes negative value f o r s o l u t i o n s d i l u t e i n the l a r g e r component, and p o s i t i v e ones f o r s o l u t i o n s con-c e n t r a t e d i n the l a r g e r component, r e p r e s e n t i n g an i n c r e a s e i n the number of d i s t i n g u i s h a b l e systems permutations i n s o l u t i o n s * l e s s the p r o p o r t i o n of these s i t e s already f i l l e d by p r e v i o u s l y placed n-mer molecules 319 d i l u t e i n the l a r g e r species i n the r e g i o n of maximum packing d i s t o r t i o n by the l a r g e r s p e c i e s , and a decrease i n d i s t i n g u i s h -able system permutations i n s o l u t i o n s concentrated i n the l a r g e r s p e c i e s . Term t ^ i s th e r e f o r e not a F l o r y - t y p e of athermal term. The e x p l i c i t NCZ n-mer entropy e f f e c t occurs i n term t^. This term was not obtained by d i r e c t g e n e r a l i z a t i o n of F l o r y ' s expression to the NCZ case. Barker had been s u c c e s s f u l i n o b t a i n i n g an expression f o r the hindered r o t a t i o n entropy e f f e c t i n a p o l a r mixture through i n c o r p o r a t i o n of the d e f i n i t i o n of the p r o b a b i l i t y of groups of simultaneous independent events i n t o the Boltzman d e f i n i t i o n of entropy*, using as 'independent p r o b a b i l i t i e s ' the 'independentized' p a i r s p r o b a b i l i t y array from h i s p a i r s p r o b a b i l i t y matrix. The author was thus prompted to i n c o r p o r a t e the d e f i n i t i o n of the dependent pro-b a b i l i t i e s of grouped events i n t o the same d e f i n i t i o n of entropy to o b t a i n the NCZ polymer entropy term appearing i n term tg i n the NCZ expression. Both the term obtained and the Flory-Huggins expression are negative f o r a l l mixtures: but the a l g e b r a i c forms are not i d e n t i c a l although the two f u n c t i o n s behave i n a * The assumption of independent, events, r e q u i s i t e to the use of Boltzman entropy, i s a l l o w a b l e i f the system considered i s that of the p a i r s whose e f f e c t i v e p a i r w i s e concentrations ( p r o b a b i l i t i e s ) have indeed been rendered independent by i t e r a t i o n of the p a i r s p r o b a b i l i t y matrix to convergence. 320 r a t h e r s i m i l a r way. While i t i s not d i f f i c u l t to i n c o r p o r a t e a composition-a l l y dependent c o o r d i n a t i o n number i n t o the Flory-Huggins de-r i v a t i o n (see Appendix 3 ) t h e c o m p a t i b i l i t y of the expression so obtained w i t h the n-mer entropy e f f e c t i n term has not yet been f o r m a l l y shown. This would come about by attempting to demonstrate the formal equivalence of the two expressions. Thus the question of the extension of the NCZ equation i n t o polymer s o l u t i o n s has not yet reached the systems-measurement stage. In the systems s t u d i e d i n d e t a i l i n the t h e s i s , none of the com-pounds are of long enough chain length to i n v o l v e s u f f i c i e n t l y l a r g e values of the term t 2 to e x h i b i t l a r g e c o n f i g u r a t i o n polymer entropy e f f e c t s compared to the magnitude of other e f f e c t s e x h i b i t e d i n the systems chosen. This aspect of the work then must s t i l l be considered an open l i n e of p u r s u i t at t h i s stage. Under the heading of r e l a t e d t h e o r i e s , the com-p a r i s o n of Flory-Huggins and H e i l expression w i t h the proposed extension should be undertaken. 3. JUSTIFICATION OF THE NCZ FREE ENERGY TERM In the matter of the e n e r g e t i c s of p a i r w i s e i n t e r a c t i o n s , the author f e e l s that he has discovered the e x i s t e n c e of an 'NCZ" f r e e energy e f f e c t , as i n c o r p o r a t e d i n the term t ^ of the general expression. The i m p l i c a t i o n s of t h i s term are that the Gibbs f r e e energy of mixing i s not i n s e n s i t i v e to packing changes, though, at l e a s t i n p r i n c i p l e , i t i s I n s e n s i t i v e t o volume changes as long as the mixture i s f r e e to assume i t s minimum-free-energy volume duri n g the mixing process. The second f a c t of i n t e r e s t regarding t h i s term i s t h a t i t pro-321 vldes an ex p l a n a t i o n of the p e r s i s t e n c e of f a r - l e f t w a r d , narrow two-phase envelopes to higher temperature than the usual temperature a t t e n u a t i o n of O /RT) parameters would permit. This i s due to the very d i f f e r e n t temperature c h a r a c t e r -i s t i c expected of the (a/RT) term, when a i s a f r e e energy r a t h e r than a heat of mixing enthalpy d i f f e r e n c e . There are strong i n d i c a t i o n s , provided by systems 5 ) , 6) and 7) that the bas i c NCZ e f f e c t - namely, a d i s t o r t i o n of packing numbers i n mixtures of unequal s i z e d molecules, - does .operate i n s o l u t i o n s , even up to pressures that are ap p r e c i a b l e f r a c t i o n s of the c r i t i c a l pressure. Thus the author f e e l s that one i n n o v a t i o n of h i s t h e s i s , which was to i n c o r p o r a t e a pu r e l y mechanistic p o s t u l a t e of packing behavior analogous to that of a bed of random packed unequal s i z e d spheres, w i t h s u i t -able m o d i f i c a t i o n s f o r e n e r g e t i c l l l y induced c l u s t e r i n g e f f e c t s i n t o the model f o r a r e a l l i q u i d mixure, i s v i n d i c a t e d by ex-perimental r e s u l t s so f a r . No formal proof has been s u p p l i e d f o r the v a l i d i t y of the packing model analogy i n r e a l s o l u t i o n s . 4. CONSISTENCY OF THE NCZ EXPRESSION WITH THE MASS LAW The author wishes to point out that the NCZ expression -as a quasichemical e x p r e s s i o n , remains c o n s i s t e n t w i t h the mass law expression f o r the minimum f r e e energy of the mixture. This statement a p p l i e s s p e c i f i c a l l y to the form of the e n e r g e t i c -a l l y Induced l o c a l c l u s t e r i n g e f f e c t , which i s dependent on u.., r a t h e r than on the ( g ^ ~ &a ) type of energy d i f f e r e n c e s c h a r a c t e r i s t i c of c l u s t e r i n g e f f e c t s of 2 - l i q u i d t h e o r i e s . 5. MULTICOMPONENT. SYSTEMS Though, no t e r n a r y or higher order systems have been run y e t , i t i s f e l t that m u l t i f u n c t i o n a l two-component systems (such as Carbon Tetrachloride-Methanol and Nicotine-Water) have been f i t t e d with, adequate success to i n d i c a t e that the ex-pected property of quasichemical s o l u t i o n t h e o r i e s , namely that of g e n e r a l i z a t i o n to the multicomponent case without the a d d i t i o n of t e r n a r y i n t e r a c t i o n parameters, i s at l e a s t to be reason-a b l y expected to hold f o r the NCZ expression. Because of the e s s e n t i a l reasonableness of t h i s s u p p o s i t i o n , the author has used the l i m i t e d time a v a i l a b l e to i n v e s t i g a t e some examples of a very broad range of b i n a r y system types, r a t h e r than a l a r g e r number of more c l o s e l y r e l a t e d or r e l a t i v e l y w e l l behaved ones (which would have been f a s t e r to a n a l y s e ) , or to i n v e s t i g a t e r e l a t i v e l y w e l l behaved t e r n a r y or higher systems. From the p o i n t of view of g e n e r a l i z a t i o n to the multicomponent case, i t can be s a i d at t h i s p o i n t that there are no formal d i f f i c u l t i e s to be expected i n t h i s regard. On the p u r e l y pragmatic l e v e l , the examination of g o o d n e s s - o f - f i t c r i t e r i a by other methods enumerated i n Appendix 2 needs to be undertaken. A study of s e v e r a l sets of b i n a r i e s i n v o l v i n g common species i s a l s o needed fo l l o w e d by multicomponent s t u d i e s of groups or a l l of these, to t e s t the i n t e r n a l c o m p a t i b i l i t y of NCZ parameters i n a network of r e -l a t e d systems. 6. DIRECTIONS OF FUTURE INTEREST F i n a l l y , a few general d i r e c t i o n s of f u t u r e i n t e r e s t might be b r i e f l y considered. In the preceding chapter, pre-d i c t i o n of phase e q u i l i b r i a has been made using the NCZ equation and the r e s u l t s compared f o r systems f o r which some experimental data were a v a i l a b l e , and others had to be estimated. Further v a l i d a t i o n of the NCZ equation w i l l p a r t l y depend on i t being t e s t e d against a c o n s i d e r a b l y l a r g e r number of systems f o r which more complete sets of the a p p r o p r i a t e pro-p e r t i e s w i l l have to be measured. Some of these measurements may be uncommon; f o r example, data f o r heat of mixing at con-stant volume r a t h e r than at constant pressure i s r e q u i r e d , or f o r the f r e e energies of the pure component l i q u i d s at high temperature. I t i s hoped however, that the success claimed f o r the wide range of system types i n d i c a t e d along w i t h the reasonably acceptable accuracy obtained, w i l l encourage such t e s t i n g . C l a s s i c a l methods most e a s i l y d e a l w i t h l i q u i d - l i q u i d e q u i l i b r i a f o r which the c r i t i c a l s o l u t i o n composition i s but l i t t l e removed from the equimolecular. The NCZ equation w i l l e q u a l l y w e l l deal w i t h t h i s situation but a l s o can account f o r the case of extreme d e v i a t i o n s from equimolecular c o n d i t i o n s towards mixtures d i l u t e In the l a r g e r molecular s p e c i e s . A u s e f u l t e s t of the proposed equation i n a r e g i o n not examined i n t h i s t h e s i s would be to f i n d c o n d i t i o n s which would p r e d i c t c r i t i c a l s o l u t i o n compositions r i c h i n the l a r g e r molecular species ( t h i s would imply a dominance of t ^ i n that r e g i o n , and t h e r e f o r e , systems might be sought i n which t h i s p a r t i c u l a r f e a t u r e was emphasized). T r a d i t i o n a l l y systems of i n t e r e s t have been those centred on compositions lean i n the l a r g e r molecule, 324 but perhaps mixtures r i c h , i n th.e l a r g e r molecule could prove to be e q u a l l y I n t e r e s t i n g . A very severe t e s t might be the p r e d i c t i o n of systems w i t h two two-phase e n v e l o p e s 3 e i t h e r "side by s i d e " ('in two composition ranges at the same temperature, or i n two temperature ranges i n d i f f e r e n t composition ranges. The examples worked out i n t h i s t h e s i s are a l l of b i n a r y systems, and c l e a r l y , r e a l I n t e r e s t l i e s i n m u l t i -component phase e q u i l i b r i a r a t h e r than i n b i n a r y , no matter how non-ideal the behavior of the l a t t e r may be. P r e d i c t i n g the behavior of mixed solvent systems i n l i q u i d e x t r a c t i o n , and e s t i m a t i o n of v a p o r - l i q u i d e q u i l i b r i a f o r multicomponent d i s t i l l a t i o n s are examples of such uses. I t i s i n t h i s f i e l d , now, that the major e f f o r t must be made. However, the time and resources of a graduate student are somewhat l i m i t e d , and at some p o i n t he i s expected to w r i t e up the work and hand i n h i s t h e s i s . In t h i s l i g h t , the author now proposes to do so, although obviously the job i s not f i n i s h e d . He hopes that the p r o j e c t i s not f i n i s h e d , even if the t h e s i s i s , on the grounds that the only l i n e s of research that ever r e a l l y f i n i s h are those which represent dead ends. 325 NOMENCLATURE SUBSCRIPTS AND SUPERSCRIPTS A Subscript. Refers to a s p e c i f i c segment on a given n-mer molecule. a Subscript. S p e c i f i c a l l y designates molecular species a. (b = another molecular species.) When molecular species are multifunctional, double subscript i a designates the i t n chemical functional group on molecular species a. ave Subscript. Denotes an overall average quantity. See for example the d e f i n i t i o n of z r aye c Subscript. Used i n Chapter 5 to designate a generic nearest-neighbor molecule. E Superscript. Denotes an excess thermodynamic property. f Subscript. Stands for 'free'. See for example the d e f i n i t i o n of free volume. The thermodynamic variable Free Energy i s not given t h i s subscript. H Subscript. Stands for " H e i l " . See d e f i n i t i o n of Q.^ . i Subscript. S p e c i f i c a l l y designates the i type of chemical functional group. (j = another species.) On a monofunctional species, however, i therefore also unambiguously designates molecular species. Subscript i i s used t h i s way i n Guggenheim's treatment of monofunctional binaries. The same applies to other standard texts. mix. Subscript. Designates the property so subscripted as being that of a mixture. mixing. Subscript. Denotes the net effect due to mixing. This effect i s the same as the "excess" e f f e c t , except i n the case of free energy. s Superscript. Denotes the saturated state. See the d e f i n i t i o n I I i s of 4. . s Subscript. Designates the s e l f - i n t e r a c t i n g (pure-component) state. See d e f i n i t i o n of z . s v Subscript. Used i n the sense of an operator, indicating that the property so subscripted was the cons