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NMR study of methyl group reorientation and relaxation in clathrate hydrates and their guests Khanzada, Abdul Wahab Khan 1972

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1 1 2 3 0 NMR STUDY OF METHYL GROUP REORIENTATION AND RELAXATION IN CLATHRATE HYDRATES AND THEIR GUESTS. by ABDUL WAHAB KHAN KHANZADA B.Sc. (Hons.), U n i v e r s i t y o f Sind, Pakistan, 1965. M.Sc, U n i v e r s i t y o f Sind, Pakistan, 1966. M.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1970. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Chemistry We accept t h i s t h e s i s as conforming to the requi r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1972. In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis fo r scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Chpmi<; try The University of B r i t i s h Columbia Vancouver 8, Canada Date February 25, 1972. - i i -ABSTRACT The proton magnetic resonance absorption and s p i n - l a t t i c e r e l a x a t i o n measurements have been c a r r i e d out on a number o f methyl groups c o n t a i n i n g guest compounds, and t h e i r c l a t h r a t e deuterates i n order to study the motional behaviour o f methyl groups and the guest molecules. The a c t i v a t i o n energies a s s o c i a t e d with these motional processes are reported. The proton second moment data f o r d i e t h y l amine, and d i e t h y l -amine deuterate over a temperature range o f 77 to 270K i n d i c a t e t h a t i n both these m a t e r i a l s , the only motion i s that o f methyl group r e o r i e n t a t i o n . The r e l a x a t i o n data f o r both compounds e x h i b i t non-exponential behaviour. These data are t h e r e f o r e analysed according to H i l t and Hubbard theory and the a c t i v a t i o n energies of 2.90 ± 0.03 and 2.34 + 0.02 kcal/mole are obtained f o r d i e t h y l -amine and d i e t h y l amine deuterate r e s p e c t i v e l y ; these energies represent the b a r r i e r s hindering methyl group r e o r i e n t a t i o n according to the above theory. The strength o f the d i p o l a r i n t e r -a c t i o n s i n the deuterate as estimated from both the second moment and the maximum i n temperature dependence o f nuclear r e l a x a t i o n o r a t e i s c o n s i s t e n t with a C-H bond length o f 1.13 A. The r e s u l t s o f proton second moment i n acetone reveal that both methyl group r o t a t e around t h e i r t h r e e f o l d axis from 77 t o 180 K. The r e l a x a t i o n data give an a c t i v a t i o n energy o f 1.33 + 0.01 kcal/mole f o r t h i s motion. The acetone molecule i n the deuterate hydrate shows i s o t r o p i c motion i n the 16-hedral c a v i t i e s o f the deuterate from ^212 to 260K. The a c t i v a t i o n energy obtained from the r e l a x a t i o n data i n the temperature range of 77 to 120K f o r acetone deuterate was found to be 0.33 ± 0.01 kcal/mole, which was not assigned. The absorption l i n e measurements of non hydrogen-bonded tert-butylamine deuterate showed a motional behaviour ranging from methyl, t e r t - b u t y l , and i s o t r o p i c r o t a t i o n o f whole molecule i n i t s deuterate c a v i t i e s i n the temperature range of 77 to 272 K. The T-j measurements o f the same compound e x h i b i t e d a broad d i s t r i b u t i o n o f c o r r e l a t i o n times among these three motions. The a c t i v a t i o n energies obtained from T^ measurements i n t h e i r upper l i m i t were 1.7 kcal/mole (methyl r e o r i e n t a t i o n ) , and 2.5 ± -1 kcal/mole ( t e r t - b u t y l group and i s o t r o p i c r o t a t i o n o f t e r t - b u t y l a m i n e ) . In the pure tert-butylamine the second moment data show a d i r e c t t r a n s i t i o n from r i g i d l a t t i c e value (77 K) to a value c o n s i s t e n t with a l l methyl as well as t e r t - b u t y l group r e o r i e n t a t i o n (^150 K). The r e l a x a t i o n measurements showed a smaller d i s t r i b u t i o n of c o r r e l a t i o n times than the deuterate. The a c t i v a t i o n energy a s s o c i a t e d with the methyl group r e o r i e n t a t i o n i n the upper l i m i t was found to be 3.2 ± -1 kcal/mole i n t h i s amine. Isopropylamine, i n i t s pure form showed r i g i d s t r u c t u r e at 77K and motion of both methyl groups at higher temperatures i n the proton absorption l i n e measurements. The a c t i v a t i o n energy f o r t h i s motion - i v -obtained from non-exponential r e l a x a t i o n measurements was 3.50 + 0.07 kcal/mole. The corresponding isopropy1 amine deuterate revealed i n a d d i t i o n to methyl r e o r i e n t a t i o n ( a c t i v a t i o n energy l e s s than 1.7 kcal/mole), a r o t a t i o n around i t s pseudo t h r e e f o l d axis ( a c t i v a t i o n energy l e s s than 1.6 kcal/mole) from absorption and r e l a x a t i o n measurements. Relaxation measurements o f trimethylamine gave an energy b a r r i e r of 5.75 kcal/mole f o r methyl r e o r i e n t a t i o n . The hexagonal deuterate of trimethylamine showed a Lorentzian l i n e shape from 77 to 100 K and non-exponential r e l a x a t i o n i n nearly the same range. T^ data gave an energy b a r r i e r of ^0.7 kcal/mole f o r r e o r i e n t a t i o n around t h r e e f o l d axis and 2.9 + -2 kcal/mole f o r t r a n s l a t i o n a l motion o f trimethylamine i n the deuterate. Some methylene group c o n t a i n i n g deuterates of cyclopropane, p i p e r a z i n e , and hexamethylenetetramine were a l s o s t u d i e d . H 1 resonance spectra of cyclopropane showed i s o t r o p i c r o t a t i o n o f cyclopropane i n i t s type I. (at high temperatures) and type II s t r u c t u r e s . No evidence of motion was found i n the r e s t o f the two deuterates of p i p e r a z i n e , and hexamethylenetetramine. - V -TABLE OF CONTENTS Page Abs t r a c t i i Table o f Contents v L i s t of Tables x i i L i s t o f Figures x i v Acknowledgements x v i i i D e dication x x CHAPTER I INTRODUCTION 1 A. General Nature of C l a t h r a t e Hydrates 1 B. Various Studies on C l a t h r a t e Hydrates 2 C. The Present Studies 8 References (Chapter I) 1° CHAPTER II STRUCTURE OF CLATHRATE HYDRATES 13 A. Introduction 13 B. von Stackelberg's Cubic Hydrates ^ 1. The host s t r u c t u r e ^ 2. . von Stackelberg's type I s t r u c t u r e hydrates ... ]6 3. von Stackelberg's type II s t r u c t u r e hydrates .. 17 C. Alkylamine Hydrates 20 1. t e r t - B u t y l amine hydrate 23 2. Diethylamide hydrate 23 3. iso-Propylamine hydrate 25 - vi -4. Trimethylamine hydrate 25 5. n-Propylamine hydrate 27 References (Chapter II) 29 CHAPTER III NUCLEAR MAGNETIC RESONANCE THEORY 31 A. Introductory Remarks .... 31 B. The Line Shape Function 34 C. Second Moment of Absorption Line Shape 38 1. Second Moment from Absorption Line Shape 38 2. Second Moment and FID Curve 39 3. E f f e c t of Molecular Motion on Second Moment and Line Width 40 4. E f f e c t of T u n n e l l i n g on Second Moment 43 D. S p i n - L a t t i c e Relaxation Time 44 1. C o r r e l a t i o n Function, S p e c t r a l D e n s i t i e s and T-j ... 44 2. S p i n - L a t t i c e Relaxation f o r 2-Spin (1/2) Systems ... 45 3. S p i n - L a t t i c e Relaxation f o r Methyl Groups 46 (a) Exponential Relaxation 46 (b) Non-exponential s p i n - l a t t i c e r e l a x a t i o n 48 4. E f f e c t of T u n n e l l i n g on T ] 49 E. D i s t r i b u t i o n of C o r r e l a t i o n Times and T h e i r E f f e c t on T, and E 53 i a References (Chapter III) 57 - v i i -CHAPTER IV APPARATUS AND METHODS OF MEASUREMENT . 60 A. Continuous Wave (cw) Measurements 60 1. cw Spectrometer 60 2. C a l i b r a t i o n of Spectrometer 61 3. Line Width and Second Moment Measurements 61 4. V a r i a b l e Temperature Assembly 63 B. S p i n - L a t t i c e Relaxation Measurements 64 1. Pulse Spectrometer 64 2. L i n e a r i t y of Receiver 65 3. V a r i a b l e Temperature Assembly 67 4. Measurement of S p i n - L a t t i c e Relaxation Time .... 68 5. C o r r e c t i o n of H^  Inhomogeneity 69 C. The Cold Box , 71 References (Chapter IV) 72 CHAPTER V DIETHYLAMINE AND DIETHYLAMINE CLATHRATE DEUTERATES ...73 A. I n t r o d u c t i o n 73 B. Experimental 74 1. M a t e r i a l s 74 2. Preparation of (C 2H 5) 2ND .... 74 3. Preparation o f Deuterate and Amine Samples 75 4. Spectrometers , 76 C. Results 76 1. Absorption Line A n a l y s i s (cw Measurements) 76 (a) Second Moment C a l c u l a t i o n s 76 - v i i i -(b) Experimental Second Moment Data 8 0 oo 2. S p i n - L a t t i c e Relaxation Measurements (a) A n a l y s i s o f Non-Exponential Relaxation ..... 83 (b) Experimental Relaxation Data D. Discussion 90 References (Chapter V) 92 CHAPTER VI ACETONE AND ACETONE DEUTERATE 93 A. Introduction 9 3 B. Experimental • 94 1. Preparation o f Acetone-Deuterate 94 2. Preparation of Acetone Sample 95 3. Spectrometers and Methods of Measurement 95 C. Results 9 6 1. Absorption L i n e A n a l y s i s 96 2. Relaxation Measurements ... 100 D. D i s c u s s i o n 103 References (Chapter VI) 1 0 6 CHAPTER VII TERTIARY BUTYLAMINE AND TERTIARY BUTYLAMINE DEUTERATE 1 0 7 A. Introdu c t i o n 1 ° 7 B. Experimental 108 1. Mate r i a l 1 0 8 2. Preparation o f (CH 3) 3CND 2 108 - i x -3. Preparation of Deuterate and Amine Samples .. 109 4. Spectrometers and Methods o f Measurements 110 C. Results • I" 1 1 1. Absorption Line A n a l y s i s I l l (a) Second Moment C a l c u l a t i o n s I l l (b) Experimental Results 114 2. T-j Measurements 119 D. Di s c u s s i o n 126 References (Chapter VII) 130 CHAPTER VIII ISOPROPYLAMINE, ISOPROPYLAMINE DEUTERATE, TRIMETHYLAMINE, AND TRIMETHYLAMINE DEUTERATE ..... 131 A. In t r o d u c t i o n 131 B. Experimental 134 1. M a t e r i a l s 134 2. Preparation of (CH 3) 2CHND 2 134 3. Preparation of Amine and Deuterate Samples 134 (a) Isopropylamine and Isopropy1 amine Deuterate ... 134 (b) Trimethylamine and Trimethylamine Deuterate ... 135 4. Spectrometer and Method of Measurements 135 (a) Isopropylamine and Isopropylamine Deuterate ... 135 (b) Trimethylamine and Trimethylamine Deuterate ... 136 C. Results 137 1. Absorption l i n e A n a l y s i s I 3 7 - X -(a) Isopropylamine and Isopropylamine Deuterate ... 137 ( i ) Second Moment C a l c u l a t i o n s 137 ( i i ) Experimental Results 138 (b) Trimethyl amine and Trimethyl amine Deuterate ... 141 ( i ) Second Moment C a l c u l a t i o n s 141 ( i i ) Experimental Results 142 2. Relaxation Measurements 144 (a) Isopropylamine and Isopropylamine Deuterate ... 144 (b) Trimethylamine and Trimethyl amine Deuterate ... 149 D. Discussion 152 1. Isopropylamine and Isopropylamine Deuterate 152 2. Trimethyl amine and Trimethylamine Deuterate 154 References (Chapter VIII) 157 CHAPTER IX SOME OTHER STUDIES, CONCLUSION, AND SUGGESTION FOR FUTURE WORK 158 158 A. Some Other Studies 1. Cyclopropane Hydrate 158 2. P i p e r a z i n e Hydrate 159 3. Hexamethylenetetramine Hydrate 160 B. Conclusion 160 C. Suggestion f o r Future Work 163 References (Chapter IX) 166 - xi -APPENDIX A Computer Programme HILT to c a l c u l a t e R ^ U ) from the tables provided by H i l t and Hubbard 1 6 7 APPENDIX B Computer Programme TIME to c a l c u l a t e R(t) at d i f f e r e n t value of t using H^  c o r r e c t i o n 180 APPENDIX C Computer programme MOMENT to c a l c u l a t e t h e o r e t i c a l second moment 182 - x i i -LIST OF TABLES Table T i t l e Page No. 1.1 Hydration number f o r some s o l u b l e n o n - e l e c t r o l y t e s 6 1.2 Some a c t i v a t i o n parameters f o r c l a t h r a t e hydrates 7 2.1 Some Known Cubic C l a t h r a t e Hydrates 19 2.2 C r y s t a l S t r u c t u r e Data on Some Alkylamine Hydrates 21 2.3 S t r u c t u r a l C h a r a c t e r i s t i c s o f Some Al kyl ami ne Hydrates 2 4 5.1 Intermolecular Second Moment M£ f o r DNH~D20 and DND-D^O 79 5.2 Second Moment f o r Diethyl amine -ND~ Deuterate (DND-D20) using r c _ H = 1.13 A and N=10 • 82 6.1 Proton Second Moment i n Acetone Deuterate q7 7.1 Intramolecular Second Moment M 2 f o r tert-Butylamine 112 7.2 Second Moment Values f o r tert-Butylamine deuterate and t e r t - B u t y l ami ne-ND 2 H 7 8.1 T h e o r e t i c a l Second Moment Values f o r (CH 3) 2CHND 2 and Isopropylamine Deuterate ^ 0 8.2 Second Moment Values f o r Trimethylamine Deuterate 144 - x m -Values of R A V ( t ) at d i f f e r e n t 2 t/T 1 f o r d i f f e r e n t ( W 0 T C ) - x i v -LIST OF FIGURES Figure T i t l e Page No. 2.1 Various polyhedra found i n von Stackelberg's type I and type II s t r u c t u r e c l a t h r a t e hydrates 18 2.2 von Stackelberg's type I s t r u c t u r e hydrate *.• 2.3 von Stackelberg's, type II s t r u c t u r e hydrate showing f a c e - s h a r i n g arrangement of 12- and 16-hedra 18 2.4 17-hedra and 8-hedron found i n t e r t -butyl amine hydrate 26 2.5 D i e t h y l amine molecule i n d i e t h y l amine hydrate 26 2.6 Isopropylamine molecule i n 14- and 16-hedra .... 28 2.7 T r i methyl amine i n i t s water cages 28 3.1 Some H i l t and Hubbard (H-H) curves f o r d i f f e r e n t U T ) 2 50 o c 3.1 Continued 51 4.1 L i n e a r i t y range o f the r e c e i v e r ... 66 5.1 Second moment versus temperature f o r d i e t h y l amine deuterate and d i e t h y l amine-ND 81 i 5-2 Dependence of t Q / T 0 h as p r e d i c t e d by H-H theory 84 - XV -LIST OF FIGURES (continued) Figure T i t l e Page No. 5.3 Temperature dependence of t i n d i e t h y l amine deuterate and d i e t h y l -amine-ND 85 5.4 T h e o r e t i c a l and experimental curves f o r non-exponential r e l a x a t i o n f u n c t i o n f o r 2 d i f f e r e n t U 0 T ) i n d i e t h y l amine deuterate and d i e t h y l amine-ND 87 5.5 P l o t of u ) 0 x c versus r e c i p r o c a l o f the absolute temperature i n d i e t h y l amine deuterate and diethylamine-ND 88 6.1 V a r i a t i o n o f second moment with temperature i n acetone and acetone deuterate 98 6.2 Some proton magentic resonance spectra o f acetone deuterate at d i f f e r e n t temperatures 99 6.3 Temperature dependence of t in acetone and acetone deuterate 101 6.4 t versus r e c i p r o c a l o f the absolute temperature f o r acetone and acetone deuterate 102 7.1 Proton magnetic resonance second moment vs temperature f o r tert-butylamine-ND 2 and tert-butylamine deuterate H 5 - xvi -LIST OF FIGURES (continued) Figure T i t l e P a 9 e N o' 7.2 Proton magentic resonance l i n e width vs temperature f o r t e r t -butyl amine-ND 2 and tert-butylamine deuterate ... n ^ 7.3 Some resonance absorption spectra o f tert-butylamine-ND 2 and tert-butylamine deuterate ... H 8 7.4 Temperature dependence o f s p i n - l a t t i c e r e l a x a t i o n time i n tert-butylamine-ND 2 and tert-butylamine deuterate 120 7.5 V a r i a t i o n o f s p i n - l a t t i c e r e l a x a t i o n time as a f u n c t i o n o f the r e c i p r o c a l o f the absolute temperature i n tert-butylamine-ND 2 and t e r t -butyl amine deuterate 124 8.1 PMR second moment vs temperature i n isopropylamine-ND 2 and isopropylamine deuterate 139 8.2 Line width i n gauss and second moment 2 i n gauss as a f u n c t i o n o f temperature i n t r i m e t h y l amine and trimethylamine deuterate 143 8.3 t vs temperature and ajQT c p l o t t e d against 1000/T (T i n K) f o r i s o p r o p y l ami ne-ND ? 145 - x v i i -LIST OF FIGURES (continued) Figure T i t l e Page No. 8.4 Proton s p i n - l a t t i c e r e l a x a t i o n time, T.| p l o t t e d a g a i n s t temperature i n isopropyl-amine deuterate 147 8.5 Proton s p i n - l a t t i c e r e l a x a t i o n time, T^ 3 p l o t t e d on a log s c a l e a g a i n s t 10 /T i n isopropylamine deuterate 148 8.6 The observed s p i n - l a t t i c e r e l a x a t i o n time, T.| as a f u n c t i o n of the absolute temperature i n trimethylamine and trimethylamine deuterate 150 8.7 Values of s p i n - l a t t i c e r e l a x a t i o n time f o r protons i n trimethylamine and trimethylamine deuterates a g a i n s t 10 3/T ( K - 1 ) 151 - x v i i i -ACKNOWLEDGEMENTS TO P r o f e s s o r C A . McDowell, who as my research s u p e r v i s o r introduced me to the f i e l d s o f broad l i n e and pulsed nmr. I am g r e a t l y indebted to him f o r his guidance, and extension o f his generous research f a c i l i t i e s . His kind help, advices and time to time encouragements are s i n c e r e l y acknowledged. TO Dr. P.S. A l l e n , teacher, f r i e n d , who t r i e d to teach me the mysteries o f pulsed nmr. His f r i e n d l y methods o f teaching, help, and c o l l a b o r a t i o n were a source o f great i n s p i r a t i o n i n the p u r s u i t o f t h i s work. TO P r o f e s s o r J.B. Farmer, f o r his s i n c e r e help i n the c o n s t r u c t i o n of c o l d box, and f o r many other h e l p f u l and rewarding d i s c u s s i o n s . TO P r o f e s s o r J . T r o t t e r i n e x p l a i n i n g some of the massive but b e a u t i f u l c r y s t a l s t r u c t u r e s o f hydrates. TO Pr o f e s s o r R.F. Snider f o r many hours o f extensive and i n f o r m a t i v e discussions. TO Dr. B. Shizgal f o r his f r i e n d l y help i n one o f the computer program, and f o r other useful d i s c u s s i o n s r e l a t i n g to t h i s work. TO P r o f e s s o r B.A. Dunell f o r some his kind help and valua b l e i n s p i r i n g comments. TO the members of e l e c t r o n i c shop f o r t h e i r i d e a l s e r v i c e i n keeping the spectrometers i n operating c o n d i t i o n , and the members of mechanical and glassblowing shop f o r c o n s t r u c t i o n of some of the equipments. - x i x -TO my colleagues Mr. T.T. Ang and Dr. S.E. U l r i c h f o r t h e i r c h e e r f u l and f r i e n d l y cooperation and to the U n i v e r s i t y o f B r i t i s h Columbia f o r f i n a n c i a l a s s i s t a n c e i n the form of teach a s s i s t a n t s h i p s and other funds. - X X -TO Hasina, K h a l i d , and T a r i q - 1 -CHAPTER I INTRODUCTION A. General Nature o f " C l a t h r a t e Hydrates The term ' c l a t h r a t e ' was f i r s t used by Powell [1.1] to d e s c r i b e a group of i n c l u s i o n compounds i n which one molecule termed the 'host' formed a c a g e - l i k e three dimensional s t r u c t u r e e n c l o s i n g another molecule c a l l e d the 'guest'. In the case of c l a t h r a t e hydrates, the host s t r u c t u r e i s that of water and the guest may be any compound which i s v o l a t i l e and has a d e f i n i t e o o s i z e (approximately from 3.8 A to 6.5 A) to f i t i n t o voids formed by the host and which does not react with i t . The m a j o r i t y o f compounds which form c l a t h r a t e hydrates are hydrocarbons, halogens and halogenated hydrocarbons. The f o r c e s which maintain these guest molecules i n voids o f the host s t r u c t u r e , are thought to be of p h y s i c a l r a t h e r than chemical i n nature. C e r t a i n p o l a r compounds such as c y c l i c e t h e r s , ketones, and a l c o h o l s form c l a t h r a t e hydrates; but i t i s not c l e a r whether the guest i n t h i s case i s hydrogen bonded to water l a t t i c e or i f i t i s held by a p h y s i c a l f o r c e . However r e c e n t l y J e f f r e y and co-workers have discovered that there i s a c l a s s of amines and a l c o h o l s which are hydrogen bonded to the host l a t t i c e [ 1 . 2 ] . The present work i s concerned mostly with amine hydrates and t h e i r guests. - 2 -B. Various Studies on C l a t h r a t e Hydrates The e a r l y work was mostly concerned with the composition, p h y s i c a l chemistry and thermodynamic p r o p e r t i e s o f hydrates. The p h y s i c a l chemistry and thermodynamic p r o p e r t i e s of hydrates were reviewed r e c e n t l y by Byk and Fomina [1.3']. The use o f s t a t i s t i c a l mechanics as a p p l i e d to hydrates reviewed by van der Waals and Platteeuw [1.4]. The c r y s t a l l o g r a p h i c aspects o f various hydrates were reviewed by J e f f r e y and McMullan [1.5] and aspects of water s t r u c t u r e i n organic hydrates by J e f f r e y [1.2]. A b r i e f account of various studies on hydrates i s covered i n the books by Hag an [1.6], Mandelcorn [1.7] and Bhatnagar [1.8]. The e f f e c t of various e l e c t r o s t a t i c f i e l d s a c t i n g on guests i n hydrates and t h e i r e f f e c t on d i e l e c t r i c p r o p e r t i e s of hydrates i s discussed i n a review type paper by Davidson [1.9]. The i n t e r e s t i n the c l a t h r a t e hydrates arose when von Stackelberg and co-workers published t h e i r f i r s t s e r i e s o f papers concerning p r e p a r a t i o n , p h y s i c a l chemistry and s t r u c t u r e o f gas hydrates[1.10]'. The work o f Claussen [1.11] helped i n formulating these s t r u c t u r e s . At the same time Pauling and Marsh [1.12] published a d e t a i l x-ray o d i f f r a c t i o n study o f c h l o r i n e hydrate. Thus h i g h l y symmetrical 12 A o (von Stackelberg's type I s t r u c t u r e ) and 17 A (von Stackelberg's type II s t r u c t u r e ) were e s t a b l i s h e d . S t a t i s t i c a l mechanical c a l c u l a t i o n s were c a r r i e d out by van der Waals and Platteeuw £1.12] and by Barrer and his co-workers [1.13]. van der Waals and Platteeuw e s t a b l i s h e d that the c l a t h r a t e hydrates - 3 -are s o l i d s o l u t i o n s o f the gas(es) i n a metastable host l a t t i c e [1.4]. These authors a f t e r making the f o l l o w i n g assumptions: 1) The c o n t r i b u t i o n o f molecules to the f r e e energy i s independent o f mode of occupation of the c a v i t i e s . 2) C a v i t i e s can never hold more than one s o l u t e molecule. 3) The mutual i n t e r a c t i o n o f s o l u t e molecules i s neglected. 4) C l a s s i c a l s t a t i s t i c s i s v a l i d , c a l c u l a t e d the d i s s o c i a t i o n pressure of hydrates at various temperatures, t h e i r heat of formation and composition of c o - e x i s t i n g e q u i l i b r i u m phases using Lennard-Jones and Devonshire (L-J-D) 12:6 p o t e n t i a l assuming that p o t e n t i a l energy of s o l u t e molecule at some d i s t a n c e i s d e s c r i b e d by s p h e r i c a l l y symmetric (L-J-D) p o t e n t i a l and that the s o l u t e molecules r o t a t e f r e e l y i n the c a v i t i e s . Kobayashi and co-workers [1.14] a f t e r some m o d i f i c a t i o n a p p l i e d van der Waals and Platteeuw theory [1.4] to methane, argon and nitro g e n and mixed methane-argon and argon-nitrogen hydrates. They found good agreement between t h e o r e t i c a l and experimental d i s s o c i a t i o n pressure as a f u n c t i o n of temperature f o r CH 4 > Ar, (CH 4 + Ar) hydrates, but some d e v i a t i o n from theory was observed f o r Nitrogen, (Ar + N,,) hydrates a t higher pressures. At the same time McKoy and Sinanoglu [1.15] showed that (L-J-D) 12:6 p o t e n t i a l was good only f o r monoatomic gases and methane but gave poor r e s u l t s f o r d i s s o c i a t i o n pressure o f rod l i k e molecules such as C 2H g, C 0 2 > N 2 and C 2H 4. T h e i r c o n c l u s i o n was that the Kihara p o t e n t i a l which took i n t o account the form and s i z e o f i n t e r a c t i n g molecules, gave b e t t e r r e s u l t s f o r rod - 4 -l i k e molecules. An i n t e r e s t i n g a p p l i c a t i o n of thermodynamics to c a l c u l a t e the formulae of hydrates from an aqueous s o l u t i o n f r e e z i n g curve of s o l u b l e n o n - e l e c t r o l y t e has been given by Glew [1.16]. Glew's equation i n s l i g h t l y modified form f o r the r e a c t i o n M + nH 20 j M-nHgO (1.1) where M i s s o l u b l e n o n - e l e c t r o l y t e (e.g. some amine or c y c l i c e t h e r ) , n i s the number of water molecules, i s .nO-xJ = -n*nx w -"^1 -"'•^1 •w R T R - [^n y M + runyw] (1.2) where x . Y are mole f r a c t i o n , a c t i v i t y c o e f f i c i e n t of water and w w i s the a c t i v i t y c o e f f i c i e n t f o r M . A H ° , A S ° J T are the standard enthalphy o f c r y s t a l l i z a t i o n , the standard entropy change and the temperature at which the e q u i l i b r i u m i s e s t a b l i s h e d . Equation (1.2) i s used with a s e r i e s o f f r e e z i n g p o i n t determinations f o r d i f f e r e n t mole f r a c t i o n s of the systems. Supposing that i n t h i s s e r i e s a fr e e z i n g temperature T . corresponds to water a c t i v i t y x .y . and M j wj wj a c t i v i t y O-x^jK,^ (because x w + x M = 1), while the other adjacent f r e e z i n g temperature T ^ corresponds to water a c t i v i t y x ^ y ^ and M a c t i v i t y (T — x w k ^ Y M k * W e g e t t w 0 e c ) u a t l 0 n s f r o m Equation (1.2) and d i f f e r e n c e of these two equations leads to - 5 -A j k mO-V/A^O/T) = - n ( A j k £ n x w / A j k ( l / T ) •(AH°/R) - [ ( A j k £ n Y M + n A j k £ r i Y w ) / A j k ( l / T ) ] (1.3) where A ^ n O - x J = A n [ ( 1 ) / ( l - x y / k ) ] , V n X w = ^ V W ' V " Y M = * n ( V W e t c . and A j k ( l / T ) = (1/T..) - (1 / T k ) The Gibbs-Duhem c o n d i t i o n d^nyM + [ x w / ( l - x w ) ] d ^ n y w = o makes the a c t i v i t y c o e f f i c i e n t d i f f e r e n c e small i n the l a s t bracket o f Equation (1.3), so i t can be neglected. By c o n s t r u c t i n g a s e r i e s o f equations s i m i l a r to Equation (1.3), n can be determined e i t h e r by s o l v i n g simultaneously or by a graphical p l o t o f A..£n(l-x )/A.. (1/T) J K W J K versus A j. kfi,n(x W)/A j. k(l/T) which r e s u l t s i n a s t r a i g h t l i n e with slope equal to n. Once n i s found, i t i s s u b s t i t u t e d i n Equation (1.2) which i n rearranged form f o r T. becomes 3 * n ( l - x w j ) + n*n x w j + [*nY M j + n i n ^ ] t ! ! . J _ AS1 (1.4) R -Tj R - 6 -from which A H° and AS° are determined by the method o f l e a s t squares. Equation (1.4) a f t e r n e g l e c t i n g the terms i n v o l v i n g a c t i v i t y c o e f f i c i e n t s reduces to * n ( l - x . ) + nan x . = (A/T.) + B (1.5) WJ WJ J which can be used as a c o r r e l a t i n g equation. Some r e s u l t s obtained by Glew using Equation (1.5) are summarized i n Table 1.1 Table l . T Hydration number f o r Some Soluble N6n-E1ectrolytes Hydrate Former n/ c a l c . n -A B (CH 3) 3N 10.22+0.34 10.0 2314 6.8526 (C 2 H 5 ) 2 N H 6.80+0.39 6.66 2519 8.1595 8.12±0.45 8.10 2780 9.0870 *C 2H 40 6.58 ± .48 6.67 * D;N. Glew, Nature,201 (1964) 922. n / c a l c . i s hydration no. c a l c u l a t e d , n found by experiment. Of the other s t u d i e s , the ones worthy of mention are those of Davidson and co-workers [1.17] and o f Davies and Williams [1.18] on d i e l e c t r i c p r o p e r t i e s of hydrates. The work o f Davidson and co-workers [1.17] p r e d i c t e d two d i e l e c t r i c d i s p e r s i o n regions f o r hydrates and the a c t i v a t i o n energies reported i n the m a j o r i t y of - 7 -the cases arose from the d i p o l e o r i e n t a t i o n o f the water molecules. Davies and Williams [1.18] f o r the f i r s t time measured the guest molecule d i s p e r s i o n i n hydrates. Some of the r e s u l t s obtained by Davies and Williams are given i n Table 1.2. Table 1.2 Some a c t i v a t i o n parameters f o r c l a t h r a t e hydrates AH AS Host Guest Temp.(°K) (kcal/mole) e.u. (sec.) H 20 Tetrahydrofuran 88 0.27+0.05 -2.3+0.3 l l x l O " 1 2 H 20 acetone 93 <0.25 -1.4 4 . 3 x l 0 " 1 2 H 20 Ethylene oxide 88 0.46+0.10 -3.2+1.0 24x10" 1 2 T In Table 1.2 AH and AS are the a c t i v a t i o n enthalpy and the a c t i v a t i o n entropy, f o r r e o r i e n t a t i o n of guest molecule as c a l c u l a t e d from the Eyri n g r a t e equation T = T q exp(.AH/RT) expfAS/R) (1.6) where T i s c o r r e l a t i o n time f o r the motion. I t i s c l e a r from the Table 1.2 that the guests which give good examples of von S t a c k l e b e r g 1 s type Methylene oxide) and type II (acetone and tetrahydrofuran) hydrates are r o t a t i n g i n s i d e the c a v i t i e s with a c o r r e l a t i o n time of the -12 order of 10 seconds. In a recent a r t i c l e Davidson [1.9] has reviewed - 8 -the e f f e c t o f e l e c t r o s t a t i c f i e l d s o f the water molecules on the r e o r i e n t a t i o n rates o f p o l a r guest molecules. He concluded that o f the d i p o l a r f i e l d due to the water molecules vanishes at the centre o f cages and th a t the quadrupolar f i e l d becomes very small because of the hydrogen bonding o f water molecules. A d i s t r i b u t i o n o f r e l a x a t i o n times i s expected because of the r e o r i e n t a t i o n o f the water molecules. There has been reported i n f r a - r e d s p e c t r o s c o p i c measurements on some of the hydrates. Harvey et al [1.19] reported that the motion of SC"2 i n SG^-hydrate was r e s t r i c t e d . In a recent work Hardin [1.20] pointed out the d i f f i c u l t y encountered i n p r e p a r a t i v e work on SO,,-hydrate f o r i r study, and drew the c o n c l u s i o n that f o r some halo-methane and halogen hydrates the motion o f guest was not f r e e . Evidence f o r hydrogen bonding between guest and host was observed i n a very recent i r study by Falk [1.21] i n trimethylamine hydrate. C. The Present Studies The use o f Nuclear Magnetic Resonance to study the molecular motion of guest molecules i n c l a t h r a t e s was f i r s t under-taken by G i l son and McDowell and those s t u d i e s were made on the motion o f guest molecules i n urea-hydrocarbon [1.22], t h i o u r e a -hydrocarbon adducts [1.23] and some Hofmann-type c l a t h r a t e s (e.g. Ni(NH 3) 2Ni(CN) 4'2M where M i s benzene, thiophene, p y r i d i n e etc.) [1.24]. Studies on Hofmann-type c l a t h r a t e s using E.P.R. and N.M.R. were continued by other workers [1.25]. Some E.S.R. and N.M.R. r e s u l t s have been reported on hydroquinone c l a t h r a t e s [1.26-1.27]. The work - 9 -on the study of motion of guest molecules i n c l a t h r a t e hydrates by nmr was s t a r t e d by McDowell and Raghunathan and i n a s e r i e s o f papers [1.28-1.31] ;von Stackelberg's type I, type II and two t e t r a -alkylammonium-salt hydrates were st u d i e d . These s t u d i e s were l i m i t e d mostly t o l i n e width and second moment measurements because of equipment l i m i t a t i o n s . In the mean time some nmr stud i e s on hydrates appeared by Davidson and co-workers [1.33-1.35], Afanas'ev et al [1.36] and El e y e t al [1.37]. There have been very few r e l a x a t i o n s t u d i e s on c l a t h r a t e s except the one on HF 2~ ion i n [A g 7 0 g ] + H F 2 ~ c l a t h r a t e by Hindermann et al [1.38] and a detailed study by B e l l and Richards [1.39] on urea-hydrocarbon adducts. The present work i s concerned with the hydrates of s o l u b l e n o n - e l e c t r o l y t e s , a l l of which contain methyl groups. In a d d i t i o n to absorption l i n e measurements, the r e l a x a t i o n measurements are a l s o presented. The hydrates s t u d i e d have t h e i r host deuterated to make the c o n t r i b u t i o n of water protons n e g l i g i b l e . The study o f guests i s al s o presented, so that a comparison between the behaviour o f guest molecule i n c l a t h r a t e d and unclathrated s t a t e can be made. The maj o r i t y o f the r e s u l t s are on amines. A l l of these have the amine molecule hydrogen bonded to water cage (except f o r t e r t - B u t y l amine hydrate). The other r e s u l t s are on acetone and acetone deuterate. The c r y s t a l structures of a l l of the hydrates are known from X-ray d i f f r a c t i o n measurements. - 10 -References (Chapter One) [1.1] H.M. Powell, J . Chem. S o c , (1948) 61. [1.2] G.A. J e f f r e y , Accounts Chem. Res., 2 (1969) 344. [1.3] S. Sh. Byk and V.I. Fomina, Russian Chem. Rev. [Eng. T r a n s l a t i o n ] , 37 (1968) 469. [1.4] J.H. van der Waals and J.C. Platteeuw, Adv. Chem. Phys., 2 (1959) 1. [1.5] G.A. J e f f r e y and R.K. McMullan, Progr., Inorg., Chem., 8 (1967) 43. [1.6] S.M. Hagan, C l a t h r a t e I n c l u s i o n Compounds, Reinhold N.Y., 1962. [1.7] L. Mandelcorn, Ed. Non S t o i c h i o m e t r i c Compounds, Academic Press, N.Y., 1964. [1.8] V.M. Bhatnagar, C l a t h r a t e Compounds, Chemical P u b l i s h i n g Co., N.Y., 1970. [1.9] D.W. Davidson, Can. J . Chem., 49 (1971) 1224. [1.10] M. von Stackelberg, Naturwiss, 36 (1949) 327, 359. and H.R. M u l l e r , J . Chem. Phys., 19 (1951) 1319. [1.11] W.F. Claussen, J . Chem. Phys., 19 (1951) 259, 662, 1425. [1.12] L. Pauling and R.E. Marsh, Proc. N a t l . Acad. S c i . U.S., 38 (1952) 112. [1.13] R.M. Barrer and W.I. S t u a r t , Proc. Roy. Soc. (London), A242 (1957) 172. and D.J. Ruzicka, Trans. Faraday S o c , 58 (1962) 2239, 2253, 2262. and A.V.J. Edge, Proc. Roy. S o c (London), A300 (1967) 1. - 11 -[1.14] S. S a i t o , D.R. Marshall and R. Kobayashi, A.11Ch.E. J o u r n a l , 10, (1964) 734. and R. Kobayashi, i b i d , 11 (1964) 96. [1.15] V. McKoy and 0. Sinano^lu, J . Chem. Phys., 38 (1963) 2496. [1.16] D.N. Glew, Trans. Faraday S o c , 61 (1965) 30. [1.17] Ref. [1.9] and references t h e r e i n . [1.18] M. Davies and K. Williams, Trans. Faraday Soc. 64 (1968) 529. [1.19] K.B. Harvey, F.R. McCourt and H.F. S h u r v e l l , Can. J . Chem., 42 (1964) 960. [1.20] A.H. Hardin, Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1970. [1.21] M. Fa l k , Can. J . Chem., 49 (1971) 1137. [1.22] D.F.R. G i l s o n and C A . McDowell, Nature, 183 (1959) 1183. [1.23] D.F.R. G i l s o n arid C A . McDowell, Mol. Phys., 4 (1961 ) 125. [1.24] D.F.R. G i l s o n , Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1962. [1.25] T. Miyamoto, T. Iwamoto and Y. Sasaki, J . Mol. S p e c , 35 (1970) 244 and references t h e r e i n . [1.26] H. Ohigashi and Y. K u r i t a , J . Mag. Res., I (1969) 464. [1.27] J.P. McTague, J . Chem. Phys.,5_0' (1969) 47. [1.28] C A . McDowell and P. Raghunathan, Mol. Phys., 13 (1967) 331 . [1.29] C A . McDowell and P. Raghunathan, i b i d , 15 (1968) 259. [1.30] C A . McDowell and P. Raghunathan, J . Mol. S t r u c t u r e , 2 (1968) 359. [1.31] C A . McDowell and P. Raghunathan, i b i d , 5 (1970) 433. [1.32] C A . McDowell and P. Raghunathan, Molecular Dynamics and Struc t u r e o f S o l i d s , N.B.S. (U.S.A.) Special P u b l i c a t i o n No: 301 (1969) 571. - 12 -[1.33] S. Brownstein, D.W. Davidson and D. F i a t , J . Chem. Phys., 46 (1967) 1454. [1.34] Y.A. Majid, S.K. Garg and D.W. Davidson, Can. J . Chem., 46 (1968) 1683. [1.35] Y.A. Majid, S.K. Garg and D.W. Davidson, Can. J . Chem., 47 (1969) 4697. [1.36] B.L. Afanas'ev, V.I. K r l i v i d z e and G.G. Malenkov, Doklady Akademii Nauk SSSR (Physical Chemistry) [Eng. T r a n s . ] , 183 (1968) 816. [1.37] D.D. Eley, M.J. Hey, K.F. Chew and W. Derbyshire, Chem. Comm., 23 (1968) 1474. [1.38] D.K. Hindermann, M.B. Robin and N.A. Kuebler, J . Mag. Res., 1 (1969) 479. [1.39] J.D. B e l l and R.E. Richards, Trans. Faraday S o c , 65 (1969) 2529. - 13 -CHAPTER II STRUCTURE OF CLATHRATE HYDRATES A. Introduction In t h i s chapter the s t r u c t u r e of c l a t h r a t e hydrates based on X-ray d i f f r a c t i o n s t u d i e s w i l l be d e s c r i b e d . The e a r l y work of von Stackelberg and co-workers [2.1], Claussen [2.2], and Pauling and Marsh [2.3] had e s t a b l i s h e d two symmetric cubic type I and type II hydrates. Only three hydrates belonging to von Stackelberg's type I and type II s t r u c t u r e s have been s t u d i e d i n d e t a i l , out o f which two belong to von Stackelberg's type I s t r u c t u r e s i . e . c h l o r i n e [2.3] and ethylene oxide hydrate [2.4] and the other be-longs to von Stackelberg's type II s t r u c t u r e i . e . hydrogen s u l p h i d e -terahydrofuran hydrate [2.5], For the other guests, detailed analyses have not been reported and the c l a s s i f i c a t i o n i s u s u a l l y based on the u n i t c e l l dimensions i . e . i f the u n i t c e l l i s cubic with a - 12A, i t belongs to von Stackelberg's s t r u c t u r e I, and i f a - 17A, von Stackelberg's s t r u c t u r e II i s assumed. In a l l these s t u d i e s of von Stackelberg's type I and type II s t r u c t u r e s , no evidence f o r chemical bonding between the guest and host water l a t t i c e i s assumed. There may be hydrogen bonding i n the type I - 14 -and type II s t r u c t u r e s f o r some ketones, a l c o h o l s and amines. Recently J e f f r e y and co-workers [2.6] have shown that there are some guests l i k e pinacol and some amines which form hydrogen bonds with water molecules i n the l a t t i c e , but with the exception of two or three amines, a l l e x h i b i t d i f f e r e n t s t r u c t u r e s from von S t a c k e l -berg's type I and type I I . We w i l l give a s l i g h t l y d e t a i l e d account of s t r u c t u r e of these amines together with a b r i e f d e s c r i p -t i o n of von Stackelberg's type I and type II s t r u c t u r e s . The des-c r i p t i o n o f per-alkylammonium s a l t hydrates where the host s t r u c t u r e i s formed by water and anions, and the guest i s a c a t i o n w i l l not be given as they were not s t u d i e d i n our work. A good account o f the X-ray d i f f r a c t i o n s t u d i e s on various c l a t h r a t e hydrates can be found i n two recent reviews by J e f f r e y and McMullan [2.7] and J e f f r e y [2.6]. B. von Stackelberg's cUbic hydrates Before a d i s c u s s i o n o f these hydrates, a b r i e f account of the host s t r u c t u r e s i s necessary. X-ray s t u d i e s have only given the d e s c r i p t i o n of the host s t r u c t u r e . These s t u d i e s have only i n d i c a t e d the motion of some guest molecules. 1. The host s t r u c t u r e In a l l of these hydrates, the 'host' s t r u c t u r e i s a poly-hedral framework of hydrogen-bonded water molecules. The s m a l l e s t polyhedron found i n these frameworks i s pentagonal dodecahedron 1 p (12-hedron,5 ),(D). D has 12 f a c e s , 20 v e r t i c e s and 30 edges and can - 15 -be described by Euler's theorem on convex polyhedra (F+V = E+2) i . e . 12F+20V = 30E+2 where F, V, E denote f a c e , vertex and edge of D r e s p e c t i v e l y . The v e r t i c e s are formed by oxygen atoms o f water molecules. Since there are 20 v e r t i c e s , 20 H 20 molecules are r e -quired g i v i n g 40 H-atoms f o r H-bonding. 30 o f these H-atoms are o used i n 0 - H — 0 (-2.8 A) bonds f o r 30 edges o f D. To make oxygen atoms coordinate roughly t e t r a h e d r a l l y 20 more 0 - H — 0 bonds are needed, and these are provided e x t e r n a l l y . The volume of D i s approx-°3 t 0 imately 170 A with a f r e e diameter o f 5.1A [2.7] so that i t can enclose an atom or molecule l i k e Ar, Kr, Xe, H 2S and CH^ but not a molecule l i k e C l 2 , S0 2, e t c . The pentagonal dodecahedron, D, alone can't f i l l the space, other polyhedra are needed to give r i s e to homogeneous s p a c e - f i l l i n g arrangement with long-range p e r i o d i c i t y . The other polyhedra are given as below: (a) Tetrakaidecahedron(14-hedron,5 .6 ) , ( T ) . This i s formed by 24 H 20 and has 12 pentagonal and 2 hexagonal faces (14F+24V = 36E+2) o o and has a volume of 216 A with a mean f r e e dimension o f -5.3 and o •^ 6.4 A [2.7]. I t can enclose molecules l i k e C l 9 , S0 o, e t c . By the word f r e e diameter we mean the diameter o f D or any other polyhedra a f t e r s u b t r a c t i n g the non-bonded diameter o f the oxygen atom which i s -2.8 A. 12 2 * This n o t a t i o n mn r e f e r s to n m-sided faces e.g. 5 .6 means 12 pentagonal and 2 hexagonal faces and i s due to Wells [ 2 . 8 ] . - 16 -(b) Hexakaidecahedron(16-hedron,5 .6 ),(H). This i s formed by 28 H 20 with 12 pentagonal and 4 hexagonal faces (16F+28V = 42E+2). °3 It has a volume of 250 A and i s nearly s p h e r i c a l i n shape with a o f r e e diameter o f -6.6 A [2.7]. I t can enclose l a r g e r molecules l i k e SFg, acetone, e t c . A l l o f these polyhedra are shown i n Figure 2.1. 2. von Stackelberg's type I s t r u c t u r e hydrates These hydrates are formed by 46 water molecules per u n i t c e l l . The space group f o r these i s Pm3n ( 0 h ) with u n i t c e l l edge o a = 12 A. The u n i t c e l l has two 12-hedra ( D ) and s i x 14-hedra ( T ) . (Figure 2.2) The centres of D are at / Q 0 Q ^ 1_ 1_ 1_\ and those o f T are a t / l n 1 3 n 1 n 1 1 n 1 3 1 1 n \ 3 n\. The hydrate-forming guest which has mean f r e e diameter o « 5.0 A w i l l form type I hydrateswith formula 8X-46 H 20 (X-5.75 H 20) where X i s a guest molecule e.g. Ar, CH^, H 2S, e t c . However i f the O o mean f r e e diameter o f the guest i s > 5.1 A and < 5.8 A, then only the l a r g e c a v i t i e s T are f i l l e d , and the two smaller c a v i t i e s D remain vacant g i v i n g r i s e to a hydrate with the formula 6Y*46H 20 (Y*7.67H 20), where Y can be C l 2 , S0 2, C 2Hg, e t c . Another i n t e r e s t i n g s i t u a t i o n a r i s e s when we have two types o f guest species X, Y r e -s u l t i n g i n a mixed or double hydrates with the formula 2X*6Y*46 H 20 (X-3Y-23 H 20) where X may be N 2 > 0 2, Ar, e t c . and Y may be S0 2, C l 2 > e t c . Some o f the type I hydrates are given i n Table 2.1. - 17 -3. von Stackelberg's type II s t r u c t u r e hydrates. These hydrates are described by space group Fd 3n (0^) o with cubic u n i t c e l l edge a - 17 A. There are 136 water molecules and 24 c a v i t i e s per u n i t c e l l . Out of these 24 c a v i t i e s 16 are 12-hedra (D) and 8 are 16-hedra (H) (Figure 2.3). The 12-hedra are s l i g h t l y d i s t o r t e d i n t h i s case. The centres o f the two polyhedra are as f o l l o w s : 12-hedra a t / I 1 1 3 3 V 3 H 133 5 5 1 7 H 7 5 3 5 7 3 I s s s ' s s s ' s s s ' s s s ' s s s ' s s s ' s s s ' s s s ' 5 1 5 7 3 5 7 H 5 3 7 3 5 7 177 155 3 8 8 8' 8 8 8 '8 8 8 ' 8 8 8 ' 8 8 8 ' 8 8 8 ; 8 8 8 ; 8 16-hedra at 8 8 j / 1 1 1 3 3 3 n n 1 1 1 3 n 1 n 1 3 1 1 n n 3 1 l \ . H y p o t h e t i c a l l y i t i s p o s s i b l e to have a hydrate with the formula 24X- 136 H 20 (X-5-67 H 20), but none has been reported. However, i f a o o molecule has a f r e e diameter > 5.8 A and < 6.6 A, a type II hydrate with the formula 8Y« 136 H 20 (Y- 17 H 20) i s formed (Y may be S F g , acetone, t e t r a h y d r o f u r a n (THF) e t c . ) . Double hydrates with the formula 16X- 8Y- 136 H 20 (2X- Y- 17 H 20) have been reported; a known example of which i s 16 H2S- 8 THF- 136 H 20 [2.5]. A l i s t o f known type II hydrates i s given i n Table 2.1 12-hedron ( 5 1 2 ) (Pentagonal dodecahedron) 14-hedron ( 5 , 2 . 6 2 ) (Tetrakaidecahedron) 16-hedron ( 5 1 2 . 6 4 ) (Hexakaidecahedron) Figure 2.1 Various polyhedra found i n von Stackelberg's type I and type II s t r u c t u r e c l a t h r a t e hydrates 00 •.(b) Figure 2.2 von Stackelberg's type I s t r u c t u r e hydrate (a) 12 A cubic l a t t i c e with 12- and 14-h.edra (b) 12-hedra with corner l i n k e d , forming 14-hedra. • Figure 2.3 von Stackelberg's type II s t r u c t u r e hydrate showing face- s h a r i n g arrangement • of 12- and 16-hedra. - 19 -Table 2.1 SOME KNOWN CUBIC CLATHRATE HYDRATES St r u c t u r e Ideal Composition Guest Molecules X,Y (a) 12 A cub i c 8 X- 46 H 20 6 Y- 46 H 20 (b) 17 A cub i c 8Y- 136 H 20 16H 2S- 8Y- 136H 20 A. Ar, Kr, Xe, C 2H 4, N 2, H 2S, PH 3 > CH^F, 0 2 > H 2Se, CH^. AsH 3, N 20, CH 3Br, B r C l , CH 3CH 2F, ^2^4' ^^3^1, COS, CHF 3, C 2Hg, C l 2 , CH 3SH, CH2=CHF, CH 3CHF 2, Cyclo-C 3Hg, C F 4 . B. C0 2, S0 2, C10 2 C. ethylene oxide, (CH 3) 2NH, C 2H 5NH 2 Trimethylene oxide A. CH 3I, CH 3 CH 2 CH 3, CH 3CH 2Br, CFC1 3, C F 2 B r 2 , CH 2C1 2, CH 3CHC1 2 J CH 3CF 2C1, CH 3N0 2, Cyclo-C 5Hg, CH 3CH 2C1, (CH 3) 3CH, C F 2 C l 2 , C F 2 C l B r , C y c l o - C 5 H 1 6 , C y c l o - C ^ g . CH 3Br, CS 2, CH 3CH 2Br, S F g , CgHg, COS, CH 3CH 2C1, CFC1 3, CH 2C1:CH 2C1, CH 3I, CH 3CHF 2, CC1 4, CH 3CH 2CH 2Br, CHF=CF 2, CH 3CH 2CH 3, (CH 3 ) 2 S , CC1 3N0 2, CHgCl 2, CHC1 3, CH 3CF 2C1, C C l 3 B r . - 20 -Table 2.1 (cont.) S t r u c t u r e Ideal Composition Guest Molecules X,Y 8Y- 136 H 20 C. CH 3CH 20H, (CH 3) 2C0, tet r a h y d r o f u r a n ( C H 3 ) 2 0 , f u r a n , dioxane, 2,5 dihydrofuran, propylene oxide, trimethylene oxide, cyclobutanone A. Hydrophobic gases, B. Water-soluble acidogenic gases, C. Water-soluble p o l a r compounds References: [2.7] and [2.9]. C. Alkylami ne Hydrates The existence of c r y s t a l l i n e hydrates of alkylamines has been known from phase st u d i e s by P i c k e r i n g [2.10] and Sommerville [2.11]. About 35 o f them have been reported with melting points ranging from -35 to 5°C. These amines g e n e r a l l y form two types of hydrates, low hydrates with one-half, one or two water of c r y s t a l -l i z a t i o n , and the hydrates with hydration number varying from s i x to eleven. J e f f r e y and co-workers have reported p r e l i m i n a r y c r y s t a l s t r u c t u r e data on high hydrates of nine amines [2.12]. These r e s u l t s are summarized in Table 2.2. The ethyl amine and dimethyl amine hydrates have s t r u c t u r e s s i m i l a r to von Stackelberg's type I and type II hydrates, but s i n c e C r y s t a l Table 2.2 Structure data on some Alkylamine Hydrates Amine A M.P. of hydrate (°c) C r y s t a l c l a s s Space group Unit c e l l dimension i n & S t o i c h i o m e t r i c formula per c e l l mA. n H 20 Hydration number from f r e e z i ng curve (n/m) 1. Ethyl amine -7.5 Cubic P43n, Pm3n a = 12.17 m = 6; n = 46 or m = 8; n = 48 5.5 2. Dimethyl amine -16.9 Cubic P23, Pm3 a = 12.55 m = 6; n = 52 or m = 7; n = 49 6.9 3. Trimethylamine 5.3 5.9 Hexagonal P6/mmm a = 12.38 c = 12.48 m = 4; n = 40 11, 10, 10.22, 10.03 4. n-Propylamine -13.5 Hexagonal a = 12.20 c = 12.38 m = 4; n = 40 or m = 4; n = 38 8.0 5. iso-Propyl amine -4.0 Hexagonal Pe^ /mmc a = 12.30 c = 24.85 m = 10; n = 80 7.5 6. n-Proplamine monoclinic P2 1/n a = 12.43 b = 20.73 c = 17.28 3 = 89.3° m = 16; n = 104 3.5 Table 2.2 (cont.) C r y s t a l Structure data on some Alkylamine Hydrates Amine A M.P. of hydrate (°C) C r y s t a l c l a s s space group Unit c e l l dimension i n A S t o i c h i o m e t r i c formula per c e l l mA. n H 20 Hydration number from f r e e z i n g curve (n/m) Diethyl amine -6.6 monoclinic a = 13.86 m = 4; n = 28 6.8 P2 1/c b = 8.44 c = 10.93 8 = 97.5° Diethyl amine -7.0 Orthorhombic a =13.44 m = 12; n = 104 8.07, 8.12 -7.3 Pbcn b = 11.77 8.10 c = 27.91 tert-Butylamine -1 cubic a = 18.81 m = 16; n = 156 I43d References: [2.12 - 2.17] - 23 -the composition i s v a r i a b l e and does not e x a c t l y correspond to type I and type II s t r u c t u r e s , i t i s b e l i e v e d that the NH 2 or NH groups are part of hydrogen bonded water framework [2.12]. Out of these nine s t r u c t u r e s f i v e have been worked out i n d e t a i l [2.13 - 2.17]. The amine molecules i n t e r a c t with the water l a t t i c e i n f i v e d i f f e r e n t ways i n these f i v e c r y s t a l s t r u c t u r e s to give r i s e to a v a r i e t y of cages not observed p r e v i o u s l y i n c l a t h r a t e hydrates. The s t r u c t u r a l c h a r a c t e r i s t i c s of these f i v e hydrates are summarized i n Table 2.3. These f i v e s t r u c t u r e s i n which the amines ate held i n the cages i n f i v e d i f f e r e n t ways w i l l be described below: 1. tert-Bulylamine Hydrate The amine molecule i n t h i s hydrate i s not hydrogen-bonded to water l a t t i c e . The host framework c o n s i s t s of f a c e - s h a r i n g 3 2 9 3 17-hedra (7.6.5.4 ) which have 3 square, 9 pentagonal, 2 hexagonal and 3 heptagonal f a c e s . The squares and pentagonal faces a l s o form 4 4\ 8-hedra (4 .5 ) which are empty and serve space f i l l i n g purpose. The amine molecules occupy 17-hedra and show r e s t r i c t e d r o t a t i o n [2.14], The 17-hedra and 8-hedra are shown i n Figure 2.4. 2. Diethyl amine Hydrate: T h i s hydrate with formula 12(C 2H 5) 2NH.104H 20 has been st u d i e d by Jordan and Mak [2.13]. The amine molecule i n t h i s case forms two types of hydrogen bonds. One of these bonds i s a donor H-bond formed by NH-group of amine and i n v o l v i n g one oxygen o f H ?0, Table 2.3 St r u c t u r a l C h a r a c t e r i s t i c s o f some Alkylamine Hydrates Name formula per u n i t c e l l Amine-water r e l a t i o n s h i p Hydrogen bonded n-hedra [ ( H 2 0 ) n = mn]< 1. t e r t - B u t y l ami ne hydrate 16(CH 3) 3.CNH 2.156 H 20 2. Diethyl amine hydrate 12(CH 3CH 2) 2NH.104 H 20 3. Trimethylamine hydrate 4(CH 3) 3N. 41 H 20 4. Isopropylamine hydrate 10(CH 3) 2CH.NH 2. 80 H 20 5. n-Propylamine 16CH 3CH 2CH 2NH 2. 104 H 20 Nonbonded 17-hedra Hydrogen bonded i n d i s t o r t e d 18-hedra and i n an i r r e g u l a r cage Hydrogen bonded i n very d i s t o r t e d 15- and 26- polyhedra Hydrogen bonded i n very d i s t o r t e d 14- and 16-hedra Hydrogen bonded i n very d i s t o r t e d 14- and 16-hedra 17- hedra ( 7 3 . 6 2 . 5 9 . 4 3 ) 8-hedra ( 4 4 . 5 4 ) + 18- hedra ( 5 1 2 . 6 6 ) I r r e g u l a r cage (6^.5 8.4 3) 15- hedra ( 5 1 2 . 6 3 ) 26-hedra ( 5 2 4 . 6 2 ) 12-hedra ( 5 1 2 ) + 14-hedra ( 4 2 . 5 8 . 6 4 ) 16- hedra ( 5 1 2 . 6 4 ) 8-hedra ( 6 2 . 4 6 ) + 12-hedra ( 5 1 2 ) + 14-hedra ( 5 1 2 . 6 2 ) 16-hedra ( 5 1 2 . 6 4 ) 11-hedra ( 4 2 . 5 8 . 6 1 ) + r o -pi-Reference [2.6] empty cages. * m means n m-sided f i g u r e [2.8] - 25 -and the other i s an acceptor H-bond between the NH-group and the oxygen ( l y i n g on opposite s i d e of cage) o f the other H 20. There are two types o f c a v i t i e s i n t h i s case, the f i r s t i s 18-hedron 12 6 (. 5. .6 ) where the H-bond d i v i d e s 18-hedron i n t o two nea r l y equal halves, each of which accomodates one ethyl group (Figure 2.5a) 6 8 3 and the other i s i r r e g u l a r cage (6 .5 .4 ). The amine molecule i n both c a v i t i e s i s shown i n Figure 2.5. 3. iso-Propylamine Hydrate: This hydrate has been studie d r e c e n t l y by McMullan et al [2.17] and provides an example where the NH 2-group of the amine forms two donor H-bonds b r i d g i n g across two adjacent oxygen atoms of water molecules. The water s t r u c t u r e i n t h i s hydrate c o n s i s t s o f 2 8 4 " f a c e - s h a r i n g " l a y e r s o f 14-hedra (4 .5 .6 ), separated by 12-hedra 12 12 4 (5 ) and 16-hedra (5 .6 ). The amine molecules l i e i n s i x 14-hedra and four 16-hedra where they are H-bonded s i n g l y i n 16-hedra and doubly i n 14-hedra. In 14-hedra, the NH 2 group makes an 0--H-N-H--0 bridge across two oxygens and these bonds form the edges o f d i s t o r t e d 14-hedrons (Figure 2.6). The d i s t o r t i o n of these polyhedra leads to ft 2 1 ? a d d i t i o n a l c a v i t i e s which are 8-hedra (4 .6 ) and 12-hedra (5 ). They are vacant. 4. Trimethylamine Hydrate. [2.15] In trimethylamine hydrate 4 (CH 3) 3N. 41 H 20, the amine nitro g e n forms acceptor H-bonds. The c a v i t i e s i n which the t r i m e t h y l -12 3 24 2 amine molecules are held are 15-hedra (5 .6 ) and 26-hedra (5 .6 ). 17-hedra with d i f f e r e n t amine o r i e n t a t i o n s i (a) C-N bond v e r t i c a l (b) C-N bond towards reader (c) empty 8-hedron ro cn Figure 2.4 17-hedra and 8-hedron found i n tert-butylamine hydrate i Figure 2.5 (a) Diethyl amine i n 18-hedron (b) Diethyl amine i n an i r r e g u l a r cage - 27 -In 15-hedra there i s only one acceptor hydrogen bond from the oxygen of a water molecule [ F i g u r e 2.7a] to the N-atom of the amine. In 26-hedron, a p a i r o f amine molecules are hydrogen bonded to an a d d i t i o n a l water molecule, which i n turn i s hydrogen bonded to 26-hedron (formed by two 14-hedra, sharing a common hexagonal f a c e ) . T his s i t u a t i o n i s depicted i n Figure 2.7b. The s t r u c t u r e o f the water l a t t i c e thus contains " f a c e - s h a r i n g " arrangements of 15-hedra, 26-hedra and the space f i l l i n g vacant 12-hedra. An i n t e r e s t i n g t hing to note i n the case of the t r i m e t h y l a -mine hydrate i s th a t i t shows a s i m i l a r c r y s t a l s t r u c t u r e i n the l i q u i d s t a t e ( i . e . at the melting point of c r y s t a l l i n e s o l i d hydrate v i z . 5°C). This s t r u c t u r e i n l i q u i d s t a t e has been s t u d i e d by F o l z e r e t al [2.18]. 5. n-PrOpylamine Hydrate The hydrate n-CH 3CH 2CH 2NH 2-6.5 H 20 has been studied i n l e s s d e t a i l [2.17]. The nitrogen i s hydrogen bonded i n the 14-hedron 12 2 (5 .6 ) and here i t makes one of common v e r t i c e s of a hexagon and a pentagon. The other cage found i s a 16-hedron(5 .6 ) where the nitrogen atom replaces a water oxygen and makes a bridge across a void forming a hydrogen-bonded dimer. The other polyhedron which h a s s o f a r 2 8 1 not been found i s 11-hedron (4 .5 .6 ) and t h i s i s not occupied [2.17], The s t r u c t u r e s o f other alkylamine hydrates have not been reported, but i t i s be l i e v e d that every amine w i l l show a d i f f e r e n t s t r u c t u r e with a v a r i e t y o f cages and may open a new branch i n chemistry 'The Hydrate Chemistry 1. 2.6a Isopropylamine molecule i n 14- and 16-hedra. i n 16-hedron s i n g l y bonded i n 14-hedron (doubly bonded with three d i f f e r e n t views. Figure 2.7 Trimethylamine i n i t s water cages a /two trimethylamine molecules 26-hedron b one trimethylamine molecule i 15- hedron with N attached to 0 by a s i n g l e H-bond. Figure 2.6 a b,c,d - 29 -References (Chapter Two) [2.1] M. von Stackelberg, Naturwiss., 36 (1949) 327, 359., M. von Stackelberg and H.R. M u l l e r , i b i d , 38 (1951) 456., M. von Stackelberg, Z. Electrochem., 58 (1954) 25., M. von Stackelberg and W. Meinhold, i b i d , 58 (1954) 40., M. von Stackelberg and H. Fruhbuss, i b i d , 58 (1954) 99., M. von Stackelberg, i b i d , 58 (1954) 104., M. von Stackelberg and W. Jahns, i b i d , 58 (1954) 162., M. von Stackelberg and B. Meuthen, i b i d , 62 (1958) 130. [2.2] W.F. Claussen, J . Chem. Phys., 1_9 (1951 ) 259, 662, 1425. [2.3] L. Pauling and R.E. Marsh, Proc. N a t l . Acad. S c i . U.S., 38 (1952) 112. [2.4] R.K. McMullan and G.A. J e f f r e y , J . Chem. Phys., 42 (1965) 2725. [2.5] T.C.W. Mak and R.K. McMullan, J . Chem. Phys., 42 (1965) 2732. [2.6] G.A. J e f f r e y , Accounts Chem. Res., 2 (1969) 344. [2.7] G.A. J e f f r e y and R.K. McMullan, Progr. Inorg. Chem., 8 (1967) 43. [2.8] A.F. Wells, T h i r d Dimension i n Chemistry, Oxford U n i v e r s i t y Press (London), 1962. [2.9] B. Morrison and D.W. Davidson, Can. J . Chem., 49 (1971) 1243. [2.10] S.U. P i c k e r i n g , Trans. Chem. S o c , 63,1 (1893) 141 . [2.11] W.C. Sommerville, J . Phys. Chem., 35 (1931) 2412. [2.12] R.K. McMullan, T.H. Jordan and G.A. J e f f r e y , J . Chem. Phys., 47 (1967) 1218. - 30 -[2.13] T.H. Jordan and T.C.W. Mak, J . Chem. Phys., 47 (1967) 1222. [2.14] R.K. McMullan, G.A. J e f f r e y and T.H. Jordan, J . Chem. Phys., 47 (1967) 1229. [2.15] D. Panke, J . Chem. Phys., 48 (1968) 2990. [2.16] C A . Shelton and D. Panke, Acta C r y s t . A25 (1969) S147. [2.17] R.K. McMullan, G.A. J e f f r e y and D. Panke, J . Chem. Phys., 53 (1970) 3568. [2.18] C. F o l z e r , R.W. Hendricks and A.H. Narten, J . Chem. Phys., 54 (1971 ) 799. - 31 -CHAPTER I I I NUCLEAR MAGNETIC RESONANCE THEORY This chapter i s a b r i e f i n t r o d u c t i o n to the p r i n c i p l e s of nuclear magnetic resonance with p a r t i c u l a r emphasis on methyl group r e o r i e n t a t i o n and r e l a x a t i o n . The subject o f NMR has been e x t e n s i v e l y covered i n many t e x t books [3.1-3.6]. The n u c l e i studied i n t h i s work are protons and a l l the compounds, c l a t h r a t e deuterates, and guest molecules are diamagnetic i n the s o l i d s t a t e . A. Introductory Remarks Nuclei i n a sample with non-zero spin possess a net macro-sco p i c magnetic moment M which i s the sum of magnetic moment of i n d i v i d u a l n u c l e i k. When such a sample with magnetic moment M_, is placed i n a la r g e magnetic f i e l d H, i t experiences a torque C_ = MX H_ equal to r a t e of change of i t s angular momentum ti(dl/dt), where I_ i s t o t a l angular momentum vector equal toV* L . Since M_ = H_ can be the sum of two f i e l d s , the s t a t i c f i e l d H and the magnetic SMJ, = Syfil, > where y i s nuclear gyromagnetic r a t i o , the motion o f k; "* k ~k magnetic moment M i n a f i e l d H_ i s given by (dM/dt) = y M X H (3.1) - 32 -vector of radiofrequency ( r f ) f i e l d r o t a t i n g with frequency w i . e . , fij = (H-jCOSoyt, -H^sinut, 0) The Bloch equations f o r an i n d i v i d u a l component of magnetization M can now be w r i t t e n [3.1, p.28] ( d i y d t ) = y(M yH Q + M ^ s i n w t ) - (M/V (dM y/dt) = Y ( M z H l C o s w t - M xH Q) - (M y/T 2) (3.2) (dM /dt) = -Y,(MvH,sina)t + M H,cosoot) + (M -Mj/T, where T-j and T 2 are c a l l e d the longitudnaJl and the transverse r e l a x a t i o n times r e s p e c t i v e l y , and M Q i s e q u i l i b r i u m magnetization. Equation (3.2) i s the simplest phenomenological d e s c r i p t i o n of approach to e q u i l i b r i u m i n a constant magnetic f i e l d i n z - d i r e c t i o n . In absence of JH-j these equations can be w r i t t e n [3.1 p.28] (dM z/dt) = (M 0-M 2)/T i ; (dM x/dt) =-Mx/T2; (dM y/dt) = -My/T2 (3.3) The s o l u t i o n of these equations gives T-j and L,. This i s somewhat s a t i s f a c t o r y f o r l i q u i d s , but not f o r s o l i d s e s p e c i a l l y i n case o f T 2 . The s o l u t i o n s o f Equation (3.2) f o r M x and M y under steady s t a t e c o n d i t i o n s gives the components x 1 and x" of nuclear magnetic s u s c e p t i b i l i t y x r e s p e c t i v e l y . I t i s the component x" which gives r i s e to NMR absorption [3.1 p. 29]. Quantum mechanically one i s i n t e r e s t e d i n the t r a n s i t i o n p r o b a b i l i t i e s computed from time-dependent p e r t u r b a t i o n theory. The - 33 -Fermi Golden Rule gives the t r a n s i t i o n p r o b a b i l i t y W a b per u n i t time induced by some perturbing Hamiltonian between s t a t e s |a> and |b> as:-W |<a|j^|b>| 26(E a-E b-fi W) (3.4) ab where 6 i s Dirac d e l t a f u n c t i o n , E Q , E^ are energies of s t a t e |a> and |b> r e s p e c t i v e l y . U s u a l l y ^ = -b xH x coswt, where y x i s the x-component of t o t a l magnetic moment. The s u s c e p t i b i l i t y equation g i v i n g r i s e to a b s o r p t i o n i n the high temperature approximation fi»=(E -E. ) « k T (k, Boltzmann's constant and T absolute temperature) a D i s given by [3.3 p.43] **mLe-E*m MP x |b>| 2 ME a -E b - i t » ) >VEb <3-5' Here Z i s j u s t c l a s s 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 eigenvalue spectrum, i . e . Z =^^e~^c^^ (c stands f o r complete spectrum) c Equation (3.5) leads to the idea of a s o - c a l l e d l i n e shape f u n c t i o n g(to) which i s given by g(u) = |<a|ux |b>| 26(E a-E b-fiu) (3.6) t Experimentally one i s i n t e r e s t e d in g(w -to) = g(A) where to i s centre of l i n e shape and A = (to -to) [3.5 Chap. IV]. - 34 -This l i n e shape f u n c t i o n i s the main essence o f cw NMR. Experimentally one can determine x"(o)) and then c a l c u l a t e g(u) from Equation (3.5) or g(w) can be c a l c u l a t e d t h e o r e t i c a l l y and then x"(o)) can be c a l c u l a t e d and compared with the experiment. B_. The Line Shape Function The c a l c u l a t i o n of the l i n e shape i s the most d i f f i c u l t task e s p e c i a l l y i n case of s o l i d s where the l i n e s are broad due to l o c a l f i e l d s produced by neighbouring d i p o l e s . Of a l l the d i f f e r e n t causes of broadening of resonance l i n e shapes, the one important one f o r our case (protons) i s d i p o l a r broadening. The Hamiltonian of i n t e r e s t can be w r i t t e n as f u n c t i o n of nucleus i . Jf^ f o r two n u c l e i i and j with magnetic moments y. and y. r e s p e c t i v e l y and j o i n e d by a ve c t o r £. . i s given by [3.3 p. 46]. (3.7) (3.8) Here H Q i s a p p l i e d f i e l d i n z - d i r e c t i o n and I z i s nuclear spin eigen-(3.9) (3.10) i <J - 35 -where A.. = i V ( l - 3 c o s 2 6 . .)r.~.3 B i a. = - 0/4)(I^ + O j)(l - 3 c o s 2e...)r:] C.. - - ( 3 / 2 ) ( i ; i J + i ; i J ) s i n e ^ c o s e . - e x p C - i ^ ) ^ E 1 d - -(3/4)l|lJ s l n ^ j exp(-2i*..) r~] + + (3.11) D U - c ! j > F i j = E i j Where r. ., 6.., <b.. are the polar coordinates of the vector between sp i n i and j , 1^ e t c . are the usual r a i s i n g and lowering operators f o r e i g e n f u n c t i o n of 1^ e t c . , and the dagger signs denote the complex conjugates of the f u n c t i o n s . I n j ^ j , the termsA and B commute with Jfz and are c a l l e d the s e c u l a r terms g i v i n g the s o - c a l l e d truncated i d i p o l a r Hamiltonian i . e . X - Y V J ^ • B f j ] (3.12) 1 ^  = - ( y V / Z j ^ d - . L . - 3 I ^ ) ( l - 3 c o s 2 e i : J ) r : ] (3.13) i<j So f a r we are only i n t e r e s t e d i n _ ^ and J ^ , the other terms C to F of J^j w i l l be discussed i n a l a t e r s e c t i o n . The Hamiltonian o f i n t e r e s t now reduces to 3i-Kz+'](t (3.14) - 36 -Since the f i r s t c a l c u l a t i o n o f the l i n e shape f o r two spin system using t h i s Hamiltonian f o r gypsum (CaS04'2H,,0) by Pake [3.1 p.152] various attempts have been made using the d e n s i t y matrix formalism and other methods e s p e c i a l l y f o r the well known cases of CaF 2 and gypsum c r y s t a l s . Current trends l i e i n p r e d i c t i n g l i n e shapes from f r e e i n d u c t i o n or Bloch decay shapes (decay of transverse magneti-z a t i o n M y i n a c h a r a c t e r i s t i c time T 2 ( c f . Equation (3.2)). Lowe and Norberg [3.7] showed that usual assumption of Gaussian l i n e shapes i n s o l i d s was no longer a good approximation. I t should be noted that the f r e e i n d u c t i o n decay (FID) shape G(t) i s j u s t a F o u r i e r transform o f absorption l i n e shape g(A)[3.7, 3.5 p.222] i . e . , g(A) = (2TT) - 1 / G(t) e x p ( i A t ) d t (3.15) where G(0) = 1,/ g(A)dA = 1 and r -oo /+oo g(A)exp(-iAt)dA (3.16) The work done on FID shape and i t s conversion to absorption l i n e shape i n two spin systems f o r c r y s t a l l i n e and p o l y c r y s t a l l i n e m a t e r i a l s i s summarized i n a paper by Gade [3.8]. Fornes et a l . have reviewed the various t h e o r i e s on l i n e shape c a l c u l a t i o n s and have a p p l i e d them to S r F 2 , CsF and NaCl [3.9] by converting usual absorption shape to FID shapes. In a more recent paper a general approach to l i n e shape c a l c u l a t i o n i s presented by the same group [3.10]. Some mention of - 37 -other work on two s p i n l i n e shapes by other methods (Monte Carl o and by numerical s o l u t i o n o f d i f f e r e n t i a l equation f o r time e v o l u t i o n of m a g n e t i z a t i o n ) i s given i n a paper by Cobb and Johnson [3.11]. Andrew and Brookman have r e c e n t l y c a l c u l a t e d the absorption l i n e shape f o r two spin systems and have a p p l i e d these r e s u l t s to conf i r m a t i o n a l changes [2.12]. The l i n e shape f o r a 3-spin system (e.g. i s o l a t e d CH^-group) was f i r s t c a l c u l a t e d by Andrew and Bershon [3.11]. This l i n e shape c a l c u l a t i o n has been r e v i s e d by Apayaditf^and Clough [3.14] by taking i n t o account the t u n n e l l i n g e f f e c t f o r CHg-group, and more r e c e n t l y by Cobb and Johnson [3.11] using d e n s i t y matrix formalism. The l i n e shape c a l c u l a t i o n f o r higher spin systems becomes extremely d i f f i c u l t although some attempts f o r four [3.1 p.158], f i v e [3.15] have been re p o r t e d , but the r e s u l t s hardly show any f i n e s t r u c t u r e due to inte r m o l e c u l a r i n t e r a c t i o n s . Some promising r e s u l t s have been depicted 1 g by S F g - c l a t h r a t e deuterate, where F absorption l i n e shows a good f i n e s t r u c t u r e [3.16] a t l i q u i d helium temperature because of l e s s i n t e r m o l e c u l a r i n t e r a c t i o n s . I t i s be l i e v e d that i f guests having other simple s p i n systems are t r i e d i n form of c l a t h r a t e s , a bett e r comparison between theory and experiment w i l l be a v a i l a b l e . For complicated systems, the well known Van Vleck's formula [3.17] f o r second moment of absorption l i n e shape i s much more useful instead of exact l i n e shape c a l c u l a t i o n s . - 38 -C. Second Moment of Absorption Line Shape 1. Second Moment from Absorption Line Shape We d e f i n e the nth moment, Mn, of a resonance absorption curve (normalized to un i t y ) by the r e l a t i o n [3.5 p.223] /«+<» M n V A n 9 ( A ) d A (A=O)O-OJ) (3.17) where the second moment M ? i s given by M 2 = J A 2g ( A)dA (3.18) - 0 0 The formula which i s obtained using the Hamiltonian as + J ^ ' f o r p o l y c r y s t a l l i n e m a t e r i a l comes out to be [3.1 p. 160] M 2 = ( 6 / 5 ) 1 ( 1 + 1 ) g 2 3 2 N _ 1 ^ , r ^ j i>j + ( 4 / 1 5 ) 8 2 N _ 1 ^ I f ( I f + l ) g 2 r ^ i , f (3.19) where I i s nuclear s p i n number f o r nu c l e i at resonance, g i s nuclear g f a c t o r , 3 i s nuclear magneton and r . . i s d i s t a n c e between i t h and j t h n u c l e i . S u b s c r i p t f r e f e r s to other n u c l e i not a t resonance. The second moment can be c a l c u l a t e d by numerical i n t e g r a t i o n of absorption curve. This i s a long and tedious procedure. Caution must a l s o be exer c i s e d i n o b t a i n i n g the absorption curve because of the danger of sa t u r a t i o n and account must be taken of the c o n d i t i o n s imposed by Provotorov theory^. A l t e r n a t i v e l y i t can a l s o be c a l c u l a t e d from Bloch decay. C a l c u l a t i o n o f second moment from the absorption l i n e +A d e t a i l e d account of Provotorov theory and i t s a p p l i c a t i o n i s presented by Goldman [3.5,Chap. I l l & IV]. - 39 -i s well known and need not any comment, but the c a l c u l a t i o n from Bloch decay (FID) w i l l be des c r i b e d b r i e f l y (although we have pre-f e r r e d the long and slow cw experimental technique). 2. Second Moment and FID curve I t i s well known now that FID curve can be expanded i n terms of even powered moments of the resonance absorption curve Equation (3.20) shows that second and f o u r t h moment can be c a l c u l a t e d from FID curve, but the main d i f f i c u l t y here i s that the important i n i t i a l p art of G(t) i s buried i n the dead time of the r e c e i v e r . However, Powles and Strange [3.18], and Mansfield [3.19], have developed a method to overcome t h i s s i t u a t i o n by using two 90° pu l s e s , the second a f t e r a time T, and 90° out of r f phase with the f i r s t (90° - x - 90gQ O), t h i s produces s o l i d echo. Using the d e n s i t y matrix formalism [3.2-3.5, 3.18,3.19] and r o t a t i n g c o o r d i n a t e s , the FID i s given by ( i n frequency u n i t s ) where Tr means t r a c e and other terms have usual meanings. Equation (3.21) can be expanded a f t e r some manipulations g i v i n g [3.7, 3.2 p. 110] i . e . G(t) =1 - ^  M 2 + j, M4~ (3.20) G(t) = T r { I x e x p ( - i ^ t ) I x e x p ( i ^ t ) } / T r ( I 2 ) (3.21) G(t) = 1 + t 2 5 1 Tr [I 2 ] + I T r [ # ' , [ f l 1 . I x ] ] 2  4 ! ' -+ (3.22) - 40 -Comparison of equations (3.20) and (3.22) gives For a 90° - T - 90gQ O pulse sequence, the FID s i g n a l a f t e r a time T' from the second pulse, the f i n a l r e s u l t of Powles and Strange [3.18] i s , where the l a s t term M^x i s a c o r r e c t i o n term shown by Powles and Man s f i e l d equal to zero [3.20] f o r an i s o l a t e d p a i r of spin 1/2 n u c l e i . The echo amplitude can be obtained by p u t t i n g T=T' and Equation (3.20) can be used with t = 2T. 3. E f f e c t of Molecular Motion on Second Moment and Line Width. In a r i g i d l a t t i c e , the r e l a t i v e p o s i t i o n s o f nuclear spins do not change i n time, and the second moment has a constant value because terms A and B i n j ^ j are constant. In case of motion which we assume to be of r o t a t i o n a l jump type, the l o c a l f i e l d seen by a s p i n , which i s due t o ^ , f l u c t u a t e s i n time. Only the average value of the l o c a l f i e l d taken over a long time compared with the duration of f l u c t u a t i o n w i l l be observed, and t h i s average i s much smaller than the instantaneous value of the l o c a l f i e l d . The time average l o c a l 1 /p f i e l d ~ l i n e width ~(M 9) ' . The r a t e of f l u c t u a t i o n of the l o c a l (3.23) - 41 -f i e l d can be described by a c o r r e l a t i o n time T . Thus the c r i t e r i o n f o r motional narrowing i s ( M 2 ) 1 / 2 T c « 1 or ( M 2 ) 1 / 2 « T c ' 1 (3.24) where M 2 i s the r i g i d l a t t i c e second moment i n angular frequency 1/2 -1 u n i t s . I f (M 2 ) ' » T , the l i n e width and second moment c o r r e s -1/2 -1 ponds to the r i g i d l a t t i c e v a l u e, however i f (M 2 ) ' « x c , a completely narrowed l i n e width and a reduction i n second moment r e s u l t s . In f a c t , the second moment i s i n v a r i a n t to motional r o t a t i o n , but according to argument of Pake [3.21] and more elaborate work o f Andrew and Newing [3.22], i t i s c l e a r that r o t a t i o n produces weak side spectra which are u s u a l l y unobservable experimentally, because they are buried i n the noise i n the wings, and hence a reduced second moment i s observed. I f the motion i s thermally a c t i v a t e d and the l i n e shape i s the same before and a f t e r the motional narrowing (which i s u s u a l l y not e x a c t l y the c a s e ) , a r e l a t i o n between l i n e width and c o r r e l a t i o n time i s given by [3.23]. T c = tan[Tr(6H 2 - B 2)/2(C 2-B 2)]/(ay6H) (3.25) where a = ( 8 £ n 2 ) ~ \ 6H i s l i n e width i n the narrowing r e g i o n , B i s narrowed l i n e width and C i s unnarrowed l i n e width. t c obeys Arrhenius a c t i v a t i o n energy r e l a t i o n s h i p T C = T oexp(E a/RT) (3.26) where E h a c t i v a t i o n energy f o r b a r r i e r hindering the r o t a t i o n , a Equations (3.25) and (3.26) can provide an estimate of the a c t i v a t i o n energy from l i n e width versus temperature measurements. - 42 -The e f f e c t of molecular r o t a t i o n on second moment can be c a l c u l a t e d from Gutowsky and Pake's formula [3.24] M 2 = M 2 [ ( l / 4 ) ( 3 c o s 2 y - l ) 2 ] (3.27) where y i s angle between i n t e r n u c l e a r v e c t o r r_ and axis o f r o t a t i o n , M 2, M 2 are reduced and r i g i d l a t t i c e second moments r e s p e c t i v e l y . I f however the motion i s o s c i l l a t o r y , Andrew's expression [3.25] should be used; (M 2 ) = i y osc where P= l - ( 3 / 4 ) [ ( l - J Q 2 ( a ) ) s i n 2 Y + (1 - J 0 2 ( 2 a ) ) s i n 4 Y ] (3.28) Here J Q i s a Bessel f u n c t i o n of f i r s t k ind, a i s the amplitude of o s c i l l a t i o n , and y i s the angle between the p a i r d i r e c t i o n and the axi s of r o t a t i o n a l o s c i l l a t i o n . For a small angle a, p reduces to P= 1 - (3/2)a 2sin 2Y (3.29) More d e t a i l s on o s c i l l a t o r y motion i s given by 01f and P e t e r l i n [3.26]. Coming to methyl group, the r i g i d l a t t i c e second moment M 2 f o r powder samples i s given by Powles and Gutowsky [3.27] as 2 M 2 = (8/5)a 2 = (8/5)(3u/2R3) (3.30) where u i s proton magnetic moment, and R i s si d e of the t r i a n g l e formed by methyl protons. In case when C 3 - a x i s o f t r i a n g l e makes an angle ^ wi th the axi s o f r o t a t i o n , the reduced second moment M 2 i s given by [3.27] M 2 = (2/5)a2[(27/8) s i n 4 ' / ' - 3 s i n 2 ^ + 1] (3.31) - 43 -On the other hand i f there i s a C ^ - r o t a t i o n as well as r o t a t i o n along sortie other a x i s which makes an angle ^ with the C^-axis of t r i a n g l e , the reduced second moment M 2 i s given by M 2 = (a 2/10)(3 c o s 2 * - I ) 2 (3.32) Equations (3.27 - 3.32) are v a l i d f o r the i n t r a m o l e c u l a r second moment. The e f f e c t o f motion on the i n t e r m o l e c u l a r second moment i s discussed i n references [3.25, 3.28]. 4. E f f e c t of T u n n e l l i n g on the Second Moment Although because of equipment l i m i t a t i o n s we were not able to reach a temperature below 77 K, where t u n n e l l i n g i s more important, but a few points are worthy of d e s c r i p t i o n . A l l e n [3.29] has derived an expression f o r M 2 f o r an i s o l a t e d t u n n e l l i n g CH^-group accounting f o r reduced second moment at low temperatures. For b a r r i e r height of < 3 kcal./mole, the second moment i s j u s t ( l / 4 ) t h r i g i d l a t t i c e value. For high b a r r i e r s (3-3.5 kcal./mole) h i s expression was M 2 = ( 9 / 4 0 ) ( Y 2 - t i 2 / r 6 ) [ l - 0 . 1 1 ( r / A ) 1 / 2 - 0.64(r/A)] ....(3.33) 2 2 3 where r= (y ti / r ) and A i s t o r s i o n a l s p l i t t i n g of t o r s i o n a l ground s t a t e . In a more recent paper Clough [3.30] has explained the i n c r e a s e of second moment due to t u n n e l l i n g methyl groups at low temperatures 2 2 2 and has shown that increase may be as great as (2y a /5) or about 5G where a i s the same as given i n Equation (3.30). - 44 -D. Spin L a t t i c e Relaxation Time 1. C o r r e l a t i o n f u n c t i o n , S p e c t r a l d e n s i t i e s and T^ We defined the spin l a t t i c e r e l a x a t i o n T-j by Equation (3.2). We a l s o assume here that T^ i s due to the d i p o l a r Hamiltonian, and the terms C to F i n [Equation (3.11)] are r e s p o n s i b l e f o r i t . We w i l l w r i t e the time dependent part C to F i n a more compact way ( s u b s c r i p t s are dropped f o r c l a r i t y ) . f-j(t) = sin6cos6exp(i<{>)r f 2 ( t ) = sin 2eexp(2i<j))r" 3 (3.34) Since the f ^ ( t ) are randomly varying f u n c t i o n s o f time, they are as c r i b e d a c o r r e l a t i o n f u n c t i o n K(T) defined by K(T) =< f ( t ) f * ( t + x ) > (3.35) where < > denotes the ensemble average and the s t a r i s the complex conjugate. A common form of c o r r e l a t i o n f u n c t i o n which i s mostly used, i s of exponential form i . e . K(T) =< f ( t ) f * ( t ) > e " x / T c ....(3.36) where T c i s c o r r e l a t i o n time c h a r a c t e r i s t i c o f motion. The s p e c t r a l d e n s i t i e s J(w) of random f u n c t i o n f ( t ) are given by Fo u r i e r transform of K ( T ) , j(o>) = J K(T) exp(io)x)dT ....(3.37) —oo = < f ( t ) f * ( t ) > 2 x / ( l + ufV 2 ) ....(3.38) - 45 -The s p i n l a t t i c e r e l a x a t i o n time T-j i s defined by [3.2 p. 291] 1/^ = ( 3 / 2 ) I ( I + l ) Y V ^ [ J i j ( a ) 0 ) + J . j (2a) o)] ....(3.39) m where CO q i s resonance frequency. 2. Spin L a t t i c e Relaxation f o r 2-Spin (1/2) Systems Equation (3.39) f o r the present case reduces to 1/^ = (9/8)Y^ Z[J 1(« 0) + J 2 ( 2 W o ) ] ....(3.40) Th i s case may a r i s e i n i s o l a t e d CH 2 or NH 2 groups where the group may r o t a t e around an axis perpendicular to H-H vector and passing through the C or N atom. Assuming t h a t the angle between the a p p l i e d s t a t i c f i e l d H Q and the a x i s of r o t a t i o n be a, and that around t h i s axis be 8, we can convert 9 and <j> of Equation (3.34) i n t o a and B as was done by S t e j s k a l and Gutowsky [3.31], f-j ( t) = [2 sinasin2B - isin2acos2B + i s i n 2 a ] / 4 r ^ 4 1 j. 2 3 f 2 ( t ) = [2icosasin2B + (1+cos a)cos28 + i s i n 2 a ] / 4 r Using the f a c t < s i n 28 > = < cos 8 > = 1/2, Equations (3.41) (3.42) reduce to f ^ t ) = [ 2 s i n 2 a + ( l / 2 ) s i n 2 2 a ] / 1 6 r 6 f 2 ( t ) = [(1/2) + 3cos 2 a + ( l / 2 ) c o s 4 a ] / 4 r 6 Upon averaging over a l l o r i e n t a t i o n s f o r ' a powder sample f 1 ( t ) f 1 * ( t ) >= ( 1 / 1 0 r 6 ) , < f 2 ( t ) f 2 * ( t ) > = (2 / 5 r 6 ) ....(3. - 46 -which gives T 1 a f t e r using values of J^ (OJ q) and J^^0) 4 2 _L _ 9. y f i T 1 40 r 6 T i 4T c 4. c 7 — ? — 7" l+o) V l+(2xox T 0 c 0 c (3.44) However i f the group i s o s c i l l a t i n g and the o s c i l l a t i o n i s simple harmonic given by expression [3.25] 6 =B sin/at + <|> (3.45) where <$> i s the angle between the H-H vector and the x-axis of e q u i l i b r i u m process, 6 Q i s the angle with respect to <f>Q during o s c i l l a t i o n . Using t h i s value of 6 and proceeding i n an analogous manner we get, ^Vosc. = H-AMl/T^) (3.46) • where (1/T-|) i s given by Equation (3.44) and A = [< cos2B 0siricot >] . Equation (3.46) reduces to (3.44) i f A « 1 which i s the case of o s c i l l a t i o n changing to r e o r i e n t a t i o n . 3. S p i n - l a t t i c e Relaxation f o r Methyl Group (a) Exponential Relaxation Hubbard [3.32] has reviewed various theories of nuclear magnetic r e l a x a t i o n using d e n s i t y matrix (quantum mechanically, or s e m i c l a s s i c a l l y ) approach which was developed by Bloch [3.33] and R e d f i e l d [3.34]. As discussed before the c a l c u l a t i o n of r e l a x a t i o n time i n v o l v e s some c o r r e l a t i o n f u n c t i o n s f o r each mutual d i p o l e - d i p o l e i n t e r a c t i o n which we term the ' a u t o c o r r e l a t i o n f u n c t i o n ' , and with - 47 -other d i p o l e - d i p o l e i n t e r a c t i o n which i s c a l l e d ' c r o s s - c o r r e l a t i o n ' . H i l t and Hubbard ( h e r e a f t e r r e f e r r e d to H-H) [3.35] have shown that i n case of 3-spin systems undergoing hindered r o t a t i o n , i f cross c o r r e l a t i o n are neglected the r e l a x a t i o n f u n c t i o n i s e x p o n e n t i a l . The s o l u t i o n of Equation (3.3) adapted to the language of pulsed NMR can be w r i t t e n as M z ( t ) - M Q = ( c o s e - l ) M o e x p ( - t / T 1 ) (3.47) and i f 6 = 180°, the r e l a x a t i o n f u n c t i o n which we denote by R(t) i s now given by R(t) = [M Q - M z ( t ) ] / 2 M Q = e x p ( - t / T 1 ) (3.48) Equation (3.48) shows that i f r e l a x a t i o n f u n c t i o n i s simple e x p o n e n t i a l , a p l o t of InR(t) versus t should give a s t r a i g h t l i n e with a slope-1/T^. The expression obtained by H-H when cross c o r r e l a t i o n are neglected i s 4 2 1 9YVT( T ^ I O r 6 4 2 4 ' 1-cos 6 1 + 6cos 8 + cos 3 u 2 2 1 + w o T c l+4w V 2 o c (3.49) where 8 i s the angle between s t a t i c a p p l i e d f i e l d H and C ^ - r o t a t i o n 0 o a x i s . Averaging over a l l 8 f o r p o l y c r y s t a l l i n e m a t e r i a l , Equation (3.49) reduces to 4 2 r 1 9 Y V R T 1 20 r 4T. 7 T T T 0 c 2—T 1 +4u T o c (3.50) which i s j u s t a f a c t o r of two greater than equation (3.44) and i s the - 48 -same as obtained by other workers using a much simpler approach, e.g. O ' R e i l l y and Tsang [3.36]. (b) Non-Exponential spin l a t t i c e r e l a x a t i o n According to H-H theory [3.35] when c r o s s - c o r r e l a t i o n are not neglected, the magnetization M 2 ( t ) given by Equation (3.47) i s not a simple e x p o n e n t i a l , but a sum of 4 exponentials given by, 4 M z ( t ) - M Q = (cose-1) M o e x p ( - q j t / T ' ) (3.51) j=l where Q. and q^ are complicated f u n c t i o n s of W Q T c and B the angle between C^-axis and H Q, and T' i s a measure of strength of i n t e r a c t i o n given by (1/T') = ( Y 2 t i / r 3 ) 2 ( l / o ) 0 ) (3.52) where r i s i n t e r p r o t o n d i s t a n c e i n CH^-group. Since i n 180°-T-90° pulse sequence, 6 = 180°, Equation (3.51) gives an analogous equation to Equation (3.48) i . e . 4 R(t) = [M Q - M z ( t ) ] / 2 M Q Cj e x p ( - q j t / T ' ) (3.53) j=l but now i t i s the sum of 4-exponentials and the concept of no longer i s v a l i d . The authors H-H have provided t a b l e s f o r C. and q. J J i n terms of 6 and OJ T . Equation (3.53) i s v a l i d f o r a s i n g l e c r y s t a l . To use Equation (3.53) f o r p o l y c r y s t a l l i n e samples one has to average over a l l 8 to get R A v ( t ) , R A v ( t ) = {[ M 0 - M z ( t ) ] / 2 M Q | = 1 / ^ c j e x p ( - q j t / T ' ) s i n B d B 0 J'=1 (3.54) We have evaluated t h i s i n t e g r a l f o r many values of t / T ' by using - 49 -Simpson's formula from the values of C. and q. provided by H-H[3.35] J J f o r a p a r t i c u l a r value of ('w T ) . We w i l l give the d e t a i l of t h i s method i n the f o l l o w i n g chapters. The t a b l e s of R A v ( t ) constructed from equation (3.54) are given i n Appendix A f o r a p a r t i c u l a r value 2 o f (COQ^) . Some non-exponential curves obtained by p l o t t i n g In R A y ( t ) versus t/T' are shown i n Figures 3.1. These curves are e x a c t l y s i m i l a r to those given by H-H. In such a case i t i s useful to take time t equal to t Q , when In R A y ( t ) = 1/2 instead of f i n d i n g T-j which i s not defined i n t h i s case. More d e t a i l s and a p p l i c a t i o n s to experiment w i l l be given i n the next chapters. 4. E f f e c t of T u n n e l l i n g on T-j Equations (3.44), (3.50) can be w r i t t e n i n a most general way known as the modified BPP Equation [3.37] I - C , T l 4 T C 1 +H)V 1 + -K\ 2 (3.55) where x c i s assumed to obey the Arrhenius equation s i m i l a r to Equation (3.26) i . e . T =.T_ exp(E /RT). Equation (3.55) shows a C O a minimum i n the temperature dependence of T-j at MQTC = 0.616 f o r one p a r t i c u l a r type of motion. A l l e n and Cowking [3.38] observed m u l t i p l e minima i n T-j i n a s e r i e s of methyl benzenes at very low temperatures. It was r e a l i z e d that the c l a s s i c a l theory of random r e o r i e n t a t i o n of the CH^-group at very low temperatures was no longer v a l i d . I t was suggested by these authors that a t r a n s i t i o n took place i n t h i s Figure 3.1 Some H i l t and Hubbard (H-H) curves f o r d i f f e r e n t ( ^ x ^ 2 . These curves are obtained from the data of Tables A2-A8 ( c f . p. 172-175). o.o H.O i i or C ~^-2.0 -3.0 (CU 0 T- C ) 2 = I O O ( c u 0 r c ) 2 = 3 . 0 ( c u 0 - r c ) 2 = I O . O ( O J 0 T c ) 2 = 5 . 0 ( O ; 0 T - c ) 2 = | . 0 J L ° 2 4 6 8 IO 12 14 16 18 2 0 Figure 3.1 continued. These curves are obtained from the data of Tables A9-A15 ( c f . p. 176-179) - 52 -temperature from a motional process a t high temperature (obeying c l a s s i c a l random jump t h e o r i e s ) to the quantum mechanical t u n n e l l i n g process dominant at l i q u i d helium temperatures. T h i s t u n n e l l i n g mechanism was r e a l i z e d before by Powles and Gutowsky [3.27] and St e j s k a l and Gutowsky [3.31] who assumed the b a r r i e r to r o t a t i o n of methyl group of the form V = (V Q/2) (1 + cos3tj>) (3.56) where V q i s the height o f p o t e n t i a l b a r r i e r and <j> i s the angular c o o r d i n a t e s d e s c r i b i n g r o t a t i o n o f the CH^-group. This p o t e n t i a l f u n c t i o n when put i n Schrodinger equation g i v e s , j ! . + [E - \ (l-cos3<j>)M<j>) = 0 ( 3 - 5 7 ) 21 6cf> 2 where ^ {$) i s wavefunction d e s c r i b i n g the p o s i t i o n of the CH^ group, E i t s energy, and I, i t s moment o f i n e r t i a about the symmetry a x i s . Powles and Gutowsky [3.27] attempted to c a l c u l a t e the average t u n n e l l i n g frequency while S t e j s k a l and Gutowsky [3.31] a f t e r w r i t i n g Equation (3.57) according to way of Koehler and Dennisson [3.39] i . e . d2M(cf>)/d<{>2 + [A - 2cos3<f)]M(cf)) = 0 ....(3.57) (which i s Mathieu equation, with M(cf>) = ^(cf>), A = (I/ft 2)(2E-V Q)) solved f o r 30 lowest eigenvalues and c a l c u l a t e d the average t u n n e l l i n g frequency from the s p l i t t i n g of t o r s i o n a l s t a t e s . More r e c e n t l y A l l e n and Clough [3.40] have c a l c u l a t e d the temperature dependence of T-| taki n g i n t o account the t u n n e l l i n g s p l i t t i n g of the t o r s i o n a l - o s c i l l a t o r - 53 -ground s t a t e of the hindered CH^-group which accounts f o r second minimum a t low temperature. The expression obtained by these authors i s [3.40] 1 , C T 4T \—z + 2 + *f((V + 47Tf (2a3o} 1+0) T 1+4(JO T 0 C 0 c (3.58) where f(u>) = 8y/{ 6TTJ[X 4 + 2 x 2 ( 2 y 2 - l ) + 1] } with x = GO/3J, y = ( 3 d r t ) - 1 . In x and y, 3Jfi i s t u n n e l l i n g s p l i t t i n g and i t i s the c o r r e l a t i o n time obeying the = T q exp(E Q/RT) equation. In Equation (3.58), the f i r s t two terms are the usual BPP terms dominant at high temperature, while the l a s t two terms are due to those t u n n e l l i n g processes important a t very low temperatures. A more elabora t e theory o f T-j due to t u n n e l l i n g motion i s presented i n a very re c e n t paper by Clough [ 3 . 4 1 ] . E. D i s t r i b u t i o n of C o r r e l a t i o n Times and Th e i r E f f e c t on T, and E, a The modified BPP Equation (3.55) i s only s a t i s f a c t o r y f o r one c o r r e l a t i o n time T . Because of the symmetry of the BPP equation about the minimum, the a c t i v a t i o n energy E, can be extracted from the a slopes on e i t h e r s i d e of minimum, e.g. i ) i n the low temperature region (^ 0T C » 1), InT^ ^  E a/ R T where slope i s + E =/R., a i i ) i n the high temperature region (W Q T c. . « ! ) , lnT-j ^ -E a/RT where slope i s - E /R., a - 54 -However, when there i s a d i s t r i b u t i o n of c o r r e l a t i o n times, a broad and f l a t minimum e x i s t s and the a c t i v a t i o n energies extracted on the assumption of the BPP equation, or from the slopes,are some-times much smaller than those reported from other measurements. T h i s d i s t r i b u t i o n of c o r r e l a t i o n times may be due to two cases which we o u t l i n e b r i e f l y . The f i r s t case i s that we may have two independent types of r o t a t i o n proceeding a t nearly the same temperature but with d i f f e r e n t r a t e s and each c o n t r i b u t i n g towards T-|. The known example of t h i s case i s tert-butyl group, which may e x h i b i t both methyl and t e r t - b u t y l group r o t a t i o n . Sometimes i t i s p o s s i b l e to see separate minima due to both groups because of d i f f e r e n t b a r r i e r s to r o t a t i o n e.g. 1-tert-butyl -4-methyl -benzene i n the work of A l l e n and Johnson [3.42]. On the other hand i n the m a j o r i t y of cases the c o r r e l a t i o n times f o r methyl and tert-bu t y l are n e a r l y the same and give a s i n g l e broad minimum. In t h i s case the two BPP equations should be combined together to account f o r t h i s minimum, each with i t s own C and T . Examples of t h i s case w i l l be seen i n the f o l l o w i n g chapters. The second case i s where the same r o t a t i n g group i n d i f f e r e n t environments and with d i f f e r e n t T C ' S gives r i s e to a broad and f l a t minimum. This case has been reviewed by Odajima [3.43] and Connor [3.44]. We w i l l very b r i e f l y o u t l i n e some aspects of t h i s case. According to Odajima [3.43] t h i s kind of d i s t r i b u t i o n i s test e d by r a t i o ( T 1 / T 2 ) a t minimum. I f t h i s r a t i o i s 1.6, the BPP equation i s v a l i d and i f the - 55 -r a t i o i s greater than 1.6, other forms o f d i s t r i b u t i o n are present. To account f o r broad minimum i n t h i s case, we have to inc l u d e a continuous d i s t r i b u t i o n of T C ' S instead o f s i n g l e T . This i s u s u a l l y done [3.44] by a de n s i t y f u n c t i o n G ( x ^ ) , where G(T ) d T „ = 1 Jo c c to give r i s e to a new expression f o r T^ from BPP equation, /OO 00 T c G < T c > d T c f T c G ( T c > d T 1+W 2 T 2 J 1+400 2 T 2 uo o c 0 0 c (3.59) Density f u n c t i o n s are defined i n terms o f S = &n(T^/T^) where T Q i s c e ntre o f d i s t r i b u t i o n on logarithmic s c a l e . Then G(x c) i s replaced by F(S) with c o n d i t i o n s , J -oc F(S);d.S= 1, G ( T c ) dx c = F(S)dS, T C G ( T c ) = F(S) ....(3.60) The new d i s t r i b u t i o n F(S) i s of two types, symmetric and asymmetric. Some of symmetric d i s t r i b u t i o n f u n c t i o n s are as under [ 3 . 4 4 ] . i ) Gaussian d i s t r i b u t i o n . This d i s t r i b u t i o n has no e f f e c t on the slopes i n the curve of lnT^ versus 1/T, except i t broadens the minimum. This case i s discussed by [3.43] and by Resing [ 3 . 4 5 ] . i i ) Rectangular d i s t r i b u t i o n . This d i s t r i b u t i o n has als o no i e f f e c t on the slope of the InT^, versus 1/T curve. An example of t h i s i s the work o f McCall e t a l . [ 3 . 4 6 ] . i i i ) Fouss and Kirkwood; Cole and Cole d i s t r i b u t i o n s . These d i s t r i b u t i o n s change the slopes o f the l n ^ versus 1/T curve, but these are symmetric [ 3 . 4 4 ] . - 56 -The only asymmetric d i s t r i b u t i o n f u n c t i o n i s that of Davidson and Cole [3.44] where the slopes on both sides of curve (lnT-j vs 1/T) are d i f f e r e n t . The r a t i o of the two slopes gives the width of the d i s t r i b u t i o n . These d i s t r i b u t i o n s are i n f a c t important i n d i e l e c t r i c work from which they have o r i g i n a t e d . I t i s very hard to use them in NMR because of the d i f f i c u l t y i n f i n d i n g the width of the d i s t r i b u t i o n . There i s some doubt from the work of Hunt and Powles [3.47] and Zimmerman and B r i t t i n [3.48] whether such kind of d i s t r i b u t i o n e x i s t s or not or the broad minimum i s due to non-exponential c o r r e l a t i o n f u n c t i o n . - 57 -REFERENCES (Chapter I II) 3.1] E.R. Andrew, Nuclear Magnetic Resonance, Cambridge Univ. Press, 1955. 3.2] A. Abragam, The P r i n c i p l e s of Nuclear Magnetism, Oxford Univ. Press, 1961. 3.3] C P . S l i c h t e r , P r i n c i p l e s o f Magnetic Resonance, Harper, N.Y., 1963. 3.4] A. Aleksandrov, The Theory of Nuclear Magnetic Resonance, Eng. Trans, by C P . Poole J r . , Academic Press, N.Y., 1966. 3.5] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance  i n S o l i d s , Oxford Univ. Press, 1970. 3.6] T.C F a r r a r and E.D. Becker, Pulsed and Fo u r i e r Transform NMR, Academic Press, N.Y., 1971. 3.7] I . J . Lowe and R.E. Norberg, Phys. Rev., J_07 (1957) 46. 3.8] S. Gade, Phys. Rev., J87 (1969) 419. 3.9] R.E. Fornes, G.W. Parker and J.D. Memory,. Phys. Rev., 1 (1970) 4228. 3.10] F. Lado, J.D. Memory and G.W. Parker, Phys. Rev., 4 (1971) 1406. 3.11] T.B. Cobb and C S . Johnson J r . , J . Chem. Phys., 52 (1970) 6224. 3.12] E.R. Andrew and J.R. Brookman, J . Mag. Res., 2 (1970) 259. 3.13] E.R. Andrew and R. Bershon, J . Chem. Phys., J8 (1950) 159. 3.14] F. Apayadin and S. Clough, J . Phys. C, 1 (1968) 932. 3.15] R. B l i n c , Z. T r o n t e l j and B. Volavsek, J . Chem. Phys., 44 (1966) 1028. 3.16] M.B. Dunn and C A . McDowell - ( t o be published) 3.17] J.H. Van Vleck, Phys. Rev., 74 (1948) 1168. 3.18] J.G. Powles and J.H. Strange, Proc. Phys. Soc., 82 (1963) 6. - 58 -i3.19] P. M a n s f i e l d , Phys. Rev., 137 (1965) A961. ;3.20] J.G. Powles and P. M a n s f i e l d , Phys. L e t t e r s , 2 (1962) 58. 3.21] G.E. Pake, J . Chem. Phys., 16 (1948) 327, S o l i d State Phys. 2 (1956) 1. !3.22] E.R. Andrew and R.A. Newing, Proc. Phys. S o c , 72 (1959) 959. 13.23] G.W. Smith, J . Chem. Phys., 42 (1965) 4229. !3.24] H.S. Gutowsky and G.E. Pake, J . Chem. Phys., 18 (1950) 162. !3.25] E.R. Andrew, J . Chem. Phys., 18 (1950) 607. [3.26] H.G. O l f and A. P e t e r l i n , J . Polym. Sc., Pt. A-2, 8 (1970) 753. "3.27] J.G. Powles and H.S Gutowsky, J . Chem. Phys., 21 (1953) 1695, 1704; 23 (1955) 1692. ;3.28] E.R. Andrew, Proc. Roy. S o c (London). A216 (1953) 398. 13.29] P.S. A l l e n , J . Chem. Phys., 48 (1968) 3031. ;3.30] S. Clough, J . Phys. C, S o l i d State Phys., 4 (1971) 1075. ;3.31] E.A. S t e j s k a l and H.S. Gutowsky, J . Chem. Phys., 28 (1958) 388. !3.32] P.S. Hubbard, Rev. Mod. Phys., 33 (1961) 249. [3.33] F. Bloch, Phys. Rev., J05 (1957) 1206 and references t h e r e i n . ;3.34] A.G. R e d f i e l d , IBM J . Research Develop. 1 (1957) 19. 13.35] R.L. H i l t and P.S. Hubbard, Phys. Rev., 134 (1964) A392. !3.36] D.E. O ' R e i l l y and T. Tsang, Phys. Rev., 1_5_Z (1967) 417. 13.37] R. Kubo and K. Tomita, J . Phys. S o c Japan 9 0954) 888. !3.38] P.S. A l l e n and A. Cowking, J . Chem. Phys., 49 (1968) 789. !3.39] J.S. Koehler and D.M. Dennison, Phys, Rev., 57 (1940) 1006. ;3.40] P.S. A l l e n and S. Clough, Phys. Rev. L e t t e r s , 22 (1969) 1351. 13.41] S. Clough, J . Phys. C. ( S o l i d State Phys.), 4 (1971) 2180. - 59 -[3.42] P.S. A l l e n and L.W. Johnson (to be published) [3.43] A. Odajima, Progr. Theoret. Phys. (Kyoto), Suppl. 10 (1959) 142. [3.44] T.M. Connor, Trans. Faraday S o c , 60 (1964) 1574, and references t h e r e i n . [3.45] H.A. Resing, Adv. Mol. Relaxation Process, 1 (1967) 109, J . Chem. Phys., 43 (1965) 669. [3.46] D.W. Mc C a l l , D.C. Douglass and E.W. Anderson, J . Chem. Phys., 30 (1959) 1272. [3.47] B.J. Hunt and J.G. Powles, Proc. Phys. S o c , 88 (1966) 513. [3.48] J.R. Zimmerman and W.E. B r i t t i n , J . Phys. Chem., 61_ (1957) 1328. - 60 -CHAPTER IV APPARATUS AND METHODS OF MEASUREMENT This chapter i s intended to give some d e t a i l s o f apparatus used and the methods of measurement of second moments M 2, and s p i n -l a t t i c e r e l a x a t i o n times T-j. The pr e p a r a t i v e methods f o r hydrates are not di s c u s s e d , because they are not general and w i l l be given i n d e t a i l i n the r e s p e c t i v e chapters. A. Continuous Wave (cw) Measurements 1. cw Spectrometer The spectrometer used f o r cw measurements was a conventional c r o s s - c o i l Varian V4200 wide l i n e spectrometer equipped with a s i x inch magnet producing good homogeneous and s t a b ! i z e d p o l a r i z i n g magnetic f i e l d . The t r a n s m i t t e r and r e c e i v e r u n i t was a Varian V4210A v a r i a b l e frequency r f u n i t operating a t a resonance frequency of 16 MHz i n a l l experiments. The usual l o c k - i n d e t e c t i o n method was used. The modulating f i e l d was s u p p l i e d by Varian V4250B sweep u n i t and i n a l l experiments the modulation frequency was kept at 80 Hz. The f i r s t d e r i v a t i v e o f the V mode or absorption s i g n a l was recorded by means of Varian V4270B output c o n t r o l u n i t a m p l i f i e r and phase s e n s i t i v e d e t e c t o r . A Varian 6-10 s t r i p c h a r t recorder was used. - 61 -2. C a l i b r a t i o n of Spectrometer The resonance f i e l d H Q was adjusted from the absorption s i g n a l obtained from a dopped water sample. The c a l i b r a t i o n o f the scan r a t e was acheived by u t i l i z i n g the s i d e band technique. To produce these s i d e bands, the main r f c a r r i e r frequency (16 MHz) was modulated by a known audiofrequency generated from H.P. model 200CD wide range o s c i l l a t o r , the frequency of which was measured by a H.P. model 3734A e l e c t r o n i c counter. Thus scan rate was c a l i b r a t e d i n gaus's per cm along the base l i n e of the recorder. The modulation amplitude was c a l i b r a t e d by d i r e c t l y recording the narrow absorption s i g n a l from a oyermodulated dopped water sample. The observed l i n e width obtained from t h i s overmodulated water sample was taken equal to 2H m, where H m i s modulation amplitude i n gauss. 3. Line Width and Second Moment Measurements The l i n e width was taken as the d i s t a n c e between peak to peak i n the f i r s t d e r i v a t i v e absorption curve. The second moment expression f o r the experimental absorption d e r i v a t i v e curve a f t e r s u b t r a c t i n g the well-known modulation c o r r e c t i o n due to Andrew [4.1] i s M 2 = \ l f + a } h 3(dg/dh)dh// +°° h(dg/dh)dh]-^H 2 — OO — CO (4.1) where g(h) = g ( H Q - H) = Y _ 1 g ( u 0 - w ) = Y ^ 9 (A ) , H q i s the resonance f i e l d , H the f i e l d away from resonance and H i s the modulation amplitude. - 62 -For numerical i n t e g r a t i o n purposes Equation (4.1) reduces to (4.2) where n i s the number of s e c t i o n s on the x - a x i s , i s the y - a x i s height o f the d e r i v a t i v e curve, N i s the maximum number of s e c t i o n s on the x-axis and S i s scan r a t e value i n gauss per s e c t i o n on the x - a x i s . A computer programme was used to c a l c u l a t e the second moments using Equation (4.2). At l e a s t four s p e c t r a were recorded a t each tempera-ture and the l i n e widths and second moments from these were then averaged by an other computer programme. The r f f i e l d used was adjusted from the r e s u l t s based on T^ and T 2 , to avoid s a t u r a t i o n . In some cases where T-j and T 2 measurements were not c a r r i e d out, and where s a t u r a t i o n was predominant e.g. i n some pure amines at 77 K, the minimum r f f i e l d a v a i l a b l e on the u n i t to produce good S/N r a t i o s p e c t r a was used. The second moments reported i n t h i s region are b e l i e v e d to be accurate to about ±20% while i n other cases the accuracy may be ±10%. The modulation amplitude and the modulation frequency were always kept much smaller than the l i n e width to s a t i s f y t h e . f i r s t two c o n d i t i o n s of Provotorov's Theory [4.2]. - 63 -4. V a r i a b l e Temperature Assembly At temperatures of 77K and 88K the spectra were recorded by d i r e c t l y immersing the sample i n l i q u i d nitrogen and l i q u i d oxygen r e s p e c t i v e l y . For temperatures from 100K and onwards the c o l d gas flow methods were used. The l i q u i d n itrogen was b o i l e d o f f with a heater immersed i n a 50 l i t r e tank. The c o l d nitrogen gas was then led from a dewar system to the probe. The temperature was v a r i e d e i t h e r by changing the voltage to the heater through a v a r i a b l e transformer or by heating the c o l d gas with an a d d i t i o n a l heater placed i n s i d e the dewar system. To get minimum temperature g r a d i e n t , the voltage on the heater immersed i n l i q u i d n itrogen was held f i x e d at a value which gives a good high flow r a t e of c o l d gas and n e g l i g i b l e thermal g r a d i e n t , and then the voltage on the other heater was v a r i e d to achieve change i n temperature. The temperature of the sample was measured using a copper-constantan thermocouple placed approximately 1/4 inch below the sample, and using a Leeds and Northrup Type G s t r i p c h a r t r e c o r d e r , or a potentiometer. For temperatures between 77K and 100K, the c o l d gas from the 50 l i t r e tank was f i r s t passed through a copper heat exchanger placed i n l i q u i d nitrogen before reaching the dewar system. With t h i s type of system the sample temperature i s held constant to ±2.OK f o r about 5 to 6 hours between 100K and onwards. The s t a b i l i t y of the temperature between 77 and 100K depends on the amount of l i q u i d n itrogen used to cool the copper heat exchanger. - 64 -B. S p i n - L a t t i c e Relaxation Measurements 1. Pulse Spectrometer The pulse spectrometer used f o r T-j measurements was a Bruker v a r i a b l e frequency (16 to 62 MHz) pulse spectrometer. The frequency on which a l l our experiments were performed was 26.46 MHz. This spectrometer contains a b a s i c 1MHz quartz o s c i l l a t o r -8 -9 with a frequency s t a b i l i t y l y i n g between 10 - 10 . A l l the measuring times are d e r i v e d d i g i t a l l y from t h i s o s c i l l a t o r and t h e i r accuracy i s of the same order as t h a t of the o s c i l l a t o r frequency. Three separate pulse channels are a v a i l a b l e to gate the high frequency i n the o s c i l l a t o r u n i t . The b a s i c 1MHz frequency from the main o s c i l l a t o r i s taken i n t o a v a r i a b l e frequency u n i t (frequency synthe-s i z i n g u n i t ) , where h a l f of the resonance frequency i s produced. This h a l f high frequency i s then fed to three channels a f t e r a m p l i f i c a t i o n . The f i r s t channel i s gate channel I, where the high frequency s i g n a l i s fed a f t e r being phase s h i f t e d , doubled and a m p l i f i e d . In the second channel, which i s gate channel I I , the high frequency i s fed d i r e c t l y a f t e r being doubled and a m p l i f i e d . In the t h i r d channel, termed the reference channel, the high frequency fed serves as a phase coherent reference frequency f o r the phase s e n s i t i v e d e t e c t o r . The gate channel I i s opened only by pulse I and the gate channel II i s opened by pulses II and I I I . The high frequency i n form of pulses a f t e r passing through the gates opened up by d.c. pulses I, II and III i s l e d through a f i v e stage a m p l i f i e r to the t r a n s m i t t e r c o i l ( s i n g l e c o i l ) i n t o the probe where i t e x c i t e s the low frequency s i g n a l . This - 65 -low frequency s i g n a l combined with the s i g n a l produced by r f pulses i s passed through a p r e a m p l i f i e r and a f t e r a ttenuation i s detected by the r e c e i v e r e i t h e r by diode or phase s e n s i t i v e d e t e c t i o n . The maximum band width of the r e c e i v e r i n the spectrometer i s 1MHz and i t can be reduced to 100kHz. The dead time of the r e c e i v e r a f t e r an r f pulse i s approximately 5-6 ysec. Most of the experiments i n t h i s work were done using a band width of 1MHz, but i n some cases where noise l e v e l was high, a band width up to 300kHz was used. The magnet used f o r p o l a r i z i n g magnet f i e l d H Q was Varian DP-60, 12-inch high r e s o l u t i o n electromagnet. The s i g n a l amplitudes were recorded on a Tektronix Type 549 storage o s c i l l o s c o p e (band width 30 MHz) with Type 1A1 Dual Trace p l u g - i n u n i t . 2. L i n e a r i t y of Receiver In a l l experiments we have used a diode d e t e c t o r . To check the l i n e a r i t y o f the r e c e i v e r we connected the output 1 of a H.P. s i g n a l generator, Model 606 A, operating at 25MHz. to the input of the r e c e i v e r . The output voltage from the r e c e i v e r was measured with Tektronix Type 549 o s c i l l o s c o p e . Figure 4.1 shows the output voltage as a f u n c t i o n of the input voltage from a H.P. s i g n a l generator. The l i n e a r i t y range depicted by these output curves i s as f o l l o w s : -curves 1 to 3 are l i n e a r between 2.5 and 9 v o l t s curve 4 i s l i n e a r between 3 and 8.5 v o l t s . - 99 -- 67 -Beyond these ranges of output voltage the diode shows n o n - l i n e a r i t y c h a r a c t e r i s t i c s . Most of the measurements were taken i n the range of l i n e a r i t y ; those which were outside were c o r r e c t e d f o r n o n - l i n e a r i t y of the diode d e t e c t o r . 3. V a r i a b l e Temperature Assembly This assembly d i f f e r s from the previous one i n th a t the dewar leads going to the probe, and the temperature c o n t r o l u n i t s which are parts of Bruker temperature c o n t r o l u n i t B-ST 100/700. A Bruker quartz made v a r i a b l e temperature probe was used i n some of the e a r l y experiments with a s l i g h t m o d i f i c a t i o n made by A l l e n * The whole probe head was close d i n a transparent l u c i t e box, to avoid condensation o f water vapour from the outside atmosphere. A c y l i n d e r -i c a l brass l i d was placed on the top of the quartz probe which had an o u t l e t f o r the out going c o l d nitrogen gas. This prevented the c o o l -ing o f t h e " l u c i t e box as well as the condensation of water vapour i n s i d e . To prevent f u r t h e r water vapour condensation, a stream of dry and warm nitrogen was blown i n s i d e the l u c i t e box and the probe head. The c o l d gas to acheive the d e s i r e d temperature was passed through the dewar system using the gas from b o i l i n g l i q u i d n i t r o g e n from a 50 l i t r e tank equipped with a heater. Temperature adjustment was acheived by varying the voltage on the heater placed i n the tank through a v a r i a b l e transformer. The temperature was monitored using a copper-constantan thermocouple placed nearly 1cm below the sample and was read d i r e c t l y i n degrees K from the Bruker * V i s i t i n g Professor from Department of Physics, U n i v e r s i t y o f Nottingham, England. - 68 -temperature c o n t r o l u n i t . This setup worked well up to 11 OK. Lat e r on, to acheive a much lower temperautre, the whole quartz v a r i a b l e temperature probe was redesigned by A l l e n . With t h i s , a temperature below 11 OK was acheived and by pre c o o l i n g the c o l d gas i n the copper heat exchanger i n the same manner as before, tempera-ture up to 77 K could be obtained. The temperature between 77 and 100K was again monitored by copper-constantan thermocopule, but determined by potentiometer measurements. The temperature i n both cases was accurate to ±0.5K. Temperature v a r i a t i o n i n the range 77 to TOOK was acheived e i t h e r by changing the gas flow or heating the flowing gas by another heater present i n Bruker dewar system. In the higher range (100K and onwards), e i t h e r the gas flow was changed (up to a c e r t a i n l i m i t to give minimum temperature gradient) or use was made of the Bruker ser v o - c o n t r o l system to get the d e s i r e d temperature. 4. Measurement of S p i n - L a t t i c e Relaxation Time S p i n - l a t t i c e r e l a x a t i o n time was measured by 180°-T-90° pulse sequence. As discussed i n Chapter I I I , i f the r e l a x a t i o n i s e x p o n e n t i a l , the r e l a x a t i o n f u n c t i o n R(x) i s given by Equation (3.48). R(T) = [ M 0 - M Z ( T ) ] / 2M q = exp(-x/T 1) (4.3) where M Q i s now pr o p o r t i o n a l to voltage of the s i g n a l a f t e r the 90° pul s e , and M z(x) i s the voltage o f the s i g n a l a f t e r a 180°-x-90° pulse sequence. For d i f f e r e n t x's, we get d i f f e r e n t values of M z(x) and a p l o t of In R(x) vesus x should give a s t r a i g h t l i n e with slope (-1/T-,). - 69 -A l t e r n a t i v e l y one can s e t 180°-T-90° pulse sequence i n such a way so as to gi v e a t a time T , M z ( T Q ) = 0. In that case (Null Method) Equation (4.3) gives R ( T q ) = 1/2 and T ] = T Q / l n 2 = 1.443T q (4.4) However, t h i s method i s not very accurate, e s p e c i a l l y when the r f f i e l d H^  i s not homogeneous. A method f o r c o r r e c t i n g the H^  inhomogeneity has r e c e n t l y been proposed by van Putte [4.3] which we w i l l o u t l i n e b r i e f l y here. 5. C o r r e c t i o n o f H-j Inhomogeneity We w i l l only give the method which i s a p p l i e d to s o l i d s where T 2 i s o f the order o f the 180° pulse length, and s h o r t e r than T-j According to van Putte [4.3] the magnetization M (T) along the x-axis i n the case o f an inhomogeneous r f f i e l d H ] a f t e r 180°-T-90° pulse sequence, i s M X (T ) = M Q(1 - A ) [1-2(1-2A) exp ( - T / ^ ) ] (4.5) where A=6MQ/M0 and 6MQ i s the decrease i n magnetization a f t e r 90° pulse due to H^  inhomogeneity. When we use the second method (Null Method), we s e t T i n the 180°-T-90° sequence such that M (T ) = 0 at a time T=T . Equation (4.5) then gives e x p ( - T 0 / T 1 ) = 1/[2(1-2A) ] 4.6 or T 1 = T Q/ln(2-4A) 4.7 The f a c t o r 4A was c a l c u l a t e d using 90°?and 270° pulses. For the 270° pulse, we used the same 180°-T-90° pulse sequence but x was - 70 -chosen i n such a way that i t was greater than T 2 (to n e g l e c t the e f f e c t o f s p i n - s p i n i n t e r a c t i o n ) and sh o r t e r than T-j. U s u a l l y f o r broad l i n e s T was chosen to be 100 ysec, but f o r narrow l i n e s a l a r g e r value (200psec) was used. Thus the d i f f e r e n c e i n the i n i t i a l magnetization a f t e r the 90° pulse (M g Q) and the 180°-T-90° pulse sequence ( M 2 7 0 ) i s given by [4.3] M90" M270 = M o ( 1 ' A ) • m 0 0-4A)(1-A) = 4A(1-A)M 0 (4.8) and so d i v i d i n g t h i s by M g Q = M (1-A) we get, ( M G O - M 2 7 Q ) / M 9 0 = 4A (4.9) This 4A when used i n Equation (4.7) gives T-j. x Q can be found from the graph o f In R(T) versus T. [In t h i s p l o t R(T) i s s l i g h t l y put i n a d i f f e r e n t way i . e . now M = M (1-A) and M (x) = M (x ) ] . U U / L A . A computer programme was used to c a l c u l a t e R(x) with c o r r e c t i o n s (Appendix B) f o r d i f f e r e n t values o f x. I t should be noted that Equation (4.7) i s the c o r r e c t expression f o r t h i s case, van Putte [4.3] uses an approximate expression f o r T-j i . e . <Vvan = VL" l n 2-l! A"J ( 4' 1 0 ) where A" = 4A and (A"/2) « 1 This method (Null Method with c o r r e c t i o n ) and the slope method gives r e s u l t s d i f f e r i n g approximately 5% from each other. - 71 -C. The Cold Box C l a t h r a t e hydrates are b e a u t i f u l compounds to study, but they are too d e l i c a t e . t o handle a t room temperatures. A l l the hydrates studied here are unstable at room temperature. Therefore a l l the p r e p a r a t i v e procedures had to be c a r r i e d out below room temperature. A c o l d box was constructed from a food f r e e z e r (Zanussi Food 'Freezer Company, I t a l y ) which had a dimension of 11 x 18 x 13 inches. In the o r i g i n a l form the lowest temperature obtained from t h i s f r e e z e r was -30°C. For handling the hydrates i n s i d e the f r e e z e r , the f r o n t door was replaced by a double walled transparent l u c i t e door having two t h i c k long rubber gloves attached to i t f o r handling the compounds. This door had the same a i r - t i g h t rubber l i n i n g as the one i n the o r i g i n a l door. This m o d i f i c a t i o n r a i s e d the o r i g i n a l temperature from -30° to -15°C. This -15°C temperature was good enough f o r amine hydrates s t u d i e s here, but was not so good f o r the s t u d i e s on acetone hydrate. The acetone hydrate i s s t a b l e below -35°C [4.4]. The box was modified f u r t h e r and two holes were d r i l l e d i n the l a t e r a l s i d e s , one near the bottom of the s i d e , and the other on top of the other s i d e . A stream of c o l d nitrogen was blown from the bottom hole by b o i l i n g l i q u i d n itrogen from a 5 0 - l i t r e tank with a heater immersed i n s i d e (same procedure as used i n the v a r i a b l e temperature assemblies). The gas escaped from the top hole. With t h i s way a temperature as low as -80°C was r e a l i z e d i n s i d e the c o l d box, and t h i s temperature could be maintained nearly constant f o r a p e r i o d o f some 6-7 hours. - 72 -REFERENCES (Chapter IV) [4.1] E.R. Andrew, Phys. Rev., 9J_ (1953) 425. [4.2] M. Goldman, Spin Temperature and Nuclear Magnetic Resonance  i n S o l i d s , Oxford U n i v e r s i t y , Press., 1970 p. 109 [4.3] K. van Putte, J . Mag. Res., 2 (1970) 174 [4.4] A.S. Qu i s t and H.S. Frank, J . Phys. Chem., 65 (1961) 560 - 73 -CHAPTER V DIETHYLAMINE AND DIETHYLAMINE CLATHRATE DEUTERATES A. Introduction This chapter describes the d e t a i l s of an nmr study of d i e t h y l amine and i t s c l a t h r a t e deuterate. A p r e l i m i n a r y d i s c u s s i o n has already been given i n Chapter I concerning amine hydrates. The c r y s t a l l o -graphic aspects o f these amine hydrates has been mentioned i n Chapter I I . Diethylamide forms two types of c l a t h r a t e hydrates, type I (not the von Stackelberg's type I s t r u c t u r e ) with formula (C 2H 5) 2NH-6.8H 20 which i s monoclinic with space group P2-j/c, melting at -6.6°C; and type II [not the von Stackelberg's type II s t r u c t u r e ) with formula (C 2H 5) 2NH-8.7H 20, which i s orthorhombic with space group Pbcn, and melting at -7°C [5.1]. The c r y s t a l s t r u c t u r e of the type II hydrate has been reported i n d e t a i l by Jordan and Mak [5.2]. They [5.2] o have shown that u n i t c e l l i s orthorhombic with edges a = 13.44 A, o o b = 11.77 A and c = 27.19 A at -30°C and t h a t space group i s Pbcn. The u n i t c e l l contains 12 d i e t h y l amine molecules and 104 water molecules. Within the u n i t c e l l there are four 18-hedra formed by 32 oxygen atoms with 48 hydrogen bonding edges. In a d d i t i o n to these 18-hedra, there are 8 i r r e g u l a r cages. The d i e t h y l amine may be contained i n s i d e e i t h e r the 18-hedra or the i r r e g u l a r cages with t h e i r nitrogen atoms - 74 -hydrogen bonded to water cages (Figure 2.5). We have s t u d i e d here the type II hydrate ( i n f a c t d e u t e r a t e ) . In the f i r s t p r e p a r a t i o n , the deuterate was prepared by mixing (C^Hg^NH and D,>0, but l a t e r on because o f the exchange of -NH protons with D^O, the deuterate was prepared from ( f ^ H ^ N D . We w i l l r e f e r to the deuterate prepared from CC'2H5I2NH as DNH-D20 and that from (C 2H 5) 2ND as DND-D20. The reason f o r studying DND-D20 was to check the c o n t r i b u t i o n of the exchanged protons to the second moment. Tt w i l l be seen l a t e r t h at t h i s exchange has an apparently n e g l i g i b l e c o n t r i b u t i o n , because the exchange proton goes to deuteeate cage, where because of l a r g e i n t e r n u c l e a r d i s t a n c e from the guest protons, i t has n e g l i g i b l e c o n t r i b u t i o n . The guest i n pure s t a t e has the -NH proton deuterated, so that, a good comparison can be made between the amine wKen pure and when c l a t h r a t e d . B. Experimental 1. M a t e r i a l s Reagent grade d i e t h y l amine was obtained from Eastman Kodak, and the D 20 (99.8% d 2) was obtained from S t o h l e r Isotope Chemicals. The reagent grade d i e t h y l amine was f i r s t d r i e d on KOH f o r several days. The i n i t i a l samples were prepared immediately from the d r i e d m a t e r i a l , but l a t e r samples were prepared a f t e r diethylamide had been i n i t i a l l y d i s t i l l e d three times from BaO. 2. Preparation o f (C 2H 5) 2ND This method i s e s s e n t i a l l y the same as used by Ross et a l . [5.3] - 75 -which i s based on the method of Hawthorne [5.4]. 50 ml of amine was added to 20 ml of 99.8% D 20 to which p r e v i o u s l y had been added 1 g of P o 0 c . The s o l u t i o n was r e f l u x e d f o r about 3 hours. The amine was c o d i s t i l l e d from the r e a c t i o n mixture and then r e d i s t i l l e d from 3 g BaO. The exchange and drying procedures were repeated three times. The f i n a l product was then d i s t i l l e d 5 times from 3 g BaO to get a constant b.p. The a n a l y s i s of deuterate on Varian T-60 high r e s o l u t i o n nmr spectrometer showed that the -NH group was more than 90% deuterated to -ND. 3. Preparation o f Deuterate and Amine Samples Three d i f f e r e n t batches o f deuterate were prepared. The i n i t i a l batch was made from the d r i e d d i e t h y l amine by mixing the type II formula r a t i o o f (CgHg^NH and D^O i n an nmr tube and f r e e z i n g i n the c o l d box at -15°C. The second batch was prepared from d r i e d and t r i p l y d i s t i l l e d diethylamide by s e a l i n g the type II formula r a t i o of (CgHg^NH and D^O i n a tube and keeping the mixture at -15°C i n the cold box. The deuterate c r y s t a l s grow very slowly. When the c r y s t a l -l i z a t i o n was complete, a p o r t i o n of the deuterate (DNH-D^O) was removed from the tube and f i n e l y powdered at -15°C before being t r a n s f e r r e d to nmr sample tubes (10 mm o.d. f o r cw and 7 mm o.d. f o r pulse experiments). The t h i r d batch DND-D^O, was prepared i n the same way as the second. The a n a l y s i s of deuterates prepared i n the c o l d box was made on the decomposed l i q u i d mixtures. Such a n a l y s i s i n d i c a t e d that the deuterate cages were 92-96% f i l l e d with diethylamide molecules. - 76 -Pure d i e t h y l amine-ND [(CgHg^ND] was t r a n s f e r r e d to nmr tubes d i r e c t l y and was sealed o f f a f t e r removing the d i s s o l v e d a i r by a f r e e z e pump thaw method. The vast m a j o r i t y of the deuterate data were taken on the second batch o f samples. However l i t t l e d i f f e r e n c e was observed i n the second moment data between any of the three batches and i n the r e l a x a t i o n data from measurements on the f i r s t and second batches. 4. Spectrometers^ These were the same as described i n Chapter IV. The maximum r f f i e l d i n the absorption measurements was 40 mG. The recovery of magnetization was obtained by using a 180 ° - x-90° pulse sequence with a 180° pulse length of 3.5 ysec. The H-j c o r r e c t i o n method as described i n Chapter IV was a p p l i e d and i t was found that the r e l a x a t i o n f u n c t i o n R(t) was non-exponential. Consequently t Q , the time taken f o r R(t) to become equal to 1/2 was used instead of T-j. H-H theory [5.5] was used to i n t e r p r e t the r e l a x a t i o n data. C. Results 1. Absorption Line A n a l y s i s (cw Measurements) (a) Second Moment C a l c u l a t i o n s For p o l y c r y s t a l l i n e samples, the Van Vleck formula given by Equation (3.19) a f t e r s u b s t i t u t i n g the accepted values f o r constants reduces to M 2 = 716.164 N + 2.216 N + 9.994 N C 5 . 1 ) - 77 -where N i s number of protons i n the amine molecule, r . . i s the ' J i n t e r p r o t o n distance between proton i and j , r ^ n i s the d i s t a n c e between nitrogen and proton i and r . ^ i s the d i s t a n c e between a deuteron and proton i . M 2 can be c l a s s i f i e d i n t o two p a r t s , i n t r a m o l e c u l a r and i n t e r m o l e c u l a r M^. Because the c l o s e s t approach of two d i e t h y l amine o molecules i n the deuterate i s ~ 4 A, the i n t e r m o l e c u l a r c o n t r i b u t i o n to the second moment i s q u i t e small and w i l l be neglected i n the f i r s t i n s t a n c e . On the other hand the c r y s t a l s t r u c t u r e of pure d i e t h y l amine i s not known and i t s i n t e r m o l e c u l a r second moment M£ cannot be c a l c u l a t e d p r e c i s e l y . Therefore the only d e t a i l e d c a l c u l a t i o n s r e f e r r e d to here are i n t r a m o l e c u l a r . An estimate of M£ i n the r i g i d l a t t i c e i s obtained by using the approximate expression given by Smith [5.6] v i z . M" = 358.1 x ^ \- (5.2) L 13 irv where np i s the number of protons per u n i t c e l l , V i s the volume of the u n i t c e l l and R i s the radius of a sphere containing one molecule. R i s taken to be the molecular radius i n pure m a t e r i a l s and the radius of a cage i n the c l a t h r a t e hydrates. V may be i n terms o f the d e n s i t y p o f the material to give „ n o N M" = 358.1 x % • -E- ^ ( 5 < 3 ) 1 0 2 4 ^ n 1 M 1 where n. i s the number of molecules of type i i n the u n i t c e l l and M. i s t h e i r gram molecular weight. N i s Avogadro's number In the - 78 -s i t u a t i o n where only one type of molecule occurs, the number of molecules per u n i t c e l l cancels from Equation (5-3), and no u n i t c e l l i nformation i s r e q u i r e d , hence the a t t r a c t i o n of t h i s method f o r estimations when no X-ray data i s a v a i l a b l e . In the case of d i e t h y l amine deuterate n D Q = 104, "^i ethyl amine = o 2 o 12, p = 1.11 g cm" and R was taken to be ^4.4 A. Such values give M£ = 0.46 G 2. _3 For pure d i e t h y l amine-ND on the other hand a value of ~0.8 g cm 2 was taken f o r p i n the s o l i d and the r e s u l t i n g value of M£ i s ^4 G . Methyl group r e o r i e n t a t i o n i s expected to reduce somewhat the i n t e r m o l e c u l a r c o n t r i b u t i o n to the second moments. However, the reduction f a c t o r i s not expected to be l a r g e and i n a d d i t i o n the methods of estimation of i t s magnitude are rather u n c e r t a i n . I t was decided t h e r e f o r e to use the estimated r i g i d l a t t i c e c o n t r i b u t i o n s as the upper l i m i t s to i n t e r m o l e c u l a r second moments i n the temperature region of methyl r e o r i e n t a t i o n . For M£ use i s made of the c r y s t a l s t r u c t u r e data of Jordan and Mak [5.2]. The c r y s t a l s t r u c t u r e data of Jordan and Mak suggest that the i n t e r n u c l e a r distances of the guest molecules i n the 18-hedral cages d i f f e r s l i g h l y from those i n the i r r e g u l a r cages. In a d d i t i o n they f i n d small d i f f e r e n c e s between i n t e r n u c l e a r distances on opposite sides o f the d i e t h y l amine molecule i n the i r r e g u l a r cage due to the "openess" of one end o f t h i s type of cage. I t was decided t h e r e f o r e to use the f o l l o w i n g mean i n t e r n u c l e a r distances and angles o o i n the second moment c a l c u l a t i o n s , N-H = 1.0 A, N-C = 1.49 A, C-C = o o 1.54 A and/£-C-N = 111'A, otherwise the t e t r a h e d r a l angles were used. - 79 -The carbon proton distances of Jordan and Mak [5.2,5.7] were subject to even more v a r i a t i o n than the distances between the l a r g e r n u c l e i , o o i . e . , ranging from 0.85 A to 1.35 A. The second moment c a l c u l a t i o n s o o o were performed f o r mean C-H distances o f 1.09 A, 1.10 A, 1.12 A and 0 1.13 A. Using these data, the coordinates of various atoms were generated according to method of Thompson [5.8] using a computer programme, and then M£ was c a l c u l a t e d by another programme (Appendix C). The r e s u l t s of the M« c a l c u l a t i o n s are summarized i n Table 5.1. Table 5.1. Intramolecular Second Moment M£ f o r DNH-D 20 and DND-D20 N rC-H 0 A M 2 ( r i g i d ) G 2 M 2 (2 x CHg r o t a t i n g ) G 2 11 1.09 21.50 11.80 10 1.09 21.52 10.96 11 1.10 20.79 11.60 10 1.10 20.47 10.47 11 1.12 19.07 10.78 10 1.12 18.64 9.62 11 1.13 18.27 10.38 10 1.13 17.77 9.20 A f u r t h e r p o i n t - o f u n c e r t a i n t y i s the r o l e played by the Hydrogen atom bonded between the nitrogen atom and the cage. Second moment c a l c u l a t i o n s were t h e r e f o r e performed, f i r s t by assuming that atom was - 80 -a proton and secondly assuming that proton-deuteron exchange had taken place and that i t was a deuteron. Such an exchange i s probably most e f f i c i e n t during the preparative process i n the l i q u i d s t a t e . These r e s u l t s are a l s o summarized i n Table 5.1. The d e t a i l s of second moment Mg c a l c u l a t i o n s f o r N = 10 and r c _ H = 1.13 A are enumerated i n Table 5.2. (b\ Experimental Second Moment Data The temperature dependence of experimental second moment i s given i n Figure 5.1. A plateau value of (9.80 + 0.05)G e x i s t s i n the deuterate from about 110 K to the decomposition temperature at about 265 K. Below 110 K the second moment increases with decreasing temperature, but f a i l s t o e s t a b l i s h a new plateau value above 77 K. The experimental value at 77 K i s 13.63 +0.82 G 2 f o r DNH-D20 and 14.19 +0.96 G 2 f o r DND-DgO. The data from DND-DgO and DNH-DgO agree w i t h i n the accuracy o f the experiment, which may j u s t be great enough to allow such agreement to lend support to the view that the amine proton has been chemically exchanged and does not c o n t r i b u t e to the proton second moment. Comparison of the experimental data with Table 5.1 and 5.2 shows that the magnitude of plateau value i s co n s i s t e n t with the mean carbon proton separation of somewhere o o between 1.12 A and 1.13 A and r a p i d random r e o r i e n t a t i o n of CH^-group i n the deuterate. 2 The plateau of —12.5 G obtained f o r pure d i e t h y l amine-ND i s a l s o c o n s i s t e n t with the r a p i d random r e o r i e n t a t i o n of Cf-L-groups. CM if) CO 3 o CD c Q) E o E "O C O o if) 2 0 -16-12-8 -4 -o-o Diethylamine-ND . Diethylamine-ND- Deuterate Diethylamine—NH-Deuterate i CO i 1 1 1 1 1 1 r 100 v' 150 T 1 1 r 2 0 0 i 1 r 1 1 r 2 5 0 Temperature (° K) Figure 5.1 Second moment versus temperature f o r diethylamide deuterate and diethylamine-ND. - 82 -Table 5.2. Second Moment f o r Diethyl amine-ND Deuterate CDND-DgO) o using r r_„ = 1.13 A and N = 10 Cont r i b u t i o n R i g i d L a t t i c e G 2 2 x CH., r o t a t i n g 2 or 1. 2 x CH 3 group 10.93 * 2.73 2. 2 x CH 2 3.64 3.64 3. C H ^ - C H ^ 0.03 0.03 4. C H 3-CH 2 2.64 ** 2.27 •kick 5. ( i ) CH 3 - D 0.02 0.02 Oi) C H 2 - D 0.03 0.03 6. CH 2 ~ C H 2 0.47 0.47 7. Nitrogen-protons 0.01 0.01 8. Intermolecular diethylamide 0.46 <0.46 9. Intermolecular deuterons 0.10 <0.10 Total M 2 = 18.33 <9.73 Gutowsky and Pake [5.9] ** Obtained by m u l t i p l y i n g 2.71 by 0.86 - a " f a c t o r " given by Smith [5.10] and Chezeau e t a l . [5.11] "kick Deuteron attached to n i t r o g e n . - 83 -2. S p i n - L a t t i c e Relaxation Measurements (a) A n a l y s i s of Non-Exponential Relaxation The r e l a x a t i o n f u n c t i o n R ^ v C t ) based on H-H theory [5.5] f o r a p o l y c r y s t a l l i n e sample i n case o f CH^-group was given by Equation (3.54) i . e . Some t h e o r e t i c a l curves f o r R A y ( t ) c a l c u l a t e d by using Simpson's formula f o r numerical i n t e g r a t i o n with the values o f C. and q. from the tables provided by H-H were shown i n Figure 3.1. From these curves and s i m i l a r ones (not shown i n Figure 3.1) constructed from the tables A l - A l 5 (Appendix A), t h e o r e t i c a l n u l l points i . e . , t Q / T ' when R A y ( t ) = R / \ v ( t 0 ) = V 2 were e x t r a c t e d f o r a p a r t i c u l a r p value of (W 0 T c) . The value of ( t Q / T ' ) corresponding to a p a r t i c u l a r u o T c a r e t h e n P l ° t t e d 7 n Figure 5.2 on a l o g - l o g s c a l e . The minimum value o f t /T' = 1.52 when uQrc = 0.68 from H-H theory. The experimental data can now be analysed i n terms o f theory. (b) Experimental Relaxation Data The temperature dependence of the nuclear r e l a x a t i o n of two sep a r a t e l y prepared samples of DNH-D20 and of pure diethylamine-ND i s shown i n Figure 5.3. Since r e l a x a t i o n f u n c t i o n R(t) i n both cases i s non-exponential, i t i s t h e r e f o r e c h a r a c t e r i z e d by t Q , r a t h e r than T-|. The experimental minimum f o r DNH-D20 occurs at 129 K and i s 31 msec, while that of diethylamine-ND i s 18.5 msec at 145 K. The o 2 k R A v " 2 1 0 0 -1 0 . 0 -0 . 0 1 . High Temperatures Figure 5.2 Dependence of tQ/T on. y , , as p r e d i c t e d by H-H theory 00 -pi 1 0 . 0 Low Temperatures 2000-lOOO-8 0 0 6 0 0 4 0 0 (A TD C 2 0 0 O O 0) (/) != TOO so + ° 6 0 4 0 H 2CH 10 « Diethylamine- N H - Deuterate o Diethylamine-ND M.R (Diethylamine-ND) —1 1——| 1 1- 1 'n 1 1 1 1 1 1 1 1 1 1 1 8 0 IOO 120 140 160 180 2 0 0 220 240 Temperature [°K] Figure 5.3 Temperature dependence of t i n diethylamine deuterate and diethylamine-ND. CO - 86 -t h e o r e t i c a l minimum from previous s e c t i o n occurs at t /T' = 1.52 at ( J J 0T c = 0.68. Tn order to compare experiment with theory f o r DNH-DgO, one has to c a l c u l a t e 1/T*. Using t Q = 31 msec and t Q / T ' = 1.52, we get 1/T' = 49.03 s e c " 1 . With t h i s value o f 1/T' =49.03 s e c " 1 , some experimental curves f o r &nR(t) E £ n R A y ( t ) are compared with the t h e o r e t i c a l curves i n Figure 5.4. In a s i m i l a r way t a k i n g 1/T' = 82.16 s e c " 1 f o r diethylamine-ND, a comparison of theory with experiment i s shown a l s o i n Figure 5.4. I t i s seen t h a t the agreement with the theory i n case o f DNH-DgO i s b e t t e r than diethylamine-ND, probably because o f l e s s i n t e r m o l e c u l a r i n t e r a c t i o n i n the deuterate. Since the second moment data reported above i n d i c a t e t h a t the only motion o c c u r r i n g at a ra t e g reater than tens o f kHz i s methyl group r e o r i e n t a t i o n , i t i s concluded that the minimum i n each case of the curves of Figure 5.3 i s als o due to methyl group r e o r i e n t a t i o n . The temperature dependence o f c o r r e l a t i o n time f o r the methyl group r e o r i e n t a t i o n i s ex t r a c t e d i n the f o l l o w i n g way. From the value of 1/T' f o r DNH-D20 and diethylamine-ND ( c a l c u l a t e d above) and value of t from experiment corresponding to an observed temperature, t Q / T ' was c a l c u l a t e d f o r DNH-DgO and diethylamine-ND. Corresponding to each t /T*, to x was obtained from Figure 5.2. Thus w x at an o' o c 3 o c observed temperature i s found. The temperature dependence o f a ) Q x c f o r CHg-reorientation can now be f i t t e d t o an Arrhenius equation [ V = x . exp(E /RT) c f . Equation (3.26)] i f the r e o r i e n t a t i o n i s c o a thermally a c t i v a t e d and obeys the Arrhenius equation. The temperature dependence o f U Q T C i s shown i n Figure 5.5. I t i s seen t h i s dependence f i t s n i c e l y to the Arrhenius equation, and the a c t i v a t i o n energies oo —I Figure 5 . 4 . 4 - 8 . 12 16 20 Theoretical and experimental curves for non-exponential relaxation function for different ( w 0 O in diethylamine deuterate (curve 1 to 4 ) and diethylamine-ND (curve 7 to 8 ) theoretical experimental - 88 -l O . C K 8 . 0 : I _j 1 1— 1 1 1 i 3.0 4 . 0 5 .0 6 . 0 TO 8 . 0 9 . 0 10.0 1 0 3 / T ( K " 1 ) Figure 5.5 pl o t o f uQrc versus r e c i p r o c a l of the absolute temperature i n diethylamine deuterate and d i e t h y l amihe-ND. - 89 -obtained from the slopes f o r methyl r e o r i e n t a t i o n are 2.34 +_0.02 k c a l / mole tn the deuterate and 2.90 +_ 0i03 kcal/mole i n pure diethylamine-ND. The pre-exponential f a c t o r s x Q f o r d i e t h y l amine deuterate (DNH-DgO) and pure diethylamine-ND are [4.5 +0.3] x 1 0 " 1 3 sec and (1.6 +0.1) x 10" sec, r e s p e c t i v e l y . The depth o f the minimum which provides information about the 2 3 2 strength of the r e l a x a t i o n i n t e r a c t i o n [1/T 1 = ( Y fi/r ) 0/u>0) c f . Equation (3.52)], may be made o f q u a n t i t a t i v e use i n the case of deuterate to check the c r e d i b i l i t y of the i n t e r p r o t o n d i s t a n c e chosen to give agreement with the experimental second moment data. In the deuterate, i t i s reasonable to assume that the d i p o l a r i n t e r a c t i o n s r e s p o n s i b l e f o r r e l a x a t i o n are intramethyl group. I t then remains to decide how many protons are relaxed by each methyl group. Taking an o i n t e r p r o t o n d i s t a n c e 1.85 A, which i s c o n s i s t e n t with a C-H bond o length o f 1.13 A and t e t r a h e d r a l angles, the computed t minimum f o r a d i e t h y l amine molecule i s 32.5 msec. I f on the other hand one assumes t h a t the amine proton may have been exchanged with a deuteron, the computed minimum value of t i s 29.5 msec. The experimental value i s 31 + .1 msec. L i t t l e q u a n t i t a t i v e use can be made of the minimum value of t i n pure diethylamine-ND because i n t e r m o l e c u l a r d i p o l a r i n t e r a c t i o n s , which reduce but do not e l i m i n a t e the non-exponential character of the nuclear r e l a x a t i o n , are impossible to formulate without c r y s t a l s t r u c t u r e data. - 90 -D. Discussion The results presented i n t h i s chapter enable one to draw several conclusions regarding the c h a r a c t e r i s t i c s o f a diethylamine molecule f i r s t as a guest i n a c l a t h r a t e hydrate and secondly i n pure diethylamine. The second moment data reveal that i n both environments the diethylamine molecule e x h i b i t s r e o r i e n t a t i o n of i t s methyl groups at a r a t e i n excess of 10 kHz well below the melting p o i n t . Further-more t h i s i s the only motion which proceeds at a r a t e great enough to a f f e c t the nuclear resonance observables. The s p i n - l a t t i c e r e l a x a t i o n measurements i l l u s t r a t e t h at such methyl r e o r i e n t a t i o n proceeds more f r e e l y i n the deuterate, indeed they enable one to e s t a b l i s h a 24% increase i n the height of the hindering b a r r i e r i n going from the deuterate to pure diethylamine. Without the detailed c r y s t a l s t r u c t u r e o f diethylamine a v a i l a b l e , i t i s d i f f i c u l t t o f o l l o w t h i s q u a n t i t a t i v e r e s u l t f u r t h e r . In the deuterate, both 1he line-shape and the r e l a x a t i o n r e s u l t s i n d i c a t e a uniformity among the methyl groups which may not have been expected from the x-ray data of Jordan and Mak [5.2]. I t i s not p o s s i b l e to detect from nmr r e s u l t s any d i f f e r e n c e s i n behaviour between the guest molecules i n the 18-hedra and i n the i r r e g u l a r cages, or . a l t e r n a t i v e l y between the opposite ends of the guest molecules i n the i r r e g u l a r cages. I f l a r g e d i f f e r e n c e s i n the r e o r i e n t a t i o n r a t e s d i d e x i s t , . t h e y would show up most r e a d i l y [5.12] i n the temperature dependence of t . However, the present data give good agreement with the theory assuming a s i n g l e c o r r e l a t i o n time. A point which i s extremely uncertain from the c r y s t a l l o g r a p h i c data [5.2,5.7], but towards which nuclear r e l a x a t i o n gives some guidance - 91 -i s the C-H bond length and as a consequence the i n t e r p r o t o n d i s t a n c e i n 2 the deuterate. Both the plateau second moment of 9.8 +0.5 G and minimum t 31 + 1 msec are c o n s i s t e n t with a mean C-H bond length o — • Q . of 1.13 A, i f one i s prepared to assume a l a r g e degree of exchange between the amine proton and the D 90. - 92 -References (Chapter V) £5.1] R.K. McMullan, T.H. Jordan, and G.A. J e f f r e y , J . Chem. Phys., 47, 0967) 1218. [5.2] T.H. Jordan and T.C.W. Mak, J . Chem. Phys., 47 (1967) 1222. [5.3] S.D. Ross, M. F i n k e l s t e i n and R.C. Petersen, J . Am. Chem. S o c , 81_ 0959) 5336. 15.4] M.F. Hawthorne, J . Am. Chem. Soc., 76 (1954) 6358. £5.5] R.H. H i l t and P.S. Hubbard, Phys. Rev., 134 (1964) A392. [5.6] G.W. S v i t h , J . Chem. Phys., 36 (1962) 3081; i b i d , 42 (1965) 4229. [5.7] T.H. Jordan ( p r i v a t e communication). [5.8] H.B. Thompson, J . Chem. Phys., 47 (1967) 3407. [5.9] H.S. Gutowsky and G.E. Pake, J . Chem.'Phys.,'18 (1950) 162. 15.10] G.W:. Smith, General Motors Corporation Research P u b l i c a t i o n GMR-858, Feb. 28, 1969; and J . Chem. Phys., 5J_ (1969) 3569. [5.11] J.M. Chezeau, J . Dufourcq and J.H. Strange, Mol. Phys., 20 (1971 ) 305. [5.12] T.M. Connor, Trans. Faraday Soc., 60 (1964) 1574. - 93 -CHAPTER VI ACETONE AND ACETONE DEUTERATE A. Introd u c t i o n The c r y s t a l l i n e hydrate of acetone was f i r s t prepared s u c c e s s f u l l y by Quist and Frank [6.1] who froze a 60 weight percent s o l u t i o n of acetone i n water. They reported that the s o l u t i o n formed good octahedral c r y s t a l s (approximately 1 mm i n edge l e n g t h ) , t h a t the hydrate was von Stackelberg's type II s t r u c t u r e c l a t h r a t e hydrate with a molecular formula C3HgO -17H 20 and that the cubic u n i t c e l l edge o was 17.16 A; the acetone molecule being trapped i n the l a r g e r c a v i t i e s (16-hedra). The existence of acetone hydrate was f u r t h e r e s t a b l i s h e d by Wilson and Davidson [6.2] who i n v e s t i g a t e d the low frequency d i e l e c t r i c p r o p e r t i e s of the acetone-water system. They obtained a l i m i t i n g high frequency d i e l e c t r i c constant of about 7 at 200 K which was explained on the grounds that the acetone molecule was r e o r i e n t i n g i n s i d e the 16-hedra c a v i t i e s . More recent d i e l e c t r i c work [6.3-6.5] on acetone hydrate has i n d i c a t e d that the acetone molecules d i s p l a y an exceptional degree of r o t a t i o n a l m o b i l i t y i n the c l a t h r a t e cages. In e a r l i e r work by us [6.6] on the nmr i n v e s t i g a t i o n of acetone-DgO system, data were presented which were i n c o n f l i c t with the d i e l e c t r i c work, i n that the cw nmr r e s u l t s suggested the existence of only CH^-reorientation at - 94 -low temperatures. The o b j e c t of the present chapter i s to present nmr data whose i n t e r p r e t a t i o n i s c o n s i s t e n t with the d i e l e c t r i c work and to show, by comparing r e s u l t s from acetone hydrate and from pure acetone, t h a t the previous nmr data were probably taken on a s o l i d acetone-ice mixture which may have been formed i n a d v e r t e n t l y by phase separation r a t h e r than on the c l a t h r a t e hydrate. The data reported here are i n f a c t on acetone deuterate, i n order to ensure that the protons of the acetone molecules were well i s o l a t e d , so t h a t the only i n t e r - p r o t o n d i p o l a r i n t e r a c t i o n s were e s s e n t i a l l y i n t r a m o l e c u l a r . B. Experimental 1. Preparation of Acetone-Deuterate The p r e p a r a t i v e method employed i s e s s e n t i a l l y the same as given by Quist and Frank [6.1]. A 60 weight % s o l u t i o n o f acetone ( F i s c h e r spectro-analyzed) was prepared with DgO (99.8% dg, S t o h l e r isotope chemicals). This s o l u t i o n was i n i t i a l l y f r o z e n s o l i d and then a f t e r melting i t was placed i n s i d e the innermost space of 3-walled vessel (having two a i r spaces, one of which was evacuated). This vessel was placed i n a bath held between -40 and -50°C. The bath was prepared by mixing appropriate amounts o f dry i c e and acetone to get the d e s i r e d temperature. The q u a l i t y of the c r y s t a l s depended on the process o f a l t e r n a t e c o o l i n g and melting of acetone-DgO s o l u t i o n . The c r y s t a l s grew i n about 5 to 6 hours. A f t e r the c r y s t a l l i z a t i o n was complete, the deuterate which had an excess of acetone was t r a n s f e r r e d to the cold box held at -40°C by a flow o f c o l d n i t r o g e n gas. The excess of acetone was removed by p r e s s i n g the - 95 -deuterate on precooled absorbent f i l t e r papers i n s i d e the c o l d box at -40°C. This removal process was one of the most c r i t i c a l o f the preparation and the composition of acetone deuterate depended very much on t h i s and the temperature o f the c o l d box. Temperatures much lower than -40°C y i e l d e d an excess of acetone i n the deuterate while temperatures greater than -40°C gave a product with a lower acetone content. Three samples whose gas chromatographic a n a l y s i s gave 12.6 +0.3, 15.6 +0.4, 15.2 +0.1% acetone by weight ( s t o i c h i o -m e t r i c 14.6% by weight acetone i n C 3HgO-170 20)were used f o r nuclear resonance work. They were f i l l e d i n 10 mm and 7 mm o.d. sample tubes and sealed a f t e r evacuating. 2. Preparation o f Acetone Sample Pure acetone ( F i s c h e r spectro-analyzed) was d i r e c t l y t r a n s f e r r e d to the nmr tubes (10 mm f o r cw and 7 mm o.d. f o r pulsed nmr work) and was sealed a f t e r removal of the d i s s o l v e d oxygen by a freeze-pump-thaw method. 3. Spectrometers and Methods of Measurement They were the same as d e s c r i b e d i n Chapter IV. The 180° pulse length t h i s time was about 3.0 usee, i n the case o f acetone and 3.5 psec i n the case of deuterate. The r e l a x a t i o n f u n c t i o n R(t) i n both cases was non-exponential, so t Q , the time taken by R(t) to become equal to R ( t 0 ) = 1/2 was used to c h a r a c t e r i z e the r e l a x a t i o n i n s t e a d of T-j. The minima were not observed i n the temperature dependence of t i n both the cases. Therefore the data are not f i t t e d - 96 -to the H-H theory, and the a c t i v a t i o n energies are e x t r a c t e d from the gradient of t versus r e c i p r o c a l temperature. C. Results 1. Absorption Line A n a l y s i s For p o l y c . r y s t a l l i n e samples, the Van Vleck formula f o r second moment Mg a f t e r s u b s t i t u t i n g the accepted values of constants i n Equation (3.19) becomes M 0 = 716.164 N" 1 V] r~.6. + 9.994 N - 1 ^ r~.6, (6.1) where N i s the number o f protons (6 i n t h i s c a s e ) , r . . i s the d i s t a n c e between proton i and j and r . ^ i s the distance between proton i and o deuteron d i n A. The s t r u c t u r e of acetone has been obtained by several authors [6.7]. However, there i s q u i t e a v a r i a t i o n i n the reported bond lengths and bond angles. We have used i n our second moment c a l c u l a t i o n s the s t r u c t u r a l data derived from the microwave work of Nelson and P i e r c e [6 .8 ] and, i n a d d i t i o n , assuming that the acetone has the same molecular s t r u c t u r e i n the c l a t h r a t e - d e u t e r a t e , we obtained those proton second moment c o n t r i b u t i o n s which are summarized i n Table 6.1. The i n t e r m o l e c u l a r second moment i n the case o f pure acetone i s estimated from the work o f Dufou/cq e t a l . [6.13] on a s i m i l a r compound i . e . , dimethylsulphoxide CgHgSO, because the c r y s t a l data on acetone s t r u c t u r e have not given any information about the arrangement - 97 -Table 6.1. Proton Second Moment i n Acetone Deuterate Type of Motion Intramolecular Intermolecular T o t a l CH 3 CH 3-CH 3 R i g i d 23.77 0.98 0.26 a 25.01 2CH 3 r o t a t i n g 5.94 0.84 b 0.20 a 6.98 2CH 3 r o t a t i n g + C 2 r o t a t i o n about >C=0 bond 1.98 C 0.21 d 0.16 2.35 I s o t r o p i c 0 0 0.15 a 0.15 a Ref. [6.9] a f t e r i n c l u d i n g deuteron c o n t r i b u t i o n b Refs. [6.9] and [6.10] c Ref. [6.11] d Ref. [6.12] of acetone i n c r y s t a l l a t t i c e . Thus the i n t e r m o l e c u l a r r i g i d l a t t i c e 2 second moment f o r pure acetone i s assumed to be ^ 5 G g i v i n g 2 r i s e to t o t a l r i g i d l a t t i c e second moment equal to 24.8 + 5.0 = 29.8 G . In case of both methyl groups r o t a t i n g the t o t a l second moment 2 based on the same reference [6.13] comes out to be ^ 8.0 G . The experimental value of the proton second momend i n acetone 2 deuterate i s 0.69 +_ 0.07 G at 77 K. I t decreases s t e a d i l y (Figure 6.1) to 0.18 +0.01 G 2 at 212 K. This l a t t e r value i s co n s i s t e n t with i s o t r o p i c r o t a t i o n o f the acetone molecule i n s i d e the 16-hedron c a v i t i e s . Some spectra of acetone-deuterate are shown i n Figure 6.2. o Acetone i Acetone Deuterate I 1 1 1 1 1 1 1 1 180 2 0 0 2 2 0 2 4 0 2 6 0 - i — — i 1 r 6 0 8 0 1 0 0 —I 1 — | 1 1 r 120 140 160 Temperature [°K ] Figure 6.1 V a r i a t i o n o f second moment w i th temperature i n acetone and acetone d e u t e r a t e . Temperature 77 K M 2 = 0.66 G 2 6H = 1,24 G Temperature 180 K M 2 = 0.41 G 2 6H = 1.11 G Temperature 202 K M 2 = 0.32 G 2 6H = 1.23 G M 2 i s the second moment and SH i s the l i n e width. ( D ( 2 ) (3) i i i i———i 1 1 - 3 - 2 - 1 O 1 2 3 GAUSS Figure 6.2 Some proton magnetic resonance spectra of acetone deuterate at d i f f e r e n t temperatures. - 100 -The behaviour of pure acetone i s i l l u s t r a t e d i n the work of Gutowsky and Pake [6.12]. These authors [6.12] have shown the tempera-ture dependence of the l i n e width. We have f o r the sake of comparison with deuterate measured the second moment of pure acetone at d i f f e r e n t temperatures. The second moment versus temperature f o r pure acetone i s p l o t t e d i n Figure 6.1. The experimental value of second moment shows a plateau value of 9.15 +0.54 G 2 at 77 K to 9.81 +0.71 G 2 at 171 K. The corresponding l i n e width i s 5.84 +0.22 G at 77 K to 5.12 +0.62 G at 171 K and i s c o n s i s t e n t with the work of Gutowsky and Pake [6.12]. From the above second moment i t i s deduced that both methyl groups are r o t a t i n g i n acetone down to 77 K and no other motion i s present from 77 to 171 K. 2. Relaxatdbn Measurements The temperature dependence o f t Q (time when r e l a x a t i o n f u n c t i o n R(t) = 1/2) i s p l o t t e d i n Figure 6.3. In both cases because of equipment l i m i t a t i o n s , the minimum i n t Q versus temperature was not reached. Therefore the data cannot be f i t t e d to H-H theory, and consequently no information about the c o r r e l a t i o n time T c i s obtained. However, a c t i v a t i o n energies can be extracted from the gradients of t Q versus i n v e r s e temperature (Figure 6.4) from the high temperature s i d e of minimum. Such a procedure y i e l d s a c t i v a t i o n energy E 3 = a 0.33 +_ 0.01 kcal/mole i n the case of acetone-deuterate, which cannot be assigned unambiguously f o r reasons to be given l a t e r . On the other hand, pure acetone d i s p l a y s an a c t i v a t i o n energy of 1.33 +0.01 kcal/mole, which i s doubtless due to hindered r o t a t i o n o f i t s C+L groups. 6 0 0 0 4 0 - | 1 1 1 1 1 1 1 | " r 1 1 1 i I ' I T~ 100 150 2 0 0 2 5 0 Temperature [°K] Figure 6.3 Temperature dependence of t i n acetone and acetone deuterate. 13.0 1 2 0 1I.O 1 0 0 9.0 8 0 7.0 6 . 0 . 5 .0 4 .0 3 .0 1 0 0 0 / T [ K _ I ] Figure 6 . 4 t Q versus r e c i p r o c a l of the absolute temperature f o r acetone and acetone-deuterate. - 103 -C. Discussion The second moment r e s u l t s f o r pure acetone i n d i c a t e that the only motion which proceeds at a rate f a s t enough to a f f e c t the nuclear resonance observables i s C H 3 - r e o r i e n t a t i o n . The a c t i v a t i o n energy f o r t h i s r e o r i e n t a t i o n was determined to be 1.33 kcal/mole from the nuclear resonance data. The gas phase microwave work o f Swallen and Costain [6.14] and more r e f i n e d work of Nelson and P i e r c e [6.8] give the b a r r i e r hindering C ^ - r o t a t i o n of CH 3 group i n acetone as 0.78 kcal/mole. I t thus appears t h a t i n s o l i d acetone the i n t e r and i n t r a m o l e c u l a r c o n t r i b u t i o n s to the b a r r i e r are of the same order. 2 The a n a l y s i s of second moment value 0.69 +_ 0.07 G at 77 K i n acetone-deuterate i s most d i f f i c u l t . D i f f e r e n t models of r o t a t i o n were t r i e d i n the same view as work of Powles and Gutowsky [6.11] ( c f . Equation (3.32)) but a l l of them give second moments gr e a t e r 2 than 1 G . I t was t h e r e f o r e concluded that t h i s i s not the plateau value but part of the decrease i n the temperature dependence o f second moment to plateau c o n s i s t e n t with i s o t r o p i c r e o r i e n t a t i o n , which i s observed above 210 K. The i n t e r p r e t a t i o n of the temperature dependence o f t i n acetone-deuterate i s not unambiguous. The temperature dependence of t i n acetone-deuterate below 120 K (Figure 6.3 and 6.4) follows an a c t i v a t i o n law which y i e l d s an a c t i v a t i o n energy o f 0.33 + 0.01 kcal/mole. However t h i s cannot be assigned to i s o t r o p i c r e o r i e n t a t i o n because the second moment';in t h i s temperature region i s s t i l l above the plateau value consistent with isotropic r e o r i e n t a t i o n . To which of the other p o s s i b l e motions i t corresponds cannot be determined unambiguously without a low temperature plateau i n - 104 -second moment or a minimum i n temperature dependence of t . However t h i s a c t i v a t i o n energy of 0.33 kcal/mole can be compared * with the a c t i v a t i o n enthalpy obtained by Davies and Williams [6.5J i n t h e i r d i e l e c t r i c work on acetone hydrate. Davies and W i l l i a m s , i n f a c t observed d i s p e r s i o n due to the acetone molecule and they found a maximum a c t i v a t i o n energy of 0.25 kcal/mole at 93 K with a c c o r r e l a t i o n -12 time T f o r the motion of the order of 4.3 x 10 sec. They a l s o found that at 93 K t h e i r data f i t t e d a Fuoss and Kirkwood d i s t r i b u t i o n f u n c t i o n with width o f d i s t r i b u t i o n $ = 0.65 [ c f . Chapter I I I , Section E ] . They concluded t h a t the e v a l u a t i o n of t h i s d i s t r i b u t i o n f u n c t i o n was of l i m i t e d u t i l i t y because of inhomogeneous sample used, and a l s o because only three frequencies were used i n the guest d i s p e r s i o n region. However, assuming the d i s t r i b u t i o n to be c o r r e c t , they -12 obtained new T which was 1.1 x 10 sec. We are unable to say anything about the d i s t r i b u t i o n of c o r r e l a t i o n times s i n c e we were unable to reach a temperature where minimum i n t occurs. I f t h i s d i s t r i b u t i o n i s present the theory of non-exponential r e l a x a t i o n i n nmr becomes more complicated. However, on the basis of t h i s d i s t r i b u t i o n the a c t i v a t i o n energy we reported i s l e s s . The value of 0.25 kcal/mole by [6.5] i s obtained without taking i n t o account the d i s t r i b u t i o n . Davies and Williams [6.5] have pointed f u r t h e r that the best value * S t r i c t l y speaking the a c t i v a t i o n energy which we have used i n our work i s a c t i v a t i o n enthalpy, but because of small d i f f e r e n c e s i n a c t i v a t i o n enthalpy and a c t i v a t i o n energy, i n the present case, nmr workers most f r e q u e n t l y c a l l i t a c t i v a t i o n energy. As a convention, we have followed the same trend. - 105 -can be about 0.3 kcal/mole i f the sample i s homogeneous. Our method of p r e p a r a t i o n i s a b e t t e r method i n regard to homogeniety and composition o f sample. On t h i s b a s i s the value o f our a c t i v a t i o n energy agrees n i c e l y with t h a t o f Davies and Williams [6.5]. The f l a t t e n i n g and subsequent f u r t h e r decrease i n t as the temperature i n c r e a s e s , could well be due to i s o t r o p i c r e o r i e n t a t i o n of acetone molecule i n the c l a t h r a t e cages. An a c t i v a t i o n energy o f ~ 2 kcal/mole i s obtained from the s l o p e i n the temperature region of 230 to 255 K< Due to very poor s i g n a l to noise r a t i o from ~170 K and onwards, the accuracy o f t h i s a c t i v a t i o n energy i s u n c e r t a i n . - 106 -References (Chapter VI) [6.1] A.S. Quist and H.S. Frank, J . Phys. Chem., 65 (1961) 560. [6.2] G.J. Wilson and D.W. Davidson, Can. J . Chem., 4V (1963) 264. [6.3] B. Morrison and D.W. Davidson, Can. J . Chem., 49 (1971) 1243. [6.4] M.I. Shakhparonov and N.V. Chekalin, J . Stru c . Chem.,11 (1970) 560 t r a n s l a t e d from Russian Zhurnal Strukturnoi K h i m i i , 1 1 (1970) 599. [6.5] M. Davies and K. Wi l l i a m s , Trans. Faraday S o c , 64 (1968) 529. [6.6] A.W.K. Khanzada and C A . McDowell, J . Mol. S t r u c t u r e , 7 (1971) 241. [6.7] C. Kato, S. Konaka, T. I i j i m a and M. Kimura, B u l l . Chem. S o c (Japan), 42 (1969) 2148, and references t h e r e i n . [6.8] R. Nelson and L. P i e r c e , J . Mol. Spectroscopy, 1_8 (1965) 344. [6.9] G.W. Smith, J . Chem. Phys., 42 (1965) 4229. [6.10] G.W. Smith, General Motors Corporation, Research P u b l i c a t i o n GMR-858, Feb. 28, 1969. [6.11] J.G. Powles and H.S. Gutowsky, J . Chem. Phys., 21_ (1953) 1704. [6.12] H.S. Gutowsky and G.E. Pake, J . Chem. Phys.,'18. (1950) 162. [6.13] J . Dufourcq, J.M. Cheazeau and B. Lemanceau, C R . Acad. Sc. ( P a r i s ) , 265B (1967) 761. [6.14] J.D. Swallen and C.C Co s t a i n , J . Chem. Phys., 31_ (1959) 1562. - 107 -CHAPTER VII TERTIARY BUTYLAMINE AND TERTIARY BUTYLAMINE DEUTERATE A. Intro d u c t i o n tert-Butylamine i s the f i r s t amine hydrate to be found wherein the amine molecule i s not hydrogen bonded to water cage. The c r y s t a l s t r u c t u r e of t h i s hydrate has been studi e d by x-ray d i f f r a c t i o n by McMullan et a l . [7.1]. A b r i e f account of the c r y s t a l s t r u c t u r e was presented i n Chapter I I . This hydrate i s c u b i c , and belongs to space group I?3d with a u n i t c e l l edge of 18.81 A at -30°C [7.1]. The u n i t c e l l contains 16 amine and 156 water molecules. The amine 3 2 9 3 molecule l i e s i n s i d e the 17-hedra. The 17-hedron (7 -7 -5 -4 ) i s formed from 30 oxygen v e r t i c e s and 45 hydrogen bond edges described by the r e l a t i o n ( c f . Chap. II) 17F + 30V = 45E + 2 (Figure 2.4). The o vertex to centre distances i n 17-hedron vary from 4.68 to 5.16 A °3 and i t s volume i s approximately 395 A . The other c a v i t i e s are 8-hedra and they are vacant. This chapter deals with the study o f motional behaviour of t h i s amine encaged i n the c l a t h r a t e deuterate and i n the ordinary s t a t e . In e a r l i e r s t u d i e s the deuterate was prepared by mixing the formula r a t i o of amine with D^O. Since D 20 i s always i n excess, i t i s b e l i e v e d that the -NH protons are exchanged with D_0 where t h e i r - 108 -c o n t r i b u t i o n to nmr observables s t u d i e d here i s nearly n e g l i g i b l e . Such behaviour we n o t i c e d i n the case of d i e t h y l amine-ND-deuterate. Therefore we d i d not study (CH^CND^-deuterate s e p a r a t e l y here, but r e s t r i c t e d ourselves to check the consistency of our r e s u l t s with d i f f e r e n t batches of samples. Later f o r study of amine i n pure s t a t e , i t s -NH,, protons were deuterated. It w i l l be seen from t h i s study that the amine molecule enjoys a considerable degree of freedom of motion ranging from methyl group r o t a t i o n to i s o t r o p i c r o t a t i o n of whole molecule. The motion i n pure amine i s r e s t r i c t e d to methyl plus t e r t - b u t y l group r o t a t i o n . We do not have any information about the motion of -NH2 group, because i n both s t a t e s i t i s deuterated. However, from the d i e l e c t r i c r e l a x a t i o n data o f K r i s h n a j i and Mansingh [7.2] on t e r t -butyl amine, the molecule i s r i g i d i n the s o l i d s t a t e . I t means t h a t the -NH2 group has no motion because d i e l e c t r i c r e l a x a t i o n i s not s e n s i t i v e to methyl group motion i n t h i s case. B. Experimental 1. M a t e r i a l s Reagent grade tert-butylamine was obtained from Eastman Kodak and D 20 (99.8% d 2) was provided by S t o h l e r Isotope Chemicals. The amine was d r i e d on KOH f o r several days. L a t e r the d r i e d amine was d i s t i l l e d three times from BaO. 2. Preparation ' o f (CH,) 3CND 2 The preparation i s s i m i l a r to t h a t described i n Chapter V. The a n a l y s i s of p a r t i a l l y deuterated amine was performed by Varian T-60 - 109 -high r e s o l u t i o n nmr and i n d i c a t e d that the -NH2 protons were about 90% deuterated to ND 2 < 3. Preparation of Deuterate and Amine Samples Three d i f f e r e n t batches of deuterates were prepared. The i n i t i a l batch was from s t r a i g h t amine sample. The second batch was from d r i e d amine on KOH and the t h i r d batch was from the r e d i s t i l l e d sample. In a l l the three batches the formula r a t i o (16 amine:156 D^O) of amine and D 20 was mixed i n a t i g h t l y closed vessel and the s o l u t i o n was cooled slowly i n the cold box. Cooling of s o l u t i o n to about -2°C as carried out by McMullan et a l . [7.1] y i e l d e d the c r y s t a l s , but sometimes i t f a i l e d even i f the s o l u t i o n s were l e f t f o r about 24 hours. However, the r e s u l t s o f c r y s t a l l i z a t i o n at about -2°C and at c o l d box temperature (-15°C) were nearly c o n s i s t e n t w i t h i n experimental u n c e r t a i n t y . The m a j o r i t y of samples were t h e r e f o r e prepared at c o l d box temperature. The deuterate thus obtained was f i n e l y crushed to powder i n a pre-cooled mortar i n the co l d box and f i l l e d i n nmr tubes. The nmr tubes were sealed a f t e r degassing. The a n a l y s i s of samples i n these batches done on decomposed samples using Varian T-60 high r e s o l u t i o n nmr i n d i c a t e d the range of 89-96% guest p u r i t y i n the deuterate. The samples l e f t at -15°C i n c o l d box probably had a tendency to decompose when l e f t f o r a few days and gave lower values of T-| when studied a f t e r some days. Therefore e i t h e r f r e s h samples were used or samples were kept at l i q u i d n itrogen temperature. A l l of these samples showed a small t r a c e of l i q u i d peak i n the cw s p e c t r a near the melting point of guest i n d i c a t i n g e i t h e r - n o -t r a c e o f amine l e f t unreacted or some o f the hydrate decomposed. A sample.was prepared with 90% guest, but the l i q u i d peak s t i l l p e r s i s t e d . The sample prepared according to the method of McMullan et a l . [7.1] showed a l s o t h i s l i q u i d peak and i t must show up because i n t h i s case amine was i n excess amount than the formulae r a t i o i . e . , 11.1 mole %. However there was' no i n d i c a t i o n of l i q u i d peak i n pulsed nmr measurements. This l i q u i d peak had however a n e g l i g i b l e c o n t r i -bution to second moment. The pure amine (CH-^CNDg s a m p i . e W a s d i r e c t l y t r a n s f e r r e d to nmr tubes and sealed a f t e r degassing the d i s s o l v e d a i r by f r e e z e -pump-thaw-method. This sample a l s o showed a narrow l i q u i d peak before the melting point. 4. Spectrometers and Methods of Measurements These equipment were the same as discussed before. The maximum r f f i e l d i n the case of cw measurements was 40 mG f o r the deuterate. In the case of the pure amine an r f f i e l d of 0.5 mG was used from 77 to about 110 K. T-j measurements were performed by using a 180° pulse length of 3 ysec using the 180°-T-90° pulse sequence. The r e l a x a t i o n f u n c t i o n R(x) was exponential i n a l l cases. Consequently T-j was c a l c u l a t e d using Equation (4.7) where T q was obtained from the p l o t of &nR(x) versus x. The other methods were the same as described i n Chapter IV. - i n -C. Results 1. Absorption Line A n a l y s i s (a) Second Moment C a l c u l a t i o n s The second moment was c a l c u l a t e d using Equation (5.1). These c a l c u l a t i o n s were performed using two sets of molecular parameters. o The f i r s t s e t i s the data of McMullan e t a l . [7.1] where C-N = 1.54 A, C-C = 1.42 A.^NCC = 112°, z_CCC = 106.8°. In t h i s s e t N-H = 1.0 A, o C-H = 1.09 A and other angles were assumed to be t e t r a h e d r a l . The second s e t i s based on the molecular parameters from microwave work on other amines [7.3-7.4] because the c r y s t a l and molecular s t r u c t u r e of tert-butylamine i s not known. This second set assumes the f o l l o w i n g bond lengths f o r t e r t - b u t y l a m i n e : C-N = 1.47 A, C-C = 1.54 A, N-H = o o 1.02 A, and C-H = 1.09 A;' A l l the angles used to generate the coordinates were assumed to be t e t r a h e d r a l . The r e s u l t s of the c a l c u l a t i o n of the i n t r a m o l e c u l a r second moment M2, performed i n a s i m i l a r way as described i n Chapter V and VI are summarized i n Table 7.1. Since t e r t - b u t y l amine i s n e a r l y s p h e r i c a l i n shape, the c a l c u l a -t i o n o f the i n t e r m o l e c u l a r second moment M 2 can be approximated i n the case of deuterate by an expression given by Smith [7.5], i . e . , M" = 358.1 N £ N. f ( h ) RT D (7.1) c i=l 1 1 where N i s the number of proton per molecule (9 i n t h i s c a s e ) , N. i s the number of i t h nearest neighbours, and R^  i s centre to centre d i s t a n c e between o r i g i n molecule and i t s i t h neighbour molecules, f ( h ) i s given by - 112 Table 7.1 Intramolecular Second Moment M£ f o r tert-Butylamine Type of Motion Mi using 1st set 2 [7.1] .2 MA using 2nd set 1 [7.3-7.4] G 2 1. R i g i d 3ChL CH^-CH^ CH 3-D 2 N-CH-Total 2. 3CH 3 Rotating 3CH 3 CH 3~CH 3 CH 3-D 2* Total 22.53 5.11 0.03 0.00 27.67 5.63 4.39 0.03 10.05 3. 3CH 3 + t e r t - b u t y l group r o t a t i n g 3CH-C H 3 " C H 3 CH 3-D 2* 0.47 1.10 0.02 Total 1.59 22.53 2.83 0.02 0.00 25.38 5.63 2.43 0.02 8.08 0.62 0.61 0.01 1.24 Deuteron attached to nitrogen i n -ND2 group, - 113 -f ( h ) = [ ( l - h 2 ) 2 + ( 5 / 3 ) h 4 ] / [ ( l - 4 h 2 ) 3 ] (7.2) o where h = R/R. with R i s equal to molecular radius i n A. D i f f e r e n t R. were c a l c u l a t e d from the 16 p o s i t i o n o f the 17-hedra using the * coordinates given by McMullan et a l . [7.1] and space group data [7.6] o p Such a c a l c u l a t i o n with R = 2.02 A gives M£ = 0.04 G which i s very s m a l l . However i f M£ i s c a l c u l a t e d using Equation (5.3) with p = 1.071 g/cc, and R = 4.63 A a value of 0.32 G 2 i s obtained. We b e l i e v e t h i s value to be more reasonable on the basis of previous r e s u l t s . On taking i n t o account the approximate c o n t r i b u t i o n due to the deuteron and the exchanged protons (17-hedron has 45 deuterons o with an approximate d i s t a n c e of 4.68 A from the centre [ 7 . 1 ] ) , the 2 o v e r a l l value of MJJ = 0.4 G i s e j e c t e d . The c a l c u l a t i o n of M£ f o r the pure amine i s most d i f f i c u l t , because f i r s t o f a l l no c r y s t a l s t r u c t u r e data are a v a i l a b l e , and secondly to use Equation (5.3) the d e n s i t y p i n the s o l i d s t a t e must be known. Some values of d e n s i t i e s at d i f f e r e n t temperatures (up to 5°C) are given i n [7.2]. E x t r a p o l a t i n g these values roughly when the s o l i d s t a t e would e x i s t a minimum value of 0.78 g/cc f o r p i s o obtained. This p value when used i n Equation (5.3) with R = 2.2 A 2 gives M£ = 7.93 G . This value makes the t o t a l r i g i d l a t t i c e 2 second moment M 2 33 G , a value c o n s i d e r a b l y higher than the 2 r i g i d l a t t i c e M 2 value f o r s i m i l a r compounds i e . , about 30 G [7.7]. The only broad l i n e nmr work on amines from which we can compare our r e s u l t s i s that of Kromhout and Moulton [7.8] and of Haigh et a l . [7.9]. Since the m a j o r i t y of compounds s i m i l a r to tert-butylamine 2 give M 7 - 30 G , we have t h e r e f o r e taken the experimental value of - 114 -M 2 = 31 G as the r i g i d l a t t i c e second moment f o r (CH 3) 3CND 2. Using t h i s basis f o r the pure amine, where we have p r e f e r r e d microwave 2 data, the i n t e r m o l e c u l a r second moment M 2 comes out to b e ~ 6 G , a reasonable value comparable to the i n t e r m o l e c u l a r second moment values on a s e r i e s of s i m i l a r compounds [7.5,7.7]. The second moment f o r i s o t r o p i c motion ( M 2 ) 1 - s o can be c a l c u l a t e d from the expression [7.5] ( M 2 ) i s Q = 358.1 N L N.R"b (7.3) where a l l the symbols have the same meaning as defined i n Equation (7.1). The value of f o r the deuterate i s obtained i n the same way 2 as before. The r e s u l t s of these c a l c u l a t i o n s give ( M 2 ) . j s o = 0.02 G which a f t e r t aking i n t o account the c o n t r i b u t i o n from deuterons and 2 from exchanged protons can be i n the range of 0.10-0.15 G at the most. The o v e r a l l r e s u l t s are thus summarized i n Table 7.2. (b) Experimental Results The experimental r e s u l t s of l i n e width and second moment determination p l o t t e d against temperature are shown i n Figures 7.1 and 7.2. The 2 pure amine second moment at 77 K i s 30.96 +1.14 G which probably corresponds to the r i g i d l a t t i c e value. The second moment value 2 approaches a plateau value of 2.2 G around 150 K which i s c o n s i s t e n t with the three methyl groups plus the t e r t - b u t y l group r o t a t i n g . The corresponding l i n e width at 77 K i s 19.37 +0.28 G and i t approaches a plateau value o f 3.7 G at about 120 K. Some sp e c t r a of (CH 3) 3CND 2 are shown i n Figure 7.4a. 1 0 0 1 5 0 Z O O 2 5 0 Temperature [°K] 2 0 (0 1 6 =3 O CD w 1 2 -+-5> 8 CD C J 4 . O • tert-Butylamine —ND2 o tert-Butylamine Deuterate 1 T 1 r - l 1 r—1 1 r— 100 150 2 0 0 Temperature [°K] i 1 r 2 5 0 Figure 7.2 Proton magnetic resonance l i n e width vs. temperature f o r tert-butylamine-ND 2 and tert-butylamine deuterate. - 117 -Table 7.2 Second Moment Values f o r tert-Butylamine Deuterate and tert-Butylami ne-ND^ Type of Motion Deuterate (CH 3) 3CND 2 M^ using 1st M 2 using 2nd M 2 using 2nd s e t set [7.1] set [7.3,7.4] [7.3,7.4] G 2 G 2 G 2 1. R i g i d 28.06 25.77 3 1 + 2 2. 3CH 3 r o t a t i n g 10.35 8.38 9 + 1 3. 3CH 3 + t e r t - b u t y l 1.79 1.44 2 + 0 . 5 group r o t a t i n g 4. I s o t r o p i c 0.10-0.15 0.10-0.15 The experimental value of the second moment f o r t e r t - b u t y l deuterate 2 at 77 K i s 19.80 + 0.97 G . I t decreases very slowly with i n c r e a s i n g temperature passing through a value c o n s i s t e n t with the 3CH 3 groups 2 plus the t e r t - b u t y l group r o t a t i n g , to give a plateau value of 0.2 G 2 around 250 K. This l a s t plateau value of 0.2 G i s c o n s i s t e n t with i s o t r o p i c r o t a t i o n of whole molecule i n s i d e the 17-hedron. Some spectra o f deuterate are shown i n Figure 7.4b. The r e s u l t s of the second moment data on the deuterate suggest a broad d i s t r i b u t i o n of c o r r e l a t i o n times between a l l the three motions i . e . , methyl group r o t a t i o n , methyl plus t e r t - b u t y l group r o t a t i o n , and i s o t r o p i c r o t a t o n . As discussed i n Chapter I I I , the magnitude of the a c t i v a t i o n energy f o r the motion causing t h i s l i n e narrowing can be extracted from the Equation (3.25). For (CH 3) 3CND 2, we took C = 20 G and B = 3.7 G t e r t - B u t y l a m i n e D e u t e r a t e - N D Curve 1 M 2 = 30.81 G 6H = 19.21 G Temperature 77 K 6H = 3.53 G Temperature = 151 K oo Curve 1 M 2 = 19.77 G 6H = 17.46 G Temperature 77 K t e r t - B u t y l a m i n e D e u t e r a t e 6H = 1.84 G Temperature = 126 K 53 G" 47 G 185 K Figure 7.3 Some H resonance absorption spectra of tert-butylamine-ND 2 and tert-butylamine deuterate. - 119 -and6H, the l i n e - w i d t h values i n the narrowing region (Figure 7.2). The r e s u l t s o f t h i s a n a l y s i s gave = 3.19 + 0.26 kcal/mole with a — -4 -7 T c ranging from 1.1 x 10 sec to 1.7 x 10 sec i n the region of t r a n s i t i o n i . e . , from 77 to 114 K. This i s i n f a c t the a c t i v a t i o n energy f o r the b a r r i e r hindering the methyl as well as t e r t - b u t y l group r o t a t i o n . The a c t i v a t i o n energy f o r the deuterate f o r the f i r s t t r a n s i t i o n cannot be c a l c u l a t e d as we do not know the r i g i d l a t t i c e l i n e width C. However, an estimate can be obtained f o r the E, involved i n the second t r a n s i t i o n i . e . , from three methyl plus a t e r t - b u t y l group r o t a t i o n t o i s o t r o p i c r o t a t i o n . Use of Equation (3.25) with C = 1.8 G and B = 1.1 G gives an a c t i v a t i o n energy of 2.14 +0.21 kcal/mole with x c varying from 6.2 x 10" 4 to 5.6 x IO" 5 i n the range of 132 to 211 K. These a c t i v a t i o n energies and c o r r e l a t i o n times T c are not very accurate, but these can be h e l p f u l i n comparing the a c t i v a t i o n energies derived from the T-j measurements. 2. T-j Measurements The temperature dependence of T-| f o r both compounds i s shown i n Figure 7.4. The data f o r (CH^CNDg show a s i n g l e broad minimum of 16 msec f o r both methyl and t e r t - b u t y l group motions. Since there i s no separate minimum detectable f o r these two motion, there i s a d i s b r i b u t i o n of c o r r e l a t i o n times and so the b a r r i e r heights f o r both motions must be roughly the same. A broader d i s t r i b u t i o n of c o r r e l a t i o n times i s e x h i b i t e d by the t e r t - b u t y l amine deuterate where the minimum i s much broader than (CH^CNDg minimum (Figure 7.4). The reason f o r t h i s i s that i n t h i s case, the minimum of about 32 +_ 1 msec a r i s e s from three types of motions i . e . , methyl, t e r t - b u t y l , and o lOO 150 200 250 Temperature ( °K ) Figure 7.4 Temperature dependence of s p i n - l a t t i c e r e l a x a t i o n time i n tert-butylamine-ND^ and tert-butylamine deuterate. - 121 -i s o t r o p i c r o t a t i o n o f the whole molecule. For such a composite motion, the observed s p i n - l a t t i c e r e l a x a t i o n time T-j i s r e l a t e d to the s p i n - l a t t i c e r e l a x a t i o n time T^. f o r each motion i by the r e l a t i o n n ( 1 / T J = L (1/T,,) (7.5) 1 i=l 1 1 where VT-| • i s given by Equation (3.55) i . e . , l i V c i . 0 c l 1 + 2 2 V c i 1 + A 2 2 4 V c . i (?.6) T c i = T o i e x P ( E a i / R T ) <7'7> Equation (7.5) requires the knowledge of i n d i v i d u a l T-^ which i n turn need a knowledge of C., T q 1 . and E ... can be c a l c u l a t e d t h e o r e t i c a l l y but t h e o r e t i c a l c a l c u l a t i o n s of t h i s type seldom agree with the experimental r e s u l t s . A l t e r n a t i v e l y i t can be c a l c u -l a t e d from the experimentally observed minimum i n the p l o t of T^ agains t temperature, (provided minimum due to that motion i s observed). At the minimum value of T^. we have u ) Q x . = 0.616 and then Equation (7.6) gives ( 1 / T l i > m i n = (V%)0.425) (7.8) Thus C . can be e x t r a c t e d . E . can be found from the slope of the £nT,. versus 1/T curve and x . can be found from minimum c o n d i t i o n 122 -0.616 = V o l e x P ( E a i / R T ) (7.9) In Equation (7.9) E . and the temperature T at minimum are known, a n a ) o T o i c a n b e c a^ c u^ a t e c i' T n e actual f i t t i n g then r e q u i r e s a d j u s t i n g of i n d i v i d u a l C., T., E . to get a best f i t to Equation (7.5). The i l a i s i t u a t i o n i s extremely d i f f i c u l t even f o r the case n = 2 i n Equation (7.5). The only work reported so f a r i s t h a t of A l l e n and Johnson [7.10]. In the present case of (CHgJgCNDg (n = 2) the s i t u a t i o n i s d i f f i c u l t because only one broad s i n g l e minimum i s observed as both motions seem to overlap. An estimate of from experiment i s not th e r e f o r e very r e l i a b l e . This broad minimum f u r t h e r a f f e c t s the ev a l u a t i o n o f E . from the slope of the >inT, versus 1/T curve. However some progress can be made i n the f o l l o w i n g way. I f we assume that on the low temperature s i d e of the broad minimum, the r e l a x a t i o n process which i s dominating T-j i s t h a t o f the methyl group r o t a t i o n , the s i t u a t i o n i s s i m p l i f i e d a l i t t l e b i t . On the low temperature s i d e of the minimum u T . >> 1. Then f o r the case when n = 2 Equations (7.5) and (7.6) give ]_ T n u b T c l _ 0 C2 (7.10) The second term i n Equation (7.10) which i n the present case a r i s e s from the t e r t - b u t y l group motion, can be neglected. Equation (7.10) and (7.7) then become - 123 -E , log = 21og % + log X q 1 + ^ R T - log 2C ] or log T q 1 = (log T ] + l o g 2C-,) - (21og u>0 + 2 > 3 Q 3 r t ) (7.11) Equation (7.11) was used to estimate the value of T - J . C-| was estimated from the experimental second moment value. C-| i s r e l a t e d to * second moment M 2 by the f o l l o w i n g r e l a t i o n Using. M 2 = 31 G 2, &Q = 2-n x 26.46 x 10 6 s e c " 1 , T ] = 1.25 sec at T = 112 K with E.^ from slope of £nT-| versus 1/T curve (Figure 7.5) which i s -13 equal to 3.18 kcal/mole, Equation (7.11) gives T - J = 8.5 x 10 sec. -13 The same equation with T-j data at 116 K gives x0-| = 7.7 x 10 sec, which means there may be some experimental e r r o r i n T-j and/or the temperature T, or t h i s i n d i c a t i o n that the second motion i s c o n t r i b u t i n g to T-j. The a c t i v a t i o n energy which i s obtained from low temperature side of minimum i n the £nT 1 versus 1/T p l o t i s 3.18 +_ 0.07 kcal/mole and agrees with the value obtained from l i n e width data namely, 3.19 + 0.26 kcal/mole. The estimation of b a r r i e r height f o r t e r t - b u t y l r e o r i e n t a t i o n from * In f a c t the modulated value of second moment should be used and t h i s r e l a t i o n i s true f o r i n t r a m o l e c u l a r value. However, because we have no knowledge of c r y s t a l s t r u c t u r e , we use the above method. - 1 2 4 -3 4 5 6 7 8 9 1 0 0 0 / T ( ° K ~ 1 ) Figure 7.5 V a r i a t i o n of s p i n - l a t t i c e r e l a x a t i o n time as a f u n c t i o n of the r e c i p r o c a l of the absolute temperature i n t e r t - b u t y l ami n e - ^ and tert-butylamine deuterate. - 125 -the high temperature side of the minimum i s p o s s i b l e i f we assume that on t h i s s i d e , the r e l a x a t i o n process involves the t e r t - b u t y l group. But i n our case, we are he l p l e s s because (CH-^CNDg melts before enough data can be obtained to enable the a c t i v a t i o n energy to be c a l c u l a t e d . In the case of the tert-butylamine deuterate the s i t u a t i o n i s high l y complicated because i n t h i s case there are three motions each having n e a r l y the same b a r r i e r height with a wide d i s t r i b u t i o n of T c ' s . Complete a n a l y s i s of i n terms of Equation (7.5) f o r n = 3 i s p r a c t i c a l l y impossible with t h i s s e t of data, because of lack of knowledge of C , T . and E . to permit the use of Equation (7.6) and i o i an (7.7). A rough estimate i s again made on the assumption t h a t at low temperature s i d e of minimum, the only motion r e s p o n s i b l e f o r r e l a x a t i o n i s methyl group motion. E , was estimated from the slope a i o f £nT-| versus 1/T curve (Figure 7.5) and i t s value i s 1.73 + 0.04 kcal/mole. The T-] value at 77 K i s 2.01 sec. C-j was again c a l c u l a t e d 2 using the M 2 value of 25.8 G f o r r i g i d l a t t i c e second moment i n the deuterate (Table 7.2). Equation (7.11) then gives T = 2.2 x 10" 1 1 sec. The corresponding value of T-j a t 108.5 K i s 352 msec g i v i n g T q . | = 1.02 x 1 0 " ^ sec. The high temperature s i d e of minimum gives an a c t i v a t i o n energy of 2.46 +0.08 kcal/mole which agrees well with the a c t i v a t i o n energy obtained from l i n e width data i . e . , 2.14 +_ 0.21 kcal/mole. Estimation of T . i n t h i s case i s not meaningful, because the process c l e a r l y i n v o l v e s more than one type of motion. - 126 -D. Discussion The i n t e r m o l e c u l a r second moment Mg f o r (CH^CNDg v/as c a l c u l a t e d using p = 0.78 g/cc and was 7.93 G . The data given i n reference [7.2] show that P increases roughly by 0.012 g/cc f o r every 10°C decrease i n temperature and on t h i s basis a d e n s i t y o f 0.78 g/cc i s expected i n the s o l i d s t a t e . But t h i s i s j u s t a crude approximation. The actual density, may be higher which i n turn w i l l lead to a higher value f o r M^. Smith has l i s t e d the values of the r i g i d l a t t i c e second moment Mg f o r a s e r i e s of s i m i l a r compounds [7.7] and these values l i e i n the range of 28-31 G . On the other hand Haigh et a l . [7.9] observed that f o r amines nearly a l l the c o n t r i b u t i o n s to Mg comes from the i n t r a m o l e c u l a r second moment Mg, and t h a t estimation of Mg i s d i f f i c u l t because o f H-bonding of the amino group. In t r i m e t h y l amine 2 2 t h e i r value f o r Mg was 2.1 G . A value o f about 4 G was estimated by Kromhout and Moulton [7.8] f o r isobutylamine. However, the s i t u a t i o n i n trimethylamine and tert-butylamine i s d i f f e r e n t because of the d i f f e r e n t C-C and C-N bond lengths. In view o f these values 2 the estimate of 7.93 G f o r Mg i s considerably higher. We t h e r e f o r e 2 have taken the value o f experimental second moment (30.96 + 1.14 G ) at 77 K as being the r i g i d l a t t i c e second moment value. We checked the s a t u r a t i o n f a c t o r by e x t r a p o l a t i n g the T-j values to 77 K and found 2 i t to be much l e s s than 1. Moreover a value o f 6 G f o r Mg derived from t h i s experimental value i s q u i t e a reasonable value f o r t h i s compound. The second moment values f o r deuterate were c a l c u l a t e d using the atomic parameters given by McMullan et a l . [7.1] and data derived from microwave work on other amines [7.3-7.4]. The main d i f f e r e n c e between - 127 -the data of McMullan et a l . and the microwave data i s i n the C-C and C-N bond lengths. In the former case [7.1] the C-C bond length i s o o 1.42 A, and t h i s increases CH 3-CH 3 i n t e r a c t i o n to 5.11 G . This value agrees well with the value of 5.1 G c a l c u l a t e d by Haigh et a l . [7.9] f o r trimethylamine which has the same order of C-N bond length i . e . , 1.47 A . We could not decide on the v a l i d i t y of these atomic parameters, as we were unable to reach the r i g i d l a t t i c e second moment. However there i s d e f i n i t e l y some u n c e r t a i n t y i n these bond lengths because the molecule i s approaching i s o t r o p i c motion at -30°C, the temperature at which the x-ray d i f f r a c t i o n s t u d i e s have been c a r r i e d out. McMullan et a l . [7.1] points out that the amine molecule behaves as a hindered r o t o r at -30°C and the molecular parameters of amine are r a t h e r u n c e r t a i n . They exclude the p o s s i b i l i t y of f r e e r o t a t i o n at -30°C. Our c a l c u l a t e d i s o t r o p i c second moment, ne g l e c t i n g the cage c o n t r i b u t i o n , i s very s m a l l , but a value of 0.1-0.15 i s expected i f cage c o n t r i b u t i o n i s i n c l u d e d . This value agrees w i t h i n experimental u n c e r t a i n t y to the experimental value of 0.2 G 2 around 250 K. At -30°C (243 K), the molecule i s d e f i n t e l y not a f r e e r o t o r , but at higher temperatures i t behaves l i k e a f r e e r o t o r . The r e l a x a t i o n i n both pure amine and the deuterate was exponential i n form. This i s because of the f a c t that now methyl groups are not well i s o l a t e d as was the case i n diethylamine and diethylamine deuterate ( c f . Chap. V). The intermethyl c o n t r i b u t i o n due to two other methyl groups introduces several terms i n c r o s s - c o r r e l a t i o n , but the c o e f f i c i e n t s of these terms are so small t h a t they can be neglected and o v e r a l l r e l a x a t i o n shows an exponential character [7.11]. - 128 -Ex i s t e n c e of a s i n g l e minimum i s c h a r a c t e r i s t i c o f the t e r t - b u t y l group. In the m a j o r i t y o f cases the b a r r i e r height and the T C ' S are nea r l y the same g i v i n g a sharp minimum e.g., i n hexamethylethane i n the work of Chezeau et a l . [7.12], while i n other cases e i t h e r separate minima [7.10] or. a broad minimum e x i s t . The broad minimum i s due to ne a r l y the same b a r r i e r h e i g h t s , but with a d i s t r i b u t i o n of c o r r e l a t i o n times between the methyl and t e r t - b u t y l group motions. T h e o r e t i c a l c a l c u l a t i o n of the x c ' s f o r such a l a r g e molecule i s extremely tedious and i s impossible because of lack o f knowledge of the c r y s t a l s t r u c t u r e . The t h e o r e t i c a l c a l c u l a t i o n o f C^'s from experimental second moment i s again doubtful because of lack of knowledge of the true second moment and the mechanism of r e l a x a t i o n . Because of the d i s t r i b u t i o n of c o r r e l a t i o n times a r i s i n g from these motions, the E a i-'s are al s o not true E a l- 's. We t h e r e f o r e regard the reported a c t i v a t i o n energies and T Q ' S as upper l i m i t s to the true values. Thus f o r methyl group i n (CH-^gCNDg the a c t i v a t i o n energy to the b a r r i e r h indering methyl group r o t a t i o n i n the upper 1imit i s 3.2 +0.1 kcal/mole with T Q = (8-9) x -13 10 I J sec. The a n a l y s i s of the T-j data i n the case of the deuterate using the above arguments has been c a r r i e d out only f o r the methyl group motion. The a c t i v a t i o n energy a s s o c i a t e d with the b a r r i e r hindering t h i s motion was estimated from the T-j data to be 1.7 kcal/mole and t h i s i s probably a large value f o r t h i s b a r r i e r . The approximate value of T q = 2.2 x T O - 1 1 sec also suggest t h a t more than one motional process i s o c c u r r i n g on t h i s s i d e of the minimum. The high temperature s i d e of minimum gave - 129 -an a c t i v a t i o n energy of 2.5 + .1 kcal/mole and t h i s b a r r i e r i s mostly due to the t e r t - b u t y l group r o t a t i o n s plus i s o t r o p i c r o t a t i o n a l motion o f the whole molecule. - 130 -References (Chapter VII) [7.1] R.K. McMullan, G.A. J e f f r e y , and T.H. Jordan, J . Chem. Phys., 47 (1967) 1229. [7.2] K r i s h n a j i and A. Mansingh, J . Chem. Phys.,'42 (1965) 2503; i b i d . , 44 (1966) 1590. [7.3] D.R. Lide and D.E. Mann, J . Chem. Phys., 28 (1958) 572. [7.4] J.E. Wollrab and V.W. L a u r i e , J . Chem. Phys., 48 (1968) 5058. [7.5] G.W. Smith, J . Chem. Phys., 42 (1965) 4229. [7.6] I n t e r n a t i o n a l Tables f o r X-Ray C r y s t a l l o g r a p h y , N.F.M. Henry and K. Lonsdale eds., The Kynoch Press, England, 1952, V o l . I, p. 329. [7.7] G.W. Smith, J . Chem. Phys., 54 (1971) 174. [7.8] R.A. Kromhout and W.G. Moulton, J . Chem. Phys., 23 (1955) 1673, and i b i d . , 25 (1956) 34. [7.9] P.J. Haigh, P.C. Canepa, G.A. Matzkanin, and T.A. S c o t t , J . Chem. Phys.^48 (1968) 4234. [7.10] P.S. A l l e n and L.W;. Johnson, (to be pub l i s h e d ) . [7.11] P.S. Hubbard, Rev. Mod. Phys., 33 (1961) 249. [7.12] J.M. Chezeau, J . Dufourcq, and J.H. Strange, Mol. Phys., 20 (1971) 305. - 131 -CHAPTER VIII ISOPROPYLAMINE, ISOPROPYLAMINE DEUTERATE, TRIMETHYLAMINE, AND TRIMETHYLAMINE DEUTERATE A. Introdu c t i o n Isopropylamine and trimethylamine belong to a s e r i e s o f alkylamines which form hexagonal amine hydrates. A short d e s c r i p t i o n of thermo-dynamic aspect of trimethylamine was given i n Chapter I. The s t r u c t u r e s of both of the hydrates were discussed b r i e f l y i n Chapter I I I . The s t r u c t u r e of isopropylamine hydrate has r e c e n t l y been studi e d by McMullan e t a l . [8.1] by X-ray d i f f r a c t i o n . This hydrate having formula 10(CH 3) 2CHNH 2.80H 20 belongs to space group P63/mmc. The u n i t c e l l dimension at -160°C are a = 12.30 A, c = 24.85 A [8.1]. The water s t r u c t u r e c o n s i s t s of fo u r types of polyhedra: hexagonal prisms, 12-hedra, 14-hedra, and 16-hedra. The amine molecules are enclosed i n s i x 14-hedra and fo u r 16-hedra. The amine molecule i s hydrogen -bonded s i n g l y i n 16-hedra and double i n 14-hedra (Figure 2.6). Trimethylamine hydrate has been studi e d by x-ray d i f f r a c t i o n by Panke [8.2]. This hydrate belongs to space group P6/mmm with molecular formula 4(CH 3) 3N.41H 20. The u n i t c e l l dimension at -30°C o o are a = 12.378 A and c = 14.480 A. The host s t r u c t u r e c o n s i s t s of three u n d i s t o r t e d 12-hedra, two d i s t o r t e d 14-hedra, and two d i s t o r t e d - 132 -15-hedra due to hydrogen-bonding of amine molecule. The amine molecule l i e s i n s i d e the two 14-hedra and two 15-hedra. The two 14-hedra i n f a c t share a common hexagonal face and give r i s e to a new polyhedron c a l l e d 26-hedron. The two amine molecules i n s i d e 26-hedron are shown i n Figure 2.7b and one amine molecule i n s i d e 15-hedron i s shown i n Figure 2.7a. There has been no broad line and pulsed nmr work on isopropylamine, but trimethylamine has been studi e d by broad!ine nmr by Fyfe and Ripmeester [8.3] and Haigh et a l . [8.4]. According to [8.3] t r i m e t h y l -2 amine behaves as r i g i d molecule at 77 K with a second moment of 30.5 G 2 and at 140 K the second moment reduces to 2.7 G . The l a t t e r value corresponds to methyl plus r o t a t i o n o f whole molecule around a second three f o l d axis ( h e r e a f t e r t h i s axis w i l l be denoted as C ^ - a x i s ) . 2 On the other hand Haigh e t a l . [8.4] observed a second moment of 29.5 G at 4.2 K and 4.6 G 2 at 140 K. The value of 4.6 G 2 i s small to account f o r a l l the methyl group r o t a t i n g (C^ r o t a t i o n ) a t 146 K, while on the other hand i t i s too large f o r C^+C^ r o t a t i o n . Usually such compounds show a s i n g l e t r a n s i t i o n f o r C^+C^ motion, and a value 2 '•' of 2.7 G i s more reasonable to account f o r such a t r a n s i t i o n . The a c t i v a t i o n energy f o r the b a r r i e r hindering t h i s motion (C^+C^) from the l i n e width data (where Fyfe and Ripmeester [8.3] and Haigh e t a l . [8.4] agree with one another) by [8.3] was (6.6-8.4) +_0.5 kcal/mole and by [8.4] to be 5.75 kcal/mole. There has been some heat c a p a c i t y measurement work on trimethylamine by Aston e t a l . [8.5] and an a c t i v a t i o n energy of 4.27 kcal/mole per methyl group has been obtained. The microwave work o f Lide and Mann [8.6] shows the b a r r i e r hindering methyl group to be 4.4 kcal/mole. - 133 -There has been some broad line nmr work on trimethylamine hydrate and deuterate [8.7]. In trimethylamine hydrate a second moment of 29 G 2 was observed at 100 K and 2 G 2 at 200 K [8.8]. The line width in trimethylamine deuterate was 1.4 G between 208 and 270 K [8.8]. Some qualitative dielectric measurement has been reported by Davidson [8.8] on trimethylamine hydrate. The present nmr study shows that in isopropylamine, the only motion involved is that of methyl group rotation. The relaxation function in this case is non-exponential. The isopropylamine deuterate shows much more molecular mobility ranging from methyl group rotation to rotation of whole molecule around pseudo three fold axis (C^)-The relaxation function in this deuterate is exponential. We have not repeated the cw work on trimethyl amine as there is enough work done Calthough the question of second moment value at higher temperatures has to be solved) by two groups [8.3-8.4]. We have therefore supplemented this work by relaxation studies on pure trimethylamine. The relaxation function in trimethylamine is exponential. The deuterate of trimethyl-amine shows entirely different behaviour than the pure amine as well as other amines studied here. The line shape of trimethylamine deuterate is nearly close to Lorentzian line shape from 77 to ~100 K. The relaxation function in this range exhibits non-exponential behaviour, while at higher temperature it is exponential. - 134 -B. Experimental 1. M a t e r i a l s Isopropylamine was obtained from two sources, Eastman Kodak (reagent grade) and Baker Chemical Co. (Baker grade). Two batches o f samples were prepared from Eastman Kodak and Baker amine samples without f u r t h e r p u r i f i c a t i o n . (99.8% d 2) was obtained from S t o h l e r Isotope Chemicals. In some cases 99.75% D,,0 s u p p l i e d by Atomic Energy Canada was used. Trimethylamine (anhydrous) was a l s o obtained from the same above-mentioned two sources and was used s t r a i g h t without f u r t h e r p u r i f i c a t i o n . 2. Preparation of (CH 3) 2CHND 2 The preparation was e x a c t l y the same as described i n Chapter V. The a n a l y s i s of (CH 3) 2CHND 2 done i n the same way as before showed that NH 2 protons were about 98% deuterated to ND2. 3. Preparation of Amine and Deuterate Samples (a) Isopropylamine and Isopropylamine Deuterate The pure (CH 3) 2CHND 2 was f i l l e d d i r e c t l y i n the sample tubes and was sealed o f f by freeze-pump-thaw method. I t was Baker Sample. Two batches of samples were prepared f o r isopropylamine deuterate, each from Eastman Kodak and Baker amine samples. The formula r a t i o 10(CH 3) 2 CHNH2.80D20 of amine and D 20 s o l u t i o n was mixed and was freezed slowly i n the c o l d box i n an a i r t i g h t g l a s s vessel at -15°C. The hydrate obtained was crushed to powder, f i l l e d i n nmr tubes and sealed a f t e r degassing the d i s s o l v e d a i r . One sample from Baker grade amine was prepared by c o o l i n g 12 mole % amine s o l u t i o n i n D ?0 - 135 -as done by McMullan et a l . [8.1] at about -6 to -7°C. The r e s u l t s of t h i s sample and the others obtained by f r e e z i n g formula r a t i o were about the same. The a n a l y s i s of decomposed sample done i n a s i m i l a r way as before showed the gjest content i n deuterate i n the range of 93-96%. (b) Trimethylamine and Trimethylamine Deuterate Trimethylamine (Baker) was d i r e c t l y f i l l e d i n 7 mm o.d. nmr tubes and was sealed by freeze-pump-thaw method. The deuterate was prepared i n two batches, one from Eastman Kodak and the other from Baker amine samples. 9 mole % of trimethylamine i n D 20 was cooled slowly i n the c o l d box (-15°C). The c r y s t a l s obtained were f i n e l y powdered, f i l l e d i n nmr tubes and sealed a f t e r removing d i s s o l v e d a i r . The a n a l y s i s of decomposed smaples were well w i t h i n the s t o i c h i o m e t r i c composition (4(CH 3) 3N.41D 20). 4. Spectrometer and Method of Measurements (a) Isopropylamine and Isopropylamine Deuterate The r f f i e l d used f o r (CH 3) 2CHND 2 was about 0.025 mG from 77 to 100 K. " In other temperature ranges i t was higher but adjusted according to r e l a x a t i o n measurements to make the s a t u r a t i o n f a c t o r well below 1. In the m a j o r i t y of experiments, the 90° pulse length i n 180°-T-90° pulse sequence was 1.5 ysec, but i n some cases the higher 90° pulse length up to 2.2 ysec was used. The r e l a x a t i o n f u n c t i o n R(t) was non-exponential from about 145 to 178 K i n case of (CH 3) 2CHND 2. Consequently t when R ( t 0 ) = 1/2 was used f o r a l l temperature ranges studied here to c h a r a c t e r i z e the r e l a x a t i o n . The data of (CrL) ?CHND ? are t h e r f o r e - 136 -analysed i n view of H-H theory. In the case of isopropylamine deuterate, the r e l a x a t i o n f u n c t i o n was exponential. T-j was t h e r e f o r e c a l c u l a t e d using Equation (4.7). (b) Trimethylamine and Trimethylamine Deuterate No absorption l i n e measurements have been c a r r i e d out f o r trimethylamine as there i s enough work by the two groups [8.3-8.4]. However, we have extended some T^ measurement on pure trimethylamine i n order to get a b e t t e r value o f a c t i v a t i o n energy f o r the b a r r i e r hindering methyl group r e o r i e n t a t i o n . The r e l a x a t i o n f u n c t i o n i n trimethylamine was exponential and 90° pulse length used i n 180°-T-90° pulse sequence v a r i e d from 1.5 to 2 ysec i n d i f f e r e n t measurements. A l l the r e l a x a t i o n data are on the Baker trimethylamine sample. Trimethylamine deuterate e x h i b i t e d a strange behaviour i n the range of 77 to about 100 K where l i n e shape was c l o s e to Lorentzian l i n e shape. The second moment reported here may not be too accurate as they are c a l c u l a t e d at about 1% cut o f f the t o t a l i n t e n s i t y of absorption l i n e shape. The f i n e s t r u c t u r e o f 3 s p i n system was present i n t h i s temperature range (77 to 100 K), but was not very well r e s o l v e d . The r e l a x a t i o n f u n c t i o n i n the range 77 to 115 K was non-exponential i n trimethylamine deuterate, while i n other temperature ranges i t showed exponential behaviour. - 137 -C. Results 1. Absorption L i lie A n a l y s i s (a) Isopropylamine nand Isopropylamine Deuterate ( i ) Second Moment C a l c u l a t i o n s The second moment was c a l c u l a t e d using Equation (5.1) and with > molecular parameters provided by McMullan et a l . [8.1] and microwave data [8.6,8.9]. The data taken from [8.1] have bond length C-C = 1.52 A, C-N = 1.40 A, N-H = 1.0 A, and C-H = 1.09 A. The angles used i n coordinate generation of protons and nitrogen were assumed to be t e t r a h e d r a l . In the second set which i s based on microwave data the o o f o l l o w i n g bond lengths have been assumed: C-C = 1.54 A, C-N = 1.47 A, o o C-H = 1.09 A, and N-H = 1.02 A. The angles f o r coordinate generation were a l s o assumed to be t e t r a h e d r a l . The i n t e r m o l e c u l a r second moment f o r (CH-^gCHNDg i s hard to c a l c u l a t e as the c r y s t a l s t r u c t u r e o f t h i s amine i s not known. Based on experimental r e s u l t s we take i n t e r m o l e c u l a r second moment M£ of about 2 2 4 G . The reason of taking M£ = 4 G w i l l be given l a t e r . Estimation of M£ ( i n t e r m o l e c u l a r second moment) f o r isopropylamine deuterate can be done i n the usual way as discussed i n Chapter VII. o Use o f Equation (7.1) with R = 2.046A(from McMullan et a l . data) gives 2 0 Mg = 0.43 G and the use of microwave data where R = 2.091A gives Mg = 2 0.39 G . For R^, the data of space group and the coordinates of c e n t r a l carbon atom (attached to -CH group) given by McMullan et a l . [8.1] have been used. As a check on these values Mg was a l s o c a l c u l a t e d using Equation (5.3). The dens i t y based on u n i t c e l l parameters at -160°C [8.1] and at -30°C [8.10] i s 1.118 g/cc and 1.081 g/cc r e s p e c t i v e l y . - 138 -The value of R = 4.27 A (-160°C) gives M£ = 0.41 G 2 and the value of o p R = 4.30 A (-30°C) gives M£ = 0.39 G . The values from both methods 2 agree very w e l l . We take M£ = 0.40 G . The deuteron and exchanged proton 2 c o n t r i b u t i o n i s approximated to be about 0.1 G , so that t o t a l M£ 0.50 G 2. The second moment f o r i s o t r o p i c r o t a t i o n O ^ i s o ^ o r t h e d e u t e r a t e 2 was c a l c u l a t e d using Equation (7.3) and i s equal to 0.13 G a f t e r i n c l u d i n g the c o n t r i b u t i o n s due to deuterons and exchanged protons. A summary of r e s u l t s f o r a l l these c a l c u l a t i o n s i s given i n Table 8.1. ( i i ) Experimental Results The experimental r e s u l t s f o r the second moment are p l o t t e d against temperature i n Figure 8.1 f o r (CH 3) 2CHND 2 and isopropylamine deuterate. (CH 3) 2CHND 2 shows r i g i d s t r u c t u r e of 3 sp i n system. The experimental value o f second moment f o r (CH 3) 2CHND 2 at 77 K i s 22.73 +0.91 G 2 and 2 i t approaches a plateau value of 9 G around 165 K. This l a t t e r value 2 of 9 G i s c o n s i s t e n t with both methyl groups r o t a t i n g i n (CH-^CHND,,. Tbe l i n e width at 77 K i s 18.78 +0.35 G and the plateau value o f 7 G i s reached around 150 K. A rough estimate of a c t i v a t i o n energy f o r the b a r r i e r hindering the methyl group r o t a t i o n can be obtained from t h i s l i n e width t r a n s i t i o n with the help of Equation (3.25). This equation with C = 18.8 G, B = 7 G gives t h i s a c t i v a t i o n energy to be 3.0 + 0.20 kcal/mole. More motional freedom i s shown by isopropylamine deuterate. The experimental value of second moment i n t h i s case at 77 K i s 2 14.24 +_1.40 G and it shows a very slow decrease reaching a value 2 8 -2 4 -CD •-c 2 0 -CD E Mo 16 -on 12 -o CD CO ' 8 -A Isopropy la m i n e - N D 2 o IsopropyIamirve< D e u t e r a t e Figure 8.1 PMR second moment vs, temperature i n , isopropy!amine-ND^ and isopropylamine deuterate. CO to O c5> 1 r~ IOO T T 1 5 0 2 0 0 T e m p e r a t u r e [°K] 2 5 0 140 -Table 8.1 T h e o r e t i c a l Second Moment Values f o r (CH 3) 2CHND 2 and Isopropylamine Deuterate Types of Motion based on Deuterate based on (CH 3) 2CHND 2 based on Ref. [8.1] G 2 Ref. [8.6,8.9] G 2 Ref. [8.6,8 G 2 R i g i d 2CH 2 19.30 19.30 19.30 CH 3-CH 3 1.45 1.35 1.35 C H 3 - C H 2.19 2.10 2.10 CH 3-ND 2 0.03 .0.02 0.02 CH-ND2 0.02 0.02 0.02 M£ 0.50 0.50 4.00 T o t a l 23.49 23.29 26.79 2CH 3 r o t a t i n g 2CH 3 4.83 4.83 4.83 CH 3 _CH 3 1.25 1.16 1.16 C H 3 - G H 1.88 1.81 1.81 CH3-ND 0.03 0.02 0.02 CH-ND2 0.02 0.02 0.02 MJ /vO.35 ^0.35 1.00 T o t a l ^8.36 ^8.19 8.84 2CH 3 + CS r o t a t i o n * 2CH 3 0.53 0.62 C H 3 _ C H 3 0.31 0.29 — CH3-CH 0.10 0.09 CH 3-ND 2 0.02 0.01 /V0.20 a,0.20 T o t a l • ^ 1.16 ^1.11 - 141 -Table 8.1 (Continued) 4. I s o t r o p i c r o t a t i o n 0.00 0.00 0.13 0.13 To t a l 0.13 0.13 * C^ r o t a t i o n means r o t a t i o n around pseudo three f o l d a x i s ( i . e . , around -CH bond). of 5.59 + 0.21 G 2 at 112 K to a plateau value of 1 G 2 around 212 K. The value of 1 G i s c o n s i s t e n t with both CH3 + C^ r o t a t i o n . The l i n e width i n the f i r s t t r a n s i t i o n region i s of no value, because at f i r s t the l i n e width at 77 K i s not a r i g i d l a t t i c e l i n e width, secondly the l i n e width of about 5.4 G i s not the completely narrowed l i n e width. However, a rough estimate can be made f o r the b a r r i e r hindering C^ r o t a t i o n . For this we assumed C = 5 G and took B as 1.9 G. Equation (3.25) thus l e d to an a c t i v a t i o n energy of 1.69 +_0.22 -4 -5 kcal/mole with x c ranging from 1.5 x 10 sec to 1.3 x 10 sec from 112 to 156 K. The cw data are thus i n agreement with the f i n d i n g of McMullan et a l . [8.1] that the isopropylamine i s r o t a t i n g around C3 axi s. (b) Trimethylamine and Trimethylamine Deuterate ( i ) Second Moment C a l c u l a t i o n s o o Here the average bond d i s t a n c e C-N = 1.472 A, C-H = 1.09 A, /iCNC = 108.7° [8.6] and other angles to be t e t r a h e d r a l have been used i n the c a l c u l a t i o n o f i n t r a m o l e c u l a r second moment Mi. The in t e r m o l e c u l a r - 142 -second moment M£ was c a l c u l a t e d using d e n s i t y p = 1.06 g/cc and R = o 4.62 A i n equation (5.3) f o r the trimethylamine deuterate. A value of 0.33 G i s obtained f o r M£ from Equation (5.3). The deuteron 2 c o n t r i b u t i o n does not increase t h i s value, and a value o f 0.35 G i s estimated a f t e r t a k i n g t h i s i n t o account. The f i n a l r e s u l t s are l i s t e d i n Table 8.2. ( i i ) Experimental Results The experimental r e s u l t s f o r trimethylamine deuterate are p l o t t e d i n Figure 8.2. Owing to Laurentzian l i n e shape the second moment are c a l c u l a t e d up to approximately 1% cut o f f the t o t a l absorption i n t e n s i t y . These values may not be as much accurate as the other values. The second moment thus c a l c u l a t e d at 77 K i s 8.46 +_0.76 G 2 and i t approaches a value o f 1.42 +0.02 G 2 at about 114 K f i n a l l y 2 reaching a plateau value of 0.60 G around 180 K. The l a s t value of 2 0.60 G i s not assigned, because t h i s value corresponds n e i t h e r to 2 Cg+Cg motion ( t h e o r e t i c a l value 1.40 G ) nor to i s o t r o p i c motion ( t h e o r e t i c a l value 0.15 G 2 ) . The other models were t r i e d , but i n each 2 case the second moment i s much l a r g e r than 0.6 G . The corresponding 2 l i n e width at 77 K i s 1.50 +_ 0.03 G and i t stays n e a r l y constant up to the me l t i n g point ( 280 K). The l i n e width agrees to the one mentioned by Davidson [8.8] (1.4 G between 208 and 270 K) w i t h i n experimental u n c e r t a i n t y f o r t h i s deuterate. The l i n e width against temperature i s a l s o shown i n Figure 8.2. For a comparison with pure trimethylamine the data o f Fyfe and Ripmeester [8.3] are als o shown i n Figure 8.2. 32- , 2 8 -"cxP 24 -1 •4— c 2 0 E J I6-| "D u d) CO 8H 4 -O ( C H 3 ) 3 N Figure 8.2 Line width i n gauss and second moment i n gauss as a fu n c t i o n of temperature i n trimethylamine and trimethylamine deuterate. S e c o n d M o m e n t L i n e W i d t h S e c o n d M o m e n t | Ref. [8.3] 3 3 2 I • L i n e W i d t h 0 •*-•a c OS ® O •0-0—o-o- -6>—O-IOO I i — I ~T—I r — 1 5 0 2 0 0 T e m p e r a t u r e [°K] - O O Q> r-20 IB hl6 14 -12 -IO " 8 - 6 -4 - 2 O c o 2 5 0 - 144 -Table 8 . 2 Second Moment Values f o r Tr imethy lamine Deutera te . Mot ion 3CH 3 I n t r a M£ (G 2 ) C H 3 - C H 3 CH 3 -N In te r MJ (G 2 ) T o t a l M 2 (G 2 ) R i g i d 22.52 4 .65 0 .03 0 .35 27.55 C g - r o t a t i o n 9 5.63 3 . 0 6 b 0.03 0 .25 8 .97 C 3 + C 3 r ° t a t i o n c 0.57 0.67 0.01 0 .20 1.40 I s o t r o p i c 0 0 0 0 . 1 5 0 .15 C 3 ~ r o t a t i o n means r o t a t i o n of 3GH.3 groups around t h e i r symmetry a x i s . Obtained by c o n c e n t r a t i n g the three CH 3 ~protons on the cent re of c i r c l e which they make d u r i n g r o t a t i o n . C i means, the second three f o l d a x i s o f whole m o l e c u l e . 2. R e l a x a t i o n Measurements (a) Isopropylamine and Isopropylamine Deuterate The r e l a x a t i o n i n the case of pure i sopropy lamine was non-exponent ia l beyond 145 K. A l l the data are t h e r e f o r e analyzed from the p o i n t of view of H-H t h e o r y . The temperature dependence of t ( t ime when R ( t Q ) = 1/2) i s p l o t t e d i n F igure 8 . 3 . The exper imenta l va lue of t at minimum i s 17 msec at 177 K. We know from H-H theory t h a t a t minimum ( t /T 1 ) = 1.52 and t h i s g ives (1/T 1 ) = 0.0894 s e c " 1 . Th is va lue of (1/T 1 ) was used t o e x t r a c t M Q T C as a f u n c t i o n of temperature i n a s i m i l a r way as done i n Chapter V. The temperature dependence of W . T i s p l o t t e d a g a i n s t 1/T(K _ 1 ) i n F igure 8 . 3 . - 145 -T e m p e r a t u r e [°K] IIO 130 150 170 190 _!__ , I • I , I , L_ i amine-ND 9. i i 1 1 1 ' 1 1 1 1 1 r 6.0 7.0 8.0 IOOO/T pK-'} - 146 -The a c t i v a t i o n energy obtained from t h i s p l o t i s 3.50 +_ 0.07 kcal/mole -13 with T = (1.78 + 0.22) x 10 sec. The second moment i n t h i s o — range corresponds to methyl group r o t a t i o n second moment. Hence t h i s i s the a c t i v a t i o n energy f o r the b a r r i e r hindering methyl group r e o r i e n t a t i o n . This a c t i v a t i o n energy agrees roughly to the one obtained from l i n e width data i . e . , 3.0 +0.20 kcal/mole. The isopropylamine deuterate shows e n t i r e l y d i f f e r e n t behaviour. There i s only a s i n g l e minimum of 45 +_ 1 msec f o r both methyl group motion and methyl plus motion (Figure 8.4). The a c t i v a t i o n energy obtained from the low temperature slope of znl^ versus 1000/T p l o t (Figure 8.5) i s 1.65 +0.03 kcal/mole, while the high temperature s i d e gives 1.61 +_ 0.05 kcal/mole. The high temperature side a c t i v a t i o n energy agrees to the one obtained by l i n e width data i . e . , 1.69 +_ 0.22 kcal/mole. A rough estimate of x Q can be made i n a s i m i l a r way as done i n Chapter VII. Use i s made of Equation (7.11) and (7.12) with M 2 values of 23.5 and 23.3 G 2 ( c f . Table 8.1). The T ] values at 96, 103.5, and 114 K are 1.251, 0.831, 0.807 sec r e s p e c t i v e l y . T h is gives T q 1 = 1.76 x 1 0 " 1 0 , 1.41 x 1 0 " 1 0 , and 1.37 x 1 0 " 1 0 sec (with M 2 = 23.5 G 2 ) ; and 1.30 x 1 0 - 1 0 , 1.42 x 1 0 - 1 0 , and 1.37 x 1 0 " 1 0 sec (with M 2 = 23.3 G ) r e s p e c t i v e l y . Tbe a c t i v a t e d s t a t e theory demands that f o r one r e o r i e n t a t i o n a l process should l i e i n the -12 -15 range of 10 - 10 sec. This means t h a t the second r e o r i e n t a t i o n a l process i s c o n t r i b u t i n g to the r e l a x a t i o n . The true a c t i v a t i o n energy thus appears to be l e s s than 1.65 kcaj/mole f o r the b a r r i e r hindering methyl groups r e o r i e n t a t i o n i n isopropylamine deuterate. For high temperature s i d e use of Equation (7.11) and (7.12) gives 10 3 J I sopropy lamine D e u t e r a t e c o o <D tf) e -pa — 1 — r 100 — I — 2 0 0 -i T 250 150 T e m p e r a t u r e [ ° K ] F igure 8.4 Proton s p i n - l a t t i c e r e l a x a t i o n t i m e , T ] p l o t t e d a g a i n s t temperature i n i sopropy lamine d e u t e r a t e . 2 0 0 0 IOOO-* 5 0 0 -c o u w F IOO-M.P —r— 4 Isopropylamine Deuterate T" 8 IOVT [°K*'] IO 00 12 Figure 8,5 Proton s p i n - l a t t i c e r e l a x a t i o n time, T-j p l o t t e d on a log s c a l e a g a i n s t 10 /T (where T i s the absolute temperature) i n isopropylamine deuterate. - 149 -x o 2 equal to 2.55 x 1 0 " 1 0 and 2.28 x I O " 1 0 sec at 220 and 213 K 2 2 r e s p e c t i v e l y (with M^ = 8.4 G ). The other value of M 2 = 8.2 G gives on these temperatures x o 2 = 2.53 x 1 0 ~ 1 0 and 2.10 x 1 0 ~ 1 0 sec r e s p e c t i v e l y . This again r e f l e c t s the second r e o r i e n t a t i o n a l process i s e f f e c t i v e at these temperatures. (b) Trimethylami he and Trimethylamine Deuterate The temperature dependence of T-j f o r t r i methyl amine and t r i m e t h y l -amine deuterate i s shown i n Figure 8.6. Trimethylamine melts before minimum i n T-| i s achieved. I f we assume that r e l a x a t i o n i s i n t r a -molecular the t h e o r e t i c a l value o f T-j minimum i n t r i m e t h y l amine from Equation (7.8) and (7.12) i s 8.4 msec. The experimental value of T^ minimum i s 35.5 msec which shows that trimethylamine melts before minimum i n T-j i s reached. The a c t i v a t i o n energy can be e x t r a c t e d from the slope of £nT-j versus 1000/T curve (Figure 8.7). The a c t i v a t i o n energy obtained i n t h i s way i s 5.75 +_0.14 kcal/mole which i s the same as the one obtained by Haigh e t a l . [8.4] c o i n c i d e n t l y , but i t i s consi d e r a b l y d i f f e r e n t from (6.6-8.4) +_0.5 kcal/mole obtained by Fyfe and Ripmeester [8.3]. This a c t i v a t i o n energy i s higher than the much accurate a c t i v a t i o n energy obtained from gas phase microwave work o f Lide and Mann [8.6] i . e . , 4.4 kcal/mole. This i n d i c a t e s t h a t some c o n t r i b u t i o n i s coming from motion. In trimethylamine deuterate, there appears to be p o s s i b l y two minima. The f i r s t , which werwere not able to achieve, which i s probably at a temperature lower than 77 K. The second i s a broad minimum of 50 + 1 msec (Figures 8.6 and 8.7). In Figures 8.6 and 8.7 - 1 5 0 -IOO 150 2 0 0 2 5 0 T e m p e r a t u r e (°K) Figure 8.6 The observed s p i n - l a t t i c e r e l a x a t i o n time, T-j as a f u n c t i o n of the absolute temperature i n trimethylamine and trimethylamine deuterate. - 151 -Figure 8.7 Values of s p i n - l a t t i c e r e l a x a t i o n tine f o r protons i n t r i m e t h y l -amine and trime t h y l amine deuterate. - 152 -although T-j i s not defined i n the region of 77 to %115 K, but the values of are c a l c u l a t e d using Equation (4.7) j u s t f o r the c o n t i n u i t y i n the curves. The p l o t of t i s a l s o shown i n both of the f i g u r e s f o r t h i s temperature range. The a c t i v a t i o n energy obtained from the p l o t of t versus 1000/T (Figure 8.7) i s 0.32 kcal/mole. The a c t i v a t i o n energy obtained from the slope of low temperature side of broad minimum i n the £hT^ versus 1000/T p l o t i s 0.73 +0.02 kcal/mole, while the high temperature s i d e gives an a c t i v a t i o n energy of 2.90 + 0.19 kcal/mole. The f i r s t a c t i v a t i o n energy of 0.23 kcal/mole i s very small and we are not able to assign i t , as we do not know the behaviour of trimethylamine i n t h i s deuterate below 77 K. The second a c t i v a t i o n energy of 0.73 kcal/mole can be due to the b a r r i e r h indering motion. The higher a c t i v a t i o n energy of 2,90 kcal/mole r e f l e c t s t h at some other motional process (other than C^+C^ r o t a t i o n ) i s o c c u r r i n g i n t h i s temperature range as the second moment i n t h i s temperature range i s much smaller than C^+C^ motion value. D. Discussion 1. Isopropylami ne and Isopropylami ne Deuterate The i n t e r m o l e c u l a r second moment M£ f o r pure isopropylamine was 2 taken to be 4 G from experimental r e s u l t s . Use of Equation (5.3) with d e n s i t y p = 0.81 g/cc and R = 2.091 A leads to a value of ^ 9 G 2 which i s d e f i n i t e l y too high. Kromhout and Moulton [8.11] assume 2 M£ = 6 G f o r isobutylamine. I f the assumption of [8.11] i s c o r r e c t , we expect M£ of the order of (7/11)6 = 3.8 G 2 f o r (CH 3) 2 CHNDg. Powles 2 and Ka i l on a s i m i l a r compound isobutylbromide obtained IC = 5.6 G [8.14]. - 153 -On t h i s basis we w i l l have a value o f (7/9)5.6 = 4.4 G 2 f o r ( C H 3 ) 2 C H N D 2 . 2 Our assumed value of 6 G f o r tert-butylamine-ND 2 leads to a value of 4.6 G 2 as M 2 value f o r (CH 3) 2,CHND 2. On these arguments a value of 4 G 2 i s q u i t e a reasonable value o f MJ of (CH 3) 2CHND 2. We are unable to say anything about the molecular parameters of isopropylamine i n isopropylamine deuterate as the temperature at which r i g i d l a t t i c e second moment i s reached, was not obtained by us. 2 The experimental value o f 1 G i s much c l o s e r to both methyl plus C 3 2 r o t a t i o n t h e o r e t i c a l M 2 value using microwave data (1.11 G ) than the 2 data of McMullan e t a l . (1.16 G ). But t h i s d i f f e r e n c e i s too small and l i e s w i t h i n the experimental e r r o r . However that the molecule shows C 3 motion i s i n agreement with the f i n d i n g s o f McMullan e t a l . The slow decrease i n second moment r e f l e c t s a broad d i s t r i b u t i o n o f c o r r e l a t i o n times. This d i s t r i b u t i o n o f c o r r e l a t i o n times a r i s e s from nearly the same b a r r i e r heights f o r the methyl and C 3 r o t a t i o n s of whole molecules. The r e l a x a t i o n f u n c t i o n i n the case o f (CH 3) 2CHND 2 was non-exponential. This non-exponential behaviour was a l s o present i n (CH 3) 2CHNH 2. I t i s t h e r e f o r e concluded t h a t the non-exponential behaviour of R(t) i s due to methyl groups. The minimum value o f 17 msec occurs at 177 K and t h i s value remains the same up to the melting point 179 K of (CH 3) 2CHND 2. This implies e i t h e r t h a t there i s a d i s t r i b u t i o n o f c o r r e l a t i o n times, or t h i s i s not a true minimum. A t h e o r e t i c a l c a l c u l a t i o n o f the minimum i s not j u s t i f i e d as the c r y s t a l s t r u c t u r e of amine i s not known. However the value o f a c t i v a t i o n energy when the data are analyzed i n terms of H-H theory i s the same as the one - 154 -obtained from the slope of £nt Q versus 1/T p l o t (3.52 +0.06 kcal/mole). This j u s t i f i e s the i n t e r p r e t a t i o n that 17 msec i s the minimum i n t Q . The same value t (17 msec) up to 179 K may be because these temperatures are very c l o s e to melting point o r due to the experimental e r r o r . In the isopropylamine deuterate, the r e l a x a t i o n f u n c t i o n was exp o n e n t i a l . This may be because of amine methyl groups come more c l o s e r towards each other because of the r e p u l s i v e f o r c e s of the deuteron cage. The broad minimum c l e a r l y a f f e c t s the a c t i v a t i o n energy. The a c t i v a t i o n energy of 1.65 kcal/mole i s t h e r e f o r e should be t r e a t e d as upper l i m i t f o r the b a r r i e r hindering methyl group r e o r i e n t a t i o n . On the same grounds 1.61 kcal/mole i s a upper l i m i t f o r C j ' r e o r i e n t a t i o n . We are not able to e x p l a i n the constant value o f T-j from 220 K. to the melting point o f the deuterate ( 270 K). P o s s i b l y t h i s i s because of the exchange process ND,, + D + — » NDg + taking place during C^ ' r o t a t i o n as suggested by McMullan e t a l . [8.1]. 2. Trimethylamine and Trimethylamine Deuterate. The T-j data i n trimethylamine give an a c t i v a t i o n energy of 5.75 kcal/mole which i s by coincidence the same as the one obtained by Haigh et a l . [8.4] and lower than the one obtained by Fyfe and Ripmeester [8.3]. The a c t i v a t i o n energy obtained from l i n e width data are not accurate, hence the accuracy of t h i s value i s not comparable to [8.3] and [8.4]. This a c t i v a t i o n energy which f o r the s o l i d s t a t e must be higher than the one obtained from gas phase microwave work i . e . 4.4 kcal/mole, but not to t h a t extent. The heat c a p a c i t y data of Aston et a l . [8.5] p r e d i c t 4.27 kcal/mole per methyl group. Goldfarb - 155 -and Kharl [8.12] i n t h e i r i r study of s o l i d trimethylamine at 77 K obtained an a c t i v a t i o n energy o f 4.73 kcal/mole when the b a r r i e r model was assumed to be p a r a b o l i c and 5.19 kcal/mole when b a r r i e r model was assumed to be p e r i o d i c . A l l these values are s t i l l l e s s than our value of 5.75 kcal/mole. This shows t h a t motion i s c o n t r i b u t i n g to the b a r r i e r . The data are d i f f i c u l t to analyze from the p o i n t of view of BPP theory as the observed minimum i s not the true minimum i n T-|. The compound melts before true minimum i s achieved. I t thus appears t h a t energy b a r r i e r f o r methyl group r e o r i e n t a t i o n i s l e s s than 5.75 kcal/mole. 2 In trimethylamine deuterate, the plateau value of 0.6 G i s hard to e x p l a i n . This value i s c e r t a i n l y not the one corresponding to C3+C3 r o t a t i o n . I f we assume that C-N bond length i s constant, then the c a l c u l a t i o n of second moment due to C^+C^ r o t a t i o n depends on the angle \> between C^-axis and C^-axis of CH^ group. In the present case t h i s angle (based on microwave data [8.6]) i s 69.8°. A smaller angle can be expected i n deuterate because of r e p u l s i v e f o r c e s of the deuteron cage. This smaller angle w i l l reduce the methyl group c o n t r i b u t i o n of second moment [ c f . Equation (3.32)] i n Table 8.2. At the same time t h i s w i l l i ncrease the CH^-CH^ c o n t r i b u t i o n i n Table 8.2. Therefore the t o t a l value of second moment f o r C^C^ r o t a t i o n w i l l be 2 hardly changed and i n any case i t w i l l not be c l o s e to 0.6 G . The question of p a r t i a l d i f f u s i o n o f trimethylamine from deuteron cage i s r u l e d out on the grounds of constant l i n e width and the plateau value of second moment. There i s a l s o not a random motion e x h i b i t e d by (CH-^N, because the second moment due to random motion can be as high as 15% - 156 -than the i s o t r o p i c second moment value of 0.15 G [8.13]. The only p o s s i b l e explanation i s that besides C 3+C 3 r o t a t i o n , (CH 3) 3N e x h i b i t s some t r a n s l a t i o n a l motion. This explanation i s supplemented from the X-ray d i f f r a c t i o n work of Panke [8.2] from the disordered p o s i t i o n of nitrogen atom represented by l a r g e thermal e l l i p s o i d ( Figure 2.7). In the case of C 3+C 3 r o t a t i o n , the p o s i t i o n of N-atom i s f i x e d , and i t s p o s i t i o n would not have been represented by a l a r g e thermal e l l i p s o i d . The a c t i v a t i o n energy of 0.32 kcal/mole i s not assigned. There are not enough data below 77 K both i n cw and r e l a x a t i o n measurements to enable a s a t i s f a c t o r y assignment to be made. The a c t i v a t i o n o f 0.73 kcal/mole i s due to C 3 r o t a t i o n , but t h i s value i s probably a higher value because of the d i s t r i b u t i o n of c o r r e l a t i o n times which give r i s e to a broad minimum. The a c t i v a t i o n energy of 2.90 kcal/mole i s c l e a r l y not due to C 3 motion. This i s t h e r e f o r e due to t r a n s l a t i o n a l motion of trimethylamine molecule i n view of the above arguments. - 157 -References (Chapter VIII) [8.1] R.K. McMullan, G.A. J e f f r e y , and D. Panke, J . Chem. Phys., 53 (1970) 3568. [8.2] D. Panke, J . Chem. Phys.,48 (1968) 2990. [8.3] C A . Fyfe and J . Ripmeester, Can. J . Chem., 48 (1970) 2283. [8.4] P.J. Haigh, P.C Canepa, G.A. Matzkannin, and T.A. S c o t t , J . Chem. Phys., 48 (1968) 4234. [8.5] J.G. Aston, M.L. Sagenkahn, G.J. Szasz, G.W. Moessen, and H.F. Zuhr, J . Am. Chem. Soc., 66 (1944) 1171. [8.6] D.R. Lide J r . , and D.E. Mann, J . Chem. Phys., 28 (1958) 572. [8.7] S. Brownstein, D.W. Davidson, and D. F i a t , J . Chem. Phys., 46 (1967) 1454. [8.8] D.W. Davidson, CIathrate Hydrates, National Research Council of Canada, D i v i s i o n o f Chemistry 1971. [8.9] J.E. Wollrab and W.W. L a u r i e , J . Chem. Phys., 48 (1968) 5058. [8.10] R.K. McMullan, T.H, Jordan and G.A. J e f f r e y , J . Chem. Phys., 47_ (1967) 1218. [8.11] R.A. Kromhout and W.G. Moulton, J . Chem. Phys., 23 (1955) 1673. [8.12] T.D. Goldfarb and B.N. Kha r l , J . Chem. Phys.i"46 (1967) 3379. [8.13] G.W. Smith, J . Chem. Phys., 42 (1965) 4229. [8.14] J.G. Powles and J.A.E. K a i l , Proc. Phys. Soc. (London), U (1959) 833. - 158 -CHAPTER IX SOME OTHER STUDIES, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE WORK A. Some Other Studies Among the other s t u d i e s c a r r i e d out were the i n v e s t i g a t i o n of some hydrates where the guest molecule contains only methylene groups. These are hydrates of cyclopropane, p i p e r a z i n e , and hexamethylenetetramine. A b r i e f d e s c r i p t i o n o f t h i s work i s given below. 1. Cyclopropane Hydrate Cyclopropane forms both types of von Stackelberg's type I and type II s t r u c t u r e hydrates. Both types of deuterates have been studied thermodynamically by Hafemann and M i l l e r [9.1]. According to these authors the type I deuterate (C 3H g-7.8D 20) i s s t a b l e below -23°C and between 5.52 and 18.34°C. The type II deuterate i s s t a b l e between the range s t a r t i n g from -23.31 to 5.52°C. The present r e v i s e d study of previous work by us [9.2] shows a second moment of 2.3 + .2 G at 77 K i n type I s t r u c t u r e . The second moment i s reduced very slowly 2 with the temperature and approaches a value of 0.8 G around 230 K. 2 This value of 0.8 G i s e i t h e r due to i s o t r o p i c or random motion of cyclopropane i n s i d e 14-hedron. The type II s t r u c t u r e deuterate s t u d i e d - 159 -i n the stabe range (-23.31 to 5.52°C) shows a second moment of about 0.30 G which i s c o n s i s t e n t with the i s o t r o p i c r o t a t i o n of cyclopropane i n s i d e much l a r g e r cage i e . , 16-hedron. An i n t e r e s t i n g f e a t u r e of cyclopropane was t h a t the r e l a x a t i o n f u n c t i o n appeared to be non-exp o n e n t i a l . Unfortunately i n the temperature range a c c e s s i b l e to us, and because of poor s i g n a l to noise r a t i o , we were not able to draw any substantive conclusions from the r e l a x a t i o n measurements. The minimum i n t appeared to be well below 77 K. 2. p i p e r a z i n e Hydrate P i p e r a z i n e i s one of the c y c l n c amines which form,a hexahydrate, the s t r u c t u r e of which has been s t u d i e d by X-ray d i f f r a c t i o n by Schwarzenbach [9.3], The amine molecule which i s i n c h a i r form, i s hydrogen-bonded, through i t s -NH groups to the water cage. The p i p e r a z i n e , which was Eastman Kodak p r a c t i c a l grade, was f u r t h e r p u r i f i e d by vacuum s u b l i m a t i o n . A f t e r p u r i f i c a t i o n i t s -NH protons were deuterated to -ND by repeated exchange with D2O. The deuterate was prepared from p a r t i a l l y deuterated p i p e r a z i n e (C^HgND2). A l l the processes of preparation and p u r i f i c a t i o n were c a r r i e d out i n an i n e r t atmosphere of n i t r o g e n , as p i p e r a z i n e and i t s deuterate are h i g h l y hygroscopic. The absorption l i n e measurements were c a r r i e d out on Varian DP-60 spectrometer using a frequency of 60 MHz. Strong s a t u r a t i o n and poor s i g n a l to noise r a t i o were c h a r a c t e r i s t i c s of the specimen at a l l temperatures (77 to the melting p o i n t o f the deuterate 317 K). In s p i t e of a l l these d i f f i c u l t i e s , some conclusions can be drawn from those noisy s p e c t r a . The p i p e r a z i n e molecule from these r e s u l t s appears to-be r i g i d at a l l temperatures i n s i d e the p a r t i a l 12-hedron. - 160 -The second moment obtained under optimum c o n d i t i o n s at 77 K was 16.3 +_ 2.0 G . This value stayed constant nearly up to the melting p o i n t . The t h e o r e t i c a l value of second moment, obtained from the data of Schwarzenbach [9.3] was 15.78 G . The cw r e s u l t s d i d not i n s p i r e us to pursue f u r t h e r s t u d i e s on r e l a x a t i o n measurements. 3. Hexamethyl e i i e t e t f ami ne Hydrate This hydrate, although s t r u c t u r a l l y d i f f e r e n t , and worthy o f study, presented s i m i l a r type of problem as that o f p i p e r z i n e . The s t r u c t u r e o f t h i s hydrate has been studie d by Mak [9.4] using X-ray d i f f r a c t i o n . The hydrate (CH 2)gN 4.6H 20 i s rhombohedral. The amine molecule i s surrounded by e i g h t (H 20)g r i n g s and i s hydrogen-bonded to three o f these r i n g s so as to hang " b a t l i k e " to the upper walls of the c a v i t y formed by (H 20)g r i n g s . Only cw studies using Varian DP-60 spectrometer ( r f frequency 60 MHz) were c a r r i e d out on (CH 2)gN^.6D 20. These st u d i e s under the obscure s i t u a t i o n o f strong s a t u r a t i o n and poor s i g n a l to noise r a t i o reveal no motion of the guest molecule i n the deuterate. The experimental second moment 2 at 77 K was 13.4 +2.0 G and t h i s value stayed n e a r l y constant up to 281 K. The t h e o r e t i c a l value of second moment based on the data o f Smith [9.5] i s 14.82 G 2 using a C-H bond length o f 1.13 A and an in t e r m o l e c u l a r second moment o f 0.52 G using Equation (5.3). B. Conclusions Of a l l the guests, and t h e i r deuterates s t u d i e d here, the f o l l o w i n g conclusions are drawn. - 161 -In the diethylamine and diethylamine deuterate, the only motion observed was that of methyl group r e o r i e n t a t i o n . The absorption o measurements were c o n s i s t e n t with a C-H bond length o f 1.13 A. The r e l a x a t i o n f u n c t i o n R(t) i n both o f these compounds was non-exponential and the a n a l y s i s of the data was t h e r e f o r e made according to H-H theory. The r e s u l t s of r e l a x a t i o n gave an a c t i v a t i o n energy o f 2.90 +0.03 kcal/mole i n diethylamine and 2.34 +0.02 kcal/mole i n diethylamine deuterate f o r the b a r r i e r h indering methyl group r e o r i e n t a t i o n . The pre-exponential f a c t o r i n Arrhenius equation T c = T o e x P ( E a / R T ) was (1.6 + .1) x 1 0 " 1 3 and (4.5 + .3) x 1 0 " 1 3 sec f o r diethylamine and diethylamine deuterate r e s p e c t i v e l y . The cw r e s u l t s of acetone i n d i c a t e d only the r o t a t i o n o f both o f i t s methyl groups throughout the temperature range a c c e s s i b l e to us (77 K. and onwards). The corresponding deuterate of acetone showed that the acetone molecule d i s p l a y e d more motion i n the deuteron cage reaching t o an i s o t r o p i c motion around 212 K. The r e l a x a t i o n i n both cases was non-exponential, but unfortunately minimum i n t was not achieved f o r both the compounds. The a c t i v a t i o n energies e x t r a c t e d from the gradient of ir\tQ versus 1000/T (T i n K) were 1.33 + 0.01 kcal/mole i n acetone a s s o c i a t e d with the b a r r i e r h indering methyl group r e o r i e n t a t i o n , and 0.33 kcal/mole i n acetone deuterate. The vaTue of 0.33 kcal/mole was not assigned due to lack o f knowledge of data at lower temperatures (below 77 K) i n acetone deuterate. The t h i r d group c o n s i s t i n g o f the guest, and deuterate was t e r t -butyl amine. The absorption l i n e measurment revealed a d i r e c t l i n e width t r a n s i t i o n from r i g i d l a t t i c e to a motion corresponding to methyl - 162 -plus t e r t - b u t y l group r o t a t i o n i n t e r t - b u t y l ami ne-ND,,. The T-| data of (CHg^CNDg where the r e l a x a t i o n f u n c t i o n was expoential showed a broad d i s t r i b u t i o n of c o r r e l a t i o n times. The a c t i v a t i o n energy i n the upper l i m i t a s s o c i a t e d with the b a r r i e r hindering methyl group r o t a t i o n from T-| data was found to be 3.2 + .1 kcal/mole with T Q = -13 (8-9) x 10 sec. The amine i n the deuterate showed a d d i t i o n a l i s o t r o p i c motion. The T-j minimum of 32 +_ 1 msec t h i s time i n the deuterate was much broader because of d i s t r i b u t i o n of c o r r e l a t i o n times among three types of motion: methyl, t e r t - b u t y l , and i s o t r o p i c . The a c t i v a t i o n energy obtained from the low temperature slope of £nT-j versus 1000/T curve was 1.7 kcal/mole and from high temperature slope i t was 2.5 +_0.1 kcal/mole. The f i r s t a c t i v a t i o n energy was assigned to be a s s o c i a t e d with the b a r r i e r hindering methyl group r e o r i e n t a t i o n , while the l a t t e r was mostly due to t e r t - b u t y l and i s o t r o p i c r o t a t i o n . Among the hexagonal amine hydrates, the hydrates of isopropylamine and trimethylamine were s t u d i e d and a l s o the guests molecules were studied s e p a r a t e l y . In isopropylamine, the only motion present was that of methyl groups r e o r i e n t i n g around t h e i r three f o l d a x i s . The b a r r i e r a s s o c i a t e d with t h i s motion from the r e l a x a t i o n measurement, analyzed a c c o r d i n t to H-H theory was 3.50 +0.07 kcal/mole with x Q = (1.78 + .22) x 1 0 " 1 3 sec. The isopropylamine deuterate in. a d d i t i o n to methyl group r o t a t i o n , d i s p l a y e d r o t a t i o n around pseudothree-fold axis at a temperature of about 212 K. The r e l a x a t i o n f u n c t i o n i n t h i s case was exponential with a s i n g l e broad minimum of 45 +_ 1 msec. The b a r r i e r a s s o c i a t e d with methyl group r e o r i e n t a t i o n was c a l c u l a t e d to be 1.7 kcal/mole, while - 163 -the one a s s o c i a t e d with pseudothree-fold axis r e o r i e n t a t i o n to be 1.6 kcal/mole. The true b a r r i e r heights should be i n f a c t be lower than these values. Only r e l a x a t i o n measurements were c a r r i e d out on trimethylamine and the b a r r i e r height f o r methyl group r e o r i e n t a t i o n was found to be 5.75 kcal/mole. This value was higher than the various values obtained by other methods, but agreed to the one obtained by Haigh et a l . [9.6], Trimethylamine deuterate showed a Lorentzian l i n e shape from 77 to 100 K and non-exponential r e l a x a t i o n i n the same range of temperature. There appeared to be two minima i n the T^-temperature curve, the f i r s t well below 77 K and the other of 50 +_ 1 msec around 190 K. The a c t i v a t i o n energies a s s o c i a t e d with the b a r r i e r heights to r o t a t i o n around t h r e e - f o l d axis and to t r a n s l a t i o n motion were c a l c u l a t e d to be 0.7 and 2.9 + 0.2 kca/mole r e p s e c t i v e l y from T-| measurements. The l a s t s e r i e s reported were deuterates of cyclopropane, p i p e r a z i n e , and hexamethylenetetramine. Cyclopropane showed i s o t r o p i c or random axis motion at high temperatures i n type I s t r u c t u r e deuterate, while i n type IT s t r u c t u r e , i s o t r o p i c motion of cyclopropane molecule was present at a l l temperatures of i t s s t a b i l i t y . Of the other two amine deuterates of p i p e r a z i n e and hexamethylenetetramine, the guest molecule showed no motion a t a l l . C. Suggesticms f o r Future Work The molecular parameters reported f o r the m a j o r i t y of the guests studied i n deuterates are q u i t e u n c e r t a i n . I t w i l l be worthwhile to extend the present study to a temperature below 77 K, where a r i g i d - 164 -l a t t i c e behaviour i s obtained. The extension of r e l a x a t i o n study perhaps would not reveal any f u r t h e r information i n the case of amine hydrates, except t h a t of trimethylamine. The deuterate of acetone and cyclopropane d e f i n i t e l y need a low temperature study both i n absorption l i n e width, and r e l a x a t i o n measurements. The problem of d i s t r i b u t i o n of c o r r e l a t i o n times w i l l s t i l l be there i n the m a j o r i t y of c l a t h r a t e deuterates as i s apparent from the e x i s t i n g low temperature cw work (up to 4.2 K) of Davidson group [9.7]. There are u n f o r t u n a t e l y l i t t l e data on the d i e l e c t r i c d i s p e r s i o n due to the guest molecule. These can be however somewhat more useful i n s o l v i n g the problem of d i s t r i b u t i o n of c o r r e l a t i o n times. Another p o s s i b l e s o l u t i o n to t h i s problem i s to deuterate various groups but t h i s i s economically not too f e a s i b l e . The technique o f high r e s o l u t i o n nmr i n s o l i d s i s now becoming a r e a l i t y from the pioneering work of Andrew [9.8], Mansfield [9.9], and Waugh [9.10] groups, but c l a t h r a t e compounds s t i l l provide a good means o f comparing experimental r e s u l t s obtained using conventional spectrometers with various e x i s t i n g nmr t h e o r e t i c a l work on s o l i d s . In the present work, the guests chosen had a high concentration of protons, although the b a s i c b u i l d i n g u n i t s o f these compounds were simple spin systems of methylene, and methyl groups. The i n t e r a c t i o n s due to others groups may to some extent s p o i l the whole idea of an i s o l a t e d two, three spin systems. S t i l l low temperature cw work on these compounds has l e d q u i t e i n t e r e s t i n g r e s u l t s [9.7]. I t w i l l be worthwhile to study the guest molecules having simpler s p i n systems, as there are many such guests which form hydrates, and which have e i t h e r not s t u d i e d at a l l or have been studi e d but not i n great d e t a i l . - 165 -The problem of poor s i g n a l to noise r a t i o .will be encountered, but t h i s can be overcome by a d d i t i o n a l equipment. Some of the guest molecules have shown very slow motion i n the deuterates. This motion needs to be s t u d i e d by r e l a x a t i o n time i n r o t a t i n g frame T^ . Another f e a t u r e which can be expected i n c l a t h r a t e s i s s p i n - r o t a t i o n i n t e r a c t i o n f o r which to date there are very few s t u d i e s i n the s o l i d s t a t e . - 166 -References (Chapter IX) [9.1J D.R. Hafemann and S.L. M i l l e r , J . Phys. Chem.,73 (1969) 1398. [9.2] A.W.K. Khanzada and C A . McDowell, J . Mol. S t r u c t u r e , 7_ (1971) 241. [9.3] D. Schwarzenbach, J . Chem. Phys., 48 (1968) 4134. [9.4] T.C.W. Mak, J . Chem. Phys., 43 (1965) 2799. £9.5] G.W. Smith, J . Chem. Phys. ,36 (1962) 3081. [9.6] P.J. Haigh, P.C. Canepa, G.A. Matzkanin and T.A. S c o t t , J . Chem. Phys., 48 (1968) 4234. £9.7] D.W. Davidson, C l a t h r a t e Hydrates, National Research Council of Canada, D i v i s i o n o f Chemistry, 1971. £9.8] E.R. Andrew, Progress i n Nuclear Magnetic Resonance Spectroscopy 8^  (1971) 1 and references t h e r e i n . [9.9] P. Ma n s f i e l d , i b i d . , 8^  (1971) 41, and references t h e r e i n . [9.10] M. Mehring, R.G. G r i f f i n and J.S. Waugh, J . Chem. Phys., 5_5, (1971) 746 and references t h e r e i n . - 167 -APPENDIX A Computer program "HILT" to c a l c u l a t e 4 £ C. e x p ( - q ; j t / T ' ) s i n 3 c l e ( A . l ) Since C. and q. are f u n c t i o n of cosg, we can w r i t e the Equation (A.l) i n the form /IT. R/\ v(t] = T / f ( c o s e ) s i n e d g where a f t e r s u b s t i t u t i n g cosg = X, and R ^ v ( t ) = Y becomes Y . = / f[X)dX (A.2) * This i n t e g r a l can now numerically be c a l c u l a t e d using Simpson's formula i . e . , Y = (H/3)[Y 0 + 4(Y 1 + Y 3 + ... + Y^-,) + 2 ( Y 2 + Y 4 +....+ Y N_ 2) + Y N ] (A.3) 2 For each ( W Q T c ) the numerical value of Y^ f o r a p a r t i c u l a r N i s * H. Margenau and G.M. Murphy, The Mathematics of Physics and Chemistry, D. Va,n Nostrand, Co. Inc., N.J., 1956, p. 477. - 168 -c a l c u l a t e d by using C. and q. from the tables of H i l t and Hubbard with J J 2 t/T 1 = 0 , 1, 2, 3, ...,25. In the o r i g i n a l program ( " 0 t c ) = WT» t/T 1 = T, cosg = XI. The other symbols are n e a r l y the same and are explained i n the program. The output given i n the form of Tables A l - A l 5 i s the r e s u l t of another simple program. > 1 C COMPUTER PROGRAM 'HILT' TO CALCULATE RAV(T) FOR NON-EXPONENT 1AL > 2 C RELAXATION > 3 DIMENSION C(k),CC(kk,k),0.(.k),W(kk,k),FACT(.k),X(ll),Y(ll),R(k) > 1+ COMMON X,Y > 5 1 READ(5,50) WT > 6 50 FORMAT(FIO.O) > 7 IF(WT.EQ.O.O) GO TO 20 > 8 WRITE(6,8) WT > 9 8 FORMAT( 5X, "ViT-1, F 1 0 . **/ ) > 10 C START WITH T=0.0 > 11 T=0.0 > 12 DO 70 L=l /26 WRITE(6,80) T > 13 > It* 80 F0RMAT(5X,'T=',F5.1) > 15 C START WITH FI R S T VALUE OF COS(BETA) DENOTED HERE BY XI > 16 X1=0.0 > 17 DO 2 1=1,11 > 18 IF(T.NE.O.O) GO TO 22 > 19 10 DO 3 J=l,k > 20 C READ THE VALUES OF C ( J ) , Q ( J ) TAKEN FROM THE TABLES OF H I L T AND > 21 C HUBBARD FOR A PARTICULAR VALUE OF X1=C0S(BETA) > 22 READ(5/U) C ( J ) , Q ( J ) > 23 k FORMAT(2F10.0) > 2k C C ( I , J ) » G ( J ) > 25 Q Q ( I , J ) = Q ( J ) > 26 3 CONTINUE > 27 22 SUM=0.0 > 28 DO 30 J=l,4 > 29 C ( J ) = C C ( I , J ) > 30 Q ( J ) = Q Q ( I , J ) > 31 R ( J ) = - Q ( J ) * T > 32 F A C T ( J ) = C ( J ) * E X P ( R ( J ) ) > 33 C CALCULATE SUM FROM J=l TO J=4 > 3k SUM=SUM+FACT(J) > 35 30 CONTINUE > 36 X ( l ) = X 1 > 37 Y( 1 )=SUM > 38 C INCREASE C0S(BETA)=X1 BY 0.1 > . 39 X1=X1+0.1 > hO 2 CONTINUE > i l l C A L L VALUES OF XI F1N1 SHED,CALL SUBROUTINE 'SIMPSN' FOR INTEGRATION > k2 CALL SIMPSN > k3 C INCREMENT T BY 1.0 FOR ANOTHER INTEGRAL AND CONTINUE IT FURTHER > kk C T I L L T=25.0 > kS T=T+1.0 > 1*6 70 CONTINUE > U7 GO TO 1 > U8 20 STOP > 49 END > 50 SUBROUTINE SIMPSN > 51 DIMENSION X ( l l ) , Y ( 1 1 ) , A ( 1 1 ) , B ( 1 1 ) > 52 COMMON X,Y > 53 C VALUE OF H IN EQUATION(A.3) IS SET =0.1 > 5k H = 0.1 > 55 SUM1=0.0 > 56 SUM2=0.0 > 57 C SET VALUE OF YO AND YN > 58 Y0=Y(1) > 59 YN=Y(11) > 60 DO 2 1=2,10,2 > 61 A ( l ) = t t . 0 * Y ( l ) > 62 C SUM ODD Y IN EQUATION(A.3) > 63 SUM1=SUM1+A(I) > 61* 2 CONTINUE > 65 DO 3 1=3,9,2 > 66 B ( 1 ) = 2 . 0 * Y ( I ) > 67 C SUM EVEN Y IN EQUATION(A.3) > 68 SUM2=SUM2+B(I) > 69 3 CONTINUE > 70 C CALCULATE Y=RAV > 71 RAV=(H/3.0)*(Y0+SUM1+SUM2+YN) > 72 C CALCULATE LN(RAV) > 73 ARAV=ALOG(RAV) > lk WRITE(6,83) RAV,ARAV > 75 83 F O R M A T C 1 0 X , , R A V ( T ) = , / F 1 0 . ^ / 5 X / > 76 RETURN > 77 END #END OF F I L E # ' L N ( R A V ( T ) ) = ' , F 1 0 . 4 ) Table Al Table A2 ( V c ) 2 « 1 . 0 0 0 t/T' . R A v ^ £riR A v(t) 0.0 1.0000 -0.0 1.0 0.3342 -1.0959 2.0 0.2159 -1.5329 3.0 0.1592 -1. 8376 4.0 0.1270 -2.0638 5.0 0.1070 -2.2349 6.0 0.0937 -2.3676 7.0 0.0843 -2.4738 8.0 0.0772 -2.5615 9.0 0.0716 -2.6360 10.0 0.0672 -2.7007 11.0 0.0634 -2.7581 12.0 0.0602 -2.8096 13.0 0.0575 -2.8564 14.0 0.0551 -2.8994 15.0 0.0529 -2.9392 16.0 0.0510 -2.9763 17.0 0.0492 -3.0110 18.0 0.0477 -3.0437 19.0 0.0462 -3.0746 20.0 0.0449 -3.1039 21.0 0.0436 -3.1319 22.0 0.0425 -3.1586 23.0 0.0414 -3.1843 24.0 0.0404 -3.2089 25.0 0.0395 -3.2327 ("Vc) = 0.001 t/T' R A v ^ £ n R A v ( t ) 0.0 1.0000 -0.0 1.0 0.9334 -0.0689 2.0 0.8746 -0.1340 3.0 0.8226 -0.1953 4.0 0.7763 -0.2532 5.0 0.7351 -0.3077 6.0 0.6983 -0.3592 7.0 0.6652 -0.4077 8.0 0.6354 -0.4535 9.0 0.6085 -0.4968 10.0 0.5840 -0.5378 11.0 0.5618 -0.5766 12.0 0.5415 -0.6134 13.0 0.5229 -0.6483 14.0 0.5058 -0.6815 15.0 0.4901 -0.7132 16.0 0.4755 -0.7434 17.0 0.4619 -0.7723 18.0 0.4493 -0.8000 19.0 0.4376 -0.8265 20.0 0.4266 -0.8520 21.0 0.4162 -0.8765 22.0 0.4065 -0.9002 23.0 0.3973 -0.9231 24.0 0.3886 -0.9452 25.0 0.3804 -0.9666 Table A3 Table A4 t/T' R A v ( t ) 0.0 1.0000 -0.0 1.0 0.8634 -0.1469 2.0 0.7587 -0.2761 3.0 0.6773 -0.3896 4.0 0.6129 -0.4896 5.0 0.5610 -0.5779 6.0 0.5186 -0.6566 7.0 0.4833 -0.7271 8.0 0.4535 -0.7908 9.0 0.4279 -0.8488 10.0 0.4057 -0.9022 11.0 0.3861 -0.9516 12.0 0.3687 -0.9976 13.0 0.3531 -1.0410 14.0 0.3390 -1.0819 15.0 0.3260 -1.1207 16.0 0.3142 -1.1578 17.0 0.3032 -1.1934 18.0 0.2930 -1.2276 19.0 0.2835 -1.2605 20.0 0.2746 -1.2924 21.0 0.2663 -1.3232 22.0 0.2584 -1.3531 23.0 0.2510 -1.3822 24.0 0.2440 -1.4105 25.0 0.2374 -1.4381 (a) x ) 2 = 0.010 x o c t/T' R A v ( t ) * n R A y ( t ) 0.0 1.0000 0.0 1.0 0.8192 -0.1995 2.0 0.6933 -0.3663 3.0 0.6029 -0.5059 4.0 0.5359 -0.6238 5.0 0.4845 -0.7247 6.0 0.4439 -0.8122 7.0 0.4109 -0.8895 8.0 0.-3834 -0.9587 9.0 0.3600 -1.0218 10.0 0.3397 -1.0798 11.0 0.3218 -1.1338 12.0 0.3059 -1.1845 13.0 0.2916 -1.2325 14.0 0.2786 -1.2782 15.0 0.2667 -1.3217 16.0 0.2558 -1.3636 17.0 0.2457 -1.4037 18.0 0.2363 -1.4425 19.0 0.2277 -1.4 799 20.0 0.2196 -1.5160 21.0 0.2120 -1.5510 22.0 0.2050 -1.5849 23.0 0.1983 -1.6177 24.0 0.1921 -1.6496 25.0 0.1863 -1.6805 Table A5 Table A6 T c ) 2 = 0.050 ( V c ) 2 = 0.100 t/T' R A v ( t ) * n R A v ( t ) t/T' R A v.(t) * n R A v ( t ) 0.0 1.0000 -0.0 0.0 1.0000 -0.0 1.0 0.6947 -0.3643 1.0 0.6465 -0.4362 2.0 0.5367 -0.6223 2.0 0.4847 -0.7242 3.0 0.4440 -0.8119 3.0 0.3949 -0.9290 4.0 0.3828 -0.9603 4.0 0.3363 -1.0898 5.0 0.3384 -1.0836 5.0 0.2935 -1.2259 6.0 0.3039 -1.1909 6.0 0.2602 -1.3465 7.0 0.2760 -1.2873 7.0 0.2332 -1.4560 8.0 0.2527 -1.3 756 8.0 0.2109 -1.5565 9.0 0.2328 -1.4575 9.0 0.1922 -1.6495 10.0 0.2157 -1.5340 </ 10.0 0.1763 -1.7356 11.0 0.2007 -1.6058 11.0 0.1628 -1.8154 12.0 0.1876 -1.6732 12.0 0.1511 -1.8895 13.0 0.1761 -1.7367 13.0 0.1411 -1.9583 14.0 0.1659 -1.7965 14.0 0.1324 -2. 0 2 2 1 15.0 0.1568 -1.8529 15.0 0.1248 -2.0813 16.0 0.1487 -1.9061 16.0 0.1181 -2.1363 17.0 0.1414 -1.9562 17.0 0.1122 -2.1874 18.0 0.1349 -2.0035 18.0 0.1070 -2.2350 19.0 0.1290 -2.0482 19.0 0.1023 -2.2794 20.0 0.1236 -2.0904 20.0 0.0982 -2.3208 21.0 0.1188 -2.1302 21.0 0.0945 -2.3595 22.0 0.1144 -2.1679 22. 0 0.0911 -2.3957 23.0 0.1104 -2.2036 23.0 0.0881 -2.4297 24.0 0.1067 -2.2374 24.0 0.0853 -2.4618 25.0 0.1034 -2.2694 25.0 0.0827 -2.4920 Table A7 Table A8 ( U 0 T C ) =0.200 t/T' R A v ( t ) . * n R A v ( t ) 0.0 1.0000 -0.0 1.0 0.6139 -0.4880 2.0 0.4507 -0.7970 3.0 0.3619 -1.0164 4.0 0.3035 -1.1925 5.0 0.2606 -1.3449 6.0 0.2273 -1.4815 7.0 0.2007 -1.6058 8.0 0.1792 -1.7195 9.0 0.1615 -1.8236 10.0 0.1468 -1.9188 11.0 0.1345 -2.0059 12.0 0.1242 -2.0855 13.0 0.1155 -2.1582 14.0 0.1081 -2.2247 15.0 0.1017 -2.2856 16.0 0.0962 -2.3415 17.0 0.0914 -2.3927 18.0 0.0872 -2.4400 19.0 0.0834 -2.4836 20.0 0.0801 -2.5240 21.0 0.0772 -2.5616 22.0 0.0745 -2.5966 23.0 0.0721 -2.6294 24.0 0.0699 -2.6602 25.0 0.0679 -2.6891 ( W Q T C ) = 0.500 t/T' R A v ( t ) * n R A y ( t ) 0.0 1.0000 -0.0 1.0 0.6042 -0.5038 2.0 0.4374 -0.8268 3.0 0.3449 -1.0646 4.0 0.2830 -1.2623 5.0 0.2376 -1.4373 6.0 0.2028 -1.5957 7.0 0.1755 -1.7401 8.0 0.1539 -1.8716 9.0 0.1365 -1.9911 10.0 0.1225 -2.0993 11.0 0.1111 -2.1972 12.0 0.1017 -2.2855 13.0 0.0939 -2.3652 14.0 0.0874 -2.4371 15.0 0.0819 -2.5022 16.0 0.0772 -2.5612 17.0 0.0732 -2.6148 18.0 0.0697 -2.6638 19. 0 0.0666 -2.7087 20.0 0.0639 -2.7501 21.0 0.0615 -2.7884 22.0 0.0594 -2.8239 23.0 0.0574 -2.8572 24.0 0.0557 -2.8883 25.0 0.0541 -2.9176 Table A9, Table Al0 (<Vc) = 0-800 (OJ T ) 2 = 1.000 t/T' R A v ( t ) * n R A v ( t ) t/T' R A v ^ * n R A v ( t ) 0.0 1.0000 -0.0 0.0 1.0000 0.0 1.0 0.6148 -0.4864 1.0 0.6232 -0.4728 2.0 0.4453 -0.8091 2.0 0.4524 -0.7931 3.0 0.3488 -1.0534 3.0 0.3538 -1.0391 4.0 0.2837 -1.2599 4.0 0.2869 -1.2488 5.0 0.2359 -1.4445 5.0 0.2376 -1.4370 6.0 0.1993 -1.6128 6.0 0.2000 -1.6093 7.0 0.1709 -1.7670 7.0 0.1707 -1.7677 8.0 0.1484 -1.9079 8.0 0.1476 -1.9131 9.0 0.1305 -2.0363 9.0 0.1292 -2.0461 10.0 0.1162 -2.1528 10.0 0.1145 -2.1670 11.0 0.1045 -2.2582 11.0 0.1026 -2.2766 12.0 0.0951 -2.3533 V 12.0 0.0930 -2.3757 13.0 0.0873 -2.4389 13.0 0.0850 -2.4651 14.0 0.0808 -2.5161 14.0 0.0784 -2.5456 15.0 0.0753 -2.5856 15.0 0.0729 -2.6182 16.0 0.0708 -2.6485 16.0 0.0683 -2.6838 17.0 0.0668 -2.7055 17.0 0.0644 -2.7433 18.0 0.0635 -2.7574 18.0 0.0610 -2.7973 19.0 0.0605 -2.8048 19.0 0.0580 -2.8466 20.0 0.0579 -2.8484 20.0 0.0555 -2.8918 21.0 0.0557 -2.8885 21.0 0.0532 -2.9334 22.0 0.0536 -2.9258 22.0 0.0512 -2.9719 23.0 0.0518 -2.9605 23.0 0.0494 -3.0078 24.0 0.0501 -2.9930 24.0 0.0478 -3.0413 25.0 0.0486 -3.0235 25.0 0.0463 -3.0728 - 177 -J f U l D H U M O K M n i ' l C O O C T l O W N O N J ' O O i n H r H H i n N O K M D O O H N d - l f l C O O l O N I ^ J t m U J I N C O O l O l O O H H N N OOCOHHHHHHNMntMMMMCNMMKlKlfAKl^M C I I I I I I I I I I I I I I I I I I I I I I I I I I > o rH rH OO i n i n j oo CM CM CO -d- o oo m m i n co j - CD oo CM CD rH o cn 00 oo rH ro oo CM rH CM i n co cn m 00 o r--m CM O OS o oo rH O m rN. CM CD CO J - CM o cn oo r--CD CO i n m i n -=s-j -o c c i n j - m CM CM r H rH rH rH rH o o o o o o o o o o o o o r H C O O O O O O O O O O O O O O O O O O O O O O O O H- o o o o o o o o o o o o o o o c o o o o o o o o o o O H c s i ^ j - i n i o r . o o o i o H c s i r A j ' i n m N c o o i c H c M K i j t n rH H H H rH rH H H H rH CM CM CM CM CM CM > o; c O O C S I H O C t O l f l l O l f l J - i n O O N H M J ' J ' O l N J - U J O i r N N O W f f i O ( M l O « ) 4 ' H S N t D H l f t O l O J f t A O ) « O N H U ) « ) N J ' H H l D M D J - H N H i n r s O l O i O l N l A M C O J - O i n c n K l N H o ^ N o i H M i n r s o o O H p g K i j - i n i D S o o o o m o o o H H N 0 o o o 1 I I I C M C M C M C N i c M C N C M C M C M C M C M r n r n r n r n r n r n I I I I i I I I I I I I I I I I i "Tj" o r H c o o i i n o o o c T i o c M c o i n i n o o o c o r n c o j - r ^ r n c M r n r ^ c M 3. o r H c o c M O i i n r n o r n r n c o c M i H C M t n o o r n c n i n c M C T j r ^ i n i ^ r H o > o i c o o o o o w H M i n K i H o c i o o M o u i n L f i i / i ^ d - j - j - j ' j <C O l O ^ t m m c M C M r H r H r H r H r H O O O O O O O O O O O O O O o; r H O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o +-> OHMKlJ- iniOM»cnOHNKUU1lOM>00 )OHMI^d-in rH r H r H r H r H rH r H r H r H r H C M C M C M C M C M C M Table A13 Table A14 ( V c ) 2 = 5 - 0 0 0 t/T' R A v ^ * n R A v ( t ) 0.0 1.0000 -0.0 1.0 0.7289 -0.3163 2.0 0.5642 -0.5723 3.0 0.4527 -0.7926 4.0 0.3712 -0.9910 5.0 0.3088 -1.1750 6.0 0.2596 -1.3486 7.0 0.2200 -1.5140 8.0 0.1879 -1.6721 9.0 0.1615 -1.8232 10.0 0.1398 -1.9677 11.0 0.1218 -2.1052 12.0 0.1069 -2.2358 13.0 0.0945 -2.3594 14. 0 0.0841 -2.4758 15.0 0.0754 -2.5851 16.0 0.0681 -2.6873 17.0 0.0619 -2.7827 18.0 0.0566 -2.8714 19.0 0.0521 -2.9537 20.0 0.0483 -3.0299 21.0 0.0450 -3.1005 22.0 0.0422 -3.1658 23.0 0.0397 -3.2262 2k.0 0.0375 -3.2822 25.0 0.0356 -3.3342 V ( V c ) 2 = 1 0 - 0 0 0 t/T' R A v ( t ) * n R A v ( t ) 0.0 1.0000 -0.0 1.0 0.7831 -0.2445 2.0 0.6342 -0.4554 3.0 0.5247 -0.6449 4.0 0.4406 -0.8196 5.0 0.3739 -0.9837 6.0 0.3199 -1.1399 7.0 0.2753 -1.2898 8.0 0.2382 -1.4345 9.0 0.2071 -1.5747 10.0 0.1807 -1.7108 11.0 0.1584 -1.8429 12.0 0.1393 -1.9711 13.0 0.1230 -2.0954 14.0 0.1091 -2.2157 15.0 0.0971 -2.3320 16.0 0.0868 -2.4441 17.0 0.0779 -2.5519 18.0 0.0703 -2.6555 19.0 0.0636 -2.7546 20.0 0.0579 -2.8493 21.0 0.0529 -2.9395 22.0 0.0485 -3.0253 23.0 0.0448 -3.1066 24.0 0.0414 -3.1837 25.0 0.0385 -3.2565 Table Al5 (v c) = 100.0 t/T' R A v ( t ) * n R A y ( t ) 0.0 1.0000 -0.0 1.0 0.9165 -0.0872 2.0 0.8427 -0.1712 3.0 0.7767 -0.2527 4.0 0.7174 -0.3321 5.0 0.6639 -0.4096 6.0 0.6155 -0.4854 7.0 0.5714 -0.5596 8.0 0.5313 -0.6325 9.0 0.4946 -0.7040 10.0 0.4610 -0.7743 11.0 0.4302 -0.8435 12.0 0.4018 -0.9118 13.0 0.3757 -0.9790 14.0 0.3515 -1.0454 15.0 0.3292 -1.1110 16.0 0.3086 -1.1758 17.0 0.2894 -1.2399 18.0 0.2716 -1.3032 19.0 0.2551 -1.3660 20.0 0.2398 -1.4281 21.0 0.2255 -1.4896 22.0 0.2121 -1.5505 23.0 0.1997 -1.6108 24.0 0.1881 -1.6707 25.0 0.1773 -1.7300 APPENDIX B > 1 $COMPILE > 2 C COMPUTER PROGRAM 'TIME' TO CALCULATE R ( T ) FROM EXPERIMENTAL > 3 C RESULTS OF 180-T-90 PULSE SEQUENCE USING R.F. F I E L D INHOMOGENEITY > 4 C CORRECTION AS SUGGESTED BY VAN PUTTE, J . MAG. R E S . , 2 ( 1 9 7 0 ) 1 7 4 . > 5 C > 6 C DEFINITIONS-INPUT > 7 C N=NO. OF POINTS IN 180-T-90 PULSE SEQUENCE IPHl=PHOTOGRAPH NO. > 8 C FOR -VE VALUES OF MAGNETIZATON IN Z-DIRECTION IPH2=PHTOGRAPH NO. > 9 C FOR t V E VALUES OF MAGNETIZATION IN Z-DIRECTION TEMP=TEMPERATURE > 10 C M90=MAGNETIZATION AFTER 90 DEGREE PULSE M2 70=MAGNETIZATION AFTER > 11 C 270 DEGREE PULSE I . E . WITH T=100 OR 150 MICROSEC. IN 180-T-90 > 12 C PULSE SEQUENCE TAKEN AS +VE T ( I ) = T I M E T IN 180-T-90 PULSE > 13 C SEQUENCE MZ(I)=MAGNETIZATION AFTER A TIME T IN 180-T-90 PULSE > 14 C SEQUENCE, +VE IN FIRST PHOTOGRAPH, -VE IN THE SECOND. > 15 C > 16 C CALCULATED QUANTITIES-OUTPUT > 17 C X l=CORRECTION FACTOR GIVEN BY EQUATI ON(4.9) X2 = 2.0-X1 X3=M90*X2 > 18 C REST TERMS ARE SELF EXPLANATORY. > 19 REAL M90,M270,MZ > 20 DIMENSION T ( 2 0 ) , M Z ( 2 0 ) , C ( 2 0 ) , D ( 2 0 ) , E ( 2 0 ) > 21 19 READ(5,1) N,I PHI,IPH2,TEMP > 22 IF (TEMP.EQ.0.0) GO TO 18 > 23 1 FORMAT(312,F10.0) > 24 READ, ( T ( I ) , I = 1 , N ) > 25 READ, ( M Z ( I ) , 1=1,N) > 26 M90=MZ(1) > 27 M270=MZ(2) > 28 X1=(M90-M270)/M90 > 29 X2=2.0-X1 > 30 X3=X2*M90 > 31 DO 2 1=3,N C(1)-M90-MZ(1) > 32 > 33 D ( 1 ) = C ( 1 ) / X 3 > 34 E ( 1 ) = A L O G ( D ( 1 ) ) > 35 2 CONTINUE > 36 WRITE(6,7) TEMP,1 PHI,1PH2,N > 37 7 FORMAT(////2X,'TEMPERATURE* > 38 1,'NO. OF POINTS**,15/) > 39 WRITE(6,21)X1,X2,X3 > 40 21 FORMAT(5X,'Xl=',F5.2,5X, 1X2 > 41 WRITE(6,25) > 42 25 FORMAT(IX, 'S.NO.',6X,'T',7X > 43 l ' L N ( R ( T ) ) ' / ) > 44 DO 2 0 1=1,N > 45 WRITE(6,8) 1 , T ( 1 ) , M Z ( 1 ) , C ( 1 > 46 8 FORMAT(I5,2F10.1,3F10.2) > 47 20 CONTINUE > 48 GO TO 19 > 49 18 STOP > 50 END > 51 $DATA > 52 15 6 7 170.0 > 53 0. 0 .1 2. 4. 6. 8. 10. 12. > 54 23. 20.5 -17. -14. -11.5 -8.; > 55 > 56 $STOP #END # OF F l LE CO 25. 30. 35. 40. 50. 60. 80. -6. -4. 4. 6.5 9.5 11.5 15. 17. 20. - 182 -APPENDIX C Computer Program "MOMENT" to C a l c u a l t e T h e o r e t i c a l R i g i d L a t t i c e Second Moment, and Second Moment due to I s o t r o p i c Motion. This program makes use of Equations (5.1), (5.3), (6.1), (7.1), (7.2), and [7.3). The in t r a m o l e c u l a r second moment i s c a l c u l a t e d using e i t h e r Equation (5.1) or (6.1). The i n t e r m o l e c u l a r second moment i s approximated e i t h e r by Equation (5.3) or (7.1) or by both equations. In using Equation [5.3) the d e n s i t y p i s c a l c u l a t e d by _ 1.66 x Mol. wt. (g) o f the deuterate p - oo volume of u n i t c e l l ( A ) and the radius R ( A ) i s approximated by the r e l a t i o n °3 4jL R3 volume of u n i t c e l l ( A ) 3 ~ (no. of molecules of g u e s t / c e l l ) The use of other equations and various symbols are e i t h e r defined i n the program or explained i n r e s p e c t i v e chapters. > 1 $COMPILE > 2 C COMPUTER PROGRAM 'MOMENT' TO CALCULATE THEORETICAL SECOND MOMENT. > 3 C INTRAMOLECULAR PART USING EQUATION(5.3) OR(6.1) > 4 DIMENSION FACT(4 0 ) , R ( 4 0 ) , A ( 4 0 ) , X ( 4 0 ) , S M ( 4 0 ) , R I ( 2 0 ) , R K ( 2 0 ) , H ( 2 0 ) , > 5 I C ( 2 0 ) , D ( 2 0 ) , E ( 2 0 ) , F ( 2 0 ) , G ( 2 0 ) , T ( 2 0 ) , P ( 2 0 ) , Q ( 2 0 ) , S ( 2 0 ) , R J K ( 2 0 ) , > 6 1 S M 1 ( 2 0 ) / S M 2 ( 2 0 ) / S M 3 ( 2 0 ) > 7 C N=NO. OF TOTAL CARDS FOR INTRAMOLECULAR, AN=MO. OF PROTONS IN > 8 C ONE MOLECULE > 9 READ(5,1) N,AN > 10 1 FORMAT(12,F10.0) > 11 DO 2 1=1,N > 12 C F A C T O ) = N O . OF EQUIVALENT INTERACTIONS, R(1)=D1 STANCE IN ANGSTROM > 13 C UNITS BETWEEN DIFFERENT ATOMS, A ( l ) = F A C T O R , 716.164 FOR PROTONS, > 14 C 9.994 FOR DEUTERONS, 2.216 FOR NITROGEN. > 15 READ(5,3) F A C T ( 1 ) , R ( 1 ) , A ( 1 ) > 16 3 FORMAT(3F10.0) > 17 2 CONTINUE > 18 SUM=0.0 > 19 DO 4 1=1,N > 20 X ( 1 ) = R ( 1 ) * * 6 > 21 S M ( I ) = ( F A C T ( 1 ) * A ( 1 ) ) / ( A N * X ( 1 ) ) > 22 SUM=SUM+SM(1) > 23 4 CONTINUE > 24 WRITE(6,11) AN > 25 11 FORMAT(5X,'NO. OF PROTONS*',F10.0//) > 26 WRITE(6,13) > 27 13 FORMATUX, *S.NO. ',8X, ' 1 NTERACTI ONS ', 16X, 'R( 1 ) ', 13X, 'R( 1 ) * * 6 ' , 12X, > 28 1C0NSTANT',7X,'SECOND MOMENT'//) > 29 DO 5 1=1,N > 30 WRITE(6,7) 1 , F A C T ( 1 ) , R ( 1 ) , X ( 1 ) , A ( 1 ) , S M ( 1 ) > 31 7 FORMAT(15,4F20.3,F20.2) > 32 5 CONTINUE > 33 WRITE(6,8) SUM > 34 8 FORMAT(//5X,'TOTAL INTRAMOLECULAR SEC. MOMENT(R1G1D)=',F10.3//) > 35 C INTERMOLECULAR PART BY USING EQUATION(5.3) > 36 C NCELL=NO.OF MOLECULES PER UNIT C E L L , VOLUME=VOLUME OF THE UNIT > 37 C CELL IN CUBIC ANGSTROM UNITS, AMOLWT=GRAM MOL. WT., RHO=DENS1TY > 38 C IN G/CC, RADIUS=RADIUS R OF EQUATI0N(5.3) > 39 READ, NCELL,VOLUME,AMOLWT > 40 RHO=(1.66*AMOLWT)/VOLUME > 41 WRITE(6,36) NCELL,VOLUME,AMOLWT,RHO > 42 36 F 0 R M A T ( / / 1 X , » N O . OF MOL./CELL*', 15,5X,'VOLUME=',F12.3,5X,'MOL.WT=' > 43 1,F10.3,5X,'RHO=',F5.3//) > 44 R3=(3.0*VOLUME)/(4.0*3.142*NCELL) > 45 R A D I U S = ( R 3 ) * * ( l . / 3 . ) > 46 D1=(35 8.1*4.0*3.14 2*AN*NCELL*RHO*6.023) > 47 D2=3.0*R3*10.0*AMOLWT > 48 SINTER=D1/D2 > 49 WRITE(6,37) RADIUS,SINTER > 50 37 FORMAT(//5X,'MOL. RAD 1US =',F10.3,5X,* 1NTERMOL. SEC. MOMENT BY RHO = > 51 l ' , F 1 0 . 3 / / ) > 52 C CALCULATION OF SEC. MOMENT DUE TO ISOTROPIC MOTION. > 53 C M=NO. OF NEIGHBOURS, IF ZERO PROGRAM TERMINATES, R1(J)=D1 STANCE > 54 C OF NEIGHBOURS FROM ORIGIN MOLECULE IN ANGSTROM UN 1TS(REFER TO > 55 C E Q U A T I O N S . 3 ) ) . > 56 READ, M > 57 IF(M.EQ.O) GO TO 42 > 58 READ, ( R l ( J ) , J=1,M) > 59 SUM3=0.0 > 60 WRITE(6,39) > 61 39 F0RMAT(1X,'S.NO.',4X,'Rl(J)',5X,'M2ISO') > 62 DO 38 J=1,M R K ( J ) = R I ( J ) * * 6 > 63 > 64 S M 3 ( J ) = 3 5 8 . 1 * A N / R K ( J ) > 65 SUM3=SUM3+SM3(J) > 66 WRITE(6,40) J , R I ( J ) , S M 3 ( J ) > 67 40 FORMAT(15,2F10.3) > 68 38 CONTINUE > 69 WRITE(6,41) SUM3 > 70 41 F0RMAT(//5X,'TOTAL ISOTROPIC SEC.M0MENT=',F10.3//) > 71 C CALCULATION OF INTERMOLECULAR PART BY USING F ( H ) OF EQUAT1ON(7.1) > 72 C Rl=MOLECULAR RADIUS IN ANGSTROM U N I T S , I F ZERO PROGRAM TERMINATES, > 73 C R l / R I ( J ) = S M A L L H OF EQUATION(7.2). > 74 READ, R l > 75 1F(Rl.EQ.O.O) GO TO 42 > 76 W R I T E ( 6 / 1 2 ) R1,M > 77 12 FORMAT(5X/'MOLECULAR RAD 1US=',F10.3,5X,'NO. OF NEIGHBOURS > 78 SUM1=0.0 > 79 C R J K ( J ) = F ( H ) / ( R I ( J ) * * 6 ) OF E Q U A T I O N ( 7 . 1 ) . > 80 WRITE(6,33) > 81 33 FORMATUX, 'S.NO. ',4X, 'Rl ',8X, '(H) ',7X, 'RJK( - 6 ) ' ) > 82 DO 30 J « 1 , M > 83 H ( J ) = R 1 / R I ( J ) > 84 C ( J ) - H ( J ) * H ( J ) > 85 D ( J ) - C ( J ) * C ( J ) > 86 E ( J ) = 1 . 0 - C ( J ) > 87 F ( J ) = E ( J ) * E ( J ) > 88 G ( J ) = ( 5 . / 3 . ) * D ( J ) > 89 T ( d ) - F ( J ) + G ( J ) > 90 P ( J ) = 1 . 0 - 4 . 0 * C ( J ) > 91 Q ( J ) = P ( J ) * * 3 > 92 S ( J ) = R K ( J ) * Q ( J ) > 93 R J K ( J ) = T ( J ) / S ( J ) > 94 S M 1 ( J ) = 3 5 8 . 1 * A N * R J K ( J ) > 95 SUM1=SUM1+SM1(J) > 96 W R I T E ( 6 / 3 4 ) J , R I ( J ) / H ( J ) / R J K ( J ) > 97 34 FORMAK 11*^10.3, F10. 3,1PE15 . 1 * ) > 98 30 CONTINUE > 99 W R I T E ( 6 / 4 4 ) > 100 44 FORMAT(IX,'S.NO.'/4X,'SMINTER') > 101 DO 43 J=1,M > 102 WRITE(6,45) J , S M 1 ( J ) > 103 45 FORMAT(15/F10.3) > 104 43 CONTINUE > 105 WRITE(6,35) SUM1 ',F10.3//) > 106 35 FORMAT(//5X,'INTERMOL. SEC. MOMENT BY F ( H ) = > 107 42 STOP > 108 END > 109 $DATA > 110 $STOP #END i OF F l LE 

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