Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A1 K33.pdf [ 8.58MB ]
Metadata
JSON: 831-1.0060117.json
JSON-LD: 831-1.0060117-ld.json
RDF/XML (Pretty): 831-1.0060117-rdf.xml
RDF/JSON: 831-1.0060117-rdf.json
Turtle: 831-1.0060117-turtle.txt
N-Triples: 831-1.0060117-rdf-ntriples.txt
Original Record: 831-1.0060117-source.json
Full Text
831-1.0060117-fulltext.txt
Citation
831-1.0060117.ris

Full Text

11230  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, P a k i s t a n , 1965. M.Sc,  U n i v e r s i t y o f Sind, P a k i s t a n , 1966.  M.Sc,  The U n i v e r s i t y o f 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 t o the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA January, 1972.  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  permission.  Department of  Chpmi<; try  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  Department or  I t i s understood that copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  Columbia  February 25, 1972.  s h a l l not be  publication  allowed without  my  - iiABSTRACT The proton magnetic resonance a b s o r p t i o n 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 t o 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 t o 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 t h a t o f methyl reorientation.  group  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 a r e t h e r e f o r e analysed  according t o H i l t and Hubbard theory and the a c t i v a t i o n energies o f 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 h i n d e r i n g methyl group r e o r i e n t a t i o n a c c o r d i n g t o the above theory.  The s t r e n g t h 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 t h a t both methyl group r o t a t e around t h e i r t h r e e f o l d a x i s 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 a b s o r p t i o n l i n e measurements o f non hydrogen-bonded t e r t - b u t y l a m i n e 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 in i t s deuterate c a v i t i e s i n the temperature range o f 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 t e r t - b u t y l a m i n e 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 s m a l l e r 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 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 w i t h 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 o f both methyl groups a t 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  - iv -  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 a b s o r p t i o n and r e l a x a t i o n measurements. R e l a x a t i o n 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 o f 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 o f 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 o f 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 o f motion was found i n the r e s t o f the two deuterates o f p i p e r a z i n e , and  hexamethylenetetramine.  -  V  -  TABLE OF CONTENTS Page Abstract  ii  Table o f Contents  v  L i s t o f Tables  xii  L i s t o f Figures  x i v  Acknowledgements  xviii  Dedication  x x  CHAPTER I A.  INTRODUCTION  General Nature o f C l a t h r a t e Hydrates  1 1  B. Various Studies on C l a t h r a t e Hydrates  2  C. The Present Studies  8  References (Chapter I) CHAPTER II A.  STRUCTURE OF CLATHRATE HYDRATES  Introduction  B. von Stackelberg's Cubic Hydrates  1° 13 13 ^  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.  17  von Stackelberg's type II s t r u c t u r e hydrates ..  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.  27  n-Propylamine  hydrate  References (Chapter I I ) CHAPTER I I I  NUCLEAR MAGNETIC RESONANCE THEORY ....  29 31  A.  Introductory Remarks  31  B.  The Line Shape Function  34  C.  Second Moment o f Absorption L i n e Shape  38  1.  Second Moment from Absorption Line Shape  38  2.  Second Moment and FID Curve  39  3.  E f f e c t o f Molecular Motion on Second Moment  4.  and L i n e Width  40  E f f e c t o f 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 R e l a x a t i o n Time 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 R e l a x a t i o n f o r Methyl Groups  46  (a)  Exponential Relaxation  46  (b)  Non-exponential  48  spin-lattice relaxation  4. E f f e c t o f T u n n e l l i n g on T E.  44  ]  49  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 and T h e i r E f f e c t on T, and E i a References (Chapter I I I )  53 57  - vii -  CHAPTER IV A.  B.  C.  APPARATUS AND METHODS OF MEASUREMENT .  60  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.  L i n e Width and Second Moment Measurements  61  4.  V a r i a b l e Temperature Assembly  63  S p i n - L a t t i c e R e l a x a t i o n Measurements  64  1.  Pulse Spectrometer  64  2.  L i n e a r i t y o f 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 o f H^ Inhomogeneity  69  The Cold Box  ,  71  References (Chapter IV) CHAPTER V  72  DIETHYLAMINE AND DIETHYLAMINE CLATHRATE DEUTERATES  ...73  A.  Introduction  73  B.  Experimental  74  C.  1.  Materials  74  2.  Preparation of (C H ) ND  3.  P r e p a r a t i o n o f Deuterate and Amine Samples  75  4.  Spectrometers  76  2  5  2  ,  Results 1.  ....  74  76  Absorption Line A n a l y s i s (cw Measurements)  76  (a)  76  Second Moment C a l c u l a t i o n s  - viii -  (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) D.  Experimental Relaxation Data  Discussion  90 References  CHAPTER VI  92  ACETONE AND ACETONE DEUTERATE  A.  Introduction  B.  Experimental  C.  (Chapter V)  93 9 3  •  1.  P r e p a r a t i o n o f Acetone-Deuterate  94  2.  P r e p a r a t i o n o f Acetone Sample  95  3.  Spectrometers and Methods o f Measurement  95  Results  9 6  1. Absorption L i n e A n a l y s i s 2. D.  94  96  R e l a x a t i o n Measurements  ...  Discussion  103 References  CHAPTER VII  100  (Chapter VI)  1 0 6  TERTIARY BUTYLAMINE AND TERTIARY BUTYLAMINE DEUTERATE  1 0 7  A.  Introduction  1°  B.  Experimental  108  1.  Material  2.  Preparation of (CH ) CND  7  1 0 8  3  3  2  108  - ix -  3. P r e p a r a t i o n o f Deuterate and Amine Samples  ..  4. Spectrometers and Methods o f Measurements C. R e s u l t s  (b)  Experimental R e s u l t s  11  Ill 114  2. T-j Measurements  119  Discussion  126 References (Chapter V I I )  CHAPTER VIII  I"  Ill  (a) Second Moment C a l c u l a t i o n s  D.  110 •  1. A b s o r p t i o n Line A n a l y s i s  109  130  ISOPROPYLAMINE, ISOPROPYLAMINE DEUTERATE, TRIMETHYLAMINE, AND TRIMETHYLAMINE DEUTERATE ..... 131  A.  Introduction  131  B.  Experimental  134  1. M a t e r i a l s  134  2. P r e p a r a t i o n o f (CH ) CHND 3  2  2  3. P r e p a r a t i o n o f Amine and Deuterate Samples (a)  134 134  Isopropylamine and Isopropy1 amine Deuterate ... 134  (b) Trimethylamine and Trimethylamine Deuterate ... 135 4. Spectrometer and Method o f Measurements (a)  135  Isopropylamine and Isopropylamine Deuterate ... 135  (b) Trimethylamine and Trimethylamine Deuterate ... 136 C. R e s u l t s 1. Absorption l i n e A n a l y s i s  137 I  3 7  -  (a)  X  -  Isopropylamine and Isopropylamine Deuterate ... 137 (i) (ii)  Second Moment C a l c u l a t i o n s  137  Experimental R e s u l t s  138  (b) Trimethyl amine and Trimethyl amine Deuterate ... 141 (i) (ii) 2.  Second Moment C a l c u l a t i o n s  141  Experimental Results  142  R e l a x a t i o n Measurements (a)  144  Isopropylamine and Isopropylamine Deuterate ... 144  (b) Trimethylamine and Trimethyl amine Deuterate ... 149 D. D i s c u s s i o n  152  1.  Isopropylamine and Isopropylamine Deuterate  152  2.  Trimethyl amine and Trimethylamine Deuterate  154  References (Chapter V I I I ) CHAPTER IX  SOME OTHER STUDIES, CONCLUSION, AND SUGGESTION FOR FUTURE WORK  A.  157  158 158  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 t a b l e s provided by H i l t and Hubbard  APPENDIX B  1 6 7  Computer Programme TIME to c a l c u l a t e R ( t ) a t d i f f e r e n t value o f t using H^ c o r r e c t i o n  APPENDIX C  180  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  - xii-  LIST OF TABLES  Table 1.1  Title  Page No.  Hydration number f o r some s o l u b l e non-electrolytes  1.2  6  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  2.2  C r y s t a l S t r u c t u r e Data on Some  19  Alkylamine Hydrates 2.3  21  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  5.1  2 4  I n t e r m o l e c u l a r Second Moment M£ f o r DNH~D 0 and DND-D^O  79  2  5.2  Second Moment f o r D i e t h y l amine -ND~ Deuterate (DND-D 0) using r _ = 1.13 A and 2  c  H  N=10  •  6.1  Proton Second Moment i n Acetone Deuterate  7.1  I n t r a m o l e c u l a r Second Moment M f o r 112  2  H7  T h e o r e t i c a l Second Moment Values f o r (CH ) CHND and Isopropylamine Deuterate 3  8.2  7  Second Moment Values f o r t e r t - B u t y l a m i n e deuterate and t e r t - B u t y l ami ne-ND  8.1  q  2  tert-Butylamine 7.2  82  2  2  ^0  Second Moment Values f o r Trimethylamine Deuterate  144  -  xm  -  Values o f R ( t ) at d i f f e r e n t A V  2  t/T  1  for different (  W  T 0  C  )  - xiv LIST OF FIGURES Figure 2.1  Title  Page No.  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  2.2  von S t a c k e l b e r g ' s type I s t r u c t u r e hydrate  2.3  18  *.•  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  2.4  18  17-hedra and 8-hedron found i n t e r t butyl amine hydrate  2.5  26  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 different U  ) o c  50  2  T  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  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 5-2  i  Dependence o f t / T h by H-H theory Q  0  as p r e d i c t e d  66  81 84  -  XV  -  LIST OF FIGURES (continued) Figure 5.3  Title  Page No.  Temperature dependence o f t i n d i e t h y l amine deuterate and d i e t h y l amine-ND  5.4  85  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 T ) i n d i e t h y l amine deuterate 0  and d i e t h y l amine-ND 5.5  87  P l o t o f u ) x versus r e c i p r o c a l o f the 0  c  absolute temperature i n d i e t h y l amine deuterate and diethylamine-ND 6.1  88  V a r i a t i o n o f second moment with temperature in acetone and acetone deuterate  6.2  Some proton magentic resonance  98  spectra  o f acetone deuterate a t d i f f e r e n t temperatures 6.3  99  Temperature dependence o f t i n acetone and acetone deuterate  6.4  101  t versus r e c i p r o c a l o f the absolute temperature f o r acetone and acetone deuterate  7.1  102  Proton magnetic resonance second moment vs temperature f o r tert-butylamine-ND and 2  t e r t - b u t y l a m i n e deuterate  H  5  - xvi -  LIST OF FIGURES (continued) Figure 7.2  Title  P a  9  e  No  '  Proton magentic resonance l i n e width vs temperature f o r t e r t butyl amine-ND and t e r t - b u t y l a m i n e 2  deuterate 7.3  Some  ...  n  ^  resonance a b s o r p t i o n s p e c t r a o f  tert-butylamine-ND and t e r t - b u t y l a m i n e 2  deuterate ... 7.4  H8  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 t e r t - b u t y l a m i n e deuterate 7.5  120  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 t h e 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 8.1  124  PMR second moment vs temperature i n isopropylamine-ND  2  and isopropylamine  deuterate 8.2  139  L i n e width i n gauss and second moment 2 i n gauss  as a f u n c t i o n o f temperature  in t r i m e t h y l amine and trimethylamine 8.3  deuterate t vs temperature and aj T p l o t t e d a g a i n s t 1000/T (T i n K) f o r i s o p r o p y l ami ne-ND Q  143  c  ?  145  - xvii LIST OF FIGURES (continued) Figure 8.4  Title  Page No.  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  8.5  147  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 l o g s c a l e a g a i n s t 10 /T i n isopropylamine deuterate 8.6  148  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 in trimethylamine and trimethylamine deuterate  8.7  150  Values o f 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 /T ( K ) 3  -1  151  - xviii 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 i n t r o d u c e d me t o 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 t o him f o r h i s guidance, and extension o f h i s generous research f a c i l i t i e s .  His kind help, advices and time t o 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 t o 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 h i s s i n c e r e help i n the c o n s t r u c t i o n o f 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 o f 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  P r 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. S h i z g a l f o r h i s 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 t o t h i s work. TO  P r o f e s s o r B.A. Dunell f o r some h i s kind help and v a l u a b l e  i n s p i r i n g comments. TO  t h e members o f 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 o f mechanical  i n o p e r a t i n g c o n d i t i o n , and the members  and glassblowing shop f o r c o n s t r u c t i o n o f some o f  the equipments.  - xix -  TO  my c o l l e a g u e s Mr. T.T. Ang and Dr. S.E. U l r i c h f o r t h e i r  cheerful 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 assistantships  and other funds.  -  XX  -  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 o f 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 o f  c l a t h r a t e hydrates, the host s t r u c t u r e i s t h a t 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 r e a c t 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 o f 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 d i s c o v e r e d t h a t t h e r e i s a c l a s s o f 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 S t u d i e s 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 o f 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 v a r i o u s hydrates were reviewed by J e f f r e y and McMullan [1.5] and aspects o f water s t r u c t u r e i n o r g a n i c hydrates by J e f f r e y [1.2]. A b r i e f account of v a r i o u s s t u d i e s 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 o f 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 d i s c u s s e d 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 S t a c k e l b e r g and co-workers p u b l i s h e d 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 f o r m u l a t i n g these s t r u c t u r e s . At the same time P a u l i n g and Marsh [1.12] p u b l i s h e d 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 S t a c k e l b e r g ' s type I s t r u c t u r e ) and 17 A (von S t a c k e l b e r g ' 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 B a r r e r and h i s 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 o f occupation o f 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 n e g l e c t e d .  4)  Classical statistics i s valid,  c a l c u l a t e d the d i s s o c i a t i o n pressure o f hydrates at v a r i o u s temperatures, t h e i r heat o f formation and composition o f c o - e x i s t i n g e q u i l i b r i u m phases u s i n g Lennard-Jones and Devonshire (L-J-D) 12:6 p o t e n t i a l assuming t h a t p o t e n t i a l energy o f s o l u t e molecule a t 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 t h a t 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 d e r Waals and Platteeuw theory [1.4] t o methane, argon and n i t r o 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 p r e s s u r e as a f u n c t i o n o f temperature f o r C H  4>  A r , ( C H + A r ) hydrates, but 4  some d e v i a t i o n from theory was observed f o r N i t r o g e n , (Ar + N,,) hydrates a t higher p r e s s u r e s . At the same time McKoy and Sinanoglu [1.15] showed t h a t (L-J-D) 12:6 p o t e n t i a l was good o n l y 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 H , C 0 2  g  2 >  N and C H . 2  2  4  Their  c o n c l u s i o n was t h a t 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 o f 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 o f 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 m o d i f i e d form f o r the r e a c t i o n M + nH 0 j M-nHgO  (1.1)  2  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 o f water molecules, i s = -n*nx  .nO-xJ  •w  -"^1  w  -  RT  -"'•^1 R  [^n y + runy ] M  (1.2)  w  where xw . Yw 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 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. AH°,  AS°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 a t 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 o f the systems.  Supposing that i n t h i s s e r i e s a  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 j  wj wj  a c t i v i t y O-x^jK,^ (because x + x = 1), w h i l e the other adjacent w  M  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 k ^ M k * Y  w  W e g e t t w 0  e c  )  u a t l 0 n s  f r o m  d i f f e r e n c e of these two equations leads to  Equation (1.2) and  -  A  5 -  j k mO-V/A^O/T) = •(AH°/R) -  [(A  £n  j k  -n(A £nx /A (l/T) w  jk  Y M  +  jk  nA £riY ) j k  w  /A (l/T)]  (1.3)  j k  where  A^nO-xJ  V  n  X  w  =  e t c . and  = An[(1)/(l-x ^ V W ' A (l/T)  y / k  V" M Y  )], =  *  n  (  V W  = (1/T..) - ( 1 / T )  j k  k  The Gibbs-Duhem c o n d i t i o n d^ny  M  + [x /(l-x )]d^ny w  w  = o makes  w  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 o r by a g r a p h i c a l p l o t o f A..£n(l-x )/A.. (1/T) J K  W  J K  versus A . fi,n(x )/A . (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 j  k  W  j  k  s l o p e equal t o 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 ) + n*n x w j  t!!.J_ R  -Tj  w j  +  AS1 R  [*nY  Mj  +nin^]  (1.4)  - 6 -  from which AH° 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.) +  WJ  WJ  B  (1.5)  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 S o l u b l e N6n-E1ectrolytes Hydrate Former  n/calc.  n  -A  B  (CH ) N  10.22+0.34  10.0  2314  6.8526  3  3  (C H ) NH 2  5  *C H 0 2  *  4  2  6.80+0.39  6.66  2519  8.1595  8.12±0.45  8.10  2780  9.0870  6.58 ± .48  D;N. Glew, Nature,201 (1964)  6.67  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 o f mention are those o f 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 o f 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 r e p o r t e d 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 W i l l i a m s [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 o f the r e s u l t s obtained by  Davies and W i l l i a m s a r e 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 Host  Guest  Temp.(°K)  AH (kcal/mole)  AS e.u.  T (sec.)  H0  Tetrahydrofuran  88  0.27+0.05  -2.3+0.3  llxlO"  H0  acetone  93  <0.25  -1.4  4.3xl0"  H0  Ethylene oxide  88  0.46+0.10  -3.2+1.0  24x10"  2  2  2  In Table 1.2 AH and AS are t h e 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 t h e E y r i n g r a t e equation T =T  q  exp(.AH/RT) expfAS/R)  where T i s c o r r e l a t i o n time f o r t h e motion.  (1.6)  I t i s c l e a r from the  Table 1.2 t h a t the guests which give good examples of von S t a c k l e b e r g s 1  type Methylene oxide) and type II (acetone and tetrahydrofuran)  hydrates a r e 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 r e c e n t a r t i c l e Davidson [1.9] has reviewed  1 2  12  12  - 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 r a t e s o f p o l a r guest molecules.  He concluded t h a t o f  the d i p o l a r f i e l d due t o the water molecules vanishes a t the centre o f cages and t h a t the quadrupolar f i e l d becomes very small because of the hydrogen bonding o f water molecules.  A distribution of relaxation  times i s expected because o f the r e o r i e n t a t i o n o f the water molecules. There has been r e p o r t e d i n f r a - r e d s p e c t r o s c o p i c measurements on some o f the hydrates.  Harvey e t al [1.19] r e p o r t e d t h a t 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 t h a t f o r some halomethane 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 in a very r e c e n t i r study by Falk [1.21] i n trimethylamine hydrate. C.  The Present S t u d i e s  The use o f Nuclear Magnetic Resonance t o study the molecular motion o f guest molecules i n c l a t h r a t e s was f i r s t undertaken 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 ) Ni(CN) '2M 3  2  [1.24].  4  where M i s benzene, thiophene, p y r i d i n e e t c . )  S t u d i e s 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 r e p o r t e d on hydroquinone c l a t h r a t e s [1.26-1.27].  The work  - 9 -  on t h e study o f motion o f 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 ;  alkylammonium-salt  S t a c k e l b e r g ' s type I , type II and two t e t r a -  hydrates were s t 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 o f equipment limitations.  In the mean time some nmr s t u d i e s on hydrates appeared  by Davidson and co-workers [1.33-1.35], Afanas'ev e t a l [1.36] and E l e y e t a l [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 ~ i o n i n [ A g 0 g ] H F ~ +  2  7  2  c l a t h r a t e by  Hindermann e t a l [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 o f 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 o f which c o n t a i n methyl groups.  In a d d i t i o n  to a b s o r p t i o n 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 t o make  the c o n t r i b u t i o n o f water protons n e g l i g i b l e .  The study o f guests i s  a l s o presented, so t h a t a comparison between the behaviour o f guest molecule i n c l a t h r a t e d and u n c l a t h r a t e d s t a t e can be made. The m a j o r i t y o f the r e s u l t s are on amines.  A l l o f these have the amine  molecule hydrogen bonded t o 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 o f a l l o f 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 S t a c k e l b e r g , Naturwiss, 36 (1949) 327, 359. and H.R. M u l l e r , J . Chem. Phys., 19 (1951) 1319.  [1.11]  W.F. C l a u s s e n , 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. R u z i c k a , 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  [1.18]  M. Davies and K. W i l l i a m s , 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.,  therein.  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 o f B r i t i s h Columbia, 1970.  [1.21]  M. F a 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. S a s a k i , J . Mol. S p e c , 35 (1970) 244 and references  therein.  [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 S t r u c 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. M a j i d , S.K. Garg and D.W. Davidson, Can. J . Chem., 46 (1968) 1683.  [1.35]  Y.A. M a j i d , 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. E l e y , M.J. Hey, K.F. Chew and W. D e r b y s h i r e , 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. R i c h a r d s , 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 o f 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 o f von S t a c k e l b e r g and co-workers [ 2 . 1 ] , Claussen [ 2 . 2 ] , and P a u l i n g 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 t h r e e 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 belongs 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 r e p o r t e d 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 c u b i c 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 S t a c k e l b e r g ' 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 o f 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 t h a t there are some guests l i k e p i n a c o l and some amines which form hydrogen bonds with water molecules i n the l a t t i c e , but with the exception o f 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 g i v e 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 o f these amines together with a b r i e f d e s c r i p t i o n o f von S t a c k e l b e r g ' s type I and type II s t r u c t u r e s . The desc 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 o f  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 o f the host s t r u c t u r e . These s t u d i e s have o n l y i n d i c a t e d the motion o f some guest molecules. 1.  The host s t r u c t u r e In a l l o f these hydrates, the 'host' s t r u c t u r e i s a p o l y -  hedral framework o f hydrogen-bonded  water molecules.  The s m a l l e s t  polyhedron found i n these frameworks i s pentagonal dodecahedron 1p (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 d e s c r i b e d by E u l e r ' 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 o f D respectively. molecules.  The v e r t i c e s a r e formed by oxygen atoms o f water  S i n c e t h e r e a r e 20 v e r t i c e s , 20 H 0 molecules are r e 2  q u i r e d g i v i n g 40 H-atoms f o r H-bonding.  30 o f these H-atoms a r e  o  used i n 0 - H — 0 (-2.8 A) bonds f o r 30 edges o f D. To make oxygen atoms c o o r d i n a t e roughly t e t r a h e d r a l l y 20 more 0 - H — 0 bonds a r e needed, and these a r e provided e x t e r n a l l y . The volume o f D i s approx°3 t imately 170 A with a f r e e diameter o f 5.1A [2.7] so t h a t i t can 0  enclose an atom o r molecule l i k e Ar, Kr, Xe, H S and CH^ but not a 2  molecule l i k e C l , S 0 , e t c . 2  2  The pentagonal dodecahedron, D, alone can't f i l l the space, other polyhedra a r e needed t o g i v e r i s e t o 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 o t h e r polyhedra a r e  given as below: (a) Tetrakaidecahedron(14-hedron,5  .6 ) , ( T ) .  T h i s i s formed by  24 H 0 and has 12 pentagonal and 2 hexagonal faces (14F+24V = 36E+2) 2  oo  and has a volume o f 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 , S 0 , e t c . 9  o  By t h e word f r e e diameter we mean the diameter o f D o r any o t h e r polyhedra a f t e r s u b t r a c t i n g t h e non-bonded diameter o f the oxygen atom which i s -2.8 A. 12 2 * T h i s n o t a t i o n m r e f e r s t o n m-sided faces e.g. 5 .6 means 12 pentagonal and 2 hexagonal faces and i s due t o Wells [ 2 . 8 ] . n  - 16 (b)  Hexakaidecahedron(16-hedron,5  .6 ),(H). T h i s i s formed by  28 H 0 with 12 pentagonal and 4 hexagonal faces (16F+28V = 42E+2). °3 It has a volume o f 250 A and i s n e a r l y s p h e r i c a l i n shape with a 2  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 S t a c k e l b e r g ' 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  cell.  The space group f o r these i s Pm3n ( 0 ) with u n i t c e l l edge h  o  The u n i t c e l l has two 12-hedra ( D ) and s i x 14-hedra ( T ) .  a = 12 A.  The centres o f D are a t / Q ^ 1_ 1_ 1_\ and those o f T are  (Figure 2.2) at/l  1  n  3  0 Q  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 0 (X-5.75 H 0) 2  2  where X i s a guest molecule e.g. Ar, CH^, H S, e t c . However i f the 2  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 s m a l l e r 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 0 2  (Y*7.67H 0), where Y can be C l , S 0 , C Hg, e t c . Another i n t e r e s t i n g 2  2  2  2  s i t u a t i o n a r i s e s when we have two types o f guest s p e c i e s X, Y r e s u l t i n g i n a mixed o r double hydrates with the formula 2X*6Y*46 H 0 2  (X-3Y-23 H 0) where X may be N 2  2>  0 , Ar, e t c . and Y may be S 0 , 2  etc. Some o f the type I hydrates are given i n Table 2.1.  2  Cl  2 >  - 17 -  3. von Stackelberg's type II s t r u c t u r e hydrates. These hydrates are d e s c r i b e d by space group Fd 3n (0^) o  with c u b i c 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 o f these 24 c a v i t i e s 16 a r e  12-hedra (D) and 8 a r e 16-hedra (H) ( F i g u r e 2.3). The 12-hedra a r e 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 t h e 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 51 7 H  7 5 3 573  I 5s1s s5 ' s7 s3 s5 ' s7 sH s ' s5 s3 s7' s3 s5 s7' s177 s s ' s155 s s ' s3 s s ' 8 8 8 ' 8 8 8 '8 8 8 ' 8 8 8 ' 8 8 8 ' 8 8 8 8 8 8 8 ;  16-hedra a t / 1 1 1  33 3  n  n  1 113  1  n  n  13 1 1  8 8  ;  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 t o have a hydrate with the formula 24X- 136 H 0 (X-5-67 H 0 ) , but none has been r e p o r t e d . However, i f a 2  2  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 0 (Y- 17 H 0) i s formed (Y may be S F , 2  2  acetone, t e t r a h y d r o f u r a n (THF) e t c . ) .  g  Double hydrates with t h e  formula 16X- 8Y- 136 H 0 (2X- Y- 17 H 0) have been r e p o r t e d ; a known 2  2  example o f which i s 16 H S- 8 THF- 136 H 0 [ 2 . 5 ] . A l i s t o f known 2  type II hydrates i s given i n Table 2.1  2  j  12-hedron ( 5 ) (Pentagonal dodecahedron) 1 2  Figure 2.1  14-hedron ( 5 . 6 ) (Tetrakaidecahedron) , 2  2  16-hedron ( 5 . 6 ) (Hexakaidecahedron) 1 2  4  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 f a c e - s h a r i n g arrangement • o f 12- and 16-hedra.  - 19 Table 2.1 SOME KNOWN CUBIC CLATHRATE HYDRATES Structure Guest Molecules X,Y  Ideal Composition (a)  12 A c u b i c 8 X- 46 H 0  A. A r , Kr, Xe, C H , N , H S, P H 2  2  CH^F, 0  4  2  2  3>  H Se, CH^.  2 >  2  A s H , N 0, CH Br, B r C l , CH CH F,  6 Y- 46 H 0  3  2  2  3  3  2  ^2^4' ^^3^1, COS, CHF , C Hg, C l , 3  2  CH SH, CH =CHF, CH CHF , 3  2  3  2  Cyclo-C Hg,  2  3  CF . 4  B. C 0 , S 0 , C 1 0 2  2  2  C. ethylene o x i d e , (CH ) NH, C H N H 3  2  2  5  2  Trimethylene oxide (b)  17 A c u b i c 8Y- 136 H 0  A. C H I , CH C H CH , CH CH Br, CFC1 , 3  2  3  2  3  3  2  C F B r , C H C 1 , CH CHC1 2  2  2  2  3  3  CH CF C1,  2J  3  2  CH N0 , Cyclo-C Hg, CH CH C1, ( C H ) C H , 3  2  5  3  2  3  CF Cl ,CF ClBr, Cyclo-C H , 2  16H S- 8Y- 136H 0 2  2  2  2  5  16  3  Cyclo-C^g.  CH Br, C S , CH CH Br, S F , CgHg, COS, 3  2  3  2  g  CH CH C1, C F C 1 , CH C1:CH C1, C H I , 3  2  3  2  2  3  CH CHF , CC1 , CH CH CH Br, CHF=CF , 3  2  4  3  2  2  2  CH CH CH , ( C H ) S , CC1 N0 , CHgCl , 3  2  3  3  2  3  CHC1 , CH CF C1, C C l B r . 3  3  2  3  2  2  - 20 -  Table 2.1  (cont.)  Structure Ideal Composition 8Y- 136  Guest Molecules X,Y C.  H0 2  CH CH 0H, ( C H ) C 0 , t e t r a h y d r o f u r a n 3  2  3  2  ( C H ) 0 , f u r a n , dioxane, 3  2  2,5  d i h y d r o f u r a n , propylene o x i d e , t r i m e t h y l e n e oxide, cyclobutanone A.  Hydrophobic gases,  C.  Water-soluble p o l a r compounds  References: C.  B.  Water-soluble a c i d o g e n i c gases,  [2.7] and [ 2 . 9 ] .  Alkylami ne Hydrates The e x i s t e n c e o f c r y s t a l l i n e hydrates o f alkylamines has  been known from phase s t u d i e s by P i c k e r i n g [2.10] and [2.11].  Sommerville  About 35 o f them have been reported with m e l t i n g p o i n t s  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 v a r y i n g 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 o f nine amines [2.12].  These r e s u l t s  are summarized i n 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  Table 2.2 C r y s t a l S t r u c t u r e data on some Alkylamine Hydrates  Amine A  1.  Ethyl amine  2.  M.P. of hydrate (°c)  Crystal class Space group  Unit c e l l dimension in &  Stoichiometric formula per c e l l mA. n H 0 2  Hydration number from f r e e z i ng curve (n/m)  -7.5  Cubic P43n, Pm3n  a = 12.17  m = 6; n = 46 or m = 8; n = 48  5.5  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  -13.5  Hexagonal  a = 12.20 m = 4; n = 40 c = 12.38 or m = 4; n = 38  8.0  -4.0  Hexagonal Pe^/mmc  a = 12.30 c = 24.85  m = 10; n = 80  7.5  monoclinic P2 /n  a b c 3  m = 16; n = 104  3.5  4.  n-Propylamine  5.  iso-Propyl amine  6.  n-Proplamine  1  = = = =  12.43 20.73 17.28 89.3°  11, 10, 10.22, 10.03  Table 2.2 (cont.) C r y s t a l S t r u c t u r e data on some Alkylamine Hydrates  Amine A  Diethyl amine  M.P. Crystal class of hydrate space group (°C) -6.6  -7.0 -7.3  tert-Butylamine  -1  References:  2  a b c 8  Orthorhombic Pbcn  a =13.44 b = 11.77 c = 27.91  m = 12; n = 104  a = 18.81  m = 16; n = 156  [2.12 - 2.17]  cubic I43d  = 13.86 = 8.44 = 10.93 = 97.5°  Stoichiometric formula per c e l l mA. n H 0  monoclinic P2 /c 1  Diethyl amine  Unit c e l l dimension in A  m = 4; n = 28  Hydration number from f r e e z i n g curve (n/m)  6.8  8.07, 8.12 8.10  - 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 t h a t the NH  2  groups are part o f hydrogen bonded water framework [2.12].  or NH Out o f  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  in 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 g i v e r i s e to a v a r i e t y o f 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 in 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 d e s c r i b e d below: 1. t e r t - B u l y l a m i n e 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 o f 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], 2.  The 17-hedra and 8-hedra are shown i n Figure 2.4.  Diethyl amine Hydrate: T h i s hydrate with formula 12(C H ) NH.104H 0 has been 2  5  2  2  s t u d i e d by Jordan and Mak [2.13]. The amine molecule i n t h i s case forms two types o f hydrogen bonds.  One o f these bonds i s a donor  H-bond formed by NH-group o f amine and i n v o l v i n g one oxygen o f H 0, ?  Table 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 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 0 ) = m ]< n  n  2  1. t e r t - B u t y l ami ne hydrate 16(CH ) .CNH .156 H 0  Nonbonded 17-hedra  2. Diethyl amine hydrate 12(CH CH ) NH.104 H 0  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  18- hedra ( 5 . 6 ) I r r e g u l a r cage (6^.5 .4 )  3. Trimethylamine hydrate 4 ( C H ) N . 41 H 0  Hydrogen bonded i n very d i s t o r t e d 15- and 26- polyhedra  15- hedra ( 5 . 6 ) 26-hedra ( 5 . 6 ) 12-hedra ( 5 )  3  3  2  3  3  2  2  3  3  2  4  2  2  17- hedra ( 7 . 6 . 5 . 4 ) 8-hedra ( 4 . 5 )  2  4  1 2  3  2  2  2  5. n-Propylamine 16CH CH CH NH . 104 H 0 3  2  2  2  2  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  14-hedra 16- hedra 8-hedra 12-hedra  (4 (5 (6 (5  empty cages.  +  6  1 2  3  2 4  2  2  .5 .6 ) .6 ) .4 ) )  1 2  2  -pi-  +  8  1 2  4  4  6  +  +  * m means n m-sided f i g u r e [2.8]  1 2  2  1 2  4  8  3  ro  14-hedra ( 5 . 6 ) 16-hedra ( 5 . 6 ) 11-hedra ( 4 . 5 . 6 ) 2  Reference [2.6]  3  8  1 2  4. Isopropylamine hydrate 10(CH ) CH.NH . 80 H 0  9  1  +  - 25 and t h e o t h e r i s an a c c e p t o r H-bond between the NH-group and t h e oxygen ( l y i n g on o p p o s i t e s i d e o f cage) o f the o t h e r H 0. There 2  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 n e a r l y equal h a l v e s , each o f which accomodates one ethyl group (Figure 2.5a) 6 8 3 and the o t h e r 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.  i s o - P r o p y l a m i n e Hydrate: T h i s hydrate has been s t u d i e d r e c e n t l y by McMullan e t a l  [2.17] and p r o v i d e s an example where t h e NH -group o f the amine 2  forms two donor H-bonds b r i d g i n g a c r o s s two a d j a c e n t oxygen atoms o f water m o l e c u l e s . 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  4  8  " 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 (5 12 ) and 16-hedra (512.64 ). The amine molecules l i e i n s i x 14-hedra and f o u r 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 group makes an 0--H-N-H--0 bridge 2  across two oxygens and these bonds form t h e 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 o f these polyhedra leads t o 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 a r e vacant. 4.  Trimethylamine Hydrate. [2.15] In trimethylamine hydrate 4 ( C H ) N . 41 H 0, t h e amine 3  n i t r o g e n forms a c c e p t o r H-bonds.  3  2  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 (a)  C-N bond v e r t i c a l  (b) C-N bond towards reader  (c) empty 8-hedron  i  ro cn  Figure 2.4  17-hedra and 8-hedron found i n t e r t - b u t y l a m i n e hydrate  Figure 2.5 (a) (b)  Diethyl amine i n 18-hedron Diethyl amine i n an i r r e g u l a r cage  i  - 27 -  In 15-hedra there i s only one acceptor hydrogen bond from the oxygen o f a water molecule [ F i g u r e 2.7a] t o the N-atom o f the amine.  In 26-hedron, a p a i r o f amine molecules a r e 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 face).  T h i s 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 o f 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 h i n g t o note i n the case o f t h e t r i m e t h y l a mine hydrate i s t h 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 t h e l i q u i d s t a t e ( i . e . a t the m e l t i n g point o f 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 a l [2.18]. 5.  n-PrOpylamine Hydrate The hydrate n-CH CH CH NH -6.5 H 0 has been s t u d i e d i n 3  2  2  2  2  l e s s d e t a i l [2.17]. The n i t r o g e n i s hydrogen bonded i n the 14-hedron 12 2 (5  .6 ) and here i t makes one o f common v e r t i c e s o f a hexagon and a  pentagon. The o t h e r cage found i s a 16-hedron(5  .6 ) where t h e  n i t r o g e n atom r e p l a c e s a water oxygen and makes a bridge a c r o s s a v o i d forming a hydrogen-bonded dimer. The o t h e r 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 o t h e r a l k y l a m i n e hydrates have not been r e p o r t e d , but i t i s b e 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 C h e m i s t r y . 1  2.6a Figure 2.6 Isopropylamine molecule i n 14- and 16-hedra. a in 16-hedron s i n g l y bonded b,c,d 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.  - 29 -  References (Chapter Two) [2.1]  M. von S t a c k e l b e r g , 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 S t a c k e l b e r g , 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 S t a c k e l b e r g , 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. C l a u s s e n , 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. W e l l s , 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. M o r r i s o n 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 t o the p r i n c i p l e s o f n u c l e a r 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 s u b j e c t 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 s t u d i e d  in t h i s work a r e protons and a l l the compounds, c l a t h r a t e d e u t e r a t e s , 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 s p i n possess a n e t macro-  s c o p i c magnetic moment M which i s the sum o f 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_, i s placed i n a l a r g e magnetic f i e l d H, i t experiences a torque C_ = M X H_ equal to r a t e o f change o f i t s angular momentum  ti(dl/dt),  where I_ i s t o t a l angular momentum v e c t o r equal toV* L . Since M_ =  SMJ, Syfil, > where  y i s n u c l e a r 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)  H_ can be the sum o f two f i e l d s , the s t a t i c f i e l d H and the magnetic  - 32 vector o f radiofrequency ( r f ) f i e l d  r o t a t i n g with frequency w  i.e., fij  =  -H^sinut, 0)  (H-jCOSoyt,  The Bloch equations f o r an i n d i v i d u a l component o f m a g n e t i z a t i o n M can now be w r i t t e n [3.1, p.28] ( d i y d t ) = y(M H + M ^ s i n w t ) y  (M/V  Q  (dM /dt) = ( M H o s w t - M H ) - (M /T ) y  Y  z  lC  x  (dM /dt) = -Y,(M H,sina)t  Q  y  (3.2)  2  + M H,cosoot) + (M - M j / T ,  v  where T-j and T are c a l l e d the longitudnaJl and the t r a n s v e r s e 2  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 i s e q u i l i b r i u m m a g n e t i z a t i o n . Q  Equation (3.2) i s the s i m p l e s t phenomenological to e q u i l i b r i u m i n a constant magnetic f i e l d  d e s c r i p t i o n o f approach 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 /dt) = ( M - M ) / T z  0  2  (dM /dt) =-M /T ; (dM /dt) = -M /T  i;  x  x  2  y  y  2  (3.3) The s o l u t i o n o f these equations g i v e s T-j and L,.  T h i s 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 and M under steady s t a t e x  c o n d i t i o n s g i v e s the components x susceptibility x respectively.  1  y  and x" o f n u c l e a r magnetic  I t i s the component x" which gives r i s e  to NMR a b s o r p t i o n [3.1 p. 29]. Quantum m e c h a n i c a l l y 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 g i v e s the t r a n s i t i o n p r o b a b i l i t y W time induced by some p e r t u r b i n g Hamiltonian  ab  per u n i t  between s t a t e s |a>  and |b> as:|<a|j^|b>| 6(E -E -fi )  Wab  (3.4)  2  a  b  W  where 6 i s D i r a c d e l t a f u n c t i o n , E , E^ are energies of s t a t e |a> Q  and |b> r e s p e c t i v e l y . U s u a l l y ^ = -b H coswt, where y i s the xx  x  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 a -E.D ) « k T (k, Boltzmann's constant and T absolute i s given by [3.3  temperature)  p.43]  **mL - * E  e  >V b  <-'  E  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  (c stands f o r complete spectrum)  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|u |b>| 6(E -E -fiu) 2  x  a  b  (3.6)  t E x p e r i m e n t a l l y one i s i n t e r e s t e d i n g(w -to) = g(A) where to i s c e n t r e of l i n e shape and A = (to -to) [3.5 Chap. IV].  m  - 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 (3.7)  (3.8) Here H 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 Q  z  i s n u c l e a r spin eigen-  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 v e c t o r £. . i s given by [3.3 p. 46]. (3.9)  (3.10) i <J  - 35 -  where A.. = i V ( l - 3 c o s 6 . .)r.~. 2  3  B . = - 0 / 4 ) ( I ^ + Oj)(l-3cos e...)r:] 2  ia  C.. - - ( 3 / 2 ) ( i ; i J i ; i J )  sine^cose.-expC-i^)^  +  E  1d  - -(3/4)l|lJ s l n ^ j e x p ( - 2 i * . . ) r~] +  D  (3.11)  +  U-c!j  >  i j=  F  i j  E  Where r . ., 6.., <b.. are the p o l a r c o o r d i n a t e s o f the v e c t o r between s p i n i and j ,  1^ e t c . a r e 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 o f 1^ e t c . , and the dagger signs denote the complex conjugates o f the f u n c t i o n s . Jf  z  I n j ^ j , the termsA and B commute with  and a r e 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 X  i.e.  -Y V J ^ 1  •B  f j  (3.12)  ]  ^  = -(yV/Zj^d-.L.  -3I^)(l-3cos e 2  i : J  )r:]  (3.13)  i<j So f a r we a r e only i n t e r e s t e d i n _ ^ and J ^ , the other terms C to F o f J^j w i l l be d i s c u s s e d 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 t o  3i-K +'](t z  (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 (CaS0 '2H,,0) by Pake [3.1 p.152] 4  v a r i o u s 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 C a F  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 t r a n s v e r s e magnetization 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 o f 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 t h a t 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 a b s o r p t i o n l i n e shape g(A)[3.7, 3.5 p.222] i . e . , g(A) = (2TT)  where G(0) = 1,/ r  -1  /  G(t) e x p ( i A t ) d t  (3.15)  g(A)dA = 1 and -oo  +oo  /  g(A)exp(-iAt)dA  (3.16)  The work done on FID shape and i t s conversion to a b s o r p t i o n l i n e shape in two s p i n 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 , CsF and NaCl [3.9] by c o n v e r t i n g usual a b s o r p t i o n shape to FID 2  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 C a r l 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 t o c o n f 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 t a k i n g 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 m a t r i x 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 r e 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 t o intermolecular interactions.  Some promising r e s u l t s have been d e p i c t e d 1g  by S F g - c l a t h r a t e d e u t e r a t e , where  F a b s o r p t i o n 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 o f l e s s intermolecular interactions.  I t i s b e 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 o f c l a t h r a t e s , a b e t t 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 V l e c k ' s formula [3.17] f o r second moment o f a b s o r p t i o n l i n e shape i s much more useful i n s t e a d 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, M , o f a resonance a b s o r p t i o n n  curve (normalized t o u n i t y ) by the r e l a t i o n [3.5 p.223] /«+<» M  nV  An  9(A)dA  (3.17)  (A=O) -OJ) O  where the second moment M i s given by ?  M  2  =J  A g(A)dA  (3.18)  2  - 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 t o be [3.1 p. 160] M  = (6/5)1(1+1)g 3 N ^ ,r^j 2  2  2  _ 1  i>j + (4/15)8 N ^ I (I +l)g r^ 2  _ 1  2  f  i,f  f  (3.19)  where I i s nuclear s p i n number f o r n u c l e i a t 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 o f a b s o r p t i o n curve.  T h i s i s a long and tedious procedure.  Caution must a l s o be  e x e r c i s e d i n o b t a i n i n g the a b s o r p t i o n curve because o f the danger o f s a t u r a t i o n and account must be taken o f the c o n d i t i o n s imposed by Provotorov theory^. Bloch decay.  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  C a l c u l a t i o n o f second moment from the a b s o r p t i o n l i n e  +A d e t a i l e d account o f 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 & I V ] .  - 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 d e s c r i b e d b r i e f l y (although we have pref e r r e d the long and slow cw experimental t e c h n i q u e ) . 2.  Second Moment and FID curve It i s well known now t h a t FID curve can be expanded i n  terms o f even powered moments o f the resonance a b s o r p t i o n curve [3.7, 3.2 p. 110] i . e . G(t) = 1 - ^  M + j,  M~  2  (3.20)  4  Equation (3.20) shows t h a t 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 t h a t the important i n i t i a l p a r t o f G(t) i s buried i n the dead time o f the r e c e i v e r . However, Powles and Strange [3.18], and M a n s f i e l d [3.19], have developed a method t o overcome t h i s s i t u a t i o n by using two 90° p u l s e s , the second a f t e r a time T, and 90° out o f r f phase with the f i r s t (90° - x - 90gQ ), t h i s produces s o l i d echo. O  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 ) G(t) = T r { I e x p ( - i ^ t ) I e x p ( i ^ t ) } / T r ( I )  (3.21)  2  x  x  where T r 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 G(t) = 1 + t 5 1  2  Tr[I 2 ]  +  I 4 !  Tr[#', [fl . I ] ] 1  x  '-  2  +  (3.22)  - 40 Comparison of equations (3.20) and (3.22) gives  For a 90° - T - 90gQ p u l s e sequence, the FID s i g n a l a f t e r a time O  T' from the second p u l s e , the f i n a l r e s u l t of Powles and Strange [3.18] i s ,  (3.23) 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  M a n 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 o f 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 o f M o l e c u l a r 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 n u c l e a r s p i n s  do not change i n time, and the second moment has a constant value because  terms A and B i n j ^ j are c o n s t a n t . In case o f motion which  we assume to be o f 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 d u r a t i o n 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 s m a l l e r than the instantaneous value o f 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 ) ' . The r a t e o f f l u c t u a t i o n of the l o c a l 9  - 41 f i e l d can be d e s c r i b e d 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 '  (3.24)  1  c  where M 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 ) ' » T , the l i n e width and second moment c o r r e s 1/2 -1 2  2  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 r e d u c t i o n 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 o f Pake [3.21] and more e l a b o r a t e work o f Andrew and Newing [3.22], i t i s c l e a r t h a t r o t a t i o n produces weak s i d e s p e c t r a which a r e u s u a l l y unobservable e x p e r i m e n t a l l y , because they a r e b u r i e d i n the n o i s e i n the wings, and hence a reduced second moment i s observed.  I f the motion i s t h e r m a l l y 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 = tan[Tr(6H - B )/2(C -B )]/(ay6H) 2  2  2  (3.25)  2  c  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 obeys A r r h e n i u s c  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 exp(E /RT) o  a  (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 h i n d e r i n g t h e r o t a t i o n , a Equations (3.25) and (3.26) can p r o v i d e an estimate o f 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 o f 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 = M [(l/4)(3cos y-l) ] 2  2  (3.27)  2  2  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 a x i s o f r o t a t i o n , M , M a r e 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 2  2  however the motion i s o s c i l l a t o r y , Andrew's e x p r e s s i o n [3.25] should be used; (M 2 )  osc  =i y  where P= l - ( 3 / 4 ) [ ( l - J ( a ) ) s i n Y + (1 - J ( 2 a ) ) s i n Y ] 2  2  2  Q  Here J  Q  (3.28)  4  0  i s a Bessel f u n c t i o n o f f i r s t k i n d , a i s the amplitude o f  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 a x i s of rotational o s c i l l a t i o n .  For a small angle a, p reduces to  P= 1 - (3/2)a sin Y 2  (3.29)  2  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 0 1 f 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 f o r powder samples i s given by Powles and Gutowsky [3.27] as 2 M = (8/5)a = (8/5)(3u/2R ) 2  3  2  2  (3.30)  where u i s proton magnetic moment, and R i s s i d e o f the t r i a n g l e formed by methyl p r o t o n s .  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 a x i s o f r o t a t i o n , the reduced second moment M  2  i s given by  [3.27] M = (2/5)a [(27/8) s i n ' / ' - 3 s i n ^ + 1] 2  2  4  2  (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 M  = (a /10)(3 c o s * - I ) 2  2  2  2  i s given by (3.32)  2  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 d i s c u s s e d 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 o f 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 p o i n t s are worthy o f d e s c r i p t i o n . A l l e n [3.29] has d e r i v e d 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 a t 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 v a l u e . For high b a r r i e r s (3-3.5 kcal./mole) h i s expression M  = (9/40)(Y -ti /r )[l-0.11(r/A) 2  2  2 2 where r= state.  2  6  1/2  was  - 0.64(r/A)]  ....(3.33)  3  (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 In a more r e c e n t paper Clough [3.30] has explained the i n c r e a s e  o f second moment due to t u n n e l l i n g methyl groups a t low 2 2  temperatures  and has shown that i n c r e a s e may be as great as (2y a /5) or about 5G where a i s the same as given i n Equation  (3.30).  2  - 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 d e f i n e d the s p i n l a t t i c e r e l a x a t i o n T-j by Equation  We a l s o assume here t h a t T^ i s due t o the d i p o l a r Hamiltonian, terms C to F i n  [Equation  (3.11)]  (3.2). and t h e  a r e 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 t h e time dependent part C to F i n a more compact way ( s u b s c r i p t s a r e dropped f o r c l a r i t y ) . f-j(t) = sin6cos6exp(i<{>)r f ( t ) = sin eexp(2i<j))r" 2  3  (3.34)  2  Since the f ^ ( t ) a r e randomly v a r y i n g f u n c t i o n s o f time, they a r e a s 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 ) > where <  > denotes the ensemble average and the s t a r i s the complex  conjugate.  A common form o f 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 o f exponential K(T)  where T  c  (3.35)  form i . e .  =< f ( t ) f * ( t ) > e "  x / T  c  ....(3.36)  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) o f random f u n c t i o n f ( t ) a r e given by F o u r i e r of  transform  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 ) V ^ [ J ( a ) ) + J . j (2a) )] ....(3.39) Y  i j  0  o  m where CO i s resonance q  2.  frequency.  Spin L a t t i c e R e l a x a t i o n f o r 2-Spin (1/2) Systems Equation 1/^  =  (3.39) f o r the present case reduces t o (9/8)Y^ [J (« ) + Z  1  0  J (2 )] 2  ....(3.40)  W o  T h i s case may a r i s e i n i s o l a t e d C H o r NH groups where the group 2  2  may r o t a t e around an axis p e r p e n d i c u l a r t o H-H v e c t o r and passing through the C o r 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 and the a x i s o f r o t a t i o n be a, and that around t h i s Q  axis be 8, we can convert 9 and <j> o f 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 ( t ) = [2icosasin2B + (1+cos a)cos28 + i s i n 2 a ] / 4 r 2  Using the f a c t < s i n 28 > = < cos 8 > = 1/2, Equations  (3.41)  reduce t o f ^ t ) = [2sin a + (l/2)sin 2a]/16r 2  2  6  (3.42)  f ( t ) = [(1/2) + 3 c o s a + ( l / 2 ) c o s a ] / 4 r 2  4  6  2  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 ( t ) f * ( t ) >= (1/10r ), < f ( t ) f * ( t ) > 6  1  1  2  2  = ( 2 / 5 r ) ....(3. 6  - 46 which gives T  a f t e r using values o f J^ (OJ ) and  1  _L _ 9. y4 f 2i T  40  1  J^^ )  q  r  Ti  c 7—?— l+o) 0 V c  6  4.  4T  0  c  (3.44)  7"  l+(2xox T 0 c  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 v e c t o r and the x - a x i s o f e q u i l i b r i u m process, 6 oscillation.  Q  i s the angle with r e s p e c t to <f> during Q  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 siricot >] . 0  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 R e l a x a t i o n f o r Methyl Group (a)  Exponential R e l a x a t i o n Hubbard [3.32] has reviewed v a r i o u s t h e o r i e s of nuclear  magnetic r e l a x a t i o n using d e n s i t y matrix (quantum m e c h a n i c a l l y , 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 d i s c u s s e d 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 t h a t  i n case o f 3-spin systems undergoing hindered r o t a t i o n , i f c r o s s 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 . s o l u t i o n of Equation (3.3) adapted to the language o f pulsed NMR  The can  be w r i t t e n as M (t) - M z  = (cose-l)M exp(-t/T )  Q  o  (3.47)  1  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  - M (t)]/2M  Q  z  Q  = exp(-t/T )  (3.48)  1  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 o f 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 c r o s s c o r r e l a t i o n are neglected is 4 2 9YVT  1 T^IO  r  (  6  4 1-cos 6 22 o c u 1 + w  T  2 4 ' 1 + 6cos 8 + cos 3 l+4w o V c  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  axis.  (3.49)  2  o  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 2R 1 9 YV T  1  20  r  r 7  4T.  T  T  0  T  c  1 +4u o 2—TTc  (3.50)  which i s j u s t a f a c t o r o f two g r e a t e r than equation (3.44) and i s the  - 48 same as obtained by other workers u s i n g a much simpler approach, e.g. O ' R e i l l y and Tsang [3.36]. (b)  Non-Exponential s p i n l a t t i c e r e l a x a t i o n According t o H-H theory [3.35] when c r o s s - c o r r e l a t i o n are  not n e g l e c t e d , the magnetization M ( t ) given by Equation (3.47) i s 2  not a simple e x p o n e n t i a l , but a sum o f 4 e x p o n e n t i a l s given by, 4 M ( t ) - M = (cose-1) M exp(- t/T') (3.51) z  Q  o  qj  j=l where Q. and q^ a r e complicated f u n c t i o n s o f  W T Q  and B the angle  c  between C^-axis and H , and T' i s a measure o f s t r e n g t h o f i n t e r a c t i o n Q  given by (1/T') = ( Y t i / r ) ( l / o ) ) 2  3  (3.52)  2  0  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) g i v e s an analogous equation to Equation (3.48) i . e . 4 R(t) = [M - M ( t ) ] / 2 M Q  z  Cj e x p ( - t / T ' )  Q  (3.53)  qj  j=l but now i t i s t h e sum o f 4-exponentials and the concept o f longer i s v a l i d .  no  The authors H-H have provided t a b l e s f o r C. and q. J  J  i n terms o f 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 t o average over a l l 8 t o g e t R ( t ) , A v  R ( t ) = {[M -M (t)]/2M | A v  0  z  Q  = 1 / ^ 0  c  j exp(- t/T')sinBdB (3.54) qj  J  '  =1  We have e v a l u a t e d t h i s i n t e g r a l f o r many values o f t / ' by using T  - 49 Simpson's formula from the values o f 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 o f t h i s method i n the f o l l o w i n g c h a p t e r s .  The t a b l e s of R ( t ) c o n s t r u c t e d A v  from equation (3.54) are given i n Appendix 2 of  (COQ^)  A f o r a p a r t i c u l a r value  . Some non-exponential curves obtained by p l o t t i n g In R ( t ) A y  versus t/T' are shown i n Figures 3.1. to those given by H-H.  These curves are e x a c t l y s i m i l a r  In such a case i t i s u s e f u l to take time t  equal to t , when In R ( t ) = 1/2 i n s t e a d of f i n d i n g T-j which i s not Q  A y  d e f i n e d 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 c h a p t e r s . 4.  E f f e c t o f 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 m o d i f i e d BPP Equation [3.37] 4  I T  where x  c  l  -C, 1+  H)V  1  T  (3.55)  C  -K\  +  2  i s assumed to obey the Arrhenius equation s i m i l a r to  Equation (3.26) i . e . T  =.T_ C  O  exp(E /RT).  Equation (3.55) shows a  a  minimum i n the temperature dependence of T-j at p a r t i c u l a r type o f motion.  MT Q  C  = 0.616 f o r one  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 o f methyl benzenes a t very low  temperatures.  It was r e a l i z e d t h a t 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 a t very low temperatures was no longer v a l i d .  I t was  suggested by these authors t h a t 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 the data of Tables A2-A8 ( c f . p. 172-175).  ( ^ x ^  2  .  These curves are obtained from  o.o  (CU T- ) = I O O 0  H.O  2  C  i  i  or  (cu -r ) 0  C  (OJ  ~^-2.0  0  c  T ) c  2  2  =  IO.O  = 5.0  (cu r ) =3.0 0  c  2  (O; T- ) =|.0 0  c  2  -3.0 °  2  Figure 3.1 continued.  4  6  8  IO  12  14  J 16  L 18  20  These curves are obtained from the data o f 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 ) t o the quantum mechanical t u n n e l l i n g process dominant a t l i q u i d helium temperatures.  This tunnelling  mechanism was r e a l i z e d before by Powles and Gutowsky [3.27] and S t 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 o f methyl group o f the form V = (V /2) (1 + cos3tj>)  (3.56)  Q  where V i s the h e i g h t o f p o t e n t i a l b a r r i e r and <j> i s the angular q  coordinatesdescribing  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 - \ 6cf>  21  (l-cos3<j>)M<j>) = 0  ( - ) 3  5 7  2  where ^ {$) i s wavefunction d e s c r i b i n g the p o s i t i o n o f 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 t o 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) a c c o r d i n g t o way o f Koehler and Dennisson [3.39] i . e .  dM(cf>)/d<{> + [A - 2cos3<f)]M(cf)) = 0 2  ....(3.57)  2  (which i s Mathieu e q u a t i o n , with  M(cf>) = ^(cf>),  A = (I/ft )(2E-V )) 2  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 t h e s p l i t t i n g o f 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 o f T-| t a k i 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 o f 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 o f 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  where f(u>) = 8y/{  4T  T  \— +  6TTJ[X  2* V +  1+0) T z 0 C  1+4(JO  T  0  c  f((  }  + 2 x ( 2 y - l ) + 1] } with x = 2  4  2  (3.58)  o  + 47Tf (2a3  GO/3J,  y = ( 3 d r ) . 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 - 1  t  c o r r e l a t i o n time obeying t h e  t  i s the  = T exp(E /RT) equation. In q  Q  Equation (3.58), t h e f i r s t two terms a r e the usual BPP terms dominant at high temperature, while t h e l a s t two terms a r e due t o those t u n n e l l i n g processes important a t very low temperatures.  A more  e l a b o r a t e theory o f T-j due t o t u n n e l l i n g motion i s presented i n a very r e c e n t paper by Clough  E.  [3.41].  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 and T h e i r E f f e c t on T, and E, a The m o d i f i e d 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 o f the symmetry o f the BPP equation about the minimum, t h e a c t i v a t i o n energy E, can be e x t r a c t e d from t h e a slopes on e i t h e r s i d e o f minimum, e.g. i ) i n the low temperature r e g i o n ( ^ T » 1 ) , InT^ ^ / where s l o p e i s + E /R., a i i ) i n t h e high temperature r e g i o n ( W T . . « ! ) , lnT-j ^ -E /RT where slope i s - E /R., a E  C  0  R T  a  =  Q  c  a  - 54 However, when 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, 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 e x t r a c t e d on the assumption o f the BPP equation, or from the s l o p e s , a r e sometimes much s m a l l e r than those reported from other measurements. T h i s 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 may be due to two cases which we outline briefly. The f i r s t case i s t h a t we may have two independent types o f r o t a t i o n proceeding a t n e a r l y 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 o f 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 o f d i f f e r e n t b a r r i e r s t o r o t a t i o n e.g. 1 - t e r t butyl -4-methyl -benzene i n the work o f A l l e n and Johnson [3.42]. On the other hand i n the m a j o r i t y o f cases the c o r r e l a t i o n times f o r methyl and tert-butyl a r e n e a r l y the same and g i v e a s i n g l e broad minimum.  In t h i s case the two BPP equations should be combined  together t o account f o r t h i s minimum, each with i t s own C and T . Examples o f t h i s case w i l l be seen i n the f o l l o w i n g c h a p t e r s . 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 ' S g i v e s r i s e t o a broad and f l a t C  minimum.  T h i s 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 t o Odajima [3.43] t h i s kind o f d i s t r i b u t i o n i s t e s t e d by r a t i o ( T / T ) a t 1  minimum.  2  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 g r e a t e r than 1.6, other forms o f d i s t r i b u t i o n are p r e s e n t . To account f o r broad minimum i n t h i s case, we have t o i n c l u d e a continuous d i s t r i b u t i o n o f T ' S i n s t e a d o f s i n g l e T . T h i s i s u s u a l l y C  done  [3.44]  by a d e n s i t y f u n c t i o n  G(x^),  where  G(T ) dT„ = Jo  c  c  1  to g i v e r i s e t o a new e x p r e s s i o n f o r T^ from BPP equation, 00  OO  (3.59)  /  c < c> c f c c> 1+S400 Density f u n c t i o n s are1+Wd e fTi n e d i n termsJ o f = &n(T^/T^) where T uo o c 0 c i s c e n t r e o f d i s t r i b u t i o n on l o g a r i t h m i c s c a l e . Then G ( x ) i s T  G  T  2  d T  T  G ( T  d T  2  2  2  T  Q  0  c  r e p l a c e d by F(S) with c o n d i t i o n s , F(S);d.S= J  1,  G ( T ) d x c = F(S)dS, T G ( T ) = c  C  F(S)  c  ....(3.60)  -oc  The new d i s t r i b u t i o n F(S) i s o f two types, symmetric and asymmetric. Some o f 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 i)  Gaussian d i s t r i b u t i o n .  [3.44].  T h i s d i s t r i b u t i o n has no e f f e c t on  the s l o p e s i n the curve o f lnT^ versus 1/T, except i t broadens the minimum. T h i s case i s d i s c u s s e d by [ 3 . 4 3 ] and by Resing [3.45].  i i ) Rectangular d i s t r i b u t i o n .  T h i s d i s t r i b u t i o n has a l s o no i  e f f e c t on t h e slope o f t h e InT^, versus 1/T curve. An example o f t h i s i s the work o f McCall e t a l .  [3.46].  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  [3.44].  -  56  -  The o n l y asymmetric d i s t r i b u t i o n f u n c t i o n i s t h a t o f Davidson and Cole [3.44] where the slopes on both s i d e s o f 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 s l o p e s g i v e s the width o f the  distribution. 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 o f the d i f f i c u l t y i n f i n d i n g the width of the distribution. 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 o f 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 correlation function.  - 57 REFERENCES (Chapter I I I )  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 o f 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 o f 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 F o 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., J 8 (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 p u b l i s h e d )  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., P t . 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 S t a t e 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., J 0 5 (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 0 9 5 4 ) 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 S t a t e Phys.), 4 (1971) 2180.  - 59 [3.42]  P.S. A l l e n and L.W. Johnson ( t o 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 r e f e r e n c e s therein. [3.45]  H.A. Resing, Adv. Mol. R e l a x a t i o n Process, 1 (1967) 109, J . Chem. Phys., 43 (1965) 669.  [3.46]  D.W. M c 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 g i v e some d e t a i l s o f apparatus used and the methods o f measurement o f second moments M , and s p i n 2  l a t t i c e r e l a x a t i o n times T-j. The p r e p a r a t i v e methods f o r hydrates are not d i s c u s s e d , because they a r e 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 c h a p t e r s .  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 V a r i a n V4210A  v a r i a b l e frequency r f u n i t o p e r a t i n g a t a resonance frequency o f 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 t h e modulation frequency was kept a t 80 Hz. The f i r s t d e r i v a t i v e o f the V  mode o r a b s o r p t i o n s i g n a l was recorded by means  o f 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 detector.  A V a r i a n 6-10 s t r i p c h a r t r e c o r d e r was used.  - 61 2.  C a l i b r a t i o n o f Spectrometer The resonance f i e l d H was a d j u s t e d from the a b s o r p t i o n Q  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 o f which was measured by a H.P. model 3734A e l e c t r o n i c counter.  Thus scan r a t e was c a l i b r a t e d  i n gaus's per cm along the base l i n e o f the r e c o r d e r . The modulation amplitude was c a l i b r a t e d by d i r e c t l y r e c o r d i n g the narrow a b s o r p t i o n 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 2 H , where H i s modulation amplitude i n gauss. m  3.  m  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 t o  peak i n the f i r s t d e r i v a t i v e a b s o r p t i o n curve.  The second moment  expression f o r the experimental a b s o r p t i o n 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] is M  2  = \lf  + a }  h (dg/dh)dh// °° 3  h(dg/dh)dh]-^H  +  — OO  —  2  CO  (4.1) where g(h) = g ( H  Q  - H)  =Y g( -w) _ 1  u  0  =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 o f 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 o f s e c t i o n s on the x - a x i s 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 f o u r 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  a d j u s t e d from the r e s u l t s based on T^ and T , to avoid s a t u r a t i o n . 2  In some cases where T-j and T measurements were not c a r r i e d out, 2  and where s a t u r a t i o n was predominant e.g. i n some pure amines a t 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 r e p o r t e d i n t h i s  region are b e l i e v e d to be a c c u r a t e to about ±20% w h i l e i n other cases the accuracy may be ±10%. The modulation amplitude and the modulation frequency were always kept much s m a l l e r 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 o f Provotorov's Theory [4.2].  - 63 -  4.  V a r i a b l e Temperature Assembly At temperatures o f 77K and 88K the s p e c t r a were recorded  by d i r e c t l y immersing the sample i n l i q u i d n i t r o g e n and l i q u i d oxygen respectively.  For temperatures from 100K and onwards the c o l d gas  flow methods were used.  The l i q u i d n i t r o g e n 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 n i t r o g e n 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 v o l t a g e 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 v o l t a g e on the heater immersed i n l i q u i d n i t r o g e n was held f i x e d at a value which g i v e s a good high flow r a t e o f 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 v o l t a g e on the o t h e r 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 p l a c e d 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 p l a c e d i n l i q u i d n i t r o g e n before reaching the dewar system.  With t h i s type o f system the sample  temperature i s held c o n s t a n t 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 o f the temperature between 77 and  100K depends on the amount o f l i q u i d n i t r o g e n used to cool the copper heat exchanger.  - 64  -  B.  S p i n - L a t t i c e R e l a x a t i o n 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 c o n t a i n s a b a s i c 1MHz quartz o s c i l l a t o r -8  with a frequency s t a b i l i t y l y i n g between 10  -9  - 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 o f 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 synthes i z i n g u n i t ) , where h a l f o f 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 f e d 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 r e f e r e n c e channel, the high frequency f e d serves as a phase coherent r e f e r e n c e 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  (single  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 t t e n u a t i o n 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 o f 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 o f 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 was V a r i a n Q  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 T e k t r o n i x 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 o f 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 of a H.P. 1  signal  generator, Model 606 A, o p e r a t i n g a t 25MHz. to the i n p u t of the r e c e i v e r . The output v o l t a g e from the r e c e i v e r was measured with T e k t r o n i x Type 549 o s c i l l o s c o p e .  Figure 4.1 shows the output v o l t a g e as a f u n c t i o n  of the i n p u t v o l t a g e from a H.P. s i g n a l generator.  The l i n e a r i t y  range d e p i c t e d 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 o f output v o l t a g e the diode shows n o n - l i n e a r i t y characteristics.  Most o f the measurements were taken i n the range  o f l i n e a r i t y ; those which were o u t s i d e 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 o f the diode d e t e c t o r .  3.  V a r i a b l e Temperature Assembly T h i s assembly d i f f e r s from the previous one i n t h 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 p a r t s o f Bruker temperature c o n t r o l u n i t B-ST 100/700. A Bruker q u a r t z made v a r i a b l e temperature probe was used i n some o f 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 c l o s e d i n a t r a n s p a r e n t l u c i t e box, to a v o i d condensation o f water vapour from the o u t s i d e atmosphere.  A cylinder-  i c a l brass l i d was placed on the top o f the q u a r t z probe which had an o u t l e t f o r the out going c o l d n i t r o g e n gas.  T h i s prevented the c o o l -  ing o f t h e " l u c i t e box as well as the condensation o f water vapour i n s i d e . To prevent f u r t h e r water vapour condensation, a stream o f dry and warm n i t r o g e n 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 t o 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 v a r y i n g the v o l t a g e 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 n e a r l y 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 P r o f e s s o r from Department o f P h y s i c s , 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 .  T h i s setup worked well up t o 11 OK.  L a t e r on, t o acheive a much lower temperautre, the whole q u a r t z 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 p r e 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 b e f o r e , temperature up t o 77 K c o u l d be o b t a i n e d . 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 a c c u r a t e t o ±0.5K. Temperature v a r i a t i o n i n the range 77 t o TOOK was acheived e i t h e r by changing the gas flow o r heating the f l o w i n g gas by another heater p r e s e n t i n Bruker dewar system.  In the higher range (100K and  onwards), e i t h e r the gas flow was changed (up t o a c e r t a i n l i m i t t o g i v e minimum temperature g r a d i e n t ) o r use was made o f the Bruker s e r v o - c o n t r o l system t o get the d e s i r e d temperature.  4.  Measurement o f S p i n - L a t t i c e R e l a x a t i o n 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 d i s c u s s e d 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 g i v e n by Equation (3.48). R(T)  =  [M -M (T)]/2M 0  Z  q  = exp(-x/T )  (4.3)  1  where M i s now p r o p o r t i o n a l t o v o l t a g e o f the s i g n a l a f t e r the 90° Q  p u l s e , and M ( x ) i s the v o l t a g e o f the s i g n a l a f t e r a 180°-x-90° p u l s e z  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 o f M ( x ) and a z  p l o t o f In R(x) vesus x should g i v e a s t r a i g h t l i n e with s l o p e (-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 t o g i v e a t a time  T  ,  M (T )  =  Q  z  0.  In t h a t case (Null Method)  Equation (4.3) gives R(T )  = 1/2  q  and T = T / l n 2 = 1.443T ]  Q  q  (4.4)  However, t h i s method i s not very a c c u r a t e , e s p e c i a l l y when the r f f i e l d H^ i s not homogeneous. inhomogeneity  A method f o r c o r r e c t i n g the H^  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 g i v e the method which i s a p p l i e d t o s o l i d s  where T  2  i s o f the order o f the 180° pulse l e n g t h , and s h o r t e r than T-j  According t o 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 ( 1 - A ) [1-2(1-2A) exp  M (T)  Q  X  (-T/^)]  (4.5)  where A=6M /M and 6M i s the decrease i n magnetization a f t e r 90° Q  0  Q  pulse due t o H^ inhomogeneity.  When we use the second method  (Null Method), we s e t T i n the 180°-T-90° sequence such t h a t M (T ) = 0 a t a time T=T . Equation (4.5) then gives e x p ( - T / T ) = 1/[2(1-2A)]  4.6  = T /ln(2-4A)  4.7  0  or  1  T  1  Q  The f a c t o r 4A was c a l c u l a t e d using 90°?and 270° p u l s e s .  For t h e  270° p u l s e , we used the same 180°-T-90° p u l s e sequence but x was  - 70 chosen i n such a way t h a t i t was g r e a t e r than T  (to n e g l e c t the  2  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 s h 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 t o 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 ) and the 180°-T-90° pulse gQ  sequence ( M M  )  2 7 0  90" 270 M  =  given by [4.3]  i s  o  M  ( 1  '  • 0-4A)(1-A) = 4 A ( 1 - A ) M  A )  m  and so d i v i d i n g t h i s by M (M  GO  -  M  2 7 Q  )/M  9 0  =  0  gQ  = M (1-A)  (4.8)  0  we get,  4A  (4.9)  This 4A when used i n Equation (4.7) gives T-j. x the graph o f In R(T) versus T.  can be found from  Q  [In t h i s p l o t R(T) i s s l i g h t l y p u t  M = M (1-A) and M (x) = M  i n a d i f f e r e n t way i . e . now  U  U  /  L  A  A computer programme was used t o c a l c u l a t e R(x) with (Appendix B) f o r d i f f e r e n t values o f x.  (x)]. .  corrections  I t should be noted t h a t  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 e x p r e s s i o n f o r T-j i . e .  <Vvan  =  VL" -l! "J ln2  A  (4  '  10)  where A" = 4A and (A"/2) « 1 T h i s 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. s t u d i e d here are unstable at room temperature.  A l l the hydrates  T h e r e f o r e 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 c o n s t r u c t e d from a food f r e e z e r (Zanussi Food 'Freezer Company, I t a l y ) which had a dimension o f 11 x 18 x 13 i n c h e s . 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 r e p l a c e d by a double w a l l e d t r a n s p a r e n t 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. T h i s 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.  T h i s 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.  T h i s -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 m o d i f i e d 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 o f the s i d e , and the o t h e r on top o f the other s i d e .  A stream of c o l d n i t r o g e n was blown from the bottom  hole by b o i l i n g l i q u i d n i t r o g e n 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 n e a r l y 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 , P r e s s . , 1970 p. 109 [4.3]  K. van Putte, J . Mag. Res., 2 (1970) 174  [4.4] A.S. Q u 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 d e s c r i b e s the d e t a i l s o f 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 -  g r a p h i c aspects o f these amine hydrates has been mentioned i n Chapter II.  Diethylamide forms two types o f 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 H ) NH-6.8H 0 2  5  2  2  which i s m o n o c l i n i c with space group P2-j/c, m e l t i n g 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 H ) NH-8.7H 0, which i s orthorhombic with space group Pbcn, and 2  5  2  2  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 r e p o r t e d i n d e t a i l by Jordan and Mak [5.2]. They [5.2] o  have shown t h a t 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 f o u r 18-hedra formed by 32 oxygen atoms with 48 hydrogen bonding edges. there are 8 i r r e g u l a r cages.  In a d d i t i o n to these 18-hedra,  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 n i t r o g e n atoms  - 74 hydrogen bonded t o water cages ( F i g u r e 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 t h e 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 o f -NH protons with D^O, t h e deuterate was prepared from ( f ^ H ^ N D .  We w i l l r e f e r t o the deuterate prepared from  CC' H I NH as DNH-D 0 and that from ( C H ) N D as DND-D 0. The reason 2  5  2  2  2  5  2  2  f o r studying DND-D 0 was t o check the c o n t r i b u t i o n o f the exchanged 2  protons t o t h e second moment. Tt w i l l be seen l a t e r t h a t t h i s exchange has an a p p a r e n t l y n e g l i g i b l e c o n t r i b u t i o n , because t h e exchange proton goes t o deuteeate cage, where because o f 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 d e u t e r a t e d , 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 0 (99.8% d ) was obtained from S t o h l e r Isotope Chemicals. The 2  2  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 s e v e r a l 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 d i e t h y l a m i d e had been i n i t i a l l y d i s t i l l e d t h r e e times from BaO.  2.  Preparation o f (C H ) ND 2  5  2  T h i s method i s e s s e n t i a l l y the same as used by Ross e t a l . [5.3]  - 75 which i s based on the method o f Hawthorne [5.4]. 50 ml o f amine was added to 20 ml o f 99.8% D 0 to which p r e v i o u s l y had been added 1 g o f 2  P c 0 o. o  The s o l u t i o n was r e f l u x e d f o r about 3 hours.  c  The amine was  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 d r y i n g procedures were repeated t h r e e 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 g e t a constant b.p.  The a n a l y s i s o f deuterate on Varian T-60 high r e s o l u t i o n nmr  spectrometer showed t h a t the -NH group was more than 90% deuterated t o -ND. 3.  P r e p a r a t i o n 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 d i e t h y l a m i d e by s e a l i n g the type II formula r a t i o o f (CgHg^NH and D^O i n a tube and keeping the mixture a t -15°C i n the cold box. The deuterate c r y s t a l s grow very s l o w l y . 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 o f the deuterate (DNH-D^O) was removed from the tube and f i n e l y powdered a t -15°C before being t r a n s f e r r e d t o 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 o f 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 t o nmr tubes d i r e c t l y and was s e a l e d 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 o f 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 d e s c r i b e d i n Chapter IV. The maximum r f f i e l d i n the a b s o r p t i o n measurements was 40 mG.  The recovery o f  magnetization was obtained by using a 180°-x-90° pulse sequence with a 180° pulse length o f 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 t h a t 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 , the Q  time taken f o r R(t) to become equal to 1/2 was used i n s t e a d o f 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 t o  M = 716.164 N 2  + 2.216 N  + 9.994 N C5.1)  - 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 d i s t a n c e between proton i and j , r ^ i s the d i s t a n c e n  between n i t r o g e n 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  i n t e r m o l e c u l a r M^.  and  Because the c l o s e s t approach o f 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 instance.  On the other hand the c r y s t a l s t r u c t u r e o f 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 o f M£ i n the r i g i d  lattice  i s obtained by using the approximate expression given by Smith [5.6] v i z . M"  =  358.1  x  ^  L  (5.2)  \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 c o n t a i n i n g 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 m a t e r i a l to give „  M"  =  358.1  x  %  n • -E-  N  o  ^ 10 ^n M  (5 ) < 3  2 4  1  1  where n. i s the number of molecules o f type i i n the u n i t c e l l M. i s t h e i r gram molecular weight.  N  i s Avogadro's number  and  In the  - 78 s i t u a t i o n where only one type o f molecule o c c u r s , 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 n f o r m a t i o n i s r e q u i r e d , hence the a t t r a c t i o n o f t h i s method f o r e s t i m a t i o n s 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 o  12,  p  = 1.11 g cm"  M£ = 0.46  and R was taken t o be  D  2  Q = 104, "^i ethyl amine  =  o  ^4.4 A.  Such values g i v e  G. 2  For pure d i e t h y l amine-ND on the other hand a value of ~ 0 . 8 g cm 2  _3  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 o f 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 r e d u c t i o n 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 o f e s t i m a t i o n of i t s magnitude are r a t h e r 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 o f 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 o f Jordan and Mak suggest t h a t the i n t e r n u c l e a r d i s t a n c e s 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 d i s t a n c e s on opposite s i d e s 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 t o use the f o l l o w i n g mean i n t e r n u c l e a r d i s t a n c e s 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 d i s t a n c e s of Jordan and Mak [5.2,5.7] were s u b j e c t to even more v a r i a t i o n than the d i s t a n c e s 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 d i s t a n c e s o f 1.09 A, 1.10 A, 1.12 A and 0  1.13 A.  Using these data,  the coordinates o f v a r i o u s atoms were  generated according to method o f 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 The r e s u l t s o f the M« c a l c u l a t i o n s are summarized i n Table Table N  5.1.  Intramolecular Second Moment M£ f o r r  C-H  M (rigid)  A  G  0  2  2  C).  5.1.  DNH-D 0 and DND-D 0 2  M  2  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 n i t r o g e n 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 t h a t atom was  - 80 a proton and secondly assuming t h a t proton-deuteron exchange had taken place and t h a t i t was a deuteron.  Such an exchange i s probably  most e f f i c i e n t during the p r e p a r a t i v e 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 o f second moment Mg c a l c u l a t i o n s f o r N = 10 and r _ = 1.13 A are enumerated c  H  in Table 5.2.  (b\  Experimental Second Moment Data  The temperature dependence o f experimental second moment i s given i n Figure 5.1. A plateau value o f (9.80 + 0.05)G  exists i n  the deuterate from about 110 K t o t h e decomposition temperature a t about 265 K. Below 110 K the second moment i n c r e a s e s with decreasing temperature, but f a i l s t o e s t a b l i s h a new p l a t e a u value above 77 K. The experimental value a t 77 K i s 13.63 +0.82 G f o r DNH-D 0 and 2  2  14.19 +0.96 G f o r DND-DgO. The data from DND-DgO and DNH-DgO 2  agree w i t h i n the accuracy o f the experiment, which may j u s t be great enough t o allow such agreement t o lend support t o t h e view t h a t the amine proton has been c h e m i c a l l y exchanged and does not c o n t r i b u t e to the proton second moment. Comparison  o f the experimental data  with Table 5.1 and 5.2 shows t h a t the magnitude o f plateau value i s c o n s i s t e n t with the mean carbon proton s e p a r a t i o n o f 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 o f CH^group i n the deuterate. 2 The plateau o f —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 o f Cf-L-groups.  CM  if) CO 3  o  CD  c  o Diethylamine-ND . Diethylamine-ND- Deuterate Diethylamine—NH-Deuterate  20-  16i CO  12-  Q)  E  o  8-  "O C O  4-  E  o  if)  o-  i  1  1  100  1  '  v  1  1  1  r 150  T  1  1  r  200  i  1  r  1  Temperature (° K) Figure 5.1 Second moment versus temperature f o r diethylamide deuterate and diethylamine-ND.  1  250  r  - 82 -  Table 5.2. Second Moment f o r Diethyl amine-ND Deuterate CDND-DgO) o  using r _ „ = 1.13 A and N = 10 r  Rigid Lattice G  Contribution  2  1.  2 x C H group  2.  2 x CH  3.  10.93  3  2 x CH., r o t a t i n g 2 or  2.73  3.64  3.64  CH^-CH^  0.03  0.03  4.  CH -CH  2.64  2.27  5.  ( i ) CH -D 3  0.02  0.02  CH -D  0.03  0.03  0.47  0.47  3  2  2  •kick  Oi)  2  *  **  6.  CH ~CH  7.  Nitrogen-protons  0.01  0.01  8.  Intermolecular diethylamide  0.46  <0.46  0.10  <0.10  18.33  <9.73  2  2  9. Intermolecular deuterons Total  **  M = 2  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 t o 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 o f Non-Exponential  Relaxation  The r e l a x a t i o n f u n c t i o n ^ C t ) based on H-H theory [5.5] f o r a R  v  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.  2  o  Some t h e o r e t i c a l curves f o r R ( t ) c a l c u l a t e d by using Simpson's A y  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 t a b l e s 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) c o n s t r u c t e d from the t a b l e s A l - A l 5 (Appendix A ) , t h e o r e t i c a l n u l l p o i n t s i . e . , t / T ' when R ( t ) = / \ ( t ) p R  Q  A y  v  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  value o f ( W T ) . The value o f ( t / T ' ) corresponding t o a p a r t i c u l a r 0  u  o c T  a r e  t h e n  Q  c  Pl°  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 u r = 0.68 from H-H theory. The Q  c  experimental data can now be analysed i n terms o f theory. (b)  Experimental Relaxation Data  The temperature dependence o f the n u c l e a r r e l a x a t i o n o f two s e p a r a t e l y prepared samples o f DNH-D 0 and o f pure diethylamine-ND 2  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 , r a t h e r than Q  T-|.  The experimental minimum f o r DNH-D 0 occurs a t 129 K and i s 2  31 msec, while t h a t o f diethylamine-ND i s 18.5 msec a t 145 K. The  k  R  Av"  2  100-  00  -pi  10.0-  0.01. High Temperatures Figure 5.2  Dependence o f t /T on. y , , as p r e d i c t e d by H-H theory Q  10.0  L o w Temperatures  2000-  « Diethylamine- N H - Deuterate o Diethylamine-ND  lOOO800 600 400 (A TD  C 200 O O 0)  (/)  !=  CO  TOO  so +°  60  M.R (Diethylamine-ND)  40H  2CH 10  —1  80  1——|  IOO  1  1  1-  120  'n  140  1  1  160  1  1  180  1  1  200  1  1  220  1  1  1  240  Temperature [°K] Figure 5.3 Temperature dependence of t i n diethylamine deuterate and diethylamine-ND.  - 86 t h e o r e t i c a l minimum from previous s e c t i o n occurs a t t /T' = 1.52 at  (JJ T 0  c  = 0.68. Tn order t o compare experiment with theory f o r  DNH-DgO, one has t o c a l c u l a t e 1/T*. Using t = 31 msec and t / T ' = Q  Q  1.52, we g e t 1/T' = 49.03 s e c " . With t h i s value o f 1/T' =49.03 s e c " , 1  1  some experimental curves f o r &nR(t) E £ n R ( t ) a r e compared with the Ay  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 " f o r diethylamine-ND, a comparison o f theory with experiment 1  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 d e u t e r a t e . Since the second moment data r e p o r t e d above i n d i c a t e t h a t t h e only motion o c c u r r i n g a t a r a t e g r e a t e r 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 t h a t the minimum i n each case of the curves o f Figure 5.3 i s a l s o due t o 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 e x t r a c t e d i n t h e f o l l o w i n g way. From t h e value o f 1/T' f o r DNH-D 0 and diethylamine-ND ( c a l c u l a t e d above) and value 2  of t from experiment corresponding t o an observed temperature, t / T ' Q  was c a l c u l a t e d f o r DNH-DgO and diethylamine-ND.  Corresponding t o  each to'/T*, tooxc was obtained from F i g u r e 5.2. Thus w o xc a t an observed temperature i s found. The temperature dependence o f a ) 3  Q  x  c  f o r C H g - r e o r i e n t a t i o n can now be f i t t e d t o an Arrhenius equation [ Vc =o x . exp(E a /RT) c f . Equation (3.26)] i f the r e o r i e n t a t i o n i s t h e r m a l l y a c t i v a t e d and obeys t h e Arrhenius e q u a t i o n . The temperature dependence o f  U  Q  T  C  i s shown i n F i g u r e 5.5. I t i s seen t h i s  dependence  f i t s n i c e l y t o the A r r h e n i u s e q u a t i o n , and the a c t i v a t i o n energies  oo —I  8 . 12 16 20 Theoretical and experimental curves for non-exponential relaxation function for different ( w O in diethylamine deuterate (curve 1 to 4 ) and diethylamine-ND (curve 7 to 8 ) theoretical experimental 4  Figure 5 . 4 .  0  - 88 lO.CK 8 . 0  :  1  I _j  3.0  4.0  5.0  1—  1  6.0  TO  10 /T(K" ) 3  1  1  8.0  9.0  i  10.0  1  Figure 5.5 p l o t o f u r 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. Q  c  - 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 d i e t h y l a m i n e ND.  The pre-exponential f a c t o r s x f o r d i e t h y l amine deuterate (DNH-DgO) Q  and pure diethylamine-ND are [4.5 +0.3] 10"  x 10"  sec and (1.6 +0.1)  13  x  sec, r e s p e c t i v e l y . The depth o f the minimum which provides i n f o r m a t i o n about the 2  s t r e n g t h o f 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  3  = ( fi/r ) Y  2 0/u> ) 0  cf.  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 o f deuterate to check the c r e d i b i l i t y o f the i n t e r p r o t o n d i s t a n c e chosen to g i v e agreement with the experimental second moment data.  In the  d e u t e r a t e , i t i s reasonable to assume t h a t 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 r e l a x e d 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 a n g l e s , 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 o t h e r hand one  assumes t h a t the amine proton may have been exchanged with a deuteron, the computed minimum value o f 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 o f the minimum value of t in 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 c h a r a c t e r o f the n u c l e a r r e l a x a t i o n , are i m p o s s i b l e 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 s e v e r a l  c o n c l u s i o n s r e g a r d i n g 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 d i e t h y l a m i n e . The second moment data reveal t h a t i n both environments 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 at a r a t e i n excess o f 10 kHz well below the m e l t i n g p o i n t .  the groups 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 n u c l e a r 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 a t 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 d e u t e r a t e , indeed they enable one to e s t a b l i s h a 24% i n c r e a s e i n the height of the h i n d e r i n g b a r r i e r i n going from the deuterate to pure d i e t h y l a m i n e .  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 quantitative result further. In the deuterate, both 1he l i n e - s h a p e 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 u n i f o r m i t y among the methyl groups which may not have been expected from the x-ray data o f Jordan and Mak [5.2].  I t i s not p o s s i b l e  to d e t e c t 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 o f 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 p o i n t which i s extremely u n c e r t a i n 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 n u c l e a r 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 o 31 —+ 1 msec are c o n s i s t e n t with a mean C-H bond length • Q . of 1.13 A, i f one i s prepared to assume a l a r g e degree o f exchange between the amine proton and  the  D 0. 9  - 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, 0 9 6 7 ) 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. P e t e r s e n , J . Am. Chem. S o c , 81_ 0 9 5 9 ) 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.  Introduction The c r y s t a l l i n e hydrate of acetone was f i r s t prepared  successfully  by Quist and Frank [6.1] who f r o z e 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 hydrate was von Stackelberg's  1 mm i n edge l e n g t h ) , t h a t the  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 C HgO -17H 0 and that the cubic u n i t c e l l edge 3  2  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  further  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 t h a t 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 t h a t the cw nmr r e s u l t s suggested the existence of only C H ^ - r e o r i e n t a t i o n  at  - 94 low temperatures.  The o b j e c t o f 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 s e p a r a t i o n r a t h e r than on the c l a t h r a t e hydrate. The data r e p o r t e d here are i n f a c t on acetone d e u t e r a t e , i n order to ensure t h a t 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 essentially intramolecular.  B.  Experimental 1.  P r e p a r a t i o n 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 i s o t o p e chemicals).  T h i s 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 m e l t i n g i t was placed i n s i d e the innermost space of 3-walled vessel (having two a i r spaces, one o f which was evacuated). was placed i n a bath held between -40 and -50°C.  This vessel  The bath was  prepared by mixing a p p r o p r i a t e 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 o f 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 m e l t i n g 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.  T h i s removal process was one o f the most c r i t i c a l o f the  p r e p a r a t i o n 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 g r e a t e r 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 HgO-170 0)were used f o r n u c l e a r 3  2  resonance work. They were f i l l e d i n 10 mm and 7 mm o.d. sample tubes and s e a l e d a f t e r e v a c u a t i n g .  2.  P r e p a r a t i o n 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 s e a l e d a f t e r removal o f the d i s s o l v e d oxygen by a freeze-pumpthaw method.  3.  Spectrometers and Methods o f Measurement  They were the same as d e s c r i b e d i n Chapter IV. length t h i s time was about 3.0  The 180° pulse  usee, i n the case o f acetone and  3.5 psec i n the case o f d e u t e r a t e . The r e l a x a t i o n f u n c t i o n R(t) in both cases was non-exponential, so t , the time taken by R(t) Q  to become equal to R ( t ) = 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 0  i n s t e a d of T-j. The minima were not observed i n the temperature dependence o f t  i n both the cases. T h e r e f o r e the data are not  fitted  - 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 g r a d i e n t of t  C.  versus r e c i p r o c a l  temperature.  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 o f constants i n Equation (3.19) becomes  M  0  =  716.164 N" V] 1  r~. . + 9.994 N 6  - 1  ^  r~. ,  (6.1)  6  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 d i s t a n c e 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 s e v e r a l 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 d e r i v e d 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 t h a t 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 in 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 i n f o r m a t i o n about the  arrangement  - 97 Table 6.1. Proton Second Moment i n Acetone Deuterate Intramolecular CH -CH CH  Type o f Motion  3  3  Rigid 2CH  3  23.77  0.98  5.94  0.84 0.21  rotating  2CH r o t a t i n g + C r o t a t i o n about >C=0 bond 3  Intermolecular  Total  3  0.26  a  b  0.20  a  d  0.16  25.01 6.98  2  Isotropic  1.98  C  0  0  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]  2.35 a  0.15  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 t o be ^ 5 G g i v i n g  2 r i s e t o t o t a l r i g i d l a t t i c e second moment equal t o 24.8 + 5.0 = 29.8 G . In case o f both methyl groups r o t a t i n g the t o t a l second moment 2 based on the same r e f e r e n c e [6.13] comes out t o be ^ 8.0 G . The experimental value o f the proton second momend i n acetone 2 deuterate i s 0.69 +_ 0.07 G a t 77 K. I t decreases s t e a d i l y (Figure 6.1) t o 0.18 +0.01 G a t 212 K. T h i s l a t t e r value i s 2  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 o f the acetone molecule i n s i d e the 16-hedron c a v i t i e s . i n Figure 6.2.  Some s p e c t r a o f acetone-deuterate are shown  o Acetone  i  Acetone Deuterate  -i——i  60  1  80  r  100  —I  120  1  — |  140  1  1  160  r  I  180  1  1  200  1  1  220  1  1  V a r i a t i o n o f second moment w i t h temperature i n acetone and acetone  1  240 260  Temperature [°K ] F i g u r e 6.1  1  deuterate.  Temperature 77 K M = 0.66 G 6H = 1,24 G 2  2  (D  Temperature 180 K M = 0.41 G 6H = 1.11 G 2  2  (2)  Temperature 202 K M = 0.32 G 6H = 1.23 G 2  2  (3)  M i s the second moment and SH i s the l i n e width. 2  i -3 Figure 6.2  i - 2 - 1  i  i———i O  GAUSS  1  1 2  1  3  Some proton magnetic resonance s p e c t r a o f acetone deuterate at d i f f e r e n t temperatures.  - 100 The behaviour o f pure acetone i s i l l u s t r a t e d i n the work o f Gutowsky and Pake [6.12]. These authors [6.12] have shown the temperature dependence o f the l i n e width. with d e u t e r a t e  We have f o r the sake o f comparison  measured the second moment o f pure acetone a t  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 o f second moment shows a p l a t e a u value o f 9.15 +0.54 G a t 77 K t o 9.81 +0.71 2  G at 171 K. The c o r r e s p o n d i n g l i n e width i s 5.84 +0.22 G a t 77 K t o 2  5.12 +0.62 G a t 171 K and i s c o n s i s t e n t with the work o f 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 t o 77 K and no o t h e r motion i s present from 77 t o 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 o f equipment l i m i t a t i o n s , the minimum i n t versus temperature was not Q  reached.  T h e r e f o r e the data cannot be f i t t e d t o H-H t h e o r y , and  consequently no i n f o r m a t i o n 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 e x t r a c t e d from the g r a d i e n t s o f t  Q  versus i n v e r s e temperature (Figure 6.4) from t h e high temperature  s i d e o f 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 o f acetone-deuterate, which cannot be assigned unambiguously f o r reasons t o be given l a t e r .  On the o t h e r  hand, pure acetone d i s p l a y s an a c t i v a t i o n energy o f 1.33 +0.01 kcal/mole, which i s doubtless due t o hindered r o t a t i o n o f i t s C+L groups.  6000  40-|  1  1  1  100  1  1  1  1  |  150  "  r  1  1  1  200  i  I  '  I  T~  250  Temperature [°K]  Figure 6.3  Temperature dependence of t i n acetone and acetone deuterate.  13.0  120  1I.O  100  9.0  80  1000/T  Figure 6 . 4  7.0  6.0  .  5.0  4.0  3.0  [K ] _ I  t versus r e c i p r o c a l of the absolute temperature f o r acetone and acetone-deuterate. Q  - 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 r a t e f a s t enough to a f f e c t the n u c l e a r resonance observables i s C H - r e o r i e n t a t i o n . 3  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 n u c l e a r resonance data. The gas phase microwave work o f Swallen and C o s t a i n [6.14] and more r e f i n e d work o f Nelson and P i e r c e [6.8] give the b a r r i e r h i n d e r i n g C ^ - r o t a t i o n o f CH group i n 3  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 t o 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 a t 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 o f r o t a t i o n  were t r i e d i n the same view as work o f Powles and Gutowsky [6.11] ( c f . Equation (3.32)) but a l l o f them g i v e second moments g r 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 p l a t e a u value but p a r t of the decrease i n the temperature dependence o f second moment to p l a t e a u 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 o f t i n acetone-deuterate below 120 K (Figure 6.3 and 6.4) f o l l o w s 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 t o 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 r e g i o n i s s t i l l above the p l a t e a u value reorientation.  consistent with isotropic  To which o f the o t h e r p o s s i b l e motions i t corresponds  cannot be determined unambiguously without a low temperature p l a t e a u 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 t o the acetone molecule and they found a maximum a c t i v a t i o n energy o f 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 o f the order o f 4.3 x 10  sec.  They a l s o  found t h a t at 93 K t h e i r data f i t t e d a Fuoss and Kirkwood  distribution  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 , S e c t i o n E ] . They concluded t h a t the e v a l u a t i o n o f 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 o f inhomogeneous sample used, and a l s o because only three f r e q u e n c i e s 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 o f 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.  If this distribution  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 b a s i s 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 t a k i n g 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 t h a t 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 t r e n d .  - 105 can be about 0.3 kcal/mole i f the sample i s homogeneous. Our method o f p r e p a r a t i o n i s a b e t t e r method i n regard t o 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 w i t h t h a t o f Davies and W i l l i a m s [ 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 , c o u l d w e l l be due t o 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 o b t a i n e d from the s l o p e i n the temperature r e g i o n o f 230 to 255 K<  Due to very poor s i g n a l to n o i s e r a t i o from ~170  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 .  K  - 106 References (Chapter VI) [6.1]  A.S. Q u i s t 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. C h e k a l i n , J . S t r u c . Chem.,11 (1970) 560 t r a n s l a t e d from Russian Zhurnal S t r u k t u r n o i K h i m i i , 1 1 (1970) 599.  [6.5]  M. Davies and K. W i 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 C o r p o r a t i o n , 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 C o s t a i n , J . Chem. Phys., 31_ (1959) 1562.  - 107 -  CHAPTER VII TERTIARY BUTYLAMINE AND TERTIARY BUTYLAMINE DEUTERATE A.  Introduction t e r t - B u t y l a m i n e 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 o f t h i s hydrate has been s t u d i e d by x-ray d i f f r a c t i o n by McMullan e t a l . [7.1]. A b r i e f account o f the c r y s t a l s t r u c t u r e was presented i n Chapter I I . T h i s 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 o f 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 d e s c r i b e d by the r e l a t i o n ( c f . Chap. II) 17F + 30V = 45E + 2 ( F i g u r e 2.4).  The  o  vertex to centre d i s t a n c e s 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 o r d i n a r y state.  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. i s b e l i e v e d t h a t the -NH  Since D 0 i s always i n excess, i t 2  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 n e a r l y n e g l i g i b l e . Such behaviour we n o t i c e d i n the case o f d i e t h y l amine-ND-deuterate. Therefore we d i d not study ( C H ^ C N D ^ - d e u t e r a t e  s e p a r a t e l y here,  but r e s t r i c t e d o u r s e l v e s to check the c o n s i s t e n c y o f our r e s u l t s with d i f f e r e n t batches of samples.  L a t e r 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 t h a t the amine molecule enjoys a c o n s i d e r a b l e 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 o f 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 rotation.  We do not have any i n f o r m a t i o n about the motion o f  -NH  2  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 . -NH  2  I t means t h a t the  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 t e r t - b u t y l a m i n e was obtained from Eastman Kodak  and D 0 (99.8% d ) was provided by S t o h l e r Isotope Chemicals. 2  2  amine was d r i e d on KOH f o r s e v e r a l days.  The  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.  P r e p a r a t i o n ' o f (CH,) CND 3  2  The p r e p a r a t i o n i s s i m i l a r to t h a t d e s c r i b e d i n Chapter V.  The  a n a l y s i s o f 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 t h a t the -NH protons were about 2  90% deuterated to ND  3.  2<  P r e p a r a t i o n o f Deuterate and Amine Samples  Three d i f f e r e n t batches o f deuterates were prepared. batch was from s t r a i g h t amine sample.  The i n i t i a l  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 t h r e e batches the formula r a t i o (16 amine:156 D^O) o f amine and D 0 was mixed i n a t i g h t l y c l o s e d v e s s e l and the s o l u t i o n 2  was cooled s l o w l y i n the cold box. C o o l i n g o f s o l u t i o n t o about -2°C as carried out by McMullan e t 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. at c o l d  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 a t about -2°C and box temperature (-15°C) were n e a r l y 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 o f samples were t h e r e f o r e prepared a t c o l d box temperature.  The deuterate thus obtained was  f i n e l y crushed t o powder i n a p r e - c o o l e d mortar i n the c o l d box and f i l l e d i n nmr tubes.  The nmr tubes were s e a l e d a f t e r degassing.  The a n a l y s i s o f 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 o f 89-96% guest p u r i t y i n t h e d e u t e r a t e . The samples l e f t a t -15°C i n c o l d box probably had a tendency t o decompose when l e f t f o r a few days and gave lower values o f T-| when s t u d i e d a f t e r some days.  Therefore  e i t h e r f r e s h samples were used o r samples were kept a t l i q u i d n i t r o g e n temperature.  A l l o f these samples showed a small t r a c e o f l i q u i d  peak i n the cw s p e c t r a near the m e l t i n g p o i n t o f guest i n d i c a t i n g e i t h e r  - no  -  t r a c e o f amine l e f t unreacted o r 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 o f 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 11.1 mole %.  i.e.,  However t h e r e was' no i n d i c a t i o n o f l i q u i d peak i n pulsed  nmr measurements. T h i s 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 t o second moment. The pure amine (CH-^CNDg  s a m  pi.  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. T h i s sample a l s o showed a narrow l i q u i d peak before the m e l t i n g p o i n t .  4.  Spectrometers and Methods o f Measurements  These equipment were the same as d i s c u s s e d before.  The maximum  r f f i e l d i n the case o f cw measurements was 40 mG f o r the deuterate. In the case o f the pure amine an r f f i e l d o f 0.5 mG was used from 77 t o about 110 K.  T-j measurements were performed by using a 180°  pulse length o f 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 was obtained from the q  p l o t o f &nR(x) versus x. d e s c r i b e d i n Chapter IV.  The o t h e r methods were the same as  - in C.  Results 1. (a)  Absorption Line A n a l y s i s 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). c a l c u l a t i o n s were performed using two sets o f molecular  These  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 C-C = 1.42 A.^NCC = 112°, z_CCC = 106.8°.  In  A,  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 o f t e r t - b u t y l a m i n e i s not known. T h i s second s e t 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 : o  C-N = 1.47 A, C-C = 1.54 A, N-H =  o  1.02 A, and C-H = 1.09 A;' A l l the angles used t o 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 o f 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 M , performed i n a 2  s i m i l a r way as d e s c r i b e d 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  in  the case o f deuterate by an expression given by Smith [7.5], i . e . , M" = 358.1 N £ N. f ( h ) RT (7.1) i=l D  c  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 t e r t - B u t y l a m i n e Mi using 1 s t s e t [7.1] .2  Type o f Motion  2  MA using 2nd s e t [7.3-7.4] 1  G  2  1. R i g i d 22.53  22.53  CH^-CH^  5.11  2.83  CH -D  0.03  0.02  0.00  0.00  27.67  25.38  5.63  5.63  3  4.39  2.43  CH -D *  0.03  0.02  10.05  8.08  3ChL  3  2  N-CHTotal 2. 3CH Rotating 3  3CH  3  CH ~CH 3  3  2  Total  3. 3CH + t e r t - b u t y l group r o t a t i n g 3  3CH3" 3 CH -D *  C H  C H  3  2  Total  0.47  0.62  1.10  0.61  0.02  0.01  1.59  1.24  Deuteron attached t o n i t r o g e n i n -ND  2  group,  - 113 f(h)  =  [(l-h ) + (5/3)h ]/[(l-4h ) ] 2  2  4  2  (7.2)  3  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 small. p  However i f M£ i s c a l c u l a t e d using Equation (5.3) with  = 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 b a s i s of previous results.  On t a k i n g 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  is ejected.  The c a l c u l a t i o n o f 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 t o 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 o f 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. T h i s value when used i n Equation (5.3) with R = 2.2 A 2 gives M£ = 7.93 G . T h i s value makes the t o t a l r i g i d l a t t i c e 2 second moment M 33 G , a value c o n s i d e r a b l y higher than the 2 p  2  rigid lattice M  2  value f o r s i m i l a r compounds i e . , about 30 G  [7.7].  The o n l y broad l i n e nmr work on amines from which we can compare our r e s u l t s i s t h a t o f Kromhout and Moulton [7.8] and o f 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 t e r t - b u t y l a m i n e 2 give M - 30 G , we have t h e r e f o r e taken the experimental value of 7  - 114 M  2  = 31 G  as the r i g i d l a t t i c e second moment f o r ( C H ) C N D . 3  3  Using  2  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 reasonable value comparable t o the  2  comes out to b e ~ 6 G , a  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 o f 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  so  can be c a l c u l a t e d  from the e x p r e s s i o n [7.5] (M ) 2  i s Q  = 358.1 N L  N.R"  (7.3)  b  where a l l the symbols have the same meaning as d e f i n e d i n Equation (7.1).  The value o f  f o r the d e u t e r a t e i s obtained i n the same way 2 as b e f o r e . The r e s u l t s o f these c a l c u l a t i o n s g i v e ( M ) . j = 0.02 G which a f t e r t a k i n g i n t o account the c o n t r i b u t i o n from deuterons and 2 2  from exchanged protons can be i n the range o f 0.10-0.15 G most.  so  at the  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 a g a i n s t 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 t o the r i g i d l a t t i c e v a l u e . The second moment value 2 approaches a p l a t e a u value o f 2.2 G with the t h r e e methyl groups  around 150 K which i s c o n s i s t e n t  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 a t 77 K i s 19.37 + 0 . 2 8 G and i t approaches a p l a t e a u value o f 3.7 G at about 120 K. are shown i n Figure 7.4a.  Some s p e c t r a o f ( C H ) C N D 3  3  2  1 0 0  1 5 0  Z O O  Temperature [°K]  2 5 0  20  (0 1 6 =3  O  • tert-Butylamine —ND2 o tert-Butylamine Deuterate  CD w  1 2 -  +-  5>  8  CD C J  4  .  O  -l 100  1  r—1  1  r— 150  T  1  r  i  200  1  r  250  Temperature [°K] Figure 7.2  Proton magnetic resonance l i n e width vs. temperature f o r tert-butylamine-ND and t e r t - b u t y l a m i n e deuterate. 2  1  - 117 Table 7.2  Second Moment Values f o r t e r t - B u t y l a m i n e Deuterate and t e r t - B u t y l a m i ne-ND^  Type o f Motion  Deuterate M^ using 1 s t M using 2nd set [7.1] s e t [7.3,7.4] G G 2  2  2  (CH ) CND M using 2nd s e t [7.3,7.4] G 3  3  2  2  2  1. R i g i d  28.06  25.77  31+2  2. 3CH r o t a t i n g  10.35  8.38  9+1  1.79  1.44  2+0.5  0.10-0.15  0.10-0.15  3  3. 3CH + t e r t - b u t y l group r o t a t i n g 3  4. I s o t r o p i c  The experimental value o f 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  plus the t e r t - b u t y l group r o t a t i n g , t o give a plateau value o f 0.2 G 2  2  around 250 K. This l a s t p l a t e a u value o f 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 o f whole molecule i n s i d e the 17-hedron.  Some  s p e c t r a o f deuterate are shown i n Figure 7.4b. The r e s u l t s o f the second moment data on the deuterate suggest 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 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 d i s c u s s e d i n Chapter I I I , the magnitude o f 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 e x t r a c t e d from the Equation (3.25).  For ( C H ) C N D , we took C = 20 G and B = 3.7 G 3  3  2  tert-Butylamine  Curve 1 M = 30.81 G 6H = 19.21 G Temperature 77 K  Deuterate-ND  2  6H = 3.53 G Temperature = 151 K oo  tert-Butylamine  Curve 1 M = 19.77 G 6H = 17.46 G Temperature 77 K  Deuterate  2  Figure 7.3  6H = 1.84 G Temperature = 126 K  53 G" 47 G 185 K Some H resonance absorption s p e c t r a o f tert-butylamine-ND and t e r t - b u t y l a m i n e d e u t e r a t e . 2  - 119 a n d 6 H , the l i n e - w i d t h values i n the narrowing r e g i o n ( F i g u r e 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 T  c  = 3.19 + 0.26 kcal/mole with a — -4 -7 ranging from 1.1 x 10 sec t o 1.7 x 10 sec i n the r e g i o n o f  t r a n s i t i o n i . e . , from 77 to 114 K. T h i s 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 h i n d e r i n g the methyl as w e l l 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,a i n v o l v e d i n the second t r a n s i t i o n i . e . , from t h r e e methyl plus 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 o f 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 o f 2.14 +0.21 kcal/mole with x  v a r y i n g from 6.2 x 1 0 " t o 5.6 x I O " 4  c  5  i n the range o f 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 a c c u r a t e , 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 d e r i v e d from the T-j measurements. 2. T-j Measurements The temperature dependence o f T-| f o r both compounds i s shown i n Figure 7.4. The data f o r ( C H ^ C N D g 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 d e t e c t a b l e f o r these two motion, there i s a d i s b r i b u t i o n o f 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 o f 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 d e u t e r a t e where the minimum i s much broader than ( C H ^ C N D g minimum ( F i g u r e 7.4). The reason f o r t h i s i s t h a t i n t h i s case, the minimum of about 32 +_ 1 msec a r i s e s from three types o f 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 t o 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/TJ = L (1/T,,) i=l 1  (7.5)  11  where VT-| • i s given by Equation (3.55) i . e . , . 1 + A V c2. 2i  Vci  1 + V 2c i2  li  T  ci  =  T  oi P( e x  E  (?.6)  0 cl  4  / )  <'>  R T  a i  7  7  Equation (7.5) r e q u i r e s the knowledge o f i n d i v i d u a l T-^ which i n turn need a knowledge o f C., T . 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 o f t h i s type seldom q 1  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 e x p e r i m e n t a l l y observed minimum i n the p l o t o f T^ a g a i n s t temperature, ( p r o v i d e d minimum due t o t h a t motion i s observed).  At the minimum value o f T^. we have u ) x . = 0.616 and Q  then Equation (7.6) gives  (  1 / T  li>min  =  (V%)0.425)  (7.8)  Thus C . can be e x t r a c t e d . E . can be found from the s l o p e o f 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( ai E  (7.9)  / R T )  In Equation (7.9) E . and the temperature T a t minimum are known, an a)  o oi T  c a n  b e  ca  ^ ^ cu  ateci  '  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., i ., l E a.i t o g e t a best f i t t o Equation (7.5). The T  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 o n l y work reported so f a r i s t h a t o f A l l e n and Johnson  [7.10]. In the present case o f (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 o n l y one broad s i n g l e minimum i s observed as both motions seem t o o v e r l a p . An estimate o f t h e r e f o r e very r e l i a b l e .  from experiment i s not  T h i s broad minimum f u r t h e r a f f e c t s the  e v a l u a t i o n o f E . from the s l o p e 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 t h a t  on the low temperature s i d e o f 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 o f 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 cl T  (7.10) _  0  C2  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 n e g l e c t e d . Equation (7.10) and (7.7) then become  - 123 log  or  log T  = 21og  q 1  + log  %  Xq1  + ^  E , R T  - l o g 2C  ]  = ( l o g T + l o g 2C-,) - (21og u> + 3 Q ]  0  2 >  3  r t  )  (7.11)  Equation (7.11) was used t o estimate the value o f T - J . C-| was estimated from the experimental second moment v a l u e .  C-| i s r e l a t e d to  *  second moment M by the f o l l o w i n g r e l a t i o n 2  Using. M = 31 G , & = 2-n x 26.46 x 1 0 s e c " , T = 1.25 sec a t T = 112 K 2  2  6  1  Q  ]  with E.^ from slope o f £nT-| versus 1/T curve ( F i g u r e 7.5) which i s -13 equal t o 3.18 kcal/mole, Equation (7.11) gives T - J = 8.5 x 10 sec. -13 The same equation with T-j data a t 116 K gives x -| = 7.7 x 10 0  sec,  which means there may be some experimental e r r o r i n T-j and/or the temperature T, o r t h i s i n d i c a t i o n t h a t 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 s i d e o f minimum i n the £ n T versus 1/T p l o t i s 3.18 +_ 0.07 kcal/mole and 1  agrees with the value obtained from l i n e width data namely, 3.19 + 0.26 kcal/mole. The e s t i m a t i o n o f 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 o f second moment should be used and t h i s r e l a t i o n i s t r u e f o r i n t r a m o l e c u l a r value. However, because we have no knowledge o f c r y s t a l s t r u c t u r e , we use the above method.  -  3  4  5  124  -  6  1000/T Figure 7.5  7  8  9  (°K~ ) 1  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 of the r e c i p r o c a l o f the absolute temperature i n t e r t - b u t y l ami n e - ^ and t e r t - b u t y l a m i n e deuterate.  - 125 the high temperature s i d e o f the minimum i s p o s s i b l e i f we assume t h a t on t h i s s i d e , the r e l a x a t i o n process i n v o l v e s the t e r t - b u t y l group. But i n our case, we are h e l p l e s s because (CH-^CNDg melts before enough data can be obtained t o enable the a c t i v a t i o n energy t o be calculated. In the case of the t e r t - b u t y l a m i n e deuterate the s i t u a t i o n i s h i g h 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 ' s . T  c  Complete a n a l y s i s o f  i n terms o f 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 o f data, because o f l a c k o f knowledge of C i, To i. and Ean. t o permit the use o f Equation (7.6) and (7.7).  A rough estimate i s again made on the assumption t h a t a t  low temperature s i d e o f 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 ai 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 a t 77 K i s 2.01 s e c . C-j was again c a l c u l a t e d 2 using the M value o f 25.8 G f o r r i g i d l a t t i c e second moment i n 2  the deuterate (Table 7.2). Equation (7.11) then g i v e s T 10"  11  = 2.2 x  s e c . The corresponding value o f T-j a t 108.5 K i s 352 msec g i v i n g  T . | = 1.02 x 1 0 " ^ sec. q  The high temperature s i d e o f minimum gives  an a c t i v a t i o n energy o f 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.  E s t i m a t i o n o f 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 o f 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 ( C H ^ C N D g 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 r e f e r e n c e [7.2] show t h a t P i n c r e a s e s roughly by 0.012 g/cc f o r every 10°C decrease i n temperature and on t h i s b a s i s 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 t o a higher value f o r M^.  Smith has l i s t e d the values o f 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 o f 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 e t a l . [7.9] observed t h a t f o r amines n e a r l y a l l the c o n t r i b u t i o n s t o 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 e s t i m a t i o n o f Mg i s d i f f i c u l t because o f H-bonding o f the amino group. In t r i m e t h y l amine t h e i r value f o r Mg was 2.1 G2 . A value o f about 4 G2 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 t e r t - b u t y l a m i n e i s d i f f e r e n t because o f the d i f f e r e n t C-C and C-N bond l e n g t h s . In view o f these values 2 the estimate o f 7.93 G f o r Mg i s c o n s i d e r a b l y 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 t o be much l e s s than 1. Moreover a value o f 6 G f o r Mg d e r i v e d 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 e t a l . [7.1] and data d e r i v e d 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 o f McMullan et a l . and the microwave data i s i n the C-C and C-N bond l e n g t h s . In the former case [7.1] the C-C bond l e n g t h i s o  o  1.42 A, and t h i s i n c r e a s e s CH -CH i n t e r a c t i o n t o 5.11 G . This value 3  3  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 o f 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  e t a l . [7.1] p o i n t s out t h a t 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 uncertain.  They exclude the p o s s i b i l i t y o f 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, n e 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 o f 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 . T h i s 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.  T h i s i s because o f 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 c h a r a c t e r [7.11].  - 128 E x i s t e n c e o f 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 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 ' S are  group.  C  n e a r l y the same g i v i n g a sharp minimum e.g., i n hexamethylethane  i n the  work o f Chezeau e t 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 t o  n e 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 o f 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.  Theoretical  c a l c u l a t i o n o f the x ' s f o r such a l a r g e molecule i s extremely tedious c  and i s impossible because o f lack o f knowledge o f 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 o f l a c k o f knowledge o f the t r u e second moment and the mechanism o f r e l a x a t i o n .  Because o f 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 -'s are a l s o not ai  t r u e E - 's. al  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 ' S as upper l i m i t s t o the true v a l u e s . Thus f o r methyl group i n Q  (CH-^gCNDg the a c t i v a t i o n energy to the b a r r i e r h i n d e r i n g 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 = (8-9) x -13 10 s e c . Q  IJ  The a n a l y s i s o f the T-j data i n the case o f the deuterate using the above arguments has been c a r r i e d out o n l y 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 h i n d e r i n g t h i s motion was estimated from the T-j data t o be 1.7 kcal/mole and t h i s i s probably a l a r g e value f o r t h i s b a r r i e r . TO  - 1 1  The approximate value o f T = 2.2 x q  sec a l s o 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 o f 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 V I I ) [7.1]  R.K. McMullan, G.A. J e f f r e y , and T.H. J o r d a n , 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, ( t o be p u b 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. S t r a n g e , Mol. Phys., 20 (1971) 305.  - 131 -  CHAPTER VIII ISOPROPYLAMINE, ISOPROPYLAMINE DEUTERATE, TRIMETHYLAMINE, AND TRIMETHYLAMINE DEUTERATE A.  Introduction Isopropylamine and trimethylamine belong t o a s e r i e s o f alkylamines  which form hexagonal amine hydrates.  A s h o r t d e s c r i p t i o n o f thermo-  dynamic aspect o f trimethylamine was given i n Chapter I. The s t r u c t u r e s of both o f the hydrates were d i s c u s s e d b r i e f l y i n Chapter I I I . The s t r u c t u r e o f isopropylamine hydrate has r e c e n t l y been s t u d i e d by McMullan e t a l . [8.1] by X-ray d i f f r a c t i o n .  T h i s hydrate having  formula 10(CH ) CHNH .80H 0 belongs t o space group P6 /mmc. The u n i t 3  2  2  2  3  c e l l dimension a t -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 o f f o u r types o f polyhedra: hexagonal prisms, 12-hedra, 14-hedra, and 16-hedra.  The amine molecules a r e enclosed  in s i x 14-hedra and f o 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 ( F i g u r e 2.6). Trimethylamine hydrate has been s t u d i e d by x-ray d i f f r a c t i o n by Panke [8.2]. T h i s hydrate belongs t o space group P6/mmm with molecular formula 4(CH ) N.41H 0. The u n i t c e l l dimension a t -30°C 3  o  3  2  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 o f 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 t o 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 f a c e and g i v e 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 i s o p r o p y l a m i n e , but trimethylamine has been s t u d i 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 o f 30.5 G 2 and at 140 K the second moment reduces t o 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 o t h e r 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 l a r g e f o r C^+C^ r o t a t i o n . U s u a l l y 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 '•' f o r such a t r a n s i t i o n . The of 2.7 G 2 i s more reasonable to account a c t i v a t i o n energy f o r the b a r r i e r h i n d e r i n g 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 o f 4.27 kcal/mole per methyl group has been o b t a i n e d . The microwave work o f L i d e and Mann [8.6] shows the b a r r i e r h i n d e r i n g methyl group t o 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 was observed at 100 K and 2 G at 200 K [ 8 . 8 ] . 2  2  The line width  in trimethylamine deuterate was 1.4 G between 208 and 270 K [ 8 . 8 ] . qualitative dielectric measurement has been reported by Davidson  Some  [8.8]  on trimethylamine hydrate. The present nmr study shows that in isopropylamine, the only motion involved is that of methyl group rotation. function in this case is non-exponential.  The relaxation  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. function in trimethylamine is exponential.  The relaxation  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 L o r e n t z i a n 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 s o u r c e s , 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 . S t o h l e r Isotope Chemicals.  (99.8% d ) was obtained from 2  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 abovementioned 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.  P r e p a r a t i o n of (CH ) CHND 3  2  2  The p r e p a r a t i o n was e x a c t l y the same as d e s c r i b e d i n Chapter V. The a n a l y s i s of (CH ) CHND done i n the same way as before showed t h a t 3  NH  2  2  2  protons were about 98% deuterated to ND . 2  3.  P r e p a r a t i o n of Amine and Deuterate Samples  (a)  Isopropylamine and Isopropylamine Deuterate  The pure (CH ) CHND was f i l l e d d i r e c t l y i n the sample tubes and 3  2  2  was s e a l e d o f f by freeze-pump-thaw  method.  I t was Baker Sample.  Two batches of samples were prepared f o r isopropylamine d e u t e r a t e , each from Eastman Kodak and Baker amine samples.  The formula r a t i o  10(CH ) CHNH .80D 0 of amine and D 0 s o l u t i o n was mixed and was 3  2  2  2  2  f r e e z e d s l o w l y i n the c o l d box i n an a i r t i g h t g l a s s v e s s e l 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 e t a l . [8.1] a t about -6 to -7°C.  The r e s u l t s o f  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 d e u t e r a t e i n the range o f 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 s e a l e d by freeze-pump-thaw method. The deuterate was prepared i n two batches, one from Eastman Kodak and the o t h e r from Baker amine samples.  9 mole % o f trimethylamine  in D 0 was cooled s l o w l y i n the c o l d box (-15°C).  The c r y s t a l s  2  obtained were f i n e l y powdered, f i l l e d i n nmr tubes and s e a l e d 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 o f 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. (a)  (4(CH ) N.41D 0). 3  3  2  Spectrometer and Method o f Measurements Isopropylamine and Isopropylamine Deuterate  The r f f i e l d used f o r (CH ) CHND was about 0.025 mG from 77 t o 3  2  2  100 K. " In other temperature ranges i t was h i g h e r but a d j u s t e d a c c o r d i n g to r e l a x a t i o n measurements t o 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 o f experiments, the 90° pulse l e n g t h i n 180°-T-90° pulse sequence was 1.5 y s e c , but i n some cases the higher 90° pulse l e n g t h up to 2.2 y s e c 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 t o 178 K i n case o f (CH ) CHND . 3  t  2  2  Consequently  when R ( t ) = 1/2 was used f o r a l l temperature ranges s t u d i e d here 0  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 o f (CrL) CHND ?  ?  are t h e r f o r e  - 136 analysed i n view o f H-H theory.  In the case o f isopropylamine d e u t e r a t e ,  the r e l a x a t i o n f u n c t i o n was e x p o n e n t i a l . 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 a b s o r p t i o n 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 h i n d e r i n g 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 t o 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 o f 77 t o about 100 K where l i n e shape was c l o s e t o L o r e n t z i a n l i n e shape.  The second moment r e p o r t e d here may not be too accurate  as they are c a l c u l a t e d a t about 1% c u t o f f the t o t a l i n t e n s i t y o f a b s o r p t i o n 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 resolved. The r e l a x a t i o n f u n c t i o n i n the range 77 t o 115 K was non-exponential i n trimethylamine d e u t e r a t e , w h i l e i n other temperature ranges i t showed exponential behaviour.  - 137 C.  Results 1. (a)  Absorption Li lie A n a l y s i s Isopropylamine nand Isopropylamine (i)  Deuterate  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 e t 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 in c o o r d i n a t e generation o f protons and n i t r o g e n were assumed tetrahedral.  t o be  In the second s e t which i s based on microwave data the o  f o l l o w i n g bond lengths have been assumed: o  o  C-C = 1.54 A, C-N = 1.47 A,  o  C-H = 1.09 A, and N-H = 1.02 A. The angles f o r c o o r d i n a t e 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£ o f about 4 G2 . The reason o f t a k i n g M£ = 4 G2 w i l l be given l a t e r . E s t i m a t i o n 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 d i s c u s s e d i n Chapter VII. o  Use o f Equation (7.1) with R = 2.046A(from McMullan e t a l . data) gives Mg = 0.43 G2 and the use o f microwave data where R = 2.091A gives Mg = 2 0  0.39 G . For R^, the d a t a o f space group and the c o o r d i n a t e s o f c e n t r a l carbon atom (attached t o -CH group) given by McMullan e t 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 d e n s i t y based on u n i t c e l l parameters a t -160°C [8.1] and a t -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 o f R = 4.27 A (-160°C) gives M£ = 0.41 G and the value o f 2  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 t o be about 0.1 G , so t h a t t o t a l M£ 0.50 G . 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 ^ 2  or  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 t o 0.13 G  after  i n c l u d i n g the c o n t r i b u t i o n s due t o deuterons and exchanged protons. A summary o f 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. (ii)  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 a g a i n s t temperature i n Figure 8.1 f o r (CH ) CHND and isopropylamine d e u t e r a t e . 3  2  2  (CH ) CHND shows r i g i d s t r u c t u r e o f 3 s p i n system. 3  2  2  The experimental  value o f second moment f o r (CH ) CHND a t 77 K i s 22.73 +0.91 G and 2 i t approaches a p l a t e a u value o f 9 G around 165 K. T h i s l a t t e r value 2 2  3  of 9 G  2  2  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 p l a t e a u value o f 7 G  i s reached around 150 K. A rough estimate o f a c t i v a t i o n energy f o r the b a r r i e r h i n d e r i n g 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 o f Equation (3.25).  This  equation with C = 18.8 G, B = 7 G g i v e s 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 d e u t e r a t e . The experimental value o f second moment i n t h i s case a t 77 K i s 2 14.24 +_1.40 G and it shows a very slow decrease reaching a value  28-  2 4 -  CD  c E  •-  A I s o p r o p y la m i n e - N D  20-  o  on  Mo  CD  16  -  CO ' 8  -  IsopropyIamirve< Deuterate  -  12  2  CO Figure 8.1 PMR second moment vs, temperature i n , isopropy!amine-ND^ and isopropylamine deuterate. to  o CD  1  r~ IOO  O  T  T  150  200  Temperature  [°K]  250  c5>  140 Table 8.1 T h e o r e t i c a l Second Moment Values f o r (CH ) CHND and 3  2  2  Isopropylamine Deuterate Types of Motion  Deuterate based on based on Ref. [8.1] Ref. [8.6,8.9] G  Rigid 2CH CH -CH  2  G  2  (CH ) CHND based on Ref. [8.6,8 3  2  G  2  19.30 1.35 2.10  M£  19.30 1.45 2.19 0.03 0.02 0.50  Total  23.49  23.29  26.79  4.83 1.25 1.88 0.03 0.02  4.83 1.16 1.81 0.02 0.02  4.83 1.16 1.81 0.02 0.02  MJ  /vO.35  ^0.35  1.00  Total  ^8.36  ^8.19  8.84  2CH + CS r o t a t i o n * 2CH 0.53 CH CH 0.31 CH -CH 0.10 CH -ND 0.02  0.62 0.29 0.09 0.01  /V0.20  a,0.20  2  3  3  CH -CH 3  CH -ND 3  CH-ND  2  2  2CH r o t a t i n g 2CH CH CH  .0.02  0.02 0.50  19.30 1.35 2.10 0.02 0.02 4.00  3  3  _  3  3  CH -GH 3  CH -ND 3  CH-ND  2  3  3  3  _  3  3  3  Total  2  • ^1.16  ^1.11  —  2  - 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 0.13 Total *  0.13  0.00 0.13 0.13  C^ r o t a t i o n means r o t a t i o n around pseudo t h r e e f o l d a x i s ( i . e . , around -CH bond).  of 5.59 + 0.21 G a t 112 K t o a p l a t e a u value o f 1 G around 212 K. The value o f 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 2  2  width i n the f i r s t t r a n s i t i o n r e g i o n i s o f no v a l u e , because a t 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 o f 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 t h e b a r r i e r h i n d e r i n g  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 t o an a c t i v a t i o n energy o f 1.69 +_0.22 kcal/mole with x ranging from 1.5 x 10-4 sec t o 1.3 x 10-5 sec from c  112 t o 156 K. The cw data a r e thus i n agreement with the f i n d i n g o f McMullan e t a l . [8.1] t h a t t h e 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 i n 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  o f 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 i n c r e a s e t h i s v a l u e , 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 a r e  l i s t e d i n Table 8.2. (ii)  Experimental Results  The experimental r e s u l t s f o r trimethylamine deuterate a r e p l o t t e d i n Figure 8.2. Owing t o Laurentzian l i n e shape the second moment are c a l c u l a t e d up t o approximately 1% c u t o f f the t o t a l a b s o r p t i o n intensity. values.  These values may not be as much accurate as the other  The second moment thus c a l c u l a t e d a t 77 K i s 8.46 +_0.76  G and i t approaches a value o f 1.42 +0.02 G a t about 114 K f i n a l l y 2 reaching a p l a t e a u value o f 0.60 G around 180 K. The l a s t value o f 2 0.60 G i s not assigned, because t h i s value corresponds n e i t h e r t o 2 Cg+Cg motion ( t h e o r e t i c a l value 1.40 G ) nor t o 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 ) . The other models were t r i e d , but i n each 2 case t h e 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 t o the m e l t i n g p o i n t ( 280 K). The l i n e width agrees t o the one 2  2  2  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 a g a i n s t  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] a r e a l s o shown i n Figure 8.2.  Figure 8.2  32-,  Line width i n gauss and second moment i n gauss as a f u n c t i o n of temperature i n trimethylamine and trimethylamine deuterate.  28-  "cxP 24-1  ( C H  3  )  3  Second Moment | Ref. [8.3] L i n e Width  N  •4— c  J  Second Moment  20  33  E  2  I  •  r-20  LineWidth  IB  0  I6-|  •*-  "D  •a  u d) CO 8H  hl6 14 -12 -IO  c  " 8 - 6  4-  -4  O IOO  I 150  OS ® O •0-0—o-o-  i — I ~T—I r — 2 0 0  T e m p e r a t u r e [°K]  - 2 -6>—O250  -OO  Q>  O  co  -  Table 8.2  Second Moment Values f o r T r i m e t h y l a m i n e  Motion  Cg-rotation + C  3  Inter  2  Rigid  3  Deuterate.  I n t r a M£ ( G ) 3CH  C  144 -  r  3  2  M  (G ) 2  2  27.55  0.03  0.25  8.97  0.67  0.01  0.20  1.40  0  0  0.15  0.15  5.63  3.06  0.57 0  Isotropic  MJ ( G )  3  0.35  4.65  c  CH -N  3  0.03  22.52 9  °tation  CH -CH  3  Total  b  C ~ r o t a t i o n means r o t a t i o n o f 3GH. groups around t h e i r symmetry a x i s . 3  3  Obtained by c o n c e n t r a t i n g t h e t h r e e CH ~protons on t h e c e n t r e o f 3  c i r c l e which they make d u r i n g  rotation.  C i means, t h e second t h r e e 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)  I s o p r o p y l a m i n e and I s o p r o p y l a m i n e  Deuterate  The r e l a x a t i o n i n t h e case o f pure i s o p r o p y l a m i n e was n o n exponential  beyond 145 K.  A l l t h e d a t a a r e t h e r e f o r e a n a l y z e d from  the p o i n t o f view o f H-H t h e o r y .  The temperature dependence o f t  ( t i m e when R ( t ) = 1/2) i s p l o t t e d i n F i g u r e 8 . 3 .  The e x p e r i m e n t a l  Q  value of t  a t minimum i s 17 msec a t 177 K.  t h a t a t minimum ( t /T ) 1  = 1 . 5 2 and t h i s g i v e s  T h i s v a l u e o f ( 1 / T ) was used t o e x t r a c t 1  We know from H-H t h e o r y  M T Q  C  (1/T ) 1  _1  1  as a f u n c t i o n o f  temperature i n a s i m i l a r way as done i n Chapter V. dependence o f W . T i s p l o t t e d a g a i n s t 1 / T ( K )  = 0.0894 s e c " .  The t e m p e r a t u r e  i n Figure 8 . 3 .  - 145 -  T e m p e r a t u r e [°K] IIO _!__  130 I  ,  150 I  •  170 I  ,  190 L_  ,  i  i  amine-ND . 9  i  1 6.0  1  1  '  1 7.0  1  IOOO/ pK-'} T  1  1  1 8.0  r  - 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 with T o = (1.78 +— 0.22) x 10 -13 sec. The second moment i n t h i s 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 h i n d e r i n g methyl group reorientation.  T h i s 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 o n l y a s i n g l e minimum o f 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 s l o p e o f znl^ versus 1000/T p l o t (Figure 8.5) i s 1.65 +0.03 kcal/mole, w h i l e  the high temperature  s i d e gives 1.61 +_ 0.05 kcal/mole. The high temperature s i d e a c t i v a t i o n energy agrees t o the one obtained by l i n e width data i . e . , 1.69 +_ 0.22 kcal/mole. A rough estimate o f x can be made i n a s i m i l a r Q  way as done i n Chapter VII. Use i s made of Equation (7.11) and (7.12) with M  2  values o f 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 . This gives T (with M  = 1.76 x 1 0 " , 1.41 x 1 0 " , and 1.37 x 1 0 " 10  q 1  = 23.5 G ) ; and 1.30 x 1 0 2  2  sec (with M  2  10  - 1 0  = 23.3 G ) r e s p e c t i v e l y .  , 1.42 x 1 0  - 10  sec  , and 1.37 x 1 0 "  10  Tbe a c t i v a t e d s t a t e theory  demands t h a t f o r one r e o r i e n t a t i o n a l process -12 -15 range of 10  - 1 0  10  should l i e i n the  sec. T h i s 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 t o the r e l a x a t i o n .  The t r u e 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 h i n d e r i n g methyl groups r e o r i e n t a t i o n i n isopropylamine d e u t e r a t e . For high temperature s i d e use o f Equation (7.11) and (7.12) g i v e s  10 J 3  Isopropylamine Deuterate -pa  c o o <D tf)  e  —1—  100  r  150  — I —  200  Temperature [°K] F i g u r e 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  ]  -i  T  250  p l o t t e d a g a i n s t temperature i n i s o p r o p y l a m i n e d e u t e r a t e .  2000  IOOOIsopropylamine Deuterate  *500-  00  c o u w  F  M.P  IOO-  —r—  4  Figure 8,5  T" 8 I O V T [°K*']  IO  12  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  and 2.28 x I O " sec at 220 and 213 K r e s p e c t i v e l y (with M^ = 8.4 G 2 ). The other value of M = 8.2 G 2 g i v e s o2  equal to 2.55 x 1 0 "  10  10  2  on these temperatures x respectively.  o2  = 2.53 x 1 0 ~  10  and 2.10 x 1 0 ~  10  sec  T h i s 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 o f 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. minimum i n T-| i s achieved.  Trimethylamine melts before  I f we assume t h a t 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 t h a t 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 c o n s i 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. some c o n t r i b u t i o n i s coming from  This indicates that  motion.  In trimethylamine d e u t e r a t e , 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 a c h i e v e , 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  -  IOO  150  -  150 200 T e m p e r a t u r e (°K)  250  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 o f the absolute temperature i n trimethylamine and trimethylamine deuterate.  - 151  Figure 8.7  -  Values o f 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 trimethyl amine  deuterate.  - 152 -  although T-j i s not d e f i n e d i n the r e g i o n o f 77 t o %115 K, but the values o f  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 o f t i s a l s o shown i n both o f  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 o f t versus 1000/T ( F i g u r e 8.7) i s 0.32 kcal/mole. The a c t i v a t i o n energy obtained from the s l o p e o f low temperature s i d e o f 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 o f 2.90 + 0.19 kcal/mole.  The f i r s t a c t i v a t i o n energy o f 0.23 kcal/mole i s  very small and we are not able t o a s s i g n i t , as we do not know the behaviour o f trimethylamine i n t h i s deuterate below 77 K.  The  second a c t i v a t i o n energy o f 0.73 kcal/mole can be due t o the b a r r i e r hindering  motion.  The higher a c t i v a t i o n energy o f 2,90 kcal/mole  r e f l e c t s t h a t 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 s m a l l e r than C^+C^ motion v a l u e .  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 o f Equation (5.3) with d e n s i t y p = 0.81 g/cc and R = 2.091 A leads t o a value o f ^ 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 o f [8.11] i s c o r r e c t , we expect M£ o f the order o f (7/11)6 = 3.8 G f o r ( C H ) CHNDg. Powles 2 and Kail on a s i m i l a r compound isobutylbromide obtained IC = 5.6 G [8.14]. 2  3  2  - 153 On t h i s b a s i s we w i l l have a value o f (7/9)5.6 = 4.4 G f o r 2  (CH ) CHND . 3  2  2  2  Our assumed value o f 6 G f o r tert-butylamine-ND  leads t o a value  2  of 4.6 G as M value f o r (CH ) ,CHND . On these arguments a value o f 2  2  3  2  2  4 G i s q u i t e a reasonable value o f MJ o f (CH ) CHND . 2  3  2  2  We are unable t o say anything about the molecular parameters o f isopropylamine i n isopropylamine deuterate as the temperature a t 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 t o 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 value using microwave data (1.11 G ) than t h e 2  2  data o f 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 t h a t the molecule shows C motion i s i n agreement with the f i n d i n g s o f McMullan e t a l . 3  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.  T h i s 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  n e a r l y the same b a r r i e r heights f o r the methyl and C r o t a t i o n s o f 3  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 ) CHND was non3  2  2  exponential.  This non-exponential behaviour was a l s o present i n  (CH ) CHNH .  I t i s t h e r e f o r e concluded t h a t the non-exponential  3  2  2  of R ( t ) i s due t o methyl groups.  behaviour  The minimum value o f 17 msec occurs  at 177 K and t h i s value remains t h e same up t o the m e l t i n g p o i n t 179 K o f (CH ) CHND . 3  2  2  This i m p l i e s 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  of c o r r e l a t i o n times, or t h i s i s n o t a t r u e minimum.  A theoretical  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 o f H-H theory i s the same as the one  - 154 obtained from t h e slope o f £ n t versus 1/T p l o t (3.52 +0.06 kcal/mole). Q  T h i s j u s t i f i e s the i n t e r p r e t a t i o n t h a t 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 t o m e l t i n g p o i n t o r due t o the experimental e r r o r . In the isopropylamine d e u t e r a t e , the r e l a x a t i o n f u n c t i o n was exponential.  This may be because o f amine methyl groups come more  c l o s e r towards each other because o f the r e p u l s i v e f o r c e s o f 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 o f 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 h i n d e r i n g methyl reorientation.  group  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. t o the melting p o i n t o f the deuterate ( 270 K). P o s s i b l y t h i s i s because o f 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 g i v e an a c t i v a t i o n energy o f 5.75 kcal/mole which i s by c o i n c i d e n c e 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 a r e not  a c c u r a t e , hence the accuracy o f t h i s value i s not comparable t o [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 t h e one obtained from gas phase microwave work i . e . 4.4 kcal/mole, but not t o t h a t extent.  The heat c a p a c i t y data o f  Aston e t 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. to the b a r r i e r .  T h i s shows t h a t  motion i s c o n t r i b u t i n g  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 t r u e minimum i n T-|.  The compound melts before t r u e 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 d e u t e r a t e , the p l a t e a u value of 0.6 G  i s hard  to e x p l a i n . T h i s 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 t h a t C-N bond l e n g t h 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 o f r e p u l s i v e f o r c e s of the deuteron cage.  This s m a l l e r angle w i l l reduce the methyl group  c o n t r i b u t i o n o f 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 n c r e a s e 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  q u e s t i o n o f 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 p l a t e a u 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 o f 0.15 G  [8.13].  The only  p o s s i b l e e x p l a n a t i o n i s that besides C +C r o t a t i o n , ( C H ) N e x h i b i t s 3  some t r a n s l a t i o n a l motion.  3  3  3  This e x p l a n a t i o n 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 d i s o r d e r e d p o s i t i o n of n i t r o g e n 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 + C r o t a t i o n , the p o s i t i o n o f N-atom i s f i x e d , and 3  3  i t s p o s i t i o n would not have been represented by a large thermal ellipsoid. 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 g i v e r i s e to a broad minimum. The a c t i v a t i o n energy of 2.90 i s c l e a r l y not due to C motion. 3  kcal/mole  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 o f trimethylamine molecule i n view o f the above arguments.  - 157 References (Chapter V I I I ) [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. S z a s z , 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, C I a t h r a t e Hydrates, National Research Council o f 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. G o l d f a r b and B.N. K h a 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 o f some  hydrates where the guest molecule contains only methylene groups. are hydrates of cyclopropane, p i p e r a z i n e , and  These  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 o f deuterates have been s t u d i e d  thermodynamically by Hafemann and M i l l e r [9.1]. According to these authors the type I deuterate (C H -7.8D 0) i s s t a b l e below -23°C and 3  between 5.52 and 18.34°C.  g  2  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 v a l u e 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 o f  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 in the stabe range (-23.31 to 5.52°C) shows a second moment o f 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 o f 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 o f  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 nonexponential.  U n f o r t u n a t e l y i n the temperature range a c c e s s i b l e to  us, and because o f poor s i g n a l t o n o i s e r a t i o , we were not able t o draw any s u b s t a n t i v e c o n c l u s i o n s from the r e l a x a t i o n measurements. The minimum i n t  2.  appeared to be well below 77 K.  p i p e r a z i n e Hydrate  P i p e r a z i n e i s one o f the c y c l n c amines which form,a hexahydrate, the s t r u c t u r e o f 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 t o 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 t o -ND by repeated exchange with D2O.  The deuterate  was prepared from p a r t i a l l y d e u t e r a t e d p i p e r a z i n e (C^HgND2).  A l l the  processes of p r e p a r a t i o n 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 o f 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 a b s o r p t i o n l i n e measurements were c a r r i e d out on  Varian DP-60 spectrometer using a frequency o f 60 MHz.  Strong s a t u r a t i o n  and poor s i g n a l to n o i s e r a t i o were c h a r a c t e r i s t i c s o f the specimen at a l l temperatures (77 to the m e l t i n g p o i n t o f the d e u t e r a t e 317 K). In s p i t e o f a l l these d i f f i c u l t i e s , some c o n c l u s i o n s can be drawn from those n o i s y 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 a t 77 K was 16.3 +_ 2.0 G . T h i s value stayed constant n e a r l y up t o the m e l t i n g p o i n t . The t h e o r e t i c a l value o f second moment, obtained from the data o f 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 t o 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 eiietetf 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 o f problem as t h a t 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 s t u d i e d by Mak [9.4] using X-ray d i f f r a c t i o n .  The hydrate (CH )gN .6H 0 i s rhombohedral. 2  4  2  The  amine molecule i s surrounded by e i g h t (H 0)g r i n g s and i s hydrogen2  bonded t o three o f these r i n g s so as t o hang " b a t l i k e " t o the upper w a l l s of the c a v i t y formed by (H 0)g r i n g s . Only cw s t u d i e s using 2  Varian DP-60 spectrometer ( r f frequency 60 MHz) were c a r r i e d out on (CH )gN^.6D 0. These s t u d i e s under the obscure s i t u a t i o n o f strong 2  2  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 o f the guest molecule i n the d e u t e r a t e . 2  The experimental second moment  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 o f second moment based on the data o f Smith [9.5] i s 14.82 G using a C-H bond length o f 1.13 A and an 2  i n 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 g u e s t s , 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  c o n c l u s i o n s a r e drawn.  - 161 In the diethylamine and diethylamine deuterate, the only motion observed was t h a t o f methyl group r e o r i e n t a t i o n .  The a b s o r p t i o n 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 o f 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 o f 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 . 0 3 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 i n d e r i n g methyl reorientation. T  c  =  T  o  e x  The pre-exponential f a c t o r i n Arrhenius equation  P ( / ) was (1.6 + .1) x 1 0 " E  group  R T  a  1 3  and (4.5 + .3) x 1 0 " s e c 1 3  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 o f 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 o f 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 u n f o r t u n a t e l y 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 g r a d i e n t o f ir\t  Q  versus 1000/T (T i n K) were 1.33 + 0.01 kcal/mole  in acetone a s s o c i a t e d with the b a r r i e r h i n d e r i n g 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 o f 0.33 kcal/mole was not assigned due to lack o f knowledge of data a t 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 r e v e a l e d 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 t o a motion corresponding t o 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 o f (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 o f 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 h i n d e r i n g methyl  group  r o t a t i o n from T-| data was found to be 3.2 + .1 kcal/mole with T = -13 Q  (8-9) x 10  s e c . 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 o f 32 +_ 1 msec t h i s time i n the  deuterate was much broader because o f 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 three types o f 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 s l o p e o f £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 h i n d e r i n g 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 separately. In isopropylamine, the only motion present was t h a t o f 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 = (1.78 + .22) x 1 0 " Q  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 a t a temperature o f  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 o f 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 a x i s r e o r i e n t a t i o n t o be 1.6 kcal/mole.  The t r u e 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.  T h i s value was higher than the v a r i o u s values  obtained by other methods, but agreed to the one obtained by Haigh et a l . [ 9 . 6 ] , Trimethylamine deuterate showed a L o r e n t z i a n 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 o f temperature.  There appeared t o be two minima i n the T^-  temperature curve, the f i r s t well below 77 K and the other o f 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 t o r o t a t i o n around t h r e e - f o l d a x i s and t o 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 r e p o r t e d were deuterates o f 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 a x i s motion a t high temperatures i n type I s t r u c t u r e d e u t e r a t e , w h i l e i n type IT s t r u c t u r e , i s o t r o p i c motion o f cyclopropane molecule was present a t a l l temperatures o f i t s s t a b i l i t y .  Of the  other two amine deuterates o f 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 r e p o r t e d f o r the m a j o r i t y o f the guests  s t u d i e d 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 t o extend the present study t o 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 r e v e a l any f u r t h e r i n f o r m a t i o n 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 a b s o r p t i o n l i n e width, and r e l a x a t i o n measurements. The problem o f 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 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]. are  There  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 u s e f u l 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 v a r i o u s 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 p i o n e e r i n g work o f Andrew [ 9 . 8 ] , M a n s f i e l d [ 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 v a r i o u s 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 c o n c e n t r a t i o n 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 s p i n systems of methylene, and methyl groups.  The i n t e r a c t i o n s  due to others groups may t o some extent s p o i l the whole idea of an i s o l a t e d two, three s p i n 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 s i m p l e r 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 s t u d i e d but not i n g r e a t d e t a i l .  - 165 The problem o f poor s i g n a l to n o i s e 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 d e u t e r a t e s . T h i s 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 t h e r e 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. , 3 6 (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. M a n s f i e l d , i b i d . , 8^ (1971) 41, and r e f e r e n c e s 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 ( - t / T ' ) s i n 3 c l e  (A.l)  q;j  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 ) in the form IT.  R/\ (t]  T  =  v  /  /  f(cose)sinedg  where a f t e r s u b s t i t u t i n g cosg  Y . =  /  = X, and R ^ ( t ) v  =  Y becomes  f[X)dX  (A.2) *  This i n t e g r a l can now n u m e r i c a l l y be c a l c u l a t e d using Simpson's formula i.e., Y = (H/3)[Y + 4(Y 0  1  + Y  3  + ... + Y^-,) + 2 ( Y + Y 2  4  +....+ Y _ )  + Y] N  N  2  (A.3)  2 For each ( W T ) Q  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 t a b l e s o f H i l t and Hubbard with J  J  2 t/T  1  t/T  1  = 0 , 1, 2, 3, . . . , 2 5 .  In the o r i g i n a l program  (" t ) 0  c  =  WT»  = 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 o f another simple program.  > > >  1 2 3  >  1+  > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >  5 6 7 8 9 10 11 12 13 It* 15 16 17 18 19 20 21 22 23  C C 1 50  C  3k  35  1  /  C  C C  80  10  k  2k  25 26 27 28 29 30 31 32 33  8  COMPUTER PROGRAM ' H I L T ' TO C A L C U L A T E R A V ( T ) FOR NON-EXPONENT 1AL RELAXATION DIMENSION C(k),CC(kk,k),0.(.k),W(kk,k),FACT(.k),X(ll),Y(ll),R(k) COMMON X,Y R E A D ( 5 , 5 0 ) WT FORMAT(FIO.O) IF(WT.EQ.O.O) GO TO 20 W R I T E ( 6 , 8 ) WT FORMAT( 5X, "ViT- , F 1 0 . **/ ) START WITH T=0.0 T=0.0 DO 70 L = l 2 6 WRITE(6,80) T F0RMAT(5X,'T=',F5.1) START WITH F I R S T VALUE OF C O S ( B E T A ) DENOTED HERE BY X I X1=0.0 DO 2 1=1,11 I F ( T . N E . O . O ) GO TO 22  3 22  C 30  DO  3 J=l,k  READ T H E VALUES OF C ( J ) , Q ( J ) TAKEN FROM THE T A B L E S OF H I L T AND HUBBARD FOR A PARTICULAR VALUE OF X 1 = C 0 S ( B E T A ) READ(5/U) C ( J ) , Q ( J ) FORMAT(2F10.0) CC(I,J)»G(J) QQ(I,J)=Q(J) CONTINUE SUM=0.0 DO 30 J=l,4 C(J)=CC(I,J) Q(J)=QQ(I,J) R(J)=-Q(J)*T FACT(J)=C(J)*EXP(R(J)) C A L C U L A T E SUM FROM J=l TO J = 4 SUM=SUM+FACT(J) CONTINUE  > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >  36 37 38 . 39  C  ill  C  hO  k2 k3 kk kS  C C 70  1*6  U7 U8 49 50 51 52 53 5k  55 56 57 58 59 60 61 62 63 61* 65 66 67 68  2  20  C  C  A(l )=tt.0*Y(l )  C 2 C  X(l)=X1 Y( 1 )=SUM INCREASE C 0 S ( B E T A ) = X 1 BY 0.1 X1=X1+0.1 CONTINUE A L L VALUES OF X I F1N1 S H E D , C A L L SUBROUTINE 'SIMPSN' FOR I N T E G R A T I O N C A L L SIMPSN INCREMENT T BY 1.0 FOR ANOTHER INTEGRAL AND CONTINUE I T FURTHER T I L L T=25.0 T=T+1.0 CONTINUE GO TO 1 STOP END SUBROUTINE SIMPSN DIMENSION X ( l l ) , Y ( 1 1 ) , A ( 1 1 ) , B ( 1 1 ) COMMON X,Y VALUE OF H IN E Q U A T I O N ( A . 3 ) IS S E T =0.1 H = 0.1 SUM1=0.0 SUM2=0.0 S E T VALUE OF YO AND YN Y0=Y(1) YN=Y(11) DO 2 1=2,10,2 SUM ODD Y IN E Q U A T I O N ( A . 3 ) SUM1=SUM1+A(I) CONTINUE DO 3 1=3,9,2 B(1)=2.0*Y(I) SUM EVEN Y IN E Q U A T I O N ( A . 3 ) SUM2=SUM2+B(I)  > > > > > > > > > #END  #  69 3 70 C 71 72 C 73 lk 75 83 76 77 OF F I L E  CONTINUE C A L C U L A T E Y=RAV RAV=(H/3.0)*(Y0+SUM1+SUM2+YN) CALCULATE LN(RAV) ARAV=ALOG(RAV) W R I T E ( 6 , 8 3 ) RAV,ARAV FORMATC10X, RAV(T)= F10.^ 5X 'LN(RAV(T))=',F10.4) RETURN END ,  ,  /  /  /  Table A2  Table Al (  V c  t/T' . 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  )  «  2  Av^ 1.0000 0.3342 0.2159 0.1592 0.1270 0.1070 0.0937 0.0843 0.0772 0.0716 0.0672 0.0634 0.0602 0.0575 0.0551 0.0529 0.0510 0.0492 0.0477 0.0462 0.0449 0.0436 0.0425 0.0414 0.0404 0.0395  R  ("Vc)  1.000  £riR (t) Av  -0.0 -1.0959 -1.5329 - 1 . 8376 -2.0638 -2.2349 -2.3676 -2.4738 -2.5615 -2.6360 -2.7007 -2.7581 -2.8096 -2.8564 -2.8994 -2.9392 -2.9763 -3.0110 -3.0437 -3.0746 -3.1039 -3.1319 -3.1586 -3.1843 -3.2089 -3.2327  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  =  0.001  Av^ 1.0000 0.9334 0.8746 0.8226 0.7763 0.7351 0.6983 0.6652 0.6354 0.6085 0.5840 0.5618 0.5415 0.5229 0.5058 0.4901 0.4755 0.4619 0.4493 0.4376 0.4266 0.4162 0.4065 0.3973 0.3886 0.3804 R  £nR (t) Av  -0.0 -0.0689 -0.1340 -0.1953 -0.2532 -0.3077 -0.3592 -0.4077 -0.4535 -0.4968 -0.5378 -0.5766 -0.6134 -0.6483 -0.6815 -0.7132 -0.7434 -0.7723 -0.8000 -0.8265 -0.8520 -0.8765 -0.9002 -0.9231 -0.9452 -0.9666  Table A4  Table A3 (a) x  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  Av 1.0000 0.8634 0.7587 0.6773 0.6129 0.5610 0.5186 0.4833 0.4535 0.4279 0.4057 0.3861 0.3687 0.3531 0.3390 0.3260 0.3142 0.3032 0.2930 0.2835 0.2746 0.2663 0.2584 0.2510 0.2440 0.2374 R  t/T'  ( t )  -0.0 -0.1469 -0.2761 -0.3896 -0.4896 -0.5779 -0.6566 -0.7271 -0.7908 -0.8488 -0.9022 -0.9516 -0.9976 -1.0410 -1.0819 -1.1207 -1.1578 -1.1934 -1.2276 -1.2605 -1.2924 -1.3232 -1.3531 -1.3822 -1.4105 -1.4381  0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  x  ) = 0.010 2  o c R (t) A v  1.0000 0.8192 0.6933 0.6029 0.5359 0.4845 0.4439 0.4109 0.-3834 0.3600 0.3397 0.3218 0.3059 0.2916 0.2786 0.2667 0.2558 0.2457 0.2363 0.2277 0.2196 0.2120 0.2050 0.1983 0.1921 0.1863  *nR (t) Ay  0.0 -0.1995 -0.3663 -0.5059 -0.6238 -0.7247 -0.8122 -0.8895 -0.9587 -1.0218 -1.0798 -1.1338 -1.1845 -1.2325 -1.2782 -1.3217 -1.3636 -1.4037 -1.4425 -1.4 799 -1.5160 -1.5510 -1.5849 -1.6177 -1.6496 -1.6805  Table A6  Table A5 T )  = 0.050  2  c  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  R (t) 1.0000 0.6947 0.5367 0.4440 0.3828 0.3384 0.3039 0.2760 0.2527 0.2328 0.2157 0.2007 0.1876 0.1761 0.1659 0.1568 0.1487 0.1414 0.1349 0.1290 0.1236 0.1188 0.1144 0.1104 0.1067 0.1034 A v  (  *nR (t) -0.0 -0.3643 -0.6223 -0.8119 -0.9603 -1.0836 -1.1909 -1.2873 -1.3 756 -1.4575 -1.5340 -1.6058 -1.6732 -1.7367 -1.7965 -1.8529 -1.9061 -1.9562 -2.0035 -2.0482 -2.0904 -2.1302 -2.1679 -2.2036 -2.2374 -2.2694 Av  </  V c  )2  =  0.100  t/T'  R .(t)  *nR (t)  0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22. 0 23.0 24.0 25.0  1.0000 0.6465 0.4847 0.3949 0.3363 0.2935 0.2602 0.2332 0.2109 0.1922 0.1763 0.1628 0.1511 0.1411 0.1324 0.1248 0.1181 0.1122 0.1070 0.1023 0.0982 0.0945 0.0911 0.0881 0.0853 0.0827  -0.0 -0.4362 -0.7242 -0.9290 -1.0898 -1.2259 -1.3465 -1.4560 -1.5565 -1.6495 -1.7356 -1.8154 -1.8895 -1.9583 -2.0221 -2.0813 -2.1363 -2.1874 -2.2350 -2.2794 -2.3208 -2.3595 -2.3957 -2.4297 -2.4618 -2.4920  Av  A v  Table A8  Table A7  =0.200  (U T ) 0  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  C  R (t). 1.0000 0.6139 0.4507 0.3619 0.3035 0.2606 0.2273 0.2007 0.1792 0.1615 0.1468 0.1345 0.1242 0.1155 0.1081 0.1017 0.0962 0.0914 0.0872 0.0834 0.0801 0.0772 0.0745 0.0721 0.0699 0.0679 A v  = 0.500  (W T ) Q  *nR (t) -0.0 -0.4880 -0.7970 -1.0164 -1.1925 -1.3449 -1.4815 -1.6058 -1.7195 -1.8236 -1.9188 -2.0059 -2.0855 -2.1582 -2.2247 -2.2856 -2.3415 -2.3927 -2.4400 -2.4836 -2.5240 -2.5616 -2.5966 -2.6294 -2.6602 -2.6891 Av  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19. 0 20.0 21.0 22.0 23.0 24.0 25.0  C  R (t) 1.0000 0.6042 0.4374 0.3449 0.2830 0.2376 0.2028 0.1755 0.1539 0.1365 0.1225 0.1111 0.1017 0.0939 0.0874 0.0819 0.0772 0.0732 0.0697 0.0666 0.0639 0.0615 0.0594 0.0574 0.0557 0.0541 A v  *nR (t) Ay  -0.0 -0.5038 -0.8268 -1.0646 -1.2623 -1.4373 -1.5957 -1.7401 -1.8716 -1.9911 -2.0993 -2.1972 -2.2855 -2.3652 -2.4371 -2.5022 -2.5612 -2.6148 -2.6638 -2.7087 -2.7501 -2.7884 -2.8239 -2.8572 -2.8883 -2.9176  Table A9, (<V )  =  c  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  (OJ T )  0-800  Av 1.0000 0.6148 0.4453 0.3488 0.2837 0.2359 0.1993 0.1709 0.1484 0.1305 0.1162 0.1045 0.0951 0.0873 0.0808 0.0753 0.0708 0.0668 0.0635 0.0605 0.0579 0.0557 0.0536 0.0518 0.0501 0.0486 R  Table A l 0  ( t )  *nR (t)  t/T'  -0.0 -0.4864 -0.8091 -1.0534 -1.2599 -1.4445 -1.6128 -1.7670 -1.9079 -2.0363 -2.1528 -2.2582 -2.3533 -2.4389 -2.5161 -2.5856 -2.6485 -2.7055 -2.7574 -2.8048 -2.8484 -2.8885 -2.9258 -2.9605 -2.9930 -3.0235  0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  Av  V  2  = 1.000  Av^ 1.0000 0.6232 0.4524 0.3538 0.2869 0.2376 0.2000 0.1707 0.1476 0.1292 0.1145 0.1026 0.0930 0.0850 0.0784 0.0729 0.0683 0.0644 0.0610 0.0580 0.0555 0.0532 0.0512 0.0494 0.0478 0.0463 R  *nR (t) Av  0.0 -0.4728 -0.7931 -1.0391 -1.2488 -1.4370 -1.6093 -1.7677 -1.9131 -2.0461 -2.1670 -2.2766 -2.3757 -2.4651 -2.5456 -2.6182 -2.6838 -2.7433 -2.7973 -2.8466 -2.8918 -2.9334 -2.9719 -3.0078 -3.0413 -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 OKMDOOHNd-lflCOOlONI^JtmUJINCOOlOlOOHHNN OOCOHHHHHHNMntMMMMCNMMKlKlfAKl^M  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  C  o  >  r H r H OO oo O cc i n j -  o cn 00 o oo r H  o  in in rH m rN. m CM  j o o CM CM CO -d- o oo m m i n m in co r H CM CD CO J - CM o oo r H r H r H r H o o o o o CM r H  ro CM oo CM CD CO  cn  r--  cn  I  CD r H c o j - CD oo CM 00 CM O OS in m in jo o o o o o o o  o r--m -=s-  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  o  H-  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 r H H H H r H r H H H H rH CM CM CM CM CM CM  >  o; c  OOCSIHOCtOlfllOlflJ-inOONHMJ'J'OlNJ-UJOirNNO 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  "Tj" 3. > <C  o;  CMCMCMCNicMCNCMCMCMCMCMrnrnrnrnrnrn  o o o  1 I I  I  I  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 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 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 r H O O O O O O O O  o + > -  I I I i I I I I I I I I I I Ii  o  o  o  o  o  o  o  o  OO  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 (  t/T' 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14. 0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0  2k.0  25.0  V  c  )  =  2  5  Av^ 1.0000 0.7289 0.5642 0.4527 0.3712 0.3088 0.2596 0.2200 0.1879 0.1615 0.1398 0.1218 0.1069 0.0945 0.0841 0.0754 0.0681 0.0619 0.0566 0.0521 0.0483 0.0450 0.0422 0.0397 0.0375 0.0356 R  -  Table A14 0  0  (  0  *nR (t) -0.0 -0.3163 -0.5723 -0.7926 -0.9910 -1.1750 -1.3486 -1.5140 -1.6721 -1.8232 -1.9677 -2.1052 -2.2358 -2.3594 -2.4758 -2.5851 -2.6873 -2.7827 -2.8714 -2.9537 -3.0299 -3.1005 -3.1658 -3.2262 -3.2822 -3.3342  V  t/T'  Av  V  0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  c  )  =  2  1  0  -  0  0  0  R (t)  *nR (t)  1.0000 0.7831 0.6342 0.5247 0.4406 0.3739 0.3199 0.2753 0.2382 0.2071 0.1807 0.1584 0.1393 0.1230 0.1091 0.0971 0.0868 0.0779 0.0703 0.0636 0.0579 0.0529 0.0485 0.0448 0.0414 0.0385  -0.0 -0.2445 -0.4554 -0.6449 -0.8196 -0.9837 -1.1399 -1.2898 -1.4345 -1.5747 -1.7108 -1.8429 -1.9711 -2.0954 -2.2157 -2.3320 -2.4441 -2.5519 -2.6555 -2.7546 -2.8493 -2.9395 -3.0253 -3.1066 -3.1837 -3.2565  A v  Av  Table Al5  (v ) c  =  100.0  t/T'  R (t)  *nR (t)  0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0  1.0000 0.9165 0.8427 0.7767 0.7174 0.6639 0.6155 0.5714 0.5313 0.4946 0.4610 0.4302 0.4018 0.3757 0.3515 0.3292 0.3086 0.2894 0.2716 0.2551 0.2398 0.2255 0.2121 0.1997 0.1881 0.1773  -0.0 -0.0872 -0.1712 -0.2527 -0.3321 -0.4096 -0.4854 -0.5596 -0.6325 -0.7040 -0.7743 -0.8435 -0.9118 -0.9790 -1.0454 -1.1110 -1.1758 -1.2399 -1.3032 -1.3660 -1.4281 -1.4896 -1.5505 -1.6108 -1.6707 -1.7300  A v  A y  APPENDIX  > > > > > > > > > > > > > > > > > > > > > > > > > > >  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  B  $COMPILE C COMPUTER PROGRAM 'TIME' TO C A L C U L A T E R ( T ) FROM E X P E R I M E N T A L C R E S U L T S OF 1 8 0 - T - 9 0 PULSE SEQUENCE USING R . F . F I E L D INHOMOGENEITY C CORRECTION AS SUGGESTED BY VAN P U T T E , J . MAG. R E S . , 2 ( 1 9 7 0 ) 1 7 4 . C C DEFINITIONS-INPUT C N=NO. OF POINTS IN 1 8 0 - T - 9 0 PULSE SEQUENCE IPHl=PHOTOGRAPH NO. C FOR -VE VALUES OF MAGNETIZATON IN Z - D I R E C T I O N IPH2=PHTOGRAPH NO. C FOR t V E VALUES OF MAGNETIZATION IN Z - D I R E C T I O N TEMP=TEMPERATURE C M90=MAGNETIZATION AFTER 90 DEGREE PULSE M2 70=MAGNETIZATION A F T E R C 270 DEGREE PULSE I . E . WITH T=100 OR 150 MICROSEC. IN 1 8 0 - T - 9 0 C PULSE SEQUENCE TAKEN AS +VE T ( I ) = T I M E T IN 1 8 0 - T - 9 0 PULSE C SEQUENCE M Z ( I ) = M A G N E T I Z A T I O N A F T E R A TIME T IN 1 8 0 - T - 9 0 PULSE C SEQUENCE, +VE IN F I R S T PHOTOGRAPH, -VE IN THE SECOND. C C CALCULATED Q U A N T I T I E S - O U T P U T C Xl=CORRECTION FACTOR GIVEN BY EQUATI O N ( 4 . 9 ) X2 = 2.0-X1 X3=M90*X2 C REST TERMS ARE S E L F EXPLANATORY. REAL M90,M270,MZ DIMENSION T ( 2 0 ) , M Z ( 2 0 ) , C ( 2 0 ) , D ( 2 0 ) , E ( 2 0 ) 19 R E A D ( 5 , 1 ) N,I P H I , I P H 2 , T E M P IF ( T E M P . E Q . 0 . 0 ) GO TO 18 1 FORMAT(312,F10.0) READ, ( T ( I ) , I = 1 , N ) READ, ( M Z ( I ) , 1=1,N) M90=MZ(1) M270=MZ(2)  28 X1=(M90-M270)/M90 29 X2=2.0-X1 > 30 X3=X2*M90 > 31 DO 2 1=3,N > 32 C(1)-M90-MZ(1) > 33 D(1)=C(1)/X3 > 34 E(1)=ALOG(D(1)) > 35 2 CONTINUE > 36 W R I T E ( 6 , 7 ) TEMP,1 PHI,1PH2,N > 37 7 FORMAT(////2X,'TEMPERATURE* > 38 1,'NO. OF P O I N T S * * , 1 5 / ) > 39 WRITE(6,21)X1,X2,X3 > 40 21 FORMAT(5X,'Xl=',F5.2,5X, X2 > 41 WRITE(6,25) > 42 25 FORMAT(IX, 'S.NO.',6X,'T',7X > 43 l'LN(R(T))'/) > 44 DO 2 0 1=1,N > 45 WRITE(6,8) 1,T(1),MZ(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 1 7 0 . 0 > 53 0. 0 .1 2. 4. 6. 8. 10. 12. 25. 3 0 . 3 5 . > 54 23. 20.5 - 1 7 . - 1 4 . - 1 1 . 5 -8.; - 6 . - 4 . 4. > 55 > 56 $STOP #END OF F l LE # >  >  1  CO  40. 6.5  50. 9.5  60. 80. 11.5 1 5 .  17. 20.  - 182 APPENDIX C Computer Program "MOMENT" t o 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 o f Equations (5.1), (5.3), (6.1), (7.1), (7.2), and [7.3).  The i n 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) o r (7.1) o r by both equations. [5.3) the d e n s i t y  p  In using Equation  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 o f 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  4jL R3 3 ~  °3 volume o f u n i t c e l l ( A ) (no. o f molecules o f g u e s t / c e l l )  The use o f other equations and various symbols are e i t h e r d e f i n e d i n the program o r explained i n r e s p e c t i v e chapters.  > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37  $COMPILE C COMPUTER PROGRAM 'MOMENT' TO C A L C U L A T E T H E O R E T I C A L SECOND MOMENT. C INTRAMOLECULAR PART USING E Q U A T I O N ( 5 . 3 ) O R ( 6 . 1 ) DIMENSION F A C T ( 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 ) , IC(20),D(20),E(20),F(20),G(20),T(20),P(20),Q(20),S(20),RJK(20), 1SM1(20) SM2(20) SM3(20) C N=NO. OF TOTAL CARDS FOR INTRAMOLECULAR, AN=MO. OF PROTONS IN C ONE MOLECULE READ(5,1) N,AN 1 FORMAT(12,F10.0) DO 2 1=1,N C F A C T O ) = N O . OF E Q U I V A L E N T I N T E R A C T I O N S , R(1)=D1 STANCE IN ANGSTROM C UNITS BETWEEN D I F F E R E N T ATOMS, A ( l ) = F A C T O R , 7 1 6 . 1 6 4 FOR PROTONS, C 9.994 FOR DEUTERONS, 2.216 FOR NITROGEN. READ(5,3) F A C T ( 1 ) , R ( 1 ) , A ( 1 ) 3 FORMAT(3F10.0) 2 CONTINUE SUM=0.0 DO 4 1=1,N X(1)=R(1)**6 SM(I)=(FACT(1)*A(1))/(AN*X(1)) SUM=SUM+SM(1) 4 CONTINUE W R I T E ( 6 , 1 1 ) AN 11 FORMAT(5X,'NO. OF PROTONS*',F10.0//) WRITE(6,13) 13 F O R M A T U X , *S.NO. ',8X, ' 1 NTERACTI ONS ', 16X, 'R( 1 ) ', 13X, 'R( 1 ) * * 6 ' , 12X, 1C0NSTANT',7X,'SECOND MOMENT'//) DO 5 1=1,N WRITE(6,7) 1 , F A C T ( 1 ) , R ( 1 ) , X ( 1 ) , A ( 1 ) , S M ( 1 ) 7 FORMAT(15,4F20.3,F20.2) 5 CONTINUE W R I T E ( 6 , 8 ) SUM 8 FORMAT(//5X,'TOTAL INTRAMOLECULAR S E C . MOMENT(R1G1D)=',F10.3//) C INTERMOLECULAR PART BY USING E Q U A T I O N ( 5 . 3 ) C NCELL=NO.OF MOLECULES PER UNIT C E L L , VOLUME=VOLUME OF THE UNIT C C E L L IN CUBIC ANGSTROM UNITS, AMOLWT=GRAM MOL. WT., RHO=DENS1TY /  /  > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >  38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74  C  36  37 C C C C  39  40 38 C C C  41  IN G/CC, RADIUS=RADIUS R OF E Q U A T I 0 N ( 5 . 3 ) READ, NCELL,VOLUME,AMOLWT RHO=(1.66*AMOLWT)/VOLUME W R I T E ( 6 , 3 6 ) NCELL,VOLUME,AMOLWT,RHO F 0 R M A T ( / / 1 X , » N O . OF MOL./CELL*', 15,5X,'VOLUME=',F12.3,5X,'MOL.WT=' 1,F10.3,5X,'RHO=',F5.3//) R3=(3.0*VOLUME)/(4.0*3.142*NCELL) RADIUS=(R3)**(l./3.) D1=(35 8.1*4.0*3.14 2*AN*NCELL*RHO*6.023) D2=3.0*R3*10.0*AMOLWT SINTER=D1/D2 WRITE(6,37) RADIUS,SINTER FORMAT(//5X,'MOL. RAD 1US =',F10.3,5X,* 1NTERMOL. S E C . MOMENT BY RHO = l',F10.3//) C A L C U L A T I O N OF S E C . MOMENT DUE TO I S O T R O P I C MOTION. M=NO. OF NEIGHBOURS, IF ZERO PROGRAM T E R M I N A T E S , R 1 ( J ) = D 1 STANCE OF NEIGHBOURS FROM ORIGIN MOLECULE IN ANGSTROM UN 1 T S ( R E F E R TO EQUATIONS. 3)). READ, M I F ( M . E Q . O ) GO TO 42 READ, ( R l ( J ) , J=1,M) SUM3=0.0 WRITE(6,39) F0RMAT(1X,'S.NO.',4X,'Rl(J)',5X,'M2ISO') DO 38 J=1,M RK(J)=RI(J)**6 SM3(J)=358.1*AN/RK(J) SUM3=SUM3+SM3(J) WRITE(6,40) J , R I ( J ) , S M 3 ( J ) FORMAT(15,2F10.3) CONTINUE W R I T E ( 6 , 4 1 ) SUM3 F 0 R M A T ( / / 5 X , ' T O T A L ISOTROPIC SEC.M0MENT=',F10.3//) C A L C U L A T I O N OF INTERMOLECULAR PART BY USING F ( H ) OF E Q U A T 1 O N ( 7 . 1 ) Rl=MOLECULAR RADIUS IN ANGSTROM U N I T S , I F ZERO PROGRAM T E R M I N A T E S , R l / R I ( J ) = S M A L L H OF E Q U A T I O N ( 7 . 2 ) . READ, R l  75 76 77 12 > > 78 > 79 C > 80 > 81 33 > 82 > 83 > 84 > 85 > 86 > 87 > 88 > 89 > 90 > 91 > 92 > 93 > 94 > 95 > 96 34 > 97 > 98 30 > 99 > 100 44 > 101 > 102 > 103 45 > 104 43 > 105 > 106 35 > 107 42 > 108 > 109 $DATA > 110 $STOP #ENDi OF F l LE >  >  1 F ( R l . E Q . O . O ) GO TO 42 WRITE(6 12) R1,M FORMAT(5X/'MOLECULAR RAD 1US=',F10.3,5X,'NO. OF NEIGHBOURS SUM1=0.0 RJK(J)=F(H)/(RI(J)**6) OF E Q U A T I O N ( 7 . 1 ) . WRITE(6,33) F O R M A T U X , 'S.NO. ',4X, 'Rl ',8X, '(H) ',7X, ' R J K ( - 6 ) ' ) DO 30 J « 1 , M H(J)=R1/RI(J) C(J)-H(J)*H(J) D(J)-C(J)*C(J) E(J)=1.0-C(J) F(J)=E(J)*E(J) G(J)=(5./3.)*D(J) T(d)-F(J)+G(J) P(J)=1.0-4.0*C(J) Q(J)=P(J)**3 S(J)=RK(J)*Q(J) RJK(J)=T(J)/S(J) SM1(J)=358.1*AN*RJK(J) SUM1=SUM1+SM1(J) WRITE(6 34) J , R I ( J ) H ( J ) R J K ( J ) F O R M A K 1 1 * ^ 1 0 . 3 , F 1 0 . 3,1PE15.1*) CONTINUE WRITE(6 44) FORMAT(IX,'S.NO.'/4X,'SMINTER') DO 43 J=1,M WRITE(6,45) J,SM1(J) FORMAT(15/F10.3) CONTINUE W R I T E ( 6 , 3 5 ) SUM1 FORMAT(//5X,'INTERMOL. S E C . MOMENT BY F ( H ) = ' , F 1 0 . 3 / / ) STOP END /  /  /  /  /  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0060117/manifest

Comment

Related Items