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NMR study of molecular motions in some clathrate hydrates. Khanzada, Abdul Wahab Khan 1970

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NMR STUDY OF MOLECULAR MOTIONS I N SOME CLATHRATE HYDRATES ABDUL WAHAB KHAN KHANZADA B . S c . ( H o n s . ) * U n i v e r s i t y o f S i n d , P a k i s t a n , 1 9 6 5 M . S c , U n i v e r s i t y o f S i n d , P a k i s t a n , 1 9 6 6 A THESIS SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF M a s t e r o f S c i e n c e i n t h e D e p a r t m e n t o f CHEMISTRY We a c c e p t t h i s t h e s i s a s c o n f i r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF BRJ.TISH COLUMBIA F e b r u a r y , 1 9 7 0 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f 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 that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department o f Chemistry  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada Date February 13, 1970 A B S T R A C T A n N . M . R . s t u d y o f t h e c l a t h r a t e h y d r a t e s o f S F ^ , C^Hg a n d ( C H ^ g C O h a s b e e n c a r r i e d o u t t o e x a m i n e t h e t y p e o f m o t i o n a g u e s t m o l e c u l e u n d e r g o e s i n t h e c l a t h r a t e d c a v i t y . 1 9 P n u c l e a r m a g n e t i c r e s o n a n c e s p e c t r a o f S u l p h u r H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e show i s o t r o p i c r o t a t i o n o r r e o r i e n t a t i o n 1 a b o u t a n a x i s a t r a n d o m o f t h e S F g m o l e c u l e . H m a g n e t i c r e s o n a n c e s p e c t r a o f C y c l o p r o p a n e D e u t e r a t e show h i g h l y r e s t r i c t e d r o t a t i o n up t o 240°K, a n d t h e n f r e e r o t a t i o n a b o u t 1 C j - a x i s a t h i g h t e m p e r a t u r e s . H m a g n e t i c r e s o n a n c e s p e c t r a o f A c e t o n e D e u t e r a t e show t h a t t h e C H , - g r o u p i s r o t a t i n g e v e n a t 77°K, a n d s e l f d i f f u s i o n o c c u r s a t 172°K. One b a r r i e r h i n d e r i n g d i f f u s i o n o f t h e (CH^JgCO h a s b e e n c a l c u l a t e d a n d i s 3 .9 k c a l / m o l e . i i i T A B L E O F C O N T E N T S P a g e A b s t r a c t i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i A c k n o w l e d g m e n t s v i i i D e d i c a t i o n i x CHAPTER ONE 1 INTRODUCTION 1 A . D e f i n i t i o n 1 B . B r i e f H i s t o r y 2 C . F o r m a t i o n a n d S t a b i l i t y o f C l a t h r a t e s 3 D . C o m p a r i s o n o f V a r i o u s S t u d i e s o n 6 C l a t h r a t e s CHAPTER TWO 10 THE STRUCTURE OF CLATHRATES 10 A . I n t r o d u c t i o n 10 B . C l a s s i f i c a t i o n b a s e d o n G u e s t 10 C . H o s t S t r u c t u r e 11 D. T y p e s o f G a s H y d r a t e s 14 CHAPTER THREE 18 N . M . R . THEORY & ITS A P P L I C A T I O N TO CLATHRATE HYDRATES 18 A . I n t r o d u c t i o n 18 B . L i n e S h a p e a n d L i n e W i d t h s 19 C . S e c o n d Moments o f R e s o n a n c e L i n e s .. . 20 D. R e l a x a t i o n M e c h a n i s m s 21 CHAPTER FOUR 26 EXPERIMENTAL DETAILS 26 A . M a t e r i a l s 26 B . P r e p a r a t i o n o f H y d r a t e s 27 ( i ) S u l p h u r H e x a f l u o r i d e H y d r a t e 27 a n d D e u t e r a t e ( i i ) C y c l o p r o p a n e D e u t e r a t e 28 ( i i i ) A c e t o n e D e u t e r a t e 28 C . The S p e c t r o m e t e r ' 2 9 D. S e c o n d Moments a n d L i n e W i d t h s 29 CHAPTER F I V E EXPERIMENTAL RESULTS A . S u l p h u r H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e f i ) S e c o n d Moment C a l c u l a t i o n s ( i i ) E x p e r i m e n t a l R e s u l t s B . C y c l o p r o p a n e D e u t e r a t e f x ) S e c o n d Moment C a l c u l a t i o n s ( i i ) E x p e r i m e n t a l R e s u l t s C . A c e t o n e D e u t e r a t e i ) S e c o n d Moment C a l c u l a t i o n s * ; ; : i i ) E x p e r i m e n t a l R e s u l t s CHAFFER S I X DISCUSSION OF RESULTS A . S u l p h u r H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e B . C y c l o p r o p a n e D e u t e r a t e C . A c e t o n e D e u t e r a t e CHAPTER SEVEN CONCLUSION & FUTURE PROSPECTS A . C o n c l u s i o n B . F u t u r e P r o s p e c t s B I B L I O G R A P H Y A p p e n d i x I A p p e n d i x I I L I S T O F T A B L E S D e c o m p o s i t i o n P r e s s u r e s a n d T e m p e r a t u r e s o f Some G a s H y d r a t e s C l a t h r a t e H y d r a t e s A c c o r d i n g t o G u e s t s 1Q T h e o r e t i c a l F 7 S e c o n d M o m e n t s f o r S F g ^ l 7 D 2 0 E x p e r i m e n t a l S e c o n d M o m e n t s f o r S F g ^ - ~ 1 7 D 2 0 D i s t a n c e s o f 7 H e x a k a i d e c a h e d r a f r o m H e x a k a i d e c a h e d r o n C e n t r e d a t (-ggj-) T h e o r e t i c a l S e c o n d M o m e n t s f o r C ^ H g - D e u t e r a 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 S e c o n d Moment f o r ( C H 5 ) 2 C O T h e o r e t i c a l S e c o n d Moments f o r ( C H ^ C O ~ 1 7 D 2 ° v i L 1 S T O F F I G-1U R E S 1 P - T D i a g r a m 4 2 V a r i o u s P o l y h e d r a l C a g e s F o u n d i n t h e 13 t h e C l a t h r a t e H y d r a t e s 3 The h o s t l a t t i c e o f 1Z~'1 a n d l f ' A g a s h y d r a t e s 16 4 M o t i o n o f i n t e r n u c l e a r v e c t o r OP a b o u t a x i s ON 22 5 B l o c k d i a g r a m o f 30 M c / s e c W i d e L i n e N . M . R . 30 s p e c t r o m e t e r 6 B l o c k d i a g r a m o f v a r i a n DP-60 N . M . R . 31 s p e c t r o m e t e r 7a S e c o n d Moment v s T e m p e r a t u r e D a t a o f 36 S F g - H y d r a t e & D e u t e r a t e 7b L i n e W i d t h v s T e m p e r a t u r e D a t a o f S F ^ - H y d r a t e 37 & D e u t e r a t e 8 ( i ) Some S p e c t r a o f S F g - H y d r a t e a t D i f f e r e n t 38 T e m p e r a t u r e s 8 ( i i ) Some S p e c t r a o f S F g - D e u t e r a t e a t D i f f e r e n t 40 T e m p e r a t u r e s 9 E x p e r i m e n t a l S e c o n d M o m e n t s a g a i n s t T e m p e r a t u r e 45 f o r C y c l o p r o p a n e D e u t e r a t e 10 Some S p e c t r a o f C y c l o p r o p a n e D e u t e r a t e 46 11 S e c o n d Moment & L i n e W i d t h v s T e m p e r a t u r e 50 P l o t o f A c e t o n e D e u t e r a t e I n S H v s 107T a n d l n ^ v s l ( r / T P l o t f o r A c e t o n e D e u t e r a t e The T e t r a k a i d e c a h e d r o n a n d O r i e n t a t i o n o f C y c l o p r o p a n e i n i t . A C K N O W L E D G B M E N I S To P r o f e s s o r C . A . M c D o w e l l , t h e s i s s u p e r v i s o r , who i n t r o d u c e d me t o N u c l e a r M a g n e t i c R e s o n a n c e a n d t o whom I am d e e p l y i n d e b t e d f o r h i s a d v i s e , m o r a l s u p p o r t , c o n t i n u o u s e n c o u r a g e m e n t a n d g e n e r a l p e r s o n a l a t t e n t i o n t h r o u g h o u t t h e d u r a t i o n o f t h i s p r o j e c t . To P r o f e s s o r B . A . D u n e l l f o r many v a l u a b l e , i n f o r m a t i v e a n d h e l p f u l d i s c u s s i o n s . To D r . P . R a g h u n a t h a n f o r v a l u a b l e a s s i s t a n c e i n some o f e x p e r i m e n t a l p r o c e d u r e s a n d c o n s t r u c t i v e c r i t i c i s m o n some o f r e s u l t s t h a t l e d t o a b e t t e r u n d e r s t a n d i n g o f t h e s u b j e c t . To M r . J . R i p m e e s t e r who t a u g h t me t h e o p e r a t i o n o f N . M . R . S p e c t r o m e t e r a n d t o whom I e x p r e s s my s i n c e r e a p p r e c i a t i o n f o r h i s h e l p i n k e e p i n g t h e s p e c t r o m e t e r i n w o r k i n g o r d e r a n d f o r h i s h e l p i n i n t e r p r e t a t i o n o f some o f t h e r e s u l t s . Ib my c o l l e a g u e s M e s s r s . T . T . A n g a n d S . Y . K a n g f o r t h e i r c o o p e r a t i o n s p e c i a l l y i n t h e , l a s t d a y s o f t h i s w o r k . To Professor Naseem Ahmed Naqvi 1 C H A P T E R O N E  INTRODUCTION A . D e f i n i t i o n Tne t e r m " C l a t h r a t e C o m p o u n d " a p p l i e s t o a s p e c i a l t y p e o f m o l e c u l a r compound i n w h i c h o n e c o m p o n e n t c a l l e d " H o s t " f o r m s a c a g e s t r u c t u r e i m p r i s o n i n g t h e o t h e r c o m p o n e n t c a l l e d t h e " G u e s t " . T h e h o s t u s u a l l y f o r m s a h y d r o g e n b o n d e d l a t t i c e a n d c o h e s i v e f o r c e s w h i c h e x i s t b e t w e e n h o s t a n d g u e s t a r e s o l e l y t h o u g h t t o b e v a n d e r W a a l s f o r c e s . I f t h e h o s t i s a h y d r o g e n b o n d e d w a t e r l a t t i c e , t h e c l a t h r a t e s a r e c a l l e d h y d r a t e s . The h y d r a t e s i n t h i s c a t e g o r y may b e f o r m e d b y g a s e s c a l l e d " g a s h y d r a t e s " , o r b y l i q u i d s , o r b y q u a t e r n a r y a l k y l - o n i u m , o r b y t e r n a r y a l k y l - o n i u m s a l t s . 2 B . B r i e f H i s t o r y The g a s h y d r a t e s h a v e b e e n k n o w n a l o n g t i m e , p e r h a p s t h e f i r s t (1) (2) r e p o r t w a s b y D a v y x ' i n 1811 a n d F a r a d a y v ' i n 1823. T h e n t h e w o r k s o f (3) de P o r c r a n d a n d V i l l a r d a r e p a r t i c u l a r l y n o t a b l e ; de F o r c r a n d v ' a n d V i l l a r d ^ f o r t h e d i s c o v e r y a n d p r e p a r a t i o n o f many o f g a s h y d r a t e s i n -e l u d i n g t h o s e o f t h e i n e r t g a s e s . R o o z e b o o m w / d e v e l o p e d t h e g e n e r a l f o r m o f p h a s e d i a g r a m f o r t h e g a s h y d r a t e s . P h a s e s t u d y w o r k w a s t h e n s u p p l e m e n t e d b y P i c k e r i n g ^ . The c r y s t a l s t r u c t u r e o f s u c h c o m p o u n d s w a s n o t e x p l a i n e d u n t i l 1947 w h e n P a l i n a n d P o w e l l r e p o r t e d t h e c r y s t a l s t r u c t u r e o f ( 7 ) q u i n o l - s u l p h u r d i o x i d e m o l e c u l a r c o m p l e x v . I n 1949 v o n S t a c k e l b e r g a n d c o - w o r k e r s s t a r t e d w o r k o n g a s h y d r a t e s a n d t h i s w o r k c o n t i n u e d t w e l v e y e a r s , c o n t a i n i n g n e a r l y a l l m a j o r i n f o r m a t i o n a b o u t t h e p r e p a r a t i o n , p h y s i c a l c h e m i s t r y , a n d s t r u c t u r e o f g a s h y d r a t e s . ( ® a " " k ) ^ n ^ s p e r i o d t h e w o r k o f C l a u s s e n ^ a n d t h a t o f P a u l i n g a n d M a r s h i s w o r t h y o f d e s c r i p t i o n ; t h e y e s t a b l i s h e d t h e 12 A a n d 17 1 c u b i c c l a t h r a t e s t r u c t u r e s f o r g a s h y d r a t e s . The w o r k o f G-.A. J e f f r e y a n d h i s t e a m i s a m a j o r r e c e n t s t e p i n t h i s f i e l d S e v e r a l e x c e l l e n t r e c e n t r e v i e w s ^ ^ , a n d two g o o d b o o k s ^ ^ i n E n g l i s h a r e now a v a i l a b l e o n C l a t h r a t e Compounds d e a l i n g w i t h d i f f e r e n t a s p e c t s , e . g . S t a t i s t i c a l M e c h a n i c s , T h e r m o d y n a m i c s a n d C r y s t a l S t r u c t u r e s . C . F o r m a t i o n & S t a b i l i t y o f C l a t h r a t e s Hie c o n d i t i o n o f f o r m a t i o n o f a h y d r a t e may h e r e p r e s e n t e d o y t h e e q u a t i o n M + n H 2 0 « ± M . n H 2 0 ( l - 1 ) (9) P r o c e e d i n g b y t h e m e t h o d o f v o n S t a c k e i b e r g ^ , i f a p l o t b e t w e e n p r e s s u r e a n d t e m p e r a t u r e i s m a d e , a p h a s e d i a g r a m o f s y s t e m a s shown i n ( F i g u r e 1 ) i s o b t a i n e d . T h i s d i a g r a m i s e x p l a i n e d a s f o l l o w s : -( a ) C u r v e I r e p r e s e n t s t h e s a t u r a t i o n v a p o u r - p r e s s u r e o f t h e h y d r a t e f o r m i n g s u b s t a n c e P ^ w h i c h i n t u r n i s s a t u r a t e d w i t h w a t e r v a p o u r . ( b ) C u r v e I I i s t h e v a p o u r - p r e s s u r e o f h y d r a t e f o r m i n g s u b s t a n c e M o v e r t h e h y d r a t e i n p r e s e n c e o f w a t e r . ( c ) C u r v e I I 1 i s t h e v a p o u r - p r e s s u r e o f M o v e r h y d r a t e i n p r e s e n c e o f i c e . ( d ) C u r v e I I I i s t h e v a r i a t i o n o f m e l t i n g p o i n t o f t h e h y d r a t e w i t h p r e s s u r e . ( e ) C u r v e I V i s t h e d e p r e s s i o n o f t h e f r e e z i n g p o i n t o f w a t e r d u r i n g d i s s o l u t i o n o f M i n i t u n d e r p r e s s u r e . F r o m F i g u r e 1 i t i s c l e a r t h a t t h e r e g i o n o f e x i s t e n c e o f h y d r a t e i s s i t u a t e d t o t h e l e f t o f t h e C u r v e I I a n d I I I . The i n t e r s e c t i o n o f t h e s e c u r v e s g i v e s t h e s o c a l l e d " c r i t i c a l p o i n t f o r t h e d e c o m p o s i t i o n o f h y d r a t e " . A t t h i s p o i n t h y d r a t e + M g Q s + l i q u i d + v ' a ' f c e r c o e x i s t . H e n c e i t i s a l s o c a l l e d t h e q u a d r u p l e p o i n t o f t h e s y s t e m . When t h e p r e s s u r e o f t h e s y s t e m e x c e e d s t h e c r i t i c a l p r e s s u r e , t h e n o n r a i s i n g t h e t e m p e r a t u r e , t h e h y d r a t e d e c o m p o s e s i n t o two l i q u i d p h a s e s HgO a n d M ^ i ( l u i ( j E a c h h y d r a t e i s c h a r a c t e r i z e d a t a g i v e n t e m p e r a t u r e b y a d e f i n i t e v a l u e o f d i s s o c i a t i o n p r e s s u r e F ^ s s « I n p r i n c i p l e P d i s s = p , M + p , H 2 0 C 1 " 2 ) Where P 1 ^ = v a p o u r p r e s s u r e o f h y d r a t e f o r m i n g s u b s t a n c e M o v e r h y d r a t e P ' J J . Q = v a p o u r p r e s s u r e o f w a t e r a b o v e c r y s t a l l a t t i c e o f same h y d r a t e c o n s t r u c t e d f r o m HgO m o l e c u l e s . 4 I n p r a c t i c e P , . = P ' ( S i n c e P ' » P ' _ . ) . T a b l e I g i v e s some o f t h e d i s s M N M H 2 0 ' k n o w n h y d r a t e s , t h e i r d i s s o c i a t i o n p r e s s u r e a t 0 ° C a n d t e m p e r a t u r e i n ° C a t w h i c h t h e d i s s o c i a t i o n p r e s s u r e i s 1 a t m o s p h e r e . T h e h y d r o p h o b i c g a s e s a n d l i q u i d s w h i c h a r e s l i g h t l y s o l u b l e i n w a t e r f o r m h y d r a t e s w h e n m i x e d w i t h w a t e r a t ' Q ° C u n d e r a p r e s s u r e w h i c h i s h i g h e r t h a n t h e d i s s o c i a t i o n p r e s s u r e . Tne e x c e s s o f t h e g a s i s a l l o w e d t o p a s s . I f t h e r e a c t i o n s u r f a c e s o f g a s a n d l i q u i d a r e r e n e w e d b y a g i t a t i o n , t h e p r o c e s s o f f o r m a t i o n b e c o m e s r a p i d b u t t h e c r y s t a l s a r e m i c r o i n n a t u r e , v o n S t a c k e l b e r g a n d M u l l e r ^ " ^ h a v e c o n c l u d e d f r o m t h e r m o d y n a m i c p o i n t o f v i e w t h a t h y d r o p h o b i c l i q u i d s h a v i n g B . P . a b o v e 60 ° C do n o t f o r m h y d r a t e s , b u t i f a i r i s i n s e r t e d , t h i s l i m i t c a n b e i n c r e a s e d . " TABLE I * D e c o m p o s i t i o n P r e s s u r e a n d T e m p e r a t u r e s o f Some G a s H y d r a t e s H y d r a t e f o r m e r M Xe N 2 0 C 0 2 H 2 S H 2 S e C I 2 P H 3 C 2 H 5 C 1 C H 2 C l 2 C H C I 3 C H 3 I C 2 H 5 B r C 2 H 3 C H 5 C H C 1 2 U n i t C e l l B * . P . D e c o m p . Decomp 1. D i m e n s i o n o f M P r e s s u r e ( a t m ) Temp ( ° C k ° C a t 0 ° C a t 1 ( a t m 11.97 -107 1.5 - 3.4 12 .03 - 89 10.0 - 1 9 . 3 12.04 - 79 12 .3 - 2 4 . 0 12.00 - 60 0.731 + 0.35 12 .06 - 42 0.346 + 8 . 0 12.03 o r 1 1 . 8 2 - 34 0.252 + 9.6. - - 87 1.6 - 6.4 17.30 + 13 0.201 7.2 17.53 - 10 - -17.31 + 42 0.116 -17.30 + 61 0.050 -- + 43 0.074 -- + 38 0.155 -- + 45 1.000 -- + 57 0.055 -* D a t a m a i n l y t a k e n f r o m t h e r e v i e w s o f G . A . J e f f r e y ( 11 ,12a) . 6 The w a t e r s o l u b l e c o m p o u n d s g e n e r a l l y f o r m h y d r a t e s s i m p l y o n c o o l i n g s o l u t i o n s w i t h a p p r o x i m a t e c o m p o s i t i o n s o f s t o i c h i o m e t r i c f o r m u l a e . I h e s i z e a n d s h a p e o f m o l e c u l e s a r e a l s o d o m i n a t i n g f a c t o r s i n f l u -e n c i n g c l a t h r a t e f o r m a t i o n . The u p p e r l i m i t f o r h y d r a t e f o r m i n g m o l e c u l e i s a b o u t 6.5 A . T h u s t h e o v e r a l l c o n d i t i o n s o f c l a t h r a t e f o r m a t i o n a r e ( a ) M o l e c u l e s o f s u i t a b l e s i z e a n d s h a p e ( b ) S u f f i c i e n t l y l o w w a t e r s o l u b i l i t y ( c ) S u f f i c i e n t l y h i g h v o l a t i l i t y . D . C o m p a r i s o n o f V a r i o u s S t u d i e s o n C l a t h r a t e s T n e s e s t u d i e s a r e m o s t l y r e s t r i c t e d t o h o s t a n d g u e s t i n t e r a c t i o n . I n b r o a d t e r m s t h e y w i l l b e d i s c u s s e d u n d e r t h r e e h e a d i n g s ( a ) X - r a y d i f f r a c t i o n m e t h o d s ( b ) S t a t i s t i c a l m e c h a n i c s a n d t h e r m o d y n a m i c s ( c ) O t h e r e x p e r i m e n t a l s t u d i e s . X - r a y d i f f r a c t i o n s t u d i e s o f h y d r a t e s a r e d i s c u s s e d i n C h a p t e r I I , b u t a s h o r t a c c o u n t o f ( b ) a n d ( c ) i s g i v e n h e r e . ( b ) S t a t i s t i c a l M e c h a n i c s S t a t i s t i c a l m e c h a n i c s w e r e f i r s t a p p l i e d t o c l a t h r a t e s b y v a n d e r W a a l s a n d P l a t t e e u w ^ 1 2 - 0 ) b u t t h e i r s t u d y w a s m o s t l y r e s t r i c t e d t o h y d r o q u i n o n e c l a t h r a t e s o f n o n - p o l a r g a s e s . A c c o r d i n g t o v a n d e r W a a l s a c l a t h r a t e i s c o n s i d e r e d t o b e a s o l u t i o n o f a g a s i n a s o l i d i n w h i c h t h e g a s m o l e c u l e i s f o r c e d t o move i n a c e l l o f f i x e d s i z e . I n s u c h a s y s t e m t h e i d e a o f c e l l (15) m o d e l o f L e n n a r d - J o n e s a n d D e v o n s h i r e x ' i s t a k e n . The p o t e n t i a l f i e l d i n w h i c h t h e g a s m o l e c u l e m o v e s w i t h i n i t s c a v i t y i s t h e n c a l c u l a t e d . The e n e r g y o f f o r m a t i o n a n d d i s s o c i a t i o n p r e s s u r e o f c l a t h r a t e c a n t h e n b e e x p r e s s e d i n t e r m s o f t h e L e n n a r d - J o n e s f o r c e c o n s t a n t s o f g a s c o m p o n e n t . The t h e o r y o f s o l i d h y d r a t e f o r m a t i o n u s i n g p r i n c i p l e o f s t a t i s t i c a l m e c h a n i c s h a s b e e n f u r t h e r d e v e l o p e d b y S a i t o a n d c o - w o r k e r s ^ . T h e s e a u t h o r s p r o p o s e d a m e t h o d o f c a l c u l a t i n g s e v e r a l p r o p e r t i e s o f g a s h y d r a t e s a t t e m p e r a t u r e a b o v e t h e l o w e r q u a d r u p l e p o i n t ( i n p r a c t i c e a t t y 0 ° C ) . Among t h e i r s t u d i e s a r e t h e h y d r a t e s o f C f l ^ , A r , a n d a l s o m i x e d h y d r a t e s i n t h e CH^ + A r + HgO, A r + + HgO s y s t e m s . They c o n c l u d e d t h a t f o r N g , A r c o m p l e t e f i l l i n g o f s m a l l a n d l a r g e c a v i t i e s o f h y d r a t e l a t t i c e i s o b t a i n e d a t v e r y h i g h e q u i l i b r i u m p r e s s u r e , o f t h e o r d e r o f 3400 a t m o s p h e r e s b u t CH^ r e q u i r e 1300 a t m o s p h e r e s . T h e i r c a l c u -l a t e d v a l u e s o f t h e h e a t o f f o r m a t i o n o f t h e a b o v e h y d r a t e s a g r e e d w e l l w i t h t h e v a l u e o b t a i n e d b y P - t r e s u l t s w i t h t h e a i d o f C l a u s i u s - C l a p e r o n e q u a t i o n f o r A r , N g , CH^ a s w e l l a s A r + N g , CH^ + A r m i x t u r e s . B a r r e r a n d c o - w o r k e r s u s i n g t h e same m e t h o d s made a q u a n t i t a t i v e e v a l u a t i o n o f i n f l u e n c e o f a " h e l p e r g a s " o n h y d r a t e f o r m a t i o n t e m p e r a t u r e ( l 7 a > b ) ^ T h e i r a p p r o x i m a t e c a l c u l a t i o n s f o r m i x e d h y d r a t e f o r m e d b y C H C l j a n d s h o w e d t h a t when i s i n s m a l l c a v i t i e s o f s t r u c t u r e I I a n d p a r t l y i n l a r g e c a v i t i e s t h e n /yi= - jut = -115.2 k c a l / m o l e ( l -3) w h e r e A/u- = d i f f e r e n c e i n c h e m i c a l p o t e n t i a l / U ? = c h e m i c a l p o t e n t i a l o f e m p t y c r y s t a l l a t t i c e o f h y d r a t e yt?'= c h e m i c a l p o t e n t i a l o f s t a b l e i c e . I n a b s e n c e o f Ng> t h e v a l u e o f £yu. i s -112 k c a l / m o l e a n d c o n s e q u e n t l y t h e d i f f e r e n c e i s 3*2 k c a l / m o l e a n d i s due t o t h e e f f e c t o f r i s e i n C H C l ^ h y d r a t e f o r m a t i o n t e m p e r a t u r e i n p r e s e n c e o f N g . 8 ( c ) O t h e r E x p e r i m e n t a l S t u d i e s Among t h e s e a r e h e a t c a p a c i t y , m a g n e t i c s u c e p t i b i l i t y , m e a s u r e m e n t o f d i e l e c t r i c l o s s , N.Q . R . , E . S . R . a n d N . M . R . H e a t c a p a c i t y m e a s u r e m e n t s ( 1 8 ) b y S t a v e l e y ^ ' o n h y d r o q u i n o n e c l a t h r a t e s g i v e i n f o r m a t i o n a b o u t t h e movement o f t r a p p e d g u e s t m o l e c u l e s , a n d t h e h e i g h t o f e n e r g y b a r r i e r r e s t r i c t i n g r o t a t i o n o f t h e g u e s t m o l e c u l e s . S i m i l a r i n f o r m a t i o n h a s b e e n ( 1 9 ) o b t a i n e d b y M e y e r N ' f r o m m a g n e t i c s u c e p t i b i l i t y m e a s u r e m e n t s o n o x y g e n , i n o x y g e n c l a t h r a t e a t . ~ - 0 . 2 5 - 2 ° K , a n d t h e b a r r i e r h e i g h t w a s e s t i m a t e d t o b e 1 2 8 k c a l / m o l e . The same a u t h o r s t u d i e d N.Q .R . o f ^ n ( l u i n ° l c l a t h r a t e . F r o m t h e t e m p e r a t u r e d e p e n d e n c e o f t h e q u a d r u p l e r e s o n a n c e f r e q u e n c y b e t w e e n 1 . 5 a n d 2 . 5 ° K M e y e r a n d S c o t t e s t i m a t e d b a r r i e r h e i g h t t o b e 940 k c a l / m o l e . D i e l e c t r i c m e a s u r e m e n t s o n v a r i o u s c l a t h r a t e h y d r a t e s h a v e b e e n ( 2 1 ) r e p o r t e d b y B r e y a n d L e g g o n t r i c h l o r o f l u o r o m e t h a n e h y d r a t e v ' , b y D a v i d s o n a n d c o - w o r k e r s ^ ^ " " ^ , a n d more r e c e n t l y b y D a v i e s a n d W i l l i a m s ^ ^ < A l l t h e s e s t u d i e s s h o w e d r a p i d r e o r i e n t a t i o n a n d more t h a n a s i n g l e r e l a x -a t i o n t i m e d i f f e r e n t f r o m t h a t f o r o r d i n a r y i c e . The r e l a x a t i o n t i m e f o r a c e t o n e h y d r a t e , e t h y l e n e o x i d e , a n d t e t r a h y d r o f u r a n d e t e r m i n e d b y D a v i d s o n a n d c o - w o r k e r s i s v e r y s h o r t o f t h e o r d e r o f l O - ^ s e c o n d s e v e n a t l i q u i d Ng t e m p e r a t u r e . I n t h e S F g h y d r a t e t h e r e o r i e n t a t i o n o f w a t e r m o l e c u l e s i s a f f e c t e d a n d b e c a u s e S F g h a s no d i p o l e moment, i t d o e s n o t f o r m h y d r o g e n b o n d s w i t h w a t e r a n d h e n c e a c t i v a t i o n e n e r g y a n d e n t r o p y o f a c t i v a t i o n o f t h e c l a t h r a t e a r e s i m i l a r t o t h a t o f i c e . The d i e l e c t r i c r e l a x a t i o n t i m e S F g h y d r a t e i s a b o u t 18 x 1 0 " ^ s e c . ^ 2 2 " ^ . I n f r a r e d s p e c t r o s c o p i c s t u d i e s o n A r , K r , X e h y d r a t e s h a v e b e e n r e p o r t e d b y R e d i n g t o n a n d M i l l i g a n ^ ^ a n d e v i d e n c e o f r o t a t i o n o f t h e g u e s t m o l e c u l e h a s b e e n f o u n d . S i m i l a r c o n c l u s i o n s w e r e d r a w n b y H a r v e y e t a l v f o r SOg h y d r a t e , b u t t h e y r e p o r t e d t h a t m o t i o n o f t h e S 0 2 m o l e c u l e s w a s l i m i t e d . The e l e c t r o n s p i n r e s o n a n c e s t u d i e s h a v e b e e n m o s t l y l i m i t e d t o o t h e r t y p e s o f c l a t h r a t e s e . g . N i ( C N ) 2 NH^ CgH^ . E v i d e n c e f o r r a p i d r o t a t i o n a l m o t i o n a r o u n d C~C b o n d i n h y d r o q u i n o n e c l a t h r a t e h a s r e c e n t l y ( 2 7 ) b e e n r e p o r t e d b y O h i g a s h i a n d K u r i t a ^ The u s e o f N . M . R . t o s t u d y t h e m o t i o n o f g u e s t m o l e c u l e i n s i d e t h e ( 2 8 a - c ) c a g e w a s f i r s t s t a r t e d i n t h i s l a b o r a t o r y b y G i l s o n a n d M c D o w e l l v ' a n d f u r t h e r e x t e n d e d b y M c D o w e l l a n d R a g h u n a t h a n ^ 2 ^ a ~ e ^ . A t p r e s e n t s e v e r a l w o r k e r s a r e t r y i n g t o u s e N . M . R . t o s t u d y t h e m o t i o n o f g u e s t m o l e c u l e s i n h o s t l a t t i c e s . The u s e o f N . M . R . a s s e e n i n t h e f o l l o w i n g c h a p t e r s i s w e l l s u i t e d f o r t h i s w o r k . B e c a u s e t h e g u e s t m o l e c u l e s i n c a g e s a r e f a r a p a r t i n c o m p a r i s o n t o t h e i r p o s i t i o n s i n n o r m a l l a t t i c e , a g o o d e s t i m a t e o f t h e l i n e w i d t h , s e c o n d moment a n d r e l a x a t i o n t i m e s c a n b e m a d e . S o m e t i m e s d i f f i c u l t i e s a r i s e when t h e h o s t a n d g u e s t b o t h h a v e same t y p e o f n u c l e i e . g . p r o t o n s , b u t t h i s c a n b e o v e r c o m e b y d e u t e r a t i n g t h e h o s t . The d e u t e r i u m i n t e r a c t i o n w i t h p r o t o n s i n s u c h a s y s t e m w h e r e h o s t a t o m s a r e f a r a p a r t i s n e a r l y n e g l i g i b l e . 10 C H A P T E R T W O  THE STRUCTURE OF CLATHRATE HYDRATES A » I n t r o d u c t i o n A s t h e name i m p l i e s , t h e m a i n b e a u t y i n c l a t h r a t e s i s t h e i r s t r u c t u r e . The m o s t d e t a i l e d a n d p r e c i s e i n f o r m a t i o n w h i c h i s o b t a i n e d a b o u t t h e s t r u c t u r e a n d g e o m e t r i c a l a r r a n g e m e n t o f a s s e m b l a g e o f a t o m s a n d m o l e c u l e s i n c l a t h r a t e h y d r a t e i s f r o m X - r a y d i f f r a c t i o n s t u d i e s . B . C l a s s i f i c a t i o n B a s e d o n G u e s t The c l a t h r a t e h y d r a t e s p r e s e n t l y k n o w n c a n b e g r o u p e d t o g e t h e r i n t o 4 c l a s s e s w i t h r e f e r e n c e t o c h e m i c a l p r o p e r t i e s o f g u e s t s p e c i e s . T h e s e f o u r t y p e s o f g u e s t s p e c i e s a r e g i v e n b e l o w ^ 2 - 8 ^ . ( 1 ) H y d r o p h o b i c Compounds w h i c h a r e g a s e s o r l i q u i d s a t 0 ° C . The g u e s t h e r e i s s l i g h t l y s o l u b l e i n w a t e r a n d t h e i n t e r a c t i o n b e t w e e n g u e s t a n d h o s t i s m a i n l y v a n d e r W a a l s f o r c e s . 11 (2) W a t e r s o l u b l e a c i d o g e n i c g a s e s : - Tnese g u e s t s f o r m i o n s w i t h a h y d r o l y t i c r e a c t i o n s a n d c l a t h r a t e c o m p o u n d s a r e f o r m e d u n d e r c e r t a i n c o n d i t i o n s . (3) W a t e r s o l u b l e p o l a r c o m p o u n d s w i t h a c c e p t o r o r d o n o r a n d a c c e p t o r h y d r o g e n - b o n d i n g p o t e n t i a l . (4) W a t e r s o l u b l e t e r n a r y o r q u a t e r n a r y a l k y l - o n i u m s a l t s . H e r e g u e s t s a r e c a t i o n s a n d t h e a n i o n s f o r m a p a r t o f h y d r o g e n b o n d e d l a t t i c e . A l l o f t h e s e t h r e e c l a s s e s a r e l i s t e d i n T a b l e I I w i t h t h e i r i d e a l c o m p o s i t i o n a n d s t r u c t u r e . The f o u r t h t y p e i s n o t g i v e n h e r e . C . The H o s t S t r u c t u r e The h o s t s t r u c t u r e o f t h e h y d r a t e i s a p o l y h e d r a l f r a m e w o r k o f h y d r o g e n b o n d e d w a t e r m o l e c u l e s . T h e s e p o l y h e d r a f o r m " c a g e s " w h i c h e n c l o s e t h e . ' i g u e s t m o l e c u l e s o r i o n s . The s m a l l e s t p o l y h e d r o n i n t h e s e s t r u c t u r e s i s p e n t a g o n a l d o d e c a h e d r o n JJ s h o w n i n F i g u r e 2(a) - a b a s i c p o l y h e d r a l u n i t f o r h y d r a t e s . I t may b e r e g u l a r o r d i s t o r t e d a n d i s f o r m e d b y 20 w a t e r m o l e c u l e s i . e . H ^ Q 0 2 q . I t h a s 12 r e g u l a r p e n t a g o n a l f a c e s , 20 v e r t i c e s a n d 30 e d g e s a n d s a t i s f i e s E u l e r ' s t h e o r e m o n c o n v e x p o l y h e d r a n a m e l y F + V = E + 2. Of t h e s e 40 h y d r o g e n b o n d s f o r m e d b y 20.H20, 30 f o r m e d g e s o f p o l y h e d r o n D b y (12F + 20V = 30E + 2). 20 more h y d r o g e n b o n d s c a n b e f o r m e d o n e a t e a c h v e r t e x g i v i n g t h e 4 - c o o r d i n a t i o n t o o x y g e n a t o m s . Of t h e s e 20 e x t e r n a l b o n d s , 10 w i l l b e d o n o r a n d 10 w i l l b e a c c e p t o r b o n d s . T h i s s i t u a t i o n g i v e s u s t h e i d e a o f t h r e e d i m e n s i o n a l l i n k i n g o f p o l y h e d r a a n d t h e s e c o n s i d e r a t i o n s l e d P a u l i n g i n 1957 t o s u g g e s t t h a t H ^ q 0 2 Q p e n t a g o n a l d o d e c a h e d r o n m i g h t b e 1 2 TABLE II Clathrate Hydrates According to Guests (12-a) Structure Ideal Composition 12 I Cubic 8 X . 4 6 H 2 0 6X .46H 20 17 A Cubic 8X . 1 3 6 H 2 0 1. Hydrophobic Guests Ar N2 0 2 AsH C2H| C2 H6 CH.J CH&l CH^CH2ei 8X . 1 6 H ?S . 1 3 6 H 2 0 CHjBr COS C H 3 I CHF=CF2 CH2G12 12JA Cubic 6X. 4 6 1 ^ 0 1 2 ' I Cubic 6X .46H20 17 A" Cubic 8X . 1 3 6 H 2 0 Kr H S H2Se N 90 C H 3 G I Cl„ C H 3 C H 2 C H 3 C H 3 C H G 1 2 (CH5)3CH c s 2 CH5CH2C1 CH5CHF2 CH3CH2CH3 CHCI3 Gases or Liquid at 0 C Xe CHI CHxBr COS CH2SH 3 CH^CH^r CH3CF2S1 CF2C12 CH5CH2Br CFQI3 C C I 4 (CH5)2S CH3CF2.CI 2. Water Soluble Acidogenic Gases CO, SO, C 1 0 R 3 « Water Soluble Polar Compounds ( c ^ ) 2 o CH3CH2OH (CH3)20 Structure unknown* (Possibly of gas-|bH3NH2 hydrate type) Rhoiabohedral Clathrate (not of gas hydrate type) (CH5)2NH CHjCHgNHg (CH,)^ (Ethylene Oxide) feco 40(Furan) (CH3)2CHNH2 (CH*)^ CHjtCHgJgNHg BrSI Br 2 CH =CHF C F 6 I 3 CH3N02 CF2ClBr SF 6 CH 2C1CH 2G1 CH3CH2CH2Br CCl3N0 2 CClJBr (CH^O (tetrahydrofuran) (CH 2 ) 4 0 2 (dioxane) (CH3CH2)2NH , (CH3)2CHCH2NH2 CHXCH0CHCH,NH0 3 2 3 2 (hexamethylene tetramine) (CH3CH2J5N CH:C©CH, c ? 3 L C ! ? % H 2 . 2 W " 3 CHjCHpF C H F 3 CH2CHF0 3 2 CF2Br2 cyclo-C^Hs cyclo-C5Hi6 C6H6 and others CH3(CH2) 20H Some of the structures have been reported by Jeffrey.' and co-workers^12_a^( X - a guest molecule. ( a ) D ( b ) T _ P e n t a g o n a l d o d e c a h e d r o n T e t r a k a i d e c a h e d r o n 12(F) + 20(V) = 30(E) + 2 14 - h e d r o n 14(F) + 24(V) = 36(E) .+ 2 ( c ) P P e n t a k a i d e c a h e d r o n 1 5 - h e d r o n 15(F) + 26(V)-= 39(E)+2. ' ( d ) H H e x a k a i d e c a h e d r o n 16 - h e d r o n 16(F) + 2 8 ( V ) = 42(E) + 2 W 4 I F o u r t e t r a k a i d e c a h e d r a 4(l4 - h e d r a ) 44(F) + 70(V) = 112(E) + 2 ( f ) 3T-1P ( g ) 2 T - 2 P T h r e e t e t r a k a i d e c a h e d r a - o n e p e n t a k a i d e c a h e d r o n Two t e t r a k a i d e c a h e d r a -3(l4 - h e d r a ) l(-15-hedra) two p e n t a k a i d e c a h e d r a 45(F) + 72(V) = 115(E) +2 . 2(l4 - h e d r a ) 2(l5 - h e d r a ) 46(F) + 74(V) = 1 1 8 ( E ) + 2 - F i g u r e 2$ V a r i o u s P o l y h e d r a l C a g e s F o u n d i n t h e C l a t h r a t e H y d r a t e s 14 ( 3 1 ) s i g n i f i c a n t s t r u c t u r e i n l i q u i d w a t e r w . The h y d r o g e n b o n d d i s t a n c e i n 3 D i s a b o u t 2 . 8 & a n d v o l u m e a p p r o x i m a t e l y 170 J r . I t c a n e n c l o s e a s p h e r e o f 3«4 r a d i u s a p p r o x i m a t e l y . Now i f we a s s o c i a t e D t o f o r m a r e g u l a r c r y s t a l l a t t i c e , we a r e l e f t w i t h v o i d s i . e . o t h e r t y p e s o f p o l y h e d r a , b e c a u s e o f g e o m e t r i c a l e n t i t y . The o t h e r p o l y h e d r a a r e l a r g e r t h a n D a n d t h e s e a r e g i v e n b e l o w : -( a ) T e t r a k a i d e c a h e d r o n T (14-hedron) w i t h 12 p e n t a g o n a l a n d 2 h e x a g o n a l f a c e s - F i g u r e 2 ( b ) 1 4 F = 2 4 V = 36E + 2 ( b y E u l e r ' s t h e o r e m ) V o l u m e 230 $}', f o r m e d b y 2 4 - H 2 0 ( b ) P e n t a k a i d e c a h e d r o n P ( 1 5 - h e d r o n ) w i t h 12 p e n t a g o n a l a n d 3 h e x a g o n a l f a c e s - F i g u r e 2 ( c ) 1 5 F + 2 6 V = 39E + 2 ( b y E u l e r ' s t h e o r e m ) V o l u m e 250 l?; f o r m e d b y 2 6 . H 2 0 ( c ) H e x a k a i d e c a h e d r o n H ( 1 6 - h e d r o n ) w i t h 12 p e n t a g o n a l a n d 4 h e x a g o n a l f a c e s - F i g u r e 2 ( d ) 1 6 F + 28V = 4 2 F + 2 ( b y E u l e r ' s t h e o r e m ) V o l u m e 2 6 0 iP; f o r m e d b y 2 8 . H 2 0 ( d ) C o m b i n a t i o n o f D , T, P a n d H - F i g u r e 2 ( e ) - ( g ) . D . |&ypes o f G a s H y d r a t e s Two s t r u c t u r a l t y p e s h a v e b e e n r e c o g n i z e d a c c o r d i n g t o i n v e s t i g a t i o n s o f C l a u s s e n ^ , P a u l i n g a n d M a r s h ( 1 0 ) , v o n S t a c k e l b e r g a n d M u l l e r ^ 1 3 " ^ \ M a n d l e c o r n ^ 2 ^ a n d G . A . J e f f r e y ^ 1 2 a ^ j a n d t h e s e a r e : -( 1 ) Type I : - T h e s e a r e d e s c r i b e d b y t h e s p a c e g r o u p Oh^ = Pm3m a n d a r e c h a r a c t e r i z e d b y t h e c r y s t a l l a t t i c e p a r a m e t e r 12 k . The u n i t c e l l i s 15 c o n s t r u c t e d f r o m 46 w a t e r m o l e c u l e s a n d t h e r e a r e 8 c a v i t i e s a v a i l a b l e f o r g u e s t m o l e c u l e s o u t o f w h i c h 2 a r e s m a l l a n d 6 a r e l a r g e . The s m a l l c a v i t i e s o f t y p e I a r e p e n t a g o n a l d o d e c a h e d r a a n d l a r g e o n e s a r e t e t r a k a i d e c a h e d r a . I f t h e g u e s t m o l e c u l e h a s a n e f f e c t i v e d i a m e t e r l e s s t h a n 5«1 J&> t h e h y d r a t e o f t y p e I i s f o r m e d . I n c a s e a l l t h e 8 c a v i t i e s a r e f i l l e d , t h e i d e a l f o r m u l a w i l l b e 8 X . 4 6 H 2 0 o r X . 5 | H 2 0 w h e r e X i s g u e s t m o l e c u l e e . g . A r , C H ^ , HgS e t c . ( T a b l e I i ) . On t h e o t h e r h a n d i f t h e e f f e c t i v e d i a m e t e r i s g r e a t e r t h a n $.1 I b u t l e s s t h a n 5 * 8 & , t h e s m a l l e r c a v i t i e s c a n ' t b e f i l l e d a n d h y d r a t e i s f o r m e d b y f i l l i n g o f 6 c a v i t i e s o n l y . I n t h i s c a s e , t h e i d e a l f o r m u l a w i l l b e 6 X . 4 6 H 2 0 o r X . 7 § H 2 0 w h e r e X i s g u e s t m o l e c u l e e . g . C g H g , C l 2 , S 0 2 e t c . 12 iv c u b i c c r y s t a l l a t t i c e i s s h o w n i n F i g u r e 3 ( a ) . (2) Type l i t - The h y d r a t e s o f t y p e I I a r e d e s c r i b e d b y s p a c e g r o u p O h 7 = F d 3 n w i t h a c r y s t a l l a t t i c e p a r a m e t e r o f 17*4 '-The u n i t c e l l i s c o n s t r u c t e d f r o m 136 E>,0 m o l e c u l e s . T h e r e a r e 24 c a v i t i e s , o u t o f w h i c h 16 a r e s m a l l a n d 8 a r e l a r g e . Tne s m a l l c a v i t i e s a r e p e n t a g o n a l d o d e c a h e d r a a n d a r e s l i g h t l y d e f o r m e d ; a n d l a r g e c a v i t i e s a r e h e x a k a i d e c a h e d r a . I f t h e e f f e c t i v e d i a m e t e r o f g u e s t i s l e s s t h a n 6.7 A ( e f f e c t i v e d i a m e t e r o f h e x a k a i d e c a h e d r o n ) a n d g r e a t e r t h a n 5 A* ( e f f e c t i v e d i a m e t e r o f d i s t o r t e d p e n t a g o n a l d o d e c a h e d r o n ) , t h e h y d r a t e o f t y p e I I i s f o r m e d . When a l l t h e 8 l a r g e c a v i t i e s a r e f i l l e d , t h e i d e a l f o r m u l a f o r t y p e I I w i l l b e 8 X . 1 3 6 H 2 0 o r X . 1 7 H 2 0 w h e r e X i s g u e s t m o l e c u l e e . g . C^Hg, S F g , ( C H j ) 2 C 0 e t c . Type I I , 17 A c u b i c c r y s t a l l a t t i c e i s s h o w n i n F i g u r e 3 ( b ) . ( a ) 12 . 1 c u b i c s t r u c t u r e ( T y p e I ) o O x y g e n = = H y d r o g e n b o n d s F i g u r e 3» The h o s t l a t t i c e o f 12 % a n d 17 A g a s h y d r a t e 17 ( 3 ) M i x e d o r D o u b l e H y d r a t e s ; - V/hen two t y p e s o f m o l e c u l e s a r e p r e s e n t s i m u l t a n e o u s l y i n g a s m i x t u r e , a m i x e d o r d o u b l e h y d r a t e i s f o r m e d . The i d e a l f o r m u l a f o r 1 2 ; , i o r t y p e I d o u b l e h y d r a t e i s 2 X ^ 6 X 2 . 4 6 ^ 0 o r ^ . 3 X 2 . 2 3 ^ 0 w h e r e X^ a n d Xg a r e g u e s t m o l e c u l e s , may b e s m a l l e r g a s e . g . H g S , HgSe e t c , a n d X g may b e l a r g e r g a s e . g . C g H g , ^ 2 ' ^ 2 e * c * ^ o r ^yPe o r 17 k d o u b l e h y d r a t e , t h e i d e a l f o r m u l a e w i l l b e 1 6 X 1 . 8 X 2 . 1 3 0 H 2 0 o r 2 X 1 . X 2 1 7 H 2 0 w h e r e X 1 may b e H g S , H 2 S e ( s m a l l e r g a s ) a n d X 2 may b e S F g , ( C H , ) 2 C 0 e t c . 18 C H A P T E R T H R E E N.M.R. THEORY & ITS APPLICATION TO CLATHRATE HYDRATES A. Introduction In this chapter we shall give a very brief account of N.M.R. theory as applied to clathrate hydrate. A detail account is avoided because (33a-f) several good booksv ' and some recent review articles are available (3'3g'*;3)< Moreover the subject is too broad to cover (nearly 550 publica-tions per month). The nuclei considered in this study are proton and flourine. -» When a constant magnetic field H q is applied to an atom with nuclear spin I, the energy of interaction is given by the Hamiltonian X 7L= -i«H oI z (in-1) where fr" is gyromagnetic ratio, I& the component of nuclear spin operator 2 1 9 I a l o n g t h e d i r e c t i o n o f H Q . I f now t h e r a d i o f r e q u e n c y f i e l d ( r f . ) 3-j i s a p p l i e d p e r p e n d i c u l a r t o H Q , r e s o n a n c e o c c u r s w h e n a n g u l a r f r e q u e n c y o f t h e r o t a t i n g r f f i e l d i s e q u a l t o a n g u l a r f r e q u e n c y o f L a r m o r p r e c e s s i o n uiQ i . e . 2TIV = 2 A V e = u>o = VKQ ( H I - 2 ) w h e r e V i s n u c l e a r g y r o m a g n e t i c r a t i o = /*•/!&, /A. t h e n u c l e a r m a g n e t i c m o m e n t , I n u c l e a r s p i n q u a n t u m n u m b e r a r i d i i = h/2fl S i n c e t h e r e a r e t w o a l l o w e d o r i e n t a t i o n s f o r a n u c l e u s w i t h s p i n a n d w h e n t h e s e n u c l e a r s p i n s a r e i n t h e r m a l e q u i l i b r i u m w i t h t h e t h e r m a l m o t i o n s o f t h e a t o m s i n a s o l i d , t h e l o w e r l e v e l w i t h s p i n (+§•) m u s t b e more h i g h l y p o p u l a t e d t h a n t h e u p p e r one i n a c c o r d a n c e w i t h M a x w e l l - B o l t z m a n D i s t r i b u t i o n Law s u c h t h a t N+/N- = e y - n H o / k T ( i l l - 3 ) w h e r e N+ a n d N - a r e n u m b e r s o f n u c l e i i n l o w e r (+) a n d u p p e r ( - ) l e v e l s , a n d i s e n e r g y d i f f e r e n c e b e t w e e n two l e v e l s . The m e c h a n i s m b y w h i c h t h i s e q u i l i b r i u m i s e s t a b l i s h e d i s k n o w n a s s p i n - l a t t i c e , o r t h e r m a l , o r l o n g i t u d n a l , r e l a x a t i o n t i m e T^. T^  w i l l b e d i s c u s s e d i n more d e t a i l l a t e r . T h e r e i s s e c o n d k i n d o f r e l a x a t i o n t i m e known a s s p i n - s p i n i n t e r a c t i o n o r t r a n s v e r s e r e l a x a t i o n t i m e T g , w h i c h c h a r a c t e r i z e s t h e e x c h a n g e o f e n e r g y b e t w e e n s p i n t h e m s e l v e s . B . L i n e S h a p e s a n d L i n e W i d t h s L i n e s h a p e s a r e d e t e r m i n e d b y t h e t y p e s o f i n t e r a c t i o n s b e t w e e n t h e s p i n s y s t e m s a n d t h e i r e n v i r o n m e n t w h i l e t h e w i d t h d e p e n d s o n t h e s t r e n g t h o f i n t e r a c t i o n a n d r e l a x a t i o n t i m e . Where c o l l i s i o n b r o a d e n i n g i s t h e m a i n 20 f a c t o r t h a t d e t e r m i n e s t h e l i n e w i d t h ( e . g . g a s e s a n d l i q u i d s ) , t h e r e s u l t i n g l i n e s h a p e s a r e L o r e n t z i a n . On t h e o t h e r h a n d i n r i g i d s y s t e m s w h e r e r e l a t i v e o r i e n t a t i o n s a n d p o s i t i o n s o f r a n d o m l y d i s t r i b u t e d a n d i n t e r -a c t i n g s p e c i e s do n o t c h a n g e w i t h t i m e G a u s s i a n L i n e s a r e m a n i f e s t e d ^ 6 P a ^ e I n a d d i t i o n t o w e l l d e f i n e d l i n e s h a p e s many s p e c t r a a p p e a r t o b e m i x -t u r e s o f L o r e n t z i a n a n d G a u s s i a n c u r v e s . The l i n e w i d t h S H i s a n i m p o r t a n t p a r a m e t e r i n s t u d y o f m o l e c u l a r m o t i o n . C o n v e r t i n g H t o f r e q u e n c y u n i t s b y r e l a t i o n 2Av = S*H we g e t Sv = ( V / 2 A ) %E a n d t h e e n e r g y w i d t h i s g i v e n b y o E = hco> = -K fc'SH. S o b y t h e H e i s e n b e r g U n c e r t a i n t y p r i n c i p l e £ E & T ^ f i , t h e l i f e t i m e o f s t a t e i s & T > < n / $ > E = 1 / V £ H i . e . S T =* 1 / V S H a n d t h i s &T i s c a l l e d t h e s p i n - s p i n i n t e r a c t i o n t i m e T g . Thus t h e l i n e - w i d t h i s a m e a s u r e o f T g . C . S e c o n d M o m e n t s o f R e s o n a n c e L i n e s Tne l i n e s h a p e f u n c t i o n f o r s i m p l e s y s t e m s l i k e 2 , 3» 4 s p i n s y s t e m s ( M a - d ) h a v e b e e n w o r k e d o u t i n t h e e a r l y d e v e l o p m e n t o f N . M . R . b y many a u t h o r s ' The l i n e s h a p e o f more c o m p l i c a t e d s y s t e m s l i k e c l a t h r a t e s b e c o m e s e x t r e m e l y d i f f i c u l t t o w o r k o u t . F o r t u n a t e l y V a n V l e e k J s c e l e b r a t e d f o r m u l a ^ ' f o r s e c o n d moments o f a b s o r p t i o n l i n e s m a k e s o u r t a s k e a s y . The s e c o n d moment 2 < A^H y o f a b s o r p t i o n l i n e f o r a s i n g l e c r y s t a l i s g i v e n b y ( A H 2 ) - | 1.(1. + ^ V 1 £ (5CO«V1)^ 1(III-4) w h e r e .II i s t h e n u c l e a r s p i n ; g t h e n u c l e a r S v a l u e ; p> t h e n u c l e a r m a g n e t o n ; N . t h e number o f n u c l e i a t r e s o n a n c e ; a n d 8. • i s t h e a n g l e x ^ g _> -» b e t w e e n i n t e r - n u c l e a r v e c t o r r ^ a n d e x t e r n a l l y a p p l i e d m a g n e t i c f i e l d H Q . . 21 The indices i and k refer to nuclei giving rise to absorption in question and k to the other magnetic nuclei in crystal. 2 2 For a polycrystalline material the angular factor (5cos Q-1 ) is replaced by spatial average 4 / 5 which converts -equation III - 4 , to the form The effect of motion on second moment can be calculated from the Gutowky and Pake formula^56) by multiplying ( Eq n.III-5'" with £(3COS2'V^ -1)' where Vx« is the angle between interaxuclear vector r. and the rotation 0 j-g axis as shown in >Figure-4°'» This formula holds for stepwise reorientation about an -n -fold axis and for classical rotation about any axis. Actually ( 3 3 - g ) the second moment remains invariant w > & J but according to the argument (57) of Andrew and Newingx '' i t is decreased. More detailed discussions of second moment calculations and the effects of molecular motion on i t will be given in Chapters V and VI. D. Relaxation Mechanisms Although this is the current aspect of broad line N.M.R., however, owing to the very short relaxation times of the compounds studied, and incapability of detection by available equipment this subject is not studied here. However a short account of relaxation is given. A recent review on N.M.R. relaxation in solids has been provided by R.E. Richards (33-g) (a) Spin-Lattice Relaxation Linder^*^ gives an expression for T| in term of the correlation timeC t F1G.4. MOTION OF 1NTERNUCLEAR VECTOR OP ABOUT AXIS ON 2 5 a s f o l l o w s V 5 " ^ i ( i + l ) l i T ^ F + ^ ^ r •(III-6) w h e r e b i s i n t e r p r o t o n d i s t a n c e . E x p r e s s i n g (III-6) i n t e r m s o f t h e s e c o n d moment we h a v e o r m o r e g e n e r a l l y may b e w r i t t e n a s , •(III-7) •(III-8) w h e r e C i s ' - a c o n s t a n t a n d c a n b e d e t e r m i n e d f r o m t h e e x p e r i m e n t a l m i n i m u m o f T j , o r c a n b e c a l c u l a t e d t h e o r e t i c a l l y f r o m t h e n a t u r e o f t h e m o l e c u l a r m o t i o n . The f i r s t e x p r e s s i o n i n (III-8) a r i s e s f r o m r a n d o m f l u c t u a t i o n s a t k> , a n d t h e s e c o n d f o r f l u c t u a t i o n a t 2to . T h i s s h o w s t h a t Tn v a r i e s o o 1 a s r ; i s c h a n g e d e . g . b y v a r y i n g t h e t e m p e r a t u r e o f s o l i d . C o n s i d e r i n g t h e f o l l o w i n g c o n d i t i o n s , we g e t 2 2 2 2 (1) w h e n U > 0 C C « 1 ; we c a n n e g l e c t wo c c i n (HI-8) a n d s e e t h a t oc Cc ( 2 ) w h e n u>24,»1 i ^ (3) Tj p a s s e s t h r o u g h a m i n i m u m w h e n w t c = 0 . 6 1 6 a n d a t t h a t p o i n t W . , - § (III.9) minimum J £oQ If t h e r e o r i e n t a t i o n p r o c e s s i s g o v e r n e d b y a n a c t i v a t i o n e n e r g y E , t h e v a r i a t i o n o f w i t h t e m p e r a t u r e w i l l h a v e t h e f o r m 2 4 C c = C e x p ( E / R T ) ( I I I - 1 0 ) 2 2 2 2 w h e n t o o c c « l ; l n ^ oC - E / R T , a n d w h e n ^ 0 r c » 1 ; l n T ^ cx + E / R T . k p l o t o f l n T ^ a g a i n s t i / T ' K" t h e r e f o r e g i v e s a s t r a i g h t l i n e w i t h s l o p e ( - E / R T ) a n d a m i n i m u m w h e n w o C c = 0 . 6 1 6 . Tnus m e a s u r e m e n t o f a s a f u n c t i o n o f t e m p e r a t u r e c a n t h e r e f o r e g i v e t h e p o t e n t i a l h 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 ( - E ) a n d t h e f r e q u e n c y f a c t o r f o r t h e m o l e c u l a r m o t i o n . ( b ) S p i n - s p i n R e l a x a t i o n a n d L i n e W i d t h ( 3 3 - « ) P a k e v ° ' g i v e s a n e x p r e s s i o n f o r Tg i n t e r m s o f t h e c o r r e l a t i o n t i m e T c , w h i c h i s we g e t j^ Tjy] r i g i d w h e n C c —> C o } a n d oc = ( 8 1 n 2 ) b y t h e o r y o f K u b o ( 3 9 ) a n d T o m i t a N . When m o t i o n i s r a p i d , C c i s s m a l l a n d we h a v e Tj = T g . Tnus f o r a s h o r t c o r r e l a t i o n t i m e ( e . g . f o r s e l f d i f f u s i o n ) we h a v e [ *c « ( T 2 ) r i g i d ] [T ] = [ f r i g i d [ 2 1 n 2 ] • ( 1 1 1 - 1 2 ) 1 T h e n b y u s i n g r e l a t i o n $E = C o n s t x ( S H l i n e w i d t h ) a n d i 2 t^ = C ^ e x p ( E / R T ) we g e t a r e l a t i o n l n _ - H = C o n s t . + E d ^ / R T ( 1 1 1 - 1 3 ) A p l o t o f {*& a g a i n s t K w i l l t h u s g i v e t h e a c t i v a t i o n e n e r g y f o r s e l f d i f f u s i o n ( E d i f f ) « 25 I f d i f f u s i o n i s a b s e n t a n d a n o t h e r t y p e o f n a r r o w i n g i s p r e s e n t , ( - L O a - O t h e n m o d i f i e d BPP t h e o r y v ' c a n b e u s e d . The c o r r e l a t i o n f r e q u e n c y b y t h i s t h e o r y i s g i v e n b y 2 * octfteftan\ •!n(SH2-B2)/2(C2-B2)l] L L i J (111-14) w h e r e C = l i n e w i d t h a t t e m p e r a t u r e b e l o w t r a n s i t i o n r e g i o n B = l i n e w i d t h a t h i g h t e m p e r a t u r e s &H = l i n e w i d t h i n t h e t r a n s i t i o n r e g i o n . T h e c o r r e l a t i o n f r e q u e n c y i s a s s u m e d t o o b e y A r r h e n i u s r e l a t i o n V = ^ o e x P ( ~ E r e o r i e n t a t i o n / R T ) . A p l o t o f ( H I - 1 4 ) t h u s c a n g i v e E r e o r i e n t a t i o n . 26 C H A P T E R F O U R  EXPERIMENTAL DETAILS A. Materials SFg gas was bought from Matheson Co. N.J. According to the catalogue i t was 9 8 $ pure but according to mass spectral analysis reported in Raghunathan's thesis^ 2^ i t was 9&?o pure. The possible impurities in (41a-cV this commercial gas reported by various authorsv^ ' are Fg, HF, HgO, SF^, SgF-^ Q and other sulphur fluorides, but the detectable impurities are SF^ and SgF^ Q (by I.R.). Mass spectral analysis can't determine the purity unless the spectrum of impurities is known ^ 2^. Tne only possible way to check impurities is by I.R. or by gas chromatography. I.R. analysis gave only a small peak which was not identified. Cyclopropane was obtained from the same source and was 9870 pure. It was used as such. Acetone was Fischer-spectroanalysed grade and Deuterium oxide (Stohler isotope chemicals) was 99.870 pure. Triply distilled and deionized water 27 w a s u s e d f o r S F g — ' 1 7 H 2 0 . B . P r e p a r a t i o n o f H y d r a t e s ( i ) S u l p h u r H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e (29) Tne s t a i n l e s s s t e e l c y l i n d e r a s u s e d b y R a g h u n a t h a n w a s f i l l e d w i t h a b o u t 15 m l o f DgO o r E>,0. Tne c e l l w a s p u t i n d r y i c e a c e t o n e b a t h a n d c o n n e c t e d t o v a c u u m l i n e f o r d e g a s s i n g HgO o r DgO. D e g a s s i n g w a s a c h i e v e d b y a l t e r n a t e c o o l i n g a n d h e a t i n g t h r e e o r f o u r t i m e s . DgO w a s h a n d l e d i n a n i t r o g e n s a t u r a t e d d r y b o x . A f t e r d e g a s s i n g t h e c e l l w a s d i s c o n n e c t e d f r o m v a c u u m l i n e a n d w a s d i r e c t l y c o n n e c t e d t o t h e g a s c y l i n d e r . S F g w a s t h u s d i r e c t l y p a s s e d i n t o t h e w a t e r u n t i l a p r e s s u r e o f 10 - 15 a t m o s p h e r e s w a s e s t a b l i s h e d i n s i d e t h e c e l l . The c e l l w a s t h e n p l a c e d i n a c o l d r o o m f o r two w e e k s . O u t o f 24 s a m p l e s p r e p a r e d i n t h i s m a n n e r , i t w a s o b s e r v e d t h a t t h e b e s t r e s u l t c a n b e o b t a i n e d b y k e e p i n g t h e c e l l i n c o l d r o o m f o r t w o w e e k s w i t h a h i g h p r e s s u r e o f g a s i n s i d e t h e c e l l . Some (43) s a m p l e s w h i c h w e r e k e p t f o r l e s s p e r i o d , g a v e t h e r e p o r t e d C o m p o s i t i o n ' o n l y a t t h e t o p l a y e r . The i n s i d e o f l a y e r g a v e s u c c e s s i v e l y a l e s s a m o u n t o f g a s s h o w i n g t h a t t h e g a s h a d n o t f i l l e d t h e c a v i t i e s c o m p l e t e l y . O n l y t h o s e s a m p l e s w h i c h g a v e t h e c o m p o s i t i o n S F g « — ' 1 7 ^ 0 w e r e t a k e n f o r N . M . R . s t u d y . The h y d r a t e w a s a n a l y s e d b y t a k i n g a s p e c i m e n i n a s p e c i a l l y b u i l t g l a s s a p p a r a t u s * w h i c h c o u l d d i r e c t l y b e a t t a c h e d t o a g a s b u r r e t t e . The h y d r a t e i n t h a t a p p a r a t u s w a s a l l o w e d t o d e c o m p o s e a n d t h e v o l u m e o f g a s e v o l v e d w a s n o t e d , c o n v e r t e d t o S . T . P . a n d g u e s t t o h o s t r a t i o w a s c a l c u l a t e d . Tne h y d r a t e S F g < —17H 2 ° w a s f i H e d i * 1 10mm o . d . a n d 5mm o . d . N . M . R . t u b e s a f t e r c r u s h i n g i t i n s i d e t h e t u b e a t d r y i c e - a c e t o n e t e m p e r a t u r e . * The a u t h o r i s t h a n k f u l t o M r . A . H a r d i n f o r t h i s . 28 ( i i ) C y c l o p r o p a n e D e u t e r a t e DgO w a s c r u s h e d t o a f i n e p o w d e r i n a m o r t a r u n d e r l i q u i d N 2 t e m p e r a t u r e . The f i n e l y c r u s h e d DgO w a s t h e n f i l l e d i n 5mm a n d 10mm o . d . N . M . R . t u b e s . T h e s e t u b e s w e r e t h e n d i r e c t l y c o n n e c t e d t o v a c u u m l i n e a n d w e r e d e g a s s e d f o r 2 - 3 h o u r s . A n a t t e m p t t o p r e p a r e Type I I d e u t e r a t e ( S t a b l e f r o m - 2 3 ° C t o +5°C.) i n s t a i n l e s s s t e e l p r e s s u r e c e l l f a i l e d a l t h o u g h t h e same w a s k e p t more t h a n t w o w e e k s . I t i s n e c e s s a r y f o r t h e DgO t o b e i n p o w d e r f o r m a c c o r d i n g t o H a f e m a n n a n d M i l l e r ^ ^ ) t o p e r m i t h y d r a t e f o r m a t i o n . C y c l o p r o p a n e , a s s e e n b y t h e e x p e r i m e n t i n t h e s t a i n l e s s s t e e l c e l l d o e s n o t f o r m h y d r a t e w i t h l i q u i d w a t e r o r DgO. A f t e r d e g a s s i n g c y c l o p r o p a n e g a s w a s p a s s e d i n e x c e s s i n t o t h e N . M . R . t u b e s . The t e m p e r a -t u r e o f b a t h i n w h i c h N . M . R . t u b e s w e r e p l a c e d was r a i s e d t o - 3 0 ° C i n o r d e r t h a t c y c l o p r o p a n e s h o u l d b e i n g a s e o u s f o r m a n d t h i s t e m p e r a t u r e w a s k e p t f o r 2 - 3 h o u r s . The e x c e s s o f g a s w a s t h e n s i m p l y pumped o f f . F o r t y p e I I , t h e s a m p l e t u b e was k e p t i n c o l d r o o m a t - 7 ° C f o r two w e e k s . The c o m p o s i t i o n f o r t y p e I w e r e d i f f i c u l t t o e s t a b l i s h b e c a u s e i t e x i s t e d u p t o - 2 9 ° C H e n c e t h e t u b e s w e r e s e a l e d o n t h e a s s u m p t i o n t h a t h y d r a t e i s f o r m e d a n d w e r e a n a l y s e d a f t e r c o m p l e t e N . M . R . r u n . The a n a l y s i s g a v e r o u g h l y s t o i c h i o m e t r i c c o m p o s i t i o n . ( i i i ) A c e t o n e D e u t e r a t e S t o i c h i o m e t r i c a m o u n t s o f a c e t o n e a n d DgO w e r e m i x e d i n a 10mm o . d . p y r e x N . M . R . t u b e a n d w e r e d e g a s s e d b y k e e p i n g t h e m i x t u r e i n l i q u i d N g . The s o l u t i o n o f a c e t o n e a n d DgO w a s t h e n m e l t e d a f t e r s e a l i n g , a n d t h e n s l o w l y c o o l e d t o n e a r - 40° C (45) ^ jn ttlis w a g t n e f e e z i n g o f a c e t o n e o n l y ( F . P . - 95° C ) w a s a v o i d e d a n d g o o d g u e s t t o h o s t r a t i o s w e r e e s t a b l i s h e d . 2 9 C . The S p e c t r o m e t e r The w o r k o n t h e S F g ' — 17H 20 h y d r a t e w a s d o n e o n a V a r i a n DP-60 N . M . R . s p e c t r o m e t e r u s i n g 56.4 M c / s a n d 30 M c / s c r y s t a l r f o s c i l l a t o r s a n d a 12" m a g n e t . The e x p e r i m e n t s o n t h e S F g / ~ 1 7 ^ 0 s p e c i e s w e r e d o n e o n V a r i a n V F-16 w i d e l i n e N . M . R . s p e c t r o m e t e r u s i n g a 30 M c / s r f o s c i l l a t o r , b u t t h e w o r k o n a c e t o n e d e u t e r a t e w a s d o n e o n t h e same s p e c t r o m e t e r u s i n g 30 M c / s a n d 16 M c / s r f o s c i l l a t o r s . C y c l o p r o p a n e t y p e I w o r k w a s p e r f o r m e d o n DP-60, b u t t y p e I I o n V F-16 u s i n g 30 M c / s r f o s c i l l a t o r . A b l o c k d i a g r a m o f b o t h e q u i p m e n t a n d v a r i o u s u n i t s a r e s h o w n i n F i g u r e s 5 a n d 6. Tne m a g n e t i c f i e l d s c a n r a t e w a s c a l i b r a t e d u s i n g t h e s i d e b a n d t e c h n i q u e f r o m a l i q u i d s a m p l e (HgO f o r 1 H a n d C g F g f o r ^ F ) u s i n g s i g n a l s t h a t w e r e g e n e r a t e d b y f r e q u e n c y m o d u l a t i o n o f t h e r f u n i t c a r r i e r f r e q u e n c y w i t h a k n o w n a u d i o f r e q u e n c y . Tne a u d i o s i g n a l w a s o b t a i n e d f r o m a H e w l e t t - P a c k a r d m o d e l 200 CD o s c i l l a t o r w h o s e f r e q u e n c y was m e a s u r e d w i t h a H e w l e t t - P a c k a r d 3734A e l e c t r o n i c c o u n t e r . The m o d u l a t i o n w i d t h w a s m e a s u r e d f r o m o v e r m o d u l a t e d l i q u i d s i g n a l s . The o b s e r v e d l i n e w i d t h i s t h e n 2 H m . The s w e e p f i e l d f r e q u e n c y w a s k e p t c o n s t a n t a t 8 0 c p s i n a l l e x p e r i m e n t s . D . S e c o n d M o m e n t s & L i n e W i d t h s The s e c o n d moment o f t h e N . M . R . a b s o r p t i o n d e r i v a t i v e c u r v e w e r e c a l c u -l a t e d o n I B M 360/67 u s i n g a p r o g r a m m e o f W . R . J a n z e n m o d i f i e d t o IBM 360/67. The s e c o n d moment o f N . M . R . a b s o r p t i o n d e r i v a t i v e c u r v e i s g i v e n b y t h e f o l l o w i n g e x p r e s s i o n : -O R V42I0A (I6MH,) V-4310 A RECEIVER R F a m p l i f i e r I F m i x e r * a m p l i f i e r d e f e c t o r g a i n l o c a l o s c i l l a t o r 30 Mc/sec T R A N S M I T T E R R F a m p l i l l i f t e r o s c i l l a t o r f i n e a t t e n u -a t o r c o a r s e a t t e n u -a t o r c r y s t a l 0) M O D U L A T I O N I N P U T P R E A M P L I F I E R V - 4 0 0 7 6-INCH I i V - 4 3 3 1 W I D E -L I N E P R O B E E L E C T R O -M A G N E T V - 4 2 9 5 S E L E C T O R P A N E L • c o p e p r e s e n t a t i o n s e l e c t o r f i l t e r s * g a i n V -4270 B O U T P U T C O N T R O L U N I T a u d i o a m p l i f i e r s y n c h r o -v e r t e r f w i n - T «- f r e q u e n c y l o w - p a s s f i l t e r s e l e c t o r f i l t e r O . C . a m p l i f i e r s y n c h r o v ; r t e r p h a s i n g V-4250 B SWEEP UNIT c o a r s e o t t e n u o l o r I P E N - A N D - C H A R T R E C O R D E R s y n e h r o v e r t e r d r i v e a m p l i f i e r s c o p e p h a s i n g f i n e a t t e n u : r o r s w e e p a m p l i f i e r • [J p h a s e i n v e r t e r f r e q u e n c y s e l e c t o r a u d i o a m p l i f i e r V-2200A P O W E R S U P P L Y c h o p p e r r e f e r e n c e 1 a m p l i f i e r b a t t e r y 4-Ju-g a t e t u b e s r e f e r e n c e r e s i s t o r s r e c t i f i e r V - 4 2 8 0 A F I E L D S C A N N I N G U N I T h e l i p o t s c a n r e s i s t o r b a t t e r y m o t o r s c a n s e l e c t o r FIGURE 5 BLOCK DIAGRAM OF 30 M c / s e c WIDE LINE NMR SPECTROMETER V-M O 31 V-2100 V-4280A MAG MET FIELD POWER SUPPLY SCANNING UNIT P R 0 3 2 PRE-AWP SWEEP AMPLIFIER V- 4250 A SWEEP UNIT V-43II TRANSMITTER RECEIVER <• » AUDIO 4 [OSCILLATORJ V » 4 2 7 0 A OUTPUT CONTROL RECOROER I SELECTOR UNIT F i g u r e 6 : B l o c k d i a g r a m o f V a r i a n D P - 6 0 N . M . R . S p e c t r o m e t e r 32 n = N (IV-1) w h e r e Y^ i s t h e c u r v e i n t e n s i t y i n a r b i t r a r y u n i t s ; n a r e i n c r e m e n t s a l o n g f r o m c e n t r e o f l i n e ; " s c a l e " i s n u m b e r o f g a u s s p e r i n c r e m e n t ; N i s t h e maximum v a l u e o f n . H m i s t h e p e a k m o d u l a t i o n a m p l i t u d e - a c o r r e c t i o n s u g g e s t e d b y A n d r e w # ^b-g moments w e r e c a l c u l a t e d f o r e a c h h a l f o f t h e c u r v e a n d t h e n a v e r a g e d . S p e c t r a f o r w h i c h t h e moments o f e a c h h a l f c u r v e w h i c h d i f f e r e d more t h a n IC70 f r o m t h e r e s p e c t i v e a v e r a g e s w e r e r e j e c t e d . L i n e w i d t h s w e r e m e a s u r e d b e t w e e n t h e s l o p e e x t r e m a o f t h e r e s o n a n c e a b s o r p t i o n d e r i v a t i v e c u r v e . C H A P I E R F I V E EXPERIMENTAL RESULTS A . S u l p h u r u H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e  ( i ) S e c o n d Moment C a l c u l a t i o n s T a k i n g t h e v a l u e o f g ^ ( f l o u r i n e g f a c t o r ) a n d ^>N = 5-05xlO""24, t h e v a n V l e c k ' s f o r m u l a g i v e n b y E q n . ( I I I - 5 ) r e d u c e s ( n e g l e c t i n g s e c o n d t e r m ) H e r e r ± g i s i n 1 a n d N ± = 6. T a k i n g S - F d i s t a n c e = 1 . 5 8 " A(47a - b ) f w e < 1 g e t F - F a n d F - S - F d i s t a n c e s 2.23 a n d 3»l6 •& u n i t r e s p e c t i v e l y . T h u s «*2>= ^¥\T—^ + ^-66 (V-2) ^ ' 6 1(2.23)6 (3.16)6 J ' 34 R o t a t i o n a b o u t a p a r t i c u l a r a x i s r e d u c e s t h e s e c o n d moment a n d t h i s 2 y 2 i s c a l c u l a t e d b y m u l t i p l y i n g w i t h t h e r e d u c t i o n f a c t o r ^-(3cos V- - l ) i g w h e r e i s a n g l e b e t w e e n t h e a x i s o f r o t a t i o n a n d t h e i n t e m u c l e a r v e c t o r r . . The i n t e r g u e s t s e c o n d moment i s c a l c u l a t e d b y t h e m e t h o d o f F r a t i e l l o a n d D o u g l o u s ^ ^ ^ a n d S m i t h ^ ^ " " ' 3 ) . £ h e v a l u e f o r i n t r a - g u e s t a r e c o m p a r e d w i t h t h a t o f M i l l e r a n d G u t o w s k y ^ ) n e g l e c t i n g P-F i n t e r a c t i o n f o r PFg ( P F £ i s a l s o o c t a h e d r a l a n d P-F d i s t a n c e i s same a s t h e S - F d i s t a n c e i . e . 1.58 A ) . The c a l c u l a t e d r e s u l t s a r e s u m m a r i z e d i n T a b l e I I I . TABLE I I I 1 9 T h e o r e t i c a l F 7 S e c o n d Moment f o r SF^- <~> 1 7 D 2 0 T y p e s o f M o t i o n I n t r a - G u e s t <AH2> G2 I n t r a m o l e c u l a r <AH2>for P F , : G2 (49) I n t e r - G u e s t <^H2> G 2 T o t a l <AH2> G 2 R i g i d M o l e c u l e No m o t i o n 10.66 10.53 0.14 10 . 8 0 R o t a t i o n a b o u t C g - a x i s 2.30 2.27 0.04 2.34 R o t a t i o n a b o u t C j - a x i s 2.58 2.55 0.02 2.60 R o t a t i o n a b o u t C ^ - a x i s 1.45 1.43 0.01 1.46 R e o r i e n t a t i o n a b o u t Random a x i s 0 - 0.17 ( a ) 0.17 I s o t r o p i c R o t a t i o n 0 - 0.11 ( b ) 0.11 ( a ) D . J . K r o o n ( b ) G . W . S m i t h (51) (40 - b ) 35 (ii) Experimental Results The experimental second moment for SFg^ 17H20 and SFg~ 17D20 are plotted in Figure 7a together with line width data Figure 7b. In the early study of SFg~ 17H 20 about 77 spectra were recorded at 77°K. The spectra for which average deviated from individual half-curve values by more than ICffo were rejected. The result seemed to be scattered. 'It was thought from these previous results that the SFg.— 17H20 clathrate was not in powder form. Tne next samples in 5mm or 10mm o.d. N.M.R. tubes were therefore fi l l e d by crushing SFg—17H 20.or SFg~ 17D20 under dry ice acetone tempera-ture in respective tubes. The results s t i l l showed some deviation but the signal to noise ratio was greatly improved. Spectra were run both on 30 and 60 MH_ spectrometers with different rf fields and. modulation amplitudes. Care was taken not to use high rf power to avoid saturation (although this was not observed) and low modulation (Hm< 0.3 SH). Some of the spectra with different rf fields and modulation are shown in Figure 8 (i) and ( i i ) . It is seen from these spectra that line shape and line width is rf frequency independent. No change in line width and second moment is observed. The scattered value of the second moment is due to some receiver noise in the wings of spectra which is characteristic of most of spectra. Some of the spectra at 77°K .+ 2 were noiseless in wings and their second moment was 1.40 - 0.10 G for S F 6 ^ 171^0. The average second moment at 77°K for SFg^ 1 7 H 20 of 51 spectra is + 2 1.39 - 0.21 G where the error quoted is standard deviation. The line width at 77°K for the same set is 2.44 - 0.14 G. The line shapes were analysed according to method of Alger^"^ and were found to be a mixture, or in between Lorentzian and Guassian, but i f we follow the method of Abragam (33e page 107)^ t h e l i n ? ± s n o t G a u s s i a n . 2 . 4 0 f CM co 3 +» 1 o a o o CD CO 2 . 0 0 1 . 6 0 1 . 2 0 0 . 8 0 O (SP 6) - H y d r a t e O ( S F ^ - D e u t e r a t e -O — o —o 0 . 4 0 0 . 0 0 -0-100 150 200 2 5 0 T e m p e r a t u r e ( ° K ) F i g u r e 7a: S e c o n d Moment v s ' T e m p e r a t u r e D a t a o f S F ^ - H y d r a t e & D e u t e r a t e ON 2.40 2.00 03 to +> •H •H 1.60 1.20 f - ® — o 0.80 0.40 O ( S F 6 ) - H y d r a t e © ( S F 6 ) - D e u t e r a t e 0.00 • • 70 f 100 J l _ — I 1~ 150 200 250 T e m p e r a t u r e ( ° K ) F i g u r e 7b: L i n e W i d t h v s T e m p e r a t u r e D a t a o f S F g - H y d r a t e & D e u t e r a t e F i g u r e 8 . £ i ) : Some S p e c t r a o f S F g - H y d r a t e a t D i f f e r e n t T e m p e r a t u r e s T r a c e Temp. S c a n ° K G/cm a 77 0.524 b 77 0.524 c 103 0.524 d 128 0.524 e 153 0.524 f 164 O.524 g 208 0.524 Time M o d u l a t i o n C o n s t G S e c . 0.132 1 0.132 0.132 3 0.132 1 0.132 3 0.132 1 0.132 3 . L i n e V & d t h ^ H 2 ) G G 2 1.15 0 . 1 8 1.04 0.17 1.07 0 . 1 6 1 . 0 8 0 . 1 6 1 . 0 2 0.15 1 . 0 0 0 . 1 8 1 . 0 2 0 . 1 6 F i g u r e 8 ( i i ) Some S p e c t r a o f S F g i f e u t e r a t e a t D i f f e r e n t T e m p e r a t u r e s N e a r 1 5 0 ° K t h e maximum s p e c t r a w e r e r e c o r d e d t o c o n f i r m w h e t h e r t h e ( 5 2 - a ) s h a r p t r a n s i t i o n i s t h e r e o r n o t a s r e p o r t e d i n o u r p r e v i o u s r e s u l t s ^ ' . I t i s s e e n t h a t f o r SF^-—' 171^0 s e c o n d moment i n r e g i o n o f t r a n s i t i o n 2 l i e s b e t w e e n 1 . 2 0 a n d 1.38 G a n d t h i s i n d i c a t e s no s h a r p t r a n s i t i o n . The s c a t t e r i n g o f d a t a i s a l r e a d y e x p l a i n e d . H o w e v e r i f t h e t r a n s i t i o n i s t h e r e , t h e n t h e d a t a o n f u r t h e r h i g h e r t e m p e r a t u r e o n SF^-~ / 17H 2 0 do n o t s u p p o r t t h i s . The s p e c t r a f o r S F g " - ' 17I>20 w e r e r u n o n l y o n 30 M H Z s p e c t r o m e t e r a n d s h o w e d n e g l i g i b l e n o i s e i n t a i l s . T h e r e f o r e t h e s e c o n d moment i s n o t a t a l l s c a t t e r e d . The s e c o n d moment d a t a f o r some o f s p e c t r a a r e s u m m a r i s e d i n T a b l e I V . TABLE I V E x p e r i m e n t S e c o n d Moments f o r S F ^ ^ 17DQQ T e m p e r a t u r e S e c o n d Moment T e m p e r a t u r e S e c o n d Moment ^ - g2" <5ic Q2 77 0 . 1 8 -* 0.02 193 0 . 1 8 -" 0.01 145 0.15 x ° * 0 2 2 0 6 ° * 1 8 i °*02 157 0.17 - 0.01 252 0.17 - 0.01 176 0 . 1 8 t 0.01 265 0.15 - 0.01 The o n l y p o s s i b i l i t y t o c o r r e l a t e t h e o b s e r v e d s e c o n d moment o f S F ^ ^ 171*2^ w i t h t h e o r y i s t h a t m o t i o n may b e e i t h e r i s o t r o p i c o r r a n d o m ( 5 1 ) a x i s . I f we u s e t h e m o d e l o f K r o o n w ' w h i c h g i v e s e x p r e s s i o n f o r r a n d o m a x i s m o t i o n a s b e l o w H = A ( 3 c o s 2 e - i ) s p h » 3 (v-3) w h e r e jU i s f l u o r i n e m a g n e t i c moment a n d D i s t h e d i s t a n c e o f o n e s p h e r e f r o m o r i g i n . A v e r a g i n g 0 o v e r a s p h e r e a n d p u t t i n g t h e v a l u e o f w e h a v e H s p h = 1.327 x 10 xJCud (D now Q § ) (V-4) D3 42 Tne u n i t c e l l o f S F g . — 1 7 H 2 0 i s g i v e n b y v o n S t a c k e l b e r g a n d J a h n w " ' a s 1 7 . 2 1 1. T a k i n g i n t o a c c o u n t t h e c o n t r a c t i o n o f 0 . 0 2 A a t -30° C a n d f i n d i n g t h e d i s t a n c e s b e t w e e n 7 H e x a k a i d e c a h e d r a , k e e p i n g 8 t h a s r e f e r -e n c e ( c e n t r e o n e ) w i t h t h e d a t e o f r e f e r e n c e we h a v e t h e f o l l o w i n g d i s t a n c e s s u m m a r i s e d i n T a b l e V . TABLE V D i s t a n c e s o f 7 H e x a k a i d e c a h e d r a f r o m t h e H e x a h a i d e c a h e d r o n C e n t r e d a t (Ma) U n i t C e l l E d g e D i s t a n c e s o f 4 D i s t a n c e s o f 3 H e x a k a i d e c a h e d r a H e x a k a i d e c a h e d r a 17.21 1 7-44 i 12.17 1 17.19 1 7.43 A 12.15 A 2 U s i n g t h e s e d i s t a n c e s i n E q . ( V-4) we h a v e H s p h = 0 . 1 2 6 G f o r b o t h ( 5 2 - c ) 1 7 . 2 1 a n d 17.19 & c e l l e d g e . B u t t h e D a v i d s o n a n d B r o w n s t e i n d a t a v ' o n t e t r a h y d r o f u r a n show t h a t i n s t e a d o f 3 t h e r e a r e 12 c a g e s a t 12.17 ( o r 1 2.15) A a p a r t . E x t e n d i n g t h i s l i m i t t o t h e s e c o n d c e l l a n d c o u n t i n g 12 c a g e s i n s t e a d 2 o f 3, we t h e n g e t H s p h = 0.17 G f o r b o t h 1 7 . 2 1 a n d 17.19 A c e l l e d g e w h i c h 2 i s i n e x c e l l e n t a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e o f 0 . 1 8 G a t l i q u i d Ng t e m p e r a t u r e . F o r t h e i s o t r o p i c r o t a t i o n c a s e we h a v e a v a r i e t y o f e x p r e s s i o n s . S m i t h ' s e x p r e s s i o n ^ ^ " ^ , g i v e s s e c o n d moment 0 . 1 1 G 2 a f t e r i n c l u d i n g t h e d e u t e r o n i n t e r a c t i o n . H e n c e i t i s c o n f i r m e d t h a t S F g i s r o t a t i n g a t r a n d o m a x i s e v e n a t l i q u i d Ng t e m p e r a t u r e . Tne r e s t o f t h e a r g u m e n t s a r e d i s c u s s e d i n t h e n e x t c h a p t e r . 43 B. Cyclopropane Deuterate Type I and Type II (i) Second Moment Calculations The calculated results based on the crystal structure determined 1(55: (54) ( 5 3 ) 1 by Bastiansen et al. ' are given in Table VI together with data of Hoch and Rushworth TABLE VI Tneoretical Second Moment for C^H^-Deuterate Types of Motion Intra-Guest 2 Intra-Guest 2 Total 2 Second Moment G Second Moment G Second Moment G (a) Type I ^ Hg-SD 2° Rigid Lattice 12.85 0.61 13.46 Reorienting 9-99 0 . 0 7 10.06 about Cj-axis Reorienting 3.40 0.15 3.55 about Cg-axis (b) Type II CjIL- 17D 2 0 Rigid Lattice 12.85 0.12 12.97 Reorienting 9.99 0.01 10.00 about C3-axis Reorienting 3 .40 0 . 0 3 3 . 4 3 about C2~axis (c) Hoch and Rushworth value for Cyclopropane Gas only Rigid Lattice 12.86 6.80 19.66 - 1.3 ^ I 3 ? 1 8 \ 10.54 0.76 11.30 rotation) ' C2-axis ) Not reported, rotation) * (ii) Experimental Results The spectra of cyclopropane deuterate were run on both 30 and 60 MH z spectrometers. For comparison some spectra of cyclopropane were 44 r e c o r d e d a t 77 K . The s a m p l e o f t y p e I c y c l o p r o p a n e d e u t e r a t e g a v e much n o i s e o n 30 MH , h e n c e i t w a s d o n e o n l y o n 60MH s p e c t r o m e t e r . I n t h e z z f i r s t s t u d y t y p e I w a s s t a r t e d f r o m 77 ° K a n d c o n t i n u e d up t o 2 9 1 ° K t o s e e w h a t c h a n g e s w e r e o b s e r v e d i n p a s s i n g f r o m t y p e I t o t y p e I I a n d f r o m t y p e I I t o t y p e I a g a i n . The e x p e r i m e n t a l s e c o n d moment a t 7 7 ° K w a s + 2 13.46 - 0 . 6 0 G w h e r e t h e e r r o r q u o t e d i s t h e s t a n d a r d d e v i a t i o n , - h i s i s 2 i n e x c e l l e n t a g r e e m e n t w i t h t h e o r e t i c a l r i g i d l a t t i c e v a l u e 13*46 G . The s e c o n d moment t h e n s t a r t s d e c r e a s i n g f r o m r i g i d l a t t i c e v a l u e a n d a p p r o a c h e s 10.60 G 2 a t 240 ° K w h i c h i s c o n s i s t e n t w i t h C ^ - r o t a t i o n o f 2 c y c l o p r o p a n e g u e s t ( T h e o r e t i c a l 10.06 G ) . The r o t a t i o n a r o u n d t h e t r i a d a x i s c o n t i n u e s up t o 291 ° K a n d no e v i d e n c e o f C g - r o t a t i o n i s o b s e r v e d . S i n c e t y p e I I i s s t a b l e b e t w e e n 2 5 0 a n d 2J9°K^\ i t w a s t h e r e f o r e d i f f i c u l t t o h a n d l e i n 60 M H z s p e c t r o m e t e r p r o b e . So i t w a s r u n o n 30 MH . The e x p e r i m e n t a l s e c o n d moment a g a i n s t t e m p e r a t u r e a r e p l o t t e d i n F i g u r e 9 a n d some o f t h e s p e c t r a i n F i g u r e 10. The s q u a r e s i n F i g u r e 9 i n d i c a t e s t y p e I I s t u d i e d w a s s e p a r a t e l y . A n a t t e m p t t o r e c o r d h i g h r e s o l u t i o n N . M . R . s p e c t r a f o r t y p e I i n t e m p e r a t u r e r a n g e 279 ° K t o 291 ° K f a i l e d b e c a u s e some c r y s t a l l i n e p h a s e w a s s t i l l p r e s e n t . H e n c e t h e s t u d y i n t h i s r a n g e i s l i m i t e d o n l y t o b r o a d l i n e m e a s u r e m e n t s a l t h o u g h c o m p o u n d i s n o t a b s o l u t e l y s o l i d i n t h i s i n t e r v a l . The s e c o n d moment i n t h i s r a n g e w a s s t i l l a r o u n d 2 I O.36 G s h o w i n g t h a t t h e m o l e c u l e i s s t i l l r o t a t i n g a r o u n d t r i a d a x i s . A s m a l l i n d i c a t i o n d u e t o l i q u i d p h a s e w a s o b s e r v e d , i h e l i n e w i d t h s w e r e d i f f i c u l t t o m e a s u r e a s i s s e e n f r o m F i g u r e 10. H e n c e t h e y a r e n o t p l o t t e d a n d t h e r e f o r e no c o n c l u s i o n i s d r a w n a b o u t a c t i v a t i o n e n e r g i e s f o r t h e t y p e o f 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 . 16 t o — o o o c P ° o O T y p e I s t u d i e d o n 60 MH ' Q Type I I s t u d i e d o n 30 MH X ± X 70 100 150 200 250 300 T e m p e r a t u r e ( ° K ) Figure $t E x p e r i m e n t a l - S e c o n d M o m e n t s . a g a i n s t T e m p e r a t u r e f o r C y c l o p r o p a n e D e u t e r a t e VJl 46 ( a ) T y p e I, T e m p e r a t u r e 7 7 ° K , S e c o n d Moment 13.78 G 2 ( b ) Type I, T e m p e r a t u r e 2 5 2 ° K , S e c o n d Moment 10.64 G 2 ( c ) Type I, T e m p e r a t u r e 2 9 1 ° K , S e c o n d Moment 10.66 G 2 ( d ) Type II, T e m p e r a t u r e 2 7 7 ° K , S e c o n d Moment 10.45 G 2 ( e ) Type II, T e m p e r a t u r e 2 6 3 ° K , S e c o n d Moment 10.76 G 2 F i g u r e 10:. Some S p e c t r a o f C y c l o p r o p a n e D e u t e r a t e . 47 C . A c e t o n e D e u t e r a t e : ( i ) C a l c u l a t i o n o f S e c o n d M o m e n t s : One o f t h e m o s t d i f f i c u l t t h i n g s i n c o r r e l a t i o n o f t h e o r e t i c a l s e c o n d moments w i t h e x p e r i m e n t i s t h e s c a t t e r e d c r y s t a l s t r u c t u r e d a t a o n t h e a c e t o n e h y d r a t e . A l l o f t h e c r y s t a l , a n d m o l e c u l a r s t r u c t u r e d a t a g i v e v a l u e s d i f f e r e n t t o e a c h o t h e r i n c l u d i n g t h e m o s t r e c e n t (59) d a t a o f 1969 • The m o s t r e l i a b l e s e e m s t o b e m i c r o w a v e d a t a o f J . D . S w a l l e n a n d C o s t a i n (-^ 0. H o w e v e r s e c o n d moments h a v e b e e n c a l c u l a t e d u s i n g a l l t h e d a t a b y d i f f e r e n t a u t h o r s . The c a l c u l a t i o n i s c a r r i e d o u t u s i n g e q u a t i o n III-5 a n d a l s o b y P o w l e s a n d G u t o w s k y ' s f o r m u l a ^ ^ a ~ ^ . P o w l e s a n d G u t o w s k y g i v e a f o r m u l a f o r t h e CH^ g r o u p a s f o l l o w s <AH2> . f « 2 . f (V-5) w h e r e /U i s p r o t o n m a g n e t i c m o m e n t ; R i s t h e s i d e o f e q u i l a t e r a l t r i a n g l e o n w h i c h 3 p r o t o n i n a CH^ g r o u p a r e s i t u a t e d . Tne H - H d i s t a n c e i n ( C H ^ ^ C O w a s c a l c u l a t e d b y a c o m p u t e r p r o g r a m m e b a s e d o n t h e m e t h o d o f Thompson T n i s p r o g r a m m e c a l c u l a t e s t h e c o o r d i n a t e s o f d i f f e r e n t a t o m s f r o m c r y s t a l s t r u c t u r e d a t a a n d w h e n u s e d w i t h a s u b r o u t i n e g i v e s d i s t a n c e s o f v a r i o u s a t o m s * . Tne p r o b l e m o f o r i e n t a t i o n o f t h e s e c o n d CH^ g r o u p i s s t i l l a q u e s t i o n o f d i s c u s s i o n e v e n i n r e c e n t l i t e r a t u r e . H o w e v e r we h a v e a s s u m e d a s t r u c -t u r e a c c o r d i n g t o r e f e r e n c e i n w h i c h t h e o r i e n t a t i o n o f 2CHy g r o u p i s j u s t r e v e r s e t o e a c h o t h e r . The s e c o n d i>H^ g r o u p g i v e s r i s e t o a d d i t i o n a l b r o a d e n i n g w h i c h P o w l e s a n d G u t o w s k y ^ 0 ^ name a s jf . T h i s ^ i s r e l a t e d * The a u t h o r i s t h a n k f u l t o M r . M . R . M u s t a f a f o r t h i s . 48 t o t h e s e c o n d moment b y t h e f o r m u l a < A f i 2 > t o t a l - [ < A h 2 > o n e + f] ~ ( V - 6 ) T h u s s e c o n d moments a r e c a l c u l a t e d f o r t h e r i g i d l a t t i c e a f t e r c a l c u l a t i n g t h e H-H D i s t a n c e s u s i n g a b o v e m e n t i o n e d p r o g r a m m e , a n d t h e y a r e s u m m a r i z e d i n T a h l e V I I . TABLE V I I T h e o r e t i c a l R i g i d L a t t i c e S e c o n d Moment f o r ( C H , ) 2 C 0 U s i n g E q . I l l - 5 G 2 (1 C H 3 ) A U s i n g E q . V - 5 G 1 C H , 5 I n t e r a c t i o n d u e t o : 2 n d C H | T o t a l A + B G2 R e f e r e n c e 23 . 6 8 23.69 4 . 4 8 2 8 . 1 6 58 23 . 2 0 23 . 2 1 3 . 9 4 27.14 59 2 2 . 5 1 2 2.52 4-03 2 6 . 5 4 55 2 1.33 2 1.34 3 - 3 8 24.71 56 2 2 . 5 8 2 2 . 5 9 6 . 1 8 2 8 . 7 6 57 O u t o f t h e s e f i v e d a t a o n l y two g i v e s a t i s f a c t o r y a g r e e m e n t w i t h e x p e r i -m e n t . A f t e r c h e c k i n g a l l t h e s e f i v e s e t s o n l y (58) ( M i c r o w a v e d a t a ) a n d (57) w e r e u s e d f o r t h e r e m a i n d e r o f t h e c a l c u l a t i o n s . C a l c u l a t i o n s b a s e d o n (57) a n d (58) a r e g i v e n i n T a b l e V I I I . The e x p e r i m e n t a l s e c o n d moment + 2 a t 77 ° K i s 7«45- 0 . 1 1 G w h e r e t h e e r r o r q u o t e d i s t h e s t a n d a r d d e v i a t i o n . 4 9 TABLE V I I I T h e o r e t i c a l S e c o n d Moment f o r ( C H , ) 2 C O ~ 17D 2 ° I n t r a C H , I n t e r m e t h y l I n t e r - G u e s t T o t a l G 2 y i n 1 ( C H , ) 9 C 0 G 2 S e c o n d Moment G' 2 5 / 2 * G 2 R i g i d l a t t i c e ( a ) 23 . 6 8 4 * 4 8 0 . 2 8 2 8 . 4 4 ( b ) 2 2 . 5 8 6 . 1 8 0 . 2 8 ? 9 * 0 4 ^ - r o t a t i o n ) (a) 17.76 3-36 ' one CH o n l y ) ( b ) 16.94 4.64 C x -  ) 7 * 7 6 6 0 . 2 8 21.40 0 . 2 8 2 1 . 8 6 C ^ - r o t a t i o n ) ( a ) 5 « 9 2 1 . 1 2 0 . 2 8 7.32 5 b o t h C H , ) ( b ) 5 . 6 5 1.55 0 . 2 8 7 - 4 8 C , - r o t a t i o n ) ( a ) I .48 0 . 2 8 0.07 1.73 5 b o t h C H , + ) ( b ) 1.41 0.39 0.07 1.87 C g - r o t a t i o n a r o u n d C=6 b o n d ( i i ) E x p e r i m e n t a l R e s u l t s The e x p e r i m e n t a l s e c o n d moment a n d l i n e w i d t h d a t a a r e p l o t t e d i n F i g u r e 1 1 . I t i s s e e n t h a t e x p e r i m e n t a l s e c o n d moment i s i n g o o d a g r e e m e n t w i t h t h e t h e o r y f o r b o t h CH^ g r o u p s r o t a t i n g a r o u n d C ^ - a x i s . The s e c o n d moment t h e n s l o w l y f a l l s a n d t h i s d e c r e a s e i s g r e a t l y a c c e n t u a t e d a r o u n d 167° K . A t 172°K t h e s e c o n d moment f a l l s t o z e r o s h o w i n g t h a t t h e a c e t o n e m o l e c u l e s h a v e d i f f u s e d t h r o u g h t h e h o s t l a t t i c e . The d e c r e a s e o f s e c o n d moment a f t e r 77° K may b e d u e t o some o s c i l l a t i o n o f ( C H ^ g C O a r o u n d t h e ^>C = 0 b o n d , a n d t h e b e g i n n i n g o f r o t a t i o n a r o u n d t h e C g - a x i s . The f a s t e r d e c r e a s e c o r r e s p o n d s t o more r a p i d o s c i l l a t i o n s . No e v i d e n c e o f C g - a x i s m o t i o n w a s n o t i c e d . The s a m p l e s h o w e d some h y s t e r i s e f f e c t a t t h e t h e r m a l t r a n s i t i o n p o i n t . A n a t t e m p t w a s made t o l o c a t e t h e e x a c t d i f f u s i o n t e m p e r a t u r e . Two s a m p l e s w e r e t r i e d a n d t h e t e m p e r a t u r e w a s v e r i f i e d s i x t i m e s . B e c a u s e o f t h i s h y s t e r i s e f f e c t a n d t h e a c c u r a c y o f p r e s e n t v a r i a b l e t e m p e r a t u r e a s s e m b l y t h e d i f f u s i o n t e m p e r a t u r e i s e s t i m a t e d t o b e 172° - 1 ° K . A n a t t e m p t 51 was made to measure t h i s extremely narrow e x i s t i n g l i n e using the lowest modulation, but the l i n e width successively decreased as the modulation was decreased showing that modulation broadening was present. rihe f i n a l attempt was made to measure the l i n e width using the 400 c/s sweep frequency (the maximum sweep frequency a v a i l a b l e on t h i s u n i t ) , but the noise l e v e l was too high.'to permit the observation. I t can't be measured on high r e s o l u t i o n N.M.R. spectrometer because compound was s t i l l i n s o l i d form. On the basis of the second moment i t was therefore assumed that l i n e width i s zero at t h i s temperature ( l 7 2 ° K ) . The a c t i v a t i o n energy f o r s e l f d i f f u s i o n was cal c u l a t e d using Eqn. 111-13 and a value of 5*93 kcal/mole was obtained from a p l o t of InSH versus ^rjr 0- p l o t (Figure 1 2 ) . Also Eqn. III-4 was used with B = 0. This was solved by a computer programme of G.W. Smith(^0-b) c o n v e r - t e ( j ^ 0 IBM 360/67 language. This programme solved Eqn. 111-14, made a l e a s t square f i t to the r e l a t i o n l n V e = l n v M - Ereor/RT, calculated , Ereor and then calculated the t h e o r e t i c a l best f i t to the l i n e width and q temperature p l o t . Using C - 5«15 G, B=0, we got vm = 1.85 x Kr cps and a c t i v a t i o n energy f o r s e l f d i f f u s i o n ( i n t h i s equation Ereor) = 3«97 - O.92 kcal/mole. Tne experimental c o r r e l a t i o n frequency Vc obtained by the above programme i s also p l o t t e d i n : Figure-12 as \nvc versus 10^/T. I t i s seen that both equations give the same r e s u l t s . Tne r e s u l t s of Eqn. 111-14 seem to be more accurate because they give the l e a s t square f i t t e d value. c! 1.60 1.50 1.40 1.30U 1.20 1.101 1.00 I 5 . 8 5.9 O I n S H vs 1 0 5 / T A . ln>> 1 0 5 / T E d i f f = 3*97 - O.92 k c a l / m o l e y> = 1 . 8 5 x 10 ^ c p s 8.4 8 . 0 7.6 7-2 6.8 6.4 6.0 6.0 6.1 _ 6.2 •• 6.3 6.4 6.5 1 0 5 / T (V 1) f i g u r e 1 2 ; I n S H v s IQ^/T a n d " ' In ^ v s 10 ^/ T P l o t f o r A c e t o n e - D e u t e r a t e 6.6 53 C H A P T E R S I X DISCUSSION OF RESULTS A . S u l p h u r H e x a f l u o r i d e H y d r a t e a n d D e u t e r a t e I t i s e v i d e n t f r o m t h e s e c o n d moment o f SFg#~' 17Hp0 a n d S F g ^ l 7 D 2 0 t h a t S F g m o l e c u l e i s e i t h e r r o t a t i n g a b o u t a s y m m e t r y a x i s a t r a n d o m , o r l e s s p r o b a b l y i s o t r o p i c a l l y . H o w e v e r f o r i s o t r o p i c r o t a t i o n v a r i o u s g o o d m o d e l s a r e a v a i l a b l e . I n t h e l a s t c h a p t e r a m e t h o d g i v e n b y G . W . S m i t h ^ ^ - 1 3 ) 2 w a s t r i e d a n d i t g a v e a v a l u e o f 0.11 G when t h e i n t e r a c t i o n d u e t o d e u t e r o n w a s i n c l u d e d . A n o t h e r e x p r e s s i o n t h o u g h n o t p r o p e r f o r . c l a t h r a t e s b u t g o o d f o r a n a p p r o x i m a t e e s t i m a t e i s < * * 2 > i s o t = 517 -0 N Q <5 N ± R - 6 ( V I . 1 } w h e r e N Q i s n u m b e r o f f l u o r i n e p e r m o l e c u l e , N^ i s n u m b e r o f i t h n e i g h b o u r s a n d R^ i s c e n t r e - c e n t r e d i s t a n c e b e t w e e n h e x a k a i d e c a h e d r a . T h i s e x p r e s s i o n w a s u s e d i n l a s t c h a p t e r a n d g a v e a v a l u e o f 0.11 G f o r i s o t r o p i c r o t a t i o n . 54 H o w e v e r i f we u s e t h e l a t t i c e sum ] > ^ N . R . ~ ^ f o r f . c . c . l a t t i c e , t h e n 'S'.N.R.-^ i s g i v e n b y J o n e s a n d I n g h a m ^ 2 ^ = 115.631 a " ^ w h e r e a i s t h e l a t t i c e c o n s t a n t (17 -21 %. h e r e ) . T h i s e x p r e s s i o n g i v e s u s a s e c o n d moment 2 2 o f 0 . 0 8 G a n d i f a p p r o x i m a t e d e u t e r o n c o n t r i b u t i o n 0.06 G i s a d d e d t o 2 i t , a v a l u e o f 0.14 G i s o b t a i n e d f o r i s o t r o p i c r o t a t i o n . A n o t h e r m o d e l g i v e n b y G . W . S m i t h i n h i s r e c e n t p a p e r ^ 6 ^ ^ i s t h a t ^ A H 2 > ± t = 2664 N o N m 2 / V 2 w h e r e N i s n u m b e r o f f l u o r i n e i n o n e m o l e c u l e , N i s n u m b e r o f m o l e c u l e o ' m x p e r u n i t c e l l a n d V i s c e l l v o l u m e . P u t t i n g N Q = 6, N f f l = 8 a n d V = 5097•33 ^  » 2 a v a l u e o f 0.39 G i s e x p e c t e d f o r i s o t r o p i c r o t a t i o n . A d i f f e r e n c e o f 5-15T° i n c a s e o f i s o t r o p i c o r r a n d o m a x i s r o t a t i o n i s p o s s i b l e i n t h e a s s u m e d m o d e l s . The s c a t t e r e d v a l u e s o f t h e s e c o n d moment f o r S F g ~ I7H2O a r e d u e t o two r e a s o n s ; f i r s t , b e c a u s e o f r e c e i v e r n o i s e i n t h e w i n g s ; a n d s e c o n d l y , d u e t o t h e n o n - u n i f o r m p o w d e r s a m p l e o f SFg—• 17Hg0. T h i s n o n u n i f o r m i t y o r f i n e n e s s o f t h e p o w d e r w a s t h e m a i n d i s c r e p a n c y i n c a s e o f K P F g a n d R b P F g s t u d i e d b y M i l l e r a n d G u t o w s k y ^ ^ . The f a c t i s t h a t i t i s p r a c t i c a l l y i m p o s s i b l e t o g e t a v e r y f i n e p o w d e r f o r S F g ~ 1 7 ^ 0 o r S F g — ' I7D2O, b u t t h i s i s r e d u c e d b y one f a c t o r w h i c h i s r e c e i v e r n o i s e i n t h e w i n g s . Tne s t a t i s -t i c a l a v e r a g e o f 51 s p e c t r a o u t o f w h i c h some a r e n o i s e l e s s a n d p e r h a p s t h e 2 b e s t o n e c a n o b t a i n g a v e a s e c o n d moment o f 1.39 G- w h i c h i s v e r y n e a r t o a v a l u e one c a n o b t a i n f r o m a g o o d s a m p l e . The s a t u r a t i o n p r o b l e m i s n o t p r e s e n t h e r e a s t h e s p e c t r a a r e r f f i e l d i n d e p e n d e n t a t l e a s t a t l i q u i d K g t e m p e r a t u r e . 2 I f we now a c c e p t a v a l u e o f 1.39 G f o r s e c o n d moment o f SFg<~-' 1 7 ^ 0 a t 7 7 ° K , t h e c o n t r i b u t i o n o f p r o t o n s t o t h e s e c o n d moment c a n t h e n b e 2 c a l c u l a t e d . T h i s v a l u e comes o u t t o b e 1 .21 G . I f a l l t h e p r o t o n s a r e 55 ' Q x 1 21 2 now replaced by deuterons, the second moment reduces to " 7*f£ Q n, *— = 0.07 G < Hence in case of SFg->' I7H2O the main contribution to second moment is due (52-a) to protons and not fluorines as was the case in the previous studyv The resolution of the discrepancy between the results given previously (52-a), ^guT^g 0£ Y.A. Majid et a l ^ 2 2 - < i ^ , and the present result depends on accurate T^  measurements. T^  measurement with the presently available equipment is not possible. However with the arrival of new pulse spectro-.. meter, i t is hoped that the sources of the discrepancy will be found. B. Cyclopropane Deuterate Cyclopropane which has a molecular diameter of 5»8 1 ^ ^ ) falls at the upper end of the l i s t of those gases which form type I hydrates (highest upper limit for type I 5»7 and at the lower end for type II (lower limit for type II 5«9 •&)• The free diameter of tetrakaidecahedron in which cyclopropane is trapped is 6 I. Tne tetrakaidecahedron is not spherical but i t is ellipsoidal with its axis 9» 9» and 6 1 as shown in ; Figure-13 ,^ 2~ a^• Tne fact that experimental second moment at 77°K equal to rigid lattice value is because of this upper limit of cyclopropane to form type I. If we look at the shape of cyclopropane, i t is not spherical, but oblong with maximum atomic distances equal to 1.858 and 2.521 1 (a dimension based on crystal (53) structure of Bastiansen et al v )• On the molecular scale this dimension corresponds to 4«2 and 5«8 k respectively. Thus two types of alignment of cyclopropane are possible in tetrakaidecahedron as shown in ''"Figure 1J- b, c; . Tnat shown in (l3-b) restricts to both types of rotation i.e. Cg and C^  -rota-tion, while that of Figure -12c opposes Cg rotation. Hence the proposed alignment of cyclopropane is that of Figure 13-c. Looking at Figure 13-c again i t is apparent that there is enough space for free rotation, but i t seems that cyclopropane molecule after commencing to move around C^-axis 57 s t r i k e s t h e w a l l o f c a g e a n d s u d d e n l y t h i s m o t i o n i s s t o p p e d . Some o f t h e s p e c t r a a t l i q u i d N 2 t e m p e r a t u r e g a v e a s e c o n d moment c o r r e s p o n d i n g t o C , - r o t a t i o n , a n d p o s s i b l y t h i s w a s d u e t o t h i s e f f e c t . A c c o r d i n g t o t h e a r g u m e n t o f J . H . v a n d e r w a a l s ^ ^ ) w n e n t h e r e i s r a t t l i n g , t h e o b l o n g m o l e c u l e a f t e r s t r i k i n g t h e w a l l o f t h e c a g e r e o r i e n t s i t s e l f p a r a l l e l t o t h e w a l l o f t h e c a g e . T h i s a r g u m e n t c a n b e a p p l i e d t o t h i s t y p e o f s i t u a t i o n . A t 240°K (-33°C) a c o m p l e t e m o t i o n s t a r t s a n d t h i s i s p o s s i b l y d u e t o e x p a n s i o n o f l a t t i c e . M c l n t y r e a n d P e t e r s o n ^ ' h a v e c a r r i e d o u t d e t a i l e d a n a l y s i s o f t h e r m a l e x p a n s i o n o f t h e l a t t i c e o f e t h y l e n e o x i d e . The u n i t c e l l e d g e o f c y c l o p r o p a n e h y d r a t e a s r e p o r t e d b y H a f e m a n n a n d M i l l e r ^ ^ ) i s 12.14 -A a t 5 ° C . I n t h e g r a p h o f M c l n t y r e a n d P e t e r s o n . u n f o r t u n a t e l y t h e d a t u m c o r r e s p o n d i n g t o 5°C i s n o t g i v e n . H o w e v e r i f t h e c o m p o s i t i o n i s s t o i c h i o m e t r i c a c e l l o f 12.06 A a t 0°C c o n t r a c t s t o 12.02 a t 240°K, o r i n o t h e r w o r d s t a k i n g i n t o a c c o u n t t h e u n c e r t a i n t y i n d e t e r m i n a t i o n o f c e l l 4. c o n s t a n t b y H a f e m a n n a n d M i l l e r i . e . 12.14 - 0.1 A , t h i s may c o n t r a c t t o 12.00 A a n d t h i s w i l l c e r t a i n l y r e d u c e t h e f r e e d i a m e t e r o f t e t r a k a i d e c a h e d r o n . B e l o w 240°K a f u r t h e r r e d u c t i o n i s p o s s i b l e , t o a r o u n d 11.94 A . T h i s r e d u c -t i o n i s t h e r e f o r e l i k e l y t o b e r e s p o n s i b l e f o r r i g i d i t y o f c y c l o p r o p a n e m o l e -c u l e i n t h e h y d r a t e . A s t h e t e m p e r a t u r e i s i n c r e a s e d , t h e f r e e d i a m e t e r o f t e t r a k a i d e c a h e d r o n i n c r e a s e s a n d h e n c e o s c i l l a t i o n o r s t r i k i n g o f m o l e c u l e w i t h w a l l i n c r e a s e s a n d t h i s c a u s e s a r e d u c t i o n o f t h e s e c o n d m o m e n t . A t 240°K t h e m o l e c u l e s t o p s s t r i k i n g t h e w a l l s a n d a r e g u l a r C ^ - r o t a t i o n t a k e s p l a c e . F r o m F i g u r e -13c i t i s c l e a r t h a t C 2 - m o t i o n i s n o t p o s s i b l e i n a n y c a s e b e c a u s e t h e f r e e s p a c e o n b o t h s i d e s i s o n l y 0.1 A . H e n c e no v a l u e o f s e c o n d moment c o r r e s p o n d i n g t o ( ^ - r o t a t i o n w a s o b s e r v e d i n o u r e x p e r i m e n t . I n t h e e a r l y s t u d y w h i c h w a s s t a r t e d f r o m 77°K a n d c o n t i n u e d t o 291°K, 58 i t w a s o b s e r v e d t h a t t h e same t y p e I when c o n v e r t e d t o t y p e I I a t 2 5 0 ° K s h o w e d C ^ - o r i e n t a t i o n a l t h o u g h some f r e e g a s m u s t b e p r e s e n t . No s i g n a l d u e t o t h i s f r e e g a s w a s o b s e r v e d p o s s i b l y i t was n o t d e t e c t e d b y s p e c t r o -m e t e r o r t h e g a s p r e s s u r e k e p t i t i n t y p e I i n t h i s s h o r t i n t e r v a l o f t e m p e r a t u r e . H o w e v e r w h e n t y p e I I w a s s t u d i e d s e p a r a t e l y , i t s h o w e d C , - o r i e n t a t i o n . S i n c e t h e s i g n a l d u e t o f r e e g a s i s n o t s e e n i n t h i s i n t e r v a l , t h i s p r o v e s t h a t t h e 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 i s t h e m a i n c o n t r i b u t i o n t o t h e s e c o n d m o m e n t . 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 i s n e g l i g i b l e t o t h e t o t a l s e c o n d moment ( w i t h i n e x p e r i m e n t a l e r r o r ) . F r o m 2 7 8 ° K t o 2 9 1 ° K , we h a v e a g a i n t h e t y p e I d e u t e r a t e , a n d i t i s s u r p r i s i n g t h a t a l t h o u g h t h i s compound i n t h i s t e m p e r a t u r e r a n g e i s s e m i l i q u i d , a s m a l l t r a c e o f l i q u i d s i g n a l was n o t i c e d a n d t h e l i n e w a s s t i l l b r o a d . T h i s s o r t o f b e h a v i o u r i s c o n s i s t e n t w i t h P a u l i n g ' s v i e w o f (66) s t r u c t u r e o f l i q u i d w a t e r ^ ' w h i c h s a y s t h a t t h e p e n t a g o n a l d o d e c a h e d r a a r e p r e s e n t e v e n i n l i q u i d w a t e r . I t seems t h a t t h e h y d r a t e f r a m e w o r k s t i l l r e m a i n s a n d c y c l o p r o p a n e i s s t i l l t h e r e i n t e t r a k a i d e c a h e d r o n , r o t a t i n g a r o u n d t h e C ^ - a x i s . C . | A c e t o n e D e u t e r a t e One o f t h e i n t e r e s t i n g p o i n t s o f o u r d i s c u s s i o n h e r e i s t h e c a l c u l a -t i o n o f t h a t p o r t i o n o f r i g i d l a t t i c e s e c o n d moment due t o s e c o n d C H ^ - g r o u p o f a c e t o n e . The m o s t o b v i o u s d i f f i c u l t y i s t h e o r i e n t a t i o n o f p r o t o n s i n C H ^ - g r o u p w h i c h i s r o t a t i n g e v e n a t l i q u i d Ng t e m p e r a t u r e . T h i s c a l c u l a t i o n i s s t i l l s u b j e c t o f some d i s c u s s i o n . I n t h e e a r l y p a p e r s o f P o w l e s a n d G u t o w s k y ( ^ ) t h e r e i s no i n d i c a t i o n o f t h i s c a l c u l a t i o n . I n a n o t h e r p a p e r P o w l e s a n d K a i l ^ p o i n t o u t how t h i s w a s d o n e f o r I s o - b u t y l b r o m i d e . T h i s w a s d o n e b y t a k i n g a v a n d e r W a a l s d i a m e t e r f o r p r o t o n s e q u a l t o 2.4 A a n d a v a l u e o f 5.6 G w a s o b t a i n e d . O u r v a l u e s b a s e d o n two d i f f e r e n t c r y s t a l s t r u c t u r e s a n d t h e m o s t p o s s i b l e a n d r e a s o n a b l e o r i e n t a t i o n o f p r o t o n s , 2 a r e 4 « 4 8 a n d 6 . 1 8 G , w h i c h j u s t b r a c k e t s P o w l e s a n d K a i l ' s v a l u e . Y u k i t o s h i e t a P ' a s s u m e d i n t h e i r c a l c u l a t i o n f o r h e x a m e t h y l d i s i l a n e t h a t m e t h y l g r o u p s h a v e t h e i r p r o t o n s r a n d o m l y o r i e n t e d o n a c i r c l e f o r m e d 2 b y 3 - h y d r o g e n a t o m s o f C H ^ - g r o u p a n d e s t i m a t e d a v a l u e o f 1.04 G f o r t h e two n e i g h b o u r i n g C H ^ - g r o u p s . We c a n ' t c o m p a r e t h i s v a l u e b e c a u s e i n t h e i r c a s e CH^ g r o u p s a r e a t t a c h e d t o S i a t o m w i t h S i - C d i s t a n c e e q u a l t o 1.88 A. O u r C B y g r o u p s a r e a t t a c h e d t o C a t o m s w i t h a C - C d i s t a n c e o f 1.515 a n d 1.52 A. T n i s s h o r t e r d i s t a n c e b r i n g s two CH^ g r o u p s c l o s e r a n d h e n c e l a r g e r 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 i s e x p e c t e d . The same a r g u m e n t c a n b e a p p l i e d t o S m i t h ' s c a l c u l a t i o n ^ 0 - 1 3 ^ f o r t h e CH^ g r o u p s o f ( C H ^ ) ^ S i . E a d e s e t a l ^ ^ c a r r i e d o u t t h i s c a l c u l a t i o n f o r 2 , 2 - a n d 2 , J - d i m e t h y l b u t a n e b y a s s u m i n g C H ^ - g r o u p s t o s t o p a t d i f f e r e n t p o s i t i o n s a n d s h o w e d t h a t i n t r a a n d i n t e r c o n t r i b u t i o n s a r e i n s e n s i t i v e t o t h e r e l a t i v e m e t h y l g r o u p o r i e n t a t i o n . T h i s d o e s n o t i n v a l i d a t e o u r m o d e l . H o w e v e r i n a l a t t e r p a p e r E a d e s e t a l ^ ^ i n t h e c a s e o f 2 , 2 - a n d 2,3 - d i m e t h y l b u t a n e , a s s u m e d t h e o r i e n t a t i o n o f C H ^ - g r o u p t o b e s u c h t h a t one p r o t o n i n e a c h m e t h y l g r o u p h a s t h e same z - c o o r d i n a t e a s t h e c a r b o n a t o m o f t h a t g r o u p . O u r 2 o r i e n t a t i o n i s e x a c t l y s i m i l a r t o t h i s . A n o t h e r e s t i m a t e o f 3.48 G b a s e d o n t h e m o d e l t h a t 3 p r o t o n s i n a C H ^ - g r o u p c o l l a p s e d i n t o t h e c e n t r e o f ( 7 1 ) e q u i l a t e r a l t r i a n g l e i s g i v e n b y H i d e a k i C h i h a r a e t a l v ' f o r t e r t i a r y b u t y l i o d i d e , b u t i n t h i s t h e G-C d i s t a n c e i s t a k e n e q u a l t o 1.54 k. A n o t h e r 2 ( 7 2 ) v a l u e o f 6.4 G i s g i v e n b y K o i d e w ' f o r h e x a m e t h y l e t h a n e u s i n g C - C d i s t a n c e , : o f 1.54 A, b u t no d e t a i l s o f t h e c a l c u l a t i o n a r e g i v e n . S i n c e b o t h o u r c r y s t a l s t r u c t u r e d a t a h a v e C - C d i s t a n c e n e a r l y t h e s a m e , s o i n 2 v i e w o f a l l t h e s e c a l c u l a t i o n s a v a l u e o f 4«48 G i s l i k e l y t o b e more r e a s o n a b l e . H e n c e i t i s shown b y o u r c a l c u l a t i o n s t h a t m i c r o w a v e d a t a ^ ^ i s more r e l i a b l e a n d a c c u r a t e t h a n t h e o t h e r e x p e r i m e n t a l r e s u l t s 5 6 , 57> 59) (59) ' i n c l u d i n g t h e m o s t r e c e n t o n e v 7 ' . 60. Some h y s t e r e s i s o c c u r e d a t t r a n s i t i o n t e m p e r a t u r e o f 172°K, w h e r e d i f f u s i o n b e g i n s . No e v i d e n c e o f h y s t e r e s i s w a s o b s e r v e d i n t h e d i e l e c t r i c C22 Q } {22 d} s t u d y o f W i l s o n & D a v i d s o n ^ ~ ' a n d t h a t o f D a v i e s a n d W i l l i a m s ^ ~ . H o w e v e r e v i d e n c e o f r o t a t i o n e v e n a t l i q u i d N g t e m p e r a t u r e i s c o n s i s t e n t w i t h N . M . R . s t u d i e s r e p o r t e d b y b o t h g r o u p s . The h y s t e r e s i s b e h a v i o u r may b e p r e s e n t b e c a u s e s u c h b e h a v i o u r i s o b s e r v e d b y P o w l e s a n d G u t o w s k y i n a l i k e s e r i e s o f c o m p o u n d s ^ ^ ) . The e n e r g y b a r r i e r t o s e l f d i f f u s i o n h a s b e e n e v a l u a t e d b y two m e t h o d s a n d b o t h g i v e same v a l u e , 3«9 k c a l / m o l e . The a c c u r a c y o f t h i s v a l u e i s d e p e n d e n t o n t h e a c c u r a c y o f t h e m e a s u r e m e n t o f t h e l i n e w i d t h . A t h i g h t e m p e r a t u r e s n o i s e l i m i t s t h e a c c u r a c y o f m e a s u r e m e n t o f l i n e w i d t h . H o w e v e r b o t h m e t h o d s g i v e n e a r l y same v a l u e . T h i s c o n f i r m s t h e a c c u r a c y o f m e a s u r e m e n t o f l i n e w i d t h a n d b o t h m e t h o d s a r e s a t i s f a c t o r y a n d h a v e b e e n u s e d b y p r e v i o u s w o r k e r s , S m i t h f o r P e n t a e r y t h r i t o l ^ ) ^ A n d e r s o n a n d (73) S l i c h t e r f o r m e t h y l s u b s t i t u t e d c o m p o u n d s N ' a n d r e c e n t l y b y Okuma e t a l f o r h y d r o g e n c h l o r i d e . ( 6'1 C H A P T E R S E V E N  CONCLUSION & FUTURE PROSPECTS A . C o n c l u s i o n T n r e e h y d r a t e s h a v e b e e n s t u d i e d o u t o f w h i c h o n e e x i s t s i n b o t h t y p e I a n d t y p e I I f o r m s . The o t h e r two a r e t y p e I I o n l y . The d a t a o n S F g a r e j u s t t o a t t e m p t t o c o n f i r m t h e t w o s h a r p t r a n s i t i o n s w h i c h w e r e r e p o r t e d i n p r e v i o u s w o r k . I n t h e p r e s e n t s t u d y o n S F g ~ ITHgO a n d SFg-—' 1 7 D 2 0 no e v i d e n c e o f t h i s s h a r p t r a n s i t i o n w a s c l e a r l y o b s e r v e d , b u t t h e r e s u l t s i n d i c a t e d a n i s o t r o p i c r o t a t i o n o r a r e o r i e n t a t i o n a b o u t a x i s a t r a n d o m . I n t h e c a s e o f c y c l o p r o p a n e i n a t y p e I h y d r a t e , t h e e x p e r i m e n t a l r e s u l t . s h o w e d t h a t c y c l o p r o p a n e d i d n o t r o t a t e f r e e l y i n t h e t e t r a k a i d e c a -h e d r o n c a v i t i e s down t o 2 4 0 ° K . A f t e r t h i s t h e m o l e c u l e s h o w e d r e o r i e n t a t i o n a b o u t a c h o s e n C ^ - a x i s o f s y m m e t r y . T h i s t y p e o f m o t i o n w a s p r o l o n g e d a g a i n b o t h i n t y p e I I a n d t y p e I h y d r a t e s . No e v i d e n c e o f m o t i o n a b o u t C g - a x i s w a s o b s e r v e d . 62 T h e t h i r d c l a t h r a t e s t u d i e d w a s t h e A c e t o n e d e u t e r a t e . M e t h y l g r o u p r o t a t i o n w a s p r e s e n t e v e n a t l i q u i d Ng t e m p e r a t u r e , a s i s u s u a l w i t h t h i s t y p e o f c o m p o u n d . Out o f f i v e c r y s t a l s t r u c t u r e d a t a w h i c h g a v e v a l u e s d i f f e r e n t t o e a c h o t h e r , t h e b e s t w a s s e l e c t e d o n t h e b a s i s o f N . M . R . The a c t i v a t i o n e n e r g y o f d i f f u s i o n w a s f o u n d t o b e 3 - 9 k c a l / m o l e f o r t h e a c e t o n e m o l e c u l e f r o m l i n e w i d t h d a t a . B . F u t u r e P r o s p e c t s S F g h y d r a t e s t i l l l e a v e s some q u e s t i o n s t o b e a n s w e r e d . I t w i l l b e n e c e s s a r y t o go t o some l o w e r t e m p e r a t u r e s a t w h i c h t h e r e o r i e n t a t i o n a r o u n d some c h o s e n a x i s o f s y m m e t r y may b e p r e s e n t . M o r e o v e r , t h e s e d a t a s h o u l d b e s u p p l e m e n t e d b y T^  m e a s u r e m e n t s . Some d o u b l e h y d r a t e s may b e i n t e r e s t i n g t o s t u d y t o s e e t h e f o r m o f m o l e c u l e b e h a v i o u r i n p e n t a k a i d e c a h e d r a . R e c e n t l y some X - r a y d a t a o n a m i n e s h a v e b e c o m e a v a i l a b l e . T h e y f o r m d i f f e r e n t t y p e s o f h y d r a t e s . I t s h o u l d b e w o r t h w h i l e t o i n v e s t i g a t e t h e m o t i o n o f g u e s t a m i n e s i n d i f f e r e n t t y p e s o f c a g e s w h i c h a r e n o t s i m i l a r t o t y p e I a n d t y p e I I h y d r a t e s . F i n a l l y t h e m o s t r e c e n t a s p e c t o f N . M . 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Phys., 16, 327 0-948). (b) E.R. Andrew and R. Bershon, J. Chem. Phy., 18, 159 ( 1 9 5 0 ) . (c) R. Bershon and H.S. Gutowsky, J. Chem. Phys., 22, 651 ( 1 9 5 4 ) * (d) K. Tomita, Phy. Rev., 8°, 429 ( 1 9 5 3 ) . 35. J.H. Van Vleck, Phy. Rev., JA> n 6 s ( 1 9 4 8 ) . 36. H.S. Gutowsky and G.E. Pake, J. Chem. Phys., 18, 162 (1950) 37. E.R. Andrew and R.A. Newing, Proc. Roy. Soc. (London), J2, 959 (1958) 38. S. Linder, J. Chem. Phys., 26, 900, (1957) 39. R. Kubo and K. Tomita, J. Phy. Soc. Japan, % 888 (1954) 40. (a) N. Bloembergen, E.M. Purcell and R.V. Pound (BPP), Phy. Rev., , x I i , 679 (1948) (b) G.W. Smith, J. Chem. Phys., 42, 4229 (1965) (c) G.W. Smith, J. Chem. Phys., -£3_, 4325 ( 1 9 6 5 ) . 41. (a) R.T. Lagemann and A.E. Jones, J. Chem. Phys., 19_, 534 (1951) (b) D. Edelson and K.B. McAfee, ibid., 1£, 1311 (1951) (c) A. Lattre, ibid., 20, 520 ( 1 9 5 2 ) . 42. D.C. Frost, Personal Communication. 43« L.D. Shortland and A.I. Robinson, Can. J. Chem. Eng., 42(l), 38 (1964) 44. D.R. Hafemann and S.L. Miller, J. Phys. Chem., _JJ> 1392» 1398 (1969) 45. A.S..Quist and H.S. Frank, ibid., 65_, 56O (1961) 46. E.R. Andrew, Phy. Rev., ^ 1, 425 (1953) 47. (a) J. Guant, Trans. Faraday Soc, jjO, 546 (1954) (b) W.H. Hackelmann and P.S. Hubbard, J. Chem. Phys., J5S_, 2688 (1963) 6:6 48. A. Pratiello and D.C. Douglous, J. Chem. Phys., 41, 974 0-964). 49- G.R. Miller and H.S. Gutowsky, ibid., J59_, 1983 (1963) .50. R.S. Alger "Electron Paramagnetic Resonance", Theory and Applications, Inter Science N.Y., 1968, page 42. 51. D.J. Eroon, Phillips. Res. Rep., 1%, 501 ( i 9 6 0 ) 52. (a) C.A. McDowell and P. Raghunathan, Mol. Phys., 1^, 331 D-967) (b) M. Von Stackelburg and W. Jahns, Z. Electrochem., £8, 163 (1954) (c) S. Brownstein, D.W. Davidson and D. Fiat, J. Chem. Phys., 46, 1454 ( 1 9 6 7 ) . 53. 0. Bastiansen. F.N. Fritsch and K. Hederberg, Acta Cryst., 1J_, 538 ( 1 9 6 4 ) . 54* M.J.R. Hoch and F.A. Rushworth, "Nuclear Magnetic Resonance in Chemistry" Ed. Pescue, page 343 ( 1 9 6 5 ) . 55« R.W. Allen, H.J.M. Boven, L.E. Sutton and 0. Bastiansen, Trans.«Faraday Soc, 41, 991 ( 1 9 5 2 ) . 56. Y. Kimura and Y. Kurita, J. Chem. Soc. Japan, _7_2, 396 ( l 9 5 l ) . 571 (a) 0. Bastiansen and H. Viervoll, Acta Chem. Scand., 2, 702 (1948) (b) 0. Bauer - quoted in Ref. (57-a). 58. J.D. Swallen and C.C. Costain, J. Chem. Phys., J51, 1562 ( 1 9 5 9 ) . 59* C. Kato, S. Konaka. T. Iigima and M. Kimura, Bull. Chem. Soc Japan, 42, 2148 ( 1 9 6 9 ) . 60. J.G. Powles and H.S. Gutowsky, J. Chem. Phys., 21, I 6 9 5 , 1704 ( 1 9 5 3 ) . 6 1 . H.B. Tnompson, ibid., 47, 3407 ( 1 9 6 7 ) . 62. J.E. Jones and A.E. Ingham, Proc. Roy. Soc. (London), A107, 636 ( 1 9 2 5 ) quoted in Ref. (40-b). 63. G.W. Smith, J. Chem. Phys., $0, 3595 (1969) 64. J.H. Van der Waals, J. Phys. Chem. Solids, 18, 82 ( 1 9 6 I ) 65. J.A. Mclntyre and D.R. Petersen, J. Chem. Phys., 4 J , 3850 (1967) 66. L. Pauling, Trans. Int. Cofn. on the H-bond, Ljubljana ( 1 9 5 7 ) 67. J.G. Powles, J.A.E. Kail, Proc. Phys. Soc, 1^, 833 (1959) 68. '' T. Yukitoshi, H. Suga, S. Seki and J. Itoh, J. Phys. Soc. Japan, 12, 506 (1957) 69. R.G. Eades, G.P. Jones, J.P. Llewellyn and K.W. Terry, Proc. Phys. Soc. (London), 124 ( 1 9 6 7 ) . 6;7 70. R.G. Ea<ies, T.A. Jones and J.P. Llewellyn, ibid., £1, 632 (1967) 7 1 . H. Chihara, M. Otsuru and H. Seki, Bull. Chem. Soc. Japan, 39* 2145 (1966) 72. T. Koide, ibid., 40, 2026 (1967) 73. J.E. Anderson and W.P. Slichter, J. Chem. Phys., 44, 3647 (1966) 74« H. Okuma, N. Nakamura and H. Chihara, J. Phys. Soc. Japan 24, 452 (1968). APPENDIX I CALCULATION OF EXPERIMENTAL SECOND MOMENTS FROM N.M.R. DERIVATIVE'CURVES C C r. COMPUTER PROGRAMME WRITTEN BY W.R.JANZEN.MODIFIED TO IBM 3 6 0 / 6 7 . THIS PROGRAMME CALCULATES SECOND MOMENTS FROM BOTH.HALVES OF DERIVATIVE CURVES AND COMPUTES INDIVIDUAL AND AVERAGE VALUES OF C c c SECOND MOMENTS. DEFINITIONS - INPUT K= TRACE NUMBER E.G. AA04,L=NUMBER OF DATA POINTS ON ABSCISSA, c c c J=NUMBER OF DATA POINTS ON ABSCISSA ON OTHER HALF OF"THE CURVE, X=SCAN IN GAUSS PER INCREMEMENT ON ABSCISSA,Y=MODULATION TN GAUSS PEAK WIDTH-A CORRECTION SUGGESTED BY E.R.ANDREW PHY. REV. ,91,425 c c c (1953),KTEMP=TEMPERATURE IN DEGREE KELVIN,N(I)=LENGTH OF ORDINATE. DEFINITIONS - OUTPUT K=TRACE NUMBER E.G. AA04,KTEMP=TEMPER ATURE IN DEGREE K E L V I N , S ( l ) = c c SECOND MOMENT DUE TO FIRST HALF OF THE CURVE,S(2)=SECOND MOMENT DUE TO OTHER HALF,SM=AVERAGE SECOND MOMENT OF BOTH HALVES. DIMENSION N ( 4 8 ) . A ( 2 ) , B ( 2 ) , S ( 2 ) 27 19 PRINT 2 7 FORMAT (1X,42H TRACE TEMP S ( l ) S<2) SECOND MOMENT) READ 20,K,L , J ,X, Y,KTEMP, ( N M ) , 1 = 1 ,48) 20 FORMAT ( A 4 , 2 1 4 , F 6 . 3 , F 7 . 3 , 15/24 13/2413) DO 1 M=l,2 A(M)=0. 1 8(M)=0. DO 21 1=1,L A ( 1 ) •= A ( 1 ) + FL 0 AT ( I * I * I* N ( I ) ) 21 B ( 1) = BI 1) + F L 0 A T ( I. * N ( I) ) CONTINUE DO 22 I=1,J O N A(2)=A{2)+ FLOAT < I * I * I*N ( I + 2 ' t ) ) C D B(?)=B(2)+ F L O A T ( I * N < I + 2 4 ) ) 22 C O N T I N U E DO 26 M = l , 2 S ( M ) = 0 . 3 3 3 * X * X ' * A - ( M ) /B{ M)-0.25*Y * Y 26 C O N T I N U E S M = 0 . 5 * ( S ( 1 ) + Si 2 ) ) P R I N T 2 8 t K t K T E M P f S ( 1 ) . S ( 2 ) t S M 28 F O R M A T . ( l X , A 5 , I 6 t l X , 2 F 7 . 2 t 5 X , F 7 . 2 ) GO TO 1 9 30 S T O P END O N A P P E N D I X I I M O D I F I E D B P P L I N E N A R R O W I N G A N A L Y S I S ' C C c I F . Y O U DO A L L T H E S E C A L C U L A T I O N S BY H A N D A N D P L O T T H E M I T I S E A S I E R TD S E E WHAT I S G O I N G O N . M O D I F I E D B P P L I N E N A R R O W I N G A N A L Y S I S c c c D I F F E R S FROM O R I G I N A L C A S E BY I N C L U S I O N OF T E R M . AG I N N U M E R A T O R OF B P P . EQN P H Y S R E V 1 9 4 8 VOL 73. . P 6 7 9 . FOR PRO TONS A G = 7 6 7 . 5 T H I S P R O G R A M U S E S T H E M O D I F I E D B P P EQN T O D E R I V E . T H E C O R R E L A T I O N c c c F R E Q U E N C Y , M A K E S A L E A S T S Q U A R E S F I T OF L N ( C O R . F R Q ) TO A S T . L I N E WHEN P L O T T E D V S . 1 / R T , D E R I V E S T H E A C T I V A T I O N E N E R G Y AND I N F . T E M P . C O R R E L . F R E Q . F R O M THE F I T . P R O G R A M T H E N R E V E R S E S T H E P R O C E S S TO G I V E c c c T H E O R E T I C A L F I T TO THE L I N E W I D T H V S . T E M P . D A T A P L O T . D E F I N I T I O N S - I N P U T A L P H A=DAT A C O M M E N T S E . G . N A M E OF C O M P O U N D , D A T E . . c c c. T E M P . I N D E G . K E L V I N . , D E L T A H = L I N E W I O T H ( G A U S S } I N N A R R O W I N G R E G I O N ( L E S S T H A N C , G R E A T E R T H A N B ) . C = A V G . L I N E W I D T H B E L O W T R A N S I T I O N . B = A V G . L I N E W I D T H A B O V E T R A N S I T I O N . AG=P AR A M E T E R D E P E N D E N T ON N U C L R c c • c S P E C I E S , = 7 6 7 . 6 FOR H . D E L M I N = M I N L I N E W I D T H I N R E G I O N OF T H E O R E T I C F I T , G R E A T E R T H A N B . O E L M A X = M A X L I N E W I D T H I N R E G I O N OF T H E O R E T I C F I T , L E S S T H A N C . D E L I NC= L I N E W I D T H I N C R E M E N T FOR T H E O R Y F I T . N = N O . OF < c c c D A T A P O I N T S . C A L C U L A T E D Q U A N T I T I E S - O U T P U T " C O R F R Q = D E R I V E D C O R R E L . F R E Q . R E C I P T = 1 / T E M P . X L N F R Q = L N - , OF C Q R F R Q . c c c R C P R T = 1 / ( R * T E M P ) , W H E R E R - 1 . 9 8 6 9 C A L / D E G - M O L E . F A C T = A C T I V A T I O N E N E R G Y C C A L / M O L E ) t E R E A C T = E R R O R I N S A M E , F R Q M A X = I N F . T E M P . C O R R E L . F R E Q . I N T H E O R . F I T . E R F R Q = I T S E R R O R . T H D E L = L I N E -c c " WIDTH- I N T H E O R . F I T . T H T E M P = T E M P . I N T H E O R . F I T . T H F R Q = C O R R E L . F R E Q I N T H E O R . F I T . o D I M E N S I O N T E M P ( 5 0 0 ) , D E L T A H l 5 0 0 ) , C O R F R Q ( 5 0 0 ) , R E C I P T ( 5 0 0 J , 1 X L N F R Q ( 5 0 0 ) , X S I N ( 5 0 0 ' ) , X C O S ( 5 0 0 ) , X T A N ( 5 0 0 ) , R C P R T < 5 0 0 ) , A L P H A ( 2 0 ) , 2 T H . F R Q 1 5 0 0 ) , T H T E M P < 5 0 0 ) , T H D E L < 5 0 0 ) , T H S N < 5 0 0 ) , T H C O ( 5 0 0 ) , T H T A N ( 5 0 0 ) , 3 X ( 5 0 0 ) , Y( 5 0 0 ' ) , X 2 < 5 0 0 ) , Y 2 I 5 0 0 ) P R I N T 1 0 1 0 F O R M A T ( I X . 4 9 H N M R L I N E W I D T H D A T A T R E A T E D A C C O R D I N G TO G . W . S M I T H ) 1 0 1 R E A D ( 5 , 2 0 ) ( A L P H A { I ) ,1 = 1 , 1 9 ) 2 0 F O R M A T ( 1 9 A 4 ) W R I T E ( 6 , 3 0 ) ( A L P H A ! I ) , 1 = 1 , 2 0 ) , 3 0 F O R M A T ( 1 X , 2 0 A 4 ) . 3 5 R E A D ( 5 , 4 0 ) N , C , B , A G 4 0 F O R M A T ( I 5 . 3 F 6 . 2 ) R E A D . ( 5 , 5 0 ) ( T E M P H ) , D E L T A H ( I ) , I = l , N ) 5 0 F O R M A T ( 1 0 F 8 . 3 ) DO 6 0 .1 = 1 , N X S I N ( I ) = S I N ( 1 . 5 7 0 7 * « D E L T A H ( I . ) * * 2 - B * * 2 ) / ( C * * 2 - B * * 2 ) ) X C O S l I ) = C O S ( 1 . 5 7 0 7 * ( O E L T A H l I ) * * 2 - B * * 2 ) / < C * * 2 - B * * 2 ) ) X T A N ( I ) = X S I N ( I ) / X C O S ( I ) C O R F R Q I I ) = A G * D E L T A H ( I ) / X T A N ( I ) X L N F R Q ( I ) = A L O G ( C O R F R 0 ( I ) ) R E C I PT ( I ) = 1 . / T E M P I I ) R C P R T ( I ) = ! . / ( 1 . 9 8 6 9 * T E M P I I ) ) 6 0 C O N T I N U E 6 5 W R I T E ( 6 , 7 0 ) N , C , B , A G 7 0 F O R M A T ( 1 H O , 5 X , 2 H N = , I 5 , 5 X , 2 5 H R I G I O L A T T I C E L I N E W I D T H = , F 1 0 . 5 , 5 X , 2 0H I N A R R O W E D L I N E W I D T H = , F 1 0 . 5 , 5 X , 4 H A G = , F 1 5 . 5 ) P R I N T 8 0 8 0 F O R M A T ( 1 H 0 , 1 X , 1 6 H T E M P ( 0 E G K E L V I N ) , 4 X , 1 6 H L I N E W I D T H ( G A U S S ) , 4 X , 1 1 4 H C 0 R . F R E Q . ( C P S ) , 1 0 X , 6 H 1 / T E M P , 1 0 X , 1 3 H L N ( C O R . F R E Q . ) , 1 I X , 4 H 1 / R T ) W R I T E ( 6 . 9 0 ) ( T E M P ( I ) , D E L T A H ( I ) , C 0 R F R 0 ( I ) , R E C I P T ( I ) , X L N F R O ( I ) , 1 R C P R T ( I ) , 1 = 1 , N ) 9 0 F O R M A T ( 1H , 2 X , F 1 2 . 5 , 8 X , F 1 2 . 5 , 7 X , E 1 4 . 6 , 9 X , F 1 0 . 7 , 9 X , F 1 2 . 7 , 9 X , F 1 0 . 7 ) SU.MX = 0 . 0 SUMY = 0 . 0 DO 2 0 0 J = 1 , N S U M X = SUMX + R C P R T ( J ) SUMY = S U M Y + X L N F R Q ( J ) 2 0 0 C O N T I N U E C A L L L S Q F I T C N , R C P R T , X L N F R Q , S U M X , S U M Y , Q , P , S T D E R Q f S T D E R P,XAVf 1 Y A V , N 0 G 0 ) F R Q M A X = E X P ( Q ) E R F R Q = E X P ( Q ) * S T D E R Q E A C T = ( - 1 . ) * P E R E A C T = ( - l . ) * S T D E R P W R I T E ( 6 , 2 1 0 ) F R Q M A X , E R F R Q , E A C T , E R E A C T . 2 1 0 F O R M A T U H 0 , 2 5 H C 0 R . F R Q . A T I N F . T E M P . = , E 1 2 . 5 , 2 X » 7 H E R R 0 R = , E 1 2 . 5 , 1 5 X , 1 2 H A C T . I V . E N . = , E 1 2 . 5 , 7 H C A L / M O L , 2 X , 7 H E R R O R = , E 1 2 . 5 , 7 H C A L / M 0 L ) R E A D ( 5 , 2 2 0 ) D E L M A X , D E L M I N , D E L I N C 2 2 0 F O R M A T { 3 F 1 0 . 5 ) N O I N C = ( D E L M A X - D E L M I N ) / D E L I N C + 0 . 0 0 1 NSW = N O I N C + 1 DO 2 3 0 K= 1 , N S W A K = K T H D E L ( K ) = D E L M I N + ( A K - 1 . ) * D E L I N C C B = C * * 2 - 8 * * 2 T H S N { K ) = S I N ( 1 , 5 7 0 7 * { T H D E L < K . ) * * 2 - B * * 2 ) / C B ) T H C O ( K ) = C O S ( 1 . 5 7 0 7 * 1 T H D E L ( K ) * * 2 - B * * 2 ) / C B ) T H T A N ( K ) = T H S N ( K ) / T H C O { K ) T H F R Q ( K ) = ( A G * T H D E L ( K ) J / T H T A N ( K ) T H T E M P l K ) = E A C T / ( 1 . 9 8 6 9 * A L 0 G ( F R Q M A X / T H F R Q ( K ) ) ) 2 3 0 C O N T I N U E P R I N T 2 4 0 2 4 0 F 0 R M A T ( 1 H O , 3 4 X , 4 7 H T H E O R E T I C A L M O D I F I E D B P P L E A S T S Q U A R E S D A T A F I T ) P R I N T 2 5 0 IV) 2 5 0 F O R M A T ( 1 H 0 , 6 X , 2 8 H T E M P E R A T U R E ( D E G R E E S K E L V I N ) , 1 7 X , 1 8 H L I N E W I D T H , G A 1 U S S ) , 1 7 X , 2 7 H C 0 R R E L A T I 0 N F R E Q U E N C Y ( C P S ) ) W R I T E ( 6 , 2 6 0 ) ( T H T E M P ( K ) , T H D E L ( K ) , T H F R Q ( K ) , K = 1 , NSW ) 2 6 0 F O R M A T ( I H , 1 4 X , F 1 1 . 5 , 3 0 X , F 1 0 . 5 , 2 8 X , E 1 4 . 6 ) 2 0 0 0 S T O P END S U B R O U T I N E L S Q F I T ( N , X , Y » S U M X , S U M Y » B t C , S T D E R B t S T D E R C f X A V t Y A V f N O G O ) D I M E N S I O N X ( 5 0 0 ) , Y ( 5 0 0 ) I F ( N - 2 ) 1 0 5 0 , 1 0 5 0 , 1 0 0 0 1 0 0 0 AN=N X A V = S U M X / A N Y A V = S U M Y / A N D I F X Y = 0 . D I F X S Q = 0 . DO 1 0 1 0 J = l , N D I F X Y = D I F X Y + ( - X ( J ) - X A V ) * Y < J ) D I F X S Q = O I F X S Q + ( X ( J ) - X A V ) * * 2 1 0 1 0 C O N T I N U E C = D I F X Y / D I F X S Q B = Y A V - C * X A V DSQ = 0 . X S Q = 0 . ' DO 1 0 2 0 J = 1 , N DSQ = DSQ + ( B + C * X ( J ) - Y ( J > ) * * 2 X S Q = XSO+ X ( J ) * * 2 1 0 2 0 C O N T I N U E Q = S Q R T ( D S Q / ( A N - 2 . ) ) D E E = A N * . X S O - < A N * X A V ) * * 2 •IF ( D E E ) 1 0 5 0 , 1 0 5 0 , 1 0 3 0 . 1 0 3 0 QOVR TD= Q / S O R T ( D E E J STDE.RC= 0 O V R T D * S Q R T ( AN) S T D E R B= 0 O V R T O * S Q R T ( X S Q ) NOGO = 1 P R I N T 1Q4Q  1 0 4 0 F O R M A T { 1 H , 5 2 H N O G O = 1 , T H E R E F O R E L S Q F I T H A S MADE A S U C C E S S F U L F I T 1) GO T O 1 1 0 0  1 0 5 0 NOGO = 2 P R I N T 1 0 6 0 1 0 6 0 F O R M A T ( 1 H t 8 4 H N 0 G 0 = - 2 t L S Q F I T U N S U C C E S S F U L DUE T O D E E L E S S T H A N 0 1R = 0, OR TO N L E S S THAN OR = 2 . ) 1 1 0 0 R E T U R N END 

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