UBC Theses and Dissertations

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UBC Theses and Dissertations

Quantum effects in dilute adsorption systems Macrury, Thomas Bernard 1967-12-31

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QUANTUM EFFECTS IN DILUTE ADSORPTION SYSTEMS by THOMAS BERNARD MACRURY B . S c , , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERS OF SCIENCE Fn t h e Depa r tment o f C h e m i s t r y We a c c e p t t h i s t h e s i s as c o n f o 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 UNIVERSITY OF BRITISH COLUMBFA S e p t e m b e r , 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag r ee t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . | f u r t h e r ag r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h.i)s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f <^-^<g_A-~N_L'^ W The U n i v e r s i t y o f B r i t i s h C o l u m b i a Vancouve r 8, Canada i ABSTRACT The a d s o r p t i o n i s o t h e r m and t h e e q u a t i o n o f s t a t e f o r t h e t w o - d i m e n s i o n a l gas a r e d e r i v e d f r om t h e g rand c a n o n i c a l e n s e m b l e . Then t h e quantum s t a t i s t i c a l e q u a t i o n ' o f s t a t e i s d e v e l o p e d and a p p l i e d t o (2 ) t h e t w o - d i m e n s i o n a l s econd v i r i a l c o e f f i c i e n t , B and t h e second g a s - s u r f a c e v i r i a l c o e f f i c i e n t , B ^ , We compare t h e o r e t i c a l l y t h e (12,6) (2 ) and (12,6 ,3 ) p o t e n t i a l mode l s f o r B , F i n a l l y . t h e a d s o r p t i o n d a t a f o r CH^ , CD^,, H_ and D_ on g r a p h i t e a r e a n a l y s e d q u a n t a l l y f o r t h e two- d i m e n s i o n a l s econd v i r i a l c o e f f i c i e n t and t h e s e c o n d g a s - s u r f a c e v i r i a l c o e f f i c i e n t . i i ACKNOWLEDGEMENT I am s i n c e r e l y g r a t e f u l t o D r , J , R . Sams J r . f o r h i s p a t i e n c e and i n v a l u a b l e g u i d a n c e d u r i n g t h e c o u r s e o f t h i s w o r k , I wou ld a l s o l i k e t o t h a n k R, W o l f e f o r h i s a s s i s t a n c e i n some o f t h e compute r p r o g r a m m i n g , TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENT t L IST OF TABLES i i L IST OF FIGURES i v CHAPTER I. I n t r o d u c t i o n v CHAPTER 2 . The A d s o r p t i o n I s o t h e r m and t h e E q u a t i o n o f S t a t e T h r e e - d i m e n s i o n a l deve l opemen t II T w o - d i m e n s i o n a l d e v e l o p m e n t 14 CHAPTER 3 . Ouantum S t a t i s t i c a l E q u a t i o n o f S t a t e G e n e r a l d e v e l o p m e n t 20 T w o - d i m e n s i o n a l second v i r i a l c o e f f i c i e n t 26 Second g a s - s u r f a c e v i r i a l c o e f f i c i e n t . 31 CHAPTER 4 . T h e o r e t i c a l C o m p a r i s o n o f t h e ( 1 2 , 6 ) and ( 1 2 , 6 , 3 ) M o d e l s (2) % for B 35 CHAPTER 5 . A n a l y s i s o f t h e Da t a Two-d imens iona I s econd v i r i a I ' c o e f f i c i e n t . 43 Second g a s - s u r f a c e v i r i a l c o e f f i c i e n t 50 APPENDIX I . 5 7 APPENDIX 2 63 BIBLIOGRAPHY - 64 IV L IST OF TABLES T a b l e Page I C u r v e f i t r e s u l t s f o r t w o - d i m e n s i o n a l CH^ and C D 4 on g r a p h i t e . 45a II V a l u e s o f A and 6 f o r CH^ and CD^ based on t h e f o u r m o d e l s . 46a II I B A S and C^s d a t a f o r H 2 and D 2 . 48a IV C u r v e f i t r e s u l t s f o r t w o - d i m e n s i o n a l ( 1 2 , 6 ) and on g r a p h i t e . ^ 49a V V a l u e s o f A and 6 f o r t w o - d i m e n s i o n a l ( 1 2 , 6 ) ht, and . 50a VI C l a s s i c a l and Quantum K i rkwood-MuI I e r f i t r e s u l t s f o r C H 4 , C D 4 , H 2 and D?, ' 53a VII Quantum S l a t e r - K i r k w o o d and London f i t r e s u l t s f o r C H ^ C D ^ , H 2 and D 2 . 53b ( 2 ) ^ ( 2 ) ^ ( 2 ) ^ * V I I I V a l u e s o f B C ( , B| , and B| j a t s e l e c t e d v a l u e s of T and n f o r t h e ( 1 2 , 6 , 3 ) p o t e n t i a l f u n c t i o n , 57 IX V a l u e s o f B ^ 1 f o r t h e ( 9 , 3 ) , ( 1 0 , 4 ) and ( 1 2 , 3 ) m o d e l s . 63 V L IST OF FIGURES F i g u r e Page 1. ' C u r v e s o f r e d u c e d t w o - d i m e n s i o n a l s econd v i r i a l c o e f f i c i e n t v e r s u s r e d u c e d t e m p e r a t u r e , 35a 2 . C u r v e s o f r educed t w o - d i m e n s i o n a I s econd v i r i a l c o e f f i c i e n t v e r s u s r e l a t i v e r e d u c e d t e m p e r a t u r e . 36a 3 . R a d i u s o f c o n v e r g e n c e c u r v e s , 38a 4 . Quantum ( 1 2 , 6 , 3 ) c u r v e s f o r d i f f e r e n t n v a l u e s , 39a 5. C u r v e s o f t h e second v i r i a l c o e f f i c i e n t i n -one, tv/o, and t h r e e d i m e n s i o n s . •'- 40a .CHAPTER- I INTRODUCTI ON B e f o r e 1954, n e a r l y a l l a d s o r p t i o n d a t a were t a k e n below t h e c r i t i c a l t e m p e r a t u r e o f t h e a d s o r b a t e . L a t e r a l i n t e r a c t i o n s , m u l t i l a y e r f o r m a t i o n , and p o s s i b l e c a p i l l a r y c o n d e n s a t i o n a r e i m p o r t a n t i n t h i s r e g i o n and s i n c e t h e t h e o r y o f t h e s e phenomena i s even more complex t h a n t h e t h e o r y of l i q u i d s , i t seemed advantageous t o i n v e s t i g a t e a d s o r p t i o n a t h i g h e r t e m p e r a t u r e s . S t e e l e and H a l s e y ' found t h a t i f t h e t e m p e r a t u r e was h i g h enough and t h e gas p r e s s u r e low, t h e i n t e r a c t i o n s between a d s o r b a t e m o l e c u l e s c o u l d be i g n o r e d and t h e problem o f t h e i n t e r a c t i o n o f s i n g l e m o l e c u l e s w i t h t h e s u r f a c e i n v e s t i g a t e d . The a d s o r p t i o n i s o t h e r m was expanded i n a v i r i a l s e r i e s , a n a l o g o u s t o t h a t used i n i m p e r f e c t gas t h e o r y . The l e a d i n g t e r m i n t h e e x p a n s i o n g i v e s t h e Henry's law c o n s t a n t o f t h e i s o t h e r m . A t somewhat lower temper a t u r e s and h i g h e r p r e s s u r e s , a d d i t i o n a l t e r m s , which a c c o u n t f o r i n t e r a c t i o n s between t h e a d m o l e c u l e s , must be i n c l u d e d . An e x a c t d e r i v a t i o n i n t h e 2 c l a s s i c a l grand c a n o n i c a l ensemble l e a d s t o a development o f t h e i s o t h e r m , e x p r e s s i n g t h e number o f m o l e c u l e s a d s o r b e d a t K e l v i n t e m p e r a t u r e T and b u l k gas p r e s s u r e p, i n terms o f t h e i r r e d u c i b l e c l u s t e r i n t e g r a l s : Na = B | ( £ ) +. (e 2 + (I.01 ) B AS P P (~) + AAS + . ( I .Ola) 8 f .dr. I 1^ o = V f|2d£ld£2 ' ( I .02) Wl2d£|d£2 2 w h e r e t h e f ' s a r e t h e u s u a l M a y e r f u n c t i o n s ' ^ l + f j = e x p [ - B<j»(r. . ) ] , (1.03) l+f. = exp C- B t^jr,) ] . i j ) ( r . j ) i s t h e p a i r p o t e n t i a l b e t w e e n m o l e c u l e s i a n d j , <j> s ( r j ) t h e i n t e r a c t i o n p o t e n t i a l o f m o l e c u l e i a t r . w i t h t h e s o l i d , a n d g = ( K T ) ' . In t h i s s t a t i s t i c a l m e c h a n i c a l d e v e l o p m e n t o f t h e t h e o r y o f p h y s i c a l a d s o r p t i o n , t h e f o l l o w i n g a s s u m p t i o n s a r e r e q u i r e d : 1. O n e c a n t r e a t t h e a d s o r b e n t a s an i n e r t s o l i d w h i c h m e r e l y f u r n i s h e s a p o t e n t i a l e n e r g y o f i n t e r a c t i o n b e t w e e n t h e a d s o r b a t e m o l e c u l e s a n d t h e a d s o r b e n t . T h e i n t e r a c t i o n e n e r g y b e t w e e n t h e i t h g a s m o l e c u l e a n d t h e a d s o r b e n t w i l l b e d e n o t e d by 4> ( r . ) . 2, T h e i n t e r a c t i o n e n e r g i e s b e t w e e n m o l e c u l e s i n t h e g a s p h a s e a r e p a i r - w i s e a d d i t i v e . T h e i n t e r a c t i o n e n e r g y b e t w e e n m o l e c u l e s i a n d j c a n b e w r i t t e n a s ^ ( r - . r . ) o r a s i f c ( r . . , R. . ) , w h e r e ° \> | ' ' V j ^ i j ^ i j r . . = r . - r . , <\,l j <V,I <\,j' R. . = r . + r . . (I.04) In many t r e a t m e n t s , t h e p r o b l e m c a n b e s i m p l i f i e d f u r t h e r by a s s u m i n g t h a t . t h e p o t e n t i a l d e p e n d s o n t h e s e p a r a t i o n b e t w e e n 1 a n d j m o l e c u l e s , i e <j> ( r . . ) . ' U 3. T h e v a r i o u s i n t e r a c t i o n p o t e n t i a l s a r e i n d e p e n d e n t o f m o l e c u l a r o r i e n t a t i o n . T h i s i n f e r s t h a t a l I g a s m o l e c u l e s w i I I h a v e s p h e r i c a l s y m m e t r y , a t l e a s t i n t h e a b s e n c e o f t h e s o l i d . F u r t h e r m o r e , any changes i n i n t e r n a l v i b r a t i o n s t a t e upon a d s o r p t i o n wiI I be n e g l e c t e d . U s i n g t h e s e a s s u m p t i o n s , t h e t o t a l p o t e n t i a l e nergy of N m o l e c u l e s i n a c o n f i g u r a t i o n r . . . . r , . and i n t e r a c t i n g w i t h each o t h e r and w i t h a s o l i d a d s o r b e n t i s N • < r , . . . r ) = I 6 ( r , ) + I * < r . ) . (1.05) I t i s c l e a r from t h e i s o t h e r m e q u a t i o n ( I . O l a ) t h a t , when terms h i g h e r t h a n t h e q u a d r a t i c a r e n e g l e c t e d , p l o t s of Na/p v e r s u s p a r e 2 l i n e a r , w i t h g r a d i e n t (C^^ / ( k T ) and i n t e r c e p t B ^ / k T . T h e r e f o r e t h e s e s o - c a l l e d g a s - s u r f a c e v i r i a l c o e f f i c i e n t s can b e . o b t a i n e d e x p e r i m e n  t a l l y f rom a d s o r p t i o n i s o t h e r m s measured a t low s u r f a c e d e n s i t i e s . The tv/o p a r a m e t e r t h e o r y d e r i v e d by S t e e l e and H a l s e y ' f o r t h e second o r d e r ( o r Henry's Law) r e g i o n o f t h e i s o t h e r m , a l l o w e d t h e e v a l u a t i o n o f t h e d e p t h of t h e g a s v s u r f a c e i n t e r a c t i o n w e l l , £j , and t h e c a p a c i t y f a c t o r Az , where A i s t h e s u r f a c e ' a r e a of t h e a d s o r b e n t and o z 0 t h e a p p a r e n t g a s - s o l i d c o l l i s i o n d i a m e t e r . From t h e o b s e r v e d e and any o f t h e f a m i l i a r e x p r e s s i o n s f o r t h e d i s p e r s i o n e n e r g y c o n s t a n t , one can c a l c u l a t e a v a l u e f o r Z q , and hence A. T h i s p r o v i d e d a new method of d e t e r m i n i n g s u r f a c e a r e a s w h i c h was i n d e p e n d e n t of any m u l t i l a y e r a d s o r p t i o n t h e o r y , o r any m o l e c u I a r c r o s s - s e c t i o n a l e s t i m a t e s . A model f o r t h e n e x t t e r m i n t h e i s o t h e r m e q u a t i o n , which a c c o u n t s f o r t h e s i m u l t a n e o u s i n t e r a c t i o n s o f two gas m o l e c u l e s w i t h t h e s u r f a c e 4 and w i t h each o t h e r was p r e s e n t e d by Freeman and H a l s e y and l a t e r r e f i n e d 5 by Freeman. E m p l o y i n g L e n n a r d - J o n e s p o t e n t i a l s f o r both g a s - s o l i d and gas-gas • i n t e r a c t i o n s , agreement w i t h e x p e r i m e n t a l r e s u l t s was deemed t o be 4 u n s a t i s f a c t o r y . I t was s u g g e s t e d t h a t t h i s might be due t o t h e f a c t t h a t t h e intermoI ecu I a r p o t e n t i a l s a r e not p a i r w i s e a d d i t i v e as assumed i n t h e model. R e c e n t l y i t has been p o i n t e d o u t ^ ' ^ t h a t t h e s o l i d has an app r e c i a b l e e f f e c t on t h e p a i r w i s e i n t e r a c t i o n between m o l e c u l e s near t h e s u r f a c e . The a t t r a c t i v e p a r t of t h e energy appeared t o be enhanced by about 20% when r . . was p e r p e n d i c u l a r t o t h e s u r f a c e and d e c r e a s e d by a p p r o x i m a t e l y t h e same amount when £.. was p a r a l l e l . S i n c e t h e e v a l u a t i o n of f i ^ ^ o r r e a l i s t i c p o t e n t i a l s t s cumbersome, an a l t e r n a t i v e method o f t r e a t i n g t h e d a t a has been d e v i s e d , The compu t a t i o n s can be s i m p l i f i e d c o n s i d e r a b l y by r e d u c i n g t h e 3N-dimensionaI c l u s t e r i n t e g r a l s i n (1.02) and (1.03) t o 2N-dimensionaI i n t e g r a l s i n t h e p l a n e normal t o t h e f i e l d d i r e c t i o n . U s i n g t h e method of s t e e p e s t 8 d e s c e n t s , S t e e l e has shown t h a t a t low t e m p e r a t u r e s t h e two d i m e n s i o n a l e q u a t i o n o f s t a t e of t h e a d s o r b a t e i s g i v e n by ^ i i - = I - C - . h B° ) Na , (1.06) NakT 7 -1 ' 2 B | 2 B, where <j> i s t h e s p r e a d i n g p r e s s u r e . T h i s e q u a t i o n has t h e form of a t w o - d i m e n s i o n a l v i r i a l e x p a n s i o n i n t h e s u r f a c e d e n s i t y Na/A, (2) - 6A = I + % \Na . NakT n ( I .07) where i n t h e r e d u c t i o n from 3N d i m e n s i o n s t o 2N d i m e n s i o n s , - ( B + 2 B 6 ° ) / 26 2 * - B ( 2 ) / A. ( I .08) 2 I I 1 5 B i s t h e two d i m e n s i o n a l second v i r i a l c o e f f i c i e n t and B /A can be measured from t h e e x p e r i m e n t a l g a s - s u r f a c e v i r i a l c o e f f i c i e n t s . 9 Sams, C o n s t a b a r i s and H a l s e y have a p p l i e d t h i s e q u a t i o n t o t h e i n t e r  a c t i o n o f argon w i t h g r a p h i t i c c a r b o n . C o r r e c t i o n terms t o a c c o u n t f o r t h e m o t i o n o f t h e m o l e c u l e s i n t h e f i e l d d i r e c t i o n can be w r i t t e n down as w e l l , and one f i n d s t h a t under t h e normal e x p e r i m e n t a l c o n d i t i o n s t h e s e c o r r e c t i o n s a r e s m a l l and (1.08) s h o u l d p r o v i d e q u i t e a n . a c c u r a t e a p p r o x i m a t i o n , Thus t h e g a s - g a s - f i e l d i n t e r a c t i o n can be t r e a t e d as an e f f e c t i v e gas-gas i n t e r a c t i o n , t h e f i e l d s e r v i n g m e r e l y t o r e s t r i c t t h e m o l e c u l a r m o t i o n s t o two d i m e n s i o n s . The p r o b l e m t h e n I s t o s p e c i f y t h e form of t h e e f f e c t i v e two- d i m e n s i o n a l i n t e r a c t i o n p o t e n t i a l , s i n c e i t i s c l e a r l y i n c o r r e c t t o employ t h e u s u a l gas phase p o t e n t i a l f u n c t i o n . S i n a n o g l u and P i t z e r ^ , and Y a r i s ' ^ have d e r i v e d t h e o r e t i c a l e x p r e s s i o n s f o r t h e t h r e e - b o d y energy i n g a s - s o l i d i n t e r a c t i o n s y s t e m s . U s i n g t h i r d - o r d e r p e r t u r b a t i o n t h e o r y , S i n a n o g l u and P i t z e r found t h a t t h e p r e s e n c e o f t h e f i e l d had t h e e f f e c t o f i n t r o d u c i n g an a d d i t i o n a l t h r e e - b o d y term v a r y i n g as t h e i n v e r s e t h i r d power of t h e m o l e c u l a r s e p a r a t i o n . M c L a c h l a n ' ' a r r i v e d a t s i m i l a r c o n c l u s i o n s u s i n g image f o r c e methods, but has a l s o found t h a t a t l a r g e v a l u e s of t h e s e p a r a t i o n t h e i n v e r s e cube dependence goes o v e r i n t o an i n v e r s e s i x t h power dependence. F o r a t w o - d i m e n s i o n a l ( m o n o l a y e r ) model t h i s a d d i t i o n a l i n t e r a c t i o n i s r e p u l s i v e , b u t i f t h e a d m o l e c u l e s a r e d i r e c t l y above and below one a n o t h e r , t h e t h r e e - b o d y t e r m becomes a t t r a c t i v e ^ . These f i n d i n g s have l e d t o t h e u s e , i n a n a l y s i n g e x p e r i m e n t a l d a t a , of two assumed forms f o r t h e e f f e c t i v e p o t e n t i a l . 6 The f i r s t o f t h e s e i s t h e i n t e r m o l e c u I a r p a i r p o t e n t i a l g i v e n by S i n a n o g l u and P i t z e r ^ , 4><r..) = 4e [ ( a / r . J 1 2 - ( a / r . J 6 ] + ( S / r . ^ . 3 ) ( l - 3 c o s 2 6 ), (1.09) where e and o a r e t h e b u l k - g a s L e n n a r d - J o n e s (12,6) p o t e n t i a l p a r a m e t e r s , 0 t h e a n q l e between r . . and t h e s u r f a c e n o r m a l , and a IJ ' S = «, E | S <4, + Y n A ) /4 t (1,10) where \ = (2 6( + 6 g ) / ( 6 , + 6s) , (I .1 I) A| = (66, + 7 f i s > / ( 6 | +6 S) , (1.12) Y = E /E. . ( I .13). 'm es Is E| s i s t h e s e c o n d - o r d e r p e r t u r b a t i o n a I energy f o r t h e i n t e r a c t i o n o f a s i n g l e a d m o l e c u l e and t h e s u r f a c e , a t h e p o l a r i z a b i I i t y o f t h e a d m o l e c u l e , and 6j and 5 t h e mean o r e f f e c t i v e e x c i t a t i o n e n e r g i e s o f m o l e c u l a r - s t a t e t r a n s i t i o n s f o r a s i n g l e a d m o l e c u l e and f o r t h e s u r f a c e , r e s p e c t i v e l y ; Y m i s t h e e l e c t r o s t a t i c f r a c t i o n o f t h e t o t a l s e c o n d - o r d e r e n e r g y . If t h e t w o - d i m e n s i o n a l model i s a p p l i e d , r ^ becomes t h e t w o - d i m e n s i o n a l s c a l e r T |2 i n a p l a n e p a r a l l e l t o t h e s u r f a c e , and 8= I T / 2 . Then (1.09) f o r t h e e f f e c t i v e p o t e n t i a l between two a d m o l e c u l e s becomes * -12 • * -6 * -3 <k< T, 2) - 4 e I T ' - T - nt 3 , ( 1 . 1 4 ) t * =• T | 2 / a ; n = "-S' _/4 e a 3 '. (1.15) 7 A l t h o u g h n c a n be c a l c u l a t e d f r om e q u a t i o n s ( I . 1 0 ) - ( 1 . 1 5 ) , i t has been 12 f ound p r e f e r a b l e t o t r e a t i t as an e m p i r i c a l l y a d j u s t a b l e p a r a m e t e r . The o t h e r p o t e n t i a l v/hich has been used i s * -12 * -6 pQ ( T | 2 ) = 4 e ( x _ £T - ) # ( 1 . 1 6 ) w h e r e , i n b o t h p o t e n t i a l s , 5 i s a c o n s t a n t w h i c h can be d e t e r m i n e d e m p e r i c a l l y by f i t t i n g e x p e r i m e n t a l d a t a t o t h e m o d e l s , and w h i c h g i v e s a measure o f t h e n o n - a d d i t i v i t y . Up t o t h i s p o i n t we have been c o n c e r n e d w i t h t h e c l a s s i c a l p i c t u r e o f a d s o r p t i o n . Howeve r , quantum e f f e c t s have been measured i n d i l u t e a d s o r p t i o n s y s t e m s and found t o be o f c o n s i d e r a b l e i m p o r t a n c e . I t has long been t h o u g h t t h a t when i s o t o p i c p a i r s a r e a d s o r b e d on a s u r f a c e , t h e h e a v i e r s p e c i e s v/i I I be a d s o r b e d p r e f e r e n t i a l l y . T h i s comes abou t because o f t h e quantum s t a t i s t i c a l mass e f f e c t on t h e v i b r a t i o n a l e n e r g y l e v e l s norma l t o t h e s u r f a c e , w i t h t h e h e a v i e r m o l e c u l e l o w e r i n g t h e e n e r g y l e v e l s s l i g h t l y . T h i s quantum e f f e c t i n p h y s i c a l a d s o r p t i o n has been c o n s i d e r e d by 13 14 15 13 s e v e r a l a u t h o r s , ' ' De M a r c u s , e t , a l . , d e v e l o p e d t h e g a s - s u r f a c e 2 c o n f i g u r a t i o n a I i n t e g r a l c o r r e c t t o t e r m s o f o r d e r ,f\ f o r a L e n n a r d - J o n e s ( 9 , 3 ) p o t e n t i a l , and a p p l i e d t h e t h e o r y t o d a t a o f S t e e l e and H a l s e y ' I 5 f o r t h e i n t e r a c t i o n o f h e l i u m w i t h c a r b o n b l a c k , Freeman employed t h e same model t o i n t e r p r e t t h e a d s o r p t i o n o f H„ and D~ on s u g a r c h a r c o a l . In this treatment the configurationaI integral was made up of two parts, the usual classical part and the f i r s t quantum correction, BAS = BAS - BAS ' ( l ' l 7 ) Both these integrals are a function of the reduced gas-surface interaction temperature e. /kT, and B ' is also inversely proportional to the mass, . I S n b In view of the expected increase in B.c for heavier species, the A b results of Constabaris, Sams and Halsey'^ for the adsorption of hL,, CH^ and CD^ on P33 (2700°) were surprising. While had the anticipated larger excess volume than W^, although the difference was smaller than expected, the opposite was true with the methane pair. These results can be explained by allowing for the fact that the hydrogenated and deuterated species have a slightly different internuclear distance, and thus the electronic distributions and the forces binding the electrons differ, This manifests itself in a polar i zab i I i'ty difference between the isotopes. Knaap and Beenakker have used this to explain the difference in the 19 measured gas-gas interactions of and D^ . Since dispersion forces are directly proportional to the polarizabiIities of the interacting species, the percentage differences in the interaction energies from a quantum f i t of the data should be commensurate with the percentage difference in the polarizabiIities. Since the quantum corrections are strongly dependent upon both the depth and shape of the potential well, such a f i t should provide a rather stringent test of the potential model chosen. Yaris and Sams2<^ developed the Wigner-Ki rkwood expansion""?^ 4 order ft and examined three different models for the interaction potential', 9 16 4 using the data of Constabaris, et.al. Due to a misprint in the (rf term of 21 Uhlenbeck and Beth , we have recalculated the quantum corrected B c and no fitted i t to the data mentioned above. The fact that significant quantum deviations were observed for gas-surface interactions suggested that i t might also be possible to measure quantum effects between molecules adsorbed on a surface. To this end, the evaluation of the tv/o-dimensionaI' second v i r i a l coefficient in terms of the (12,6) and (12,6,3) potentials for a quantum degenerate gas at moderately high temperatures, employing the Wigner-Kirkwood expansion of the Slater sum will be considered. In Chapter 2 of this thesis, the equation of state and the adsorption isotherm are derived through the.grand-canonicaI ensemble, (2) In Chapter 3, B and B^ <. are developed quantally for a general potential function and are evaluated numerically for certain specific potentials. Chapter 4 contains a comparison of the (12,6) and (12,6,3) models for the evaluation of In Chapter 5, the data^on the third-order interactions . of H^ , D^ , CH^, and CD^ in the external f i e l d provided by a very uniform graphite surface P33 (2700°) are analysed, Classically both the (12,6) and (12,6,3) models have been app I ied'^' 2^' 2 7to the experimental data for Ar,Kr,Xe,CH4 and CD^, Either model led to an effective pair interaction energy which was somewhat smaller than the gas phase value, For the (12,6,3) potential, this reduction amounted to only about 8-\0% of the bulk gas inter action, while for the (12,6) model the effect was approximately twice as great, Despite this difference, both models appeared to f i t the" data almost- equal ly well, and i t was impossible to choose between them. We hope that 10 the quantum treatment will provide a better test of the model. Also in Chapter 5, the data for H^ , D^ , CH and CD^ are analysed quantally using three different potential models in an attempt to see which model f i t s the data best. II CHAPTER 2 THE ADSORPTION ISOTHERM AND THE EQUATION OF STATE If one considers a gas at given u and T, where y is the chemical potential, in a vessel of volume V under conditions such that the gas does not interact with the walls of the container, except to undergo perfectly elastic collisions, then the grand partition function is o = e^^^, If one considers the same gas and vessel, but now allows the gas to interact with one wall of the container,(e,g. the adsorbent) pV+<£A having area A, then the grand partition function is =• = e kT , where <»> is the spreading pressure. The difference in the number of molecules under the two conditions we define as the number of molecules adsorbed: Na = N - No . (2.01) An equation for Na is readily written down; Na = A In E* \ , (2.02) V3ln /J, /T,V i = 5/ E , The activity of the gas is given by y/kT 3 ^ = e / X (2.03) where where 1/2 \- h/(2TrmkT) , (2.04) m is the molecular mass .and h is Planck's constant. 28 From Hil l , one can write the two partition functions as E(T'V'A^) = I / m , (2.05) N>0 N T 12 and E°(TiV,/J) N >0 £ M / • ( 2 . 0 6 ) where and Z ° a r e t h e N p a r t i c l e c o n f i g u r a t i o n a I i n t e g r a l s wh i ch a r e g i v e n f o r a c l a s s i c a l gas w i t h p a i r i n t e r a c t i o n s as Z ° N exp [- 3 I cb(T., ) ] . d ^ N , !!•<J<N i j ( 2 . 0 7 ) and exp C- BC I * (jr.) + X d,(r ) ) ] d r N . (2..08) i = l s 1 l < i < j < N I J One can expand t h e l o g a r i t h m s o f ( 2 . 0 5 ) and ( 2 . 0 6 ) i n a s e r i e s f o r (pV + <j>A)/kT and pV/kT r e s p e c t i v e l y : = -pV+cfrA. = V I b fl/ , ~kT j > l J ( 2 . 0 9 ) where I ! Vb | l\ Vb. 31. Vb- e t c . Z 2 - Z . ' Z 3 -3Zj Z 2 + 2Zj ( 2 . 1 0 ) and -pV kT ( 2 . 1 1 ) J> 13 where 2} Vb, = V - Z, 3: Vb, O O 0 7 -32, Z 2 + 2Z, (2.12) etc. Performing the appropriate partial integrations on (2.02), one gets Na = I Vj (b.-b. )Af j> I J J J • (2.13) By expanding this equation in the activity to the second power and' 0 3 writing the b.'s and b. 's in terms of the.Mayer f functions , one gets J J Na = f l d £ l ^ tef „f + f f f ) dr.dr 0 +,. 1 / 2 1 2 1 2 %\ (2.14) Substituting (1.02) into (2.14), gives us the adsorption isotherm in terms 2 3 of the activity and the irreducible cluster integrals ' : Na = + (p_ + 20. 3, )r 2 2 I I ^ + ' " = BAS ^ + CAAS fif + (2.15) (2.15a) By inverting (2.11) to get the activity in a power series in the pressure, and then substituting the new series into (2,15), one can write the adsorption isotherm with the pressure as the independent variable rather than fi^ : Na = ( p ) + ( 6 2 + 2-6 ( 0, ) (i<r} + (2.16) AS (-ii ) + C kT AAS (-ii)2 + kT (2.16a) 14 Equations (2.15) and (2.16) are both exact isotherms, but (2.16) is more useful as a working isotherm. At sufficiently high temperatures, a l l the f.. vanish and only B^ <. is important: Na = B.c '{-£-) . (2.17) kT At lower temperatures, assumes f i n i t e values and the isotherm is no longer linear in the pressure. However, at these low temperatures the probability of finding an adsorbed molecule anywhere except in the immediate vicinity of a potential minimum becomes quite small, In this case, the three-dimensional 29 30 integrals become unnecessarily cumbersome. It has been shown" ' that the configurationaI integrals can be s p l i t into two parts: three-dimensional integrals Z ^ 5 ^ representing single molecules on the surface; and a reduced configurationaI integral in the two dimensions parallel to the surface which give the contributions of lateral interactions to the properties of the adsorbed monolayer. We consider a uniform solid surface which is made up of M identical elements of area A which will be termed r s s sites. Within each element, ^(^j ) will vary with z., the perpendicular distance between the ith gas molecule and the surface of the solid, and will ordinarily vary with a two-dimensional vector, lying in a plane parallel to the surface. If a l l N elements have identical <j> (r. ) , then s s ' Z . ( S ) = N Z , (2.18) I s s' where Z^ is the configurationaI integral for a single molecule over a site. (2) If one introduces a dimension I ess quantity/^ as 15 / ^ ( 2 ) = Z, ( S )/^ , (2.19) x z (s) ; + h e n ~ = I- N (A, ( 2 )) N . (2.20) N > 0 N ( ( Z < s ) } N ' At temperatures near the boiling point of the adsorbate, the value of -4>s(r.)/kT are expected to be rather large (> 10) at distances corres ponding to the position of maximum attractive energy. The exponential of this function will be quite large in this region, so that one can make the approximation 7 (s) I Az s exp C - A ( r . ) / kT] d T. dz. , (2.21) S 'Vl r \ i \ j ' where z g is the distance over which <j> is important. Also, since the probability of finding a gas atom at a distance z from the surface will have a very large maximum at or near z=z , where z is the position at • ^  m m v which the maximum interaction energy is to be found, one is actually saying that the adsorbed film approximates a two-dimensional phase. 8 Steele approximates ^ ( r ) ' n the region of its minimum by *s(V = e|s + I k (z-z ) 2 + e ( T) (2.22) 2 z m T 'v. where z, is the potential minimum at the center of a site (t=0), I s % ' e (T) is the variation in e, as an atom moves parallel to the surface. T O . I 5 i > and the second term is a harmonic oscillator approximation for motion in the z direction. If and the force constant, k , are large, the z integrations in the configurationaI integrals thus involve functions which have large maxima and are rapidly varying in the region of the maxima. 16 It is therefore appropriate to use the method of.steepest descents to evaluate such integrals. Steele has shown** that the ratio Z^ 5^ / ( Z , ^ ) ^ can be written as ZN (s) ( Z , ( S ) ) N (N) l<i<j<N e I J (2.23) where C ( T , . . . T M ) is a correction term for non-pIanarity, ie. z^z , (N) is equal to the probability of finding N non-interacting molecules in elements of unit area at points ^[•••J^.j o n the surface, and is given by N (N) ^ l - ' - t N 5 = G X P C " I E T ( ^ i } / k T ] A .N (2.24) exp C- T e /KT3 d j . . . . d T If E ^ ( T . ) , the potential barrier to motion across the surface, is zero, (M) —N P ( T . . . . T . , ) = A . Nectlectinq the correction term for non-p I anar ity. e M 'MN • ~ - (2.23) becomes an equation for the 2N-dimensionaI configurationaI integral which corresponds to a generalized tv/o-dimensionaI gas constrained to move in the plane defined by z=z . . ' m The two-dimensional configurationaI integral is now given by (2) P e x p L ' I *e ^..) /kT], (2.25) ° i < i < i<N J and the grand partition function becomes 17 The adsorption isotherm is obtained from Ma = r"3 I n (2,27) 13 I n / ^ ( 2 ) / T,V (2) and i t is now evident that the quantity denoted by does play the role of an activity for a two-dimensional system. The logarithm of - is given by In = * = Z . ! 2 V 2 > + <Z,<2> - < Z , ( 2 V 2 ) , >+..., (2.28) (2) with Z| = I, Performing the partial integrations indicated by (2,30), the adsorption isotherm becomes r. B (2) (2) . n (2) . (2).2 ^  „ (2) . (2).3 ^  „ 0, Ma = B j /y + ^ (/^  ) + Bj (/J, • ) +..., (2,29) where (2) p l B2 %(2) = I (2) P 0  f l 2 d ^ l d ^ » (3) (2.30) (2.31) t ^ Z ' V < f I 2 f I 3 f 2 3 + ' 3 f I 2 f 13 } d ^ l d f e % ' ( 2 ' 3 2 ) and f.. are now defined in two dimensions as U f . . = exp C- BA (T- - )H -1. i j e i J (2.33) (2) Equation (2.29) can be inverted to give/y as a power series in Na: /y(2) = Na - B 2 ( 2 ) Na2+ ( 2 < 6 2 ( 2 ) ) 2 _ ' B ^ 2 ^ ' N a 3-. .. . (2.34) From equations (2.09) and (2,11), one has for the spreading pressure 18' vA • - * — In 5 (2,35) kT Then by substituting (2,28) into (2,35), one gets an expression for the spreading pressure. This can be converted to a two-dimensional equation (2) of state by eliminating with equation (2.34). The result is NakT * A " ' • - ( 2 ) 'Na\ + ( 3 B 0 ( 2 ) -.-2B,(2).> ( R a j 2 + (2.36) '2 I \ 2 . ^3 or —£A = I + B ( 2 ) (Na) •+ C ( 2 ) -,Na 2 + (2,37) NakT A~ ~ where (2) B (2) A U > (2'38) 2 (2) - = ( B 9 ( 2 ) ) 2 - -2 R ( 2 ) , (2,39) 2 ~~ 3 A 3 J The two-dimensional v i r i a l coefficients are closely analogous to the three-dimensional coefficients and, just as in the three-dimension al case, are the ,: irreducibIe cluster integrals" f i r s t introduced by Mayer3, One can thus rewrite (2.29) as (?) (?) (?) ? (?) ? ? (?) ? Na =. - (2B^ ; /A) </J, > + 3 ( 2 ( B k / V /A -• J_ C^' / A ) (2)3' (/J J + .... (2.40) 19 If we compare this with the exact high temperature isotherm equation, (2,15), i t is readily seen that B AS -± exp [- B(fr s ( r ) ] d;rdz = Z, ( s : (2.41) •'A AS •(2B ( 2 ) /A ) ( Z , ( S ) ) 2 (2.42) These equations can be rearranged to give ,2 CAAS / 2 BAS B /A, (2.43) 20 ^CHAPTER .3 . • -21,23,25,31,32 QUANTUM STATISTICAL EQUATION OF STATE For a closed system in equilibrium at temperature T, one can N N write the classical probability density P N(£ , £ ) of the canonical ensemble as PN ( £ N ' £ N ) 1 exp (-HN/ kT), (3.01) ^ \ where H is the Hamiltonian for an N particle system and Q^  a normalizing constant, the canonical partition function. It can be determined by the normalizing condition P N d r N d ^ N = l , (3.02) or Njh 3M exp <-H / k T ) d^ N dj^ N. (3,03) Classically, one writes the Hamiltonian as a sum of the kinetic energy 2 ,„.. , , , . , , , N, i i of a l l the molecules of the system. Carrying out the integrations over p • N I p." /2m, and a potential energy <M£ ) which depends on the configuration the momenta yields the configurationaI distribution function P N { £ N ) = J . e x P C~ * ( - C N ) / k T ) ' (3.04) ZN where is the configurationaI partition function, Z N = Ni X 3 N Q n . (3.05) M Defininq a function W,. (r ) by S N a, ' 21 W, N -3M = X exp (- ) AT) (3.06) the configurationaI distribution function becomes (3.07) M and from the normalising condition we have: N N I W dr (3,08) Thus far the development has been restricted to classical systems. However there are two types of quantum effects which have to be considered under certain conditions: (i) diffraction effects, which result from the wave nature of the molecules and are important when the de Broglie wave length associated with the molecules is of the order of magnitude of the molecular diameter; ( i i ) symmetry effects, which result from the st a t i s t i c s of the particles and are of importance when the de Broglie wave length of the molecule is of the order of magnitude of the average distance between the molecules in the gas. At room temper ature the diffraction effects are measurable in helium and hydrogen but unimportant for heavier gases, At lower temperatures the quantum deviations associated with these effects are quite appreciable for helium and hydrogen and of some importance for the heavier gases, Symmetry effects are important only at high densities or very low temperatures. Deviations from classical behavior depend upon the magnitude of the * 1/2 quantity A = h/o (me) . This quantity is the ratio of the de Broglie wavelength (corresponding to a system of reduced mass y and energy e) to the c o l l i s i o n diameter a, and is characteristic for each substance, 22 Q u a n t a ! l y t h e r o l e o f t h e c l a s s i c a l p r o b a b i l i t y d e n s i t y i s t a k e n o v e r by t h e p r o b a b i l i t y d e n s i t y m a t r i x , J N N N ! * N* • N The d e n s i t y m a t r i x i s independent o f t h e p a r t i c u l a r s ystem of v/ave * N' N f u n c t i o n s c h o s e n , s i n c e i t i s t h e a v e r a g e v a l u e <V ) y(£ )> o v e r a l l systems of t h e ensemble. T h i s p r o p e r t y of i n v a r i a n c e of t h e d e n s i t y f) , N N i n terms o f any orthonormaI s e t o f b a s i s f u n c t i o n s s p a n n i n g t h e whole m a t r i x i s a v e r y u s e f u l one i n t h a t i t a l l o w s , £ ^ "t"° ^ e w r i t t e n s p a c e . The p r o b a b i l i t y d e n s i t y m a t r i x i s a f u n c t i o n of t h e two s e t s of N p o s i t i o n v e c t o r s r ^ and r ^ , The c l a s s i c a l i n t e a r a t i o n o v e r momenta. N l e a d i n g t o t h e c o n f i g u r a t i o n a I p r o b a b i l i t y d e n s i t y P.,(r ), i s r e p l a c e d i n *0 N N t h e quantum c a s e by. t h e t a k i n g o f t h e d i a g o n a l element J^K, > £ ^' ^e assume t h e d e n s i t y m a t r i x t o be n o r m a l i s e d , so t h a t ' *D i N N, , N . ... K., ( r , r ) d r = I . (3,10) * If we d e f i n e a p r o b a b i l i t y o p e r a t o r ^ s u c h t h a t t h e p r o b a b i l i t y d e n s i t y m a t r i x e l e m e n t s can be w r i t t e n as f) r N N \ r * , N 1 , 0 . N. ... K . , ( r , r ) = ) 4» ( r 5 • J , ( r ^ # (3,11) V V - 23 t h e n i t can be shown by means of t h e quantum L i o u v i l l e e q u a t i o n t h a t £) -u, N N y must be a f u n c t i o n o n l y o f t h e H a m i l t o n i a n o p e r a t o r 7 ^ j ( r , p ): I exp (- 34- 7 k T ) , (3,12) f 23 where (P. = -iH 3/9r. is the momentum operator, and the operator J J exp (-3V/kT) is defined as £ -(3V/kT)n /n . From (3.11) and (3.12), the density matrix is %v and the diagonal elements are P N ^ ' ^ H I *V > E X P ( " ^ N / K T ) V t N ) ' ( 3 J 4 ) This expression is the quantum analogue of the classical configurationa I distribution function. (3.14) can be put into a form analogous to (3.07) by defining ^ J N (£ N) d£ N = Ni Tr exp (-3^/kT), (3.15) whence NlQ N k, (3.16) From (3.10), ( 7 <j> ( A exp •(-"H-./kT) >^ ( r N ) ) d r N \ -\ N N U N (r/) dr n , (3.17) QN = --I NI corresponding to (3.07). • 33 34 -A i N The Slater sum ' hj^ (^  ) is the exact quantum mechanical analogue of the Boltzmann Factor W ^ ^ ) , and at high temperatures, where quantum deviations are small ,0/ J M — * " ! / ' ' M» This can be proven by using the invariance property of *P ^ for the system of orthonorma.l eigenf unctions N used. Since the <J> ) form a complete orthonormal set, 24 Y $ (r!) <j> Cr, > = n 6(r' - r ), (3.18) v \<i where 6(r' - r„ ) is the Dirac volume 6 function. The Slater sum can therefore be written as lO N (r N) = Cexp (- 3+/kT) I < + ,)P n 6(r' - r )] , (3.19) P " £ = where "H-^  operates only on the r^, and after this operation all r| are N set equal to the corresponding . The <j> (r ) have been properly symmetrized by summing over a l l permutations P of the indices, jr indicates the 3N-dimensional vector resulting from the application of this permutation to the vector £ . The +1 is for the symmetrical case (Bose-Einstein statistics) and -I for the antisymmetricaI case (Fermi- Dirac s t a t i s t i c s ) . We shall consider f i r s t the ideal gas, for which the Hamiltonian is just the kinetic energy operator, ~HN = t ) N = /2m) I v."" , (3.20) w i J 2 where V. is the Laplacian of the jth particle. Writing the 6 function J in terms of its Fourier integral, one gets 2, 2 r exp (- "D/kT) <5 (r - r ) = exp [ - ( x k /4ir) +2* ik-(r -rJdk = X"3 exp [- T r ( r ' - r ) 2 / X 2 ] , (3.21) and from (3. 19), N. . -3No / ,• , , P r r. ,2 " 0 J N </f)=X (+ l> P e x p E - ^ f r - r p / ] , (3.22) Throughout we shall adopt the convention that the upper sign refers to . 25 Bose-Einstein s t a t i s t i c s . Since the identity permutation operator causes the argument of ° —3N the exponent to vanish, = X " i f the symmetry of the eigenf unctions -3N has no influence. This value X is the value of for an ideal gas ( ^Cr/V = o). The distance over which the molecules influence each other due to the symmetry of the wave functions is of the order of magnitude 1/2 X = h/(2mkT) which, except for a numerical factor, is the de Broglie wavelength of the molecular motion at a temperature T. As this wavelength become much smaller than the molecular diameter at high temperatures, these deviations due to statis t i c a l effects become very small. Thus for an Ideal gas at high temperatures W ^ (r ) f (r ). For a real gas, the complete Hamiltonian must be substituted into. (.3,19). The proof that the Slater sum now approaches the Boltzmann factor at high temperatures has been given by Kirkwood, who obtained a series expansion for 10^ ; lO,(r N) = W M(r N) { l + X 2 I w + X 4 \ W <J} + } N % I H v 2 h 4 J J + U (rjS (3.23) in which j W 4 ( J ) = i & { V j V #C2 V.2 ( V.6 ) 2 + 8 960TT2 • ^ J ^ * J 26 +5- (V . 2 ( j> ) 2 ] -+ 6 2 [5V. 26 (V.A ) 2 +3 V.A-V. ( V.A) 2] B3 ( V.A)4 } . (3.25) 24 % J 2 The series (3,23) converges when the factor h /m kT, which is of the order of the square of the thermal de Broglie wavelength is small. Thus (3.23) offers a good approximation to I'lj) (r^) when the temperature is high and the quantum deviations small. We now consider the evaluation of the two-dimensional second (2) v i r i a l coefficient, B . This case is completely analogous to the 23 three-dimensional one if the following changes are made: 1. r becomes the tv/o-d imens iona I vector T . where T is a cylindrical coordinate. Hence, the Laplacian of the ith particle becomes V j 2 = T " 1 ( ) T (3/3£ )-. (3.26) 2. The three-dimensional volume is replaced by the area element 2-m dT. The two dimensional v i r i a l coefficient can then be written as B ( 2 ) = - V 2A W 2(x (, x2) -OJ, (^)\3,(T2)3 dx .. d T , (3.27) For a perfect two-dimensional gas one obtains immediately M l = A " 2 N I ( i ) P 6 XP I % - ^ . ) ] (3.28) P x2 .1 For a real gas one must employ the complete Hamiltonian to obtain the 27 two-dimensional analogues of (3.23) - (3,25), Writing the Slater sum for two particles and substituting into (3.27), one finds for the second v i r i a l coefficient, n(2) (2) , p (2) _ (2) (2) B = ( B . +6. +B.. + . , , ) - B , cl I II + perf (3.29) (2) . (2) and B (2) I I the f i rst where B . is the classical contribution, cl ' 2 4 and second quantum corrections proportional to X and X respectively, and B r^ the quantum perfect gas term arising frorrDv^ (2) We want to evaluate B using a completely general power-1 aw 35 potential function, which can be written as *( T ) ye + BT -b +C r C+... -ZT Z ] (3.30) The perfect gas term, which is independent of the potential, is done f i r s t . Using (3.27) and (3,23), and integrating, one gets, B (2) perf 2A MX' - 2 2 exp C- 0 (T, - T 0 ) 3 dx.dT 0 , r 2 M %2- ^ 1 ^ 2 ' A = + Ng-.h 8-TT m (3.31) The remaining terms are found by substituting the two-dimensional anal ogues of (3.23) - (3.25) into (3.27): (2) B cl i r N [exp (- Bcj)) -I] xdx , (3.32) 28 (2) _ T,tijl' 12 u o 2 2 e x p (- 3(f)) [ V A -. 6 ( Vd> ) ] x d r . (2) _ .-^m 4 3 240 u ' exp c- v% - l 2 & 2 ( ^ 5 2 (3.33) •+ 8 V<f> -V3cj> + 5 (V2<{>)2] +"i [5V2(j) (7* ) 2 6 + 3 X** X(^)23 B 3 (Vcf, ) 4 } T d T , 24 (3.34) where p is the reduced mass, It is desirable to write the second v i r i a l coefficient in reduced form: B « > * -CB'fV* B « > * B , ' 2 1 * + . . . ] + - / 2 B ' 2 ^ p e n where ,(2) 2 (2)* * 1/2 TiN a Q ; A =h/o (me)1' , (3.35) (3.36) Then if we introduce a reduced temperature, defined by T = (Be ) , equation (3.31) becomes n ( 2 ) ' * 2 * _| B p e r f = (4, T ) , (3.37) and after a series of partial integrations, equations (3.32) - (3.34) become (3.38) (2)' cl -So -2 exp ( -04>. ) $ T d t , M " - R 2 4 „ 2 T V e x p ( - B<|> ) 7 xdT, , (3.39) 29 (2) ^  2 ? * -2 B|V*' = - (So /20) (24TT T ) Z exp (-B6 ) { -56 12 " 12 ' " 2 + 6 - 1 | $ + 7 B(6 ) } TdT . (3.40) T T When equation (3.30) for the potential is introduced, (3.38) - (3.40) can be integrated analytically to give B ( 2 ) * = - y - ^ ^ ' y • z - ( a J + b k + c | + « • -- 2 ) j . k , l , . . . > 0 jl kj M z (I ) ( z ~ a ) , J + (z-b) k + (z-c) I + . .-. +2 T - r f i " + b k + C ' + r2-) (3.41) z B ( ( 2 ) ^ = [ z + I' U ( 2 ) / z ) A W . . . j,k, I >0 j l k t |_\ . . . -(aj+bk+cl+. ..) (z-a)'j+(z-b)k+(z-c)•! + . . . >- Y T r^aj+bc+cl+- ^ -j ^  z (3.42) where X - I nS:n (z-n) , n=a,b,c, n S n = j , k , l , . . . . , (3.43) 30 B | L ( 2 ) ^ G - ( 2 4 1 f V ) - 2 J (M ( 2 ) + v ( 2 ) + P ( 2 > + 5 ( 2 ) ) 20z j,R,l,,..> 0 -<aj+pk+ct+...+2) (z-a) j+(z-b)k+(z-c) I.+. . .-2 A W . . . Z Z \ J i k ' l l . . . T r ^aj+bk+cl.+ . • .+2 ^  (3.44) (2) 2 where y = 7 I L~mS (z-m)D m m v ( 2 > = j mS (7(z+l) 2 (z+4) - 7(m+l)[m(m+l)+4(z+l)] m m + I2(z-m) -5[(z+l )(z+2)(z+3)-(m+l ) (m+2) (m+3-)]\ (3.46) f?) p = 14 y nr S S [(z+I)(z-n-r-I) + (n+l)(r+l)] (3.47) h n r r/n (2) 5 .= (z+l) (z+2) t H (z+l) -10(2+3)1 +24 (z+2) (3.48) The prime on the summation indicates that the term j=o,k=o, etc, is omitted, which is necessitated by the fact that HO) is not defined. This leading zeroth term is a constant, and just gives the z appearing before the summation. (2)*' The various contribution to B have been computed at values of reduced temperature in the region 0.3<T < 100, using the two effective potent ia Is (1.14) and (1.16) and numeri caI tables are g i ven in Append ix I. For (1.14) several different values of n in the range 0<|n|< 0.05 have been used. It is easy to show that by renormaIizing the parameters in (1.16), this equation can be written in the usual (12,6) form, 31 -12 - t 2 ), (3.49) with 2 _ -1/6 E 2 " ? £ ' "2 " 5 o = V °2- (3.50) Therefore in using this Appendix, i t is to be understood that effective parameters must be employed for this potential. We now wish to consider the evaluation of the second gas-surface 4 v i r i a l coefficient. The total quantum , taken to order H in the Wigner-Kirkwood expansion, is given by c BAS = B A S ' + B A S + B A S t (3.51 or in reduced form, cl * * 2 I * * 4 I I -V * nerf * BAS = ( BAS + A BAS + A BAS t A B AS > ( 3 ' 5 2 ) where = Az 0 B^ <, . Substituting the correct Slater sum, one gets for the various contributions to B^: B cl AS [exp (- ) - I] dV, geo AS AS 12m K4 B 3 — ^ 240 m~ v exp (- 3* s){V% s - |[2V 2(V4) s) 2 geo (3.53) exp (- 6<J> ) [v2cj> -(£) ( V(j) ) 2] dV, (3.54) V ^ s s 2 ^ s ' geo •M-8 V V V\ +5 ( V 2 6 s ) 2 ] + ± 15,% ( V , s ) 2 o + 3VA - V(Vd> ) 2 ] - ~ (VA ) 4 }dV; 24 where B~ l/kT and m is the molecular mass. <j> is the interaction potential between an isolated gas molecule and the solid in the differential volume element-dV, and we shall use the following approximation for this quantity. If one assumes that the gas molecules interact with individual atoms of the solid through a general Lennard-Jones potential, then m/(n-m) «j>(r) = Cn/Cm-n)] (n/m) E . [ t a / r ) m - ( a / r ) U ]. (3.56) We now assume the solid to be semi-infinite (i.e., infinite in x and y, and bounded by the plane 2=0) and to obey the continuum model (i.e., uniform distribution of matter). Equation (3.5I) can then be integrated over the three-dimensions of the solid to yield, <j>s (2) = <B/(B-a>> ( 3 / a ) a / ( B " a ) e , s C ( z 0 / 2 ) A - ( z 0 / z ) 5 ] . (3.5?) Now, £ j is the maximum energy of gas-surface interaction, z 0 is the distance between a gas atom and the plane surface at zero net interaction energy, and a and B are equal to n-3 and m-3, respectively. Substituting (3.52) into (3.48) - (3.50), and performing the analytic integrations, gives = Az 0 I ( B j ! r ! (Y/T*) J ( S ^ ) + I r ( j g - l ) , (3.58) j>o B g B . ' * = - A z . '! (24, W I (ja+b) (y/T*) i !52±± r(Ja+l) , (3.59) 33 Where a = a(6-a) and b = 6+1, J l * . -. i 2 * ~ 2 Y . .2 -j(6-a)-3 B = Az 0 (24TT T ) I y j a + • jb + c N . * 160 j>o B J | (_, } FCja+3) ( 3 ' 6 0 ) where a = -7a 2.(a-6) 2 6 (3.61) b = 5a {(6+1) (6+2) (6+3) - (a+l) (a+2) (a+3)} -7 (6+1)a [(6+1) (6+6) - (a+l) 6] +7a2 (a+l) 2, (3.62) c = 3(6+1) (6+3) [3-26], . • _ (3.63) and V = (• 6/(6-a)) (6/a) a / ^ ~ a ) , (3,64) Three different models for the interaction potential will be examined to analyse the data available for gas-solid interactions. The f i r s t of these is an inverse ninth-power repulsion, inverse cube attraction law (9,3) which results from a three-fold integration of a Lennard-Jones (12,6) potential. The layer structure of graphite suggests that maybe a better model for this solid could be obtained by integrating over the atoms of the surface plane only. The large distance between basal planes, compared to the relatively short in-plane distance between carbon atoms makes the contribution from the f i r s t layer of atoms of much the greatest importance at distances close to the surface, while contributions to the interaction energy from the underlying planes become relatively important only when the total interaction energy becomes small anyway. Such an integration of a (12,6) potential over an infinite plane leads to a (10,4) potential. Finally, since the repulsive forces are really caused by orbital overlap, and are thus short ranged, when an adsorbed atom is directly over a surface atom there is only a repulsion due to the single-34 pair interaction. Thus the third model used is a (12,3) potential, corr esponding t o a three-fold integration of the attractive part of the potential only. cl * I * 20 and B^^ values have been published previously and I I * * * a table of B ^ values, computed over a T range of .I<T <l.0, is given i n Append ix I I . 35 CHAPTER 4 THEORETICAL COMPARISON OF THE (12,6) AMD (12,6,3) MODELS FOR B ( 2 ) From the tables in Appendix I, one can construct curves of the (2)-^ reduced two-dimensional second v i r i a l coefficient B as a function of * . * reduced temperature T , once values for the quantum parameter A and the perturbationaI parameter n have been chosen. For the purpose of com paring the (12,6) and (12,6", 3) potent i a I energy models i t is convenient to choose a system which displays f a i r l y large quantum deviations. The three Bose-Einstein gases which display the largest quantum effects are He, H^ , and D^. Therefore we consider a hypothetical Bose-Einstein gas . (HBEG) with a A value of 1.8, which is the average value of A for the three gases mentioned above. Accordingly, Figure I presents classical (2)^ * and Bose-Einstein quantum curves of log (l+IB I) versus T for HBEG, as predicted by the two potential models. However, i t is not appropriate to use the same value for the (12,6) model since the potential parameters 24 have been renorma1ized. Theoretical estimates for predict that * £2~°.9e which results in an effective A 2 value of 1.9. To choose an appropriate n. value we have noted that although theoretical predictions of the magnitude of this quantity for the rare gases f a l l in the range 12 0.04<|n|<.07, empirically determined values are always somewhat less , and we have therefore arb i t r a r i l y chosen n=.-0.05. Vie feel this to be a reasonable upper bound for this parameter. As shown in Figure I, the quantum curves for both models converge to the classical limit at high temperatures as expected. The quantum 35a Reduced two-dimensionaI hypothetical Bose-Einstein gas as Comparison of (12,6) and (12,6,3) FIGURE I second v i r i a l coefficient for a a function of reduced temperature, potential models. 36 deviations increase as the temperature is lowered and the quantum curves lie above the classical ones. The maximum quantum corrections occur in the vic i n i t y of the classical Boyle points for the two models, which are at reduced temperatures of 1.56 and 1.11 respectively for (12,6) and (12,6,3) potentials, V/ith further lowering of the temperature the net quantum corrections decrease and eventually become negative. Thus the quantum curves cross the classical ones and lie below them, For our hypothetical system one finds that the cross-over points occur at T - values of 0.85 and 0,74 for the (12,6) and (12,6,3) models, respectively. Owing to the fact that the Boyle temperature T^ depends quite strongly on n (see Appendix I), i t is not really correct to make detailed comparisons between the models at identical values of T . It can be 12 * * shown for any n that z^fz is in the same ratio as T^ (12,6) /T^ (12,6,3), so that one should compare the models at corresponding values of reduced temperature relative to the Boyle point, i.e, at'the same values of * * i ( 2)^1 T /Tg , Figure 2 is a plot of log (l+|B |) versus relative reduced temperature, and the difference between the models is more obvious. It would appear from this figure that the quantum corrected curves differ more in shape than do the classical ones. It is clear that at low temperatures the Wigner-Kirkwood expansion to M . 4 no longer converges and the present treatment breaks down. We wish to show that one can establish reliable limits within which the treat ment used here can be applied. However, i t should be noted that the following 4 discussion applies only to the Wigner-Kirkwood'expansion to order . 36a FIGURE 2 Reduced two-dimensionaI second v i r i a l coefficient in classical and Bose-Einstein quantum stati s t i c s as a function of relative reduced temperature. Comparison of (12,6) and (l'2 6,3) potentials for HBEG. 37 21 25 Although recursion formulas ' have been presented to calculate higher order terms, the amount of work involved is extremely tedious. For this reason a l l previous second v i r i a l coefficient studies using the Wigner- 4 Kirkwood expansion have been truncated at the ft term. Therefore our' discussion of the convergence of this series is of a practical nature only, and in no way describes the convergence properties of the complete untruncated expansion. There are two c r i t e r i a for physical reasonable quantum deviations which can be used to set such limits. F i r s t l y , for any value of the * (2)*" (2)^" quantum parameter A , B - B c| should be positive and monotonically increasing with decreasing temperature. From Figures I and 2 i t is obvious that this criterion is not met on the attractive branch -for either model, (2)^ (2)"^ but that B - B . exhibits a maximum and then declines. In order c I to use this criterion to set a low-temperature limit for the treatment, one must solve the equation [ 9 ( B V Z ; - B , ) / ST ].„ =0 , (4.01) c 1 A *, n (2)# which requires a computation of the temperature derivatives of Bj . , (2) ¥ (2) 4" B., and B , . The aloebra is straicht-forward but formidable l I pert ^ ?n quantity, especially for the (12,6,3) potential, where the calculations must be repeated for every n value of interest. Moreover, we feel that (4.01) probably sets too low a limit, for a reason mentioned below. It would probably be safer to use the expansion only down to the point Id (B^ Z J - B , ; ) / 3T 1 ] A X = 0, (4.02) cl A*,n 38 and the labour would be further increased (e.g., the second derivative ( 2 ) * of B| for the (12,6,3) model involves the evaluation of 30 integrals). However, there is a second criterion which is considerably simpler and appears to set a bound lying between those given by (4.01) * and (4.02). That is, at fixed T , the quantum deviations should be a * (2) V (2) %r monotonically increasing function of A . But if one plots B -B | versus A at constant T i t is again found that a maximum exists at some * value A , which can be determined through max, [ 3 ( B V Z ; - B z ; T ) /3A ] * •= 0, (4.03) T , n whence, for the tv/o-dimensionaI gas, A^ =C(B <2)V2)*> / 2 B < 2 ) V 3 ' f . (4.04) max perf I II * 1 #n * .. x A has the physical significance that for any T there exists an upper bound on the quantum parameter such that the quantum corrections are mono- tonic in A . One can therefore construct a curve of the the type shown in Figure 3, plotting relative reduced temperature against A m a x . Such a curve can be thought of as giving the radius of convergence in the plane x x x (T / Tg , A ), since for a l l points lying within the area bounded by 4 the ordinate and the curve, the Wigner-Kirkwood expansion to order converges. As seen in equation (4.03) the radius of convergence is a function of n, and i t is evident from Figure 3 that increasing n serves to decrease the convergence limits. As the curve for n=0 corresponds to the pure (12,6) potential, we .can now say that for any value of A the quantum (12,6) 38a FIGURE 3 Radius of'convergence of the V/K expansion in two and three dimensions. Curve' I, two-dimensionaI (12,6,3) potential with n = -0,05. Curve 2, two-dimensionaI (12,6) potential. Curve 3, three-dimensional (12,6) potential, 39 model is applicable over a wider range of temperatures than is the (12,6,3), Similarly, at a given temperature the (12,6) model can accom odate the more degenerate case. This observation is modified to some extent by the fact that the parametric renormaIization required for the (12,6) potential invariably increases the value of A . This increase may or may not be sufficient to offset the difference in models, depending upon the n. value required. For example, at the Boyle point the (12,6) model can be used for <2.0, while the (12,6,3) model with n=-0.05 is only applicable for A <l,6 so that the (12,6) potential can be used for HBEG at this temperature whereas the (12,6,3) cannot. However, the (12,6,3) case is convergent at this temperature for HBEG so long as |n|<0.03. To investigate more fully the dependence of the quantum corrections (2) (2) ^  on n, we have constructed for our hypothetical gas curves of B - B j * versus T /J for differance values of n. (See Figure 4). The dependence of the convergence limit on this parameter is evident from the shift in the position of the maximum. We should mention that in a l l cases the temperature limit of convergence found from Figure 3 lies to the right of the corresponding maximum in Figure 4, and appears to be slightly to the left of the inflection point. It should now be clear why we feel equations (4.02) and (4.03) provide more r e a l i s t i c convergence limits than does equation (4.01). This is brought out even clearer by comparing curves I and 3 in Figure 4, the difference here being caused only by different A values. At the intersection of these curves, which lies on the high temp erature side of both maxima, we have the very unreasonable situation of the 39a FIGURE 4 Dependence of the quantum corrections to the two-dimensional second v i r i a l coefficient on the perturbationaI parameter n. Curves 1,2,4,5, are for the (12,6,3) potential with -n values of 0,0.01',0.03,0.05, respectively. Curve 3 is for the (12,6) potential. 40 same potential function yielding the same net quantum corrections at the * same temperature for two different values of A . 4 (2) * (2) + In the region where the expansion converges, B - B j increases monotonicaIly with |n| at any relative reduced temperature, so that the larger the perturbation by the f i e l d the more the quantum defects will be enhanced. However, the question of which potential model predicts the Iargerquantum deviations at a given temperature is d i f f i c u l t to answer, the situation being complicated by two competing effects. On the one hand, as ]n| increases the corrections for the (12,6,3) potential are enhanced, but on the other hand this increase in the perturbation will also serve to increase the corrections for the (12,6) potential, since as the magnitude of the perturbation increases, decreases and there is a concomitant rise in A 2« '+ therefore appears that in order to say which mode! will produce the larger quantum deviations one must know both n and A 2» which effectively precludes an a priori estimate. Concerning the effect of reducing the dimensionality of the problem, Figure 5, showing the classical and Bose-Einstein quantum curves, ( 3 ) * ( 2 ) + ( I ) ^  for B , B , B as functions of reduced temperature, provides a striking comparison. These curves are computed for hydrogen molecules interacting via a (12,6) potential. As the dimensionality of the problem decreases the relative reduced temperature increases at which a treatment 4 of the H expansion f a i l s . In three dimensions the treatment appears to be satisfactory down to a relative reduced temperature as low as 0,4, whereas in two dimensions below about 0,9 the quantum corrections begin to decrease 40a FIGURE 5 Second v i r i a l coefficient in one,two, and three dimensions, In each case, curve I is the Dose-Einstein quantum curve for HBEG, and curve 2 is for a classical gas. .8 41 with decreasing temperature and the curves eventually cross. Note that 36 in one dimension there appears to be no temperature at which the treatment is satisfactory, the net quantum corrections being always negative. As was done in two-dimensions, we can calculate for the three- dimensional case the value of A at which the quantum correction cease being monotonic is A at any given temperature. In this case we find A* = L~3B ! 5 ) * + C 9 B P ^ 2 - 3 2 B t ( 3 ) 4 t B | ! 3 ^ ) , / 2 ] / 8 B 1 ! 3 ) ^ n „ max perf perf. I II J II (4.05) 23 Using the tabulations of Hirschfelder, Curtiss and Bird we can then construct a radius of convergence curve for the three-dimensional (12,6) gas, having employed a Wigner-Kirkwood ft^ expansion for the quantum corrections. This has been included in Figure 3, from which i t is apparent that the three-dimensional results are usable over a much wider range than the two-dimensional ones. When the one-dimensional results are examined i t is found that there is no value of A v/hich leads to positive quantum corrections at any temperature whatever. If one compares, at a fixed relative reduced temperature, the terms which make up the Wigner-Kirkwood quantum corrections to order 4 ft , one finds that, as the dimensionality of the gas decreases, the 4 magnitude of each contribution increases, with Bj| increasing relatively 4 more than Bj . For example, at a relative reduced temperature of about (3)^f (2)*^ ( I ) * 7, the following ratios are found: Bj : Bj : Bj ^1:2:3; and (3)^f (2) 4 (1)4? Bj i : Bj j : B | | <\J:6:20. Of course, the differences are even larger at lower temperatures. The net effect is that the series expansion to order ft^ diverges in one and two dimensions at temperatures, where i t is 42 s t i l l quite rapidly convergent in three domensions. From another point of view, equation (3.23) can be regarded as a development in powers of the operator V. Hence the requirement for conver gence of the series is that the differential quotients of <j> are small compared to <j>, and that the higher orders become successively smaller, Clearly, this requirement is met less and less satisfactorily as the dimensionality is reduced. However, the real "offender" in the lower-dimensional cases is the perfect gas contribution, owing largely to the altered temperature dependence: o (3)* * -3/2 (2)* *-| (|)* *-l/2 perf ' perf ' perf Thus for two-dimensional and especially one-dimensional gases, the effects due to symmetry of the wave functions f a l l off rather slowly with increasing (i) ¥ temperature, In fact, for the one-dimensional gas, BpQrf dominates over the Slater sum contributions at a l l temperatures above K 0,6, while 4 below this temperature the Wigner-Kirkwood expansion to order is every where divergent. The following comparison is instructive. In the vicinity of the Boyle point, B p e r j 3 ) ^ : B p e r j 2 ) * B p e r j n * ^1:20:450, while at a relative reduced temperature of about 7 the corresponding ratios are 'v,! : 50:3000. 43 CHAPTER 5 -.ANALYSIS OF THE DATA TWO-DIMENSIONAL' SECOND VIRIAL COEFFICIENT We have seen that one can determine experimentally the ratio (2) • " of B /A from the intercept and i n i t i a l gradient of a plot of Na/p versus p, It v/as shown in Chapter 3 that for the (12,6,3) model one has the theoretical form (?) ^  * * B ^ ; = f (T , n, A ) . - (5,01 ) For any given system e and o , hence A , are fixed, so that a comparison (2) of the temperature dependence of B /A with the theoretical curve of (2) ^  * * B versus T yields "best-fit values" of the parameters n andA . By "best-fit values", we mean those values of the f i t t i n g parameters which minimize the standard deviation between the experimental points and the theoretical curve, i.e., one proceeds by finding that value of n which minimizes the standard deviation in A. This f i t t i n g procedure is quite analogous to that used in determining force constants from gas phase v i r i a l coefficient data. In the case of the renormalized (12,6) function, ( 2 ) * * . * B^' f (T 2 , A2 ), ' (5,02) * • * ' 1/2 T 2 = kT/e2, A2 •=: h/o2 ( I H E ^ 1 ^ . (5.03) The f i t t i n g procedure now is a bit more complicated. One f i r s t f i t s (2) (2)*^ * the B /A versus T data to B . versus T to obtain a f i r s t estimate of c I * - 1 2 the parameters and A = A/ j u N , From o and the ratio e/£ 2,° 2 can be computed from (3.50) and an estimate of A2 obtained. The data are 44 ( 2 ) # * now refitted to the f u l l quantum corrected curve of B versus T , commencing a series of successive approximations to consistent values of E 2 , <^2> A 2 and A. Note from the form of the (12,6,3) potential in equation ( 1 . 2 0 ) , that once n is known one can obtain the effective well depth t^i the c o l l i s i o n diameter a 2, a n c' the position of the potential minimum T 0 for this model: *(o 2 ) = 0 (5.04) 4>(T0) = - e 2 , (5.05) ( 8<J)/3T) = 0. (5.06) T o Hence, for both models one can estimate the perturbation of the gas-gas pair potential by the f i e l d from a comparison of the effective parameters with those of the bulk gas. In this connection i t should be mentioned that, the n' values found using the (12,6,3) potential and thus the effective parameters, are 12 quite strongly influenced by one's choice of bulk gas force constants. A variation in the bulk-gas parameters will affect the results in the foI I — (2) owing ways.- F i r s t l y , the experimental B /A values will change slightly owing to differences in the corrections for bulk gas imperfection Swhich must be applied to the data. Secondly, the two f i t t i ng . parameters (2) n and A will vary, both by virtue of the altered B /A values and because of their inherent dependence upon e/k and 0;n is primarily dependent, upon the energy parameter, andA upon the c o l l i s i o n diameter, although there is also some slight interdependence. Therefore, this limitation 45 should be kept in.mind in discussing the significance of the results, For CH^ we have used the bulk gas parameters given by Michels and Nederbragt"^: o o CH 4 : e/k = 148.2 K , a =,3.82 A. The energy parameter for CD^ has been taken to be 0.9% lower than that for CH^, in order to agree with the findings of Thomaes and van SteenwinkeI" However, we have retained the same a values for both gases, as the d i f f e r  ence in this parameter between CH4 and CD^ has not been reliably determined and in any case should be quite small, Thus, the CD^ bulk gas parameters used are: CD4 : E/k = 146.9°K , o = 3.82°A. 19 For H^  and , we have used the values found by Michels, et.at,, : H 2 : e/k = 36.7°K , o= 2.959°A D 2 : E/k = 35.2°K , cr= 2.952A. These values are the best f i t quantum parameters. For the classical 3° 40 parameters, those values found by Michels and Goudeket J* are used: H 2 : E/k = 29.2 K , o= 2.87 A >2 2 D 9 : e/k = 31.I K , o= 2.87 A, 41 Table I presents the results of the present analysis of data for CH^ and CD^ interacting with graphite, together with the previously 12 27 determined classical results ' , which have been included for purposes of comparison. For both potentiaI.mode Is i t is seen that the quantum corrections are quite small (as was anticipated), amounting to no more •Table \\ Curve f i t results for two-di.mens.Ional-.CH and CD. on graphite, (12,6) Potential (12,6,3) Potential Molecule -e2/k( K) 'og(A) -A2 -Std, dev, c 2/k( K) <o2<A) •^n Std, dev, CH4 129,2 -3,86 CD4 |29,3 3; 86 . I>; Classica I F i t 0,425 139,2 3,84 0,428 138,6 3,84 0,0216 0,433 0,0201 0,435 CH, CD, 129,8 3,86 0,244 129,8 3,86 0,222 2, Quantum Ftt 0,424 139,5 3,84 0y23l 0,0208 0,431 0,426 138,8 3,83 0,211 0,0194 0^33 46 than 0.5% of the pair interaction energy. These corrections are larger for the (12,6) potential than for the (12,6,3). In Chapter 4 i t was pointed out that i t seemed impossible to determine a priori which model would yield the larger corrections in any given case because of two competing effects. At the same relative reduced temperature, the quantum deviations increase with increasing |n|, but on the other hand the (12,6) model appears always to provide the larger A value. Hence, each case will differ, depending on the difference in A values and on the magnitude of n. Mote that the best-fit value of |n| decreases by about 4% with the introductions of the quantum corrections. Since n is a measure of the perturbation of the gas phase potential by the f i e l d , this reduction is then reflected in the slightly increased effective interaction energies. However, these effective energies are quite insensitive to changes in n, rising by only about 0.2$, Unfortunately, the change in standard deviation of the f i t when the quantum defects are included is essentially the same for both models, so that i t is s t i l l impossible to make any definitive choice. This of course owes to the fact that the deviations from classical behaviour are very small. We have not included values of the other f i t t i n g parameter, A, in Table I , as they are effectively unchanged from the classical results 12 27 which have been discussed elsewhere. .' From the bulk gas and effective two-dimensional parameters we can calculate the apparent nonadditivity induced in the pair interaction by the surface.fieId. Table II l i s t s values of the quantity A= (e-c„)/ e. 46a Table II, Values of A=.(E-E2) /e and 6?= Ce2(H) - e^DJj/ e CD) for CH^  and CD4 based on the four models. Model Molecule A x I0 2 6 x I0 2 CI(I2,6) CH4 - 12.8 _ G ] CD4 12.0 •Cl(12,6,3) . CH4 6.1 CD4 5,7 Qf12,6) CH4 12,4 CD4 ||,6 QC12,6,3) CH4 5.9 CD4 5.5 +0,4 +0.0 +0.5 47 it is clear from these values that one could draw rather different conclusions concerning the magnitude of the perturbation on the basis of the different models, since the effect for the (12,6) potential is 2,1 times that shown by the (12,6,3). However, the small change indicated by the (12,6,3) values probably should not be taken too seriously for the following reason. If one applies a (12,6) model to both the bulk phase and the two-dimensional layer, the shape of the potential wells will be the same and the entire perturbation must be absorbed in changing just the depth of the well and the position of the minimum. But introduction of the inverse cube term appreciably alters the shape of the potential curve, and this apparently means that the well depth does not have to change to the same extent. We have pointed out that a different choice of bulk parameters would change the values given in Table II, but we can state with reasonable assurance that i t would not alter the significance of the results, An 12 examination of the results for Ar,Kr and Xe reveals an interesting coincidence. For each of these gases, four different sets of bulk pot ential parameters were employed in the analysis, which led to four values of n ,e 2 and A. Yet in every case the ratio A(12,6)/A(12,6,3) for a given e turn out to be 2.11+ 0,02. Why this should foe so is not readily apparent, but the results seem too general to be entirely fortuitous. It is interesting that for both models the A values for CD^ are 6-7$ lower than those for CH^. This would indicate that for the lighter molecule the pair . interaction is more strongly perturbed. This seems entirely reasonable on two accounts. F i r s t l y , there lis a polarizabi I ity difference of 1.45^ between the isotopes'"^, that for CD. being the lower. 48 Presumably, the less polarizable a molecule, the less i t will be perturbed by the external f i e l d . Secondly, the gas-solid interaction energy is 20 <v I .2% greater for CH4 than for CD4 (see Table VI), Finally we should comment on the values of S=L~e2(H) -e2(D)~]/e2(D) appearing in Table II. Whereas the (12,6) model suggests that this quantity is effectively n i l , the (12,6,3) values give an indication of a definite change in e 2 with deuterium substitution. For the methanes in three- 38 —2 dimensions , .6= +0,9x10 , so the effect in two dimensions appears to be only about half that found in three, although the signs are.the same. Yaris'^ has derived a theoretical expression for the difference e ~Z2> which may be used together with values of e to calculate 6. This provides -2 the estimate .5= +0.4x10 , in good agreement with the (12,6,3) values. (2) The B^g, C^g, and B /A values for and D2 on graphite 4 I are computed from the isotherms of Constabaris, et.al. are given in Table (2) From the B /A values for H 2 i t isobvious that the data are very scattered since this quantity should be a monotonicaI Iy increasing function of temperature. This explains the unreasonable best f i t z^lY. values for the (12,6) potential: o classical : e /k = 54.0 K o quantum : ^/k = 63.5 K. These values imply an enormous additional attraction between H 2 molecules due to the presence of the fi e l d created by the surface. This attraction implies multilayer adsorption but in the temperature region of the data, one has monolayer adsorption since the adsorbed phase is about b% of the monolayer coverage. Therefore the assumption that there could be co-op erative adsorption is unreasonable.' 48a Table III BAS a n d CAAS data for H 2 and D 2 adsorbed on P33 (2700°) HYDROGEN DEUTERIUM T • BAS.. •B ( 2 )/A BAS "CAAS B ( 2 )/A 9.0.057 .4245 1218,68 6763,88 ,4380 II 1 1.02 5792,06 97.122 ,2853 610.599 7501.58 .2927 550.884 6428.74 104.156 .2014 322.071 7940.22 .2093 340.784 7781,53 109.903 .1554 182.936 7575.25 - ,1590 171 . 107 6772.46 117.049 .1183 1 17.761 8413.15 ,1206 106,054 7296.60 124.128 .0928 73.867 8586,63 .0938 63.902 7269.08 131.069 .0724 31.734 6055,75 .0752 44,309 7826,98 138.128 .0605 15.175 4144,52 .0614 29.503 7818.17 49 A plot of log C^Ag v e r s u s should be nearly linear, but when this criterion is applied to the data i t is found that the two highest temperature points deviate widely. We have therefore refitted these o • o data using six points only (131 K and 138 K omitted). The data were analysed using a l l 8 data points. Although the log versus I/T plot for these data showed considerable scatter, there was no firm basis for preferentially omitting one or more points. Table IV contains the results of the f i t s for the two-dimensional (12,6) o model for and on P33 (2700 ). Error limits on the e2/k values have 9 been established by applying the estimated experimental uncertainties to the original isotherm data, computing new B f l C and C.._ values, and ^ n o A A o f i t t i n g these to the model. These results indicate the e2/k values are o • o uncertain to + I .0 K for h'2 and + I .4 K for D^. Several remarks should be made here regarding the values presented in Table IV. To begin with, the quantum effects for are considerably larger than those for D^. If one calculates P= (e^-e^ )/e^ for both gases, one finds that P H / P Q = 1.74. Classically, the two gases yield essentially the same interaction energy. However, when quantum effects are . included, hydrogen appears to have the larger value. This inverse effect is to be expected on the basis of the polarizabiIity difference between the isotopes. The surface area values obtained are in good agreement with those found from a similar analysis of data for the rare gases adsorbed 2-1 on this surface, which lie In the range 9.5 - IIm g . Comparing the stand ard deviations in Table IV, the H 2 data f i t significantly better than those for D2. This was anticipated from the greater scatter in.the D 2 C A A S values. 49a Table IV Curve f i t results for two-dimensionak12,6) H 2 and on graphite ° 2-1 * Molecule e 2/k(°k) ° 2 ( A ) A(m g ) A 2 Std.dev. 1, CI ass icaI f it . H 2 31.0 3.001 I I,0 .21 D ? 32.0 2.976 12.0 I.26 2. Quantum f i t H 2 37.0 2.957 10.7 1,714 ,21 D 2 35.5 2.950 11.8 1.240 1.26 50 Table V contains values of A= (E-E^I/E and 6 2 = ( e 2 ^ - ^ ^ - ^ Z ^ ^ for the (12,6) potential. Also included for comparison are the three- dimensional 6 values. The A values in Table V are zero within experim ental error. Both classically and quantally, the uncertainties in 6,, are sufficiently large that this quantity could have either sign, although i t is likely to be positive for the quantum corrected f i t . A (12,6,3) analysis for hydrogen and deuterium has been attemp ted but again the results are ambiguous. The experimental uncertainties are such that n, which is very small ( |n|<0.0l), could be either positive or negative. Hence we have not included the results for this model. It is disappointing that these analyses of the data for the hydrogen isotopes have provided so l i t t l e information. Any definitive analysis of the two- dimensional (12,6) and (12,6,3) models will have to wait until more reliable data have been obtained. SECOND GAS-SURFACE VIRIAL COEFFICIENT 16 ° Here we analyse the data of H2, D2, CH, and CD4 on P33 (2700 ) by f i t t i n g the experimental B ^ values to the quantum corrected theoretical expression for the second gas-surface v i r i a l coefficient. Three different models for the interaction potential were used, the (9,3), (10,4), and (12,3). The f i t t i n g procedure used was to compare the temperature depend ence of the experimental B^^ values with the theoretical quantum expression (3.52). In the work of Constabaris, e t . a l . J ^ data for a s 3 function of c l ^ temperature were fitted to B.c by.adjusting the two parameters c /k and Az no I S " . ; 50a Table V Values of A = (e-e 2) /e and 6 2 = (^(H) - e 2(D)) /e2(H) for the two-dimensional (12,6) H 2 and C^. Model Molecule AxlO 2 S„xl0 2 6xl0 2 CJ C12,6) H 2 -6.2+3,4 D 2 -2.9+4,5 0(12,6) H 2 - .82+2.7 D„ - .85+4.0 '2' -3,2+7.6 -6,5 +4.1+6.2 +4.08 51 thus obtaining best-fit values for these parameters. One then calculates z 0 as described below, and a classical estimate of A . The B c values S A o are then fitted to the quantum corrected gas-surface v i r i a l coefficient, where B A S = f (T*, A s*), (5.07) * * 1/2 Is ' "s " " ° v , l ,"ls'' T = kT /£,„ , A„ = h / z 0 (me,.) 1^. (5.08) One again finds best-fit values for e| 5/k and Az c, and calculates a new A g value. This process is repeated until one obtains seIf-consistent ej s/k, AZo and A^ values. The principal d i f f i c u l t y in the analysis of second-order inter action data lies in the evaluation of the apparent gas-surface c o l l i s i o n diameter z 0. Not only must this quantity be known to obtain the apparent surface area, but also, as seen from (3.52) and (5.08), the quantum ' 20 corrections will depend strongly on the exact value of z 0. In the past , the gas-surface attractive potential has been identified with the London forces attraction of two isolated systems, 6 e.. = C/r. . . (5.09) C, the constant of proportionality, has been evaluated by means of the Kirkwood-MuIler formula, 2 CKM = ( 6 m e C a | a 2 ) ^ a | / x | + a2 /x2 ^' (5.10) where the a's and the x's are polarizabiIites and dsamagnetic susceptibilities, 52 respectively, of the molecules, mg the electronic mass, and c the velocity of light. In addition to following the above procedure, two other equations are also employed here for calculating C, The f i r s t is due to Slater and v L ,/4,45 Kirkwood ' , C S K = (3etf n m ^ 2 ) ^ Cl/{(a,/n V / 2 + (^/n,,)172} ], (5.11) 44 45 and the second due to London ' , C L = ( 3 a|°2 | ( l 2 ) / 2( |'+l2) . (5.]:2) In (5.11) and (5.12), e is the electronic charge, the I's the ionization potentials and the n's the number of electrons in the outer shell of the molecules. Comparing one of the C values with the experimental values of e ) s then gives z 0. For the (9,3) and (12,3) models, equation (5.09) and the generalized Lennard-Jones potential, m/(n-m) 6(r) = (n/(m~n)) (n/m) e [ ( r 0 / r ) n - ( r c / r ) m ], (5.13) must be consistent at large separations. One therefore identifies C = (n/(m-n))(n/m) r n / ( n- m ) e r 0 n (5.14) Then integrating (5.09) over a semi-infinite solid yields a/ (B-a) -I C B (B-a) (B/a 6z Q Ts il°£ 3 C B (B-a) (B/a) ] , (5.15) where, again, a-n-3 and B= m-3 for the (9,3) potential, a=n and B-m-3 for the (12,3) potential and N0 is the number of atoms per cm?, in the 5 3 solid. For the (10,4) model, integration of (5.09) over a single infinite plane results in the expression = -N0 C/2z Q 4 (5.16) e Is and so for this potential, e Is [ B(S-a) (B/a) a/(8-a) -,-1 (5,17) 2 v/here a= n-2, S- m-2, and NQ is the number of atoms per cm , of the surface. It is seen that the quantum st a t i s t i c a l mass corrections have the effect of reversing the order of interaction energies with the hydrogen isotopes. That i s , when the mass effect on the vibrational levels normal to the surface are taken into account, one finds that the lighter isotope exhibits a 46 slightly stronger interaction with the surface. Olivier and Ross have made a harmonic osc i l l a t o r zero-point energy approximation for the mass corrections, using the data employed here, and find the same order of isotopic inter action. However, these authors failed to consider the cause of this difference in interaction energy for isotopic pairs, and thus did not explain the reversal in the amount of gas adsorbed between the hydrogen and methane pairs. As we have seen, this reversal is brought about by the quantum mechanlea I.effect of isotopic substitution on the dispersion forces, which serves to increase the attractive potential of the hydrogen sub stituted species. This effect is large enough in the case of the methanes, to cancel out the small mass effect, and thus CH. is adsorbed more readily The parameters of best f i t are presented in Tables VI and VII. 53a Table VI Comparison of classical and quantum f i t results using the KIrkwood-Muller Formula, Note: 6 g =. [e(H)-e(D)]/e(H), Gas E I s / k ( ° K ) A(m2g"') . z 0(A) Std .dev.(109) « s(10 2) Cl 0 Cl 0 Cl 0 Cl 0 5.72 (9,3) potentia1 «2 578 647. 12.5 9.53 2.081 2.004 .705 .197 V 581 610 12.5 II .2 2.076. 2.043 .521 .314 CH. 4 1450 1462 10.9 10.7 2.224 2.220 .186 .181 CD4 1435 1444 l l . l 1 I .0 2.231 2.228 .876 .854 ( 10,4) potential H2 568 616 12.4 10.0 2.974 2.914 1 .07 .509 D2 571 593 12.4 1 1.3 2.969 2.941 .716 .517 CH4 1435 1441 10.5 10.4 3. 120 3.117 .373 .363 CD4 1418 1423 10.9 10.8 3. 129 3. 126 1.91 1 ,86 (12,3) potentia1 H2 579 679 13.5 9.07 2.226 2.1 12 .931 .199 D2 582 619 13.6 1 1 .8 2.222 2. 177 .688 .341 CH4 1454 1462 1 1 .7 1 1 .6 2,379 2.375 .248 .240 CD4 1438 1445 12.0 1 1 .8 2.387 2.383 1 .17 1.14 1.13 3.73 .25 8.72 I . 16 53b Table VI I Quantum f i t results using the SIater-Kirkwood CSK) and London (L) formulas, Gas /k(°K) A(m2g z 0(A) Std, dev.CIO9 SK I SK L (9,3) SK potentia1 "l SK L H2 -721 - 8 - 3 1 ,613 .-305 D2 626 12.7 1.685 .517 CH4 1459 1462 12.4 14,5 1.910 1 .621 .179 .178 CD4 1442 1444 12.8 15.0 (10,4) 1 .910 potentia 1 ,619 1 .849 .844 H2 638 10.6 2,522 .354 D2 601 12.5 2.553 .348 C H4 1442 1444 1 1.7 13.1 2.785 2.463 ,362 .360 C D4 1424 1425 1.2.1 13.6 (12,3) 2.785 potentia 2.461 1 1 ,85 1 .84 H2 °2 CH4 1464 1468 13.4 15.6 2.042 1 ,733 .238 .235 CD4 1447 1450 13.8 16.2 2.042 1 .731 1.13 1.12 SK 13.2 1,17 1.23 5.73 1.25 1.32 1.16 I .23 54 than CD^. Clearly, for two isotopes which are identical in every respect 3 4 except their masses (eg. He and He ), the dispersion forces would be the same and the differences in adsorption properties could be explained solely by the quantum sta t i s t i c a l mass effect. Thus, He4 would be more readily adsorbed than.He3, owing to its lower lying position in the potential well. As seen in Tables VI and VII, the mass corrections cause quite sizeable differences in the interaction energies and areas obtained for the hydrogen pair, but the differences are, of course, small in the case of the methanes. It is interesting to note that virtually a l l the change in I 4 e|s/l< is due to the f i r s t quantum correction, , When the hydrogen l I x data are fitted to the (9,3) model' omitting the B term-, the inter- o action energy using the Kirkwood-MuI Ier formuI a is found to be 649 K. This is due to the fact that Ag is small (TO.6), so that the higher terms contribute l i t t l e . It should also be mentioned' that the perfect term, B^g , has no effect on the f i t and therefore can be neglected. The very large increase in the interaction energies when quantum corrections are included for the hydrogens is responsible for the impossibility of f i t t i n g the SIater-Kirkwood (12,3) model and a l l the London * models for these gases, using the present computer programme. For T < 0.125, machine overflows prevent the calculation of Q^. While this could be overcome by additional programming, i t was f e l t that on the basis of the apparent areas found t a t ' these low T , the SI.ater-Ki rkwood and London formulas would not yield particularly meaningful results in any case. 55 The area estimates obtained using the Kirkwood-MuI Ier formula are in quite close agreement with those from the two-dimensional analysis, and lends further support to the use of this equation in the analysis of Henry's law adsorption data. Of the three potential models examined, the (9,3) function seems to be the most successful as judged from goodness of f i t , although the differences between the models are not as marked as had been hoped. The standard deviations for the (9,3) and (L2,3) models are f a i r l y similar, while that for the (10,4) is much worse. It is of interest to note that the quantum corrected f i t s for a l l the models are better than the classical ones. The differences are very small in the methanes, of course, since the quantum deviations are small. Methane f i t s a l l the models approximately five times better than tetradeutero methane. For the hydrogens, the improvement in f i t are quite apparent. The (10,4) model seems to f i t the quantum corrected H^  and equally well, whereas in a l l other cases, H 2 f i t s slightly better than D2, The percentage energy differences between isotopes in Tables VI and VII are not in the same order as the polarizabi1ity differences (1.32$ and 1,45/5 respectively., for the hydrogens and methanes'7). Moreover, the differences are much larger than one would expect on the basis of the gas phase values. If we assume the combininq rule.e, ^(e e ) and use the a ' is g s measured 1 9' 3 7 bulk gas A values, i t follows that [e(H) - e(D)] / e(H) is 2.0? for the hydrogens and 0.41% for the methanes. Although this combining rule- may not be especially accurate, i t does not seem plausible that the difference 56 should he larger for the gas-surface case than for the bulk gas, It is f e l t this nay indicate that the Wigner-Kirkwood expansion when applied to gas- surface interactions, and/or the model adopted for the solid are inadequate, 47 Also if there is.hindered rotation on the surface, as has been suggested , the use of a spherical potential may-not be j u s t i f i e d , We intend to examine these po s s i b i l i t i e s in the near future. 57 APPENDIX I (2) ^ £ (2) ^  (2) In this Appendix tables of B C | ( B | and B are given at * selected values of T and n for the (12,6 3) potential function. Also (2)^f included is a table of E p 6 r ^ values, which are of course independent of the potential, Note: 0,032I62 = 0.0002162, 58 Table VIII '•* T R ( 2 ) * B Cl -n. = 0 B 1 - B II 0.30 -f'4.127 15.412 21.793 0.40 - 6.4884 4.3959 : 3.8634 0.50 - 3.8113 1 .9353 1.1866 0.60 - 2.5112 1.0726 0,49269 0.70 - 1 .7578 0.68303. 0.24667 0,80 - 1.2708 0,47640 0.14008 0,90 - 0,93193 ' 0.35403 0.087025 1.00 - 0.68339 0.27555 0.057806 1,20 - 0.34469 0.18393 0,029452 1.40 - 0.12581 0.13407 0.017156 1.60 0.026518 0.10363 0.010955 1.80 0,13819 0,083493 0.0274772 2.00 0.22325 0,069367 • 0.0253665 . 2.50 0.36656 0.047865 0,022734l 3.00 0,45461 0,036009 0.02|6I60 3.50 0.51333 0.028620 0.02|0522 4.00 0.55473 0,023622 0.0373320 5,00 0.60801 0.017356 0.0340870 6.00 0,63965 0,013627 0.0325760 8.00 0.67288 0,02944I9 0,03|2752 10.00 0.68778 0.027|769 0.0475287 15.00 0.69734 0,0244389 0.0429904 25.00 0.68771 0.0224835 0.0598258 50.00 ' . ' • 0,65272 • C.02|1633 0.0523I8I 100,00 0.60557 - 0,0355705 0,0657686 59 Table V I I I-B T » B ( 2 ) * Cl -n = 0.0| B ( 2 ) * ( 2 ) ^ " B II 0.30 -12.766 13,906 19,406 0.40 - 5.8999 4'.0630 3.5273^ 0,50 - 3.4574 1.8153 1.1001 0.60 - 2.2622 ' 1.0163 0.46160 0.70 - 1,5666 0.65185 0.23288 0.80 - 1,1157 0,45716 0,13302 0,90 , - 0.80150 0,34121 0.083022 1 .00 - 0.57083 0.26650 0.055354 1,20 0.25623 0", 17884 0.028366 1.40 - 0,052856 0,13086 0,016594 1.60 0.088656 0.10144 0.010631 1 ,80 0,19235 0.0SI9I0 0,0272749 2,00 0,27128 0,068175 0.0252325 2.50 0,40407 01.047I96 0,0226764 3,00 0,48544 0', 035584 0.0215863 3.50 0,53954 0,028326 0.02I0350 4,00 0,57755 0;023407 0.0372235 5,00 0^62618 0.017227 0,0340359 6,00 0,65477 0.013541 0,0325479 8.00 0^68425 0,0293956 ' 0,03I2640 10.00 0,69692 0.027I479 0,0474727 15,00 0,70351 0,0244262 0.0429739 25,00 .0,69150 • 0,0224789 0.0597889 .50.00 0.65469 • 0,02|1621 0,0523I29 100 ',00 0,60661 0;035567l 0.0657609 60 'Table VI I I-C .-V = o ;o3 T rJ-2) * 8 Cl o.(2)* . 8 1 6.30 -10.327 l'r.328 15.392 0.40 - 4.8058 3.4727 . 2.9408 0,50 -. 2.7862 1.5979 0.94572 0,60 - 1,7842 0,91259 0.40519 0^70 - 1, 1966 0.59390 0.20756 0.80 - 0,81409 0.421 13 0.11995 0.90 - 0,54686 0.31704 0.075558 1.00 - 0,35041 0.24936 0.050759 1,20 - 0,082297 0,16911 0.026313 1 .40 0,090981 0,12468 '•O-.O 15524 1 .60 0,21141 0.097203 0.010011 t .80 0,29950 0,078845 0.0268866 2.00 0.36641 0.065860 0.0249745 2,50 0,47852 , 0,045891 0.0225647 3.00 0,54672 0.034751 0.02I5285 3.50 0,59168 0.027750 0.02|OOI4 4,00 0,62297 0,022985 0.0370l14 5.00 0,66237 0.016973 0.0339355 6,00 0;68491 0,013371 0.0324926 8.00 0.70692 0;0293039 0.03|24I9 10,00 0,71515 0,0270904 0.0473620 15,00 0,71583 0,02440I0 0.0429414 25,00 . 0,69907 :0,0224697 . •. 0.0597I58 .50,00 ,'• 0,65863 • .0,02|1596 .. 0.0.523027 100,00' 0,60867 .. 0.0355604 0.0657457 61 # t (2)*' B Cl Tab 1e V11T-D -n = 0.05 R ( 2 ) * ~ B 1 1 0.30 -'8.2142 9.2358 12.213 0,40 -3.8109 2.9702 2.4522 0.50 -2.1598 1.4074 0.81305 0.60 -1.3311 • 0.81995 0.35569 0,70 -0,84238 0.54137 0.18499 0.80 -0.52334 0,38811 0.10816 0,90. . -0.30016 0.29471 0,068763 1 ,00 -0,13605 0.23340 0.046542 1.20 0,08778 0.15996 0.024408 1 .40 0,23214 0,11883 0.014523 1 ,60 0.33219 0,093170 0.029470 1 ,80 0,40513 • 0,075910 0.0265I88 2.00 0,46034 0.063636 0.0247289 2.50 0,55222 0.044628 0.0224576 3,00 0.60748 0.033942 0.02|4728 3,50 0,64344 0.027188 0.039689l 4.00. 0,66809 0,022573 0,0368053 5^00 0,69838 0.016724 0.0338376 6,00' 0,71492 0.013203 0.0324385 8,00 0,72952 0.0292134 0.03|220l 10,00 0,73332 0,0270335 0.0472529 15,00 0,72812 0,0243760 0.0429092 25,00 0,70663 0.0224605 0.0596433 50,00 0,66257 ^ 0,02|1572 0.0522924 100,00 0.61073 0.0355536 0.0657305 62 (Table VI I l-E Perfect Gas Contribution &<2! * perf .. f § <2> * perf 0,30 0.084435 2.50 0.010132 0,40 0,063326 3.00 0,0284435 0,50 0,050661 3.50 0.0272372 0,60 0,042217 4,00 0,0263326 0,70 0,036186 ' 5.00 0,025066l 0,80 0.031663 6.00 0.02422I7 0,90 0.028145 8.00 0,023I663 1 ,00 0', 025330 10.00 0.0225330 1.20 0,021109 15.00 0,0216887 1,40 0.018093 25.00 0.02|0I32 1 ,60 0,015832 50.00 0,035066l 1 ,80 0,014072 100.00 0.0325330 2,00 0.012665 63 APPENDIX II Table IX I I* Values of B . for the (9,3),(|0,4) and (12,3) models, T (9,3) (10,4) (12.3) .10 1432.5 2716.0 2458,2 .15 13.870 26.094 23.952 .20 1.0625 1.9750 1.8470 .25 .19721 .36262 0.34476 .30 .058547 .10661 • .10287 .35 .023077 .041650 .040726 .40 .010959 .019619 .019417 .45 ,0259274 . .010532 .010540 .50 ,0235262 .0262224 ,0262908 .55 ,0222546 .0239529 ,0240343 .60 ,0215250 .0226576 ,022736l .65 ,02|0788 ,0218694 ,02|9405 .70 ,0379I56 .0213644 ,02I427I .75 ,035986l ,0210266 ,02|08I5 ,80 .034643l .0379245 ,0384056 .90 .0329705 ,0350250 ,0353969 1 .00 ,0320268 ,0 3340l| ,0336940 64 BIBLIOGRAPHY. 1, W.A.Steele and G.D.Halsey, J.Chem,Phys., 22,979 (1954). 2. 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