"Science, Faculty of"@en . "Chemistry, Department of"@en . "DSpace"@en . "UBCV"@en . "Macrury, Thomas Bernard"@en . "2011-08-06T17:27:45Z"@en . "1967"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "The adsorption isotherm and the equation of state for the two-dimensional gas are derived from the grand canonical ensemble. Then the quantum statistical equation'of state is developed and applied to the two-dimensional second virial coefficient, B\u00E2\u0081\u00BD\u00C2\u00B2\u00E2\u0081\u00BE, and the second gas-surface virial coefficient, B[subscript]AS, We compare theoretically the (12,6) and (12,6,3) potential models for B\u00E2\u0081\u00BD\u00C2\u00B2\u00E2\u0081\u00BE. Finally the adsorption data for CH\u00E2\u0082\u0084, CD\u00E2\u0082\u0084, H\u00E2\u0082\u0082 and D\u00E2\u0082\u0082 on graphite are analysed quantally for the two-dimensional second virial coefficient and the second gas-surface virial coefficient."@en . "https://circle.library.ubc.ca/rest/handle/2429/36546?expand=metadata"@en . "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 <^-^ 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 \u00C2\u00B0 \> | ' ' V j ^ i j ^ i j r . . = r . - r . , <\,l 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 \u00E2\u0080\u00A2 < 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 , \u00C2\u00A3j , 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 \u00E2\u0080\u00A2 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 \u00C2\u00A3.. 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\u00C2\u00B0 ) Na , (1.06) NakT 7 -1 ' 2 B | 2 B, where 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 \u00C2\u00B0 ) / 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> / ( 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 \u00E2\u0080\u00A2 * -6 * -3 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\u00C2\u00B0(TiV,/J) N >0 \u00C2\u00A3 M / \u00E2\u0080\u00A2 ( 2 . 0 6 ) where and Z \u00C2\u00B0 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 \u00C2\u00B0 N exp [- 3 I cb(T., ) ] . d ^ N , !!\u00E2\u0080\u00A2A)/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 \u00E2\u0080\u00A2 (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 \u00C2\u00A3 l ^ tef \u00E2\u0080\u009Ef + 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. ) , 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 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 \u00E2\u0080\u00A2 ^ 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 + - < 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 ^ \u00E2\u0080\u009E (2) . (2).3 ^ \u00E2\u0080\u009E 0, Ma = B j /y + ^ (/^ ) + Bj (/J, \u00E2\u0080\u00A2 ) +..., (2,29) where (2) p l B2 %(2) = I (2) P0 f l 2 d ^ l d ^ \u00C2\u00BB (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 \u00E2\u0080\u00A2 - * \u00E2\u0080\u0094 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 \" ' \u00E2\u0080\u00A2 - ( 2 ) 'Na\ + ( 3 B 0 ( 2 ) -.-2B,(2).> ( R a j 2 + (2.36) '2 I \ 2 . ^3 or \u00E2\u0080\u0094\u00C2\u00A3A = I + B ( 2 ) (Na) \u00E2\u0080\u00A2+ 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) + 3 ( 2 ( B k / V /A -\u00E2\u0080\u00A2 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 -\u00C2\u00B1 exp [- B(fr s ( r ) ] d;rdz = Z, ( s : (2.41) \u00E2\u0080\u00A2'A AS \u00E2\u0080\u00A2(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 . \u00E2\u0080\u00A2 -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(\u00C2\u00A3 , \u00C2\u00A3 ) of the canonical ensemble as PN ( \u00C2\u00A3 N ' \u00C2\u00A3 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 ,\u00E2\u0080\u009E.. , , , . , , , N, i i of a l l the molecules of the system. Carrying out the integrations over p \u00E2\u0080\u00A2 NI p.\" /2m, and a potential energy 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 , \u00C2\u00A3 ^ \"t\"\u00C2\u00B0 ^ 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, > \u00C2\u00A3 ^' ^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\u00C2\u00BB ( r 5 \u00E2\u0080\u00A2 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 \u00C2\u00A3) -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 \u00C2\u00A3 -(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 (\u00C2\u00A3 N) d\u00C2\u00A3 N = Ni Tr exp (-3^/kT), (3.15) whence NlQ N k, (3.16) From (3.10), ( 7 ( A exp \u00E2\u0080\u00A2(-\"H-./kT) >^ ( r N ) ) d r N \ -\ N N U N (r/) dr n , (3.17) QN = --I NI corresponding to (3.07). \u00E2\u0080\u00A2 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 \u00E2\u0080\u0094 * \" ! / ' ' M\u00C2\u00BB This can be proven by using the invariance property of *P ^ for the system of orthonorma.l eigenf unctions N used. Since the ) form a complete orthonormal set, 24 Y $ (r!) Cr, > = n 6(r' - r ), (3.18) v \ (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 \u00C2\u00A3 . 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 / ,\u00E2\u0080\u00A2 , , P r r. ,2 \" 0 J N 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 \u00C2\u00B0 \u00E2\u0080\u00943N 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 ) 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\u00C2\u00A3 )-. (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) \u00E2\u0080\u00A2+ 8 V -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 \u00C2\u00AB > * -CB'fV* B \u00C2\u00AB > * 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 \u00E2\u0080\u009E 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 \u00E2\u0080\u00A2 z - ( a J + b k + c | + \u00C2\u00AB \u00E2\u0080\u00A2 -- 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)\u00E2\u0080\u00A2! + . . . >- 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 - = 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 ( 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 \u00E2\u0080\u0094 ^ 240 m~ v exp (- 3* s){V% s - |[2V 2(V4) s) 2 geo (3.53) exp (- 6 ) [v2cj> -(\u00C2\u00A3) ( V(j) ) 2] dV, (3.54) V ^ s s 2 ^ s ' geo \u00E2\u0080\u00A2M-8 V V V\ +5 ( V 2 6 s ) 2 ] + \u00C2\u00B1 15,% ( V , s ) 2 o + 3VA - V(Vd> ) 2 ] - ~ (VA ) 4 }dV; 24 where B~ l/kT and m is the molecular mass. 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) \u00C2\u00ABj>(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, s (2) = > ( 3 / a ) a / ( B \" a ) e , s C ( z 0 / 2 ) A - ( z 0 / z ) 5 ] . (3.5?) Now, \u00C2\u00A3 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\u00C2\u00B1\u00C2\u00B1 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 + \u00E2\u0080\u00A2 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], . \u00E2\u0080\u00A2 _ (3.63) and V = (\u00E2\u0080\u00A2 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 / 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 are small compared to , 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) \u00C2\u00A5 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) \u00E2\u0080\u00A2 \" 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) * \u00E2\u0080\u00A2 * ' 1/2 T 2 = kT/e2, A2 \u00E2\u0080\u00A2=: 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/\u00C2\u00A3 2,\u00C2\u00B0 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) ( 82 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 \u00E2\u0080\u00A2Table \\ 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) ; 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\u00E2\u0080\u009E)/ 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 \u00E2\u0080\u00A2Cl(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 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\u00C2\u00B0) HYDROGEN DEUTERIUM T \u00E2\u0080\u00A2 BAS.. \u00E2\u0080\u00A2B( 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 \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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 \u00C2\u00B0 2-1 * Molecule e 2/k(\u00C2\u00B0k) \u00C2\u00B0 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 \u00C2\u00B0 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\u00E2\u0080\u009Exl0 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\u00E2\u0080\u009E - .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 \" \" \u00C2\u00B0 v , l ,\"ls'' T = kT /\u00C2\u00A3,\u00E2\u0080\u009E , A\u00E2\u0080\u009E = 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|\u00C2\u00B02 | ( 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\u00C2\u00B0\u00C2\u00A3 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 ( \u00C2\u00B0 K ) A(m2g\"') . z 0(A) Std .dev.(109) \u00C2\u00AB s(10 2) Cl 0 Cl 0 Cl 0 Cl 0 5.72 (9,3) potentia1 \u00C2\u00AB2 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(\u00C2\u00B0K) 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 \u00C2\u00B02 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) ^ \u00C2\u00A3 (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 '\u00E2\u0080\u00A2* 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 \u00E2\u0080\u00A2 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 ' . ' \u00E2\u0080\u00A2 0,65272 \u00E2\u0080\u00A2 C.02|1633 0.0523I8I 100,00 0.60557 - 0,0355705 0,0657686 59 Table V I I I-B T \u00C2\u00BB 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 \u00E2\u0080\u00A2 0,0224789 0.0597889 .50.00 0.65469 \u00E2\u0080\u00A2 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 '\u00E2\u0080\u00A2O-.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 . \u00E2\u0080\u00A2. 0.0597I58 .50,00 ,'\u00E2\u0080\u00A2 0,65863 \u00E2\u0080\u00A2 .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 \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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 \u00C2\u00A7 <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 \u00E2\u0080\u00A2 .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. T.B.MacRury, B.Sc. Thesis, University of British Columbia, Vancouver (1965), 3. J.E.Mayer and M.G.Mayer, \"Statistical Mechanics\", John Wiley and Sons Inc., N.Y., (1940), Chapter 13. 4. M.P,Freeman and G.D.Halsey, J.Phys.Chem.,59,181 (1955). 5. M.P,Freeman, J.Phys.Chem., 62, 729 (1958). 6, 0.Sinanoglu and K.S.Pitzer, J.Chem.Phys,, 32, 1279 (I960), 7. J . R . Sams, G.;. Constabaris, and G.D. Halsey, J .Chem.Phys., 36, 1334 (1962), 8. W.A. Steele, to be.published. 9. - J,R.Sams, G.Constabaris, and G,D,HaIsey, J,Phys.Chem., 64,1689 (I960). 10. R. Yaris, thesis, University of Washington, Seattle, (1962), 11. A.D.McLachlan, Mo I.Phys., 7, 387 (1964). 12, R. Wolfe and J.R. Sams, J.Chem,Phys., _44, 2181 (1966). 13, W.C.DeMarcus, E.H.Hopper, and A.M.Allen, U.S. Atomic Energy Commission, KI222 (1955). 14, R.S. Hansen, J.Chem.Phys,, 6^ , 743 (1959), 15, M.P,Freeman, J.Phys.Chem., 64, 32 (I960), 16, G,Constabaris,J.R.Sams, and G.D.Halsey, J,Phys,Chem, 65, 367 (1961). 17. R,P.Bel I, Trans.Faraday Soc,, j38, 422 (1942), 18, H.F.P.Knaap and J,J.M.Beenakker, Physica,Z7, 523 (1961). 19, A.Michels, W, de Graaff, and C.A, Ten Seldam, Physica,_26, 393 (I960), 20, R. Yaris and J.R. Sams, J. Chem,Phys.,' _37, 571 (1962). 65. 21. G. UhIenbeck and E.Beth, Physica, J 5 , 729 (1936). 22. J. De Boer and A. Michels, Physica^ 5_, 945 (1933). 23. J.O. Hirschfelder, C.F. Curtiss* and R.B.Bird, \"Molecular Theory of Gases and Liquids\", John Wiley and Sons, N.Y,,' C1959), Chapter 6. 24. J.R. Sams, Mol. Phys. ,_9, 17 (1965). 25. J.Kirkwood, Phys. Rev.t_44, 31 (1933). 26. J.D.Johnstone and M.L.Klein, Trans.Faraday Soc., 60, 1964 (1964). 27. J.R.Sams, J.Chem,Phys,,_43, 2243 (1965). 28. T.L.Hill, \"Statistical Mechanics\", McGraw-Hill Inc., M.Y,, (1959) Chapter 5 and Appendix 10. 29. J.A. Barker and D.H.Everett, Trans.Faraday Soc, 58, 1608 (1962). 30. W.A. Steele and M.Ross, J .Chem.Phys.', 35, 850 (1961). 31. J. De Boer, Rep.Prog,Phys,, YZ, 305 (1949). 32. D, ter Haar, \"Elements of Statistical Mechanics\", Holt, Rinehart, and Winston, N.Y., (1954), Chapter 8. 33. J.C.Slater, Phys. Rev., 38, 237 (1931). 34. J.C.Slater, J .Chem. Phys., j_, 687 (1933). 35. J.R.Sams, to be published, 36. J.R.Sams, Mol.Phys., 9, 77 (1965), 37. A. Michels and G.W.Nederbragt, Physica,_2, 1000 (1935). 38. G. Thomaes and R.van Steenwinkel, Mol.Phys., 5, 307 (1962). 39. A. Michels and M. Goudeket, Physica,_8_, 347 U94I). 40. A. Michels and M. Goudeket, Physica_8, 353 (1941). 41* G. Constabaris, J.R. Sams, and G.' Ha I sey, J . Chem. Phys, 37, 915 (1962), 42, J ,G,KI rkwood j Z.Physi k, 3_3, 57 (1932). 66 43. A . M u l l e r , Proc.Roy.Soc. (London) . AI 54, 624 ( 1 9 3 6 ) . 44. K.S. P i t z e r , Advances i n C h e m i c a l P h y s i c s V o l . I I f I n t e r s c i e n c e P u b l i s h e r s , I n c . , New York ( 1 9 5 9 ) , Page 59. 45. H. Margenau, Rev, Modern Phy s , , _ M , I ( 1 9 3 9 ) , 46. J,P, 0 1 i v i e r and S, Ross, P r o c , Roy, Soc, ( L o n d o n ) , A265, 4 7 7 ( 1 9 6 2 ) , 47. A.L.Myers and J . M . P r a u s n i t z , T r a n s , Faraday Soc., 61, 755 ( 1 9 6 5 ) . "@en . "Thesis/Dissertation"@en . "10.14288/1.0062277"@en . "eng"@en . "Chemistry"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Quantum effects in dilute adsorption systems"@en . "Text"@en . "http://hdl.handle.net/2429/36546"@en .