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UBC Theses and Dissertations

Study of equilibrium in exchange between HC1 gas and a KBr surface Koga, Yoshikata 1969

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A STUDY OF EQUILIBRIUM IN.EXCHANGE BETWEEN HC1 GAS AND A KBr SURFACE by YOSHIKATA KOGA B.Eng., The U n i v e r s i t y o f Tokyo, i960 M.Eng., The U n i v e r s i t y o f Tokyo, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS.FOR THE DEGREE OP DOCTOR OF PHILOSOPHY i n the Department of Chemistry We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1969 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 degree a t the U n i v e r s i t y o f B r i t i s h C o lumbia, I a g r e e 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 . I f u r t h e r agree t h a 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 rposes may be g r a n t e d by the Head o f my Department 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 o r 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 not 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 . . Department o f Ch^M /s tru*^y The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date Od. , f1*1 i i A b s t r a c t In the main p a r t o f t h i s work, an e q u i l i b r i u m i n anion exchange between HC1 gas and vacuum-sublimed KBr powder has been s t u d i e d from 0 ° to 87°C. For t o t a l amounts o f exchange co r r e s p o n d i n g to l e s s than one s u r f a c e l a y e r , the e q u i l i b r i u m isotherms c o u l d be r e p r e s e n t e d by a very simple e m p i r i c a l e q u a t i o n of the form, 1/RpX = l / R p X m + 1/K'Xm', ( l ) where X = amount exchanged n o r m a l i z e d to BET s u r f a c e , Xm and Xm' are paramaters o f exchange c a p a c i t y with the same dimensions as X, K' = a d i m e n s i o n l e s s parameter r e l a t e d i n some way to an e q u i l i b r i u m constant f o r exchange, and Rp = P^Br^^Hd ^ n & a s phase at e q u i l i b r i u m . I f Xm were independent of temperature and equal to u n i t y or some whole number o f l a y e r s , eq. ( l ) with Xm = Xm1 would correspond to an i d e a l chemical e q u i l i b r i u m e x p r e s s i o n , K = R p [x/(Xm - X)] . — -(2). In p r a c t i c e Xm was found to be temperature-dependent (Xmc<exp(-4,600/RT)) and ranged 0 .5 a t 9.4°C to 3 at 87°C ( r a t h e r s c a t t e r e d e x p e r i m e ntal p o i n t s render the 0°C value u n c e r t a i n ) . The q u a n t i t y d e s i g n a t e d (K'Xm») i n eq. ( l ) i s o b t a i n e d e x p e r i m e n t a l l y as a s i n g l e constant, and i t s s e p a r a t i o n Into an l i l e q u i l i b r i u m c o n s t a n t a n d a n e x c h a n g e c a p a c i t y p a r a m e t e r s d e p e n d s on t h e i n t e r p r e t a t i o n o f t h e r e s u l t s . (K' Xm') h a s a p p r o x i m a t e l y t h e same t e m p e r a t u r e d e p e n d e n c e a s Xm , b u t i t i s n o t c l e a r i n t h e f i r s t i n s t a n c e w h e t h e r t h i s i s t o be a s c r i b e d t o K' o r t o Xm'. V a r i o u s m o d e l s a r e e x a m i n e d w h i c h may g i v e r i s e t o t h e a b o v e e x p e r i m e n t a l r e s u l t s . Among them; ( a ) one i n w h i c h c o - o p e r a t i v e i n t e r a c t i o n s o f f i r s t a n d s e c o n d a n i o n n e i g h b o u r s i n a s i n g l e l a y e r a r e t r e a t e d by a m o d i f i e d f o r m o f t h e q u a s i - c h e m i c a l a p p r o x i m a t i o n seems t o be a b l e t o e x p l a i n i s o t h e r m s f o r 0° a n d 9.h°C, ( b ) a n o t h e r i n w h i c h t h r e e l a y e r s a r e t a k e n i n t o a c c o u n t i n a s i m p l e manner i s c a p a b l e o f e x p l a i n i n g i s o t h e r m s f o r 5 0 0 t o 8 7 ° C . T h i s l a t t e r i s c a p a b l e o f e x p l a i n i n g r e s u l t s a t a l l t e m p e r a t u r e s i f t h e c o r r e l a t i o n o f BET s u r f a c e w i t h e x c h a n g e c a p a c i t y o f one l a y e r i s a s s u m e d t o be i n e r r o r by a f a c t o r o f — 2 a n d t h e e x p e r i m e n t a l i s o t h e r m f o r 0 ? C i s d i s c a r d e d on t h e b a s i s t h a t p l o t s a c c o r d i n g t o e q . ( l ) a r e s c a t t e r e d . A l t e r n a -t i v e l y , some c o m b i n a t i o n o f t h e m o d e l s ' u s e d i n ( a ) a n d ( b ) w o u l d p r o b a b l y a l s o a c c o u n t f o r t h e c o m p l e t e s e t o f r e s u l t s . I n p a r t I I o f t h i s w o r k , t h e s y s t e m i n w h i c h t h e r e a c t i o n 2IBr = I 2 + B r 2 a n d g a s c h r o m a t o g r a p h i c s e p a r a t i o n o f I B r , I g a n d B r 2 o c c u r s i m u l t a n e o u s l y on t h e c o l u m n h a s b e e n s t u d i e d - e x p e r i m e n t a l l y . ' V a r i o u s m e t h o d s o f m a t c h i n g t h e e x p e r i m e n t a l d a t a b y c o m p u t a t i o n a r e d i s c u s s e d , a n d a c o m p u t a t i o n a l m e t h o d b a s e d on a " p l a t e t h e o r y " m o d e l i s f o u n d s a t i s f a c t o r y . i v Table o f Contents Page T i t l e Page . . I A b s t r a c t . i i Table of Contents i v L i s t o f T a b l e s v i L i s t o f F i g u r e s v i i Acknowledgements i x A Study o f E q u i l i b r i u m i n Exchange between HC1 Gas and a KBr Sur f a c e Chap. 1 I n t r o d u c t i o n 1 1 E q u i l i b r i u m s t r u c t u r e o f s u r f a c e s 3 2 The nature of s u r f a c e s as r e v e a l e d by k i n e t i c s t u d i e s . . . - 6 Chap. 2 E x p e r i m e n t a l Study of E q u i l i b r i u m 8 1 E x p e r i m e n t a l 9 2 R e s u l t s . 23 Chap. 3 D i s c u s s i o n . ... 53 1 I n t r o d u c t i o n , 5^ 2 D e t a i l e d d i s c u s s i o n based on ( i ) 57 3 D e t a i l e d d i s c u s s i o n based on ( l l ) 60 Appendices I Surface Area of KBr and Chemical S i n t e r i n g 71 I I Exchange K i n e t i c s or D i f f u s i o n of Anions .. 79 I I I Quasi-Chemical Treatment 85 V Table of Contents (Cont'd) Page Appendices IV Revised Quasi-Chemical Treatment 92 V Quasi-Harmonic Theory of V i b r a t i o n s 109 V I ' Three Layer Theory 113 B i b l i o g r a p h y 123 P a r t I I (Supplement ) A Study of the R e a c t i o n 2IBr = I 2 + B r 2 O c c u r r i n g i n a Gas-Chromatographic Column I n t r o d u c t i o n S - l Chap. 1 Review o f P r e v i o u s T h e o r e t i c a l Work S - 4 1 I n t r o d u c t i o n S - 5 2 Rate theory S-9 3 P l a t e theory S -12 4 R e a c t i o n on chromatographic column S - l 8 Chap. 2 P l a t e Theory Approach to R e a c t i o n on Chromatographic Columns S - 2 1 1 Theory S -23 2 Computation and r e s u l t s S -27 3 D i s c u s s i o n S-46 Chap. 3 The IBr - I G - B r 2 System S -49 1 E x p e r i m e n t a l S -51 2 R e s u l t s and d i s c u s s i o n S -57 B i b l i o g r a p h y S - 7 0 Appendix A S -73 v i List of Tables Table Page 1 . . . 23 2A . . 27 2B . . 28 2C 29 3 34 4 36 5 48 6 51 .1-1 77 IV-1 94 IV-2 98 IV-3 99 Vl - 1 , . 116 Part II 1 S-5 2 S-57 3 S-57 4 S-58 5 S-58 v i i F i g u r e 1 2 3 4 5 6 7 8A 8 B . . 8 C . 9 I O A . 1 0 B . I O C . 1 1 1 2 13 "•• . 14 15 . 16 . 17 18 . 19 . L i s t o f F i g u r e s Page F i g u r e Page 1 4 74 15 1-2 75 16 1-3 76 x 7 . 1-4 78 18 I I - l 82 19 I I - 2 83 20 I I - 3 8 4 30 IV - 1 101 31 IV - 2 102 5 2 IV - 3 103 33 IV - 4 1 0 4 38 IV-5 105 39 V I - 1 117 4 0 V I - 2 118 4 1 V I - 3 119 43 V I - 4 120 44 P a r t I I 45 1 S - 6 49 2 S - 1 2 52 3 S - 1 4 61 4 S - 1 6 65 5 S - 2 3 69 6 S - 2 8 v l i i L i s t of Fig u r e s (Cont'd). Figure 7 8 9 10 i l 12 13 14 15 16 Page Figure Page S -20 17 S-35 18 « » . . . S -45 S -56 19 S-37 20 S -38 21 . . . . . . • ... S-63 S -39 22 S -4o 23 S-41 24 S-42 25 S-43 26 • • . . 27 i x Acknowledgements I wish to express my s i n c e r e thanks to P r o f e s s o r L.G.Harrison f o r h i s constant help and encouragement d u r i n g the course o f my r e s e a r c h . Thanks are due to Mr. P.Madderom f o r h i s h e l p i n w r i t i n g the computer program, and to the t e c h n i c i a n s i n the g l a s s - b l o w e r s shop, the e l e c t r o n i c s shop and the mechanical shop i n the department of Chemistry f o r t h e i r e x c e l l e n t jobs i n c o n s t r u c t i n g the apparatus. F i n a l l y I wish to thank the U n i v e r s i t y of B r i t i s h Columbia and the K i l l a m f o u n d a t i o n f o r the s c h o l a r s h i p s . A STUDY OP EQUILIBRIUM IN EXCHANGE BETWEEN HC1 GAS AND A KBr SURFACE Chapter 1 I n t r o d u c t i o n . - 2 -The main p a r t of t h i s t h e s i s i s concerned with e q u i l i b r i u m p r o p e r t i e s o f the exchange r e a c t i o n HC1 + B r " ^ = HBr + C l ' ^ g j , where B r " ^ denotes B r " i o n i n the outermost o r subsurface l a y e r s o f the vacuum sublimed KBr powder. I t i s t h e r e f o r e c o n s i d e r e d d e s i r a b l e to begin with a summary of the p r e s e n t s t a t e o f knowledge about the nature of s u r f a c e s of a l k a l i h a l i d e c r y s t a l s . - 3 -1 E q u i l i b r i u m s t r u c t u r e of s u r f a c e s . ? \ A c c o r d i n g to Burton e t . a l . ', f o r low Index f a c e s of a c r y s t a l i n g e n e r a l i n the presence o f o n l y i t s own vapor, there e x i s t s a c h a r a c t e r i s t i c temperature, T , o n l y above which s u r f a c e atoms can jump onto the top of a c r y s t a l plane, producing a m o l e c u l a r roughness. T c r e p r e s e n t s a k i n d of s u r f a c e m e l t i n g p o i n t and f o r (100) s u r f a c e s of simple c u b i c or FCC l a t t i c e s i s 5) l i k e l y to be c l o s e to the bulk m e l t i n g p o i n t . Cabrera l a t e r p o i n t e d out the p o s s i b i l i t y of lowering of the s u r f a c e m e l t i n g p o i n t i n the presence of an adsorbed monolayer of f o r e i g n molecules. Since the bulk m e l t i n g p o i n t s f o r a l k a l i h a l i d e s are commonly about 800°C, a t around room temperature the e q u i l i b r i u m s u r f a c e should be f a i r l y f l a t . The next q u e s t i o n a r i s i n g i s by which face a l k a l i h a l i d e c r y s t a l s i n assemblies of very s m a l l p a r t i c l e s are bounded. I t has been widely c o n s i d e r e d to be mainly the (100) f a c e . Benson et.' a l . c a l c u l a t e d s u r f a c e e n e r g i e s at 0°K of d i f f e r e n t f a c e s f o r a l k a l i h a l i d e s on the b a s i s of a p o t e n t i a l f u n c t i o n between two i o n s which Includes the Coulomb energy, the van der Waals a t t r a c t i o n i n the form of d i p o l e - d i p o l e and d i p o l e - q u a d r u p o l e i n t e r a c t i o n s , and the e l e c t r o n i c r e p u l s i o n . R e s u l t s f o r KBr, f o r i n s t a n c e , are 151 erg/cm and 317 erg/cm f o r (100.) and (110) r e s p e c t i v e l y . Although they d i d not i n c l u d e the c o r r e c t i o n term due to the expected d i s t o r t i o n of the s u r f a c e l a y e r which Benson e t . a l . - ^ d i d f o r (100) l a t e r on, the r a t i o of s u r f a c e - 4 -e n e r g i e s between (100) and (110) might not.be a l t e r e d s i g n i f i -c a n t l y even i f the c o r r e c t treatment i s done. T h i s r a t i o , being > / i " , i n d i c a t e s , a c c o r d i n g to Wu l f f ' s theorem 1-^ that (110) f a c e s should not appear i n e q u i l i b r i u m . In r e g a r d to d i s t o r t i o n o f the s u r f a c e geometry, Benson e t . 3) a l . c o n s i d e r e d the Verwey type o f d i s t o r t i o n i n which anions and c a t i o n s are d i s p l a c e d d i f f e r e n t d i s t a n c e s from t h e i r l a t t i c e s i t e s i n the d i r e c t i o n p e r p e n d i c u l a r to the s u r f a c e . They c a l c u l a t e d the c o r r e c t i o n term, the extent of displacement o f ion s and d i p o l e moments of d i s p l a c e d i o n s . The c o r r e c t i o n term to s u r f a c e energy turned out to be, f o r example, - 2 9 . 4 erg/cm f o r (100) face o f KBr, which made the c o r r e c t e d s u r f a c e energy f o r (100) of KBr to be 123 erg/cm 2. The cor r e s p o n d i n g value f o r NaCl i s 104 erg/cm . The ot h e r s i g n i f i c a n t r e s u l t s showed that the Verwey d i s t o r t i o n c o u l d extend down to the f i f t h l a y e r from the s u r f a c e . The c a l o r i m e t r i c a l l y determined s u r f a c e energy o f NaCl powder, s u r f a c e s o f which were c o n s i d e r e d to be ( 1 0 0 ) , however, was 276 erg/cm 2, although there i s a doubt whether the s u r f a c e s o f .the sample are In thermal e q u i l i b r i u m o r not. At the same time the above-mentioned t h e o r e t i c a l c a l c u l a t i o n seems to c o n t a i n u n c e r t a i n t i e s perhaps, i n ch o i c e o f p o t e n t i a l f u n c t i o n s and i n how to approximate the s u r f a c e d i s t o r t i o n c o n t r i b u t i o n . Another u s e f u l approach to e l u c i d a t e the nature o f s u r f a c e s i s to measure a s u r f a c e excess heat c a p a c i t y . Morrison e t . ' a l . ^ found a s u r f a c e excess heat c a p a c i t y at low temperatures, i n d i c a t i n g the e x c i t a t i o n of a v i b r a t i o n a l motion wi t h E i n s t e i n - 5 -temperature o f 43°K f o r NaCl powder, markedly d i f f e r e n t from the t h e o r e t i c a l value on the b a s i s o f the f r e e s u r f a c e boundary c o n d i t i o n . They t e n t a t i v e l y a t t r i b u t e d t h i s d i f f e r e n c e to s u r f a c e roughness. They suggest t h a t the s u r f a c e s o f t h e i r sample (vacuum sublimed powder) are not l i k e l y to be i n thermal e q u i -l i b r i u m . I f one c o u l d measure a s u r f a c e excess heat c a p a c i t y o f e q u i l i b r i u m s u r f a c e s , i t might g i v e a good p i c t u r e as to e q u i -l i b r i u m s u r f a c e s t r u c t u r e s . H a r r i s o n e t . a l . ^ developed a u s e f u l method to study the the nature o f s u r f a c e s . They s t u d i e d the anion exchange r e a c t i o n between gaseous HC1 and s o l u t i o n - p r e c i p i t a t e d NaBr and found t h a t the r a t e was high and an e q u i l i b r i u m was reached before the e n t i r e s u r f a c e exchanged, so that s e v e r a l specimens of HC1 gas were r e q u i r e d to complete the excange of s u r f a c e . Thus they c o u l d o b t a i n e q u i l i b r i u m constants i n terms o f the amount exchanged, and c a l c u l a t e d the standard entropy change from Br" to C l " to be about - 8 e.u., a f t e r s u b t r a c t i n g the c o n f i g u r a t i o n a l entropy. T h i s l a r g e entropy decrease was p o s t u l a t e d to be due to an a p p r e c i a b l e degree of c o v a l e n t c h a r a c t e r i n the bonding of a s u r f a c e B r " i o n to the u n d e r l y i n g N a + i o n , which would be decreased to g i v e a h i g h e r f o r c e constant when Br~ i s r e p l a c e d by the s m a l l e r and l e s s p o l a r i z a b l e C l " Ion. I f such i s the case, t h e o r e t i c a l c a l c u l a t i o n s o f the s u r f a c e energy mentioned above should be a l t e r e d at l e a s t i n the e s t i m a t i o n o f e l e c t r o n i c r e p u l s i o n s . - 6 -2 The nature of surfaces as revealed by k i n e t i c studies. It has been widely recognized that the surface layer d i f f e r s from the bulk i n r e a c t i v i t y . The extent, however, to which difference in r e a c t i v i t y has been shown, depends on the type of preparation of c r y s t a l used. Harrison et. a l . ^ ) showed, on studying the exchange reaction of Cl isotope between gaseous Clg and NaCl powder evaporated into dry nitrogen at atmospheric pressure, that only the f i r s t layer could be exchanged at room temperature, the second layer above 80°C, and the t h i r d layer at l80°C, with a gradual t r a n s i t i o n to bulk d i f f u s i o n above that temperature. A c t i v a t i o n energies calculated were 1 8 Kcal/mole f o r the second layer and 2 6 Kcal/mole for the fourth layer. Similar r e s u l t s were reported by Takaishi et. a l . for K C l - C ^ 1 1 ^ and f o r R b C l - C l 2 1 2 ^ (both a l k a l i halides were pre c i p i t a t e d from so l u t i o n ) , but they were able to d i f f e r e n t i a t e a more-active domain from a l e s s - a c t i v e domain in t h e . f i r s t layer, giving a c t i v a t i o n energies for exchange 2 . 9 Kcal/mole and 6.0 Kcal/mole respectively for KCl-Clg and 6 Kcal/mole and 17 Kcal/mole for RbCl-Clg. Ac t i v a t i o n energy fo r the fourth layer of KC1 was 40 Kcal/mole and was close to that of bulk d i f f u s i o n , 42 Kcal/mole, whereas that of t h i r d layer of RbCl was 28 Kcal/mole, and was close to 3 1 Kcal/mole for bulk d i f f u s i o n . The difference between two a l k a l i chlorides was explained in terms of the extent of the Verwey type d i s t o r t i o n discussed by Benson et. a l . ^ ) Namely the greater the difference in i o n i c r a d i i between anion and cation, - 7 -the deeper the e f f e c t of the surface penetrates i n t o the bulk. Vacuum-sublimed NaCl, on the other hand, was found to be markedly d i f f e r e n t i n that the bulk of p a r t i c l e s exchanged r e a d i l y at room temperature. This has been a t t r i b u t e d to the formation of e l e c t r o n i c d e f e c t s on i n t e r a c t i o n of the vacuum-sublimed p a r t i c l e s w i t h C l g . ^ ^ CHAPTER 2 E x p e r i m e n t a l Study of E q u i l i b r i u m i n Exchange between HC1 gas and a KBr S u r f a c e . - 9 -1 E x p e r i m e n t a l 1-1 M a t e r i a l . KBr Powder : AR grade ( A n a l a r ) KBr was p r e v i o u s l y made molten i n a p l a t i n u m c r u c i b l e , and the l a t t e r was p l a c e d at the bottom o f the powder producer ( F i g . 3 ) with the powder c o l l e c t o r a t t a c h e d at the j o i n t Q. The assembly was evacuated down to 10"^mmHg. Then, the i n d u c t i o n h e a t e r was a p p l i e d f o r a short p e r i o d of time ( ~ 1 min) and KBr was d e p o s i t e d on the i n n e r w a l l of the powder producer which was co o l e d by a i r blown from o u t s i d e the w a l l . The d e p o s i t e d KBr powder was then scraped down by t u r n i n g the s c r a p e r blade d r i v e n by the handle a t the top through the B-19 j o i n t and the f l e x i b l e j o i n t , and c o l l e c t e d i n t o the powder c o l l e c t o r by t a p p i n g . The f l e x i b l e j o i n t was used to s o l v e the d i f f i c u l t y o f c e n t e r i n g the s c r a p e r p r o p e r l y . Care was taken to p l a c e the c o i l o f the i n d u c t i o n h e a t e r a few m i l l i -meters below the top of the p l a t i n u m c r u c i b l e to a v o i d an e l e c t r i c d i s c h a r g e i n KBr vapor out of the c r u c i b l e , decomposing KBr vapor (which i s otherwise expected to c o n s i s t o f monomers, dimers, and -i o \ t r i m e r s o f KBr molecules x o ' ) to form excess K suspended c o l l o i d a l l y i n KBr powder. The procedure was repeated about ten times to o b t a i n about 0.1 g of KBr powder c o l l e c t e d i n the powder c o l l e c t o r . The bottom p a r t o f the powder c o l l e c t o r was s e a l e d o f f a t p o i n t R ( F i g . 3 ) i n vacuo, and a t t a c h e d to an a d s o r p t i o n measurement system at p o i n t T (see F i g . 1-1 i n Appendix I) to measure a BET s u r f a c e area, and to study s i n t e r i n g i n the presence - 10 -of HC1 (see Appendix I ) . I t was then s e a l e d o f f a t p o i n t S i n vacuo and a t t a c h e d to the r e a c t o r ( F i g . 2) and the KBr powder was t r a n s f e r r e d i n t o the r e a c t o r through the break s e a l . -; HC1 gas : AR KC1 (Analar) was p l a c e d i n the HC1 producer Y ( F i g . 4 ) , and mixed w i t h Reagent Grade cone. H2S0i|(B & A) i n vacuo by t u r n i n g HgSO^ arm V/ alo n g j o i n t N, and the HC1 gas evo l v e d was condensed i n X over the mixture o f KC1 and P2O5 a ^ l i q . n i t r o g e n temperature. When the r e a c t i o n was over, the gas phase over condensed HC1 s o l i d at l i q . n i t r o g e n temperature was evacuated, and then with stopcock L c l o s e d and with the t r a p Z co o l e d with a dry i c e acetone bath, the condensed HC1 s o l i d i n X was v a p o r i z e d by warming up to dry i c e temp, and c o l l e c t e d i n the 5 1 bulb In F i g . 1, which was p r e v i o u s l y evacuated. HBr gas : HBr (99.8$) from Matheson of Canada was used d i r e c t l y from a l e c t u r e b o t t l e ; i t was condensed i n X over P 2 0 5 through the j o i n t M ( F i g . 4 ) . The gas phase over condensed HBr was evacuated, and the l a t t e r was warmed up to dry i c e temperature and the HBr e v o l v e d was c o l l e c t e d through the t r a p Z a t dry i c e temperature i n t o the 5 1 bulb i n F i g . 1. - 11 -1-2 Gas chromatographic a n a l y s i s of a gaseous mixture o f HC1 and HBr The technique developed by Bergmann e t . a l . ^9). w a s a p p l i e d here to analyze a gaseous mixture of HC1 and HBr with a s l i g h t ' m o d i f i c a t i o n , i . e . s i n c e the r a t i o of HC1 and HBr here i s about 100^1,000:1 and HBr i s e l u t e d out l a t e r than HC1, p a r t i c u l a r a t t e n t i o n has to be p a i d to d i m i n i s h the t a i l i n g of the HC1 peak. T h i s was reasonably achieved by washing toluene with cone. HgSO^ s e v e r a l times and by l e a d i n g n-heptane through the column (20mm I.D.and 50mm long) f i l l e d w ith PgO^ and NagSO^. The idea was t h a t a s m a l l amount of b a s i c i m p u r i t i e s i n toluene and n-heptane were r e s p o n s i b l e f o r t a i l i n g , and the treatment above was to e l i m i n a t e p o s s i b l e b a s i c i m p u r i t i e s , H 2 0 and thiophene. Column : U-shaped Pyrex g l a s s t u b i n g (5mm I.D. and 60cm long) was f i l l e d with about 15 g of 4 0 ^ 6 0 mesh Chromosorb T ( t e f l o n powder with a s u r f a c e area of 7 .8 m 2/g) which was coated with the mixture of n-heptane (5$ by weight of Chromosorb T) and toluene (2 .5$) and cone.' HgSO^ (15 d r o p s ) . Toluene (Reagent A.C.S form B & A) and n-heptane (Spectro grade from Eastman o r g a n i c chemicals) were p r e t r e a t e d i n the manner mentioned above. C o a t i n g was achieved by shaking a stoppered erlenmeyer f l a s k with a p p r o p r i a t e amounts of chromosorb T and the mixture f o r s e v e r a l hours. The column thus prepared gave a reasonable r e p r o d u c i b i l i t y i n a q u a n t i t a t i v e a n a l y s i s a f t e r about three c o n s e c u t i v e days' use i n the same way as a n a l y s e s were c a r r i e d out i n exchange - 12 -runs and the c a l i b r a t i o n curve was v a l i d u n t i l the end of one s e r i e s (~-two weeks l o n g ) . Not u s i n g the column f o r a prolonged p e r i o d (a few weeks) seemed to degrade the f u n c t i o n of the column, but about three c o n s e c u t i v e days' use again brought back a reasonable r e p r o d u c i b i l i t y with a d i f f e r e n t c a l i b r a t i o n curve than b e f o r e . A f t e r s e v e r a l months, however, depending on the column, three c o n s e c u t i v e days' use d i d not b r i n g the r e p r o d u c i -b i l i t y back. In such cases then a new column was prepared. Apparatus : The main p a r t of the chromatograph i s schematic-a l l y o u t l i n e d i n F i g . 5 . I t i s c o n s t r u c t e d mainly of Pyrex g l a s s t u b i n g except t h a t the d e t e c t o r block i s made from s t a i n l e s s s t e e l (SS316), and f i l a m e n t s are t e f l o n c l a d tungsten wires from Gow-Mac. K e l - F # 9 0 grease was used where necessary. The s i g n a l was r e c o r d e d on a Speedomax H r e c o r d e r (Leeds and Northrup) through the a t t e n u a t i o n p a r t of Aerograph A-90P3. Operating C o n d i t i o n s : D t e c t o r c u r r e n t ; 200 mA Flow r a t e o f C a r r i e r gas (He) ; 70 ml/mln Temperature Column ; -78°C De t e c t o r ; 23°C C a l i b r a t i o n : A t y p i c a l chromatogram Is shown i n F i g . 6. As p o i n t e d out p r e v i o u l y , the HBr peak i s s i t u a t e d on the t a i l i n g p a r t o f the HC1 peak. T h e r e f o r e , the r a t i o of hatched areas of HBr over HC1 i n F i g . 6, R^, was c a l i b r a t e d a g a i n s t the molar r a t i o o f HBr over HC1 i n j e c t e d , R p. Areas i n chromatograms were determined by means of the d i s c i n t e g r a t o r a t t a c h e d to the - 13 -recorder used. A t y p i c a l c a l i b r a t i o n curve Is shown i n P i g . 7 The e r r o r l i m i t s estimated by re p e a t i n g the analyses are shown i n F i g . 9. - Fig . 1 Outline of Apparatus --15 -The rmo-- w e l l Powder C o l l e c t o r Melt botriV^p ends togeth Powder Feed Arm S S S A To Stopcock G P ^ 1 X—Convection Heater Mixing Arm > .Hammer M a t e r i a l : Pyrex Glass Volume : 150ml - F i g . 2 Reactor -- 16 -- Powder C o l l e c t o r -Q B-10 cone - Powder Producer -- F i g . 3 Powder Producer and C o l l e c t o r -- 1 7 -- F i g . 4 HC1 Producer -- 18 -Dry Ice-Acetone Bath - F i g . 5 Gaschromatograph .-- P i g . 6 T y p i c a l Chromatogram F i g . 7 Typical C a l i b r a t i o n Curve -RA = Area of HBr/Area of HC1 RP = PHBr / PHC1 -e-5.0 R. 10.0 * 10 -3 - 21 -1-3 Procedure. . R e f e r r i n g to F i g . 1, and F i g . 2, KBr powder was t r a n s f e r r e d through the break s e a l i n t o the r e a c t o r , p r e v i o u s l y evacuated down to 10" mtnHg. HC1 was then i n t r o d u c e d i n t o the d o s i n g r e g i o n p a r t i t i o n e d by stop cocks B,C,D,E,F,G,K,H and the column of the mercury manometer, the volume of which was p r e v i o u s l y c a l i b r a t e d to be 123 ml by gas expansion u s i n g the c a l i b r a t i o n bulb M as a standard volume. The. p r e s s u r e of HC1 i n the d o s i n g r e g i o n was then measured (l~10 c m H g ) . The c o r r e c t i o n i n volume was made by knowing the s e c t i o n a l area of the manometer. HC1 i n the d o s i n g r e g i o n , the molar amount of which was now known, was then t r a n s f e r r e d i n t o the r e a c t o r (/v.150 ml) by opening stop cock G and c o o l i n g the mixing arm Z with l i q u i d n i t r o g e n , then c l o s i n g stop cock G and warming Z with.a t o r c h . The pressure i n the r e a c t o r was then 1-^lOcmHg; Mixing i n the r e a c t o r was assured by the c o n v e c t i o n h e a t e r ( F i g . 2) and the mixing arm Z. The purpose of the l a t t e r was to mix gas i n the side-arms with t h a t i n the main body of the r e a c t o r by condensing the whole gas i n the r e a c t o r i n t o Z with l i q u i d n i t r o g e n f o l l o w e d by v a p o r i z i n g i t w ith heat a p p l i e d by a t o r c h . T h i s was done every 5 min. o r so d u r i n g the r e a c t i o n p e r i o d . The temperature of the water bath was c o n t r o l l e d w i t h i n ± 0.1°C by a p r o p o r t i o n a l temperature c o n t r o l (from F i s h e r ) . HC1 gas, thus i n c o n t a c t with KBr powder was allowed to stand f o r h a l f an hour to one hour, which was s u f f i c i e n t time f o r e q u i l i b r i u m to be reached (see Appendix I I ) . At the end of the r e a c t i o n p e r i o d , the gas sample was c o l l e c t e d - 22 -i n t o the sampler by c o o l i n g the l a t t e r w ith l i q u i d n i t r o g e n f o l l o w e d by t u r n i n g the four-way stop cock I by 90° and h e a t i n g the sampler with a t o r c h . The gas sample In the sampler was i n t r o d u c e d i n t o the gas chromatograph by t u r n i n g the four-way stop cock J by 9 0 ° , and a n a l y s e d . The r e a c t o r with p a r t l y exchanged KBr powder i n i t was evacuated f o r about 15 min. Then the next sample o f pure HC1 was taken i n t o the r e a c t o r , and f o l l o w i n g the same procedure mentioned above a n a l y s i s of the sample o f the e q u i l i b r i u m gas mixture was c a r r i e d out. T h i s procedure was repeated s e v e r a l times to make a s e r i e s o f runs, u n t i l the r a t i o of HBr over HC1 i n gas samples, Rp, became about 2 XlO --^, which was about the l i m i t i n o b t a i n i n g a reasonable accura c y (^10$) i n a n a l y s i s . A f t e r each s e r i e s , KBr powder p a r t l y exchanged with C l ~ was t r e a t e d with about 5 cmHg o f pure HBr twice f o r about h a l f an hour, and evacuated o v e r n i g h t -6 (~10 mmHg) p r i o r to the next s e r i e s . With v a r y i n g temperature, s e v e r a l s e r i e s were c a r r i e d out u s i n g the same KBr powder. - 23 -2 R e s u l t s 2-1 I n t r o d u c t i o n . A l l the s e r i e s o f runs are summarized i n Table 1. Table 1 S e r i e s o f runs KBr powder A B c Weight KBr Powder (g) 0.1395 0.1129 0.0762 S u r f a c e Area F r e s h (m 2/g) 43.4±0.8 44.3±0.8 34.2±0.7 A f t e r S i n t e r i n g (m2/g) 10±1 Temperature ( ° c ) 0 .A 4000 C7000 C8000 canon 9A A 7 0 0 0 28 A6000 ClOOO CQOOO 50 A5000 B8000 BllOOO C2000 CQOOO 72 B9000 C3000 86 BIOOOO BllOOO C4000 C5000 C - s e r i e s was c a r r i e d out wit h the KBr powder C whose s u r f a c e area i s expected not to change d u r i n g the whole s e r i e s (see Appendix I ) . The C9000 s e r i e s was s t a r t e d at 0.0°C, and the temperature was r a i s e d to 28.3°C; runs then f o l l o w e d up to C905O, a f t e r which the temperature was r a i s e d and runs from C9060 were c a r r i e d out at 50°C. For the case o f the A and B - s e r i e s / K B r - 24 -powder was p l a c e d i n the r e a c t o r with no p r e v i o u s attempt to s i n t e r i n HC1. For the f i r s t few hours d u r i n g the A - s e r i e s runs, t h i s powder i s expected to be s u b j e c t e d to a s i n t e r i n g p rocess -(see Appendix I ) . Runs up to AJ5000 s e r i e s were t h e r e f o r e d i s c a r d e d . The t o t a l exposure time to HC1 at the end o f AJOOO was about 15 hrs and l/RpC e vs. l/Rp p l o t s ( s e e below) e x h i b i t r e l a t i v e l y s c a t t e r e d s t r a i g h t l i n e s f o r runs A1000~A3000. From A4000 on, s t r a i g h t l i n e s are e x c e l l e n t , as i n F i g . 10A. F o r B - s e r i e s , the t o t a l exposure time was 20 h r s . up to run B7000. The exchange r e a c t i o n seemed to reach e q u i l i b r i u m w i t h i n 10 min. and the p r e s s u r e r a t i o R p = P H B r / P H C l s t a v e d p r a c t i c a l l y c o nstant f o r about 100 min., except f o r runs beyond the "bend" i n the isotherms (see Appendix I I ) . A l l the e q u i l i b r i u m v a l u e s are t a b u l a t e d i n T a b l e s 2A, 2B and 2C. 2-2 Data A n a l y s i s . Since the r a t i o o f HBr over HC1 i n the i - t h gas sample, Rp , was of the o r d e r o f 10 ^ 10 J , the amount of HBr i n the gas phase a^, n e g l e c t i n g the amount o f HC1 adsorbed on s u r f a c e or assuming that the r a t i o o f HBr and HC1 i n adsorbed phase i s not s i g n i f i c a n t l y d i f f e r e n t from t h a t i n gas phase, can be w r i t t e n , a i = n i V"' where n^ i s the amount of HC1 i n i t i a l l y taken i n t o the r e a c t o r f o r the i - t h run. Then the t o t a l amount of exchange a f t e r the i - t h run, Cg 1, which i s the amount o f C l " i n the exchangeable - 25 -region of KBr i n equilibrium with the i - t h gas phase, i s ; n i f C e's are tabulated i n the sixth column of Tables 2A, 2B, and 2C, and plotted against l/Rp i n F i g . 8A, 8B, and SC. For C-series in F i g . 8C, the amount corresponding to BET'surface area, Cg, (calculated on the basis that a l l the surfaces are (100)) i s indicated by the arrow on the C e axis. Numbers attached to p l o t s i n F i g . 8A, 8B, and 8C are run numbers. Error was estimated from error i n Rp (Fig. 9) and those in pressure and volume measurement {^1% each) and tabulated i n column 8, 9, and 10 of Table 2A, 2B, and 2C, and i l l u s t r a t e d i n F i g . 8B f o r BllOOO ser i e s . 2-3 Re p r o d u c i b i l i t y and R e v e r s i b i l i t y Good r e p r o d u c i b i l i t y among runs with the same powder KBr, i s well demonstrated i n the pairs of series, ClOOO and-C9000, C4000 and C5000, C7000 and C8000,(see F i g . 8C), B8000 and BllOOO, and BIOOOO and BllOOO (see F i g . 8B). It should be noted that the HBr treatment a f t e r each series brings the state of the KBr powder back to the i n i t i a l one completely. This i s so even for the case of series BIOOOO where the t o t a l amount of exchange at the end of the series (BlOljJO, see Table 2B) i s about three times the BET surface (see below), and the following series (BllOOO) a f t e r the usual HBr treatment agrees f a i r l y well with series B8000, and BIOOOO within experimental error. - 26 -2-4 Pressure Independence Comparing s e r i e s B8000 and s e r i e s B11000, (the former runs were c a r r i e d out with the p r e s s u r e about 3 . 5 cmHg, and the l a t t e r 10 cmHg, (see Table 2B)), both isotherms e x h i b i t no s i g n l g i c a n t d i f f e r e n c e d e s p i t e a s m a l l temperature d i f f e r e n c e . Run C 1 0 2 0 , (see Table 2C and F i g . 8C), which was done with the pr e s s u r e o f 6 cmHg, i s on the same curve as runs C1010, C9030, C9040 and C9050, which were c a r r i e d out at about 2 . 5 cmHg. A s i m i l a r s i t u a t i o n can be seen-in the group o f runs, C4010, C 4 0 2 0 , C5010, and C5020, and a l s o i n the group o f s e r i e s C7000, and C 8 0 0 0 . I t seems reasonable to conclude that the isotherms ( F i g . 8A, 8B and 8 c ) are independent o f the t o t a l p r e s s u r e o f gas phase. -27 -Table 2A Rum T e m p . or PT Re«c "t\me. Ce Y* >^ct 4 % 6 P * c ? 4 4Y/ Y F7lte«l f e Mo. ° K cm Hj x lO* mote •/•• •A •A 255 47 1.42 1.67 0. S5 i . 7 4.7 £.4 0.07? A-4-°2c nv 2.55 36) 2.50 2.63 0.75 5.9 5-5 il.4 0.20 0-123 A4030 273.2'k 2.55 4ff 3.21? 3.37 0.?75 6.? 6.0 i2.9 1.025 0.15? 2.65 30 4.14 3.11 l.ol .7.? 6-5" 143 I.2I 0.1*6 A-4-05D 2.58 30 5.36 4.4-1 1.21 6-7 15.2 l-3l 0.202 A-4060 2.55 30 £.22 4-7**- /.3I 10 5.3 143 1.44 0222 A-5010 2.<?5 60 .0.51 J.?3 0.150 1.5 2.5" 4.0 I.l1 0.1 ?3 A-5-020 3. IS 30 HO 6.20 0.177 3.0 5.1 0.291 A-5030 52. rc 2.15 30 1.42 7 . ? l 0.\2Z 3.7 3.3 y.O 2.3? 0.366 A-5W 3253°* 3.O67M0 2.15 30 l-?7 ?. 00 0.212 4.9 3.7 7 6 2.74 0.421 3.20 30 2.22 0.2/? 4-.0 13 3.°? 0.475 A-S»60 3.35 30 2.52 11.2 0.225 4.3 IDA 3.4I 0.525" A - » 3.40 30 2.12 12.I 0.240 £.4 4.5 I 0.4 3.6? 0.566 A-50W 3.30 30 3.35 12.1 0.251 7 0 4.7 H.7 3.92 D.M A-5o% 3S0 30 13.6 0.H1 1.2 4.1 (2-7 4.I3 0.635" A-6010 3.2S 55 o.?4 3.2? 0.2B9 2.3 3.3 5.6 o.m • 0.153 A - » 2$.0'c 3.'5 30 1.50 5.07 0.211 3.? 3.? 77 I-54 0.237 A-6030 3ol2'K 3.40 30 2.1? 6: 41 0.310 5.3 4.3 7.6 l.?5 0.300 3.32x\0* 3.20 30 2.7? 0.37? 6.2. 4.7 2.25" 0-34£ A-ftSo 3.13 30 3.3? 0.4/5 7.0 5.0 i2.o 24? 0.2?2 3.35 30 4.23 0.47? 2.0 5.3 13.3 2.6? 0.412 A-^7C 3A° 30 5-. 02 742 0.532 i s 56 14.1 iH 0.444 A-7010 2826° k 3.20 45 1.10 2 i4 0.415 2.1 3.7 S.9 0.12^ A-7020 3. OS 2.0 3.11-0.5/9 5.0 4.6 1.6 1.22 A-7030 3.10 30 3.l6 4r?1 0.645 6.7 5 2 il.1 1-41 320 30 4-.I5" 560 0.740 7 ? 5.6 I3.5 1.70 a 262, A- 7050 3. 10 30 5.12 6."f 0.?4( f.5 5 4 J 3.S" l.?7 0.2?? - 28 -Table 2B Run "few p. FT React, '/Rp 77 oC °K cm Hj_ t w e x IO1 *io~ t mole rviole 1 r .'/. K s — B-P0IO 3 P7 4-5" .0. 1% 3 7 4 (5.191 i t •z7 JT2yc 3. 45 30 1.27 0.224 3A 3.4 C7 2. /7 o.iH 3.4$ 30 1-61 7.2? 0.232 4.3 3 7 1 2.7^ B- W 3.0^X|DS 3. S3 30 2.0? 0.242 r-i 4.2 ?-3 3.30 0S07 3.32 30 2.77 4-7 / 0.7 3.U 0SS3 3.25 3S 3.11 I o.t-3 S.S 4-7 11.3 3tf 0M2 25 ?.6? H.I6 0.317 7 3 5.0 12.3 0.t>55 3-foio 3.£o 33 0.49 O.oU 1.5 2.5 4.0 /. 90 26 o.?6 D.I Of 2.3 2.9 1 3.02 B- 3.70 j>0 (.IS 0.1123 3.0 3. 1 6.1 3.V. O.b05 B - 9 W 3.70 2 4 1.55- 0.12? 1. 0 3.3 7.3 i.si O. 7/0 3-77 1? 2.06 / 3.44 0J536 5. 1 3.6 7 - 7 5.13 0.79P 3.SO 15 2-14- I4.tf 5". 2 J 7 $-.10 0.?S\ B - f "7o 3D 2 * 3 OJ r7 4-.I 6.10 B- <?<>$ 3. ^  35" 2.5S 5.? 4.4 10.2 b.76 \H B- ? W 3.£6 3 6 2.52 zl.o I 3 | B - f > O.U? O.Ob fl /. 1 2.1 7 . 1 2.32 0.351 B-1020 27. rc no- n 3.JO 30 0.750 ISO 2- 1 2:5- 3.U &-(<&> 3. se 30 i.o£ 12.DZ o.olfc 2 7 2.7 * 7 4.tf 0<7DS B-/cf# 3S5 S3 1.32 13 12 3.4 2.9 67 ^71 oM i-l 1.35 3.5" 3. 1 67 S.oS 0.932 J.C? 30 IH / 7.^  0.0W 47 3.3 7.3 l-o3 3.S5 3 2 IV O.Otfl 4.3 3.5" 7 7 7-2f I.12 14. 7? 3<7 M r 257 0. Oitfl 4.2 3.1 '•4? &-/P/30 \\X5 £0 /7' MS 1? .2 3 -MPI0 'i.ir 1.2s 553 o.m 3 iU 4-7 2.4.? 0M 34 2 AS 0.2SZ * 7 r. 1 lO.f 3 7 2 0.57Z (3-//PJ0 B.ff /2.1Z 7 ^ i-7 '13 0,723 (3-110*0 31 5-71 11M 6.2 15. I 5.21 o.hi - 29 -Table 2C Te»*f». °c °K React. x i o 2 xlO* .1 4 % ./. Ce C-IOIO C-l020 C-1030 c-l mm 1. T2 2x1 2. Z\ 2A\ too* r r 15-30 L30 V7 2.n 1.61 2.^  2 J 0 3.10. 0. Pi isi o.W 1.24 |.# 3.4 7A 4-5 £.3 l l 4.4 9-2 £ 2 ' £.2 IS-. 1 0.24-? om 0.-2.71 0:415-C-2D10 C-202.0 z-im 32?.2'\< 3.tow* l . W 2.2JT 2.ir J.07 ILO 3D 30 30 42 2r 30 1.61 2 SO 4 3 \.n 3-10 3.72 4.2.(9 3.U i .W 0.412 o.srt 0172 1.023 o.(>4 o.?9 l.IS 2.0 4 ' !>.* 9A 1* f 3 9-4 . f-7 to / P . ! i l - 4 l £ 2 O.zN 0.4-77 o.?72 0.^23 C-]Dlt) c-3020 C-joJo C-30W C-JDW 2.SJ IK U.2 30 2? 30 [.20 l.5"2 LP 2.3? H-.79 $A3 7:17 0.2.10 0.311 0.31? o.4& /.? (L.O 2.? i.4-3.7 1-0 S.O 7:7 /2J 0.732 |./94 C-liOlo c-mo c-$oio c-mo fio't .2.19 3.00 J .00 US' 4. £2 3 39 2.H 24 30 [0 7$* 24-29 on I i i l.ftT 2.22 9.{0 7.9? 3 AO 9.10 yS4 U2 p.l to oW 0.210 0.211 0.170 O.iV 0.2(4 2.3 i4 41 IS 3.Z 4.1 3.3 3.9 3.? 4-1 2.5" J.3 SS 7.7 ?./ o .m l.lfO Lift 0S23 Lie l.2f c-70/0 C-7D$ c-m C-folo O.o'c-\S\ ISO \.\o \.?s 0.99 I2S-i-4 30 30 91 90 So 30 2.^ 3 j.4r / . * $ -2.0f 1.03. 2.0<? ojt 1.0*1 \.2l O.ft \.D0 L32 0.44 o n 2.2.2 249 2.97 2.l4-J2.J0 7' 2.<P 7.7 4.<? 3.i 3.r f > 111 13 2 /D.l 13.9 91 11 oM o.m ooiti O.ISi 0.2.03 •D-D tiff 0.13i * see Aooendix I I . - 30 -- F i g . 8A -Isotherms f o r A - s e r l e s . 5090 52.1C 6070 2 8 . 0 ° C 0 ' 7050 9 . 4 C 4060 4050 0" C 5 . 0 X 10 l/Rp 87.3 °C - 32 -- P i g . 8 -C -2 0 5 . 0 X 1 0 l/Rp - 33 -- Fig . 9 -- 34 -2-5 Behaviour above "Bends" For series B9000, BIOOOO (Fig. 8 B ) , C 3 0 0 0 , C4000 and C 5 0 0 0 (Fig. 8C), "bends" appear at higher C g (roughly at C e = C s f o r C ser i e s , see below) i n isotherms. Table 3 time (min) ( X 10 ->) time (min) ( * 1 0 " 3 ) B11050 3 1 6.I5 B10070 5 2 5 . 9 3 B11060 3 1 5 . 9 ^ B10080 3 0 6 . 0 9 B11070 1 1 5 7 , 8 7 B10090 3 0 6 . 0 1 B11080 1 2 5 . 6 . 5 5 B10100 2 8 5 . 0 7 B11090 6 9 0 8 . 6 0 B10110 6 0 6 . Q 9 BIO120 3 0 4 . 6 3 B 1 0 1 3 0 6 0 5 . 4 9 Table 3 shows the influence of the reaction time on R p f o r runs a f t e r the "bend" i n the isotherms, which otherwise i s not seen c l e a r l y . It seems that some k i n e t i c e f f e c t sets i n . This point i s checked in d e t a i l i n Appendix I I . Therefore, further consider-ation about equilibrium i s r e s t r i c t e d to isotherms below these bends. - 35 -2 - 6 Langmuir-type plot If exchanged C l " and unexchanged Br" form an id e a l solution i n the exchangeable region of KBr powder, the equilibrium equation would take the form, < PHBr/ PH01>< X'/ 1- X'> = K • X' : the f r a c t i o n of the exchangeable anion s i t e s occupied by C l , K : the equilibrium constant. or % c e / ( c m - C J = K , where X' = 0Q/Cm and C m i s the maximum exchangeable amount, or l / R p C e = l / R p C m + l/KC m . (1) In other words, i f the solution were i d e a l , l/RpC e vs. l/Rp plot should exhibit a straight l i n e with the slope of l / C m and the intercept of l/KC , and C would be indeoendent of temperature m m " and K would depend on temperature. l/RpC g was plotted against l/R p i n F i g . 10A, 10B and IOC. The plots are in fact straight l i n e s except f o r some rather scattered points at 0°C, but C m (the inverse of the slope) c l e a r l y depends on temperature. C m and K were calculated from the slope and the intercept according to eq. ( l ) , and tabulated i n Table 4, and plotted i n F i g . 11. - 3 For a l l A,B, C series, K seems to be constant, K = 3.5^10 , and - 56 -C can be written in the form : m C m o<exp(-4,600/RT). Table 4 Series No. Temp. °C K xlO~ 5 C m -6 xlO mole X m A7000 9 . 4 5 . 5 9 . 6 0 . 45 A6000 28.0 3 . 4 5 1 5 . 0 0 . 7 0 A5000 52.1 2 . 6 ~ 3 . 9 21—28 0 . 9 8 - 1 . 3 1 B8000 5 2 . 7 2 . 8 - 3 . 8 19 -22 1.10 -1 . 2 9 B9000 7 2 . 5 3 . 1 - 4 . 3 2 9 . 5 - 3 8 1 .73 -2.2 BIOOOO 8 7 . 3 2 . 3 - 4 . 2 42 ~64 2 . 4 7 - 3 . 7 BllOOO 5 3 . 3 3 . 5 5 20 . 5 1.20 ClOOO C9000 2 8 . 3 3 . 3 - 5 . 5 3 . 6 - 4 . 7 O .56 ~ 0 . 73 C2000 09000 50.0 3.2 - 5.2 6 . 6 - 8 . 2 1.02-1.26 C3000 71 . 8 2 . 0 - 3 . 9 12 . 5-20.0 1 . 9 3-3.1 C4000 C5000 86.0 2 . 3 - 6.0 14.0-29.0 2. 1 6 ~ 4 . 4 5 . From t h i s point on, therefore, eq. ( l ) i s considered to be an empirical equation with C and K as constants which may have d i f f e r e n t physical meanings from those given above. Note, however, that C m s t i l l has the same dimension as C e, and K i s dimensionless. It w i l l i n fact appear in one of the suggested - 37 -models (see Chap. 3) t h a t C^ may r e p r e s e n t two d i f f e r e n t q u a n t i t i e s i n the slope and the i n t e r c e p t . The e m p i r i c a l e q u a t i o n should then be w r i t t e n with three c o n s t a n t s , VVe = l / R p C m + l / K , C m ' ' ' where C ' s t i l l has the same dimension as C~. and K' i s m e' d i m e n s i o n l e s s . The d i s t r i b u t i o n o f the temperature dependence between K' and C m' i s no l o n g e r n e c e s s a r i l y the same as i n the i d e a l case ; one p o s s i b l e i n t e r p r e t a t i o n i n d i c a t e s C ' as roughly independent of temperature and a t t r i b u t e s the temperature de-pendence of the i n t e r c e p t to K', t h i s dependence being roughly the same as that o f C (from the s l o p e ) o n l y by c o i n c i d e n c e . - 38 -0°C - F i g . 10A -Langmuir-type p l o t s f o r A - s e r i e s . 8 (* 1 0 m o l e - 1 ) • 1.0 ( * 10 1.3 mole h 1.2 . 9.4QC I - 1 VRpCe 1.0 0.9 8.o°c 52.1PC 0 5.0 X 10^  l/R - 3 9 -(X i o 8 i / R p c e 0 . 3 0 . 2 0 . 1 - P i g . I0B -Langmuir-type p l o t s f o r B - s e r i e s , 5 2 . 7°C 53.3°C 5 . 0 X 10 - 40 -1/R (xlO' mole - 1) 2 . Q L 0°C - Fig. IOC -Langmuir-type plots i/R ( x 10 "3.0 ?or C. e mole 28.3 °C 1. 0.0 °C 2.5 2.0 0 5i.O XlO 2 1/Rr - 42 -2-7 R e p r o d u c i b i l i t y among three KBr Powders To compare A, B, and C - s e r i e s , one has to no r m a l i z e CQ (amount exchanged ) with a c e r t a i n q u a n t i t y . The f i r s t c h o i c e , of such a q u a n t i t y would be s u r f a c e area of KBr powder w i t h s u r f a c e s assumed to be (100). The s u r f a c e areas of KBr powders A and B a f t e r s i n t e r i n g are not known. These s e r i e s show the same dependence of l o g on l / T as s e r i e s C and the same value o f K. I f C f o r d i f f e r e n t KBr samples a t the same temperature m i s c o n s i d e r e d to be p r o p o r t i o n a l to the s u r f a c e area, one can f i t A and B s e r i e s to C by m u l t i p l y i n g a p p r o p i a t e f a c t o r s to each C g of A and B s e r i e s r e s p e c t i v e l y so t h a t the l i n e s o f l o g C m versus l / T of A and B merge to t h a t of C s e r i e s . Thus f i t t e d C 's are t a b u l a t e d i n the e l e v e n t h column of Table 2A and 2B. F i t t e d C e vs. l / R p i s p l o t t e d i n F i g . 12. R e p r o d u c i b i l i t y among three KBr samples seems q u i t e good c o n s i d e r i n g the l i m i t o f e x p e r i m e n t a l e r r o r i n v o l v e d (see F i g . 13). These f i t t e d C ' s f o r A and B s e r i e s and C „ 1 s f o r C were e e a l l n o r m a l i z e d with ( t o t a l amount of B r " co r r e s p o n d i n g to BET s u r f a c e area) of KBr powder C, and t a b u l a t e d i n the t w e l f t h column i n Table 2k, 2B and 2C. X (= f i t t e d C /C ) i s p l o t t e d e t> a g a i n s t l / R p i n F i g . 14. C^'s a r e a l s o n o r m a l i z e d i n the same manner as above and t a b u l a t e d i n the f i f t h column of Table 4 . Then eq. ( l ) and (!') can be r e w r i t t e n as, 1/XRp = l/X mR p + l/KX m, (2) and l/XR p = l/x mR p + l/K»Xm' . - (2>) - 43 -- F i g . 12 R e p r o d u c i b i l i t y Among three KBr Powders 67.3°C 1 0 . 0 ;© 86.0°C. _ ? 7 l . 8 ° C 0 A © B C C 0 (Xio 5 . 0 -6 mole) 53.2°C $ 5 2 . 7 °C '"j^-'O 53.1°C ^v^ s o . o°c 28.3°C 28.0°C - 44 -- F i g . 13 Same as F i g 12 with the e r r o r l i m i t s 10.0 (X 10 8 6 - 8 7 . 3 ° C mole ) 5.04 7 1 . 8 — 72.5°C e B 0 6 B H " a 0 E 5 0 ~ 5 3°C 28° c 9.4°C o°c 0 5.0 X 10 1/Rr - 4 5 -- F i g . 14 X vs. l/Rr Q A s e r i e s C> B s e r i e s O C s e r i e s 5.0 X l O 2 l/Rp - 4 b -2-8 Thermodynamic quantities It i s possible to evaluate the p a r t i a l molar enthalpy change for the present reaction from Isotherms (e.g. Pig. 14), by modifying the Clausius-Clapeyron equation. Following Prigogine et. a l . ' the a f f i n i t y of the reaction HC1 + Br" - HBr + C l " , can be written as A = /^HCl + M B r » " ^HBr " Mci» ' where jU. 1 s are chemical potentials of each components. At equilibrium A = 0, and the change in the a f f i n i t y , along the equilibrium l i n e i s also zero, (5 A = 0. 0 = (5(A/T) = 5 ( M H C - L A ) + <5(// Br 'A) " 6 ( / > W / T > - 6(Mci</ T ) ' and (5 (fX j_/T) = - (h J_/T5. ) £)T + (v ±A) <5P + (l/f) (d JLL±/d 6 ^, where h i s the p a r t i a l molar enthalpy, v^ _ the p a r t i a l molar volume and the mole f r a c t i o n of i - t h component in the phase. (Note that x G 1 , and X are not necessarily equal, since the maxium exchangeable amount in the true physical sense i s not known, but they are proportional each other). - 47 -. 0 = 6 (A/T) .= ( - l / T 2 ) ( h H C 1 + h B r , - h H B r - h c l,)5T + • (1A)(vHC1 - vHBr + vBr, - vcl,)<5P. + (iA-)[('d 'M Hci/dx H C 1)6x H C 1 - (d//H B r/dxH B r)(5xH B rJ + ( i / T ) [ ( d ^ B r , / d x B r , ) 6 x B r I - (e>M c l,/dx c l,)6x c l,"-] . I f one keeps the compos i t ion o f B r " and C l " cons tan t , and assumes tha t the gas phase i s a p e r f e c t gas m i x t u r e , and n e g l e c t s vBr>-, V G l " c o m P a r e d t 0 VHC1 J V H B r ' t h e n <5xBr« = - 6 XC1« = °> ^ M H C l / d x H C l ) = R T/ XHC1 > ( ^ / / H B r / d x H B r ) = R T / x H B r , VHC1 " V HBr + v Br» " v C i ' * V HC1 " V HBr = 0 • The l a s t equa t ion means t h a t , ( u A/5P) = 0, i . e . the a f f i n i t y o f the r e a c t i o n does not depend on t o t a l p re s su re , which i s c o n s i s t e n t w i t h the exper imenta l f a c t mentioned p r e v i o u s l y . .\ 0 = - ( l / T 2 ) ( h H C 1 + h B r , - h H B r - h c l,)(5T 4- R(6lnx H C 1 - 5lnxH B r) } / . ^ l n ( X H B r A H C l ) / 6 T ] x B r I = ( ^ ^ V r + h C l ' " ^ H C l - h^,) - (1 / R T 2 ) A h . ' ( lnRp) = -( A h / R T ) + C. r x B r » - Table 5 -°c 0 9.4 28.0 - 2 8 . 3 5 0 - 5 3 . 3 7 1 . 8 - 7 2 . 5 86 - 8 7 . 3 Temp. °K 273.2 282.2 301 325 345 359 3.66 x i o " 5 3.54x i o " 5 3.32 x 1 0 " 5 3.08 X10~ 5 2.90 x 1 0 " 5 2 .78x 1 0 " 5 X Rp (xl0~2> Rp (xlO -^ R P (xio - 5; Rp (xlO - 5; RP (xio~5; RP (xl0-3; 0.1 5.88 12.8 20 .0.15 3.92 6.67. 11.7 23.0 0.2 2.22 4.25 8.33 16.6 0.3 4.65 9.7 18.8 31.2 0.4 2.66 6.45 13.0 21.7 0.5 4.45 10.2 16.1 0.6 3.3 7.8 13.0 0.7 2.3 6.35 10.6 0.8 5.25 8.9 0.9 4.45 7.5 - 50 -From F i g . 14, Rp was read o f f with X constant a g a i n s t temperature, and t a b u l a t e d i n Table 5 and p l o t t e d i n F i g . 15 . In F i g . 15 ( l o g R p ) x vs. l / T e x h i b i t s s t r a i g h t l i n e s except f o r 0°C ( l / T = 3'.66x'l0" 5 ( ° K ) - 1 ) . Slopes y i e l d Ah's, and they are t a b u l a t e d i n Table 6. I t then f o l l o w s that from the r e l a t i o n Ah - T As = 0, where A S i s p a r t i a l molar entropy d i f f e r e n c e (= s ^ B r + s ^ , SHC1 " SBr« A s = A h/T. A s ' s are a l s o t a b u l a t e d i n Table 6 f o r 51°C. Since the gas 21) phase i s a p e r f e c t gas mixture, from thermochemical data , hHBr " h H C l = 1 1 * 6 (Kcal/mole), sHBr " SHC1 = 2 . 8 0 - Rln ( P H B r / P H C 1 ) (e . u. ) a t ^ 3 5 0 ° K . These l e a d one to c a l c u l a t e h ^ , - h B r , and S Q - J J - s B r , as t a b u l a t e d i n Table 6 , and p l o t t e d i n F i g . 16 . For a n o n - i d e a l mixture o f C l 1 and Br', with ' f and 7 Br, t a s a c t i v i t y c o e f f i c i e n t s , d i f f e r e n c e s i n the p a r t i a l molar q u a n t i t i e s e v a l u a t e d above are , SC1« " S B r - = SC1> " SBr< " d [ R T l n < X C 1 • 7 c i ' / x B r ' TBV • >] / D T> h c l , - h B r , = h ° x , - h B r , - R T 2 [ d l n ( 7 c l , / 7 B r ' ) / d T ] ' where s ° 1 s and h°'s are standard s t a t e q u a n t i t i e s . - 51 -- Table 6 -X Temp. Range A h AS h d . - h B r » S C l ' " s B r « ° c Kcal/mole ( 5 i ° c ) e. u. Kcal/mole (51°C) e.u. 0.1 9.4 28 4.3 13.2 - 7 . 3 0.15 9.4 50 5 . 0 15.4 - 6 . 6 4.1 0 . 2 9.4 50 5 .8 17.9 - 5 . 8 7 .0 0.3- 28 86 7.1 21.9 -4 .5 9 .9 0.4 28 86 . 7 .9 24.4 - 3 . 7 11.6 0 . 5 50 86 •8.7 2 6 . 8 - 2 . 9 13.3 0 .6 50 86 9.2 28.4 -2.4 14 . 2 0.7 50 86 10.0 3 0 . 8 -1 .6 15.9 To o b t a i n d i f f e r e n c e s i n standard s t a t e q u a n t i t i e s one has to ev a l u a t e the t h i r d terms on the r i g h t of the above eq u a t i o n s , which can be done o n l y by c o n s t r u c t i n g a model g i v i n g r i s e to the dependence o f h c l , - h B r , and s c i , - s B r , on X (o<;x c l i ) a s i n F i g . 16. The a l t e r n a t i v e way i s to s t a r t with e v a l u a t i n g the e q u i l i b r i u m constant (= R p ( x c l I / x B r » ^ a s H a r r i s o n 1 ^ e t . a l . d i d , but f o r the present case, s i n c e the t o t a l exchangeable amount i n a true p h y s i c a l sence i s not known, the term ( x c i ' / x g r i ) can not be c a l c u l a t e d . N e v e r t h e l e s s , the va l u e s o b t a i n e d i n Table 6 were f o r m a l l y c a l c u l a t e d from the isotherms g i v e n i n F i g . 14, so i n the next chapter v a r i o u s models w i l l be d i s c u s s e d which might g i v e r i s e to isotherms g i v e n i n F i g . 14 and the e m p i r i c a l equations ( l ) or ( 1 ' ) , and (2) o r ( 2 ' ) . - 52 -CHAPTER 3 Discussion - 54 -1 I n t r o d u c t i o n The o b j e c t i v e s o f t h i s d i s c u s s i o n are (a) to attempt to-, e x p l a i n why the e q u i l i b r i u m behaviour of the system g i v e s reasonably l i n e a r p l o t s a c c o r d i n g to the equ a t i o n l / R p C e = l / R p C m + lA'Cm-and (b) to attempt to e x p l a i n the s i g n i f i c a n c e o f the . t h r e e c o n s t a n t s w r i t t e n as C , C ' and K' i n t h i s e q u a t i o n . On the m m b a s i s of the exp e r i m e n t a l evidence a t pr e s e n t a v a i l a b l e , n e i t h e r o f these e x p l a n a t i o n s can be c o m p l e t e l y . u n e q u i v o c a l . In re g a r d to p o i n t ( a ) , many d i f f e r e n t models i n v o l v i n g exchange of one, two or three l a y e r s , and a l l o w i n g f o r v a r i o u s c o - o p e r a t i v e i n t e r a c t i o n s , have been found capable o f g i v i n g l i n e a r Langmuir-type p l o t s o f ( l / R p C e ) a g a i n s t l/Rp. In regard to p o i n t (b), i t must f i r s t be r e c o g n i z e d t h a t the l i n e a r p l o t s , as u s u a l , g i v e o n l y two parameters, the slope and the i n t e r c e p t . The l a t t e r i s a combination of an apparent e q u i l i b r i u m constant K' and an apparent exchange c a p a c i t y C ' ; but the way .in which these are to be separated, and whether the C from the slope m ^ i s the same q u a n t i t y as the C 1 i n the i n t e r c e p t , depends on the assumed model. I t i s not p o s s i b l e , f o r example, on an e m p i r i c a l b a s i s o n l y and before p o s t u l a t i n g a model, to s t a t e whether the e q u i l i b r i u m constant K' depends on temperature o r not. I f i t i s assumed, as i n the case o f an i d e a l e q u i l i b r i u m , t h a t C = C ', then K' = ( s l o p e / i n t e r c e p t ) ; and the v a l u e s o f - 5 5 -K' (= K i n e q . ( l ) c h a p . 2) t h u s f o u n d a r e a l m o s t i n d e p e n d e n t o f t e m p e r a t u r e . The p a r a m e t e r C m t h e n e x h i b i t s , h o w e v e r , t h e e n t i r e l y u n u s u a l p r o p e r t y o f b e i n g i t s e l f t e m p e r a t u r e - d e p e n d e n t . The most s e r i o u s d i f f i c u l t y i n i n t e r p r e t i n g t h e r e s u l t s i s t h e q u e s t i o n o f i d e n t i f y i n g t h e r e g i o n o f t h e s o l i d p h a s e w i t h w h i c h e x c h a n g e t a k e s p l a c e . T h i s p r o b l e m i s o f c o u r s e c l o s e l y r e l a t e d t o t h a t , o f d e t e r m i n i n g t h e s i g n i f i c a n c e o f C m a n d C 1 m I n t h e p r e s e n t w o r k , v a l u e s o f C m f r o m t h e s l o p e s o f t h e L a n g m u i r - t y p e p l o t s r a n g e d f r o m 0 .5 t o 3 t i m e s t h e B.E.T. s u r f a c e , a r e a ( o n t h e b a s i s t h a t a l l f a c e s a r e ( 1 0 0 ) ) . T h e s e r e s u l t s w ere c a l c u l a t e d , h o w e v e r , f r o m p o i n t s a t C g v a l u e s w h i c h were a l w a y s l e s s t h a n t h e B.E.T. s u r f a c e a r e a ; and when e x c h a n g e was c a r r i e d b e y o n d t h e e q u i v a l e n t o f one l a y e r , a b e n d a p p e a r e d i n t h e r e a c t i o n i s o t h e r m s c l o s e t o t h e e q u i v a l e n t o f one l a y e r e x c h a n g e d . T h e r e i s t h u s o b v i o u s l y an a p r i o r i c o n f l i c t i n t h e most o b v i o u s ways o f i n t e r p r e t i n g t h e e x p e r i m e n t a l e v i d e n c e ; a n d i t i s n o t a t p r e s e n t p o s s i b l e t o r e s o l v e u n e q u i v o c a l l y t h e q u e s t i o n o f w h e t h e r o n e , two o r t h r e e l a y e r s p a r t i c i p a t e i n t h e e x c h a n g e r e a c t i o n b e l o w t h e b e n d . I n t h e f o l l o w i n g d i s c u s s i o n , o n e - l a y e r a n d t h r e e - l a y e r m o d e l s a r e e x a m i n e d i n d e t a i l . The t y p e s o f m o d e l w h i c h n e e d t o be c o n s i d e r e d may be s u m m a r i z e d i n t h e most g e n e r a l t e r m s a s f o l l o w s : -( I ) The a p p a r e n t d e p e n d e n c e o f on t e m p e r a t u r e i s g e n u i n e , a n d . . w i t h i n e x c h a n g e a b l e amount t h e C l " a n d B r " i o n s f o r m an - 56 -i d e a l s o l u t i o n occupying a d e f i n i t e s et of l a t t i c e s i t e s ; i . e . the i n d i c a t i o n of i d e a l behaviour i n c o n f i g u r a t i o n a l entropy suggested by the f a c t o r C /(C - C ) i s genuine, not s p u r i o u s . ( l l ) The appearance of i d e a l i t y i n eq. ( l ) i s s p u r i o u s . There i s some c o - o p e r a t i v e phenonemenon i n the p a r t l y exchanged r e g i o n which i s capable of g i v i n g the apparent Langmuir-type behaviour and the apparent dependence of on temperature. The exchanged r e g i o n c o n s i d e r e d may be e i t h e r : -(a) one l a y e r o r (b) three l a y e r s . - 57 -2 Detailed discussion based on ( i ) The p o s s i b i l i t y of a genuine reversible v a r i a t i o n with temperature of the amount of B r - available f o r exchange cannot be completely eliminated on the basis of any of the experimental data. The problem now i s reduced to finding an appropriate mechanism with which the available amount of B r - f o r exchange, C m, depends on temperature in,the manner described i n the p r e v i -ous chapter. The large r e v e r s i b l e v a r i a t i o n , however, i s d i f f i -c u l t to explain with models outlined i n the following subsections: (a) Genuine reversi b l e v a r i a t i o n of surface structure with temperature. Apart from possible e f f e c t s of adsorbed gases (discussed i n (b) below), the thermal equilibrium structure of a (100) surface i s expected to be geometrically s i m i l a r to that of a correspond-ing plane i n the i n t e r i o r of the c r y s t a l , with l i t t l e thermal 2) disorder below the bulk melting-point . D i s t o r t i o n of the surface layer i s l i k e l y to consist c h i e f l y of unequal displace-^ ments of anions and cations in the d i r e c t i o n perpendicular to the surface-^), and, i n the dynamic properties of the l a t t i c e , the appearance of some new v i b r a t i o n frequencies i n the same d i r e c t i o n ^ ) . In the previous work on s o l u t i o n - p r e c i p i t a t e d NaBr"^, non-ideality i n the equilibrium behaviour was discussed i n terms of t h i s type of d i s t o r t i o n . But those r e s u l t s did not show anything resembling the temperature dependence of C m found in the present work. - 5 8 -Gross d e f e c t s such as steps are commonly present on most s u r f a c e s , but they are not i n thermal e q u i l i b r i u m , and c o u l d not g i v e r i s e to the l i n e a r p l o t s of l ° g C m a g a i n s t 1/T found i n t h i s work. I t would n e v e r t h e l e s s be a u s e f u l e x t e n s i o n of the present work to c a r r y out experiments designed to determine whether there i s any dependence of s u r f a c e area on temperature ; e.g. to quench samples r a p i d l y from d i f f e r e n t i n i t i a l temperatures to 7 7 ° K before a B.E.T. masurement, i n the hope of " f r e e z i n g - i n " the s u r f a c e s t r u c t u r e a t d i f f e r e n t h i g h temperatures, (b) E f f e c t of adsorbed l a y e r on the v a r i a t i o n . Except i n u l t r a - h i g h vacuum c o n d i t i o n s , i t i s always nec e s s a r y to take i n t o account p o s s i b l e d r a s t i c e f f e c t s of an adsorbed l a y e r on the thermodynamic p r o p e r t i e s of the u n d e r l y i n g s u r f a c e . Cabrera^) has shown that d i s o d e r i n g temperature of a s u r f a c e may be g r e a t l y lowered by adsorbed gases. A d s o r p t i o n measurements i n the p r e l i m i n a r y work show t h a t a KBr s u r f a c e , i n the c o n d i t i o n s of the exchange experiment, i s almost completely covered with an adsorbed l a y e r , which v a r i e s very l i t t l e i n composition. At a l l e x p e r i m e n t a l c o n d i t i o n s , i t c o n s i s t s almost e n t i r e l y of HC1, with o n l y a minute f r a c t i o n of HBr. In these circumstances, there i s l i t t l e chance of an adsorbed l a y e r b r i n g i n g about the l a r g e v a r i a t i o n . F u r t h e r evidence a g a i n s t such an e f f e c t i s that when the gas phase was removed i n three stages from a system at e q u i l i b r i u m , a l l three gas samples had c l o s e l y s i m i l a r a n a l y s e s (see - 59 -Appendix I I ) . T h i s , however, i s not very strong evidence, since i t might be pressure r a t i o , and not t o t a l pressure, which i s important, and the removal of the f i r s t two samples would s t i l l leave the composition of the adsobed l a y e r e f f e c t i v e l y u n a l t e r e d . - 60 -3 D e t a i l e d d i s c u s s i o n based on I I The true e q u i l i b r i u m constant, K g, i n t h i s case, may be w r i t t e n i n the form ; K e = ^ H B r A H C ^ ^ c i ' ^ B r ' ) = V ^ C / T B ^ C e / ( C T " C e ) * — --0) where a's are a c t i v i t i e s , 7 s are a c t i v i t y c o e f f i c i e n t s , and C^ i s a t o t a l amount of the assembly. Rearranged In the form a p p r o p r i a t e to a Langmuir-type p l o t , eq. ( j ) becomes ; l / R p C e = l / R p C T + ( 7c/7 B)(lA eC T) , — — -(4) where a c c o r d i n g to the d e f i n i t i o n s above, C T i s constant, 7's are f u n c t i o n s of C Q and T, and K o f T and C Q. The s u f f i c i e n t e ' e e c o n d i t i o n f o r a l i n e a r p l o t of ( l / R C ) a g a i n s t ( l / R p ) i s ; ( 7 c / 7 B ) ( i / K e ) = a/Rp +J3 , — — — — — — ( 5 ) where (X and are f u n c t i o n s of T o n l y . The problem now i s to f i n d a model which g i v e s r i s e to the c o n d i t i o n (5) or the e m p i r i c a l e q u a t i o n ( l ) or ( l 1 ) and (2) or (2') under v a r i o u s assumptions. The f i r s t type of model to be examined w i l l be t h a t i n which ( l ) the exchanged C l " (the amount o f which i s C e i n eq. ( l ) ) are a l l i n the outermost l a y e r . In t h i s case, the problem i s reduced to c o n s i d e r i n g a two-dimen-s i o n a l r e g u l a r s o l u t i o n of C l " and B r " on l a t t i c e s i n e q u i l i b r i u m with a p e r f e c t gas mixture of HC1 and HBr. A s t a t i s t i c a l - 61 -mechanical c a l c u l a t i o n was done i n the conventional manner with the further assumption : (2) The configurational p a r t i t i o n function can be separated from the i n t e r n a l one. Later in t h i s section assumptions ( l ) and (2) are examined, and cases where eithe r one of the two i s removed i s discussed i n subsection (b) and (c). (a) Two-dimensional l a t t i c e gas (under assumptions ( l ) and (2)). The experimental isotherm (Fig. 14) shows that, on the analogy of the problem of adsorption or mixture, there must be a repulsive i n t e r a c t i o n between l i k e species on adjacent anion s i t e s . At the same time, however, the fact that isotherms behave as i f Langmuir-type suggests that there e x i s t s some kind of compensating Interaction. • • x : cation s i t e s § i O : anion s i t e s X V I2 I I - Fig. 17 (100) face -- 62 -T h e r e f o r e a model i s examined i n which the n e a r e s t neighbour anion-anion i n t e r a c t i o n ( d e s i g n a t e d 1 i n P i g . 17) i s taken to be r e p u l s i v e between l i k e s p e c i e s whereas the second n e a r e s t neighbour one (2 i n F i g . 17) i s taken to be s l i g h t l y a t t r a c t i v e . The model then has some resemblance to t h a t of an i m p e r f e c t l a t t i c e gas, i n which the e f f e c t i v e s i z e of each "molecule" i s g r e a t e r than t h a t of one anion s i t e , but the molecules, as u s u a l , a t t r a c t each o t h e r . I t may be noted t h a t some of the a d s o r p t i o n 22) isotherms d i s c u s s e d , f o r example, by de Boer f o r i m p e r f e c t l a t t i c e gases show the same p e c u l i a r i t y of the r e a c t i o n Isotherms i n t h i s work i n t h a t have a s p u r i o u s resemblance to Langmuir isotherms. In F i g . 17, i f the 1st C l " i o n exchanges B r " on the s i t e marked with a double c i r c l e , then the second one tends to s i t on s i t e s marked with a b l a c k c i r c l e , and there Is l i t t l e chance to exchange Br~ on white c i r c l e s i t e s . T h i s may g i v e r i s e to about h a l f a B.E.T. s u r f a c e o f apparent exchangeable amount of Br a t low temperature. The f i r s t attempt to c a r r y out d e t a i l e d s t a t i s t i c a l mechanical c a l c u l a t i o n has been done on the b a s i s of the conven-t i o n a l q u a s i - c h e m i c a l approximation c o n s i d e r i n g o n l y p a i r s of anions as b a s i c u n i t s . A d e t a i l e d d i s c r i p t i o n i s i n Appendix I I I . ( i n c i d e n t a l l y , the Bragg-WIlllams approximation does not handle a second n e a r e s t neighbour i n t e r a c t i o n p r o p e r l y ) . But l a t e r , on r e c o g n i t i o n t h a t the above approximation has no power to d i f f e r e n t i a t e b l a c k c i r c l e s i t e s from white ones, because the 1s t C l " o c c u p i e s b l a c k s i t e s o r white s i t e s i n a e q u a l l y - 6 5 -probable manner, a g e n e r a l i s e d q u a s i chemical approximation due 6 ) to H i l l i s used, i n which the u n i t i s taken as a square of f o u r anion s i t e s i n c l u d i n g a c a t i o n s i t e In the middle. The o u t l i n e of the treatment i s as f o l l o w s : ( D e t a i l e d d e s c r i p t i o n i s In Appendix I V ) . C o n s i d e r a two di m e n s i o n a l mixture of B and C, the number of each being and N^ (Ng + NQ = N t o t a l anion s i t e s ) , then the. t o t a l p a r t i t i o n f u n c t i o n of the system can be w r i t t e n under assumptions ( l ) and (2), Q - (q B) % (q°)NGr , 1 3 C _ ( 6) P=[ E £ „ £ S ( N V N 2 S N " N , , N N ) exp(-W/RT), U1 N 2« N 2" N^ 1 d d * B 0 where q°, q° are the i n n e r p a r t i t i o n f u n c t i o n s of i s o l a t e d B O s p e c i e s , P i s the c o n f i g u r a t i o n a l p a r t i t i o n f u n c t i o n , Nj_'s are number of u n i t s which B and C o c c u p i e s i n a c h a r a c t e r i s t i c manner, and W i s the i n t e r a c t i o n energy g i v e n i n eq. (IV-9) i n Appendix IV Prom eq ( l V - 1 0 ) , eq. ( 6 ) becomes, Q - ( q / B (q 0) NC f l = V ^ T l . " - ( 7 ) where and are the p a r t i t i o n f u n c t i o n f o r the assembly of N f i of pure B, and NQ of pure C r e s p e c t i v e l y . On d i f f e r e n t i a t i n g eq. (7) with .respect to Ng and N c, one o b t a i n s chemical . :> p o t e n t i a l s f o r B and C (jUL^ and jbiG) r e s p e c t i v e l y . - j U B A T = - l n q B + b i n R L / ^N B , (8) -jULcAT = l n q c + d i n ^/b^C - 64 -where q , q are f u n c t i o n s o f T onl y , and the second terms on B C the r i g h t are f u n c t i o n s o f T and Ng. The r e a c t i o n c o n s i d e r e d i s o f the type, • • HC + B 5 HB + C . The e q u i l i b r i u m c o n d i t i o n , M B + M H C = M C + M H B > y i e l d s , assuming that the gas phase i s p e r f e c t , d l n Q / 6 ^ - dlnrydNg = l n R p / C ( T ) , (9) where JC (T) i s a f u n c t i o n o f temperature o n l y . With g i v e n X c ( = N c/N), g i v e n T, and t r i a l v a l u e s f o r i n t e r a c t i o n e n e r g i e s , R p K ( T ) Is computed a c c o r d i n g to eq. (9). The computed isotherms are p l o t t e d i n F i g . 1 8 t o g e t h e r with experimental ones. As can be seen i n F i g . 1 8 , a reasonable f i t was obtained f o r isotherms a t 0 ° C and at 9.4°C, but v a r i a t i o n w i t h temperature i n the apparent X m i s much too s m a l l to cover the whole range ( 0 . 5 ~ 3 ) found i n experiments (eg. exp e r i m e n t a l data f o r 5 0 ~ 5 3 ° C are matched by computed i s o t h e r m f o r 9727°C). There i s no p o s s i b i l i t y at a l l f o r the apparent X to exceed 1 , the amount co r r e s p o n d i n g m to the s u r f a c e area, i n t h i s model. The experimental f a c t t h a t both isotherms f o r 0 ° C and 9.4°C seem to head f o r about h a l f a B.E.T a u r f a c e area and the l a t t e r behaves as i f Langmuir type whereas the former shows a l i t t l e anomaly a t the e a r l y stage o f the exchange (as i n F i g . 1 0 A and IOC), although p l o t s are Computed ( Revised Q-C ) and Experimental Isotherms. -computed isotherm. 2 8°C O 0 $ - 9 . 4°C 0 - 0°C e x p e r i m e n t a l isotherms. o o Temp, f o r computed l i n e s 9 7 2 7 °C . - 66 -s c a t t e r e d , might suggest i n t h i s range (X<0 . 3 ) t h a t the mixture of B r - and C l " i s an o r d e r - d i s o r d e r mixture i n the outermost l a y e r , and the t r a n s i t i o n temperature i s somewhere i n between .„ 0°C and 9.4°C. Indeed, the o r i g i n a l i n t e n t i o n to set up the present model was to g e n e r a l i z e the c o n v e n t i o n a l treatment o f an o r d e r - d i s o r d e r mixture i n the hope of e x p l a i n i n g the whole range s t u d i e d . (b) Quasi harmonic theory o f v i b r a t i o n s (assumption (2) i s removed). The s t a t i s t i c a l mechanics o f the system with assumption (2) removed Is a d i f f i c u l t one. L e a d b e t t e r e t . alT^, however, 8) used a "Quasi harmonic theory o f v i b r a t i o n " developed by S a l t e r 9) and Leadbeter e t . a l . ' to analyze the vapour p r e s s u r e o f the s o l i d s o l u t i o n o f a l k a l i h a l i d e s . They wrote down f o r m a l l y the t o t a l p a r t i t i o n f u n c t i o n o f the assembly c o n s i s t i n g o f Ng of B and N c of C as. 3N R 3N C 3N 0= (N// N B(N C/ ) TT 'Qi TT q«, = (N.'/N R|N cl) TT q k , V i = i 1 0=1 . a k=l — — ( 1 0 ) where i s the p a r t i t i o n f u n c t i o n of a s i n g l e harmonic o s c i l l a t o r o f frequency CO ^ . D e r i v i n g Helmholtz f r e e energy i n 3N -1/3N a c o n v e n t i o n a l manner, they i n t r o d u c e d LO = ( TT 60 *) , s i = l 1 a geometric mean o f a l l f r e q u e n c i e s and c o n s i d e r i n g that OJ i s S a f u n c t i o n o f composition, evaluated, d In odg/ o I H X Q i n terms o f composition to f i t the exp e r i m e n t a l data. - 6? -T h i s method c o u l d be a p p l i c a b l e f o r the pr e s e n t work, but the mechanism which g i v e s r i s e to the be h a v i o r o f the e v a l u a t e d q u a n t i t y ((b In od / 6 l n x R - blnLO / c Y l n x ) f o r the pr e s e n t g -D g C work, see Appendix V) i s i n no way i n d i c a t e d by such a treatment. (c) Three l a y e r theory (assumption ( l ) removed). As mentioned i n Chap. 1, some anomaly of subsurface l a y e r s of a l k a l i h a l i d e s has been r e p o r t e d . In p a r t i c u l a r , T a k a i s h i 11) l ? ) e t . a l . ' d i s c u s s e d the d i f f e r e n c e i n k i n e t i c behaviour between RbCI and KC1 towards exchange r e a c t i o n with gaseous C l g i n terms of the d i f f e r e n c e i n i o n i c r a d i i of c o n s t i t u e n t components. In the pr e s e n t case, s i n c e the d i f f e r e n c e i n the i o n i c r a d i i o f the c o n s t i t u e n t components i s l a r g e r than those o f above two a l k a l i h a l i d e s and moreover the KBr powder was prepared by vacuum s u b l i m a t i o n , i t seems probable t h a t the exchanged C l i s not o n l y i n the outermost l a y e r but i s populated to some ext e n t i n subsurface l a y e r s , even i n the range where the t o t a l exchanged amount C g i s l e s s than t h a t c o r r e s p o n d i n g to B.E.T. s u r f a c e area. Indeed, the ex p e r i m e n t a l f a c t t h a t even a f t e r the bend ( C e > l x B E T s u r f a c e ) , the exchange r e a c t i o n i s very q u i c k f o l l o w e d by a slow p r o c e s s , i s not a g a i n s t the s t a t e -ment above a t a l l . In f a c t , the d i f f e r e n c e i n r a t e , one r e a c h i n g e q u i l i b r i u m w i t h i n 10 min. and the oth e r g i v i n g , . D = 3 X l O " 9 exp(-20,000/RT) cra 2/sec (see Appendix I I ) c o u l d w e l l be an a l y s e d i n terms o f the d i f f e r e n c e i n a c t i v a t i o n e n e r g i e s between one l a y e r and another, although which l a y e r a c t s as the - 6 8 -b a r r i e r i s not known y e t . But the f a c t t h a t the h i g h e s t value f o r the apparant C m found e x p e r i m e n t a l l y i s about three times the B.E.T s u r f a c e area, might suggest that the b a r r i e r e x i s t s between the t h i r d and the f o u r t h l a y e r . The formal problem now i s to e v a l u a t e under v a r i o u s assumptions the p a r t i t i o n f u n c t i o n o f a l a t t i c e three l a y e r s t h i c k c o n s i s t i n g of Ng of B and N Q o f C s p e c i e s , and to equate the chemical p o t e n t i a l d i f f e r e n c e of B and C s p e c i e s i n the assembly to t h a t o f HB and HC i n the gas phase. But f i t t i n g curves computed to the e x p e r i m e n t a l ones i s an enormous task, s i n c e the number of a d j u s t a b l e parameters such as i n t e r a c t i o n e n e r g i e s i s more than three times t h a t o f the s i n g l e l a y e r model. T h e r e f o r e , d i s c u s s i o n i s r e s t r i c t e d to the case where the popu-l a t i o n of Cl'exchanged i n the second and the t h i r d l a y e r i s no l o n g e r n e g l i g i b l e , f o r which purpose a simple three l a y e r model i s i n t r o d u c e d , with c o n d i t i o n s which correspond c l o s e l y to an analogue of the d e r i v a t i o n of the B.E.T. equation (but with s u c c e s s i v e l a y e r s of C l b u i l d i n g up inwards from the s u r f a c e ) . T h i s model g i v e s v a l u e s of the apparent C m with a minimum of one l a y e r , but a maximum of a t l e a s t e i g h t l a y e r s i n p a r t i c u l a r c o n d i t i o n s . D e t a i l e d d e s c r i p t i o n i s g i v e n i n Appendix VI and computed isotherms t o g e t h e r with experimental ones i n F i g . 1 9 . As i n F i g . 19 , a good agreement i s o b t a i n e d f o r isotherms 50°C, 72°C, 86°C. F u r t h e r p u r s u i t combining the r e v i s e d q u a s i - c h e m i c a l treatment f o r one l a y e r and the p r e s e n t model might p o s s i b l y e x p l a i n the whole range of the experimental data. -69 -- P i g . 19 Computed and Experimental Isotherm -— X — experimental p l o t — • ~•computed p l o t * / K 2 / K l = 0 - 5 ( 8 6 - 8 7 . 3 ° C ) X or X K2 / K i = 0 - 5 (71.'8^ 72.5°C) t o t Kg/K^O.l 0 . 0 7 5 0 . 0 5 ( 5 0 ~ 5 3 ° C ) f - 70 -The system s t u d i e d thus seems more complicated than the appearance of the simple e m p i r i c a l e q u a t i o n ( l ) might at f i r s t have l e d one to suppose. To continue the e l u c i d a t i o n of the pre s e n t system, f u r t h e r s t u d i e s such as NMR of C l " ( i f p o s s i b l e ) with v a r i o u s amounts exchanged, and k i n e t i c s of the qu i c k r e a c t i o n w i t h i n 10 min., are n e c e s s a r y . Appendix I Su r f a c e Area of KBr and Chemical S i n t e r i n g - 72 -Rudham p o i n t e d out th a t the presence o f HC1 o r C l 2 a c c e l e r a t e s the s i n t e r i n g o f an evaporated f i l m o f NaCl. Since i n the presen t study, the evaporated powder o f KBr i s exposed to HC1 and HBr f o r a t o t a l o f some 20 h r s . a t temp, up to 87°C, i f the s i n t e r i n g of KBr continues throughout because o f the presence of HC1 and HBr, the i n t e r p r e t a t i o n o f the r e s u l t s becomes very c o m p l i c a t e d because the r e g i o n i n which exchange i s t a k i n g p l a c e has a c o n t i n u a l l y changing s t r u c t u r e . The s i n t e r i n g behaviour was checked as f o l l o w s . KBr powder was made and c o l l e c t e d i n the powder c o l l e c t o r ( F i g . 3 ) , i n vacuo (^10"^mmHg) and s e a l e d o f f a t R. The c o l l e c -t o r w i t h powder was at t a c h e d to the BET apparatus ( F i g . I - l ) a t T, and the BET system was evacuated a t room temperature down to —6 10 mmHg befo r e the break s e a l was broken. The BET s u r f a c e was measured i n a c o n v e n t i o n a l manner with Kr a t l i q . Ng temp, and wit h He f o r d e t e r m i n i n g a dead volume. The r e g i o n p a r t i t i o n e d w i t h stop-cocks B, C and D to g e t h e r with the manometer i n F i g . I - l i s a do s i n g d e v i c e , the volume o f which was c a l i b r a t e d g r a v i m e t r i c a l l y to be 2.40 ml e x c l u d i n g paths i n stopcocks. Upon measuring isotherms the l i q u i d n i t r o g e n t r a p was p l a c e d between the c o l l e c t e r and the BET system to a v o i d mercury and grease ( K e l - F grease.was used f o r every stopcock) vapor from d e p o s i t i n g on the powder KBr. The blank isotherm i s shown i n F i g . 1-2, which i s a t y p i c a l i sotherm f o r porous m a t e r i a l . Since the c o l l e c t o r had been s u b j e c t e d to c o n s i d e r a b l e g l a s s b l o w i n g , i t i s expected to have a somewhat porous s u r f a c e . - 73 -Then the KBr powder was t r e a t e d with HC1 or HBr at d i f f e r e n t temperature and pr e s s u r e and the BET s u r f a c e was measured a f t e r each treatment. T y p i c a l BET p l o t s are i n F i g . I-3. Kr monolayer value ( n m ) and C are t a b u l a t e d i n Table I - l to g e t h e r with the HC1 and HBr treatments. Poor accura c y i s due to the f a c t t h a t a f t e r the l a s t BET measurement the c o l l e c t o r was s e a l e d o f f a t p o i n t S and i t was a t t a c h e d to the r e a c t o r a t p o i n t P i n F i g . 2, so t h a t the blank BET measurement c o u l d not be c a r r i e d out with the same c o l l e c t o r as the one with the KBr powder i n i t . Kr monolayer value ( n m ) was p l o t t e d a g a i n s t time i n F i g . 1 -4, which shows t h a t a f t e r a few hours exposure to HC1, s i n t e r i n g slows down and the s u r f a c e s t a y s constant w i t h i n experimental e r r o r up to some 40 hours. T h i s p a r t i c u l a r KBr powder was used i n the run s e r i e s C. I t i s reas o n a b l e , then, to expect t h a t f o r a f u r t h e r 20 hours the s u r f a c e area s t a y s constant with a value about 1 0 + 1 m2/g. - 74 -P - F i g . I - l BET Apparatus -- 76 -- F i g . 1-3 T y p i c a l isotherm and BET p l o t -- 77 -Table I - l n m (xlO ^ mole) C s p e c i f i c s u r f a c e Area ( l ) F r e s h l y made KBr 22 . 0±1 3.3 34.2 (2) HC1 5.3cmHg-78°C+HBr5cm-+HC15.2cmHg22°C+HBr2 . 5cr r 78°C+HC17.0cmHi 22°C t o t a l 5. 5+HBr6.3 2hrs. cmO°C (3) BET measurement 7 . 4 ± 0 . 7 6 .6 12 . 0 (4) HC1 5.3cm at 25°C 24 . 5 h i s (5) BET 6 . 5 ± 0 . 7 6.2 1 0 . 6 (6) HC1 5.2cmHg a t 9 5 - 9 6 ° C ] 2hrs. (7) BET 5 . 7 ± 0 . 6 7 (8) HBr 5cmlig a t 25°C 0.3'hrs (9) BET 6.1± 0 . 6 7 .6 1 0 . 0 Appendix I I Exchange K i n e t i c s o r D i f f u s i o n o f anions - 80 -Procedure With 5 ~ 7 cmHg o f HC1 i n the r e a c t o r and with mixing a s s u r e d by a c o n v e c t i o n h e a t e r and the mixing arm Z i n P i g . 1, and P i g . 2 , a p a r t o f the r e a c t i o n gas was taken i n t o the dos i n g r e g i o n (manometer-F-E-D-C-B-H-K-G i n F i g . l ) , and c o l l e c t e d i n t o the sampler c o o l e d w i t h a l i q . Ng c o l d t r a p and a n a l y s e d by gaschromatograph. T h i s procedure was f o l l o w e d two to three times at i n t e r v a l s , and the l a s t sample was taken by c o l l e c t i n g the whole remaining r e a c t i o n gas d i r e c t l y from the r e a c t o r i n t o the sampler. R e s u l t s F i g . I I - l shows the amount exchanged, n , vs. time and P i g . I I - 2 shows the amount exchanged vs. square r o o t o f time. Numbers a t t a c h e d to curves r e f e r to the run-numbers i n s e r i e s C i n Chap. 2 . Except # 3 0 6 0 , and # 4 0 5 0 , which are runs a f t e r the bend, the amount exchanged reaches an e q u i l i b r i u m i n l e s s than 10 min. and st a y s constant w i t h i n experimental e r r o r a f t e r t h a t . F o r runs # 3 0 6 0 and # 4 0 5 0 , a q u i c k change was f o l l o w e d by a slow p r o c e s s , which seemed to obey ft law as i n F i g . I I - 2 . D i f f u s i o n c o n s t a n t s were c a l c u l a t e d from s t r a i g h t l i n e s i n F i g . I I - 2 , and p l o t t e d i n F i g . I I - 3 a g a i n s t l / T , from which, D = 3 * 1 0 " 9 exp ( - 2 0 , 0 0 0/RT) (cm 2/sec), was o b t a i n e d . F i g . I I - 3 a l s o shows apparent d i f f u s i o n c o e f f i c i e n t s c a l c u l a t e d from n g vs. J~t l i n e s f o r runs which had not passed the bend. These l i n e s c o u l d have been drawn h o r i z o n t a l , w i t h i n l i m i t s o f ex p e r i m e n t a l e r r o r , so these p o i n t s i n d i c a t e the l i m i t - 81 -of a c c u r a c y i n measuring D. C o n c l u s i o n (1) Below the bends i n e q u i l i b r i u m isotherms, an e q u i l i b r i u m i s reached w i t h i n 10 min. (2) A f t e r p a s s i n g the bends, some k i n e t i c p r o c e s s f o l l o w i n g a square r o o t law i n time s e t s i n , which i s very l i k e l y to be a d i f f u s i o n i n t o a bulk, and g i v e s d i f f u s i o n constant D = 3 X 1 0 ~ 9 exp ( -20 ,000/RT) . (3) Whether t a k i n g a p a r t o f the r e a c t i o n gas i n t o the dosing r e g i o n and then c o l l e c t i n g i t i n t o the sampler or whole gas d i r e c t l y i n t o the sampler does not matter w i t h i n e x p e r i m e n t a l e r r o r . In o t h e r words, the l a t t e r method, which was mostly a p p l i e d i n the e q u i l i b r i u m study s e c t i o n , does not upset the e q u i l i b r i u m s i t u a t i o n . | 4 . 9 4 . 8 4010(86°C) "(X10""6mole) -4- 5020 (86* C ) - F i g . I I - l n e vs. time 4050(86°C) Q 3050(71.8° C) 3 0 6 0 ( 7 1 . 8°C) 1020 ( 3 0 ° C) 5Q 100 reaction time (min) 0 - 83 -0 5 . 0 1 0 . 0 (min) 1 / 2 - 84 -l o g -21 D - F i g . I I - 3 l o g D vs. l / T #4050 ft 20.60 0 #4010 0 # 4050 - 2 2 ^ 0#3O5O 6 0. & 1 0 2 0 2 . 5 3.0 3 . 5 x 1 0 " 5 1/T(l/°K) Appendix I I I Quasi-Chemical Treatment - 86 -Consider a two-dimensional square l a t t i c e with N s i t e s , of which Ng i s occupied by B species and NQ by C species (N = Ng + N Q ) . In the figure below, there are two kinds of anion-anion , interactions designated ( l ) and ( 2 ) . Q , ^ Q ^ Q O : anion s i t e s , t o t a l N . J(2) X : cation s i t e s . X O X O X; Interaction ( l ) ; N I ' anion-anion d i r e c t O X" Xi X O i n t e r a c t i o n . / | ^ Interaction (2) ; . x cf x No X i anion-anion through I 1 • • cation i n t e r a c t i o n . O X O X O T o t a l p a r t i t i o n function, Q, i s then written as, Q = (qg)% (qg) N C r > — — ( I I I - l ) P= Jexp(-W/kT) = ^ T2 ^ e x p ^ A T ) ' J]exp(-W 2AT) ( I I I - 2 ) W = wx + w2 , (III - 3 ) W l = NBB WBB + NBC WBC + N C C WCC * — - ( I I I - 4 ) W 2 = NBB' WBB* + NBC' WBC' + NCC* W C C ' ' — ( H I - 5 ) - 87 -where : qg and q c f w T and T 1 1 1 P and Wg N B B e t c . wBfi e t c . NBB' E T C -wBB- e t c . i n n e r p a r t i t i o n f u n c t i o n s of i s o l a t e d B and C, c o n f i g u r a t i o n a l p a r t i t i o n f u n c t i o n , c o n f i g u r a t i o n a l i n t e r a c t i o n energy, c o n f i g u r a t i o n a l p a r t i t i o n f u n c t i o n o f k i n d ( l ) and (2) i n t e r a c t i o n , c o n f i g u r a t i o n a l I n t e r a c t i o n energy o f k i n d ( l ) and (2) i n t e r a c t i o n , number of p a i r s B-B e t c . of k i n d ( l ) , i n t e r a c t i o n energy o f a p a i r B-B e t c . o f k i n d ( l ) , number of p a i r s B-B e t c . o f k i n d (2), i n t e r a c t i o n energy of a p a i r B-B e t c . o f k i n d (2), Let Z be number of neighbors connected with k i n d ( l ) i n t e r a c t i o n and Z> with k i n d (2) i n t e r a c t i o n ( i n t h i s case Z = Z' = 4), and and ZN B = 2N B B + N R n , BC ZN C - 2N C C + N B C Z'N R = 2N ' + N ' , B BB BC Z'N C = 2NCC' + N B C . ( I I I - 6 ) ( H l - 7 ) Eq. ( I I I - 6 ) and ( I I I - 7 ) l e a d eq. ( I I I - l ) * t o , Q = Q BB E g ( N B C ) [exp( W l/2kT)] NBC NBC X £ g'(N •) [exp(w /2kT) BC INBC' ( I I I - 8 ) - 8 8 -N B N B where = j^qB exp(- Zw B B + Z'w B B')/2kT j i s the p a r t i t i o n f u n c t i o n o f the assembly of N B of pure B , and 2w, Y = W B B + W C C " C"BC W 2 = W B B ' + WCC' " 2 W B C ' * ( H I - 9 ) I f w i s n e g a t i v e , which means t h a t w g B + w^ i s more n e g a t i v e than 2w , l i k e s p e c i e s are a t t r a c t i v e and u n l i k e s p e c i e s BC r u p u l s i v e . S i m i l a r l y i f w-^  i s p o s i t i v e , l i k e s p e c i e s are r e p u l s i v e and u n l i k e s p e c i e s are a t t r a c t i v e . A c c o r d i n g to the c o n v e n t i o n a l q u a s i - c h e m i c a l approximation, the s t a t i s t i c a l weights g ( N B C ) and S ' ^ - g g ' ) i n eq. ( I I I - 8 ) can be w r i t t e n as, g ( N B C ) = P1 C ] L , Px = ( Z N / 2 ) f / ( N B B ) l ( N c c ) / [ ( N B C / 2 ) / ] 2 , P2 > (ZN/2)//(N B B- )/ (N C C- ) / [ ( N B C ' / 2 ) / ] 2 > (111-10) where C.^ and Cg are n o r m a l i z a t i o n f a c t e r s determined by the r e l a t i o n , £ g ( N „ R ) - ^ S ' C N R C ' ) = N / / N / N p/ . — ( 1 1 1 - 1 1 ) N ' B C j7 , ° B C ' ~ " - - ' " B ' " C N B C N B C - 89 -Instead of summing the l e f t hand s i d e o f eq. ( i l l - l l ) , i t i s r e p l a c e d by the maximum.term In each sum as u s u a l , which leads t o , , C1C2 = ( N V / N g / N c / ) 1 - ^ 2 ' ) (111-12) Eq. ( I I I - 8 ) i s now w r i t t e n as, Q = q " ° 0XC2 E A f e x p t w . / a k T ) ] ^ T. A, r e x p ( w 2 / 2 l ( T ) l N B ? ' NBC (111 -13 ) The sums i n eq. ( I I I - 1 3 ) are then r e p l a c e d by t h e i r maximum terms, and by v i r t u e o f eq. ( I I I - 6 ) , ( I I I - 7 ) , ( I I I - I O ) and ( H I - 1 2 ) the t o t a l p a r t i t i o n f u n c t i o n , Q, now becomes Q = q ^ B q ^ C C x C a P^Ngc*) [ exp ( W l/2kT) ] NBC XP 2< NBC'*) [exp(w 2/2kT) ] NBC'* , w i t h N B C* = Z N ( 1 - [ l - exp ( -w A T ) ] , '* =. Z N ( l - / 3 2 ) / 2 [ l - e x p ( - w 2 A T ) ] , 1/2 N. BC where = 1 - 4 X BX G [ l - e x p ( - W l A T ) ] J3 2 = 1 -^ XB XC [ 1 _ E X P ( - w 2 A T ) ] 1/2 (111 -14 ) V — ( 1 1 1 - 1 5 ) where X B = N B/N and X C = N C/N. - 90 -The type of the r e a c t i o n considered here i s HC + B ^ HB + C , and at e q u i l i b r i u m , , ( M B - M C ) A T - ( M H B " ^ H C ) A T = ( M ° H B - : M H C ) A T + ^ ( P m / ? K C ) = i n [(A 0 H B/A.°HC)(PHB/ PHC)] A (111-16) where ^ B and j_i c are chemical p o t e n t i a l s of B and C i n the assemply and // H B and H C are those of gas phase, which i s assumed to be a p e r f e c t gas mixture. Prom the r e l a t i o n s , - M B A T = ( 6 l n Q / d N B ) N c j T , - Me A T = ( d i n Q / 6 N C ) N •;• ; , B J i t f o l l o w s t h a t , In ( A . ° H B / A 0 H c ) ( q B A c ) ( P H B A H c ) = l n [ R P ^( T ) ] = (Z/2 + Z ' A - l ) l n ( X c A B ) + ( z A)ln [(J31 - 1 + 2 X B ) / ( / ? 1 - 1 + 2X C) ] + (Z>A)ln[ {J32 - 1 + 2X B ) / ( /3 2 - 1 + 2 X C ) ] , — (111-17) - 91 -where O . Q /<(T) = ( AH B/AH (0 ^ B ^ c ) ' a n d l t l s a f u n c f c l o n o f temperature only. , For given X c, and t r i a l values f o r w.^  and wg, eq. ( l H - 1 7 ) can be computed. Appendix IV Revised Quasi-Chemical Treatment - 93 -C o n s i d e r a two-dimensional square l a t t i c e with N s i t e s , o f which Ng i s occupied by B and N c by C s p e c i e s (N = N B + N Q ) . As i n the f i g u r e below take the square as a u n i t (e.g. ABCD). O X anion s i t e s t o t a l N c a t i o n s i t e s Designate the i n t e r a c t i o n s between anions through c a t i o n by w's with prime and those along the s i d e of the square byzw's without prime, as the f i g u r e below. Q*.._2-W__^jpQ There are s i x p o s s i b l e manners f o r •w'' s p e c i e s B and C to occupy the u n i t /' • square, which are t a b u l a t e d i n the <~f o t a b l e below (Table I V - l ) . Then, N = N Q + N x + N 2* + N 2" + + Njj , - ( I V - l ) 4 N B = 4N 0 + 3N X + 2 N 2 ' + 2 N 2 " + N ^ , ( l V - 2 ) 4N Q = N x + 2 N 2 ' ' . . + 2 N 2 " + 3N^ + 4N 4 . (IV-3) Only two of the above three equations are independent. - 9 4 -- Table IV - 1 -Type of square No. of equiv. arrang. No. of u n i t s of t h i s type Energy of the u n i t 0 B B B B 1 N o 4 W B B + 2 W B B ' • 1, B C .B-B 4 N i 2 W B B + 2 W B C + WBB' + WBC' 2 I B B C. c 4 V W B B + WCC + 2 W B C + 2 W B C ' 2" B C C B 2 V 4 W B C + WBB' + WCC' C C B C 4 h 2 W C C + 2 W B C . + WCC' •+ w B C' 4 C C • c c 1 N 4 4w + 2W 1 CC CC H i l l ' s approximation i s that t h e " s t a t i s t i c a l weight f o r the arrangement of N Q of type 0, N-j_ of type 1 e t c . i s given by, g ( N r N 2», N2", N 5, Nfi, N c) = / i i " , (IV - 4 ) P = N B ! N c l / [ N Q J I C N X A ) ! } 4 { ( N 2 ' / 4 ) ! } 4 { ( N 2 " / 2 ) / } 2 * { ( N 5 A ) ! y 4 N4I] , -(IV -5 ) - 95 -R = N B/N C//(N B+ N c)/ (IV-6) The t o t a l p a r t i t i o n f u n c t i o n i s now w r i t t e n as, (IV -7 ) r - ' t g exp 1 3 - [ V % B + 2wBB.'> + N (2w + 2w + w '+w 1 ) 1 BB BC BB BC + V ( W B B + WCC + 2 W B C + 2 WBC') + N 2 * ' ( 4 w B C + w B B< + w c c » ) + N 3 ( 2 w c c + 2 W b c •+ w c c - + w B C.) cc (IV-8) = £ S exp(-W/kT). N _~N_ 1 J? By v i r t u e o f eq. (IV-2) and (IV-3), the t o t a l i n t e r a c t i o n energy W can be w r i t t e n as, W = V 4 w B B + 2 W W R , ) + N - ( 4 W H C + 2 W P P ' ) BB c v " " c c CC - N (w + w>/2) - N 1(w + w») - N 2"(2w) - N^(w + w«/2)* ( I V - 9 ) where w = w B B + w c c - 2w B C , w . = w B B » + wcc- - 2w B C» . - 96 -Then the t o t a l p a r t i t i o n f u n c t i o n Is now, (IV-10) r: -• i g exp 1 3 N-^w+w'A) + N '(w+w»-) + N 2"(2w) + N^' (w + w«/2)]/kT (IV-11) The sum i n ( i V - l l ) i s then r e p l a c e d by i t s maximum term. The maximum term appears a t c o n d i t i o n s , N Q = N B - 0 % + 2N 2' + 2N 2"+N N^ = N c - ( N x + 2N2-» + 2N 2" + 3N 3 ) / 4 , N , - 4(M 0) 5 /V 4)iA a* , a. = 2 ( y 1 / 2 < v 1 / 2 / 3 2 , V = 2(N 0) 1''' 2(N 1 () 1/ 2r 2 , N j - 4 (N 0) 1/ 4 ( N//' ( oe , (IV-12) where exp (w + w'/2)/kT , fl = exp (w + w' )/kT , 7= exp (2w/kT) . - 97 -S o l v i n g eq. (IV-12) one can p a r t i a l l y d i f f e r e n t i a t e eq. (IV-10) with r e s p e c t to and N c to o b t a i n jX^ and JXQ r e s p e c t i v e l y . In the s i m i l a r manner as i n Appendix I I I , the isotherm e q u a t i o n Is o b t a i n e d as, (dm r V d V ^ - ( dm r i / o N B ) N c j T - i n [ (P HB / P H c)^ 0HE / A 0H C ) ( V qC }] = l n [ R P K < T > ] • ( I V - i 3 ) By v i r t u e o f eq. (IV-12), the Isotherm i s f i n a l l y w r i t t e n as, m [ RpK(T)] = m(xc/xB) + ( i / 2 ) m ( N 0 / N 4 ) . — ( i v - i 4 ) In computation with IBM 360/67, eq. (IV-12) was s o l v e d n u m e r i c a l -l y and then Rp X. (T) was c a l c u l a t e d from eq. (IV-14) f o r g i v e n X , w, and w'. The program i s g i v e n below. - 98 -Computation R e s u l t s F i g . I V - 1 shows t y p i c a l computed p l o t s of l / R p K X c vs. l / R p K , wi t h w'/w = - 1 . 7 5 and w/kT from 0 .01797 to 0 . 5 3 9 and X c from. 0 , 1 to 0 . 5 . There i s no d i f f i c u l t y f i n d i n g the r e g i o n where p l o t s show a s t r a i g h t l i n e . As w/kT becomes h i g h e r , (or temperature becomes lower), the i n i t i a l p a r t o f l / R p K X g vs. l/RpK p l o t s tends to d e v i a t e upwards from the s t r a i g h t l i n e , and at even h i g h e r w/kT value, a minimum appears. R e f e r r i n g back to F i g . 10A and IOC, t h i s d e v i a t i o n from the s t r a i g h t l i n e i n computed p l o t s seems to correspond to the exp e r i m e n t a l p l o t s f o r 0°C. With t h i s i n mind, the value s of. w' and w were searched so t h a t l / R p y C X c vs. l / R p K p l o t s f o r 9-4°C c o u l d g i v e a s t r a i g h t l i n e i n the range XQ = 0 . 1 2 5 ~ 0.3 (the range covered e x p e r i -m e n t a l l y , c f . F i g . 14) with the slope y i e l d i n g X-m = 0 . 5 , whereas those f o r 0°C c o u l d g i v e a concave curve at the range X^ = 0 . 0 5 /~ 0 . 1 2 5 and a s t r a i g h t l i n e with the slope y i e l d i n g = 0 . 5 a t the range XQ = 0 . 1 2 5 ^ 0 . 2 2 5 . Some reasonable values o f w and w ' thus found are i n Table IV - 2 (see F i g . I V - 2 ) . - Table IV - 2 -w 1 /w w(cal/mole) X m of 9.4°C - 0 . 8 500 0.64 - 0 . 5 650 0.49 -0.3 1000 0.42 F i n a l l y w'/w = - 0 . 5 , w. = 650 cal/mole was chosen. With these - 99 -v a l u e s , temperatures, which g i v e s t r a i g h t l i n e s i n l / R p K X vs. l / R p K p l o t s with s l o p e s same as those o b t a i n e d f o r 28°C, and 5 0 ~ 5 3°C e x p e r i m e n t a l l y , were found to be 1 ,000°K and 10 ,000°K r e s p e c t i v e l y (see F i g . I V - 3 ) . K ( T ) , the s h i f t i n g f a c t o r i n R p s c a l e , was then o b t a i n e d so t h a t one p o i n t on the computed is o t h e r m merges to th a t on the exp e r i m e n t a l one at the same X Q v a l u e . K ( T ) ' s are t a b u l a t e d i n Table I V - 3 . - Table IV - 3 -Computed Temp. Corre sponding e x p e r i m e n t a l Temp. K (T) S 0°C 0°C 6 . 6 *10 4 5 . 5 * 1 0 ~ 2 9 .4°C 9 .4°C 2 . 6 8 x i o 4 6 . 2 * 1 0 ~ 2 727 °c 28 °C 8 . 0 X 1 0 2 0.75 9727°C 5 0 - 53 °C 2 . 2 5 x l 0 2 1.81 With these K ( T ) ' S , f i n a l l y , computed isotherms were obta i n e d , which are shown i n F i g . IV - 4 t o g e t h e r with experimental ones. In F i g . I V - 5 , 1 A p X c vs. l / R p p l o t s are shown f o r 0°C and 9 .4°C. Agreement f o r 0°C and 9 .4°C i s q u i t e good. As f o r o r d e r - d i s o r d e r , i t i s d i f f i c u l t to d e f i n e e x p l i c i t l y the o r d e r - d i s o r d e r parameter i n the pr e s e n t model, but S = N 2'/N 2" (see Table I V - l ) , which i s expected to be some i n d i c a t i o n o f o r d e r l i n e s s , are t a b u l a t e d i n Table I V - 3 . I f the system i s completely i n order there i s no arrangement o f type 2 ' , - 100 -t h e r e f o r e S = 0, whereas i f i n complete d i s o r d e r , S = 2, ( i n s t e a d o f 1 because the number of e q u i v a l e n t arrangements f o r type 2 ' Is 4 and t h a t f o r type 2 " i s 2 ) . As i n Table I V - 3 , the systems at 0°C and 9 . 4°C are very c l o s e to complete o r d e r whereas t h a t a t 9727°C i s almost i n d i s o r d e r . So there might not be a o r d e r -d i s o r d e r t r a n s i t i o n between 0°C and 9 . 4°C as was p o s t u l a t e d e a r l i e r . - 1 0 2 -- F i g . IV - 2 -J r± 0 0 . 0 1 0 . 0 2 l/R P>C - 103 -- F i g . I V - 3 ~ Computed l / R p X Q vs. l / R p w '/ w = - 0 . 5 . w = 650cal/mole 10.0 .0 ,000°K slope l / l ' 1 , 0Q£T K 1 /0 .72 5.0" 5 .0 10'. 0 1/Rp/C - 104 -- Fig..IV-4 Computed and experimental isotherms -* - computed isotherm. 0 - 50~53°C 0 - 28°C - A - 9.4°C ) experimental isotherms. - x - 0°C - 105 -Q : « 1 :_J i 5.0x i o 2 l/Rp - 106 -C TW0 KINDS 0F NEAREST NEIGHBORS INTERACTION Wl & W2 C REVISED Q-C TREATMENT BY T.L.HILL C PHB/PHC*QB/QC*L0HB/L0HC IS CALCULATED. DOUBLE PRECISION X,SI,X2,X3,X4,F ,F1,Y,XNEW DIMENSION WW(l0),AW ( l 0),XXNC (30),TT(l0),CM(l0) R=l.98719 READ 1,NWW,NA W,NTT,NXC,NNCM 1 FORMAT(512) READ 2,(WW(l),I=l,NWW) READ 2,(AW(J),J=1,NAW) READ 2 ,(TT(K),K=1,NTT) READ 2 ,(XXNC(L),L=1,NXC) READ 2,(CM ( L L ) , LL=1,NNCM) 2 FORMAT(8F10.0) , D0 1000 1=1,NWW W1=WW(I) D0 1001 J=1,NAW AW12=AW(J) W2=W1*AW12 . PRINT 3,W1 3 FORMAT (1H1, ' Wl = ',F10.2,' (CAL/M^fLE) ' ) PRINT.4,W2,AW12 4 FORMAT (IHO, 1 W2 = ',F10.2,'(CAL/M0LE)«,5X,1W2/W1 = l',F10 . 5 ) D^ 1002 K=1,NTT T=TT(K) . GT=1.0/T GRT=GT/R WTA=(Wl+0.5*W2)*GRT WTB(W1+W2)*GRT WTC=2.0*W1*GRT WKT=W1*GRT EA=EXP(WTA) EB=EXP(WTB) EC=EXP(WTC) EA2=EA*EA EB2=EB*EB EC2=EC*EC PRINT 5,T,GT,WKT 5 FORMAT(IHO, ' T = •,F10.3, 'DEGREE K',5X,'l/T = 1',E12.5,5X,'Wl/RT = !,F10 . 5 ) D^ 1003 L=1,NXC XNC=XXNC(L) XNB=1.0-XNC XNCB=XNC/XNB A1=XNC A2=EA2*(3.0*XNC-XNB) A3=(2.0*EB2+EC2)*(XNC-XNB). ' A4 =EA 2* (XNC -3 . 0*XNB) A5=-XNB - 107 -F1=-XNB X1=0.0 ~D0 800 JJ=1,NNCM K l = l 801 IF(K1.GT.20) G0 T0 901 X=X1+CM(JJ) . X2=X*X X3=X2*X X4=X3*X F=A1*X4+A2*X3+A3*X2+A4*X+A5 I F ( F * F 1 . L T . 0 . 0 ) G0 10 800 X1=X K1=K1+1 F1=F G0 T0 801 800 CONTINUE M=l 50 CONTINUE I F ( M . G T . 3 0 ) G0 T0 900 X2=X*X X3=X2*X -X4=X3*X F=A1*X4+A2*X3+A3*X2+A4*X+A5 FDX=4.0*Al*X3+3.0*A2*X2+2.0*A3*X+A4 103 XNEW=X-F/FDX SNX=SNGL(X) SXNW=SNGL(XNEW) IF(ABS(SNX-SXNW).LT.1 .0E-6) G0 T0 60 X-XNEW M=M+1 G0 T0 50 60 CONTINUE X=XNEW X2=X*X Y=XNB/(X2+3.0*EA2*X+EA2/X+2.0*EB2+EC2) •SNX=SNGL(X) SNY=SNGL(Y) PRINT 11_,XNC,SNX,SNY,M 11 FORMAT(1H0,'NC = « ,F10 . 5 , 5X, 1 X = ' , E 1 2 . 5 , 5 X , •Y = 1 1 , E 1 2 . 5 , 5 X , I 2 ) I F ( X . L T . O . O . ^ R . Y . L T . O . O ) G0 T0 1003 X3=X2*X XN0=X2*Y XN4=Y/X2 XN01=XN0**0.25 XN02=SQRT(XN0) XN4l=XN4**0.25 XN42-SQRT(XN4) XNl=4.0*XN01*XN02*XN4l*EA*EA XN21=4.0*XN02*XN42*EB*EB XN22=2.0*XN02*XN42*EC*EC XN3=4.0*XN01*XN4l*XN42*EA*EA XN=XNO+XN1+XN21+XN22+XN3+XN4 - 108 -C THUS N0,.N1,N2',N2",N5,N4 WERE OBTAINED. PRINT 6, XN0,XN1,XN21,XN22,XN3,XN4,XN 6 F^RMAT(15X,»N0=',P8.5,3X,'Nl-',F8.5,3X, l»N2l=',P8.5,3X,'N22=«,F8.5,3X,•N5=i,P8.5,3X,'N4=',F8.5,3X, 2fN4=»,F8.5,3X,»N=»,F8.5) ZK0TAE=AL0G (XN'CB)+0.5*AL0G (XN0/XN4 ) RP=EXP(ZK0TAE) GRP=1.0/RP TA TE=GRP/XNC EQIL=RP*(XNC/XNB) PRINT 14,RP,GRP,TA TE,EQIL 14 F^RMAT(1H0?15X,,RP=',E12.5,5XJ'l/RP=',E12.5,5X, 11l/(RP*NC)=»,E12.5,5X,•K=PHB/PHC*NC/NB=',E12.5) G / T0 1003 900 PRINT 12,XNC 12 F^feMAT(lH0,'NC=».,P10.5,5X,'M = 30 ff K0ETA • ) G0 T0 1003 901 PRINT 13,XNC 13 FORMAT(1H0, LNC=',F10.5,5X,'Kl = 20 0 K0ETA') 1003 CONTINUE 1002 CONTINUE 1001 CONTINUE 1000 CONTINUE ST0P END Appendix V Quasi-Harmonic Theory of V i b r a t i o n s . - 110 -C o n s i d e r t h e a s s e m b l y o f N t h r e e - d i m e n s i o n a l h a r m o n i c o s c i l l a t o r s , o f w h i c h N a r e B k i n d s a n d N n a r e C k i n d s . T o t a l p a r t i t i o n f u n c t i o n o f t h e a s s e m b l y i s t h e n , I D I n 3N Q = ( N / / N B ] N C I ) TT q ± TT Qj = ( N ! / N B 1 N C ! ) TT q k , 1 X J—X K! — X ( V _ D where i s t h e p a r t i t i o n f u n c t i o n o f a s i n g l e h a r m o n i c o s c i l l a t o r o f f r e q u e n c y u)^ , q ± = e x p ( - h c o ^ k T ) / ^ - e x p ( - n l O j / k T ) ] . — — -(V-2) f o \ A c c o r d i n g t o S a l t e r , t h e H e l m h o l t z f r e e ' e n e r g y o f t h e a s s e m b l y i s t h e n w r i t t e n a s , 3N F = UB C + £ l n ( q i ) - k T l n ( N / / N B ! N c l ) 3N r = U B C - k T l n ( N / / N B l N C J ) + kT £ [ l n t n ^ / k T ) oo 0 + E ( " I ) " [ B 2 n / 2 n ( 2 n ) / } (n ^ / k T ) ^ ] , - — (v-3) where B g n i s B e r n o u l l i number, a nd U BQ i s t h e s t a t i c l a t t i c e e n e r g y o f t h e a s s e m b l y , a n d i s w r i t t e n a s U B C = N B U B + N C U C + A U ' — -(V-4) where Ug e t c . ; s t a t i c l a t t i c e energy o f pure B, A u j to a good approx imat ion , exper imen ta l en tha lpy of m i x i n g . - I l l -According to Leadbetter et. a l . ' f o r T ^ Q , where i s the Debye temperature f o r the heat capacity, (fi u^/kT) term can be n e g l i g i b l e . f o r KC1 i s 2?0°K and for KBr i s 177°K, so , th i s assumption i s applicable f o r the.present case. Then eq. (V-3) becomes F = U B C - kTln(N!/N B!N cl ) + 3NkTln (fi ^ g / k T ) , (V-4) 3N where ln 60 „. = ( Y In60.)/3N, the geometric mean frequency of & .. i = l 1 . the assembly. Since PV term can be neglected f o r s o l i d phase, MB = < ^ / ^ B ^ c T = U B + A U B + K T L N X B + 3kT I" Inn 60 /kT - d In 60. / d lnx 1 . , L g g C - l M C " ^ P / ^ N C > N B , T = UC + A U C + k T l n X C +"jkT. [ lnti 60g/kT - d ln 60 g/ d lnx f i ] , ~ — (V-5) where Au_, = (dAU/dN_) and i s the p a r t i a l molar enthalpy B B NQ,T of solution f o r B, and x f i = Ng/N. From ( M B " M C ) A T - ln(A 0 H B/A ° H c)( PHB / P H c ) ^ l n [ K ( T ) ( P H B / P H C ) U c / x B ) ] = ( A U B - A U c ) k T + ( d l n W / d l n x - d l n W p / d l n x J , (V-6) g B & U where lnX(T) = lnA°H B/A°H C - (U B - U ° )/kT, and Is a function of temperature only. - 112 -( A U - A U P ) can be e v a l u a t e d from isotherms (see Chap. 2 s e c t i o n 2), and with a assumption as to N , ( P ^ / P H C ) (x^/x HB C can be e v a l u a t e d from e x p e r i m e n t a l data. Thus d In ojg/b lnx-g - d In ^ g/tO l n x Q c a n b e e v a l u a t e d i n terms composition and temperature. Appendix VI Three l a y e r theory. - 114 -In the outermost l a y e r , the f r a c t i o n exchanged (X-j_) i s assumed to be g i v e n by a Langmuir-type e x p r e s s i o n , X 1 = ^ R p " 1 / ( l + K ^ p " 1 ) , (VI-1) with the p r e s s u r e r a t i o R p ^ a c t i n g as the analogue o f the pressure, i n a simple Langmuir a d s o r p t i o n s i t u a t i o n . Computations were made i n the f i r s t i n s t a n c e with = 1, f o r convenience ; s i n c e K-^  and Rp ^ occur o n l y as the product (K-^  Rp "*"), a d i f f e r e n t value of Kj_, can a f t e r w a r d s be accommo-dated simply by changing the s c a l e o f Rp"''" i n the computed r e s u l t s . For the. second l a y e r , i t i s assumed t h a t exchange takes p l a c e o n l y immediately beneath an anion which has a l r e a d y exchanged i n the f i r s t l a y e r . Of t h a t p a r t (X-^) of the second l a y e r which i s a v a i l a b l e f o r exchange, i t i s assumed t h a t the f r a c t i o n Xg exchanged i s a g a i n g i v e n by a Langmuir-type e x p r e s s i o n , X 2 = K 2 R p _ 1 / ( l + K g R p - 1 ) . — (VI-2) The c o r r e s p o n d i n g assumptions are made about the t h i r d l a y e r ; t h a t i t exchanges o n l y below p o r t i o n s of the second l a y e r a l r e a d y exchanged, and with X^ = K^Rp" 1 /(1+K^Rp - 1) . (VI-3) I f the process stops at the t h i r d l a y e r , the t o t a l amount - 115 -exchanged (expressed as a f r a c t i o n o f one l a y e r ) Is : -x t o t = + x 2 ( 1 + V ) • — - ' ( V I " 4 ) In the p r e s e n t computation, the number of independent parameters was f u r t h e r reduced by the assumption = K^, which i s analogous to another of assumptions made i n d e r i v i n g the B.E.T. eq u a t i o n . F i g . V I - 1 shows the r e s u l t s o f a number of computations with Kg/K-^ value s between 0 and 1, p l o t t e d i n the u s u a l Langmuir-type f a s h i o n ( l / R p X t Q t vs. l / R p ) . The Langmuir p l o t s are f a i r l y l i n e a r , with a tendency to s l i g h t l y convex cur v a t u r e at low Kg/K-^ and i n c r e a s i n g concave c u r v a t u r e as .Kg/K^ approaches to 1. (At h i g h e r Kg/ K-^ the c u r v a t u r e would u l t i m a t e l y decrease, s i n c e K^/K^ = co g i v e s the exact Langmuir s i t u a t i o n f o r a l a y e r expanded three times. ) The minimum value o f X m i s 1 at Kg/K-^=0 but i n the i n i t i a l p a r t of concave p l o t s f o r Kg/K^ = 1 .0 , s t r a i g h t l i n e with the slope c o r r e s p o n d i n g to X m value of 8 can be drawn. In attempting to match t h i s f a m i l y of curves to the experimental data, the range of X ( c o r r e s p o n d i n g to X^0^. i n the p r e s e n t model), which produced the Langmuir-type p l o t s has to be c o n s i d e r e d . Relevant e x p e r i m e n t a l runs, which can be examined i n terms o f the p r e s e n t model, are, from Table 4 , those a t 5 0 ~ 53°C, 7 I . 8 ~ 7 2 . 5 ° C and 8 6 . 0 ~ 8 7 . 3 ° C . Thses are t a b u l a t e d l n Table V I - 1 t o g e t h e r w i t h the range of X s t u d i e d , and i n t e r c e p t s of l / R p X vs. l / R p p l o t s . - 116 -- Table V I - 1 -Temp. (°C) X m Range of X I n t e r c e p t s l / X m K 1/K 1 K2/K-. 5 0 - 5 3 . 3 7 1 . 8 - 7 2 . 5 8 6 ^ 8 7 . 3 1.1 ± 0 . 2 ~ 2 ~ 3 0 . 2 - 0 . 7 0 . 3 - 1 . 0 o . 3 5 - 1.0 2 . 6 * 1 0 2 1.4 x 1 0 2 0 . 9 5 K 1 0 2 ~ r 2 2 . 6 ^ 1 0 1 . 4 x l O 2 2 0 .95*10 0 . 1 0 .3 0 . 5 In F i g . V I - 1 , the range of X as i n Table V I - 1 which g i v e s a reasonable s t r a i g h t l i n e with the slope y i e l d i n g matching X m was searched and r e p l o t t e d i n F i g . V I - 2 . As i n F i g . V I - 2 , l i n e s are f a i r l y s t r a i g h t with d e s i r e d s l o p e s and with very i n s e n s i t i v e i n t e r c e p t s to Kg/K^. I t should be noted, however, t h a t i f K value i s changed, then the f a m i l y o f l i n e s s h i f t s v e r t i c a l l y , g i v i n g the same s l o p e s f o r the same Kg/K^ v a l u e . Since i n t e r -cepts i n F i g . V I - 2 are c l o s e to that o f pure Langmuir p l o t w i t h i n 10$ e r r o r , and the l a t t e r i s f o r m a l l y l/K^, the e x p e r i -mental v a r i a t i o n i n i n t e r c e p t s i s o n l y due to the temperature dependence o f l/K^. l / K 's are then t a b u l a t e d In the f i f t h column of Table V I - 1 . By matching X 's and s l o p e s i n F i g . V I - 2 m one can ro u g h l y a s s i g n K^/K^ value f o r each run. These are a l s o t a b u l a t e d i n the s i x t h column of Table V I - 1 . K-^  v a l u e s thus a s s i g n e d g i v e then the s h i f t i n g f a c t o r s f o r R p - 1 s c a l e , which f i n a l l y g i v e the computed isotherms i n F i g . V I - 4 . In F i g . V I - 4 e x p e r i m e n t a l isotherms are a l s o g i v e n . Agreement with e x p e r i -mental isotherms i s q u i t e good. - 117 -- P i g . VI-1 -Computed l / R p X t Q t vs. l/Rp - F i g . VI-2 -- 119 -- 120 -- P i g . VI-4 -Computed and Experimental Isotherm X experimental p l o t • computed p l o t X or X Kg/K^O. 5 ( 8 6 - 8 7 . 3 ° C ) 8 - 7 2 . 5 ° C ) 2Ai=0.1 . 075 ( 5 0 - 5 3 ° . 05 5.0X10 l/Rr - 121 -Some remarks c o u l d be made upon the above treatment : -(1) The s i t e immediately below an exchanged anion i s u n i q u e l y d e f i n e d o n l y i f the s u r f a c e s are (110) ; for'(1.00) s u r f a c e s , there are f o u r s i t e s i n the second l a y e r s i m i l a r l y r e l a t e d to each s u r f a c e anion s i t e . T h i s i s i g n o r e d i n the pr e s e n t c a l c u l a t i o n . The model, however, does not say p r e c i s e l y that l a y e r s taken i n t o account should be the f i r s t , the second and the t h i r d l a y e r s , but i n s t e a d they c o u l d as w e l l be the f i r s t , the t h i r d and the f i f t h l a y e r s . I f the l a t t e r i s the case, then the s i t e i n the next l a y e r (3rd) below an exchanged anion i n the 1st l a y e r Is u n i q u e l y d e f i n e d f o r (100) s u r f a c e s , and moreover the s i t u a t i o n i s c o n s i s t e n t with t h a t i n " r e v i s e d q u a s i - c h e m i c a l treatment" : I.e. i n the l a t t e r model, l i k e s p e c i e s are taken to be r e p u l s i v e when they are the n e a r e s t neighbours, and a t t r a c t i v e when they are the second n e a r e s t neighbours i n the outermost l a y e r . I f t h i s ' argument i s extended to i n t e r a c t i o n s between l i k e s p e c i e s i n the d i f f e r e n t l a y e r s , i t becomes q u i t e a probable s i t u a t i o n t h a t C l - i o n s exchanged are popu l a t e d i n the t h i r d l a y e r not i n the second l a y e r below the exchanged r e g i o n i n the f i r s t l a y e r . (2) Although t h i s treatment i s very much s i m i l a r to that o f B.E.T. m u l t i l a y e r a d s o r p t i o n , the p h y s i c a l s i t u a t i o n i n v o l v e d i s q u i t e d i f f e r e n t . In m u l t i l a y e r a d s o r p t i o n a s t a c k o f ad sobbed molecules, as i t b u i l d s up, should become a condensed phase, 24) which B.E.T. treatment f a i l s to handle p r o p e r l y as H i l l 1 p o i n t e d out, whereas i n the presen t case exchanged C l " are on - 122 -w e l l d e f i n e d l a t t i c e s i t e s , no matter how deep the stack extends. So the treatment i s c l o s e r to the present system than m u l t i l a y e r a d s o r p t i o n . But as H i l l p ointed out, ne g l e c t of h o r i z o n t a l i n t e r a c t i o n s i n a l l l a y e r s could be the weak p o i n t of the model f o r both m u l t i l a y e r a d s o r p t i o n and the present system. B i b l i o g r a p h y . - 124 -(1) L.G.Harrison and R . A . S i d d i q u i , Trans. Faraday Soc. 5_8, 982 ( 1 9 6 2 ) . R.A. S i d d i q u i , . Ph.D. T h e s i s , Univ. o f B.C. ( 1 9 6 l ) . (2) W.K.Burton, N.Cabrera and F.C.Frank, P h i l . Trans. Roy. S o c , A243, 299 ( 1 9 5 1 ) . (3) G.C.Benson, P.I.Freeman and E.Dempsey, J . Chem. Phys., 3_£, 302 ( 1 9 6 3 ) . (4) J.A.Morrison and D.Patterson, Trans. Faraday S o c , 5 2 , 764 ( 1 9 5 6 ) . (5) N.Cabrera, Z.Elektrochem. 5 6 , 294 ( 1 9 5 2 ) . (6) T . L . H i l l , J.Chem.Phys., 18 , 988 ( 1 9 5 1 ) . (7) A . J . L e a d b e t t e r and L.L.Makarov, Trans. Faraday S o c , 64, 3224 ( 1 9 6 8 ) . (8) L . S . S a l t e r , Trans. Faraday S o c , 5jb 657 ( 1 9 6 3 ) . (9) A . J . L e a d b e t t e r and D.M.T.Newsham, Trans. Faraday S o c , 6 1 , 1646 ( 1 9 6 5 ) . (10) L.G.Harrison, J.A.Morrison and G.S.Rose, J . Phys. Chem., 6 1 , 1314 ( 1 9 5 7 ) . (11) T . T a k a i s h i and Y.Sensui, Trans. Faraday S o c , 63_, 1003 ( 1 9 6 7 ) . (12) T . T a k a i s h i and Y. Sensui, i b i d , 6 5 , 131 (1969) . (13) F.van Zeggeren and G.C.Benson, J . Chem. Phys., 2 6 , 1077 ( 1 9 5 7 ) . (14) G.C.Benson, H.P.Schreiber, and F.van Zeggeren, Can. J.Chem., 24, 1553 ( 1 9 5 6 ) . (15) G.Wullf, Z . K r i s t . , j_4 44? . (1.901). ••' .\ - 125 -(16) (17) (18) (19) (20) (21) (22) (23) (24) L.G.Harrison, I.M.Hoodless and J.A.Morison, D i s c . Faraday Soc., 2 8 , 103 ( 1 9 5 9 ) . R.J.Adams and L.G.Harrison, Trans. Faraday S o c , £ 8 , 1792 ( 1 9 6 4 ) . M.G.Inghram e t . a l . , "Proc. o f I n t e r n a t i o n a l Syp. on High Temp. Technology", P.219 ( 1 9 5 9 ) . J.G.Bergmann e t . a l . , A n a l . Chem. 3jj_, 911 ( 1 9 6 2 ) . I . P r i g o g i n e and R.Defay, t r a n s l a t e d by D.H.Everett, "Chemical Thermodynamics", Longmans ( 1 9 5 4 ) . JANAF Thermochemical Data. J.H.de Boer, "The Dynamical C h a r a c t e r o f A d s o r p t i o n " , Oxford ( 1 9 5 3 ) . R.Rudham, Trans. Faraday S o c , £9 . , 1853 ( 1 9 6 3 ) . T . L . H i l l , "Advances i n C a t a l y s i s " , v o l . IV, P.227, Academic Press, ( 1 9 5 2 ) . SUPPLEMENT PART I I A STUDY OF THE REACTION 2IBr = I 2 +-Br£ OCCURRING IN A GAS-CHROMATOGRAPHIC COLUMN. I n t r o d u c t i o n - S-2 -The o r i g i n a l i n t e n t i o n was to study c a t a l y t i c a c t i v i t y of e l e c t r o n i c d e f e c t s i n potassium bromide with p a r t i c u l a r r e f e r e n c e to a n i o n excess d e f e c t s . For that purpose, the combination of -. i o d i n e and bromine to form i o d i n e bromide was chosen ( I g + Br^ = 2 I B r ) . The reasons f o r s t u d y i n g the above mentioned c a t a l y s i s are; (1) A l k a l i h a l i d e s are i n g e n e r a l i n a c t i v e f o r c a t a l y s i s 19) of any gas-phase r e a c t i o n s . So, they might be a s u i t a b l e medium f o r attempts to form c a t a l y t i c a l l y a c t i v e d e f e c t s and study them i n a simple system which otherwise l a c k s i n t r i n s i c a c t i v i t y . • . (2) Anion excess d e f e c t s i n a l k a l i h a l i d e s have been w e l l TON s t u d i e d . T h i s knowledge might, help to e l u c i d a t e the mechanisms of the c a t a l y s i s . To study t h i s system, one has to analyse the composition of the gas mixture I ^ - I B r - B ^ to f o l l o w the k i n e t i c s . R e c e n t l y v a r i o u s authors have done gas chromatographic a n a l y s i s 21) 22) of gaseous h a l i d e s ' a n d i n t e r h a l o g e n s , and the technique i s w e l l e s t a b l i s h e d . For the present purpose, gas chromatography was chosen. In the process, however, the standard sample of IBr showed three peaks (as i n F i g 21). Moreover, when gaseous I2 and gaseous B r 2 were mixed and the change of the composition of Ig and IBr and B r 2 was followed with r e f e r e n c e to time, i t was found that from the f i r s t sampling (5min. a f t e r the mixing) to-s e v e r a l .months, the composition c a l c u l a t e d from the chromatogram - s-3 -was constant and the apparent e q u i l i b r i u m constant was of the 25) order of 5 , whereas the r e p o r t e d e q u i l i b r i u m constant i s 1 6 9 . 0 a t the temperature used. I t seemed p o s s i b l e that r e a c t i o n was o c c u r r i n g on the. gas chromatographic column. Thus a study was c a r r i e d out on the r e a c t i o n 2IBr = 1^ + Br^ o c c u r r i n g on the gas chromatographic column which i s K e l - F grease # 90 coated on t e f l o n powder. T h i s study proved to be of i n t e r e s t i n r e l a t i o n to the development of computational methods f o r chromatographic s e p a r a t i o n w i t h simultaneous r e a c t i o n . P a r t I I of t h i s t h e s i s t h e r e f o r e d e a l s w i t h chemical r e a c t i o n on the gas chromatographic column. V a r i o u s t h e o r e t i c a l approaches to chromatographic s e p a r a t i o n and the shape o f the chromatogram have been proposed. I n t h i s t h e s i s t h e o r i e s of chromatographic s e p a r a t i o n , i n c l u d i n g p r e v i o u s work on those w i t h chemical r e a c t i o n on columns i s reviewed, and a new way o f t r e a t i n g chemical r e a c t i o n on chromatographic columns i s proposd. Chapter 1 Review.of previous t h e o r e t i c a l work on gas chromatographic separation and r e a c t i o n o c c u r r i n g simultaneously. 1 INTRODUCTION The. c l a s s i f i c a t i o n y of the behaviour o f chromatography i n g e n e r a l i s t a b u l a t e d i n Table 1. - Table 1 -I d e a l chromatography N o n i d e a l chromatography L i n e a r i s o t h e r m I I I I N o n l i n e a r " isotherm I I IV By " l i n e a r isotherm", i t i s meant that the e q u i l i b r i u m c o n c e n t r a t i o n s i n the two phases ( i . e . mobile and s t a t i o n a r y phases) are p r o p o r t i o n a l . " I d e a l chromatography" i s the case where the e q u i l i b r i u m between the two phases i s immediate and l o n g i t u d i n a l d i f f u s i o n and other processes having a s i m i l a r , e f f e e t (see s e c t i o n .2) can be i g n o r e d . For p a r t i t i o n chromatography, which i s the main concern of t h i s p a r t of the t h e s i s , the l i n e a r i sotherm i s u s u a l l y a good approximation. Case I and case I I I w i l l . b e c o n s i d e r e d . Case I covers the s i m p l e s t theory s t i l l p o s s e s s i n g the e s s e n t i a l f e a t u r e s of chromatography. Consider the continuous column as i n F i g 1. - S-6 -uC 3 C 9 t a c C+ x dx) mobile phase 9 C, a t s t a t i o n a r y phase .0 x+dx volume f r a c t i o n c o n c e n t r a t i o n l i n e a r of component C v e l o c i t y mobile phase F C u s t a t i o n a r y phase F s C s 0 - F i g . 1 -Taking m a t e r i a l balance between x and x+dx d u r i n g the sh o r t p e r i o d of time dt, F ^ C + F ^ C S 3 1 s a t = - Fu a c a x Since C = K C ( l i n e a r i s o t h e r m ) , s c Kc being the p a r t i t i o n c o e f f i c i e n t , •(1) •(2) 3-£ - F . "3 C a t F + K C F S a x - S-7 -S> C -u 9 c (3) The g e n e r a l s o l u t i o n f o r eq. (3) has the form: C = G ( t - ( x fic )/u) . U ) The f u n c t i o n form of G would be chosen by i n i t i a l and boundary c o n d i t i o n s . Eq. (4 ) shows t h a t the e l u t i o n band of the component C moves along the column with the e f f e c t i v e v e l o c i t y u//]c » e l u t i o n bands of d i f f e r e n t components having c h a r a c t e r i s t i c p a r t i t i o n c o e f f i c i e n t s separate from each other. Eq. (4) a l s o shows t h a t the shape of the e l u t i o n band remains constant throughout the column. Consequently the width of the e l u t i o n band i s equal to that of the feed p u l s e (see curve B i n F i g 4 ) . I n p r a c t i c e , however, i t i s u s u a l l y found 1^ that e l u t i o n bands broaden markedly with i n c r e a s i n g column l e n g t h . Thus case I I I (Table 1) must be c o n s i d e r e d . There are two p r i n c i p a l ways to handle case I I I t h e o r e t i c a l l y : (a) by the " r a t e theory" approach i n which the d i f f e r e n t i a l e quation (3) i s m o d i f i e d by a d d i t i o n of e x p l i c i t terms to r e p r e s e n t assumed mechanisms of broadening which destroy the i d e a l i t y of chromatography. T h i s w i l l be d e a l t with i n s e c t i o n 2., (b) by a s e m i - e m p i r i c a l approach u s i n g the " p l a t e theory" which w i l l be d e s c r i b e d i n s e c t i o n 3 . In t h i s approach, both the improvements i n s e p a r a t i o n and the band broadening with hence on K c* T h i s e x p l a i n s why the - S-8 -i n c r e a s i n g length of column are r e l a t e d to the f i c t i t i o u s q u a n t i t y n, the number of t h e o r e t i c a l p l a t e s , and no p h y s i c a l mechanism f o r broadening i s put i n t o the equations. - S-9 -2 RATE THEORY Va r i o u s authors ~ ' ' h a v e proposed the sources of broadening, which can be summed up i n three c a t e g o r i e s as f o l l o w s : " . (1) D i f f u s i o n When t a k i n g the m a t e r i a l balance as i n F i g 1, the d i f f u s i o n term i n the mobile phase i s taken i n t o account. Eddy d i f f u s i o n i s a l s o i n c l u d e d by m o d i f i c a t i o n of the d i f f u s i o n c o e f f i c i e n t . The equation corresponding to (1) becomes: DV: d i f f u s i o n c o e f f i c i e n t ( e f f e c t i v e ) , (2) Mass t r a n s f e r Up to t h i s p o i n t , the p a r t i t i o n e q u i l i b r i u m between the mobile phase and the s t a t i o n a r y phase has been considered to e s t a b l i s h i t s e l f i n s t a n t a n e o u s l y . In t h i s category, the r a t e of mass t r a n s f e r between the mobile phase and the s t a t i o n a r y phase' i s taken i n t o account. Then, assuming l i n e a r mass t r a n s f e r r a t e , the m a t e r i a l balance i n the mobile phase and. the s t a t i o n a r y phase becomes:-i n mobile phase F - f r -= -V ! § - + C < ( C B - K c ° > • - s-io -i n s t a t i o n a r y phase 9 C s s a t - O C ( c s - Kc c ) , •(6) where OC- r a t e constant of mass t r a n s f e r . One has to s o l v e these simultaneous d i f f e r e n t i a l equations. (3) Pressure drop along the column Gas chromatographic columns t y p i c a l l y operate with atjsfmospheric e x i t p r e s s u r e and an i n l e t p r e s s u r e i n the range of 1.5 to 3 atm. The. pressure drop causes gas expansion as gas flows through the column. The m a t e r i a l balance i n . t h e mobile phase then becomes, 9 C a t = -F (uC). •(7) I f d i f f u s i o n i s a l s o taken i n c o n s i d e r a t i o n , s i n c e the d i f f u s i o n D* c o e f f i c i e n t i s i n v e r s e l y p r o p o r t i o n a l to p r e s s u r e , (D 1 P where D* depends on temperature o n l y ) , the m a t e r i a l balance i n the mobile phase should be, _a_c Q t -F (8) Lapidus and Amundson v / f i r s t t r e a t e d c a t e g o r i e s (1) and (2) t o g e t h e r , and l a t e r van Deemter et. a l ] " ^ approximated i n the Gaussian form.to o b t a i n more p r a c t i c a l s i g n i f i c a n c e from the - S - l l -s o l u t i o n . J.C. Giddings' work H i s e s s e n t i a l l y on the l i n e o f category ( 2 ) , although he i n c l u d e d any p o s s i b l e r a t e process, such as chemical r e a c t i o n . R e c e n t l y , Yamazakl/' obtained the moments of the d i s t r i b u t i o n f u n c t i o n f o r the case (1) and ( 2 ) , and expressed the e l u t i o n band c h a r a c t e r i s t i c s (not only r e t e n t i o n time and peak width, but a l s o peak asymmetry and d e v i a t i o n from Gaussian d i s t r i b u t i o n ) i n terms of p h y s i c a l constants l i k e D,CX> u> e t c . Kambara's^ 7) and Olson's'' work are on the l i n e of c a t e g o r i e s (2) and ( 3 ) . Kambara approximated the s o l u t i o n by a Gaussian d i s t r i b u t i o n f u n c t i o n whereas Olson, by u s i n g the L a p l a c e transform, e x p l a i n e d the occurrence of " t a i l i n g " . - S-12 -5 PLATE THEORY J . J . van Deemter et.al"P have extended the p l a t e theory p) developed and a p p l i e d by M a r t i n e t . a l . ' to i n c l u d e the case where the i n f l u e n c e of the feed volume i n chromatography i s not n e g l i g i b l e . Consider the model that the g a s - l i q u i d chromatographic column c o n s i s t s of a s e r i e s of p l a t e s . w i t h equal volumes i n which g a s - l i q u i d p a r t i t i o n e q u i l i b r i u m i s e s t a b l i s h e d (see F i g 2 ) . V G i > V L C c. 1 V L X V G V L p l a t e no. mobile volume phase s t a t i o n a r y phase mobile coneen- phase t r a t i o n s t a t i o n a r y phase i - l °i-l °i-l G i+1 i+1 s i+1 - F i g 2 - s -13 -The p a r t i t i o n c o e f f i c e n t K i s assumed to be independent of the c o n c e n t r a t i o n s ( l i n e a r isotherm) and i s thus g i v e n by Ct = C i K c ' — — — " <9) Consider the m a t e r i a l balance a t the i - t h p l a t e ( i = 0,1,2, n, n being the t o t a l number of p l a t e s , and i =0 meaning the feed pulse) i f a s m a l l amount of volume dV of the moving phase passes through i t : C i - l d V " C i d V = VG d C i + V L d C i ; « (10) (9) and (10) w i l l g i v e : dC. . <VG + K c V - ^ r •= ci-i - c i - > .. — dC. , vc -ar- = c i - i " c i . — — — d a ) Here v c = v Q + K c v L (13) Let the feed p u l s e have a c o n c e n t r a t i o n C° and a volume V . x o o Then the i n i t i a l and boundary c o n d i t i o n . a r e : at V = 0, C-. = C0 .= '--- = C = 0 1 2 n o i v J v 0 > C Q = C° (Ik) V v - o0 = 0 , - S-14 -S o l v i n g (12) under the c o n d i t i o n (14) C i = f [ ( x / v c ) i " 1 / ( i - D ! ] e x p ( - x / v c ) d x f o r 0 < V < V Q — ( 1 5 ) C i = (°o / vc ) f [ ( x / v c ) i " 1 / ( i - l ) ! ] exp(-x/v c)dx J V-V Q f o r V > V Q - •(16) At the end of the column (i=n), eq. (16) r e p r e s e n t s the s i t u a t i o n , s i n c e the e l u t i o n band has a value f o r V>VQ. I f the number of p l a t e s i s not too s m a l l (say n = 100), the i n t e g r a n d of eq. (16), the P o i s s o n d i s t r i b u t i o n f u n c t i o n , i s approximated w e l l by a Gaussian d i s t r i b u t i o n f u n c t i o n . C = n C o «V v c V 2 7 T n exp , (x/v - n)d , ] dx (17) v-v: The e l u t i o n curve of the band a c c o r d i n g to (17) i s shown g r a p h i c a l l y i n F i g . 3 . n 0 tangent a t p o i n t of i n f l e c t i o n p o i n t of i n f l e c t i o n - F i g . 3 -- s - 1 5 -The c o n c e n t r a t i o n i s a maximum f o r V 1 ^ = n v c + - T J — V Q . The d i f f e r e n c e i n if„ ( i . e . d i f f e r e n c e i n K , see. eq (13)) w i t h a c o d i f f e r e n t m a t e r i a l causes the d i f f e r e n c e i n V ' R C . T h e r e f o r e , i f the d i f f e r e n c e i n V 1 ^ f o r two m a t e r i a l s i s l a r g e enough, one c o u l d expect to see two bands separated. The width W of the e l u t i o n band ( d e f i n e d as the d i s t a n c e between the i n t e r s e c t i o n s of the tangents a t the i n f l e c t i o n p o i n t s with the a b s c i s s a ) can be c a l c u l a t e d from eq . ( 1 7 ) and i s shown i n F i g . k (curve A), which agrees with, experimental data e x c e l l e n t l y " ^ . F i g . k shows that f o r ° <0. 5, v c V n W = LY v c V n . Consequently i n t h i s r e g i o n , s i n c e = V - ^ p = nv ( VRC n v c W 4V-4 v c V ^ RC x2 so that n = ( ^ ) ^ — —(18) Eq. (18) g i v e s the t h e o r e t i c a l b a s i s f o r c a l c u l a t i n g . t h e number of t h e o r e t i c a l p l a t e s . On t h i s model, although mechanisms f o r the band-broadening are l e f t u n c e r t a i n , the occurrence of - S-16 -- S-17 -gas chromatographic s e p a r a t i o n and band broadening can be w e l l simulated i n computation. - s -18 -4 REACTION ON CHROMATOGRAPHIC COLUMN Sinc e Emmett et a l . ' developed the s o - c a l l e d " m i c r o c a t a l y t i c technique" to study c a t a l y t i c r e a c t i o n s , the r e a c t i o n on a chromatographic column has drawn wide a t t e n t i o n i n theory and i n p r a c t i c e . As r e a c t o r s , chromatographic columns have e n t i r e l y d i f f e r e n t c h a r a c t e r from o r d i n a r y steady flow r e a c t o r s . Sometimes ,the c o n v e r s i o n becomes much high e r than expected from the e q u i l i b r i u m c o n s t a n t s , which i s understood e a s i l y s i n c e r e a c t a n t s and products are separated as they flow through a chromatographic r e a c t o r . Dehydrogenation-type r e a c t i o n s are the best i l l u s t r a t i o n of t h i s , s i n c e the hydrogen reduced u s u a l l y has l e s s a f f i n i t y - f o r a column and flows f a s t e r than the r e a c t a n t and the other product. Indeed p a t e n t s 9 ^ h a v e been awarded f o r t h i s type of r e a c t o r . V a r i o u s t h e o r e t i c a l treatments''"*"^ -17) have been developed f o r the extent of c o n v e r s i o n , the e l u t i o n curve p r o f i l e , the e f f e c t of. the shape of the feed pulse, e t c . The b a s i c mathematical model f o r a l l the above treatments i s e s e n t i a l l y the same, and i s summed up as f o l l o w s : . Assumption : (1) The change i n temperature and volume due to r e a c t i o n i s n e g l e c t e d . (2) The r e a c t a n t s and products i n the mobile phase flow at the same l i n e a r v e l o c i t y as the c a r r i e r gas. (3) L i n e a r isotherms f o r a l l s p e c i e s . - S-19 -(4) P a r t i t i o n e q u i l i b r i u m i s e s t a b l i s h e d i n s t a n t a n e o u l y . (5) D i f f u s i o n , mass t r a n s f e r and pressure drop ( c f . s e c t i o n 2) are ignored. Under assumption ( l ) - ( 5 ) j the m a t e r i a l balance corresponding to eq. (3) now becomes, 9 C u_ 9 C r c ,, Q a t - j3c a x ^ • at t=0, x > 0: C=0, where r c i s the ra t e of the production of the component C i n the s t a t i o n a r y phase. The problem now i s to solve the equation (19), under the boundary c o n d i t i o n s , which vary with the shape of the feed p u l s e . As c l e a r l y seen here, the broadening of the e l u t i o n band i s l e f t out of c o n s i d e r a t i o n . I f the separation of two e l u t i o n bands i s incomplete, so that two e l u t i o n bands p a r t l y overlap, the broadening becomes s i g n i f i c a n t and t h i s mathematical model i s f a r from d e s c r i b i n g the true s i t u a t i o n . . . Recently K o c i r i k ^' in c l u d e d d i f f u s i o n and mass t r a n s f e r terms i n the treatment of f i r s t order i r r e v e r s i b l e r e a c t i o n on the column. The basic equations i n K o c i r i k 1 s work are, F -ff - =FD' "=rVFu -fr- + C X ( 0 s - * cc> - k i c . X - S -20 -0 ( ( C S - K cC) - k 2 C s , — — ( 2 0 ) where and kg are f i r s t order i r r e v e r s i b l e disappearance r e a c t i o n r a t e constants i n the mobile and the s t a t i o n a r y phases r e s p e c t i v e l y . I n theory, one can add pressure drop e f f e c t e a s i l y to eq. ( 2 0 ) . This l i n e of approach i s much b e t t e r than that mentioned above g i v i n g eq. ( 1 9 ) . But one encounters complicated mathematical forms, and moreover when the comparison with experimental data i s attempted, one has to use a s e r i e s of t r i a l values not only of k^ and kg ( t a k i n g eq. (20) as an example) but a l s o of D' and OC . I n the next chapter a new and more p r a c t i c a l approach w i l l be proposed to overcome t h i s d i f f i c u l t y . <D c. Chapter 2 P l a t e theory approach to r e a c t i o n s on chromatographic columns. - S-22 -In the previous chapter, i t was hinted that the use of r a t e theory to i n c l u d e the chemical r e a c t i o n term r e s u l t s i n complicated mathematical form and too many adj u s t a b l e parameters. In t h i s Chapter the a p p l i c a t i o n of p l a t e theory w i l l be proposed. The advantage i s that e s s e n t i a l l y the q u a n t i t y n, the number of t h e o r e t i c a l p l a t e s , i s the only parameter necessary to describe the broadening of e l u t i o n bands. - S-23 -1 THEORY Consider a r e a c t i o n of type 2 B = A .+ C. Take the m a t e r i a l balance a t the i - t h p l a t e d u r i n g the s h o r t p e r i o d of time dt (see F i g . 5 ) . ( i - l ) t h . i t h p l a t e volume flow rate' w ^» V — > V G mobile phase V L V L s t a t i o n a r y phase C o n c e n t r a t i o n s mobile phase < A i - 1 B i - V . c i - l A. l B ± °i A i = KA A i C j = K c C. s t a t i o n a r y phase < • A s , i - l i - l C S i - l 4 '• q - F i g 5 -Assuming that the r e a c t i o n occurs i n the s t a t i o n a r y , phase, - S-24 -and l e t t i n g r ^ ( A ) , r^CB), r^(C),-be the r a t e of p r o d u c t i o n of component A, B and C r e s p e c t i v e l y per u n i t volume of the i - t h s t a t i o n a r y phase, the m a t e r i a l balances now becomes, dA. ( V G + KA VL> ~dT-= w ( A i - l " V + V L r i ( A ) > ] dB, ( V G + KB VL> - d l T - =^\-l ~ V + V L r i ( B>' dC, < V G + K c V - a r - = w < c i - i - ci> V L R± ( c ) - J •(21) Here, s i n c e the mixing i n the p l a t e i s complete, r ± ( A ) = r ± ( C ) = _ i . r . ( B ) = K [ ( B S ) 2 _ K L A S C S J # _ . ( 2 2 ) : chemical e q u i l i b r i u m constant i n i - t h s t a t i o n a r y phase, k : 2nd order r a t e constant f o r decomposition of B. L e t chemical e q u i l i b r i u m c o n c e n t r a t i o n s i n the mobile phase and i n ' the s t a t i o n a r y phase be A , B , C , and A s, B s, C s, then 6 O 6 6 6 6 ( B ^ ) 2 K = — s L »s „s B Ae" C e K A K C ( A e C e > " K A K C V •(22a) where, ? B e K - — e " A C e e chemical e q u i l i b r i u m constant i n the" mobile. phase. This.and the p a r t i t i o n e q u i l i b r i u m l e a d eq. (22) to: - S-25 -r ± ( A ) = - - i - r ± ( B ) = r i ( C ) ; = k Kg ( B 2 - K ^ ^ ) •(23) Then eq , (21) becomes v d A i ? " d T " = w < A i - l " V + ko< BI " K e A i C i > , dB, ?  V B " d T - = w < B i - l " V " 2 V B I - ViCi), dC\ C dt -= *<Ci-l " C i > + k o < B i - K e A i C i > „ <2h) where, k = k K B VA = V G + K A V L , V B = V G + K B V L , V C = V G + K C V L , VT •(25) E i t h e r by f e e d i n g each component s e p a r a t e l y , or by choosing the c o n d i t i o n so that k i s n e g l i g i b l y s m a l l , the number of t h e o r e t i c a l p l a t e s n and h e n c e . e t c . can be obtained a c c o r d i n g to eq. (18) of chap. 1^ s e c t i o n 3 . Consequently eq. (24) can be solv e d f o r a given'k and a g i v e n nature of feed p u l s e . Note t h a t : VRA = n V A + "2 VRA = n V A V e t c . e t c . •(25a) - S-26 -where e t c . are r e t e n t i o n volumes, and e t c . are e x t r a p o l a t e d values of V^ A to V Q = 0. - S-27 -2 COMPUTATION AND RESULTS : The s o l u t i o n of eq. (24) v/as computed with the IBM 7044 computer under the f o l l o w i n g c o n d i t i o n s : t = 0 : A. = B. = C. = 0 f o r i-= 0 , 1, 2, — . - r i i = 0 , 0 < t < T : A„ = A° jrf (t) ' = = 0 0 0 ' B 0 = B° j6 ( t ) o 0 » c 0 =-C° * ( t ) t > T : A = B = C = 0 o o o o » where / ( t ) (r e p r e s e n t e d as GO i n the computer programme) i s a f u n c t i o n of the form shown i n F i g 6. T h i s i s a step f u n c t i o n with i n f i n i t i e s a t the steps avoided by rounding o f f i n an a r b i t r a r y manner, f o r which purpose a cosine curve was used :-• 0 < t < Z T Q ; 6 = 0 . 5 [ l - cos ( 7 T t/zT 0)] (1 - - L ) T Q < t < T Q ; £ = 0 . 5 [ 1 + co-s^7T(t - T Q + Z T ^ / T T J ] (The a r b i t r a r y width-z. i s CC i n the computer programme i n Appendix (A) .) - S-28 -( t ) t Constants chosen were : BQ = 1 0 . 0 , \ ( a r b i t r a r y u n i t s ) C° = 1 . 0 , ~L = 0 . 0 5 , K = 100, e ' k Q =5.0, 1 0 . 0 , 5 0 . 0 , 1 0 0 . 0 , 2 0 0 . 0 , (ml/min c o n c e n t r a t i o n u n i t ) , w = 30 .0, 76 .0 , , 120.0 (ml/min), VRAIR = n V G = 36 .8ml , . VRA = n ( V G + W = 5 4 . 0 m l , V R B = n ( V Q + K B V L ) = 7 4 . 8 m l , . V R C = n ( V Q + • KCV'L) = 1 0 8 . 0 m l , - S-29 -n = 200 V Q = wTQ = 0 .5 - 20.Oral , where e t c . i s the extrapolated r e t e n t i o n volume of A e t c . to V = 0 when no r e a c t i o n i s occurring, and V n s T n i s the r e t e n t i o n o °' RAIR volume of a i r which i s considered to have no a f f i n i t y w i t h the s t a t i o n a r y phase i . e . the p a r t i t i o n c o e f f i c i e n t i s zero, v^ e t c . can be c a l c u l a t e d as : V. VA = VG + K A V L = n RA etc The computation program i s given i n Appendix (A) as program No.l. A t y p i c a l r e s u l t of computation of e l u t i o n bands i s given i n F i g . 7. To describe the behaviour of the column the f o l l o w i n g , q u a n t i t i e s were chosen (see F i g . 7) '• K e f f CO ( I B ( t ) d t ) ' A ( t ) d t isq— C ( t ) d t K M 1 ( A ( t ) + B ( t ) + C ( t ) ) d t J e f f f M l (A(t) + B(t) + C ( t ) ) d t r°°(A(t) + B ( t ) + C ( t ) ) d t o JM 2 (26) and W* (defined as the distance i n volume u n i t s between the i n t e r s e c t i o n of tangents.at the i n f l e c t i o n p o i n t s of the center band of the curve A ( t ) +B(t) + C(t) with the a b s c i s s a ) . - S-31 -^ e f f "*"S ^ * i e " a P P a r e n t e q u i l i b r i u m constant" c a l c u l a t e d from the e l u t e d bands: but i f these bands o v e r l a p , cannot be found d i r e c t l y , from experimental data.' K^^* i s a q u a n t i t y r e l a t e d to both K „„ and the extent of band o v e r l a p , and i t can be obtained both from computation and from experimental r e s u l t s . . . . The d i f f e r e n c e between and K g f j * can be q u i t e l a r g e (see F i g . 8 , and 9 ) . N e i t h e r K e f f nor K e f f * i s a t a l l c l o s e to the tr u e e q u i l i b r i u m constant, and the choice of which to use i n comparing experiment and theory i s an a r b i t r a r y one, i n which experimental convenience d i c t a t e s the use of K-»„*. e f t The program N o . i seemed to have the f o l l o w i n g l i m i t a t i o n s on i t s a p p l i c a t i o n . (a) When k Q > 100.0 and V /V > 0.15, the t o t a l amount of output becomes much smaller, than that of the i n p u t , a p p a r e n t l y because a stage i n v o l v i n g s u b t r a c t i o n of tv/o very l a r g e and ne a r l y equal terms i s becoming i n a c c u r a t e . (b) When k Q > 2 0 0 , one of the terms exceeds the c a p a c i t y of the computer. The program No.2 was w r i t t e n f o r k Q = oo , i . e . i n such a way that a t the e x i t of the i - t h p l a t e A^, and C^:are i n . t h e e q u i l i b r i u m composition ( i . e . (B^) /Aj^C^ = K g) f o r a l l i . The b a s i c i d e a f o r t h i s program i s s i m i l a r to Magee's treatment 15) f o r i n s tantaneous r e a c t i o n s and i s as f o l l o w s : Instead of eq. (21), the f o l l o w i n g equation has to be so l v e d ; - S -32 -TA "dV" v B dV C dV = ( c . ^ - c±) + 6 , •(27) B, A i C i = K e ' where dV = w d t , the s m a l l volume of c a r r i e r gas p a s s i n g through the p l a t e , and (5 i s the amount of c o n v e r s i o n to e s t a b l i s h the e q u i l i b r i u m i n s t a n t a n e o u s l y . Rote t h a t the assumption here i s the i n s t a n t a n e o u s establishment of the e q u i l i b r i u m , so the s o l u t i o n of eq. (27) has no dependence on the v e l o c i t y of c a r r i e r gas..' Rewrite eq. (27) as: dA, v A- dV + A i " A i - 1 = 6 dB, v B dV dC-~ + B i - B i - i = - 26 > V C "dV~~ + C i - C i _ i =6 , •(28) - s-33 -I n i t i a l and boundary c o n d i t i o n s are the same as those of the program No.1.except that the feed p u l s e i s of the step form w i t h no r o u n d i n g - o f f : 0 < V < V Q A = A° o o B = BX o o C„ = cx 0 o > v 0 < v A„ = B„ = C^ = 0. 0 0 0 To s o l v e eq. ( 2 8 ) , f i r s t l e t (5= 0 and s o l v e eq. (28) as, A_! = e " V / v A \ V e x p ( V / v A ) ( A j _ 1 / v A ) d V 0 B_! = e " V / v B fV e x p ( V / v B ) ( B i _ 1 / v B ) d V } 0 c! = e " V / v C e x p ( V / v c ) ( C i _ 1 / v c ) d V J 0 •(29) , i . p , i i and check whether (B^) /A^ C^ i s equal to the e q u i l i b r i u m - s -34 -constant or not : i f not, c a l c u l a t e X from (B^ - 2 X ) 2 (A[ + X)(C± + X) = K. and then rep l a c e A^, B^ and by A.- = B, = A ± + X B ±.- 2X C± = C± + X . (30) Feed these A^, B^, and to the ( i + l ) t h p l a t e and c a l c u l a t e t A i + 1 e i : c , » a n c* f o l l o w the same procedure u n t i l the end of the column. The program No.2 i s a l s o given i n the Appendix ( A ) . R e s u l t s of computations using both programmes are p l o t t e d i n F i g . 8 - F i g . 1 5 , as discussed i n the next s e c t i o n . For comparison, r e s u l t s f o r the case where the column was extended by a f a c t o r of 2 . 5 , i . e . , n = 500, V R A I R = 82 .3 ml , = 135-0ml, V R B 199 .0ml , V RC.= 270 .0ml , i s shown i n F i g . 16 - F i g . 18. - S-35 -- F i g . 8 .... K f f v s . FEED PULSE VOLUME -o. 1 : 1 o r y V V R A I R - S-36 -- S - 3 7 -- F i g . 10 CENTER BAND WIDTH vs. FEED PULSE VOLUME 0 V V B Y * 1.0 2 . 0 3 . 0 W*/V RAtfR 1.0 0 . 9 H 0 . 8 0.7-0 . 6 -0 . 5 4 . 0 n = 200 K = 1 0 0 e w = 76 .0ml/min k_.= oo 5 0 . 0 1 0 . 0 0 V V R A I R io.q 9.0-8 . o i 7.0-W/v 6.0-J 5 .0H 4 . 0 0 . 5 - S-38 -- F i g . 11 HEIGHT OF THE CENTER BAWD vs. FEED PULSE VOLUME. - S -39 -4 0 . 0 e f f or e f f 3 0 . 0 20.OH 1 0 . 0 F i g . 12 K f f and K g f f * vs. FLOW RATE n = 200 V V R A I R = ° ' 2 7 2 k = 1 0 . 0 o K = 100 . e K e f f 2 0 . 0 4 0 . 0 60.0 8 0 . 0 1 0 0 . 0 1 2 0 . 0 v/ (ml/min) - S-40 -- S-42 -F i g . 15 BAND WIDTH vs. K «.* er r • 7 n = 200 K = 100 e w = 76.0 ml/rain — , —, , , — 0.7 0 . 8 0 . 9 1.0 w*/v lk / VRAIR • - S-43 -- S -45 -- F i g . 18 K «.* vs. BAND WIDTH n = 500 = 1Q0 = 76 .0 ml /min 0 .4 0 .5 0 .6 0 .7 0 . 8 W*/V W / VRAIR - S - 4 6 -• 5 DISCUSSION F i g . 8 and F i g . 9 show the same.trend f o r both K „„ and K ~~*. e i f e f f As the r e a c t i o n r a t e constant, k , becomes higher, K and K „„* ' o' ° ' e f f e f f become s m a l l e r w i t h the same V which i s expected. With the same k Q , K e ^ j and K f ^ * show minima a l o n g V Q a x i s . There seem to be two competing f a c t o r s to determine (or K e f f * ) , i .e., the c o n c e n t r a t i o n s of the three components and the degree of o v e r l a p p i n g of the three peaks. G e n e r a l l y , the c o n c e n t r a t i o n s of the three components a t any p l a t e , A^, B^, and C^, become l a r g e r A S VQ goes higher, which tends to gi v e a high e r r e a c t i o n r a t e term, 2 • k (B. - K A. C. ) and to decrease K »„ ( o r K - , , * ) . At the same o i e i I e f f e f f time, the band widths i n c r e a s e as i n F i g . 4 , so that the degree of o v e r l a p p i n g becomes g r e a t e r , which h i n d e r s the decomposition of B by i n e f f i c i e n t removal of the products A and C, and tends to i n c r e a s e It ( o r K „„*). C o n c e n t r a t i o n s i n c r e a s e l i n e a r l y with, e f f e i f J V Q up to V0/Yc-\/n r 1 a n d then deviate' downwards from l i n e a r i t y " ^ , whereas the band width i n c r e a s e s s l o w l y a t the b e g i n i n g and from V Q / v c \fn = 2.5 i n c r e a s e s l i n e a r l y as i n F i g . 4 . Therefore one co u l d expect that when V Q i s s m a l l e r , the c o n t r i b u t i o n of c o n c e n t r a t i o n e f f e c t i s l a r g e r than that of o v e r l a p p i n g e f f e c t and a s V Q goes hig h e r the s i t u a t i o n i s r e v e r s e d . Consequently K e^^ (o r K shows a minimum at some value of V . For the e f f o case of kg = co , no matter how smal l the c o n c e n t r a t i o n s , A^, B/, and C^, are, the r e a c t i o n i s completed a c c o r d i n g to the r e l a t i o n , - S-47 -B. = K A . C . So t h e o v e r l a p p i n g i s t h e m a i n f e a t u r e . T h e r e f o r e 1 e 1 X . rr o t h e r e s h o u l d n o t be a minimum a n d KQff ( o r K e r-f-*) s h o u l d o n l y i n c r e a s e w i t h V Q , . ' w h i c h a g r e e s w i t h t h e c u r v e i n F i g . 8 a n d F i g . 9. A l s o i n F i g . 11, f o r t h e l a r g e r V , t h e h e i g h t o f t h e c e n t e r b a n d d e v i a t e s downwards f r o m l i n e a r i t y , w h i c h i s c o n s i s t e n t w i t h ' t h e d i s c u s s i o n a b o v e . The b a n d w i d t h i n F i g . 10 I s W* d e f i n e d i n t h e p r e v i o u s s e c t i o n , i . e . , i t i s t h e b a n d w i d t h o f t h e c e n t e r b a n d o f t h e c u r v e A ( t ) + B ( t ) + C ( t ) . S o , a s i s c l e a r l y d e m o n s t r a t e d i n F i g . 7, i t i s l a r g e r t h a n t h a t o f B ( t ) b e c a u s e o f o v e r l a p p i n g . W i t h c o n s t a n t V , W* i n c r e a s e s w i t h k Q , w h i c h i s r e a s o n a b l e , s i n c e f r o m F i g . 7, t h e d e v i a t i o n o f A a n d C b a n d s f r o m symmetry (the c a s e o f no r e a c t i o n ; d o t t e d c u r v e i n Fig.7) due t o t h e r e a c t i o n c o n t r i b u t e s t o W* t o a l a r g e e x t e n t . A s k Q becomes h i g h e r , t h i s d e v i a t i o n i n c r e a s e s , a n d t h u s W* becomes l a r g e r . A s s e e n f o r t h e c a s e k = 50.0 more c l e a r l y t h a n o t h e r s i n F i g . 10, t h e W* v s . V Q c u r v e seems 'co c o n s i s t o f two p a r t s : a t a s m a l l e r ' V ^ , W* i n c r e a s e s q u i c k l y w i t h V'. and l e v e l s o f f t o t h e l i n e w i t h a s m a l l e r s l o p e , a l i t t l e s m a l l e r t h a n t h a t o f t h e l i n e f o r k = 0.0, a s V Q g o e s h i g h e r . P o i n t P i n F i g . 10, w h i c h i s t h e p o i n t w h ere K ~„ a s i n F i g . 8 shows a minimum, l i e s i n t h e t r a n s i t i o n e f f ° ' a r e a b e t w e e n t h e s e two p a r t s . B e f o r e t h i s p o i n t P, ^-eff d e c r e a s e s w i t h V , w h i c h means p e a k h e i g h t s o f A a n d C i n c r e a s e r e l a t i v e t o t h a t o f B, t h e r e f o r e t h e c o n t r i b u t i o n o f t h e d e v i a t i o n f r o m symmetry i n b a n d s A a n d C t o W* i n c r e a s e s . C o n s e q u e n t l y W* i n c r e a s e s more q u i c k l y t h a n t h e c a s e f o r k =0. A f t e r t h e - s - 4 8 -p o i n t P, as K g f f i n c r e a s e s with V Q, i . e . , peak h e i g h t s of A and C decrease r e l a t i v e to that of B., so that the i n t r i n s i c i n c r e a s e of W* with VQ i s depressed. T h e r e f o r e the slope of the second p a r t becomes l e s s than that of the case f o r k Q = 0, where no r e a c t i o n i s occurring, and hence there i s no d e v i a t i o n from symmetrical bands. The f a c t t h a t p o i n t P's f o r k ' = 10 and k = 50, k Q = 200 seem to l i e roughly a t the same value of W*, suggests the importance of V/*, and hence the extent of o v e r l a p p i n g i n the dependence of K »» on'V„. * e f f o The flow r a t e dependence i s shown i n F i g . 12 and F i g . 13. As i s expected, the h i g h e r the flow r a t e , the higher the (and K , and t h i s dependence becomes s m a l l e r as k i n c r e a s e s , e f f o Obviously, f o r the case of k = 00 , K „„ (and K »~*) does, not J ' o . ' e f f e f f depend on w, the flow r a t e . The feed p u l s e volume V Q i s r a t h e r d i f f i c u l t to evaluate e x p e r i m e n t a l l y , because, f o r example, the k i n e t i c s o f e v a p o r a t i o n of the l i q u i d sample i n the i n j e c t o r are not d e f i n i t e l y known.' F i g . 10 shows the s i n g l e - v a l u e d f u n c t i o n a l r e l a t i o n s h i p between V and W*, whenk i s f i x e d , and W* i s e a s i l y evaluated from the o ' 0 • ' J chromatogram obtained. So, K g £ £ and K g f f * were p l o t t e d a g a i n s t V/*, i n F i g . 14 and F i g . 15. The l a t t e r i s of p r a c t i c a l value f o r comparison with the experimental r e s u l t s presented i n the next chapter. CHAPTER.. 3 The IBr - I P - B r P system - S - 5 0 -The theory developed i n the a p p l i e d to e x p l a i n the behaviour a column wi t h a Kel-F s t a t i o n a r y previous chapter i s here of the system Br^ - I B r - ^ on phase supported.on t e f l o n powder. - s-51 -•1.. EXPERIMENTAL APPARATUS F i g . 19 (a) F i g . 19 (b) and F i g . 20 . The apparatus•was e s s e n t i a l l y a gas chromatograph, but the whole system was made of pyrex g l a s s except that the detector block was s t a i n l e s s s t e e l s s 3 l 6 . The s i g n a l was recorded on a Leeds and Northrup Speedomax H recorder through the attenuator c i r c u i t of a V a r i a n AEROGRAPH A - 9 0 - P 3 gas chromatograph. Detector : Thermal c o n d u c t i v i t y c e l l w i t h t e f l o n c l a d tungsten filaments (Gov; Mac). In.jector : F i g . 20 ( a ) . Column : 5m/m I.D.x l±.5m pyrex tubing. 10% Kel-F grease No. 90 (halofluorocarbon) coated on chromosorb T ( t e f l o n powder). This was made as f o l l o w s : Kel-F grease No.90 was d i s s o l v e d i n chloroform, and chromosorb T was added and mixed. The mixture was . d r i e d over a water bath, and a mixture of acetone and water (1:1) was added to make a s l u r r y . The s l u r r y was passed through pyrex tubing p r e v i o u s l y bent i n t o shape f o r f i t t i n g i n t o the oven of the gas chromatograph w i t h a g l a s s wool plug at the end of the tubing, and w i t h s u c t i o n a p p l i e d by means of an a s p i r a t o r . Dry chromosorb T with Kel-F grease i s d i f f i c u l t to pack evenly i n the column probably because of e l e c t r o s t a t i c charge on the surface. - S-52 -4 way (1} cock c f . F i g I n j e c t o r Oven - (a) -<F-(He) -' Detector r I n j e c t o r ^ - ( H e ) Oven - Fig..19 Apparatus - S-53 -He+Sample \ Pyrex Heater &amJ?^-e—Silicone septum J v T e f l ° n F e r r u l e ] J ^ ( H e ) ^ P y r e x # Swagelok bulk head •Swagelok Nut Union o° ^ T e f l o n F e r r u l e Pyrex Tubing Sketch M a t e r i a l ; Brass Pyrex G l a s s g l a s s powder, pass #25 s i e v e (a) I n j e c t o r c 7\ C J Inner p a r t T e f l o n no grease B a r r e l Pyrex (b) 4-way Cock - F i g . 20 -- S-54 -The mixture of acetone and water (1:1) i s g u s t • enough to wet the powder, yet i t doesn't d i s s o l v e Kel-F grease on the surface. ." MATERIALS B r 2 : Baker Analysed Reagent grade. 1 2 : Baker Analysed-Reagent•grade. Ethylene dibromide : B.D.H. reagent grade. These were used without f u r t h e r p u r i f i c a t i o n . Samples : S o l u t i o n s of equimolar bromine and i o d i n e i n ethylane dibromide were prepared v o l u m e t r i c a l l y . A : 5 . 0 3x 10" 4mole/ml B : 1 . Q 2 5 * 10"^mole/ml C : 1.375 x 1 0 - Z fmole/ml D : 0 . 9 6 3 ^ 10" 4mole/ml STANDARD' OPERATING CONDITION OF GAS CHROMATOGRAPHY Filament current : 200.0mA Flow r a t e : 76.0ml/rnin at the e x i t (room temp . ,1 atm) Temperature I n j e c t o r : 180°C Column : 9 3 - 9 5 ° C Detector : 130°C Pressure before the column : 1 .5atm. CALIBRATION S o l u t i o n s of bromine and i o d i n e separately i n ethylene dibromide were q u a n t i t a t i v e l y i n j e c t e d by means of Hamilton microsyringes and the peak areas were measured by a d i s c ' ' i n t e g r a t e r attached to the Leeds and Northrup Speedomax H recorder. - S-55 -Both l a y on the same l i n e a r c a l i b r a t i o n curve of peak area versus moles i n j e c t e d . The mixture of B r ^ and 1^ was i n j e c t e d q u a n t i t a t i v e l y . This time three overlapping peaks appeared on the chromatogram- (see F i g . 2 1 ) . The f i r s t and the t h i r d c o r r e -sponded to. B r 2 and I 2 r e s p e c t i v e l y , and the center one was assigned to I B r . The t o t a l area of the three.peaks was p l o t t e d against the t o t a l molar amount.of bromine and i o d i n e added. These -plots are on the same l i n e that was obtained f o r I 2 and. B r 2 s e p a r a t e l y . Therefore the c a l i b r a t i o n . c u r v e , f o r I B r was concluded to be .the-same s t r a i g h t l i n e as B r 2 and I 2 . PROCEDURES (1) Samples A,B,C, and D were introduced q u a n t i t a t i v e l y through the i n j e c t o r shown i n F i g . 19 ( a ) . The flow.rate was v a r i e d from 20.7ml/min to. 122.0ml/min, the temperature at the i n j e c t o r , . 150°C-189°C, and the temperature of the column 88°C-100°C. : '. • • (2) A s e r i e s of experiments was c a r r i e d out usi n g a gas phase sampler. The equimolar mixture of B r 2 and I 2 was warmed up to 60°C, and the vapor phase over t h i s mixture, (which was considered to be almost completely I B r ) , was mixed w i t h N 2 i n the bulb B i n F i g . 1 9(a). This mixture of N 2 and IBr was l e d to the gas sampler at v a r i o u s pressures, and then t h i s sample was fed i n t o the gas. chromatograph by t u r n i n g cock (1) f i r s t and then cock (2). by 90°. (3) TO demonstrate unequivocally that r e a c t i o n was o c c u r r i n g on the column, the apparatus i n F i g . 19(b) was used to r e c y c l e a pa r t of the eluted m a t e r i a l through the column. 1 0 ^ 1 of - s-56 -s o l u t i o n (A) was i n t r o d u c e d through the i n j e c t o r . The s t a r t of the appearance of the s i g n a l was observed and the middle part of each peak was trapped i n the sample loop by t u r n i n g the four-way cock (2) by 9 0 ° at the appropriate moment. A f t e r the e l u t i o n was over, the sample trapped i n the sample loop was fed i n t o the gas chromatograph again by t u r n i n g cock (1) by 90° . The volume of the sample loop was 5 .7ml and the volume of each peak was about 60ml. For a l l procedures described above, the r e s u l t i n g chromatogram of three overlapping peaks was d i v i d e d at the two minima, K e f f * was c a l c u l a t e d according to equation (26) and W* was obtained at the same time. - S-57 -' 2 ' RESULTS AND DISCUSSION A t y p i c a l chromatogram i s shown i n F i g . 21 . Retention volumes at the standard c o n d i t i o n s extrapolated to V Q = 0 are tabulated i n Table 2 . - Table 2 Retention Volumes -Components A i r B r 2 I B r X 2 Retention Volumes (ml) 82.3 135.0 199,0 270.0 The number of t h e o r e t i c a l p l a t e s was c a l c u l a t e d to be 500 t, 80 from the 1^ peaks of the c a l i b r a t i o n according to eq. ( 1 8 ) , f o r the standard c o n d i t i o n . Random e r r o r s i n c a l c u l a t i n g K g f f * from chromatograms are f a i r l y l a r g e due to the d i f f i c u l t y of p i c k i n g the exact minimum between I B r and I ^ peaks. This e r r o r i s estimated to be about + 15%. The e r r o r i n c a l c u l a t i n g W*-, the band width of the center peak, i s a l s o f a i r l y l a r g e and i s about * 10%. ...After the. p r e p a r a t i o n of s o l u t i o n s , 10fj,l. of s o l u t i o n (B) was i n j e c t e d from time to time, and there was found to be no time dependence as i n Table 3 . - Table 3 Time dependence of s o l u t i o n (B) -Time a f t e r p r e p a r a t i o n (hrs) 2, 0 2.5 3 23 K * e f f 4 .5 4 .5 4 . 7 4 . 6 - s-58 -Flow r a t e e f f e c t and i n j e c t o r temperature e f f e c t on K „„* . e f f w i t h 10^1 s o l u t i o n (B) are tabulated i n Table 4 and Table 5. The r e s t of the c o n d i t i o n s were kept standard. - Table 4 Flow r a t e e f f e c t -Flow r a t e (rnl/min) 20.7 28.5 37.4 40.0 53.9. 76.0 96.7 122.0 K * e f f 4.2 3.8 4.4 4.6 4.0 4.5 4.3 5.2 - Table 5 I n j e c t o r temperature e f f e c t -I n j e c t o r temp. °C 150 160 172 189 * e f f .5.5 5.2 4.7 . 6.0 W i t h i n experimental error,, n e i t h e r flow r a t e nor i n j e c t o r temperature has e f f e c t on &eff*' The dependence of K^^* on the c o n c e n t r a t i o n of the sample and on the t o t a l amount of the sample i s p l o t t e d i n F i g . 22(a) f o r the l i q u i d sampling (procedure 1). For the gas sampling (procedure 2) the n i t r o g e n peak was used as an i n n e r standard, and K e££* was p l o t t e d i n F i g . 23(a) against the peak height of Ng, which was shown by c a l i b r a t i o n to be l i n e a r to the amount of Ng. F i g . 22(b) and F i g . 23(b) are to demonstrate the l i n e a r i t y of the t o t a l area of three peaks to the t o t a l molar amount, which assures that the column i s i n working c o n d i t i o n . For both the l i q u i d sampling and the gas sampling, concentrations of samples have no - S - 5 9 -e f f e c t on K ~ . p * , - but the t o t a l amount of samples a f f e c t s K „„* e f f ' • e f f very s t r o n g l y . . K e j f * i s p l o t t e d against V/* i n F i g . 2 7 . For both the l i q u i d sampling and the gas sampling, p l o t s are on a s i n g l e smooth l i n e , showing that t h i s phenomenon i s governed by the center band width which i s c l o s e l y r e l a t e d to the overlapping of the three peaks. F i g . 2 4 shows no e f f e c t of the column temperature, very l i t t l e i f any. F i g . 2 5 i s the schematic r e p r e s e n t a t i o n of the r e s u l t s of the experiment (procedure 3 ) using the apparatus shown i n F i g . 1 9 ( b ) . ' The sample c o l l e c t e d from the f i r s t peak, as i n F i g . 2 5 ( a ) , gave only a Br^ peak, when i t was r e - f e d . S i m i l a r l y the t h i r d peak gave only a 1^ peak. The sample from the center peak, however, showed, as i n F i g . 2 5 ( c ) , three peaks again w i t h a diminished value of K ,,.,* ( 1 . 5 i n place of 5 ) . This and a l l the other r e s u l t s e f f x s t r o n g l y suggests that the decomposition r e a c t i o n of I B r i s o c c u r r i n g on the column,- that-the r e a c t i o n r a t e i s so high that. . the flow r a t e does not a f f e c t K^^* much and at any i n s t a n t three components, Br^, I B r and 1^, are i n r e a c t i o n e q u i l i b r i u m ; t h e r e f o r e incomplete s e p a r a t i o n . ( o v e r l a p p i n g of three peaks) i s the main f a c t o r to determine. %-eff*> a n c* the a c t i v a t i o n energy of t h i s r e a c t i o n (IBr.= 1^ + Br^) i s very s m a l l . K gf£* was c a l c u l a t e d according to the theory developed i n the previous chapter f o r i n f i n i t e r e a c t i o n r a t e . For r e t e n t i o n volumes of ea.cn gas, the values i n Table. 2 Were used, and f o r the number of p l a t e s , 5 0 0 . For K ., the Gibbs free energy change of - S-60 -the r e a c t i o n , I 2 ( g ) + B r 2 ( g ) »2IBr(g), was c a l c u l a t e d to be AG°= -2 ,720 - 2.80T (cal/mole), by using standard thermochemical data ^'. For the standard o operating temperature, 93 C, K g = 1 6 9 . 0 . W i t h i n f a i r l y wide l i m i t s , the composition of the three components i n the feed pulse does not a f f e c t K „„* much. For the feed pulse with A = 5 .00 , BQ•= 5 . 0 0 , C Q = 5 .00 , and that w i t h A Q = 1 .00 , B Q = 14-75 , C Q = 1.00 ( e q u i l i b r i u m composition at the column temperature), K ^ ^ ' s were 1.95 and 2 .09 r e s p e c t i v e l y f o r V Q A J ^ J P = 0 .0983 . For the r e s t of the c a l c u l a t i o n s , the equlibrium composition was used f o r the feed pulse. K „ * and K against V^/V„» T r i, the volume of c e f f e f f 0 o RAIR' the feed pulse normalized w i t h the r e t e n t i o n volume of a i r , i s i n F i g . 26 and the bandwidth dependence of K g f j * i s i n Fig. 2 7 . The c a l c u l a t e d band width was obtained by drawing the tangents at i n f l e c t i o n p o i n t s on the c a l c u l a t e d e l u t i o n curve. So there i s about 5% e r r o r i n the c a l c u l a t e d W*. For c a l c u l a t e d K -,,*, e f f ' there i s about 5% systematic, e r r o r due to the approximation i n i n t e g r a t i o n e t c . i n computation, judging from the d e v i a t i o n of the t o t a l amount i n the e l u t i o n curve from the t o t a l amount i n the feed pulse. C a l c u l a t e d values and obseved values i n F i g . 27 agree e x c e l l e n t l y . To get the order of magnitude of the lower l i m i t of the r e a c t i o n r a t e constant k, i t i s safe to say; (ml) k > 200 ( (min)(concentration u n i t ) - s-61 -Comparing F i g . 22 and F i g . 26, 10 JU.1 of the solution corresponds to 0 .21 i n V /V D, T O unit, i . e . , V = 0 .21x V D i T D = 0 .21% 82.3ml o RAIR ' ' o RAIR = 1 7 . 3 m l . Then the actual concentration i n the feed pulse i s for solution D ; 0.963% 10"^(mole/ml) % 10 * 10~ 3(ml ) / 17 .3(ml) = 5-3 x 1 0" 5(rnole/l). A l l the information necessary i s then :- from eq. ( 2 5 ) , and Table 2 , k Q = k K 2 V L . , VRAIR = 8 2 ' 3 = * y > VRB = 1 9 9 - ° = n ( V G + W ' nK BV L = 116,7ml , . K _ 116.7 1VB " . nV L nV^ i s . the t o t a l volume of Kel-F grease coating, which i s estimated as 3ml , since t o t a l weight of Kel-F grease was 7 g r . (density of Kel-F grease No..90 being 2 .4g/ml). K B = 39 , k Q = k K | V l = k ( 3 9 ) 2 ( ^ g % > 200( m i n ( c gL.unit))> • * k > 2 2 . ( min (cone, unit) ^' In the c a l c u l a t i o n the concentration of feed pulse was: A Q = 1.00 (cone.unit), '•B = 14.75 (cone.unit) , CQ•= 1.00 (cone.unit), - S-62 -Total 16.75 (cone.unit). This corresponds to 5-5*10(mole/1) k > 2 2 • : , . (60sec)( mole/1) k > l x l o ^ (1/sec.mole). Second.order rate constants for reactions i n l i q u i d solutions 25") are commonly y ' of the order, k = 1 0 1 1 exp(-E A/RT) (1/mole sec) , and the value obtained as a lower l i m i t for k i n t h i s process indicates E^ <C 8.8 (kcal/mole) for the decomposition ZTBv-^l^r Br~ i n solution i n Kel-F grease. - S - 6 3 -IBr Attenuator change x. - Fig.21 TYPICAL CHROMATOGRAM -- S - 6 6 -5^0 ~ ' 1 0 . 0 ^ . 1 - S - 6 7 -- F i g . 25 Results from Procedure (3) Signal B r i ( a r b i t r a r y unit) (a) Br, REFEED Tiiis portion Trapped i n Gas iSA MPLER (c) ^1.5 ^In 10^1 of solution A EFEED RETENTION TIME (arb i t r a r y units) - s-68 -- S - 6 9 -F i g . 2 7 Band width vs. K * -° e f f r — • — l i q u i d sampling Observeds [ — x — g a s sampling c a l c u l a t e d -@ - k = 0 0 O.k 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 W*^HAIH BIBLIOGRAPHY - S -71 -( 1 ) J . J . van Deemter e t . a l . , Chem. Eng. S c i . , 5_, 2 7 1 ( 1 9 % ) . ( 2 ) A.J.P. M a r t i n e t . a l . , Biochem.J3_5_, 1 3 5 9 ( 1 9 4 1 ) . ( 3 ) L.Lapidus e t V a l . , J . Phys. Chem.,5_6, 9 8 4 ( 1 9 5 2 ) . ( 4 ) J . C. Giddings,. J . Chromatog., 3_, 4 4 3 (I960). ( 5 ) H.Yamazaki, J . Chromatog., 2J7_, 14 ( 1 9 6 7 ) . ( 6 ) T.Kambara, J . Chromatog., 1Q_, 4 7 8 ( 1 9 6 5 ) . T.Kambara et. a l . , I b i d , 2 1 , 3 8 3 ( 1 9 6 6 ) . ( 7 ) J.H.Olson, I b i d , 2 £ , 1 ( 1 9 6 7 ) . ( 8 ) R.T.Koks et . a l . , J . Am. Chem. Soc.,2Z» 5 8 6 0 ( 1 9 5 5 ) ( 9 ) J.A.Einwiddie (to Esso Research and Engineering Co.), U.S.Patent, 3 , 2 4 3 , 4 7 2 (March 2 9 , 196: ) . . ( 1 0 ) J.A.Dinwiddle e t . a l . , (to Esso Research and Engineereng Co.,), I b i d , 2 , 9 7 6 , 1 3 2 (March 2 1 , I96I). ( 1 1 ) E.M.Magee (to Esso Research and Engineereng Co.), Can. Patent, 6 3 1 , 8 8 2 (Nov. 2 8 , 1 9 6 1 ) , ( 1 2 ) S.Z.Roginsky e t . a l . , K i n . i Kat., 3_, 5 2 9 ( 1 9 6 2 ) . ( 1 3 ) D.W.Bassett e t . a l . , J . Phys. Chem.,.64_, 7 6 9 (I960). ( 1 4 ) G.A.Gosief e t . a l . , K i n . i Kat., 4_, 6 8 8 ( 1 9 6 3 ) . ( 1 5 ) E.M.Magee , Ind. Eng. Chem./Fund., 2 , 3 2 ( 1 9 6 3 ) . .. (16) H.Saito e t . a l . , Chem. Eng. (JAPAN), 29_, 5 8 5 ( 1 9 6 5 ) . ( 1 7 ) F.E.Gore , I & EC/Process Design & Dev., 6 , 10 ( 1 9 6 7 ) . ( 1 8 ) J.H.Schulman and W.D.Corapton, "Color Centers i n S o l i d s " Chapter IV, Pergamon Press, ( 1 9 6 3 ) . ( 1 9 ) W.B.Innes, " C a t a l y s i s " , v o l I I , P.H.Emett ( E d i t e r ) , P.I-P . 1 0 3 , Reinhold, ( 1 9 5 5 ) . - S -72 -(20) E.Mollwo, :Ann. der Physik, 2 £ , 394 ( 1 9 3 7 ) . (21) For example, I.Lysyj e t . a l . , A n a l . Chem., 3_5_, 90 ( 1 9 6 3 ) . (22) J . F . E l l i s and G.Inuson, "Gas chromatography 1958" P.300 Butterworths ( 1 9 5 8 ) . (23) F.D.Rossini e t . a l . , " S e l e c t e d values of Thermodynamic P r o p e r t i e s " , N . B . S . c i r c u l a r 5 0 0 , ( 1 9 5 2 ) . -JANAF Thermochemical Data. J.McMorris e t . a l . , J.Am. Chem. S o c , 5J5, 2625 ( 1 9 3 1 ) . (24) M.Kocirik, J.Chromatog., 3_0, 459 ( 1 9 6 7 ) . (25) E.A.Moelwyn-Hughes, "The k i n e t i c s of Reactions i n S o l u t i o n s " , P.76, Oxford, ( 1 9 4 7 ) . Appendix A Computer Programs - S-74 -Program No.l ( w r i t t e n by P.Madderom) PLATE THEORY f o r the r e a c t i o n on chromatographic columns wi t h a f i n i t e r e a c t i o n r a t e Problem : To ob t a i n a numerical s o l u t i o n to the set of equations: dA. l 1 dt KA d B i 1 dt ' K B d C 1 1 dt " KC v(A . _ 1 - A ±) + R v ( B i _ 1 - B j L) - 2R V ( C i - l ) - C.) + R , where R = K Q(B 2. - K e A i C i ) , fo r i = 0,1,...... ,N where N i s the number., of p l a t e s . Boundary c o n d i t i o n s : A Q ( t ) = f A ( t ) B Q ( t ) = f B ( t ) > f o r a l l t > O C Q ( t ) = f c ( t ) , f o r t ^  0, A. = B. = C. = 0 , ' S o l u t i o n . .: Hamming method was used to solve the d i f f e r e n t i a l equation. For a s i n g l e equation, y 1 = f ( x , y ) , the method i s : 112 'j+l " A j+l ' 121 ( C j " P j ) 1 c j + i = u [9Y. - Y._2 + 3 h ( f ( m j + 1 , X . + 1 ) + 2Yt - Y j ^ ) ] - S -75 -Y j + 1 " C j + 1 " 121 ( C j + l " P j + 1 ) . I J + 1 '= * C i j + 1 , x j + 1 ) That i s , given 4 consecutive p o i n t s ( i n c l u d i n g the d e r i v a t i v e s ) on the s o l u t i o n curve, the f i f t h p o i n t i s obtained using the above equations. The extension to more equations i s s t r a i g h t forward. The Computer Program: . The program i s a FORTRAN IV main r o u t i n e w i t h 5 subprograms.. There are three l a b e l l e d .COMMON areas: /CHEAT/, /PARAM/,. and /FUDGE/. They are of lengths 1, 3006 , and 2 r e s p e c t i v e l y . Deck: M I N . Core: l^OOOg = 6 1 4 4 1 0 This program c a l l s INPUT, HAM, and OUTPUT, i t a l s o sets I I i n /CHEAT/. V A R : the l i m i t below which outputs are considered to be zero: Deck: INPUT. Core: 6313 = W l 0 SUBROUTINE INPUT (I,N,M,H,) ' Where: I i s an i n t e g e r v a r i a b l e whose allowed value range i s from 1 to 5 i n c l u s i v e . I t s p e c i f i e s the statement number at which the subprogram begins. N i s the number of p l a t e s - S-76 -M i s the number of time steps H i s the time step s i z e . CNNNN i s the factor which determines the maximum value of time axis. The subroutine sets a l l variables i n /FUDGE/ and /PARAM/. Note: Array A(3,1000) holds the values of A ^ , a n d C^ __-^ . That i s , the concentrations at the previous plate. Time zero i s A ( I , 5 ) • statement reads. format 1 VRAIR> VRA> VRB> VRC> K e . 8 F 1 0 ' ° 2 N 15 3 A Q, B Q, C Q 8F10.0 4 K Q, v 8F10.0 5 v Q , CC, M 2F10.0, 15 CC = the r i s e f r a c t i o n used by GO. This i s the fra c t i o n of the pulse width i n which the concentration r i s e s from 0 to i t s i n i t i a l value. Deck: HAM. Core: 471g = . 3 1 3 1 0 SUBOUTINE HAM (N,H,X,Y,DY,T,F,) where: N = number of equations H = step size X = independent variable Y = array of dependent variables DY = array of derivatives at previous points. T = temporary work space F = function name - S - 7 7 -HAM expects Y ( I , J ) J = 1,4 Dy(I,J) J = 1 ,3 1 ( 1,J) J = 1 ,2 to be p r o p e r l y set on entry. • • on e x i t , Y(I,5) contains the next s o l u t i o n p o i n t DY and T are f i x e d up f o r the next c a l l . Deck: FUN. Core: 237g = 1 5 9 1 0 SUBROUTINE FUN(N,Y,MY,DY,MDY,X) . . • N,Y, and X are the same as i n HAM. . FUN uses /PARAM/ and /CHEAT/ to c a l c u l a t e DY(1,MDY), DY(2,MDY), and DY(3,KDY). Deck: G$ Core: l ^ g = • FUNCTION Gj6 (T,I,H) . This f u n c t i o n uses /FUDGE/ to give an input pulse of the form: 1 / CC*TO CC*T0 >• TO — - S-78 -Deck: OUT. Core: 1146g = 6 1 4 1 0 SUBROUTINE OUTPUT(M,H,) This subroutine p r i n t s the concentrations at the l a s t p l a t e as a f u n c t i o n of time. I t c a l c u l a t e s the areas under the curves and the r a t i o : SB 2/(SA*SC) I t a l s o adds a l l concentrations and c a l c u l a t e s a r e v i s e d r a t i o . - S-79 -SIBFTC MAIN. DIMENSION A ( 3 , 3 5 0 0),B ( 3 , 3 5 0 0),DY ( 3 , 3),T ( 3 , 5 ) ' EXTERNAL FUN LOGICAL D0D0 C0MM0N/CHEAT/ II C0MM0N/PARAM/ AA,AB,AC,A,V,AKO,AKE MMM=1 READ(5,100) VAR 100 F0RMAT(E1O.5) 1 CALL INPUT(MMM,N,M,H) WRITE(6,101) VAR 101 FORMAT(7H0VAR =,E10.5) D0 6 1=1,N . . D0Dj0=. FALSE. D0 3 J=l,3 D0 2 JJ=1 ,4 2 B(J,JJ)=0.0 D0 3 JJ = l , 3 DY(J,JJ)=0.0 3 T(J,JJ)=0.0 ' LIM=FLOAT(I)/(AC*V*H> D0 4- J=5,M II=J IF(DODO) GO TO 30 IF(A(1,J).LE.VAR) GO TO 31 D0D0=.TRUE. G0 T0 30 31 B(1,J)=0 .0 B(2,J)=0.0 B(3,J)=0.0 Gfl T0 4 30 CALL HAM(3,H,X,B(l,J-4),DY,T,FUN) IF(J.GT.LIM.AND.B(3,J).LE.VAR) GO TO 40 4 CONTINUE 40 CONTINUE . D0 5 J=1,M D0' 5 J J = l , 3 - s-8o -5 A(JJ,J)=B(JJ,J) 6 C0NTINUE CALL OUTPUT(M,H) ST0P . ' END • -. SIBFTC INPUT. • SUBROUTINE INPUT(I,N,M,H) DIMENSION A(3,3500) C0MM0N/FUDGE/ Cl, C2 C$4M0N/PARAM/ AA, AB, AC, A, V, AKO, AKE G0 T0 (1,2,3 ,4,5),I 1 READ(5,100) VRAIR,VRA,VRB,.VRC,AKE WRITE ( 6,, IQL) VRAIR, VRA, VRB, VRC, AKE 2 READ(5,102) N ,CNNNN WRITE(6,103) N . ,CNNNN 3 READ(5,100) AO,BO,CO WRITE(6,104) AO,BO,CO 4 READ(5,100) AKO,V WRITE(6,105) AKO,V 5 READ(5,106) V0,CC,M WRITE(6,107) VO,CC,M AN=N . . . AKO=AKO/AN A. A=AN/VRA AB=AN/VRB AC=AN/VRC TMAX=CNNNN*VRC/V H=TMAX/FLOAT(M) MM=M+1. T0=V0/V C1=T0*CC C2=T0*(1.0-CC) . 6 11=1, MM DJZJ 6 J=l,3 - S-81 -6 A(J,I)=0.0 NN=5+INT(T0/H) D0 7 11=5, NN C = G / ( T O , I I - 5 , H ) A(1,II)=A0*C A(2,II)=B0*C 7 A(3,II)=C0*C WRITE(6,108) H,TO RETURN 100 FORMAT(8F10.0) 101 FORMAT(1H1, 8HVRAIR = ,F10.1,11H ML VRA = ,F10.1,11H ML VRB= , 1F10.1,11H ML VRC = ,F10.1,3H ML/6H0KE = ,F10 .5) 102 F,0RKAT(I5, F10.0) 103 F0RMAT(17H0N0.OF PLATES = ,I5,9HCNNNN= ,F10 .5) 104 FJ2TRMAT(6H0A0 = ,F10.1,5H BO = ,F10.1, 1611 CO = ,F10.1) 105 F0RMAT( 6H0K0 = ,F10.4,5H V = ,F10.2,7H. ML/MIN) 106 FjZfRMAT(2F10.0,I5) • 107 F0RMAT(6HOVO = , F10.3, 21H ML . RISE FRACTION =,F10.3,6H M,= , 15) 108 F0RMAT(5HOH = ,E16.7,HH MIN TO = ,El6.7,4H MIN ) END SIBFTC HAM. SUBROUTINE HAM(N,H,X,Y,DY,T,F) EXTERNAL F DIMENSION Y(N,5),DY(N,3),T(N,5) 111=1.3333333*H II2=3.0*H C PREDICT D0 1 1=1,N 1 T(I,3)=Y(I,1)+H1*(2.0*(DY(I,1)+DY(I,3))-DY(I,2)) C • MODIFY Dl5 2 1=1, N ' 2 T(I,4)=T(I,3)+0 .92561983*(T(I,2)-T(I,1)) C CORRECT CALL F(N,T,4,T,5,X) D0 3 1=1,N - S -82 -3 T(I , 5 ) = 0 . l 2 5 * ( 9 . 0*Y(I J 4)-Y(I,@)+H2*(T(i , 5 ) + 2 . 6*DY(I , 3)-DY(I , 2 ) ) ) C FINAL ANSWER D0 k 1=1,N V h Y(I , 5)=T(I , 5)-0 . 0 7 4 3 8 0 1 7*(T(I , 5)-T(I , 3 ) ) C . CLEAN UP C0 5 I=1,N DY(I , 1)=DY(I , 2 ) 5 DY(I , 2)=DY(I , 3 ) CALL F(N,Y,p,DY,3,X) D0 6 1=1 ,N . . . T ( I , 1 ) = T ( I , 3 ) 6 T(I,2)=T(I , 5 ) RETURN '• . END SIBFTC FUN. SUBROUTINE FUN(N,Y,MY,DY,MDY,X) DIMENSION Y(N,MY),DY(N,MDY) DIMENSION A ( 3 , 3 5 0 0 ) C 0MM0N/ PA RAM/ AA,AB,AC,A,V,AKO,AKE C0MM0N/CHEAT/ I I R=AK0*(Y(2,MY)**2-AKE*Y(1,MY)*Y(3,MY)) . ... D Y (1, MDY ) = A A * ( V * ( A (1 , 1 1) -Y (1, MY.) ).+R ) DY(2,MDY)=AB * (V*(A(2,II)-Y(2,MY))-2 . 0*R) DY (3,MDY)=AC * (V*(A (3 ,H)-Y (3,MY))+R ) RETURN END $IBFTC G0 FUNCTION G0(T,I,H) DATA PI / 3 . 1 4 1 5 9 2 7 / COMMON/FUDGE/ -01,02 • X=FLOAT(I)*H IF(X.LT.Cl) IF(X.GT.C2) GO TO 2 G0=1.O RETURN 1 ' -. G0=O.5*(1.O-COS(X*PI/C1)) RETURN - S-83 -2 G0=O.5*(1.O+COS((X-C2)*PI/C1)) RETURN END 8IBFTC OUT. SUBROUTINE 0UTPUT(M,H) DIMENSION A (3 ,3500),S1 (3),TM (3),IM (3),T (3 ) : C0MMJ0N/PARAM/AAA,AB,A C,A,V,AKO, AKE WRITE(6,100) (A(1,J),J=1,M) WRITE(6,101) (A(2,J),J=1,M) WRITE(6,102) (A(3,J)|J=1,M) D0 1 1=1,3 S1(I ) = 0 . 0 D0. 1 J=1,M 1 S1(I)=S1(I)+A(I,J) D0 3 1=1,3 TM(I)=A(I,1) IM(I)=1 D0 2 J=2,M IF(TM(I) .GE.A(I,J)) GOTO 2 IM(I)=J TM(I)=A(I,J) 2 CONTINUE 3 CONTINUE D0 4 1=1,3 S1(I)=H*S1(I) 4 T(I)=FL0AT(IM(I ) - 5)*H R1=S1(2)**2/(S1(1)*S1 (3 ) ) ' , WRITE (6 , 104)(S1(I),T(I),I=1 , 3),R1 D0 5 1=1,M 5 A(l,I)=A(l',I)+A(2,I)+A ( 3,D WRITE(6,103) (A(1,I),I=1,M) IM1=IM(1) IM2=IM(2) IP1=IM1 AM=A(1,IM1) . - S - 8 4 -D0 6 I=IM1,IM2 .. IF(A(1,1).GE.AM ) G0 6 IP1=I AM=A(1,I) 6 CONTINUE IM1=IM.(.2) IM2=IM(5) IP2=IM1 AM=A(1,IM1) D07 I=IM1,IM2 IF(A(1,I).GE.AM) G0 Tjtf 7 .. IP2=I AM=A(1,I) 7 CONTINUE •T1=FL0AT(IP1-5)*H T2=FL0AT(IP2-5)*H WRITE(6 ,105) T1,T2 S1(1)=0.0 W 8 l = l , I P l 8 S1(1)=S1(1)+A(1,I) Sl ( 2 ) = 0 . 0 W 9 I=IP1 , IP2 9 S1(2)=S1(2)+A(1,I) Sl ( 3 ) = 0 . 0 W 10 I=IP2,M 10 SI (3)=S1 (3)+A (1,I) S l ( l ) = H * ( S l ( l ) - p . 5 * A ( l , I P l ) ) S1 ( 2)=H*(S1 ( 2 ) - 0 . 5*U ( 1 , I P 1)+A ( 1 , I P 2 ) ) ) S1(3)=H*(S1(3)-0.5*A(1,IP2).) R1=S1(2)**2/(S1(1) *S1 (3 ) ) WRITE(6,106) Sl(l),S1 ( 2 ),S1 ( 3 ),R1 RETURN . 100 F)2RM/ iT(5H0A(T)/(lX ,10F12 .5)) 101 F / i a R M A T(5H0B(T)/(lX , 1 0 F 1 2 . 5 ) ) 102 Fj2RMAT(5HOC(T)/(lX ,10F12 .5 ) ) 103 F/0PJ-'IAT(9HOA+B+C(T)/.(1X,1OF12.5)) 104 FJ0AMAT(6HOSA = ,E20.8,6H TA = ,F12.5.,4H MIN/ - S -85 -1. 6H0SB = ,E20.8,6H TB = ,F12.5,4H MIN/' 2 6H0SC = ,E20.8,6H TC = ,F12.5,4H MIN/9H0RATI0 = ,E20.8) 105 . FORMAT(6H0T1 = ,F12 .5,HH. MIN T 2 = ,F12 .5 ,4-H MIN ) 106 F0RMAT(7HOSSA = ,E20.8,7H SSB = ,E20.8,7H SSC = ,E20 .8, 19H RATIO = , E20 . 8 ) END - S-86 -Program No.2 ( w r i t t e n by Y.Koga) PLATE Theory f o r the r e a c t i o n on chromatographic columns w i t h i n f i n i t e r e a c t i o n r a t e . Problem ': To o b t a i n a numerical s o l u t i o n f o r the set of equations : dA. v A dV dB, - + \ - A i - i =6 • ( l ) B d V dC h+ B i - B i - i = -2<5 •(2) v C d-V r - + c i - c i - i =6 •(3) B J =.K eA i G. -(4) f o r i = 0,1,2, KK, where KK i s the number of t h e o r e t i c a l p l a t e s . Boundary and I n i t i a l Conditions : at V = 0 : o=*v=*v„ A. i — A o B o - < C o -A o = 0 B o = 0 0 - S-87 -S o l u t i o n : F i r s t , l e t (5= °> a n d solve ( 1 ) , ( 2 ) , ( 3 ) , and then adjust the s o l u t i o n s to s a t i s f y eq. ' (4)- . A i(V) = exp(~V/v A) | V exp(V/v A) A ± - 1 — ( 5 ) B±(V) = exp(-V/v B) fV exp(V/v B) B±mm± (6) Jo B C±(V) = exp(-V/v c) ^ exp(V/v c) C ± - 1 - - - (7) o Ca l c u l a t e X according to : ( B ± - 2 X ) 2 ( A ± + X)(C j_ + X) ~ K e , and replace A. e t c . w i t h : i A i(V) = A i(V) + X * B i(V) = 3j_(V) - 2X C ±(V) = C ±(V) + X , and then go to A i + 1 , B i + 1 , C j _ + 1 c a l c u l a t i o n . For the numerical, c a l c u l a t i o n of ( 5 ) , ( 6 ) and ( 7 ) , Simpson's r u l e was used. D e f i n i n g V ( l ) = V(2) = 0 . 0 , A ( I , 1 ) = A(I,2) = B ( I , 1 ) = B(I,2) =-C(I,l) = C(L2) = 0 . 0 and s e t t i n g the o r i g i n of V at J = 3 y A ( I , J ) x exp(V(J)/v.) = \ J Y. d V 3 k v A - s-88 -AV 3 [(Y 3: + Yj) + k(Yk + - + Y j _ x ) +. 2 ( Y 5 + - + Y J : _ 2 ) ] ; J " 2 dV A(I,J - 2 ) x exp(V(J - 2)/v A) = \ Y R — f J 3 A = ~ ¥ " [ ( Y 3 + W + + - + Y J - 3 y + 2 ( Y 5 " + W l where, Yfc. = exp(V(K)/v A) A(I - 1,K) A ( I , J ) x exp(V(J)/v A) - A(I,J - 2 ) x exp(V(J - 2)A A) = [ e x p ( V ( J ) / v A ) A(I - 1,J) + Zfexp(V(J-l)/v A) A(I - 1,J - 1 ) + exp ( V ( J - 2 ) / v A ) A ( I - l , J - 2 ) ] .*, A ( I , J ) = exp ( - 2 A v/v A) A(I,J - 2 ) + [ A(I - 1,J) + 4exp(-Av/v A) A(I - 1,J - 1 ) + exp ( - 2 A V/v A) A ( I - l , J - 2 ) ] . S i m i l a r l y f o r B ( I , J ) and C ( I , J ) . For I = 1, equations ( 1 ) , ( 2 ) , and (3) can be solved with (5=0. For V(J) = V Q A ( I , J ) = A° [ 1 - exp(-V(J)/v A) J e t c . For V Q < V( J) A ( I , J ) = A° [ 1 - exp(-V 0/v A) ] e x p [ - ( V ( J ) - V Q ) / v A ] e t c . - S-89 -SIBFTC YOSHI. C REVISED PR0GRAM F0R PLATE THE0RY ..INFINITE REACTI0N RATE C SIMPS0N,S RULE F0R INTEGRATION C PLATE THE0RY WITH REACTI0N -C A STANDS F0R BROMINE C B STANDS F0R I0DINEBR0MIDE C C STANDS F0R I0DINE C AO, BO, CO, — I N I T I A L CONCENTRATION C VO —PULSE WIDTH 0F INJECTED SAMPLE C KK = NUMBER 0F THE0RETICAL PLATES C CNNN = FACT0R THAT DETERMINES UPPER LIMIT 0N V-AXIS C VRAIR ETC. = RETENTION V0LUMES 0F AIR ETC. . DIMENSI0N V ( 2 5 1 0 ) , A(2 , 2 5 1 0 ) , B(2,2510),C(2,2510) .. DIMENSI0N SS (2510) L0GICAL D0D0 READ(5,60) VRAIR,VRA, VRB, VRC 60 F0RMAT(4F1O.O) WRITE(6,6l) VRAIR, VRA, VRB, VRC 61 F0RMAT(9H0VRAIR = ,F10.3, 5X, 6HVRA = ,F10.3, 5X, 6HVRB = , F10.3 1 , 5 X , 6HVRC = , F10.3) READ(5,50) KK 50 F0RMAT(I3) WRITE(6 , 5D KK 51 F0RMAT(/1X,12HPLATE N0. =,IZf) READ (5,121) AO,BO,CO 121 F0RMAT(3F1O.O) WRITE (6,122) AO, BO, CO 122 F0RMAT(6HOAO = ,F10.6,5X,5HB0 = ,F10..6,5X,5HCO = ,F10.6) 1 READ(5,52) VO,DELTA,VAR ,AKE, CNNN 52 F0RMAT (5F10.0) WRITE(6,53) VO,DELTA,VAR,AKE 53 FORMAT(///IX, 1WULSE WIDTH = ,F6.1,5X,12HINCREMENT= ,F8.3 15X,6HVAR = ,E10.5 , 5X, 4HK = , F8.3/) AAKE=(AKE-4.0)*2..Q BAKE =( AKE-4.0 ) *4 • 0 • FKK=KK . • - s-90 -V A = V R A / F K K V B = V R B / F K K V C = V R C / F K K I N.C=I N T ((CNNN * V R C ) / D E L T A ) NN=INC+3 F N N = N N V M A X = D E L T A * ( F N N - 3 . 0 ) W R I T E ( 6 , 5 4 ) V M A X 54 F0RMAT (1X,7HVMAX = ,F10 . 4 / ) D0 1000 J=1 , N N A ( 1 , J ) = 0 . 0 B ( 1 , J ) = 0 . 0 C ( 1 , J ) = 0 . 0 A ( 2 , J ) = 0 . 0 B ( 2 , J ) = 0 . 0 C ( 2 , J ) = 0 . 0 1000 C0NTINUE D0 10 J = 3 , N N F J = J V ( J ) = D E L T A * ( F J - 3 . 0 ) I F ( V ( J ) . L E . V O ) GO T O 220 I F ( V ( J ) . G T . V O ) GO T O 221 220 A ( 1 , J ) = A 0 * ( 1 . 0 - E X P ( - V ( J ) / V A ) ) B ( 1 , J ) = B 0 * ( 1 . 0 - E X P ( - V ( J ) / V B ) ) C ( l , J ) = C 0 * ( 1 . 0 - E X P ( - V ( J ) / V C ) ) G0 T0 30 221 A ( 1 , J ) = A 0 * ( 1 . 0 - E X P ( - V 0 / V A ) ) * E X P ( - ( V ( J ) - V O ) / V A ) B ( 1 , J ) = B 0 * Q . 0 - E X P ( - V 0 / V B ) ) * E X P ( - ( V ( J ) - V O ) / V B ) G ( 1 , J ) = B 0 * ( 1 . 0 - E X P ( - V 0 / V C ) ) * E X P ( - ( V ( J ) - V O ) / V C ) I F ( C ( 1 , J ) . L E . V A R ) G0 T0 1001 30 D = A K E * A ( 1 , J ) + A K E * C ( 1 , J ) + 4 . 0 * B ( 1 , J ) E = A K E * A ( 1 , J ) * C ( 1 , J ) - B ( 1 , J ) * B ( 1 , J ) X = ( - D + S Q R T ( D * D - B A K E * E ) ) / A A K E A ( 1 , J ) = A ( 1 , J ) + X - s-91 -B (1,J)=B (1,J)-2 . 0*X C(1,J)=C(1,J)+X 10 C0NTINUE 1001 C0NTTr]UE AX=DELTA/VA BX=DELTA/VB CX=DELTA/VC EXA=EXP(-AX) EXB=EXP(-BX) EXC-EXP(-CX) EX2A=EXP(-2.0*AX) EX2B=EXP(-2.0*BX) EX2C=EXP(-2.0*CX) • FEXA=Af.0*EXA FEXB=4.0*EXB FEXC=4.0*EXC TAX=AX/3.0 TBX=BX/3.0 TCX=CX/3.0 D0 116 K=2,KK D0D0=.FALSE. LIM=FL0AT(K)/CX KP2=NN D0 115 J=3,NP2 IF(D0D0) G0 T0 15 IF(A(1,J).LE.VAR) G0 T0 16 D0D0=.TRUE. G0 T0 15 16 A(2,J ) = 0 . 0 B(2,J ) = 0 . 0 C(2,J )=0 .0 G0 T0 115 15 C0MTNUE A(2,J)=EX2A*A(2,J-2)+TAX*(A(1,J)+FEXA*A(1,J-1)+EX2A*A(1,J-2)) B(2,J)=EX2B*B(2,J-2)+TBX*(B(l,J)+FEXB*B(l,J-l)+EX2B*B(l,J-2)) - S -92 -C(2,J)=EX2C*C(2,J-2)+TCX*(C(l,J)+FEXC*C(l,J-l)+EX2C*C(l,J-2)) IF( 'J.GT.LIM. AND. C(2, J) .LE.VAR) G0 T0 150 DX=AKE *A(2,J)+AKE*C(2,J)+4.0*B(2,J) EX=AKE*A(2,J)*C(2,J)-B(2,J)*B(2,J) XX=(-DX+SQRT(DX*DX-BAKE*EX))/AAKE A(2,J)=A(2,J)+XX B(2,J)=3(2,J)-2.0*XX C(2,J)=C(2,J)+XX 115 C0NTINUE 1501 C0NTINUE D0 105 J=3,NP2 A(1,J)=A(2,J) B(1,J)=3(2,J) C(1,J)=C(2,J) . 105 C0NTINUE 116 C0NTINUE C STJMMATI0N 0F CHR0MA T0GRAM D0 21 J=3,NP2 21 SS(J)=A(2,J)+B(2,J)+C(2,J) AMAX=0.0 D0 27 J=3,NP2 • . IF(A(2,J).LE.AMAX) G0 T0 27 MA=J AMAX=A(2,J)" 27 C0NTINUE AMAX=0.0 D0 28 J=3,NP2 IF(B(2,J).LE.AMAX) G0T0 28 MB=J • AMAX=B(2,J) 28 C0NTINUE AMAX=0.0 D$ 29 J=3,NP2 IF(C(2,J).LE.AMAX) G0 T0 29 MC=J - S -93 -AMAX=C(2,J) 29 C0NTINUE . SA=0.0 SB=0.0 SC=0.0 D0 20 J=3,NP2 SA=SA+A(2,J) SB=SB+B(2,J) 20 SC=SC+C(2,J) SC=SC*DELTA SB=SB*DELTA SA=SA*DELTA T0T=SA+SB+.SC ' WRITE(6,106) SA.SB,SC,T0T R=SB*SB/(SA*SC) WRITE(6 ,107) R WRITE(6,103)(A(2,J),J=3,NP2) WRITE(6 ,103)(B (2,J),J=3,NP2) WRITE(6 ,103)(C(2,J),J=3 ,NP2) C DEVIDING PEAKS AMIN=SS(MA) J1=MA D0 22 J=MA,MB IF(SS(J).GE.AMIN) G0 T0 22 AMIN=SS(J) J1=J 22 C0NTINUE AMIN=SS(MB) J2=MB . • D0 23 J=MB,MC IF(SS(J).GE.AMIN) G0 T0 23 AMIN=SS(J) J2=J - S -94 -23 C 0 N T I N U E ' SA=0.0 SB=0.0 SC=0.0 D0 24 J=3,J1 24 SA=SA+SS(J) D 0 25 J=J1,J2 • . 23 SB=SB+SS(J) D0 26 J=J2,NP2 26 . SC=SC+SS(J). SA=(SA-0.5*SS(J1))*DELTA SB=(SB-0.5*(SS.(J1) + SS(J2)))*DELTA SC=(SC-0.5*SS(J2))*DELTA R=SB*SB/(SA*S.C) . T0T=SA+SB+SC WRITE(6,106) SA,SB,SC,T0T WRITE(6,107) R . WRITE(6,110) J1,J2 WRITE(6,103) (SS(J),J=3,NP2) 103 F 0 R l M A T ( l H O / ( l X , l O F 1 3 . 6 ) ) 106 F 0 R M A T (17H0INTEGRAL 0 F A = ,E15.8,5X, 117H INTEGRAL 0 F B = ,E15.8,5X, 217H INTEGRAL 0 F C = ,E15.8/7HOSUM = ,E15.8) 107 • F0RMAT(9HORATI0 = ,E15.8) 110 F0RI4AT(23HOB0UNDARIES ARE AT J = ,I4,5H AND ,14) G 0 T 0 1 END 

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