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UBC Theses and Dissertations

Studies of the binary fluid nitrobenzene-heptane Shelton, Jay Danny 1972

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STUDIES OF THE BINARY FLUID NITROBENZENE-HEPTANE by JAY D„ SHELTON B . S c , C o l o r a d o C o l l e g e , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n PHYSICS i n t h e Department of PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g ' t o the r e q u i r e d s t a n d a r d from c a n d i d a t e s f o r the degree o f MASTER OF SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA J u n e , 1972 In present ing th i s thes is in pa r t i a l f u l f i lmen t of the requirements fo r an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying o f th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i c a t i on o f th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i t y of B r i t i s h Columbia Vancouver 8, Canada Date ( i i ) A B S T R A C T The c o e x i s t e n c e c u r v e and d i f f u s i o n c o e f f i c i e n t f o r the b i n a r y l i q u i d s ystem h e p t a n e - n i t r o b e n z e n e have been o b t a i n e d u s i n g an o p t i c a l t e c h n i q u e i n w h i c h l a s e r l i g h t i s r e f r a c t e d t h r o u g h a q u a r t z sample c e l l . A n a l y s i s o f the r e s u l t i n g f r i n g e p a t t e r n s shows t h a t t he m o l a r c o n c e n t r a t i o n d i f f e r e n c e AX between the c o e x i s t i n g l i q u i d phases i s e q u a l t o („24+<,01)x ( T c - T ) ° 3 3 3 ± ° 0 0 5 o v e r t h e t e m p e r a t u r e range .0001° < (T -T) < 11° C , and t h a t t he d i f f u s i o n c o e f f i c i e n t D a t c r i t i c a l con-c e n t r a t i o n i s e q u a l t o (3.7+1.5)xlO" 7(T-T ) ° 7 0 ± ° 0 5 C m 2 / s e c o v e r the range .005° < (T-T ) < 2.7° C . I t was f o u n d t h a t the c o e x i s t e n c e c u r v e i s symmetric about X , and t h a t the d i f f u s i o n c o e f f i c i e n t as a f u n c t i o n o f c o n c e n t r a t i o n a t a g i v e n t e m p e r a t u r e i s a l s o s y mmetric about X^9 n e a r X^, The c r i t i c a l t e m p e r a t u r e T c = 19.07+ o05 °Co Temperature d i f f e r e n c e s n e a r T were known t o w i t h i n .0001 ° C „ The c r i t i c a l m o l a r con-c c e n t r a t i o n o f n i t r o b e n z e n e X = .470+.003. I n terms o f dimen-c — c~ L — = (3.40+.15) x ^ .d^+.uub o v e r the range 3.4x10-7 < £ < 3 . 7 x l 0 " 2 . ( i i i ) TABLE OF CONTENTS I „ I n t r o d u c t i o n , , 1 I I . B i n a r y F l u i d s . 5 I I I . G l a d s t o n e ' s Law. 9 I V . E x p e r i m e n t a l T e c h n i q u e . 13 V. C o e x i s t e n c e C u r v e . 24 V I . D i f f u s i o n C o e f f i c i e n t I . 31 V I I . D i f f u s i o n C o e f f i c i e n t I I . 40 V I I I . S e p a r a t i o n Time. 50 I X . C o n c l u s i o n . 55 1 I„ INTRODUCTION-; The c r i t i c a l p o i n t was d i s c o v e r e d i n 1869 when A n d r e w s ^ f o u n d t h a t c a r b o n d i o x i d e d i d n o t have any g a s - l i q u i d t r a n s i -t i o n s above BO^K. Over the l a s t c e n t u r y s c i e n t i s t s have s t u d i e d a wide range o f phenomena around v a r i o u s c r i t i c a l p o i n t s o Today we know t h a t many d i f f e r e n t s u b s t a n c e s show s u r p r i s i n g l y s i m i l a r c r i t i c a l b e h a v i o r , f o r i n s t a n c e : gas-l i q u i d t r a n s i t i o n s o f a s i n g l e f l u i d ; phase s e p a r a t i o n o f b i n a r y f l u i d m i x t u r e s ; o r d e r - d i s o r d e r t r a n s f o r m a t i o n s i n b i n a r y a l l o y s ; t e m p e r a t u r e dependence o f ( a n t i ) f e r r o m a g n e t i s m , p a r a -magnetism, and ( a n t i ) f e r r o - e l e c t r i c i t y ; s u p e r c o n d u c t o r - n o r m a l c o n d u c t o r t r a n s i t i o n s , and s u p e r f l u i d - n o r m a l f l u i d t r a n s i t i o n s . The g a s - l i q u i d t r a n s i t i o n i s o f c o u r s e the b e s t known. The e q u a t i o n o f s t a t e f o r an i d e a l g a s , pV = RT, (1) cannot e x p l a i n i t . Van d e r Waals* e q u a t i o n f o r an i m p e r f e c t g a s , 2 (p+a/V ) ( V - b ) = RT, (2) does g i v e a q u a l i t a t i v e l y c o r r e c t e x p l a n a t i o n , p r o v i d e d we r e a l i z e t h a t the c u r v y i s o t h e r m s i n s i d e t h e c o e x i s t e n c e c u r v e a r e u n s t a b l e and c o l l a p s e i n t o (2) h o r i z o n t a l l i n e s , a c c o r d i n g t o the M a x w e l l e q u a l a r e a r u l e . (See F i g . 1) . I f we l e t t h e d e n s i t y ^>=l/V, van d e r Waals' e q u a t i o n becomes (p+ao ) (l/r> — b) = RT.(3) A t the c r i t i c a l p o i n t , ^ = Cj c , p.= p^, and T = T c . By s o l v i n g the r e s u l t i n g c u b i c e q u a t i o n f o r £>c , and by r e q u i r i n g t h a t a l l t h r e e r o o t s be , 2 i d e n t i c a l , we o b t a i n b = l / 3 , a = 3 p c / r ^ , and R T c = 8a/27b. I n s e r t i n g t h e s e i n t o (3) and r e a r r a n g i n g y i e l d s 2 F i g . 1. P-V Diagram Thus we see that van der Waals' equation i s the same f o r a l l gases, i f expressed i n terms of ^ /c>c> T/T , and P/P c» This i s c a l l e d the "law of corresponding states." Expanding (4) i n a double power serie s about £ / p c = T/T c = 1 y i e l d s where p' = p ( ^ c , T ) . On the coexistence curve p = p'; the square bracket vanishes, and The two roots correspond to gas and l i q u i d phases. This construction i s consistent with the Maxwell equal-area r u l e . Hence, Also, according to van der Waals' equation, (5a) 1^—L- = 2 (5b) where and are the densities of the coexisting l i q u i d and vapor phases, and £ = |(TQ-T) /T J. However, a large (in mass of data^ ^ i s w e l l described by the equations = 3.5- (6a) and ^ * (V ^ = ^ + J . s e . (6b) c 4 I n the case of spontaneous ferromagnetism a s i m i l a r s i t u a t i o n p r e v a i l s . The Weiss theory p r e d i c t s t h a t M = ^ Q € ^ z but i n f a c t the exponent i s about o n e - t h i r d I t turns out t h a t the c l a s s i c a l t h e o r i e s of van der Waals and Weiss t a c i t l y assume that the i n t e r - m o l e c u l a r f o r c e s are long-range, whereas (2) i n f a c t they are short-range. ' We t u r n to the l a t t i c e - g a s model, i n which space i s d i v i d e d i n t o c u b i c l e s , each of which may c o n t a i n 1 or 0 p a r t i c l e s . Only adjacent p a r t i c l e s i n t e r a c t . This model i s mathematically i d e n t i c a l to the 3-dimensional I s i n g model, i n which each s p i n on a cubic l a t t i c e may be up or down, and only adjacent spins i n t e r a c t . This c o n c e p t u a l l y simple but mathematically d i f f i c u l t model has been s o l v e d n u m e r i c a l l y and p r e d i c t s an exponent equal to .313+.004."^ This i s c l o s e to the experimental v a l u e , and most p h y s i c i s t s f e e l t h a t the l a t t i c e - g a s model i s b a s i c a l l y c o r r e c t . 5 I I . _ BINARY FLUIDS. A b i n a r y f l u i d s u c h as h e p t a n e - n i t r o b e n z e n e s h o u l d be w e l L d e s c r i b e d by the l a t t i c e - g a s model, s i n c e n i t r o b e n z e n e has a p o l a r m o l e c u l e and heptane a n o n - p o l a r m o l e c u l e , c o r r e s -p o n d i n g t o f i l l e d and empty c u b i c l e s as f a r as i n t e r a c t i o n s a r e c o n c e r n e d . Our measurements have s u p p o r t e d t h i s assertion» The b e h a v i o r o f a b i n a r y (two) f l u i d system i s i n d i c a t e d i n F i g . 2. We l e t X denote the m o l a r c o n c e n t r a t i o n o f n i t r o b e n z e n e , u n l e s s o t h e r w i s e i n d i c a t e d . I f one a t t e m p t s t o c r e a t e a m i x t u r e w i t h c o n c e n t r a t i o n and t e m p e r a t u r e g i v e n by p o i n t A, t h e m i x t u r e s e p a r a t e s i n t o two phases whose c o n c e n t r a t i o n s a r e g i v e n by p o i n t s B and C. The c o e x i s t e n c e c u r v e i s d e f i n e d as the boundary o f the u n s t a b l e r e g i o n . Any m i x t u r e whose c o o r d i n a t e s l i e o u t s i d e the c o e x i s t e n c e c u r v e , and i n p a r t i c -u l a r any m i x t u r e a t t e m p e r a t u r e g r e a t e r t h a n T c > i s s t a b l e . The p r e s s u r e c o o r d i n a t e i s removed from c o n s i d e r a t i o n by . a l l o w i n g the s y s tem t o r e m a i n i n e q u i l i b r i u m w i t h i t s own v a p o r (no a i r ) . Thus the system i s c o m p l e t e l y s p e c i f i e d by knowledge o f X and T. Some b i n a r y f l u i d s have c o e x i s t e n c e c u r v e s w h i c h open upward, and a few even have c l o s e d c o e x i s t -ence c u r v e s . B u t t h e m a j o r i t y a r e as shown. Note t h a t the b i n a r y f l u i d d i a g r a m r e s e m b l e s the p-V d i a g r a m f o r a s i n g l e f l u i d . ' A c t u a l l y I t i s more n e a r l y l i k e a p — ^ d i a g r a m , s i n c e the d e n s i t y i s bounded, b u t volume i s n o t . 6 Suppose we w i s h t o f i l l a c e l l w i t h heptane and n i t r o -benzene ( a b b r e v i a t e d H and N) 0 Suppose the m i x t u r e s e p a r a t e s i n t o upper and l o w e r phases (U and L ) 0 The f r a c t i o n r i c h i n n i t r o b e n z e n e w i l l be the l o w e r , s i n c e n i t r o b e n z e n e i s d e n s e r than h e p t a n e . L e t f be the volume f r a c t i o n o f N, be the t o t a l volume o f N, and V,r be the t o t a l volume o f H, Then, ri — H N V, V V, i f the f l u i d s a r e i n c o m p r e s s i b l e , f u V + f L V L = V N C7) ( l - f u ) V u . + ( l - f L ) V L = V H . (8) M u l t i p l y .(7) by ( l - f L ) and (8) by f , Q~fL> f U V U + ( 1 " f L ^ f L V L = ^ I ? V N ' (9) f L ^ - V V U + ^ - f L > f L V L = f L V H (10) S u b t r a c t (10) f r o m (9) V u = f L ^ N + V - V N  f L " f U (11) B u t we must have 0 < Vy < V N + V R . (12) T h i s y i e l d s f U < A V N + V H (13) From t h i s we see t h a t any f r a c t i o n a l s e p a r a t i o n i s 8 p o s s i b l e , as long as one f r a c t i o n i s greater than the t o t a l average f r a c t i o n , and the other l e s s . Thus, i f the f l u i d s are mixed i n the c r i t i c a l concentration, the two layers can adjust themselves i n any way such that one layer i s r i c h e r than c r i t i c a l and the other poorer. In other words, we can map out the en t i r e coexistence curve using one c e l l f i l l e d with a c r i t i c a l mixture. I f the c e l l i s not f i l l e d properly, the meniscus w i l l r i s e or f a l l u n t i l i t reaches the end of the c e l l , as the temperature approaches T . 9 I I I o GLADSTONE'S LAW, S i n c e we i n t e n d t o e x t r a c t i n f o r m a t i o n by o p t i c a l means, we, must know the i n d e x o f r e f r a c t i o n o f a mixture„ T h i s i s p r o v i d e d by G l a d s t o n e ' s law, w h i c h says t h a t t he i n d e x o f r e f r a c t i o n o f a l i q u i d m i x t u r e i s p r o p o r t i o n a l t o the volume f r a c t i o n o f I t s components •> T h i s law may be e s t a b l i s h e d as f o l l o w s : suppose we have a v e s s e l c o n t a i n i n g a m i x t u r e o f N and Ho Suppose we m e n t a l l y s e p a r a t e the m i x t u r e i n t o i t s componentso Assume the o p t i c a l p a t h remains unchanged. N + H H N Now we see t h a t t he o p t i c a l p a t h l e n g t h t h r o u g h the v e s s e l i s p r o p o r t i o n a l t o n^V^ + n^V^. B u t . , r. . . _ o p t i c a l l e n g t h _ ^ ^ N + " H ^ H i n d e x o f m i x t u r e = v -,—., °,_ = r f — — - — • (14J r e a l l e n g t h V N + V H Twenty samples o f m i x t u r e were made and the i n d e x o f r e f r a c t i o n measured w i t h a r e f r a c t o m e t e r a t t e m p e r a t u r e s o f 19°, 2 2 ° , 2 3 ° , and 25 °Co The c h e m i c a l s were w e i g h e d w i t h a p r e c i s i o n b a l a n c e and the volume f r a c t i o n s were c a l c u l a t e d . L e t m and ^ denote the mass and d e n s i t y . v w t v H 1 + JUiL&L (is) 10 G l a d s t o n e ' s law i s V M + V H v - v o r When the d a t a were compared t o the p r e d i c t i o n s o f G l a d -s t o n e ' s l a w , i t was f o u n d t h a t the d e v i a t i o n s were n e v e r more th a n two p e r c e n t o f ( n ^ - n ^ ) . When the d i f f e r e n c e i s t a k e n between the i n d i c e s o f the l o w e r and upper p h a s e s , most o f t h i s d e v i a t i o n c a n c e l s o u t . There was no e v i d e n c e o f i l l e f f e c t s as the t e m p e r a t u r e approached T c from above. T h e r e -f o r e we a c c e p t G l a d s t o n e ' s law as adequate. I n o ur e x p e r i m e n t , l a s e r l i g h t i s p a s s e d t h r o u g h a q u a r t z sample c e l l and i s r e f r a c t e d i n t o a sequence o f f r i n g e s . I t i s easy t o see t h a t > i f f i s the number o f f r i n g e s , An = £1 , 0?) t where A n i s the d i f f e r e n c e i n i n d e x o f r e f r a c t i o n between the upper and l o w e r p h a s e s , 1 i s the t h i c k n e s s o f the c e l l , and ^ i s the w a v e l e n g t h o f the l a s e r l i g h t . We must use A n t o c a l c u l a t e A X . L e t UJ, ^ , and M denote the mole w e i g h t , d e n s i t y , and number o f m o l e s . Then, ( i s ) v ^ I I 11 where G l a d s t o n e ' s law becomes n = n H + i n ) P u t = 1-X^ and drop the s u b s c r i p t N„ The d i f f e r e n c e between l o w e r and upper phases i s We want t h i s i n terms o f A X and X c „ We have e x p e r i -m e n t a l l y e s t a b l i s h e d the symmetry o f the c o e x i s t e n c e c u r v e . We use t h i s f a c t . X L - X u = AX X L + X u - 3 X t X L = X. + P u t t i n g t h i s i n t o (20) g i v e s An -12 o r b 4 X A n - ~ x • a Ah AX2 - lo AX 4 * 4 * = O . b - M fc>*~ U c a n 4 A x - — A n We r e j e c t the p o s i t i v e square r o o t because A X must approac h z e r o as An approaches z e r o . S i n c e A f t £ . 1 , we can expand the r o o t . AX = : — ; 2 ' b . A Y - 4: A » + A n 3 b fc 3 -M A X - ^ . 3 U n ( l - /. 3 3 d r ) » J _ ( 3 The second term i n p a r e n t h e s e s i s always l e s s t h a n .01 i n our e x p e r i m e n t s . 13 I V . EXPERIMENTAL TECHNIQUE. L i g h t f r o m a h e l i u m - n e o n l a s e r (A. = 6328 A) was a t t e n -u a t e d t o the p r o p e r i n t e n s i t y f o r photography by p a s s i n g i t t h r o u g h a p o l a r i z e r . O n l y one p o l a r i z e r was needed, s i n c e our l a s e r l i g h t i s a l r e a d y p o l a r i z e d . The beam then passed' t h r o u g h a c o n v e r g i n g l e n s , a p i n h o l e , a second l e n s , and an i r i s a r -r a n g e d so t h a t t h e r e s u l t i n g beam was p a r a l l e l and about one c e n t i m e t e r i n d i a m e t e r , as shown i n F i g . 3. The beam t h e n p a s s e d t h r o u g h d o u b l e windows i n the h e a t b a t h , t h r o u g h the meniscus r e g i o n o f the sample c e l l , o u t t h r o u g h a n o t h e r p a i r o f d o u b l e windows, t h r o u g h a c o n v e r g i n g l e n s , and onto f i l m a t t h e f o c a l p o i n t o f t h i s l e n s . The h e a t b a t h was c o n s t r u c t e d as shown i n F i g . 4. A c o m m e r c i a l Forma w a t e r b a t h pumped w a t e r whose t e m p e r a t u r e was r e g u l a t e d t o w i t h i n .05 OC t h r o u g h a s p i r a l c opper p i p e s o l d e r e d onto a b r a s s c a n . I n s i d e the can was a s t y r o f o a m i n s u l a t o r . I n the c e n t e r was a copper c y l i n d e r wound w i t h a h e a t i n g c o i l and embedded w i t h t h e r m i s t o r s . The sample c e l l r e s t e d s e c u r e l y i n s i d e the copper c y l i n d e r . G l a s s windows were p r o v i d e d a t t h e o u t s i d e o f the c a n , and a t the o u t s i d e , o f t h e i n n e r c y l i n d e r . The t e m p e r a t u r e - r e l a x a t i o n time f r o m the o u t e r can t o t h e i n n e r c y l i n d e r was about 3.5 h o u r s . V a r i a t i o n s i n the room t e m p e r a t u r e o r i n the w a t e r temper-a t u r e ( a t a g i v e n Forma t e m p e r a t u r e s e t t i n g ) had no d e t e c t a b l e i n f l u e n c e on the sample t e m p e r a t u r e , t o w i t h i n . 0 0 0 1 ° C o P r e c i s e t e m p e r a t u r e c o n t r o l was a c h i e v e d e l e c t r o n i c a l l y , F i g . 3 O p t i c a l Apparatus Schematic h-1 15" JX c e U . W I M O O W S TweRMIitORS I;CAT S I N K - C O P P E R C Y U M P i i R W O U M D w a i l III;A.TII-IC c o n . , E I-MiEROtD W I T H T H e R M l G T O t i S . BRASS OUTiZfZ CAM Vn" COPPSR. PIPS' CAEe-ONc TH31M0STATT£D V/AT6R 16 as shown i n F i g . 5. A t h e r m i s t o r embedded i n the i n n e r c y l i n d e r s e r v e d as one arm o f a b r i d g e . A mercury b a t t e r y s u p p l i e d the v o l t a g e o The o t h e r arm c o n t a i n e d a p r e c i s i o n t e m p e r a t u r e -compensated decade b o x . Any i m b a l a n c e i n the r e s i s t a n c e o f the b r i d g e arms appeared as a v o l t a g e w h i c h was a m p l i f i e d by a H e w l e t t - P a c k a r d D.C. n u l m e t e r and f e d t o a power a m p l i f i e r , t o i n c r e a s e o r d e c r e a s e the c u r r e n t s u p p l i e d t o a t e f l o n -e n c ased t u n g s t e n h e a t i n g w i r e wrapped around the i n n e r c y l i n d e r . The power a m p l i f i e r was e q u i p p e d w i t h f e e d b a c k t o make i t v e r y s e n s i t i v e t o any change i n i n p u t „ ^ By t u r n i n g the n u l m e t e r t o a h i g h a m p l i f i c a t i o n ( c . 10^) the r e s i s t a n c e s o f the t h e r -m i s t o r and decade box were e q u a l i z e d t o w i t h i n .01 ohm a t 19°C 0 S i n c e the r e s i s t a n c e o f our t h e r m i s t o r ( c . 2500 ohms) o changes by about 100 ohms p e r degree a t 19 C, the t e m p e r a t u r e was c o n t r o l l e d t o w i t h i n about .0001° C„ A t h i g h e r a m p l i f i -c a t i o n , o s c i l l a t i o n s e t s i n . The t e m p e r a t u r e of' the w a t e r b a t h and o u t e r can were m a i n t a i n e d about 1° o r l e s s l o w e r t h a n the i n n e r t e m p e r a t u r e . A p l a t i n u m thermometer was u s e d t o c a l i b r a t e the t h e r m i s t o r . I t was f o u n d t h a t the r e s i s t a n c e o f the t h e r m i s t o r v e r y a c c u r a t e l y obeys the e q u a t i o n R = R exp(T /T) o ^ v o o v e r the e n t i r e t e m p e r a t u r e range o f i n t e r e s t . F o r a t y p i c a l t h e r m i s t o r o f the t y p e we u s e d , R = .0146, and T = 3528„0°K„ J e o o The two f i x e d b r i d g e r e s i s t o r s were t h e r m a l l y a t t a c h e d t o the o u t e r can o f the h e a t b a t h so t h a t t h e i r r e s i s t a n c e w o u l d n o t be a f f e c t e d by changes i n room t e m p e r a t u r e . 18 Two sample c e l l s were used. These commercially a v a i l -able c e l l s were constructed of f l a t p a r a l l e l quartz faces spaced .2 cm and 1„0 cm apart, and were f u r n i s h e d w i t h g l a s s neckso Research-grade nitrobenzene and heptane were obtained from the F i s c h e r Chemical Company. The nitrobenzene was handled w i t h great care, s i n c e the absorption of seven drops i s f a t a l . The heptane was a l s o accorded re s p e c t , since i t i s one of the main i n g r e d i e n t s of gasoline„ This p a i r was chosen because of i t s convenient c r i t i c a l temperature, and l a r g e d i f f e r e n c e of r e f r a c t i v e index. The chemicals were mixed.in c r i t i c a l p r o p o r t i o n s (52.0% nitrobenzene by weight, 47.0% by mole) and s e a l e d i n the c e l l e x c l u d i n g a i r . The c o r r e c t p r o p o r t i o n was not known i n i t i a l l y , so the f i r s t f i l l i n g r e s u l t e d i n the meniscus going out the bottom of the c e l l as the temperature approached T c # By n o t i n g the number of f r i n g e s obtained as the meniscus approached the bottom, the proper c o n c e n t r a t i o n c o u l d be c a l c u l a t e d . The c e l l was r e - f i l l e d w i t h t h i s concen-t r a t i o n , and the meniscus then remained near the c e l l center even at .0001°C below c r i t i c a l . The c e l l s were f i l l e d u s i n g the f o l l o w i n g technique. Bu I b 19 A g l a s s b u l b was a t t a c h e d t o the c e l l as shown. The c h e m i c a l m i x t u r e was p o u r e d i n t o the b u l b t h r o u g h o p e n i n g A. Then A was s e a l e d o f f by m e l t i n g t h e g l a s s . The b u l b was t h e n immersed i n l i q u i d n i t r o g e n and the c h e m i c a l s were f r o z e n s o l i d . The system was t h e n pumped t o remove the a i r . The pump was s t o p p e d , the c h e m i c a l s thawed o u t and r e f r o z e n , and pumped a g a i n t o remove any a i r t r a p p e d i n the f r o z e n s o l i d m i x t u r e . P o i n t B was s e a l e d o f f . The c h e m i c a l s were thawed o u t and maneuvered i n t o the c e l l . F i n a l l y p o i n t C was s e a l e d o f f . The c e l l i t s e l f c o u l d n o t be f r o z e n because t h e g l a s s - q u a r t z c o n n e c t i o n s between the f a c e s and s i d e s w o u l d have b r o k e n . The c e l l was s n u g l y mounted and i n s e r t e d i n t o the c e n t e r o f t h e i n n e r copper c y l i n d e r . The t e m p e r a t u r e was th e n c o n t r o l l e d a t a g i v e n T < T c u n t i l t h e r m a l e q u i l i b r i u m was r e a c h e d , w h i c h t o o k about f i f t e e n m i n u t e s . The c e l l was shaken up, by i n v e r t i n g and s h a k i n g t h e e n t i r e h e a t b a t h . The f l u i d t h e n s e p a r a t e d i n t o two p h a s e s . A t our n e a r e s t a p p r o a c h t o T , t h i s s e p a r a t i o n r e q u i r e d about an h o u r . F a r t h e r from T c , the time d w i n d l e d down t o a few m i n u t e s o r even a few s e c o n d s . Some o f the measurements were r e p e a t e d a f t e r w a i t i n g much l o n g e r , and t h e same r e s u l t s were o b t a i n e d . F u r t h e r remarks w i l l be made on t h i s s u b j e c t i n c h a p t e r V I I I . As l o n g as t h e t e m p e r a t u r e i s m a i n t a i n e d a t a s u b c r i t i c a l v a l u e , the meniscus a t e q u i l i b r i u m i s s h a r p , and the m o l a r c o n c e n t r a t i o n o f n i t r o b e n z e n e (X) v e r s u s the c e l l h e i g h t (z) appears as i n F i g . 6a. I f the t e m p e r a t u r e i s th e n i n c r e a s e d t o a s u p e r c r i t i c a l v a l u e , the two phases b e g i n t o mix by T < r c 21 d i f f u s i o n , and a f t e r a time we have the s i t u a t i o n i n d i c a t e d i n F i g . 6 b . Since the co n c e n t r a t i o n and therefore the index of r e f r a c t i o n v a r i e s w i t h h e i g h t , the coherent l a s e r l i g h t i s r e f r a c t e d i n t o a sequence of f r i n g e s , as i n d i c a t e d i n F i g . 7. The f r i n g e s are produced by i n t e r f e r e n c e of d i f f e r e n t bundles of coherent monochromatic l i g h t , and of course would not be obtained u s i n g o r d i n a r y l i g h t . The lens causes p a r a l l e l rays from the c e l l to meet on the f i l m at the F-plane. Two methods of photography were used. I n the f i r s t method, f i l m was s l o w l y drawn past a s l i t by a c l o c k motor. As time passes, the two phases mix together by d i f f u s i o n . The gr a d i e n t i s reduced, the l i g h t i s bent l e s s , and the f r i n g e s appear to c o n t r a c t and squeeze together. A t y p i c a l photograph i s shown i n F i g . 8 a . I n such a photograph i t i s easy to f o l l o w and count a l l the f r i n g e s . I n the second method, a 1-meter f o c a l l e n g th lens produced a l a r g e image which was recorded a t one i n s t a n t on a 5 x 7" sheet of f i l m ( F i g . 8b) . This method was used only to gather data to c a l c u l a t e the d i f f u s i o n c o e f f i c i e n t . This c a l c u l a t i o n r e q u i r e s a c a r e f u l a n a l y s i s of the f r i n g e p a t t e r n . , F i g . X F u r a n a t i o n o f t h e F r a u n h o f e r P a t t e r n F i g . 8 . Drawing of Fringe Photogra 24 V. COEXISTENCE CURVE. To obtain the coexistence curve i t suffices merely to count the fringes» Tables 1 and 2 show our r e s u l t s . The 1 cm c e l l was used close to c r i t i c a l , and the .2 cm c e l l f a r from c r i t i c a l , so as to obtain a convenient number of frin g e s . A gas chromatagraph showed no impurities i n the chemicals. The chemicals i n the .2 cm c e l l were d i s t i l l e d through a packed column. The change i n T c of only .001°C tends to confirm the purity of the materials. An equation of the form (3 A X = A(T c-T) was f i t t e d to the data. In logarithmic form we have log AX = log A + Plog(T c-T), which should y i e l d a straight l i n e of slope P and intercept log A on a graph of log AX versus 3 log (Tc-T) . To f i n d T c > a graph of f versus T was. constructed, where f denotes the number of fringes. The extrapolated value of T at which f =0 must be the c r i t i c a l temperature. I f the value of the exponent of f i s s l i g h t l y more or less than 3, the extrapolated value of T c i s n e g l i g i b l y affected. This value of T c had the e f f e c t of making the l i n e on the log-log plot as str a i g h t as possible. To determine log A and P, a least-squares f i t was employed. For the f i r s t 16 points of Table 1, we found A =.241 + .010, P = .3337 + .0150. For the l a s t seven points A = .242 + .010, P = .3336 + .0020. For a l l 23 points A = .241 + .010, P = .3332 + .0050. T A B L E l o C O E X I S T E N C E C U R V E D A T A F O R 1 C M C E L L T = 2 5 6 6 . 9 4 . f l = 1 9 . 0 6 7 ° C c R of thermistor T - T c f A x 2 5 6 6 . 9 5 . 0 0 0 0 9 4 2 8 . 0 1 2 0 2 5 6 6 . 9 6 . 0 0 0 1 8 8 3 6 . 0 1 4 3 2 5 6 6 . 9 7 . 0 0 0 2 8 3 3 9 . 0 1 5 5 2 5 6 6 . 9 8 . 0 0 0 3 7 7 4 5 . 0 1 7 9 2 5 6 7 . 0 0 . 0 0 0 5 6 5 5 0 . 0 1 9 9 2 5 6 7 . 0 2 . 0 0 0 7 5 4 5 3 . 0 2 1 1 2 5 6 7 . 0 6 . 0 0 1 1 3 5 9 . 0 2 3 5 2 5 6 7 . 1 0 . 0 0 1 5 1 6 7 . 0 2 6 6 2 5 6 7 . 1 4 . 0 0 1 8 8 7 2 . 0 2 8 7 2 5 6 7 . 1 9 . 0 0 2 3 5 8 3 . 0 3 3 1 2 5 6 7 . 2 6 . 0 0 3 0 2 8 7 . 0 3 4 6 2 5 6 7 . 3 4 . 0 0 3 7 7 9 5 . 0 3 7 9 2 5 6 7 . 4 4 . 0 0 4 7 1 1 0 3 . . 0 4 1 0 2 5 6 7 . 5 4 . 0 0 5 6 5 1 1 1 . 0 4 4 3 2 5 6 7 . 6 4 . 0 0 6 6 0 1 1 5 . 0 4 5 8 2 5 6 7 . 7 4 . 0 0 7 5 4 1 1 9 . • . 0 4 7 4 2 5 6 7 . 9 4 . 0 0 9 4 2 1 2 8 . 0 5 1 1 2 5 6 8 . 9 4 . 0 1 8 8 4 1 6 0 . 0 6 3 9 2 5 7 2 . 0 0 . 0 4 7 6 5 2 1 9 . 0 8 7 3 2 5 7 7 . 0 0 . 0 9 4 6 2 2 7 2 . 1 0 8 5 2 5 9 2 . 0 0 . 2 3 4 9 3 7 0 . 1 4 7 5 2 6 1 7 . 0 0 . 4 6 6 6 4 7 4 . 1 8 8 9 2 6 6 7 . 0 0 . 9 2 2 5 5 8 6 . 2 3 3 4 T A B L E 2 . C O E X I S T E N C E C U R V E D A T A F O R „ 2 C M C E L L T = 2 5 6 6 . 8 5 A = 1 9 . 0 6 6 ° C c R of thermistor T - T c f LX 2 5 6 7 o 5 5 . 0 0 6 5 9 2 3 . 0 4 5 8 2 5 6 7 . 7 6 . 0 0 8 5 7 2 5 . 0 4 9 8 2 5 6 8 o 0 2 . 0 1 1 0 2 2 7 . 0 5 3 8 2 5 6 8 . 8 7 . 0 1 9 0 3 3 3 . 0 6 5 8 2 5 6 9 . 9 8 . 0 2 9 4 8 3 8 . 0 7 5 8 2 5 7 1 . 0 1 . 0 3 9 1 8 4 2 . 0 8 3 7 2 5 7 4 . 4 5 . 0 7 1 5 2 5 1 . . 1 0 1 7 2 5 7 8 . 0 1 . 1 0 4 9 5 8 . 1 1 5 6 2 5 8 3 . 0 0 . 1 5 1 7 - 6 5 . 1 2 9 6 2 6 0 0 . 1 0 . 3 1 1 1 8 3 . 1 6 5 4 2 6 5 0 . 1 6 . 7 7 0 9 1 1 2 . 2 2 3 0 2 7 5 0 . 0 0 1 . 6 5 8 4 1 4 5 . 2 8 8 5 2 9 5 0 . 0 7 3 . 3 2 9 0 1 8 4 . 3 6 5 5 3 3 5 0 . 0 2 6 . 3 0 4 9 2 2 8 . 4 5 2 0 4 1 0 0 . 1 5 1 0 . 9 1 0 2 7 3 . 5 3 9 7 28 For the data of Table 2, we found A = .243 + .010, P = .333 + .001. Other data not shown gave similar results. The lowest temperature was near the freezing point of nitrobenzene. The errors given for P are equal to twice the standard deviation, divided by the interval in log(T c~T). This gives a reasonable estimate of the error in the slope on a log-log plot, i f one assumes that the slope is constant over that interval. From these data, i t would appear that P is constant and equal to 1/3. Any residual errors in Gladstone's law turn up mainly in A and give rise to the estimated uncertainty given. Fig. 9 shows a plot of In AX - 1/3 In AT versus In AT. Here a horizontal line would correspond to P = l/3.^ The errors shown denote an uncertainty of +1 fringe and +.0001° in T £ - T f c So far we have measured X^- X^. I t later occurred to me that i t would be possible to determine the absolute values of XT and X,, by measuring the height of the meniscus as a Li U function of temperature. This i s seen as follows. Let q denote the total volume fraction of N. Knowing the c r i t i c a l proportions, q i s calculated to be .3808. Let f^ and f^ denote the lower and upper volume fractions of N. Let If = f L - f, U (34) But qV tot = f V + f V = f V + f V ' - f V + f V L L 1 1 ) L L L U L U U U (35) 29 * V t o t = f L V t o t - A f V U ' (36) VU fj = q + y— Af, and similarly, (37) tot f = q - ^ Af„ (38) tot But we already know that AX = .24(T Q-T)* 3 3 = 6 o 3 l A n = 6 .31 Af (n^-r^) , ' (39) from which Af = 023(T c-T) * 3 3 0 (40) This equation is used as an empirical formula to in effect predict the number of fringes, to save the trouble of measuring them again, and does not depend on the assumption that the coexistence curve is symmetric. Knowing f T and f.., i t is easy to calculate Xr and X . Li U Li U A calibrated rule was attached to the 1 cm c e l l and observed with a small telescope. Table 3 shows the results. I t is seen that (X L + Xy)/2 = .470 + .003. We conclude that the coexistence curve i s symmetric about X c within experimental error, and certainly i s not symmetric in the volume fraction. TABLE 3. SYMMETRY OF COEXISTENCE CURVE T -T c H e i g h t o f men. f L X L *U i ( X L + Xy) .0061° 1.55cm .4066 .3645 .386 .4956 .4512 .473 .0168 1.7.5 .4140 .3550 .384 .5032 .4410 .472 .0276 1.81 .4189 .3493 .384 .5083 .4349 .472 .0491 1.86 .4259 .3416 .384 .5154 .4266 .471 .0918 1.90 .4353 .3315 .383 .5250 .4155 .471 .1770 1.90 .4486 .3194 .384 .5385 .4023 .471 .3254 1.91 .4635 .3052 .384 .5533 .3865 .470 .6401 1.91 .4844 .2861 .385 .5739 .3650 .470 1.0520 1.90 .5036 .2697 .387 .5926 .3462 .470 2.0476 1.90 .5341 .2421 .388 .6217 .3141 .468 4.256 1.86 .5801 .2076 .394 .6645 .2730 .469 5.715 1.84 .6030 .1910 .397 • .6850 .2523 .469 10.400 1.80 .6567 .1551 .406 . .7328 .2083 .470 13.659 1.77' .6870 .1377 .412 .7589 .1863 .472 o 31 V I . DIFFUSION COEFFICIENT I . By o p t i c a l methods i t was p o s s i b l e to measure the coef-f i c i e n t of d i f f u s i o n , denoted D. I n a system such as ours c o n s i s t i n g of two f l u i d s H and N, there are r e a l l y two coef-f i c i e n t s of d i f f u s i o n ; the c o e f f i c i e n t of d i f f u s i o n of H i n t o N, and of N i n t o H„ But i t can be shown that these two are equal For conceptual c l a r i t y we henceforth t h i n k of D as the coef-f i c i e n t of d i f f u s i o n of N i n t o H„ Now D i s a f u n c t i o n of temperature and c o n c e n t r a t i o n . One may consider t h a t D i s negative i n s i d e the coexistence curve, zero on the coexistence curve, and p o s i t i v e o utside i t . This i n f a c t i s what causes the phase s e p a r a t i o n . One method of measuring D i s as f o l l o w s . The c e l l was maintained a t a temperature only s l i g h t l y below T c > so t h a t the c o n c e n t r a t i o n of upper and lower phases d i f f e r e d only s l i g h t l y from X^. Then the c e l l was suddenly heated w e l l above T c . Under these c o n d i t i o n s we may n e g l e c t the c o n c e n t r a t i o n dependence of D and s e t D = D(Xc,T)„ I t then becomes p o s s i b l e to c a l c u l a t e e x a c t l y the shape of the curve X = X ( z , t ) ( F i g . 6b) . Since the c o n c e n t r a t i o n near the top or bottom of the c e l l i s s c a r c e l y a f f e c t e d by d i f f u s i o n over s h o r t time i n t e r v a l s ( l e s s than a few days), we may consider the c e l l to be a s e m i - i n f i n i t e system, meaning t h a t i t s t r e t c h e s to i n f i n i t y on both s i d e s of the meniscus. Since there e x i s t no c h a r a c t e r i s t i c lengths i n such a system, the only way the v a r i a b l e s can combine i n a dimensionless way i s as 32 y = z= » O i ) Y Dt where z = 0 i s the p o s i t i o n of the d i s c o n t i n u i t y at s t a r t i n g time t = 0. We assume the v a l i d i t y of the d i f f u s i o n equation, which i s D ^ 2 X £ t I n terms of y t h i s becomes J * X _ _ _L, <*X Let u = dX/dy. Then X = M,J e-*/f1 + canst, (42) - - i y , ' C44) (45) U ^ - ^ . (46) 0^ T" ? ^ a^ j 0 c ? (47) (48) 33 Determining the constants from the i n i t i a l conditions f i n a l l y y i e l d s ( 8 ) (49) We now must calculate the deviation of a l i g h t ray which passes through the region of varying concentration„ We imagine the c e l l to be divided into h o r i z o n t a l layers with d i f f e r e n t indices of refraction» Snell's law i s t) Sen <b - const, y G - l u s s f a (50) D i f f e r e n t i a t i n g with respect to z and using the chain rule gives - O (51) But d 2 (52) Hence, (53) In our experiments, the c e l l was t h i n and the gradient small enough so that the angle of deviation was small. In t h i s case. 34 A<p= ± ilL . (5„) n d 2-On passing through the f i r s t glass w a l l , the ray suffers no deviation since i t i s normally incident„ But on i t s passage through the second glass w a l l , S n e l l T s law requires (putting sinA.0 = Putting n . =1 and l e t t i n g B = (£$>) . , we have & a i r r a i r 9= ^ 4^ - (56) Taking the derivative of equation (49) gives ~ fNo M o t -—- ^ (57) and hence axl ^x, i a* J - * (58) B u t And 35 2»L| _ (""-^shM _ (?K-nH\s d x l° ~ T TTT^F " — (61) Note X = X c when z = 0. So u s i n g equation (32), and where f denotes the number of f r i n g e s obtained from an i n i t i a l d i s c o n t i n u i t y AX. Hence, o Q £ _ * Z X 1 a n d (63) D - /* X ' <* . (64) The o p t i c a l system was c a l i b r a t e d w i t h a Ronchi r u l i n g to provide a conversion from f i l m measurements to d e v i a t i o n angles. A p o i n t near the top of the most de v i a t e d f r i n g e was considered to represent the most de v i a t e d r a y . This assumption s t i l l holds true i n an elaborate w a v e - o p t i c a l (91 treatments 1 The c e l l was brought to e q u i l i b r i u m a t a sub-c r i t i c a l temperature, then q u i c k l y heated to a chosen T > T c and h e l d t h e r e . A continuous r e c o r d i n g of the f r i n g e s was made. The f i l m was developed and © m measured a t h o u r l y _2 i n t e r v a l s . F i g . 10 shows a p l o t of versus time taken from one such f i l m , which happened to have 65 f r i n g e s . I t i s 36 h a p p i l y seen t h a t — ™ i s v e r y n e a r l y c o n s t a n t i n t i m e , a t a t a f i x e d t e m p e r a t u r e , as p r e d i c t e d by e q u a t i o n (64) . S e v e r a l r u n s were made a t v a r i o u s t e m p e r a t u r e s above T c w i t h 65 f r i n g e s , and a g a i n w i t h 31 f r i n g e s . (The number o f f r i n g e s o f c o u r s e depends on the i n i t i a l s u b - c r i t i c a l t e m p e r a t u r e ) . The d a t a i s shown i n T a b l e 4 and i s p l o t t e d i n F i g . 11. The b r e a k i n s l o p e as T c i s approached i s v e r y l i k e l y due t o the i n c r e a s i n g s i g -n i f i c a n c e o f c o n c e n t r a t i o n dependence. Note t h a t the b r e a k i s l e s s s e v e r e f o r f = 31. The f a c t t h a t D i s l e a s t a t X c causes the g r a d i e n t t o p i l e up a t X c > c a u s i n g g r e a t e r d e v i a t i o n o f l i g h t , w h i c h g i v e s the appearance o f a s m a l l e r D. From the u p p e r , more r e l i a b l e p a r t o f the g r aph we f i n d D = (2.4 + 1) x 1 0 " 7 ( T - T c ) * 7 5 - ° 0 5 cm 2/sec from the f = 65 d a t a , and D = (2.5 + 1) x 1 0 ~ 7 ( T - T c ) o 7 ° - * ° 5 cm 2/sec fr o m the f = 31 d a t a , w h i c h i s p r o b a b l y more r e l i a b l e . I t was n o t f e a s i b l e t o f u r t h e r reduce the f r i n g e number. 38 TABLE 4. DIFFUSION COEFFICIENT I N T-T D c 65 2.706 5.43 x 1 0 ~ 7 cm 2/sec 65 1.146 2.70 65 .6400 1.72 65 .3017 .948 65 .1338 .522 65 .05381 .193 65 .02632 .090 31 1.072 2.37 31 .6182 L.-97 31 .2816 .940 31 . .1407 .629 31 .06022 .378 31 .02349 .152 31 .006981 .0579 40 V I I . DIFFUSION COEFFICIENT I I . A second method o f c a l c u l a t i n g D r e q u i r e s a c a r e f u l a n a l y s i s o f the f r i n g e p a t t e r n . We assume t h a t the concen-t r a t i o n i s symmetric about X f i a t z = 0. ( T h i s a s s u m p t i o n was l a t e r v e r i f i e d ) . Rays p a s s i n g t h r o u g h symmetric p o i n t s z + and z w i l l be d e v i a t e d e q u a l l y and w i l l i n t e r f e r e a t the f i l m - p l a n e . The d i f f e r e n c e i n o p t i c a l p a t h between t h e s e r a y s i s ( l e t t i n g s i n 9 = 9 > 0) Aopt = l ( n _ - n + ) - ( z + - z ) 9. (65) A minimum o c c u r s when Aopt = ( s - l / 4 ) A , s = 1, 2, 3 . . . . N a i v e l y one wo u l d have e x p e c t e d i t t o be ( s - | ) ^ . The s o - c a l l e d (9) " q u a r t e r wave anomaly" a r i s e s from a w a v e - o p t i c a l t r e a t m e n t . S i n c e 9= l d n / d z , we have A o P t = + Jl I - . (66) 2 + ' T h i s f o r m u l a has a s i m p l e g e o m e t r i c a l i n t e r p r e t a t i o n . 41 On a plot of z versus -ldn/dz, the term l(n -n^J = 1 / (dn/dz) dz represents the area under the curve between z and z+„ The term l ( z + - z ) dn/dzl represents the area of the rectangle bounded by z, and z and the lines dn/dz = 0 and dn/dz = dn/dzl „ The " '2i difference between these areas i s equal to the area contained in the "snout" of the c u r v e S e e Fig, 12 „ When this area increases by A, a new minimum is formed. Since we are considering concentrations near X^, we expand n = n c + n^(X-X c), where n^ = l/6„31. Symmetry requires z ,-z = 2z . , and n -n. = -2(n.-n ) <• We have - ^ , f * ( v ^ + 5 s * / j f [ J . (67) '2 + Converting to phase difference and letting K = 211© = ~ L — /f l i i . / and a = —~ « we have A £>.3\ ' Minima occur when = (2s--|) TT . Solving for z,, (68) 3 + - " j£ [ * ( ^ * c ) + ^ 1 TT J „ (69) This i s the equation of a straight line„ A set of such lines having different values of s forms the envelope of the curve of a(X L-X ) versus z,. See Fig„ 13. The coordinates of the + c + intersection of two such lines s and s + 1 are 1+2 da F i g o 12„ Geometric I n t e r p r e t a t i o n of F r i n g e P a t t e r n 43 F i g . 13. Envelope Construction 44 (70) TT We assume that the curve runs through the midpoints ^ J > (72) (73) Knowing the concentration as a function of height, i t i s possible to calculate D by the following method. D may be defined as the rate of flow of material past a given point, divided by the gradient. See F i g . 14. D ( X ) - * , A t — / DCx) - ^ at (75) F i g . 14, Change of Concentration w i t h Time 46 (76) AJLl 4z I v But i f X = X(y) , where y = z/^Dt' , then when X = const, y = const. Then z = y ^ , ( ^ J y = (^J^ = ± ~f j DCx) = !_ . (77) d>2- I X /•o-i This i s c a l l e d Boltzmann's formula. A computer program was written which, when given measure-ments of fringes photographed on a 5 x 7" sheet f i l m at a given time t a f t e r heating to a temperature T > T c > calculates the curve of X versus z, and f i n a l l y calculates D(X) according to Boltzmann Ts formula. The integration near the asymptote X = X Q was c a r r i e d out by assuming that D varies smoothly i n that r e g i o n o The measurement of D at high temperatures i s d i f f i c u l t because the length of time required to heat the c e l l makes i t d i f f i c u l t to determine when the run "began." Near T c > things are d i f f i c u l t because any s l i g h t v i b r a t i o n causes more mixing than occurs by d i f f u s i o n . As a function of concentration at a given temperature, D(X) increases as one moves away from X^, but our data are not good enough to try to f i t a,formula. Table 5 shows D (X C) as a function of temperature. The p l o t of F i g . 15 shows that D(X ) = (3.7 + 1) x 10" 7 (T-T ) * 7° - * ° 5 2 cm /sec. I t i s encouraging to note that the exponent agrees with that obtained by the f i r s t method. The discrepancy i n 47 TABLE 5. DIFFUSION C O E F F I C I E N T I I R ( f i n a l ) T - T C D ( X Q ) 2567.40 J l .0047° .0729 x 1 0 " 7 2567.00 .0085 .143 2566.00 .0180 .229 2564.00 .0368 .411 2560.00 .0746 .536 2550.00 .169 .992 2530.00 .360 1.64 2500.00 .650 2.84 4 8 4 An 49 the c o e f f i c i e n t of p r o p o r t i o n a l i t y (from 2,5 to 3.7) i s c a u t i o u s l y a t t r i b u t e d to the n e g l e c t of co n c e n t r a t i o n dependence i n the f i r s t method. R e s u l t s of computation show th a t the g r e a t e s t p o s s i b l e e r r o r i n the determination of the zero-angle leads to an e r r o r of only 10% i n D„ The probable zero-angle e r r o r i s l e s s than h a l f t h i s . To t e s t the symmetry of D about X^, the apparatus was arranged so that the image of the c e l l was recorded on f i l m , w h i l e a t the same time d i r e c t l i g h t from the l a s e r i n t e r f e r e d . The r e s u l t was an image w i t h h o r i z o n t a l bands, each formed as the o p t i c a l path i n s i d e the c e l l changed by A. Measurement of the p o s i t i o n of these bands at v a r i o u s times and temperatures confirmed t h a t they were always p r e c i s e l y symmetric about the ce n t e r . Boltzmann's formula then i m p l i e s t h a t D Is symmetric about X . c 50 V I I I o SEPARATION T IME . I n the cou rse o f our e x p e r i m e n t s , i t was n e c e s s a r y to know how much t ime was r e q u i r e d f o r the phase s e p a r a t i o n . The f o l l o w i n g a n a l y s i s was made. I t i s assumed t h a t D i s n e g a t i v e i n s i d e the c o e x i s t e n c e a** = Ay at a n e g a t i v e D i s e q u i v a l e n t to t ime r e v e r s a l . I t i s assumed c u r v e . A c c o r d i n g to the d i f f u s i o n e q u a t i o n , D N n J L - — , t h a t the l i q u i d s e p a r a t e s by d i f f u s i o n i n t o two k i n d s o f d r o p l e t s . The heavy d r o p l e t s f a l l , and the l i g h t d r o p l e t s r i s e , u n t i l a boundary i s formed i n the- c e n t e r o f the c e l l . I t i s known t h a t f o r a sphere o f m a t e r i a l d i f f u s i n g ou tward . 2 2 — where T = R / I T D i s the c h a r a c t e r i s t i c d i s p e r s a l t i m e , X i s the average c o n c e n t r a t i o n i n s i d e the sphere o f r a d i u s R, and X . and X- a re the i n i t i a l and f i n a l c o n c e n t r a t i o n s a t t = 0 i f and t = oo . S tokes* law f o r a sphere f a l l i n g i n a v i s c o u s medium i s F = (* rr 1? 9 v ' (79) where F i s the f o r c e on a sphere o f r a d i u s R and v e l o c i t y v , i n a medium o f v i s c o s i t y ^ . P u t t i n g t h i s e q u a l to the buoyant f o r c e due to g r a v i t y , we have 51 URr,v s !L UR3 Ap g } (80) (81) Now i f d i s the d i s t a n c e from the c e l l end t o the m e n i s c u s , and t t h e s e p a r a t i o n time^ t - f = • l 2 ± - - (82, S i n c e the drops grow as t h e y f a l l , and s i n c e t h e i r growth i s n o t s t e a d y due t o the e x h a u s t i o n o f the medium, we t a k e an average v a l u e o f ~f - t / 3 . Hence, i r - - . (83) From the handbook we f i n d t h a t f o r heptane. »7 = .461 cp a t 17°C ^ = .409 cp a t 20°C and f o r n i t r o b e n z e n e ty = 2.24 cp a t 15°C >7 = 2.03 cp a t 20°C. Here " c p " means " c e n t i p o i s e " = .01 p o i s e . The d i m e n s i o n s o f a p o i s e a r e gm/sec cm. We t a k e 52 d = 2 cm, g = 980 cm/sec , D = 3.7 x 1 0 ~ 7 ( T - T c ) c m ^ / s e c from previous expto, &o = (^ N- f H ) A X = .52 x .24 ( T c - T ) / 3 3 gm/cm3 and f i n d t = 37.0 sec/ ( T c - T ) ° 5 1 . (84) The s i t u a t i o n analysed assumes a . c r i t i c a l mixture. (The analysis of o f f - c r i t i c a l mixtures became very complicated and was abandoned. Attempts to measure o f f - c r i t i c a l separation times were also abandoned, because the fog very gradually cleared, making i t impossible to state a d e f i n i t e separation time) . The separation time of the c r i t i c a l mixture was measured experimentally at various temperatures. F i g . 16 shows a log-log p l o t of the data. The data are reasonably w e l l f i t t e d by the formula t = 3 7 sec/ (T -T) ° 5 5 . (85) exp c The agreement i s made even better i f we account f o r the anomalous v i s c o s i t y near the c r i t i c a l p o i n t . This means t h a t ^ ^ the v i s c o s i t y at the c r i t i c a l point i s about 25% larger than the normal temperature dependence would p r e d i c t . A f t e r a lengthy c a l c u l a t i o n we f i n d S 3 54 t = 38 s e c / ( T - T ) • n e a r (T Q-T) = 1. (86) T h i s r e m a r k a b l e agreement i s no doubt p a r t l y l u c k , b u t even so i t i s s t r o n g e v i d e n c e t h a t our assu m p t i o n s a r e e s s e n t i a l l y correct«, 55 IXo CONCLUSION. Our results are in general agreement with the findings of other workers ^ 2~' L^ Values of P for various binary fluids range from .33 to .38. Chu and Schoenes^^ report an equation of the form D = d(T/T c-l + f (X)) Y where f (Xc) = 0, and 7 = .70 + .04. They did not report a value for d, but other workers find values in the neighborhood of 10~7 cm 2/sec^^ Our optical system, although f i r s t discovered by Gouy in 1880£^ seems to have attracted l i t t l e attention and merits wider use. I am indebted to Dr. David Balzarini for suggesting and fig") helping with this work, and to Mr. Kenneth Ohrn for his assistance„ 56 REFERENCES 1. Ro Lo S m i t h , Contemp. Phys., 10, 305, (196 9 ) , 2. D„ S e t t e , R i v i s t a D e l Nuovo Cimento, 1, 4-03, (1 9 6 9 ) . 3. F„ W. S e a r s , "Thermodynamics," 2nd E d i t i o n , pp. 101-102, A d d i s o n - W e s l e y , R e a d i n g ^ Mass., ( 1 9 5 3 ) . 4. -E. A. Guggenheim, J „ Chem. -Phys., -13,- 2-53, -('19-45-) . 5. L. P. K a d a n o f f e t . a l . , Rev. Mod. P h y s . , 3_9, 395, (1 9 6 7 ) . 6. D„ B a l z a r i n i , Ph.D. T h e s i s , Columbia U n i v e r s i t y , ( 1 9 6 8 ) . 7. J . 0. H i r s c h f e l d e r e t . a l . , " M o l e c u l a r Theory o f Gases and L i q u i d s , " W i l e y , J o h n & S o n s , 0 I n c . j Somerset, N. J . (1 9 6 4 ) . 8. W„ J o s t , " D i f f u s i o n i n S o l i d s , L i q u i d s , and Gases," Academic P r e s s , I n c . , New Y o r k , ( 1 9 5 2 ) . 9. G. K e g e l e s and L. G o s t i n g , J . Am. Chem. S o c , 6£, 2516, ( 1 9 4 7 ) . 10. Handbook o f C h e m i s t r y and P h y s i c s , 4 8 t h E d i t i o n , C h e m i c a l Rubber Co., C l e v e l a n d , O h i o , (1967) . <. 11. M„ S. Green and J . V. S e n g e r s , E d . , " C r i t i c a l Phenomena," N a t i o n a l B u r e a u o f S t a n d a r d s M i s c . Pub. 273, Wash i n g t o n , D„ C , (1966) . 12. A. Wims, D. M c l n t y r e , and F„ Hynne, J . Chem. Phys., 50, 616, ( 1 9 6 9 ) . 13. D. R. Thompson and 0. K. R i c e , J . Am. Chem. S o c , 86, 3547, ( 1 9 6 4 ) . 14. P. C h i e u x and M. S i e n k o , J . Chem. Phys., S3, 566, ( 1 9 7 0 ) . 15. G. D'Abramo and F. R I c c i , P h y s . Rev. L e t . , 2!8, 22, ( 1 9 7 2 ) . 16. B. Chu and F. Schoenes, P h y s . Rev. L e t . , 21, 6, ( 1 9 6 8 ) . 17. R . Haase and M. S i r y , Z . P h y s i k Chem., 57, 56, ( 1 9 6 8 ) . 18. Gouy, Compt. Rend., 90, 307, ( 1 8 8 0 ) . 19. K. E. Ohrn, M>Sc. T h e s i s , U n i v e r s i t y o f B.C., ( 1 9 7 2 ) . 

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