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The Lorentz-Lorenz function of ethane Burton, Michel Alan 1973

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( i ) c THE LORENTZ-LORENZ FUNCTION OF ETHANE by MICHEL ALAN BURTON B.A., De Pauw U n i v e r s i t y , 1968 Ed.D., U n i v e r s i t y o f Massachusetts, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard f o r can d i d a t e s of the degree of MASTER OF SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA June, 1973 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ( i i ) ABSTRACT T h i s study measured the Lorentz-Lorenz f u n c t i o n f o r 1 n 2 - 1 ethane as determined by = s ~o~— l 0 r d e n s i t i e s from " TT+ 2 1 t o 2 0 0 amagats, where i s the Lorentz-Lorenz f u n c t i o n , £ i s the d e n s i t y , and n i s the index of r e f r a c t i o n f o r a wavelength of 6 3 2 8 £. <^f was found to be constant to w i t h i n lt% with an average value of . 3 7 9 * * * . 0 0 0 3 cc/gm f o r v a l u e s of c o r r e s p o n d i n g t o |^-f c l ^ «2 • The index of r e f r a c t i o n of ethane was measured by u s i n g a l i g h t - w e i g h t c e l l which had two sapphire windows set a t an angle to each o t h e r and through which a plane wave of monochromatic l i g h t was passed. A micrometer-driven m i r r o r was used t o d i r e c t the l i g h t i n t o a c o l l i m a t i n g t e l e s c o p e and the angle of d e f l e c t i o n was t h e n r e l a t e d t o n. ^ was measured by weighing the c e l l and i t s c o n t e n t s on a p r e c i s i o n b a l a n c e . Ethane c o u l d be r e l e a s e d from the c e l l t o vary £ . The c o e x i s t e n c e curve was measured and t h r e e parameters used t o f i t the curve. T h i s enabled e x t r a p o l a t i o n f o r the c r i t i c a l temperature which was found to be 3 2 . 0 7 9 - . 0 0 1 °C . E x t r a p o l a t i o n of the r e c t i l i n e a r diameter gave a c r i t i c a l d e n s i t y of . 2 0 6 2 ± . 0 0 0 3 gm/cc. ( i i i ) TABLE OF CONTENTS I. Introduction 1 I I . The Local F i e l d and the Clausius-Mossotti Equation 4 I I I . The Lorentz-Lorenz Equation 12 IV. Literature Review 17 A. Experimental Research 17 B. Theoretical Research 20 V. The Experiment 26 A. General Considerations 26 B. Temperature Control 26 C. Optics 29 D. The C e l l 35 E. The Sample 37 F. The Weighing 37 G. Data C o l l e c t i o n 40 VI. Data Analysis 45 VII. Error Analysis 54 VIII. Conclusions 57 IX. Bibliography 58 X. Appendices 60 ( i v ) LIST OF TABLES Table I Page 47 (v) LIST OF FIGURES FIGURE PAGE l a IMAGINARY SPHERICAL SURFACE IN A DIELECTRIC 5 lb SPHERICAL CAVITY IN A DIELECTRIC 5 2 THE CELL CONTAINERS 2? 3 TEMPERATURE CONTROL 30 4 THE OPTICAL SYSTEM 31 5 REFRACTION ANALYSIS 32 6 THE CELL 36 7 SYSTEM FOR FILLING THE CELL 38 8 DATA COLLECTION PROCEDURE 41 9 REGION OF DATA COLLECTION 43 10 VARIATION OF 3C WITH £ FOR ETHANE 49 11 VARIATION OF 2£ WITH ^ FOR SF 6 AND ETHANE 51 12 VARIATION OF 5 < WITH ^ FOR XENON AND ETHANE 52 13 COEXISTENCE CURVE AND RECTILINEAR DIAMETER OF ETHANE 14 REFRACTION ANALYSIS - WEDGED WINDOW 60 1 I. INTRODUCTION In recent years i n t e r e s t i n c r i t i c a l phenomena has been increasing. Among those concerned with liquid-vapor c r i t i c a l (1) points, B a l z a r i n i has studied phase t r a n s i t i o n s using o p t i c a l means. The v a r i a t i o n of the c r i t i c a l point parameters of temperature, pressure, and density are studied i n an e f f o r t to determine what relationships exist between them. T-T £ ~ m — = "vc) i s an example of one such r e l a t i o n s h i p , where T i s the temperature and ^ i s the density, ft and a are to be determined and the subscript c denotes the c r i t i c a l value. ^ i s determined by using the Lorentz-Lorenz equation which i s 2 n2 ~ = f ^ where n i s the index of r e f r a c t i o n , i s the n +2 Lorentz-Lorenz function, and ^ i s the density of the substance, The findings of such o p t i c a l measurements are therefore d i r e c t l y linked with the v a l i d i t y of the Lorentz-Lorenz equation and a knowledge of the Lorentz-Lorenz function. (2) Measurements by Chapman, Finnimore, and Smith v '. of the Lorentz-Lorenz function suggest that ^ i s constant to within ± 1%, that no large anomaly exists near the c r i t i c a l point, that the temperature dependence i s s l i g h t but not n e g l i g i b l e , and that a s l i g h t decrease occurs i n when xenon becomes a s o l i d . Because of the necessity of having accurate values of ^ to support recent work on sulfur-hexafluoride, SF^ , by B a l z a r i n i and 0 h r n w / , B a l z a r i n i and P a l f f y v undertook 2 measurements of f o r SP^ . Their preliminary findings indicated that was constant to within . 5 $ over a density range of 200 amagats. This experiment seeks to measure fo r ethane over a density range of 250 amagats. Not only i s the c r i t i c a l temperature, 32.08 °C, experimentally convenient, but more importantly the molecule lacks the spherical symmetry of xenon and i s therefore a good choice for further c r i t i c a l phenomenon studies to test the a p p l i c a b i l i t y of the c r i t i c a l exponent / i n cases of molecules which lack spherical symmetry. Studying related research t h i s author i s concerned that density values were apparently obtained through the use of compressibility data. To avoid the error possible i n such determinations the present experiment seeks to determine ^ d i r e c t l y by weighing the contents of the c e l l . The goal of the t h e o r e t i c a l l i t e r a t u r e review i s to indicate the approach taken by various authors i n attempting to solve the problem. The degree to which theory predicted experimental findings was also considered. A lengthy exposition of the methods involved i s beyond the scope of t h i s work, however a detailed discussion of the Lorentz-Lorenz and Clausius-Mossotti equations w i l l be conducted. The Lorentz-Lorenz equation was derived separately by H.A. Lorentz and by Lorenz i n 1880. I t re l a t e s the 3 r e f r a c t i v e index of a substance to the density through r? - 1 No n 2 + 2 ' 3 M where n i s the r e f r a c t i v e index and i s frequency dependent, N Q i s Avagadro's number, ^ i s the density, M i s the molecular weight, and c* i s the atomic or molecular p o l a r i z a b i l i t y . An analogous equation for the l i m i t of low frequencies i s the Clausius-Mossotti equation, k - 1 _ N 0 e ° < k + 2 3 M where k i s the d i e l e c t r i c constant and i s frequency dependent. A solution of Maxwell's equations i n a d i e l e c t r i c substance having /* = 1 gives vph=Jt ' w ^ e r e v p h ^ s ^ e P^ a s e v e l o c i t y of the wave and c i s the speed of l i g h t . Since n=— we 2 V p h have k=n and the two expressions are i d e n t i c a l . The relationships depend on what l o c a l e l e c t r i c f i e l d acts on a molecule. The c a l c u l a t i o n of the l o c a l f i e l d which was used to derive the Lorentz-Lorenz and Clausius-Mossotti equations neglected s t a t i s t i c a l f luctuations and density e f f e c t s . The molecular p o l a r i z a b i l i t y was assumed constant and independent of temperature and density which i s also incorrect. Nevertheless i t i s useful to follow through these derivations since they approximate experimental data to within a few percent and serve as a foundation from which improvements may be made. 4 I I . THE LOCAL FIELD AND THE CLAUSIUS-MOSSOTTI EQUATION An essential part of the derivation of the Clausius-Mossotti equation i s the c a l c u l a t i o n of the l o c a l f i e l d . This i s the f i e l d that a single molecule f e e l s and i s not the space-time average of the atomic e l e c t r i c f i e l d s . To calculate t h i s f i e l d we ref e r to Figure l a which shows a homogeneous isot r o p i c d i e l e c t r i c substance which lacks any permanent dipole moments. Following J a c k s o n ^ a n d Panofsky and (6) P h i l l i p s ^ ' an imaginary spherical surface of radius r i s constructed with a single molecule at i t s center, r i s large compared to molecular dimensions, allowing s t a t i s t i c a l averages to be made over the molecules inside the sphere, but very small compared to the external dimensions of the —» d i e l e c t r i c , allowing E to be considered constant throughout the spherical region. An e l e c t r i c f i e l d E Q i s applied to the d i e l e c t r i c which causes p o l a r i z a t i o n of the molecules. Charge therefore accumulates on the surface of the d i e l e c t r i c - E* which reduces the f i e l d inside the d i e l e c t r i c to E = o , where k i s the d i e l e c t r i c constant of the material. E i s along the x axis and so E = E x i . We may consider that a charge d i s t r i b u t i o n e xists on the inside of the spherical surface due to p o l a r i z a t i o n of the molecules. The charge density at a given point on the surface w i l l depend on the charge movement normal to the surface. I f P i s the p o l a r i z a t i o n per unit volume, q the amount of charge which crosses a surface of unit area normal Figure l a . Imaginary spherical surface i n a d i e l e c t r i c , contents of the sphere are not altered. Figure l b . Spherical cavity i n a d i e l e c t r i c , contents of the sphere are removed. 6 t o E , and A X i s the d i s t a n c e through which q moves, then P = ^ V Q2 1 * L e t t i n g dA he an element of s u r f a c e a r e a on the sphere, then dA • A X 1 i s a volume of charge which has moved a c r o s s the s u r f a c e . So P = or 1 A X ax.* P • dA = q . I f n i s a u n i t v e c t o r normal t o the s u r f a c e t h e n P • n = ^ = c r , where c r i s the s u r f a c e charge d e n s i t y , — * We want to f i n d E a t the c e n t e r of the sphere where -* C —• E = -1 I cr- dA» r I n s p h e r i c a l c o o r d i n a t e s £ o J *^cT J r 3 p dA = r s i n 9 d9 df>, - (T = P s i n 9 cos <j» , and E = E s i n 9 cos <)> . From the symmetry we can e l i m i n a t e -4 any y component of E , so i n t e g r a t i n g ' f o r E we have n IT E =-TT-^— (8)( | (P cos d> s i n 9 ) ( r s i n 9 cos & ) ( r 2 s i n 9 d9 d i ) x T r € o J I 2 --Wo r 2 P f 2 2 1 * 3 So E = cos <J> d$ sin^9 d9 which i n t e g r a t e d i s x £ 0J 0 Jo E - 2 J P + i x T T £ 0 I ^ 2 at „ N R > Q , cos^9 -cos 9 + — r — 7 P Evaluation gives E = -5-7- . Contributions to the l o c a l f i e l d from molecules inside the sphere are obtained by summing over a l l the i n d i v i d u a l dipole moments, p , of the molecules. For the x component of the f i e l d at the center of the sphere due to the surrounding dipoles we have -1 E x " 4TT € G JZ P x 3(p x x 2+ p y x y + p z x z) r3 r5 However since we assumed an i s o t r o p i c homogeneous d i e l e c t r i c , the x , y , and z directions are equivalent. This means that f o r any p„ x z there i s a -p_ x z and s i m i l a r l y for Py x y , which means y ' (p^ x y + p 2 x z) = 0 . We also 2 have YZ * = y 2 = XZ ^ = ^ " j " 3 0 o 2 ^ — x 5 = 0 . Thus there i s no contribution r3 r to the l o c a l f i e l d from the molecules inside the sphere. The e l e c t r i c f i e l d acting on a charge at the center —* -* P —» of the sphere i s then ^eff = E + y ^ - , where E e f f i s the ef f e c t i v e f i e l d , also c a l l e d the l o c a l f i e l d or Lorentz f i e l d . 8 The r e l a t i o n s h i p P = ^ " X E , where % i s the e l e c t r i c s u s c e p t i b i l i t y and £ Q i s the p e r m i t t i v i t y of empty space, i s used i n the derivation of the Clausius-Mossotti equation. This r e a l t i o n s h i p has been experimentally v e r i f i e d to be a very good approximation to the data. That P can be considered a l i n e a r function of E i s reasonable since experimentally applied f i e l d s are generally small compared 11 to the interatomic f i e l d s which are roughly 10 volts/meter. -* This means that E can be large and s t i l l be only a f i r s t order perturbation of the interatomic f i e l d s , implying l i n e a r i t y . I t i s i n s t r u c t i v e to calculate the f i e l d inside a spherical cavity i n a d i e l e c t r i c . Comparison of t h i s f i e l d with the l o c a l f i e l d helps c l a r i f y the differences between , the two. Figure lb shows the s i t u a t i o n where a spherical hole has been cut i n a homogeneous i s o t r o p i c d i e l e c t r i c and the contents removed. As before the applied f i e l d E causes -» ° F p o l a r i z a t i o n which gives E = __o inside the d i e l e c t r i c . k However unlike before E i s along the x axis only inside the cavity and very f a r from the c a v i t y . This i s because material i s actually removed i n making the cavity which a l t e r s the f i e l d on the spherical boundary, while the l o c a l f i e l d c a l c u l a t i o n involved only an imaginary surface. Calculating the cavity f i e l d we have the following boundary conditions. (1) (2) out 3 9 |r=a k ^ o u t r=a and M i n i 2>r |r=a ^ r Jr=a From solving Laplace's equation i n spherical coordinates we know that the answer can he expressed i n terms of Legendre polynomials, P n . n=0 If E = Zo i s the f i e l d as r-»oo , then <f n i , + — * -E x . 4. = y 2" 7 A rn P and i , = V^"7 B r 1 1 P + C r • i n I x n n 'out £ 4 n n n n=0 ,-(n+l) out This means B Q = 0 , B 1 = -E , and B n = 0 f o r n 2 . c c c From (1) we have A^ = o , A1 = -E + __1 , A, = _2 , 0 1 . 3 . 5 and i n general A = n f o r n £ 2 . 2n+l From (2) we have C Q = 0 which means A Q = 0 from (1) k± = k -E - 2C. , A ? = k(-C 2) 1 _1_ d • * 2 5 a^ —k C and i n general A = n (n+1) n 2n+l n fo r n i 2 10 Combining these r e s u l t s we see that C n = A n = 0 f o r n>2 . S o c = E a 3 (1-k) D A = -B (3k) b 0 0 i _ ( 1+2k) a n a A l " (l+2k) Since P 1 = cos t we have, where = -y- - e <b . = " Eo r c o s t + E o a 3 (1-k) (cos t Tout tl+^k) "T2" -» -» This gives a f i e l d of E a - - = E r t 3 inside the hole. e 1 1 0 l+2k —» P ~* ~* -v. However, the Lorentz f i e l d of gives E e f f . = E Q ( 1 + •^ •) k since P = e ^ E ^ , and k = 1+X gives E f f = E Q(2*k) . k T 3 Setting P=_p_ P o< £ E f o r the two cases we get M 1 fej = No K P f o r the Lorentz f i e l d and K ^ 3 M (k-l)(2k+l) = N0oC p f o r t h e o t h e r . Thus we have the y 3 M Clausius-Mossotti equation and the "single-hole" equation. For k very close to 1 we have = y ~ f o r both cases, where oC = N o °^  and i s c a l l e d the Lorentz-Lorenz function. 3 M For k > 1 the two expressions d i f f e r however with the Clausius-Mossotti expression giving values of at from experiment which are nearly constant while the other 11 expression gives an ^ which increases with density. In working on corrections to the Clausius-Mossetti ( 7 ) equation f o r dipolar l i q u i d s however, Onsager v' used the cavity f i e l d but supplemented i t with what he termed the reaction f i e l d , which was the f i e l d due to the p o l a r i z a t i o n of the medium by the molecule i t s e l f . As we are not concerned with polar substances we s h a l l not consider theories concerning permanent dipole moment contributions to the p o l a r i z a b i l i t y . 12 I I I . THE LORENTZ-LORENZ EQUATION A derivation of the Lorentz-Lorenz equation involves considering the charges i n a molecule to behave l i k e harmonic o s c i l l a t o r s . Following Feynmanv ' we consider a p a r t i c l e of charge q and mass m i n a sinusoidally time varying e l e c t r i c f i e l d . We choose E along the x axis so F = q E i and E v = E e l u > t . Using F = m( r + r* + u> 2 r ) where r i s the displacement of the p a r t i c l e from i t s equilibrium position, 2 i s the damping c o e f f i c i e n t , and C O Q i s a measure of the restoring force, since r = x l and x = x Q e 1 0 3^, upon d i f f e r e n t i a t i o n and substitution we have 2 2 q E Q = m xQ(-cj + C O q + i ojX) . Solving f o r x Q we have E q/m E q/m x = 5 p , so x = p 5 and OJ 0 - to + i c o X 0 J 0 - OJT+ i t o * l e t t i n g p be the p o l a r i z a t i o n which i s due to a single _» /» —* 2 / charge displacement p = q x i or p = E q /m 2 2 . v G-> - CO + i c o o 2 -» q A e which can be rewritten as p = £ J*E where oc = 5 = O » 2 J2. , . y U > Q - CO + 1 GO a Writing Maxwell's equations, S 7 - E = f / £ 0 (1) c 2 V * B = J / ^ C + - | f ( 2) V - B = 0 (3) V * E = - - | f 13 Since the charge density i s due to free charge or charge due to p o l a r i z a t i o n we have p = Pp0l+ Pfree Likewise ? = J p o l + J f r e e . But -V - ? = p p o l and Noting that i n a d i e l e c t r i c which lacks any free charges — > — » ? = ?pol a n d J = J p o l Maxwell's equations become upon sub s t i t u t i o n * . S - - S f (5) c 2 V ^ = i 0 | | + i (6) V-B = 0 (7) VxE* = ~|| (8) Taking the c u r l of (8) and substituting from (6) y > 7 x g = -^V*B ) g 4 - ^ ( g £ o + (9) Using the vector i d e n t i t y y x V " E = V ( V-E) - V 2 ] S (10) and substituting (5) into (10) and then into (9) gives v 2 g _ ^ i i | = _ V ( V . ^ +15—^1 (11) o 0 Assuming the f i e l d i n the d i e l e c t r i c to be of the form E v = E e 1 s> where = v . . x o k phase 14 Defining n through n = £ , we have k = Vph C i u ( t f ) and so E x = E o e ^ 1 2 ^ 7 -\T> But P = P = 0 and since the d i e l e c t r i c i s homogenous d x =0 y 2 3>x so V • p = o . Assuming the p o l a r i z a t i o n follows the driving f i e l d P = P Q e l w t i and (11) becomes V 2 K - i 2 ^ 2 = - % ^ .™ (13) c ot C o Substituting (12) into (13) and d i f f e r e n t i a t i n g gives 2 2 - k E + ^ En = - H— P^ where P 0 = e « N E n ° c 2 0 c2€-o 0 o o o , P Q being the macroscopic p o l a r i z a b i l i t y and N the number 2 2 2 of molecules involved. This y i e l d s - k + ^ — = --^g °* N c c 2 which i s k 2 = ( 1 + <* N ) . Substituting f o r k gives c 2 n = 1 + O<N and substituting f o r oc we have 2 N ( l 2 / m G o n = 1 + 5 § • ^ a s been assumed o that the f i e l d l o c a l to a molecule i s just the applied f i e l d E* , but we know that i n fact, c l a s s i c a l l y , we must —V —* —* p use the Lorentz f i e l d E e f f = B + 3~£~ ' 15 N<* * -1 Using t h i s f i e l d P = N & 0 ° < E e f f s<> p 0 = N e ^ E ^ l -upon solving f o r P Q . Using t h i s value f o r P Q we have , 2 GJ -k + k 2 c 2 6 . N £ 0 C< ( 1 — ) This gives 1 + N 1-N , substituting f o r k gives 2 N OL . n - 1 N C X n = 1 + M,* w h i c h may b e r e w r i t t e n as — = — ~ - . 1- n^+2 N O 2 « Since N = — ^ — we have M n2+2 3 M 3 M where X i s usually c a l l e d the Lorentz-Lorenz function. This simple derivation can be extended to take into account a molecule having several resonant frequencies by writing 04 V " 7 —5 -^5 T T — where f . i s c a l l e d the o s c i l l a t o r strength f a c t o r . Considering tx to be the sum of i n d i v i d u a l cx^ contributions i s equivalent to considering a mixture of two or more types of molecules, each type having a d i f f e r e n t o<k . I f N 1 molecules have a p o l a r i z a b i l i t y oc^ , and Ng molecules have a p o l a r i z a b i l i t y cx^ , then 2 1 3 n 2 ~ = N„ Oci + N o <xo . n + 2 1 1 "2 2 16 (8) To check the additive nature of the p o l a r i z a b i l i t y Feynmanv ' calculates cXg f o r various concentrations of sucrose i n water, cx^ i s the p o l a r i z a b i l i t y of water, c * 2 i s the p o l a r i z a b i l i t y of sucrose, and both are assumed to be constant for a l l concentrations of the sucrose solution. Feynman's calcu l a t i o n s give an cx^ which i s nearly constant and therefore supports the assertion that N cx = y ^ o<^ . k Comparison of t h e o r e t i c a l and experimental values of n 2 where n = 1 + 0<N shows that the harmonic o s c i l l a t o r model fo r oc explains observed phenomena. When t*> < C O q the predicted decrease i n n as t j decreases i s experimentally observed. For co2*uJ 0, n has a large imaginary term which gives a factor corresponding to a decay of E with distance where E = E , e ^ " c . This i s c a l l e d absorption and i s x o experimentally observed. For to>Cu>o, n i s les s than 1 which means v _ r , „ _ Q > c, but the group v e l o c i t y u^> , where k i s pnase ^ the wave number, remains le s s than c i n agreement with the requirements of special r e l a t i v i t y . 17 IV. LITERATURE REVIEW A. EXPERIMENTAL RESEARCH A l l research, which was reviewed by t h i s author, of the Lorentz-Lorenz and Clausius-Mossotti equations revealed a decrease i n <5£ as density increased. Michels and Botzen^^ measured the r e f r a c t i v e index of argon f o r six wavelengths i n the o p t i c a l range at 2 5 °C f o r pressures from 1 to 2 3 0 0 atm. The density was calculated using compressibility data. seemed to have a peak value around 2 0 0 amagats and then decrease with increasing density. This was independent of the wavelength of l i g h t considered. The change was small however and was roughly a 1% decrease. Later work by Michels, ten Seldam, and O v e r d i j k ^ * ^ measured the d i e l e c t r i c constant of argon at a frequency of 3 2 0 0 kHz and along the 2 5 °C and 1 2 5 °C isotherms. Pressures ranged from k atm. up to 2 7 0 0 atm. The Clausius-Mossotti function, denoted here as ^ c m » w a s calculated and a decrease was found to exist as the density increased and was about 1% over the experimental range. No change i n a£ c m was found to exist with temperature. A plot of 5 £ vs. o£ c m was made which gave a straight l i n e with a slope of one, i n d i c a t i n g that the variat i o n s i n 5£ and ^Ccm are s i m i l a r . (11) Eatwell and Jones have measured the r e f r a c t i v e index of s o l i d argon i n the o p t i c a l range from 2 0 °K to 8 3 . 8 °K, ( 1 2 ) the t r i p l e point. Regarding e a r l i e r work by Jones and Smiths Eatwell and Jones summarized the findings as follows: 18 " ( i ) The tendency for the Lorentz-Lorenz function to decrease slowly with increasing density, already known to occur i n highly compressed gases and l i q u i d s , was continued i n the sense that values obtained f o r the s o l i d were generally s l i g h t l y lower than those f o r the l i q u i d , ( i i ) However, i n the range of temperature of the s o l i d phase i t s e l f , the Lorentz-Lorenz function increased with increasing density. We have recently extended measurements of the r e f r a c t i v e index of s o l i d argon over a much wider range of temperature, 20 °K to 83.8 °K , with a r e s u l t which appears to reverse the second of the above conclusions." There are some problems with s o l i d argon however since changes occur i n the p o l y c r y s t a l l i n e structure a f t e r i n i t i a l specimen preparation. There i s also a lack of accurate compressibility data with which to calculate the density knowing the temperature and pressure. (11) Amey and Cole v J measured the d i e l e c t r i c constants of argon, krypton, xenon, and methane i n the l i q u i d and s o l i d range near the melting point. Their r e s u l t s indicate f o r each substance that the Clausius-Mossotti function i s les s i n the l i q u i d and s o l i d states than i n the gaseous states by from .Jfo to k.0% depending on the substance. (lk) Work by Abbiss, Knobler, Teague, and Pings v shows an increase i n of about 2% when crossing the phase boundary isothermally from the saturated gas to the saturated l i q u i d , which i s i n t e r e s t i n g . However ^ i s at least 1% lower f o r l i q u i d argon, methane, and carbon t e t r a f l u o r i d e than for the d i l u t e gaseous forms. A temperature dependence i s reported f o r argon which gives a 2% lower value of 3£ 19 at low temperature, 90 K , than at room temperature. However t h i s was a comparison of t h e i r data with data of Johnston, Oudemans, and C o l e ^ - ^ . Garside, M^lgaard, and Smitlv 'measured the r e f r a c t i v e index of xenon l i q u i d and vapor i n coexistence over a temperature range of -59 °C to 16.5 °C and a density-range of .001 to .02 mol/cm3 . They report X to be constant to * 3$ over the range measured, but the indicated change was toward a decrease i n <£f as density increases. Since t h e i r density range was a factor of 20 while Michels and Botzen had a range factor of 600 and only detected a Vfo decrease i n ©C for argon, perhaps xenon would show a greater change i n i f the density range were extended. From the experimental side there i s l i t t l e doubt that the Clausius-Mossotti and Lorentz-Lorenz equations indicate a decrease i n oC as density increases. Although the experimental values of density were determined from compressibility data and the pressure i n the c e l l which would tend to compound errors, the tendency toward a decrease i s c l e a r . However there i s disagreement regarding the v a l i d i t y of i d e n t i f y i n g c< with Does cx i n fact decrease with density such that the correctness of the Lorentz-Lorenz equation i s implied? Does the Lorentz f i e l d vary with density i n such a way that c< i s t r u l y a constant? Does a combination of the two occur, or i s the molecular approach to p o l a r i z a b i l i t y fundamentally inadequate? 20 A. THEORETICAL RESEARCH Considerable t h e o r e t i c a l energy has been expended on the problem of atomic and molecular p o l a r i z a b i l i t y . (17) Ten Seldam and de Groot v ' used a spherical box to contain a helium atom and compressed i t by shrinking the sphere. They reported a 5i?° decrease i n the p o l a r i z a b i l i t y as the pressure increased to 1000 atmospheres. They had calculated the wave functions and energy l e v e l s of helium as a function of r Q , the sphere's radius, and the corresponding pressure. (1Q) Using Kirkwood's^ ' formula i n Hartree units, the p o l a r i z a b i l i t y of spherically symmetric atoms i s CX = ( k/9 k a Q ) ( y7^ r 2 ) 2 where k = 2 i s the number of electrons, a Q = fi/m e i s the f i r s t Bohr orbit f o r hydrogen, -2 - - j^U^4ll * = r„ + r 0 = dv 1'' ' " dv where r^ and are the electron coordinates r e l a t i v e to the nucleus. The problem i n comparing t h i s theory with experimental data i s that the only available data at the time only went to 65 atmospheres f o r helium. (19) In another work published by ten Seldam and de Groot v 7 1 they made simi l a r c a l c u l a t i o n s f o r argon and only rough agreement with experimental r e s u l t s was found f o r the 21 p o l a r i z a b i l i t y of argon as a f u n c t i o n of p r e s s u r e . De Boer, van der Maesen, and t e n S e l d a r a ^ 2 0 ^ used (21) (22) Kirkwood's^ ' s t a t i s t i c a l model and work by van V l e c k v ' (23) and Brown v J l t o develop the C l a u s i u s - M o s s o t t i e q u a t i o n u s i n g both a H e r z f e l d and a Lennard-Jones p o t e n t i a l f i e l d . They s t a r t with the assumption t h a t p i «• D - °< y ^ -^ik' Pjc expresses the dipole moment induced inithe i — molecule. ocDis the molecular p o l a r i z a t i o n due t o the e x t e r n a l f i e l d where «. i s the mole c u l a r p o l a r i z a b i l i t y . So - cx \ ' T ^ . p k i s the p o l a r i z a t i o n due t o the d i p o l e s of a l l the surrounding i*y-3 ." molecules where T., I K U " 3 r i k . r i k 2 r i k b e i n g the u n i t t e n s o r i i + j j + kk . T h i s i s n o t h i n g more than the f a m i l i a r E ( f ) = 3 n " p which i s |r*-r13 the f i e l d at r due t o a d i p o l e a t r , n = r - r Taking the average value of p^ , denoted p^ , they wrote p*. = C D - « ^ T i k . p * k + <* YZ < T i k ' P k " T i k - P k ) 22 Using T 7 = ^ r i j /where n'= number of molecules to / v l k v 3 volume occupied kTi and y ' ( T i k - p - p f c ) = ^ (£+2) E S where E i s the a p p l i e d f i e l d and S i s a p r o p o r t i o n a l i t y constant which i s a f u n c t i o n of n' and the temperature T . Using n'"p~ = P = (€-1) and D - ^ y - P = | (€+2) E — * g i v e s (€-1) j^p- = | (€+2) E n V + i (6+2) E* S n'<* or | ^ g = ^y 1 n'o.( l + s ) which i s the C l a u s i u s - M o s s o t t i e q u a t i o n with an added d e n s i t y and temperature dependence. For the sake of comparison w i t h experiment they e v a l u a t e the c o e f f i c i e n t s B 2 and C 2 i n £ - 1 4 TT €+ 2 n'ot( 1 + B 9 n'+ C 9 n*2 + 0(n / 3) ) u s i n g the Lennard-Jones p o t e n t i a l f i e l d and the s i m p l e r H e r z f e l d p o t e n t i a l f i e l d . While t h i s t heory seems to r e p r e s e n t the data w e l l f o r lower d e n s i t i e s , the t h e o r e t i c a l v a l u e s of o< are l a r g e r than the experimental v a l u e s f o r h i g h e r d e n s i t i e s . They attempt an e x p l a n a t i o n of t h i s t r e n d by the f a c t t h a t c a l c u l a t i o n s by M i c h e l s , de Boer, and B i j l ^ 2 ^ show t h a t cx, decreases with i n c r e a s i n g d e n s i t y . Since they used a constant c< , a theory which lowered cx with i n c r e a s i n g d e n s i t y would more n e a r l y r e p r e s e n t the data. 23 However Y a r i s and Kirtman v c a l c u l a t e the change i n p o l a r i z a b i l i t y due to many body i n t e r a c t i o n s t o be an i n c r e a s e v a r y i n g from .OkZfo f o r helium t o 3 . 5 ^ f o r xenon. Jansen and M a z u r ^ 2 ^ and Mazur and J a n s e n ^ ^ work out j u s t such a theory f o r hydrogen and helium atoms u s i n g the same T ^ k f o r the i n t e r a c t i o n p o t e n t i a l due t o d i p o l e s as was used before by de Boer, van der Masen, and t e n Seldam^ 2 0 ^ . T h e i r m o d i f i e d C l a u s i u s - M o s s o t t i e q u a t i o n i s a l s o the same w i t h the e x c e p t i o n t h a t now cx becomes cx^(rf) where cx^iO =OC 0 ( 11+0(2 C l £ T . k . T f e i + « 2 C 2 ^ ( T ^ T ^ - U ) k / i W i Where c<0 i s the p o l a r i z a b i l i t y of a f r e e molecule, and C^ , Cg , are co n s t a n t s which are d e n s i t y dependent. So they w r i t e = ^ ~ n'<xo ( 1 + R'(n',t) ) -where R'(n't) has the f u n c t i o n a l dependence of cx i n c l u d e d and can be expanded i n powers of C*0 . Ex t e n d i n g t h e i r theory t o argon they compare argon data t o the theory and f i n d t h a t i t e x h i b i t s the c h a r a c t e r i s t i c r i s e t o a maximum around 200 amagats and a decrease t h e r e a f t e r with i n c r e a s i n g d e n s i t y , but an exact f i t t o the data i s not evidenced. Because of the s i m i l a r i t y between the C l a u s i u s - M o s s o t t i (28) and Lorentz-Lorenz equations Mazur and Mandel v ' extended the theory developed f o r the s t a t i c e l e c t r i c f i e l d t o the case of time v a r y i n g f i e l d s i n order t o see what the 2k Lorentz-Lorenz e q u a t i o n becomes when OC i s a f u n c t i o n of d e n s i t y and the Lorentz f i e l d i s a l t e r e d due to s t a t i s t i c a l f l u c t u a t i o n s . A f t e r a quantum mechanical c a l c u l a t i o n of the mole c u l a r p o l a r i z a b i l i t y and a s t a t i s t i c a l c a l c u l a t i o n of the average p o l a r i z a t i o n and l o c a l f i e l d , they o b t a i n m o l e c u l a r p o l a r i z a b i l i t y and R and D r e p r e s e n t i n t e g r a l s having d e n s i t y , temperature, and frequency dependence. Making a c t u a l c a l c u l a t i o n s f o r helium are q u i t e d i f f i c u l t and assumptions have t o be made which l i m i t the accuracy t o the p o i n t t h a t e f f e c t s are g i v e n only to an order of magnitude. The important f e a t u r e s of t h e i r r e s e a r c h are t h a t the L o r e n t z -Lorenz f u n c t i o n i s shown t o be not simply a s u b s t i t u t i o n of 0< 0(GJ) f o r ( X 0 ( 0) i n the C l a u s i u s - M o s s o t t i equation, and t h a t when —g- = .1 , where CJ»0 i s the s o f t e s t a b s o r p t i o n ^ o frequency, the Lorentz-Lorenz f u n c t i o n c o r r e c t i o n i s roughly 15% l a r g e r than f o r the cor r e s p o n d i n g C l a u s i u s - M o s s o t t i case. The next c o m p l i c a t i o n i n t h e o r e t i c a l c a l c u l a t i o n s of the C l a u s i u s - M o s s o t t i f u n c t i o n i s undertaken by Jansen and (29) Solenr '. They expand the i n t e r a c t i o n H a m i l t o n i a n t o i n c l u d e not only d i p o l e - d i p o l e , but dip o l e - q u a d r a p o l e and quadrapole-quadrapole i n t e r a c t i o n s . They s t a t e t h a t i n a subsequent paper h i g h e r m u l t i p o l e s w i l l be e v a l u a t e d and the C l a u s i u s - M o s s o t t i f u n c t i o n f o r a x i a l l y symmetric molecules w i l l be d e r i v e d . 25 When J a n s e n w p u b l i s h e d the subsequent paper he s t a t e d , "A theory of the s t a t i c d i e l e c t r i c constant i s developed on a quantum mechanical b a s i s , s t a r t i n g from the Lorentz m i c r o s c o p i c f i e l d e q u a t i o n s . By i n t r o d u c i n g the concept of ' l o c a l f i e l d ' , a mo l e c u l a r v e r s i o n of the g e n e r a l theory i s obtained and a p p l i e d t o compressed n o n - ( d i ) p o l a r gases a t low d e n s i t i e s . T h i s l e a d s t o a v i r i a l s e r i e s f o r the d i e l e c t r i c c o n s t a n t . I t i s shown t h a t such a mo l e c u l a r theory i s fundamentally i n e f f e c t i v e i n ac c o u n t i n g f o r observed r e s u l t s w i t h i n e x p erimental accuracy." "The q u a n t i t y « i n the C l a u s i u s - M o s s o t t i e q u a t i o n does not r e p r e s e n t the p o l a r i z a b i l i t y t e n s o r of an i s o l a t e d molecule; n e i t h e r may i t be c o n s i d e r e d as a ' d e n s i t y -dependent* p o l a r i z a b i l i t y and t r a n s f e r r e d t o the a n a l y s i s of d i f f e r e n t p h y s i c a l phenomena. I t occurs i n t h i s form only i n c o n n e c t i o n with d i e l e c t r i c p r o p e r t i e s and i s simply a v a r i a b l e , unknown u n t i l the s o l u t i o n has been found, and t h e r e f o r e of no use f o r f i n d i n g the s o l u t i o n . " T h i s author would tend t o concur with Jansen. I t seems t h a t the t h e o r e t i c a l attempts t o improve on the simple forms of the Lorentz-Lorenz and C l a u s i u s - M o s s o t t i equations f a i l . I t i s c l e a r t h a t while the simple e x p r e s s i o n s f a i l at h i g h d e n s i t i e s , the e r r o r i s l e s s than 5$ • While improvements on the simple theory use l a r g e amounts of mathematical machinery, the r e s u l t s g i v e only q u a l i t a t i v e agreement and not true q u a n t i t a t i v e agreement. Since wave f u n c t i o n s f o r complicated molecules are not w e l l known, molecular d i s t r i b u t i o n f u n c t i o n s d i f f i c u l t t o use, and mo l e c u l a r i n t e r a c t i o n p o t e n t i a l s only approximate, i t i s c l e a r t h a t a t h e o r e t i c a l s o l u t i o n i s by no means t r i v i a l . 26 V. THE EXPERIMENT A. GENERAL CONSIDERATIONS The basic considerations i n t h i s experiment were the maintenance of a fix e d temperature that could be changed eas i l y , the construction of an o p t i c a l system capable of measuring s l i g h t changes i n the angle of r e f r a c t i o n , and the construction of a c e l l that would be strong enough to withstand the pressure of up to 1000 p . s . i . exerted by the gas, l i g h t enough to be weighed on a pre c i s i o n balance, large enough to hold a useful sample volume, transparent to allow l i g h t measurements to be made, and capable of being emptied of a portion of i t s contents during the course of the experiment, B. TEMPERATURE CONTROL As the maintenance of a constant temperature i s esse n t i a l considerable attention was given to the control system. C e l l i s o l a t i o n was achieved by the construction of three containers, each insulated from the other with a layer of styrofoam. The inner container was an aluminum cylinder i n which a hole was bored to house the c e l l . (See Figure 2) The outer surface of the cylinder was grooved to hold a heating wire which was wrapped about the surface and a small hole was tapped i n one end f o r the i n s e r t i o n of a thermistor which was epoxied into a screw. The aluminum cylinder was then encased i n styrofoam and inserted i n a copper cylinder about which copper tubing I S Side View End View threaded cap heating wire hole for l i g h t .aluminum cylinder .thermistor i n a screw hole f o r alignment pin hole f o r c e l l "body hole f o r c e l l windows THE CELL CONTAINERS Figure 2 copper tubing copper cylind e r — _ h o l e f o r l i g h t beam styrofoam i n s u l a t i o n plywood cover external 1 — tubing f o r connection to water bath 28 had been wound to provide f o r further temperature control by regulating the temperature of water which flowed i n the tubing. The copper tubing was then encased i n styrofoam and the entire assembly housed i n plywood. Temperature control was obtained i n two steps. F i r s t a Forma S c i e n t i f i c Model 2095 was used to r e f r i g e r a t e water down to . 5 °C and pump i t through the copper jacket. This was necessary f o r the portion of the experiment during which the c e l l temperature was below room temperature. The Forma was capable of maintaining constant temperatures to within i .05 °C . Secondly, the heating wire around the aluminum cylinder and the thermistor embedded i n i t were used to provide a heating-feedback system with which the temperature could be maintained to within .0002 °C . The thermistor was used as part of a DC bridge with a decade resistance box as the other side. A difference between the decade box and thermistor resistance caused a voltage to appear which was detected with a Hewlett-Packard model 419A DC n u l l voltmeter. The output of the n u l l voltmeter was then fed into a Kepco model OPS 7-2 amplifier which was wired to function as an integrator i n order to provide a zero average o f f s e t . The temperature value used f o r data purposes was taken from an i d e n t i c a l bridge network which was wired to a chart recorder. The thermistor used i n the network was c a l i b r a t e d using 2 9 a quartz c r y s t a l thermometer as a standard. Since the resistance of the thermistor obeys the equation R = R Q , a knowledge of R and T f o r various temperatures permitted a least squares f i t of the data to f i n d R Q and o< . For the thermistor i n use R Q = . 0 1 4 9 8 5 8 3 and o< = 3 5 4 4 . 0 6 l . (See Figure 3 ) The thermistor s t a b i l i t y was not checked during the course of the experiment. G. OPTICS The purpose of the o p t i c a l system was to provide a means of measuring the index of r e f r a c t i o n of the sample, ethane. In order to do t h i s a plane wave was constructed which passed through the c e l l and was r e f l e c t e d by an adjustable mirror into a collimating telescope. (See Figure 4 ) Following the optics step by step; a monochromatic beam of wavelength 6 3 2 8 °. was produced by a Metrologic model 3 6 0 helium-neon la s e r . Crossed p o l a r i z e r s were used to attenuate the beam to a l e v e l safe f o r direc t observation. A microscope objective focused the beam to a point and a 1 0 micron pinhole was positioned at the f o c a l point. The beam was then converted into a plane wave by placing a lens such that the pinhole was at i t s f o c a l point. The outer part of the beam was then masked o f f and the central portion sent through a beam s p l i t t e r which d i r e c t s the beam through the c e l l and around the box to another beam s p l i t t e r which d i r e c t s Hewlett-Packard Null Voltmeter Decade resistance box Kepco OPS 7-2 DC A m p l i f i e r TEMPER A T U R E CONTROL Figure 3 1. Metre-logic Laser, model 360 2. Crossed P o l a r i z e r s 3. Microscope O b j e c t i v e k. 10 micron pinhole 5. Lens f=135mm 6. Mask 7. Beam S p l i t t e r 8. The C e l l 9. The C e l l Holder 10. Penta Prism 11. A d j u s t a b l e M i r r o r 12. C o l l i m a t i n g Alignment Telescope a 10 8 9 THE OPTICAL SYSTEM Figu r e 4 11 10 12 e 32 the inner and outer beams toward the mirror. The mirror, a Lansing model 10.253 adjustable mirror with a ca l i b r a t e d d i f f e r e n t i a l drive, then d i r e c t s the beam into a Davidson Optronics alignment telescope model D2?5 which allows the col l i m a t i o n of the beam and v i s u a l alignment with a cross-hair. The mirror has a re s o l u t i o n of .1 arc-sec, r e s e t t a b i l i t y of .6 arc-sec, and a t o t a l angular sweep of ~ k degrees. To obtain the index of r e f r a c t i o n the following analysis was used. \ —»/ / °7 / n. REFRACTION ANALYSIS Figure 5 9 i s the angle of deviation from the o r i g i n a l beam, n^ s i n <* = rig s i n ^  = n^ s i n "V where Y = 8 S o n = n ? s i n y or n = ^ f" b o n l sinoc o r n l sin-ot [_ sine* cos 9 + s i n 9 cos "1 - » 3 [ o o s« +ffe|-] . Where i s the index of r e f r a c t i o n of the sample, n 2 i s the index of r e f r a c t i o n of the windows, and n^ i s the index of r e f r a c t i o n of a i r . 33 T h i s a n a l y s i s d i s p l a y s the b a s i c o p t i c a l system but n e g l e c t s the f a c t t h a t the two f a c e s of the windows are not p a r a l l e l but i n r e a l i t y are s l i g h t l y wedge shaped. The f i r s t window which i s normal t o the l i g h t beam has such a s l i g h t wedge t h a t i t can be ignored. I t i s l e s s than 1' . The second window c o n t a i n s a wedge angle of 4 ' 49" or .08034° . The complete a n a l y s i s i s g i v e n i n Appendix A and i s r a t h e r t e d i o u s . The f i n a l r e s u l t i s e a s i l y understood and i s the f o l l o w i n g , where <J> i s the wedge angle of the window. r, - r , [ i . ^ 2 i « ™«= A . S I N 9 " S I N 9 A \ . s i n _ i | f ( e ) - f ( e . ) l n 1 = n 3 1+cos * ^cos 0 - cos « A + J + A'J where <x i s the angle between the two i n n e r s u r f a c e s o f the windows. I f we a l l o w f o r the worst case and set <* = 20° , o o 2 0- = 3 » and <j) = .08 then cos <j> i s e s s e n t i a l l y u n i t y and s i n ct { f ( e ) " f ( e A ) ] = • 0 0 0 0 8 8 7 3 where the s u b s c r i p t A i s used t o denote the d e v i a t i o n of the l i g h t beam which i s produced wi t h the c e l l i n place and f i l l e d with a i r . I n the case of no wedge angle t h i s would o b v i o u s l y be z e r o . For t h i s experiment 0^ was .0031° . N e g l e c t i n g second order c o r r e c t i o n s , n l = n 3 s i n 0 - s i n 0. 1 + cos 0 - cos 0 which A t a n ot reduces t o the e a r l i e r p a r a l l e l face a n a l y s i s i n the case t h a t © A i s z e r o . 34 The m i r r o r i s c o n s t r u c t e d such t h a t the micrometer d i v i s i o n s are p r o p o r t i o n a l t o s i n 0. That i s , s i n 0 = a+bx where x i s the micrometer r e a d i n g . When the c e l l i s removed s i n 0=0 so a = -b x Q where x Q i s the r e a d i n g c o r r e s p o n d i n g to zero angle, s i n 0^  - s i n 0 = a + b x^ -( a + b x Q ) and s i n 0^  = b ( x^ - x Q ) . To o b t a i n b a Ronchi r u l i n g h a v ing 50 l i n e s / i n c h , Edmund S c i e n t i f i c stock number jj05ll, was p l a c e d i n the c o n t a i n e r . The o r d e r s of d i f f r a c t i o n are g i v e n by s i n 0 m = ~ - where X i s the wavelength of l i g h t i n the medium, d i s the d i s t a n c e between the l i n e s , and m i s the number of the maximum and i s an i n t e g e r . Since the X Q g i v e n f o r the l a s e r i s i n vacuum, we c o r r e c t t h i s by X = — ~ . ' n3 m X 0 We now have s i n 0„ = - — . Since m n^ a s i n 0 m + 1 - s i n 0 m = b ( x m + 1 - x m ) we have ( xm+l " xm * b = r i ^ d ( ^ + 1 - m ) . S o l v i n g f o r b , D = ~"Z • D w a s found t o n3 a *m+l " xm. be very s e n s i t i v e t o the p o s i t i o n of the f i n a l lense t h a t c o n s t r u c t e d the plane wave but i n s e n s i t i v e t o the v e r t i c a l plane of the m i r r o r , b was c a l c u l a t e d t o be ,000064?6 f o r t h i s experiment u s i n g a l e a s t squares f i t t o f i v e s e t s of data, each c o n t a i n i n g 60 orders, t o determine x - x . The l i n e m+1 m spac i n g d was v e r i f i e d t o be 50 l i n e s / i n c h by u s i n g a microscope which had a micrometer d r i v e n t a b l e . 35 D. THE CELL The c e l l was an aluminum c y l i n d e r turned down t o match the i n n e r diameters of the c e l l h o l d e r i n order t h a t good thermal c o n t a c t would be made. ( See F i g u r e 6 ) The i n s i d e o f the l o n g c y l i n d e r was d r i l l e d out t o h o l d the b u l k of the sample. The other end was t h e n m i l l e d t o h o l d two sapphire windows w i t h t h e i r planes a t an angle of ^20° t o each o t h e r . The windows were t h i c k , 1" i n diameter on one s i d e and i n diameter on the o t h e r c r e a t i n g an edge b e v e l of 45° . The windows were seated i n indium wire t o form a s e a l and aluminum r e t a i n i n g r i n g s c o v e r i n g paper gasket m a t e r i a l h e l d the windows i n p l a c e . The end of the c e l l was tapped t o take a pipe t h r e a d and a b r a s s valve seat was made t o f i t the h o l e . The v a l v e seat was s p e c i a l l y made t o cut down on the t o t a l weight of the c e l l . A s t a i n l e s s s t e e l needle v a l v e was used t o s e a l the h o l e . To i n s u r e t h a t the c e l l was r e p l a c e d i n the same p o s i t i o n i n the h o l d e r each time, two p i n s were a t t a c h e d t o the c e l l and matching h o l e s were d r i l l e d i n the c e l l h o l d e r . A tapped hole i n the top of the c e l l allowed f o r simple removal and replacement of the c e l l i n i t s c o n t a i n e r . r e t a i n i n g r i n g sapphire window r e t a i n i n g r i n g screws alignment p i n c e l l body O k F r o n t View brass valve seat tapped hole f o r pumping l i n e c o n n e c t i o n upper end of needle valve THE CELL F i g u r e 6 Side View ON 37 E. THE SAMPLE Research grade ethane was obtained from Matheson Company, Inc. I n f r a r e d and mass spectrometer d e t e r m i n a t i o n s by the s u p p l i e r on a r e p r e s e n t a t i v e sample showed i t to be 99.99 mol per cent ethane, the most probable i m p u r i t y b e i n g e t h y l e n e . The ethane i s shipped under pressure and i t was t h i s p r e ssure t h a t caused the ethane t o flow i n t o the c e l l . ( See F i g u r e 7 ) The c e l l and gas d e l i v e r y l i n e s were f i r s t pumped on f o r a minimum of 96 hours t o evacuate the system of a i r and any i m p u r i t i e s t h a t may have been present i n the system. The c e l l and l i n e were then f l u s h e d out with ethane and the c e l l f i l l e d such t h a t the volume was almost e n t i r e l y l i q u i d a t room temperature. Room temperature was lowered t o about 15 °C to permit f i l l i n g the c e l l t o a h i g h e r d e n s i t y and thereby expanding the d e n s i t y range of the experiment. F. THE WEIGHING A Sauter Monopan balance was used to weigh the c e l l . T h i s balance had minimum d i v i s i o n s of .1 m i l l i g r a m and c o u l d accomodate weights up to 200 grams. S l i d i n g windows perm i t t e d the weighing pan t o be p r o t e c t e d from dust and other f o r e i g n matter which might accumulate on the pan between weighings o f the c e l l and thus a l t e r the t r u e weights. Research Grade Ethane valve Welch Vacuum Pump and Veeco Di f f u s i o n Pump K, Veeco Vacuum Gauge valve valve SYSTEM FOR FILLING THE CELL Figure 7 3 9 A p l a s t i c glove was used t o handle the c e l l t o a v o i d g e t t i n g the c e l l d i r t y and a threaded r o d was used to remove the c e l l and r e p l a c e i t i n i t s c o n t a i n e r . To o b t a i n the sample weight we must c o n s i d e r the buoyant f o r c e t h a t e x i s t s on the c e l l due t o the surrounding a i r . I f i s the volume of the c e l l on the i n s i d e which the sample has a v a i l a b l e t o i t , and V Q i s the outer volume , or volume occupied by the c e l l , then we have P W = — ( m + V. P - V P ) , where W i s the weight r e g i s t e r e d g i V s o v a ° ° by the s c a l e , m i s the mass of the c e l l , P G i s the d e n s i t y of the sample, o the d e n s i t y of the a i r , F i s the f o r c e a c t i n g on the c e l l , and g i s the a c c e l e r a t i o n of g r a v i t y . I f the c e l l i s evacuated P = 0 so we set W ' = m - Vn P and \ s o o s a we then have W = W - VL where W i s the weight of the s o s sample. However i t i s d i f f i c u l t t o o b t a i n WQ s i n c e P G = 0 means t h a t the c e l l must be evacuated and i n the process of c o n n e c t i n g the pumping l i n e the weight of the c e l l might be a l t e r e d from what i t was d u r i n g the experiment. To a v o i d t h i s problem the c e l l v a l v e was allowed to remain open a t the end of the experiment t o permit a i r to d i f f u s e i n t o the c e l l and set P = P . T h i s g i v e s W = m + V- 9' - V P . Then we have W o = W o ~ V i P a ' W o w a s d e t e r m i n e d t o be 179.410 gm ± . 0 0 1 by a v e r a g i n g 7 measurements. V. P was determined t o be . 0 1 6 gm X ^SL by f i n d i n g V. and P independently and a l s o by weighing the X ^ EL c e l l evacuated and t h e n l e t t i n g i n a i r and re-weighing the c e l l . 40 T h i s g i v e s WQ = 179.394 gm • V\ was determined t o be 13.108 - .003 cc by averaging 3 weighings of the c e l l f i l l e d w i t h d i s t i l l e d water and then f i l l e d w ith a i r . C o r r e c t i o n s were made f o r the d e n s i t y of water at room temperature and a l s o f o r the buoyant e f f e c t of the a i r on the c e l l . Water was evacuated from the v a l v e before weighing and l a t e r from i n s i d e the c e l l by p l a c i n g the c e l l i n a b e l l j a r which was connected to a vacuum pump. G. DATA COLLECTION The measurement of the r e f r a c t i v e index f o r a g i v e n d e n s i t y means t h a t the c e l l must c o n t a i n only l i q u i d o r only vapor s i n c e only then i s the volume of the l i q u i d or the vapor known. The c e l l i s f i r s t f i l l e d so t h a t almost the e n t i r e volume i s l i q u i d a t room temperature. ( See F i g u r e 8 ) T h i s temperature i s T^ and the c o r r e s p o n d i n g d e n s i t y of the l i q u i d and vapor combined i s ^' . The l i g h t beam which i s passed through the c e l l i s r e f r a c t e d by the l i q u i d more than by the vapor and as a r e s u l t two dots of l i g h t are observed i n the t e l e s c o p e , one c o r r e s p o n d i n g t o ^ , the d e n s i t y of the l i q u i d , and the other c o r r e s p o n d i n g t o ^ y , the d e n s i t y of the vapor. As the temperature i s i n c r e a s e d t o T 2 , ^ and ^ y change as the l i q u i d o c cupies more of the c e l l volume and the vapor l e s s . At T^ the l i g h t beam through the vapor i s not v i s i b l e s i n c e i t s volume i s so s m a l l and the bubble of vapor T E M P E R A T U R E Figure 8 - DATA COLLECTION PROCEDURE k2 l i e s above the upper edge of the windows. S t i l l , i t i s known t h a t the c e l l c o n t a i n s some vapor s i n c e when the temperature i s i n c r e a s e d t o T' the dot of l i g h t i s seen t o move, i m p l y i n g a change i n the d e n s i t y of the l i q u i d . As the temperature i s f u r t h e r i n c r e a s e d t o T^ the dot i s not observed t o move which means t h a t the c e l l must have c o n t a i n e d only l i q u i d a t T' and t h e r e f o r e the f u r t h e r i n c r e a s e i n temperature t o T^ c o u l d not a l t e r the d e n s i t y of the l i q u i d i n the c e l l , and hence i t c o u l d not a l t e r the r e f r a c t i v e index and cause the dot t o move. I t i s a t T^ t h a t a data p o i n t i s obtained as i t i s here t h a t the r e f r a c t i v e index c o r r e s p o n d i n g t o a known d e n s i t y of the sample can be found, s i n c e the c e l l volume i s known and only l i q u i d o c c u p i e s the c e l l . F i g u r e 9 shows the r e g i o n of data p o i n t s t h a t were obtained d u r i n g the course of the experiment. The r e g i o n was r e s t r i c t e d t o l i e j u s t above the c o e x i s t e n c e curve s i n c e i t i s i n t h i s r e g i o n t h a t r e c e n t s t u d i e s of c r i t i c a l p o i n t phenomena r e q u i r e a knowledge of o£. . The departure from the c o e x i s t e n c e curve near >^c was made t o a v o i d the d e n s i t y g r a d i e n t s i n the sample caused by g r a v i t y as T-»T . I n order to assure t h a t the c e l l c o n t a i n e d a sample of homogeneous d e n s i t y the temperature had t o be i n c r e a s e d t o a p o i n t above T where the d e n s i t y g r a d i e n t s no l o n g e r e x i s t e d . T E M P E R A T U R E x X * X X X X X DENSITY Figure 9 - REGION OF DATA COLLECTION 44 Once past ^ c a s i m i l a r procedure was used t o ensure t h a t the c e l l c o n t a i n e d only vapor. I n t h i s case the temperature was reduced t o see i f the dot changed p o s i t i o n . I f so then some l i q u i d had formed and the temperature had t o be i n c r e a s e d s l i g h t l y t o be sure t h a t the c o n t e n t s were e n t i r e l y vapor. A c h a r t r e c o r d e r was used throughout the experiment t o keep a r e c o r d of temperature v a r i a t i o n s . T h i s enabled the experimenter t o judge the time r e q u i r e d f o r the system t o come t o e q u i l i b r i u m . Although t h i s was not important i n the d e t e r m i n a t i o n of oC s i n c e i t was only necessary t o be above the c o e x i s t e n c e curve, i t was important i n determining the c o e x i s t e n c e curve and the r e c t i l i n e a r diameter. T h i s data was obtained by l e a v i n g the c e l l f i l l e d a t near c r i t i c a l volume and sweeping over temperature. The dots from both the l i q u i d and vapor phases were then v i s i b l e as the meniscus was i n the middle of the c e l l windows. While P,^  and £ v were not known and oC-^  and<*° v were not known, a n d P v ^ v w e r e c a l c u l a t e d s i n c e n^ and n v can be c a l c u l a t e d and we have 2 2 ru - 1 _ K ' 1 v„ - F — = a n d = ? v * v ' While t h i s i n f o r m a t i o n i s on l y of secondary importance i n t h i s experiment and i s obtained t o much g r e a t e r p r e c i s i o n near the c r i t i c a l p o i n t by ot h e r means, i t does provide good data f u r t h e r away from the c r i t i c a l p o i n t . 45 VI. DATA ANALYSIS The data a n a l y s i s was made by computer and the adage, "Never t r u s t a computer" was found t o be not e n t i r e l y i l l founded. A l l c a l c u l a t i o n s were checked by hand f o r t y p i c a l data v a l u e s t o i n s u r e accuracy. The computer programs used are c o n t a i n e d i n Appendix B. Constants used were as f o l l o w s : rig = 1.66 index of r e f r a c t i o n of sapphire n^.= 1.00029 index of r e f r a c t i o n of a i r V = 13.108 ± .003 cc volume of the c e l l WQ = 179.394 * .001 gm weight of the evacuated c e l l v o o h micrometer / • -, -, ,, \ X A = - 8 . 3 4 r e a d i n g ( a l r i n c e l 1 " n 0 c e l l ) 0< = 20.088° angle between i n n e r window f a c e s (J) = - .080° wedge angle of the o b l i q u e window b = .00006476 s i n 6 i = b ( X i - X Q) rig and n^ are taken from the 5 2nd e d i t i o n of the Handbook of Chemistry and P h y s i c s . A d i s c u s s i o n of the o r i g i n of the c o n s t a n t s except , \ , and X A i s g i v e n i n d e t a i l i n e a r l i e r s e c t i o n s of t h i s work. ot and § were determined by p l a c i n g the c e l l on a r o t a t i n g t a b l e having degrees marked o f f on the c i r c u m f e r e n c e . L i g h t was r e f l e c t e d from the s u r f a c e s of the windows and the a n g u l a r p o s i t i o n r e c o r d e d . The v a l u e s are the averages of f i v e r e a d i n g s and are accurate t o - .008° . X A was the average of 11 v a l u e s taken with a i r i n the c e l l . n^ i s the index of r e f r a c t i o n of ethane and was c a l c u l a t e d a c c o r d i n g t o n, = ru ( 1 + cos 6 - cos 6 . + s i n 6 ~ 3 * n 6 A ). 1 3 A t a n &. 46 T h i s a n a l y s i s was used i n s t e a d of the f u l l a n a l y s i s which 2 i n c l u d e d the c o r r e c t i o n term and a f a c t o r o f cos § because cos2(j> was .999998 and the c o r r e c t i o n term "gin fr( f ( e ) - f ( e A ) ) was t y p i c a l l y .00009 • These were n e g l i g e a b l e compared t o the e r r o r i n t r o d u c e d by the u n c e r t a i n t y i n the micrometer r e a d i n g and the weight. W*N was determined a c c o r d i n g t o = « >  — < TL + 2 Table I g i v e s the r e s u l t s of the a n a l y s i s f o r g i v e n v a l u e s of c e l l mass and micrometer r e a d i n g . Since the constancy of i s the major c o n s i d e r a t i o n a value of was c a l c u l a t e d 2 near ^>c by t a k i n g a l l v a l u e s of ^ f o r which ( ^  - ^ c ) < .04 and a v e r a g i n g these v a l u e s t o give£^,. While t h i s i s not to be construed as at T=T c t f= P c s i n c e d e n s i t y g r a d i e n t s prevent our s e t t i n g T= T » i t does g i v e a value o f i n the r e g i o n of the c r i t i c a l p o i n t and i f «£(T) as was i n d i c a t e d by e a r l i e r r e s e a r c h , but r e f u t e d by l a t e r f i n d i n g s , ( M i c h e l s , t e n Seldam, and O v e r d i j k ^ and Abbiss, Knobler, Teague, and Pings ) then t h i s average should be a good approximation t o . The constancy of i s shown i n F i g u r e 10 which i s a p l o t o f off- pC vs ? . The t o t a l v a r i a t i o n i n o ^ i s seen <*c Sc t o be about 1% . T h i s i s i n c l u d i n g v a l u e s of 3 ^ f o r ^ near 0 which means W = WQ . I t i s i n t h i s r e g i o n t h a t d i f f e r e n c e s between the c e l l weight when empty and when f i l l e d w i t h s m a l l Table I DATA AND RESULTS ( * = 6328 X ) k? Mass Micrometer Density Index Lorentz (gm) (gm/cc) ( c c / 184 . 0 5 3 6 1 1196.52 .3555 1.2107 .3781 184.00258 1181.79 .3516 1.2082 .3779 183.93675 1161.72 .3^65 1.2047 .3773 183.89613 1151.61 1.2030 .3777 183.84352 •1137 .17 .3394 1.2005 .3777 183.78652 1122.05 .3351 1 . 1 9 7 9 .3778 183.75072 1112.04 .3323 1.1961 .3778 183.70305 1098.98 .3287 1.1939 .3778 I83.65760 I O 8 5 . 6 7 .3252 1.1916 .3775 I 8 3 . 5 7 2 8 8 1063.49 .3188 I . I 8 7 8 .3778 183 . 5 3 3 0 6 1 0 5 3 . 3 0 . 3 1 5 7 1.1860 .3780 183.49588 1043.02 .3129 1.1842 .3780 183.45684 1032.58 .3099 1.1824 .3780 183.41906 1022.54 .3070 1.180? .3781 183.37975 1011.98 .3040 1.1789 .3781 183.31190 993.30 . 2 9 8 9 1 . 1 7 5 6 .3780 I 8 3 . 2 5 2 6 1 977.91 . 2 9 4 3 1.1730 .3783 183.19706 962 . 7 1 .2901 1.1703 . 3 7 8 2 183.14800 947.21 .2864 1 . 1 6 7 7 .3773 183.11060 939.03 .2835 1.1662 .3780 183.06472 928.85 .2800 1.1645 .3788 I 8 3 . O I 6 5 2 915.20 . 2 7 6 3 1.1621 .3786 182.94860 897.50 .2712 1.1591 .3788 182.84800 871.07 . 2 6 3 5 1.15^5 .3790 182.78902 8 5 4 . 9 6 .2590 1 . 1 5 1 7 .3788 182.73998 842 . 0 0 .2552 1.1494 .3789 182.68606 828.04 .2511 1.1470 .3790 182 . 6 3 5 5 0 814.16 .2473 1.1446 .3788 182.59896 804.49 .2445 1.1429, .3789 182 . 5 4 9 2 2 792.57 .2407 1.1408 .379^ 182.51468 782.61 .2381 1.1391 .3791 182.45106 765.38 .2332 1.1361 .3789 182.37164 746.14 .2271 1.1328 .3797 182.33394 735.37 .2243 1.1309 .3793 182 . 3 0 1 0 2 726 . 6 7 .2218 1 . 1 2 9 4 .3793 182.27076 718.73 .2194 1.1280 .3793 182.23302 709.01 .2166 1.1263 .3794 182.18898 697.60 .2132 1.1243 .3795 182.14100 684.81 . 2 0 9 5 1.1221 .379^ 182.10010 6 7 4 . 0 3 .2064 1.1202 .3793 182.06508 664.64 .2038 1.1186 . 3 7 9 2 182.04030 658.39 .2019 1.1175 .3793 181.98806 644.81 .1979 1.1151 .3793 181.93390 630.77 .1937 1.1127 .379^ 181.88646 618.26 .1901 1.1105 .3793 Table I (cont.) Mass Micrometer Density Index Lorentz (gm) (gm/cc) (cc/i 181.84694 608.11 .1871 1.108? .379^  181.78708 592.65 .1825 1.1060 .3794 181.73338 578.71 .1784 1.1036 .379^  181.67764 564.15 .1742 1.1010 .3793 181.63812 55^.35 .1712 1.0993 .3796 181.58498 540.30 .1671 1.0969 .379^ 181.52576 524.80 .1626 1.0942 .3792 181.48830 515.18 .1598 1.0925 .3792 181.42338 498.56 .15^8 1.0896 .3793 181.38990 490.09 .1522 1.0881 .3794 181.34260 477.76 .1486 1.0859 .3792 181.29532 465.25 .1450 1.0837 .3789 181.23776 450.31 .1406 1.0811 .3787 181.17370 434.66 .1358 1.0784 .3793 181.12364 421.46 .1319 1.0761 .3789 181.07504 408.98 .1282 1.0739 .3788 181.02886 396.71 .1247 1.0717 .3784 180.98216 385.13 .1211 1.0697 .3786 180.9^216 37^ .92 .1181 1.0679 .3785 180.89380 362.91 .1144 I .O658 .3788 180.85550 353.25 .1115 1.0641 .3788 180.80800 340.68 .1079 I . 0 6 1 9 .3782 180.75396 327.38 .1037 1.0595 • 3785 180.70250 313.96 .0998 1.0572 .3780 180.65644 302.60 .0963 1.0552 .3782 180.60348 289.00 .0922 1.0528 .3779 180.55738 277.59 265.74 .0887 1.0508 .3780 180.51108 .0852 1 .0487 .3777 180.47022 255.86 .0821 1 .0470 .3782 180.42192 243.49 . 0 7 8 4 1.0448 .3777 180.37790 232.83 .0750 1.0429 .3782 180.33630 222.08 .0719 1.0410 .3776 180.28656 209.83 .0681 1.0388 .3778 180.22740 194.91 .0636 1.0362 .377^ I 8 O . I 8 2 5 6 183.88 .0601 1.0343 .3776 180.14342 1 7 4 . 2 2 .0572 1.0326 .3777 180.09006 161.02 .0531 1.0302 .3777 180.04108 148.47 .0493 1.0280 .3767 179.98406 134.92 .0450 1.0256 .3780 179.94180 124.54 .0418 1.0238 .3782 179.88868 111.38 .0377 1.0215 .3781 179.81136 92.01 .0318 1.0181 .3770 179.76842 81.38 .0285 1.0162 .3766 179.71290 67.5^ .0243 1.0137 .3755 179.66080 55.05 .0203 1.0115 .3767 179.61404 42.94 .0168 1.0094 .3720 179.56532 31.66 .0130 1 . 0 0 7 4 .3762 179.50588 16.87 .0085 1.0048 .3720 179.46234 6.53 .0052 1.0029 .3753 179.^0892 -7.21 .0011 1.0005 .2927 .005 -.01 h + 0.0 \- + -.005 r V ++ + + E T H A N E 0.0 0.5 1.0 1.5 2.0 Figure 10 - VARIATION OF ^ WITH io FOR ETHANE 5 0 amounts of ethane are q u i t e s m a l l and produce e r r o r s t h a t i n c r e a s e as W-»W . To produce a "better s c a l e f o r the graph the f i v e data p o i n t s n e a r e s t WQ were omitted as the v a r i a t i o n became extreme. The main body of the data shows a v a r i a t i o n of t h a t i s w i t h i n and the v a r i a t i o n f o r ^ near ^ c i s l e s s than .lfo . F i g u r e 1 1 shows a comparison of r e c e n t unpublished work on s u l f u r h e x a f l u o r i d e by B a l z a r i n i and P a l f f y . The two s e t s of data are s i m i l a r f o r hi g h d e n s i t i e s but the SF^ data i s more n e a r l y constant f o r low d e n s i t i e s . F i g u r e 1 2 shows a comparison with data presented by Smith f o r xenon. The ethane data i s c l e a r l y more constant over the e n t i r e data range. The c o e x i s t e n c e curve and r e c t i l i n e a r diameter a n a l y s i s was done by Mr. P a l f f y . The a n a l y s i s i n v o l v e d a three parameter f i t t o the data and e x t r a p o l a t i o n t o g i v e ^C^CC = . 0 7 8 2 - . 0 0 0 1 and T = 3 2 . 0 7 9 - . 0 0 1 °C . Us i n g the c a l c u l a t e d value of c e£sc = . 3 7 9 4 ± . 0 0 0 3 g i v e s a value of P Q = . 2 0 6 2 ± . 0 0 0 3 gm/cc . F i g u r e 1 3 shows the c o e x i s t e n c e curve and r e c t i l i n e a r diameter. I t was not ab l e t o determine from t h i s data t h a t the c o e x i s t e n c e curve diameter d i f f e r e d from a s t r a i g h t l i n e . . A Ba lza r in i and Palffy unpublished + present investigation 0.0 0.5 1 1.5 * SF 6 + Ethane i_ 2.0 Figure 11 - VARIATION OF X WITH ^ FOR SF 6 AND ETHANE .01 .005 0.0 -.005 -.01 • Smith et al PRL 21.1306.(1968) • + present invest igat ion • • a • • • • • + * + a + + a a + • • + + + • • • • • • a Xe + Ethane 0.0 0.5 1.0 1.5 2.0 (V) Figure 12 - VARIATION OF 2£ WITH ^ FOR XENON AND ETHANE T E M P E R A T U R E C 1—» r o r o CO CD CO > ) r o c n CO • o • r o CD o 00 4N o c n cn 54 V I I . ERROR ANALYSIS Assuming t h a t i s i n f a c t constant t h i s e r r o r a n a l y s i s seeks t o e x p l a i n any departures from constancy. To b e g i n the a n a l y s i s the known e r r o r i n c a l c u l a t i n g o£ due t o e r r o r i n weighing and r e a d i n g the micrometer i s i n v e s t i g a t e d . A o£ AW_ AX / X - A . o A ^ _ ^ + y _ v where W i s the c e l l weight. WQ i s the empty c e l l weight, X i s the micrometer r e a d i n g , and X^ i s the micrometer r e a d i n g f o r the c e l l f i l l e d w ith a i r . A ^ £ i s the e r r o r i n ^ and s i m i l a r l y f o r AW and AX. The e r r o r i n r e a d i n g the micrometer was .3 d i v i s i o n s and the e r r o r i n the weighing was 1 m i l l i g r a m . The r e s u l t i n g e r r o r i n v a r i e s over the range of data. F o r hig h d e n s i t i e s i s about 10" 3 .3 or .0598 . T h i s i s f o r P~ . 3 4 g m/cc and the co r r e s p o n d i n g accuracy i n ^  i s ± .0002 . F o r low d e n s i t i e s c o r r e s p o n d i n g t o £ 06gm/cc the e r r o r i - n Aj£ i s .5$ which l i m i t s the accuracy o f o£ t o - .002 . F o r d e n s i t i e s near ^ c which corresponds t o ^ ^ . 2 gm/cc we have Ao? _ .085$ which means ^ i s known to - .0003 . A measured value o f pj which i s s l i g h t l y l a r g e r than i t s a c t u a l value would tend t o g i v e a value of 3*s which i s lower than i t s t r u e v a l u e . One e x p l a n a t i o n f o r the departure 55 of ^ from o£ c would be that the temperature f o r ^ on e i t h e r side of p c i s lower than that at P c > This means that i t i s possible that some moisture from the a i r i s being weighed with the c e l l f o r high and low densities that i s not being included at c r i t i c a l density. This seems reasonable i n view of the fact that weighings of the c e l l hot and cold produced a consistent increase i n weight of 2 mg. However t h i s i s only .05$ of the sample weight f o r high densities and cannot account f o r the observer decrease of .5$ The c e l l volume would increase as the temperature increased and since the c e l l volume was measured at room temperature t h i s would mean that near T c the c e l l volume used i n the c a l c u l a t i o n was l e s s than the actual volume. Increasing the volume would increase the value of and cause the peak around pc to be more pronounced. Since the temperature range of the experiment was about 15 °C and the c o e f f i c i e n t of l i n e a r expansion of aluminum i s 2 x 10"^ we have V " V o = 2 x 3 x 15 x 10~5 or .09$ . V o This means that the e f f e c t of thermal expansion i s much less than errors introduced i n measurements at low densities but i s comparable to measurement errors at medium and high d e n s i t i e s . 56 E r r o r s i n t r o d u c e d by the s h i f t i n g of the l i g h t beam were c o r r e c t e d f o r by measuring the p o s i t i o n of the beam with the c e l l removed each time a data p o i n t was ta k e n and c o r r e c t i n g the observed beam d e f l e c t i o n with the zero beam re a d i n g , X Q . The s h i f t i n g of the zero beam d u r i n g the course of the experiment was l e s s than f o u r d i v i s i o n s . During the measurement of the c o e x i s t e n c e curve a beam s p l i t t e r was used t o d i r e c t the beam around the c e l l c o n t a i n e r and m a i n t a i n a r e f e r e n c e beam d u r i n g the data c o l l e c t i o n . T h i s allowed the c e l l t o remain i n place f o r the sweep over temperature and permitted thermal e q u i l i b r i u m t o be maintained more e a s i l y . ( See F i g u r e 4 ) The balance was equipped with s l i d i n g g l a s s windows t o p r o t e c t the pan from c o l l e c t i n g dust while not i n use and the c e l l was handled w i t h p l a s t i c g l o v e s i n order t h a t the weight not be a l t e r e d by contamination. Before the s t a r t of the data c o l l e c t i o n the c e l l f u l l of ethane was p l a c e d i n a b e l l j a r t o remove any moisture t h a t may have been on the surface or i n s i d e the v a l v e . E f f o r t s t o minimize the weighing e r r o r can only be assumed e f f e c t i v e as the a c t u a l e f f e c t i s not known. 57 V I I I . CONCLUSIONS 1 2- 1 T h i s work demonstrates t h a t f o r ethane o£ = n 7 , " n +2 where i s c o n s i d e r e d constant, i s an e x c e l l e n t approximation f o r d e n s i t i e s of 200 amagats and l e s s s i n c e the data showed a v a r i a t i o n of %fo or l e s s i n . T h i s j u s t i f i e s the use of the Lorentz-Lorenz e q u a t i o n f o r o p t i c a l s t u d i e s of c r i t i c a l p o i n t phenomena which r e l a t e the index of r e f r a c t i o n t o d e n s i t y . A s l i g h t decrease i n was found f o r ^ on e i t h e r s i d e of P c hut f u r t h e r s t u d i e s are needed to v e r i f y the decrease. The o p t i c a l system and the temperature c o n t r o l proved e n t i r e l y adequate, hut the method of determining the weight of the c e l l c o u l d be improved. Future s t u d i e s should pay p a r t i c u l a r a t t e n t i o n t o the c o n t r o l of a l l f a c t o r s c o n c e r n i n g the weight of the c e l l and i t s d e t e r m i n a t i o n as t h i s was probably the g r e a t e s t source of u n c e r t a i n t y . Care should be taken t o i n s u r e t h a t the weight of the c e l l does not vary w i t h temperature. E x t r a p o l a t i o n of the c o e x i s t e n c e curve data gave P =.2026 ±.0003 gm/cc and T =32.079 ±.001 °C . The c o e x i s t e n c e curve diameter was not found to vary from a s t r a i g h t l i n e . 58 BIBLIOGRAPHY (1) B a l z a r i n i , D.A.; Ph.D. T h e s i s , Columbia U n i v e r s i t y , (1968). (2) Chapman, J.A., Finnimore, P.C., and Smith, B.L.; Phys. Rev. L e t t . , 21, 1306, (1968). (3) B a l z a r i n i , D.A. and Ohrn, K.E.; Phys. Rev. L e t t . , 29_, 840, (1972). (4) B a l z a r i n i , D.A. and P a l f f y , P.5 unpublished, U n i v e r s i t y of B r i t i s h Columbia, (1973). (5) Jackson, J.D.; " C l a s s i c a l E l e c t r o d y n a m i c s " , John Wiley and Sons, Inc., New York, (1962). (6) Panofsky, W.K.H. and P h i l l i p s , M.; " C l a s s i c a l E l e c t r i c i t y and Magnetism" , Addison-Wesley, Reading, Mass., (1955). (7) F r e n k e l , J . ; " K i n e t i c Theory of L i q u i d s " , p. 260, Dover P u b l i c a t i o n s , Inc., New York, (1955). (8) Feynman, R.P., "The Feynman L e c t u r e s on P h y s i c s " , v o l . I I , Addison-Wesley, Reading, Mass., (1964). (9) M i c h e l s , A. and Botzen, A.; Physica, 15, 769, (1949). (10) M i c h e l s , A., Ten Seldam, C.A., and O v e r d i j k , S.D.J.; Phy s i c a , 17_, 781, (195D. (11) E a t w e l l , A .J. and Jones, G.O.; P h i l . Mag., 10, 1059, (1964). — (12) Jones, G.O. and Smith, B.L.; P h i l . Mag., 5_, 355, (i960). (13) Amey, R.L. and Cole, R.H . i J . Chem. Phys., 40, 146, (1964). (14) Abbiss, C P . , Knobler, CM., Teague, R.K., and Pings, C . J . j J . Chem. Phys., 42, 4l45, (19&5). (15) Johnston, D.R., Oudemans, G.J., and Cole R.A.; J . Chem. Phys., 22., 1310, (I960). (16) Garside, D.H., M^lgaard, H.V., and Smith, B.L.j Proc. Phys. S o c , 1, 449, (1968). 59 ( 1 7 ) Ten Seldara, C.A. and De Groot, S.R.; Physica, 1 8 , 905, (1952). ( 1 8 ) Kirkwood, J.G.; Phys. Z., 12, 5 7 , (1932). ( 1 9 ) Ten Seldam, C.A. and De Groot, S.R.; Physica, 1 8 , 9 1 0 , (1952). ( 2 0 ) De Boer, J . , Van Der Maesen, F., and Ten Seldam, C.A.; Phys i c a , 1 £ , 265, ( 1 9 5 3 ) . ( 2 1 ) Kirkwood, J . G . j J . Chem. Phys., 4 , 592, ( 1 9 3 6 ) . ( 2 2 ) Van Vleck, J.H.; J . Chem. Phys., £, 320, ( 1 9 3 7 ) . (23) Brown, W.F.,j J . Chem. Phys., 1 8 , 1 1 9 3 . 1 2 0 0 , (1950). ( 2 4 ) M i c h e l s , A., De Boer, J . , and B i j l , A.; Ph y s i c a , 4 , 9 8 1 , ( 1 9 3 7 ) . (25) Y a r i s , R. and Kirtman, B. ; J . Chem. Phys., ^ Z , 1 7 7 5 . ( 1 9 6 2 ) . (26) Jansen, L. and Mazur, P.? Physica, 2 1 , 193, ( 1 9 5 5 ) . (27) Mazur, P. and Jansen, L . j Physica, 2 1 , 2 0 8 , ( 1 9 5 5 ) . ( 2 8 ) Mazur, P. and Mandel, M.5 P h y s i c a , 2 2 , 289,299, ( 1 9 5 6 ) ( 2 9 ) Jansen, L. and Solem, A.D.; Phys. Rev., 1 0 4 , 1 2 9 1 , ( 1 9 5 6 ) . (30) Jansen, L.; Phys. Rev., 1 1 2 , 4 3 4 * ( 1 9 5 8 ) . 60 APPENDIX A. The a n a l y s i s of the case of the o b l i q u e window being s l i g h t l y wedge shaped i s due t o D. B a l z a r i n i and i s as f o l l o w s . 1 / / V / V n^ n. REFRACTION ANALYSIS - WEDGED WINDOW Figur e 14 sinoc = s i n p and s i n ^ = n^ s i n (ot + 0 + $ ) where n^ = index of r e f r a c t i o n o f ethane, = index of r e f r a c t i o n of sapphire, and n^ = index of r e f r a c t i o n of a i r . Since £ = y- <J> , by expanding s i n fi and s u b s t i t u t i n g f o r s i n uv we have n^ s i n o t = n^ s i n (t*+ 0 + | ) c o s § - n 2 s i n <|) cosH*. Expanding s i n («+ 6 + $>) g i v e s , n^ sincx= n^ cos2<^> sin(c*+0) + s i n <j) f" n^ cos $ cos(«x+0)-n 2 c o s ^ J 2 i Expanding cos(«+9) and l e t t i n g c o s f = (1 - s i n y ) 2 n^ cos <|) cos(ot+0) - ix, cos»V .which we c a l l f ^ , becomes , n-cos (|>(coso(cos e - sine* s i n 6) - n2C 1 \ sin 2(c*.+ 6 + n 2 Denoting the q u a n t i t y under the r a d i c a l f 2 we expand sin(ot+ 0 + as ((*.+ 0) + § and square i t . i. 2 61 Using double angle formulas we have, n 2 f 2 = 1- 2 ( sin 2(«+ <j>) + s i n 2 9 cos 2(tx + + s i n 2(<x+$)sinGcosG "2 2 2 n~ g n T factoring out 1 g s i n (c*+ <J>) we now have, l e t t i n g —g = a "2 "2 f 2 = ( l - a sin2((*+<]))) ( l + x) where x i s - a ( s i n 2 6 cos 2(M+<|)) + s i n 2(o<+<t>) s i n 9 cos e) a n d i s a 1 - a sin2(o<+(J)) small quantity. For the case of o< = 20°, <J> = .08°, and 9 = 3 ° i x i s only .003 • Since we need ( f 2 ) 2 we make use of the - x approximation ( 1 + x ) a = ( l + g ) f o r small x and write ( f 2 )== ( l - a sin 2(*+<j>))^l + |) . Since s i n 0 i s always small, 9 i s about 3° at i t s maximum, we use the same 2 approximation i n substituting 1 - s i n 9 fo r cos 0 . The 2 difference between the two fo r 0=3 i s .0000009 and .000003 f o r 0=4° , so the approximation i s quite good. Multiplying out a l l the factors we now have f o r f^ , 2 n~ cos <f> coso(- n,, cos <b cosoc s i n 0 - n~ cos § sin<x s i n 0 3 3 2 3 - n 2 ( 1 - a s i n 2 ( < K + i ) ) - ng( 1 - a sin2(oc+$>)) J f^ can be seen to contain two terms which have no 0 dependence and three terms which have 0 dependence. Thus we can write f± = P(oc,o) + F(0) 62 Solving now f o r n^ we have , n., = 1 I n^ cos 2(j) s i n (o<+9) + s i n <t» F(«,<ji) + F(9) s i n c< L ^ J s i n <x L J which upon expanding s i n (<x+9) becomes = n 3 cosh (cos 8 + + ^ [««.•) + P(6)] . When there i s a i r i n the c e l l n^ = n^ and the angle made by the deviation of the beam from i t s o r i g i n a l d i r e c t i o n , denoted © A , gives ; n 3 = n 3 c o s 2 * (cos 6 A + + [ « « . • ) + F(8 A>] Subtracting we have, 2 • s i n 9 - s i n 0.\ n i - n 3 = n 3 cos 4>(cos 9 - cos 9 A + ) Solving for n^ gives the f i n a l form which i s , 2 * s i n 9 - s i n 0. v | os 4>(cos 9 - cos 9 A + i s r 5  For the experiment the window was a c t u a l l y wedged i n the opposite d i r e c t i o n to that which i s shown, however t h i s does not change the analysis as <|> i s replaced by , This analysis was not needed as i t turned out because cos 2$ was .999998 and s i n <t> f F(9) - F(0 A) was .00009 s i n «x L A J n l = n 3 1 + C ( 63 at the most, which was l e s s than the e r r o r introduced i n the weighing and the reading of the micrometer and was t h e r e f o r e neglected. This g i v e s s i n 6 - s i n ©, n l = n 3 1 + cos 6 - cos 6. + A t a n oc which was the equation f i n a l l y used i n the data a n a l y s i s . 64 APPENDIX B. The following program was used to calculate n, 65 I M P L I C I T REAL*& (A*H,Q«Z) DIMENSION -X (S00.)\Y (500ji, RH0 (§00), XE C500), XES (500 J, CL0RZ C500),. XAIR=1,00029 'v.6;i v=i3, loeo XQ = »8»34 WQ=179,3943 AL=20,088 PH=»f080 ~ — — — r -ALP3,iai5936S*AU/180,0 PH=3,i4l59265*PH/180,0 .a FORMAT (//,iX» IBs1»F10,6) 3 FORMAT (2X,P10,5,3X,F8,2,6X FF7,«,«X,F8,5,7X,F7,4) 1 FORMAT (2X,2F10,S) -rTtmttAi'—t/77V**7^^ P L O R E N T Z C O E F F . i , / ) N = 9S DELX = 19—2~3 B=C6,328*S,0*,000n/(2,339998*XAIR) B=B/DELX W-R^E-C-feT^-B • WRITE (6,6) TNsDTAN(AL) TCBDARSINCBOXQ) — F1=XAIR*XAIR/(1,66*1,66)*DSIN(AL>PH)*DSIN(AL+PH) F l = l , 0 s F l — F 5 « »*~AI-R **-A-I-R7-(~i-i-6 6*-lT66-)-*-(WWWmmWWS«TH^ W+^  1N(2,0*(AL+PH))*DSIN(TC)*DC0S(TC))/F1 V~ F6= •XAIR*(DC0S(PH)*OSIN(TC)*DSIN(AL)tDC0S(PH)*DC0S(AL)*0SIN(TC)*D lSIN(TC)/2,0)tl,66»P3QRT(Fl)*F5/g,Q — DO 2 I=1,N READ(5,1) X U ) , Y ( I ) —Rtt-0-m-=-eY-m-«»*fi-)7-v — — X2=B*X(I) X1=B*X0 X31 = X1»X1 ^ XS=X2*X2 TH=DARSIN(B*X(I)) tN(2,0*(AL + PH))*0SINCTH)*DCO8-(TH) ) / F l F3= •XAIR*(DC0S(PH)*DSIN(TH)*0SIN(AL)+DC0S(PH)*DC0S(AL?*D8IN(TH)*D lSIN(TH)/2,0 3-»l,66*DQQRT(Fl)*F2/'g,0 ^ CTERM = B-DSIN(PH)/DSIN(AL)*(F3i!F6) j D=DC0$(PH)*DC0S(PH) > —-X£ (I) eXA IR *~(~lxO^G-0£~(~W • — ~ — — — — ~ t ~ — XESCn = X E ( I ) * X E ( I ) 7 ClORZ(I)=(XES(I)«l S0)/((XES(n*2,Q)*RHQ(l)) WRITE (6,3) Y(n,X(n,RH0(I),XE(I),CL0ftZ(I) fe XE(I)=XAIR*(l,0+(DCOS(TH)»DCOS(TC)*(X2sXl)/TN)*D)+CTERM V XES(I)=XE(I)*XE(I) V-ebORZ-(-BM-XE~8-(-I~)M WRITE (6,3) Y ( I ) , X ( I ) , R H O ( I ) , X E ( I ) , C L Q R Z ( I ) i 2 CONTINUE S-T-0-p _ END • •'---*" 66 This program i s used to calculate £Cc by averaging a l l values of <£f f o r which ) 2< .04 . D I M E N S I O N X C 1 0 0 ) r Y ( t 0 0 ) \ F O R M A T ( 2 X , 2 F t O , 0 ) 19 F O R M A T C I X » » C t L = f | F 1 6 , 6 ) N s 9 0 • — 0 0 6 I = J # N & R E A D ( 5 , U X ( J ) , Y ( I ) R H Q C = , 2 0 8 = 0,0 A V E = 0 s 0 0 0 6 0 1 = 1,N D * ( X ( I ) » R H 0 C ) / R H 0 C J P ( D * D « , 0 4 ) i7f\7,i& 17 A V E s Y ( I ) + A V E B s B * l , 18 C O N T I N U E — (}Q C O N T I N U E -C L U A V E / B W R I T E C 6 , 1 ) B W R J T E C 6 , 1 9 ) C L U S T O P E N D This program does a least squares f i t to a straight l i n e f o r the c a l c u l a t i o n of b using the Ronchi r u l i n g . I M P L I C I T R£AL*8 (A*M>Q*Z) DIMENSION, X ( 8 0 0 ) i Y C 2 0 0 ) # X l l ( 2 0 0 ) J FORMAT ( H 2 , < U 5 FORMAT(12) 3 FORMAT ( 6X,'SLOPE•?,7X,»Y INTERCEPT') U FORMAT ( I X , a F l 2 8 4 ) BA=0,0 SLOA=0,0 00 7 J c l , S READ(5,1) A READ(5,5) N SX=G,0 SY=0,0 SXY=0,0 SX2P0.0 C-sN DO 2 I=l#N READ(5 r 1 ) Y d ) SX=SX+A S Y a S Y + Y ( I ) S X Y s S X Y + A * Y ( I ) SX2=SX2tA*A A=A+1,0 2 CONTINUE SLOPE?(SX*SY«C*SXY)/(SX*$XsC*SX2) B = ( S Y B S L 0 P E * 5 X ) / C WRITE(6,3) WRITE(6»<0 S L O P E D BA=BA+B SLOAsSLOA+SLOPE 7 CONTINUE BA=BA/5,0 SLOA=SLOA/S.O WRITE(6,3) -W R I T E («»•»<•) SL0A,BA STOP END 68 The following program was used to produce the plots of X - £ 0 v s _ £ <5Cc ^c DIMENSION X ( 5 0 0 ) , Y ( 5 0 0 ) 1 FQRMAT(2X,2F10,5) 11=90 L 2 ? 3 2 =?r-K=Li+L2 J=L1+L2+L3 D'Q-9-0"t=-lTiJ" READ(5,1) X ( I ) , Y ( I ) 90 CONTINUE tP=i N=U1 RHOCa»20 ——C-U-S-=T^'9^2 D O 6 I = L , N X ( I ) = ( X ( I ) * R H O C ) / R H O C Y-(I)aCY(I)»CL3)/Cl8 : 6 CONTINUE L=L1+I N s K  RHQC=8,42 CLS = 1 0 8 5 2 7 DO 91 1 = 1,N X(I)=(X(I)«RHOC)/RHOC Y ( I ) = ( Y ( I ) * C L S ) / C L S 91 CON-T-INUE LBK + 1 N = J C L 3 ? 7 6 9 , 8 RHOC=,7357 DO 92 I = L , N X-H^=-C^m-irRtlE^eiV-R+t«^ Y ( I ) = ( Y ( I ) * C L S ) / C L S 92 CONTINUE CALL PL0T(0 8Q,0,0,»3) DATA SX,SY/8»G,6,0/ CALL S C A L E ( X , J , S X , X M I N , D X , l ) C-AL-L—S&A-LE (Y, J rS-Y-rTtWrfr^rl-1 CALL AXIS(0,0,0,0,3HRH0,»3,SX,0,0,XMIN,DX) CALL AXIS(0,0,0.0,2HLL,2,SY,90,0,YMIN,OY) tr=i N a L i DO 93 I ? L , N e ^ t L ^ ^ M & 0 ^ - ( ^ H - ^ H ^ T T ^ T % ^ ' » - H 93 CONTINUE L s L U l N « K D O 60 I = L , N CALL SYMBOL ( X ( I ) , Y ( X ) , , l , 0 , 0 , 0 , a l ) -6-0-eO'N^ H-NU'E L=K + 1 N P J DO 80 I°?L,N CALL SYMBOL (X ( I ) , Y ( I ) , , 1 , 2 , 0 , 0 , * 1 ) 80 CONTINUE STOP END 70 The f o l l o w i n g programs were used to f i t the c o e x i s t e n c e curve and the r e c t i l i n e a r diameter w i t h three parameters and to e x t r a p o l a t e t o f i n d T c and £ c ^ c • Programs are a l s o g i v e n which p l o t the c o e x i s t e n c e curve and the r e c t i l i n e a r diameter. 71 I M P L I C I T REAL*8 (A*H,0"Z) DIMENSION Y N L U 0 O ) , Y N V ( J 0 Q ) , 9 2 ( 1 0 0 ) , X M 2 Q U O O ) DIMENSION FNV(1OO),FNLC10O),Q1UQQ) DIMENSION CQ1(100),CQ2(2Q0) DIMENSION R ( 1 0 0 ) ,XMVUQQ),XMLUOQ),XMZ( J 00 ) , T (1 00 ) , XO (1 00 ) DATA B,XK/,Q0006«77748DO,»9,99DO/ DATA X AIR, ALP, GAM/1 ,0002900,0, 350300,2,8500/' DATA RQ,BET/0,QH99D0,3S44,061DQ/ READ(5,1) N FORMAT(I«) TN=DTAN(ALP) DO 2 1=4,N READ(5,3) R ( I ) , X M V ( I ) , X M L ( I ) , X M Z O ( I ) T ( I ) = B E T / D L O G ( R ( I ) / R O ) T ( I ) = T U ) * 2 7 3 , 16 X O ( I ) = X M Z O ( I ) * X K Y N L ( I ) = X A I R + B * ( X M L ( I ) * X O ( I ) ) / T N lm((B*(XML(I)*XO(I)))**2)/(2,*XAlR) Y N V ( I ) = X A I R + B * ( X M V ( I ) B X O ( I ) ) / T N l « ( ( B * ( X M V ( I ) * X 0 ( I ) ) ) * * 2 ) / ( 2 , f r X A l R ) F N V ( I ) = ( Y N V ( ! ) * Y N V ( I ) * J , 0 ) / ( Y N V ( I ) * Y N V U ) * 2 , 0 ) F N L ( I ) = ( Y N L ( I ) * Y N L ( I ) * l , 0 ) / ( Y N L ( I ) * Y N L ( I ) t 2 t 0 ) Qi ( I ) = ( F N L ( I ) e f ? N V ( I ) ) ^ * e A M Q 2 ( I ) = ( F N L ( I ) * F N V ( I ) ) / 2 , 0 CONTINUE F0RMAT(4D15,8) CALL 3 0 F T ( N , Q l , T , U I , V l , W | i , E R | ) CALL SQFT(N,Q2,T,U2,V2,W2,ER2) WRITE(6,S) UI,V1,WI,ER1 WRITE(6,6) U2,V2,W2,ER2 T C 2 = U 8 / ( 2 . * W l ) ) * ( * V I « D 3 Q R T ( V l * V I « 4 , * U i * W i ) ) WRITE(6,«7) TC2 FC=U2*(V2^W2*TC2)*TC2 WRITE(6,50) FC FORMAT(2X,»RCLC S',F15,8) FORMAT ( I X , ITCS-I ,F15,8) F ORMATdX, I ( N L » N V ) * * B E T , COEFFS ARE',SX, 1tA=l,FI5 (8,5X,»B=I,F15 88,5X,»C=«,F15,8,SX,•E=»,F15,8) FORMATdX, I (NL + NV) VS, T COEFFS, ' 5X, 1 !A?M,F15,8,5X, I Be •, F 1 5,8, SX , I C=;' , F 15 ,8, 5X , « E= •, F1 5,8) F O R M A T ( i X , 9 ( F i O , 5 , 2 X ) ) DO 8 1=1,N CQ1 ( I ) ? U l * ( V l t W l * T ( I ) ) * T ( I ) C Q 2 ( I ) = U 2 * ( V 2 + W 2 * T ( I ) ) * T ( I ) WRJTE(6,9) T ( I ) , Y N L ( I ) , Y N V ( I ) , F N V ( I ) f F N L ( I ) , Q i ( I ) , C Q t ( I ) f Q 2 ( n , I , C Q 2 ( I ? CONTINUE CALL GRAPHi(N,FNL,FNV,T,Q2,FC) CALL GRAPH2(N,T*TC2,FNL,FNV) STOP END 72 SUBROUTINE SQPT<M,Y,X,C1 ,G2 fC3#ESQ) I M P L I C I T REAL*8 < A*H , Q « Z ) DIMENSION Y U O Q ) , X ( 1 0 0 ) TS = 0,0 Al=M A2=0,0 A3=0 t0 A«=O eO A S = G,0 Bt=0,0 B2=0,0 B3=0,0 DO 10 I 8 1 , M T S = T S t Y ( I ) * Y ( I ) X 2 = X C I ) * X C I ) A2=A2+X(I) A3=A3*X2 A4=A«*X(I)*X2 ASoA5+X2*X2 B 1 = B 1 * Y ( I ) B 2 s B 2 + Y ( I ) * X ( I ) B 3 * B 3 * Y ( I ) » X 2 . CONTINUE DETaAlA(A3*A5«A4*Aa)eA2*(A2 *AS t iAa*A3)*A3*(A2*A«eA3*A3) C l = ( l (/DET)*(Bl*(A3*A5 * A a*A4)BA2*(B2*A5HB3*A«)fA3*(B2*Aa*B3*A3>-3, C2=( J ,/DET)*(Al*(B2*AS»B3*A«)wBi*(A2:M5eA3 *A«>+A3*(A2*B3-A3*B2)) e3=(r,/DET)*(Al*(A3*B3wAa*B2)oA2*(A2*83«A3*B2)+Bl*tA2*A«eA3*A3)) E S Q = C i * C l * M B 2 l * C i * B l + T S + A 2 * 2 l * C I * C 2 + A3*(C2*e2-t2,*Cl*C3) 1 +A4*2,*C2*C3+AS*C3*C3«2,*C2*B2s2 P*G3*e3 RETURN END 73 SUBROUTINE GRAPH 1CN,FNL,FNV,T,Q2,FC) REAL*8 FNL,FNV,T,Q2,PC INTEGER SYMC3)/3,«,U/ DIMENSION FNL U 00) * FN V ( H 0) rT CIO 0 ) , Y 1 C 1 0 0 ) ,Y2 d 0 0 ) , X ( | 0 0 ) , lY3UOQ),YSl (1OO),YS2UOO),TS(1OO),XSC10O),YS3U0Q),Q2(1OO3 DATA S l , S 2 / 8 , 0 , 6 , 0 / DO 1 1=1,N Y l C n=FNL(I) Y2(I)=PNVm - X t l ) s T ( I ) Y3(I)=Q2(I) CONTINUE CALL RLQT8 D L 3 a ( Y l ( t ) » Y 2 ( l ) ) / 8 1 DLT=(XCN)wX(n)/S2 DO- 2 1 = 1, N YS1 ( I ) B - ( Y l ( I > « Y 2 . ( l ) ) / D L S Y 8 2 ( I ) » ( Y 2 ( I ) » Y 2 ( 1 ) ) / 0 L 8 Y 8 3 ( I ) s ( Y 3 t I ) « Y 2 . ( J ) ) / D L 8 X S ( I ) = t X ( I ) e X ( l ) ) / D U T CONTINUE C A L . L . ' - P U O T ( 0 , 0 , 0 , Q # « 3 ) CALL AXIS(0,0,0,0,»TEMPERATURE»,+11,82,90,0,X(1)#DLT) CALL AXISCO,0,0,0,I R V L V & R L U ', «9, S 1 , Q,0, Y2 C 1 ) , D t S ) 00 3 1=1,N CALL S Y M B O L C Y S 1 ( I ) ; X S < I ) , , H , S Y M U ) , 0 , Q r » 1 ) CALL SYMBOL(YS2(I),XS(I),,t«,SYM(1),0,0,»1) CALL S Y M B O L ( Y S 3 U ) , - X 9 U > M i « # 3 Y M U > f 0 , 0 r « * j CONTINUE XM=SJt2,Q CALL PL0T(XM,0,0,«3) RETURN END 

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