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

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c  (i)  THE  LORENTZ-LORENZ FUNCTION OF  ETHANE by  MICHEL ALAN BURTON B.A., De Pauw U n i v e r s i t y , Ed.D., U n i v e r s i t y  1968  o f Massachusetts,  1970  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER  OF SCIENCE  i n t h e Department o f PHYSICS  We a c c e p t required degree  THE  t h i s t h e s i s as conforming  t o the  standard f o r candidates of the  o f MASTER  OF SCIENCE  UNIVERSITY OF BRITISH COLUMBIA June,  1973  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r  an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  study.  c o p y i n g of t h i s  be g r a n t e d by the Head of my  thesis  Department or  I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t written permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  my  (ii)  ABSTRACT This  study measured the L o r e n t z - L o r e n z  1 n ethane  as determined  = s  by  ~o~—  "  1 £  2 0 0 amagats, where  to  i s t h e d e n s i t y , and  wavelength lt%  of  6 3 2 8 £.  with an average  of  monochromatic was  used  and  the angle  found  t o be  constant  .379** *  .0003  cc/gm  2  two  was  m e a s u r e d by  passed.  A micrometer-driven  the l i g h t  the c e l l  then related  and  ±  .0003  the c e l l  m e a s u r e d and  T h i s enabled  w h i c h was  found  gm/cc.  to  n.  i t s c o n t e n t s on a  r e l e a s e d from  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 .2062  an  mirror  into a collimating telescope  o f d e f l e c t i o n was  temperature  using  s a p p h i r e windows s e t a t  light  c o e x i s t e n c e c u r v e was  of  f o r values  w h i c h a p l a n e wave o f  t o f i t the curve.  density  to within  through  was  for a  c  E t h a n e c o u l d be  The  function,  |^-f l ^ « •  w h i c h had  m e a s u r e d by w e i g h i n g  critical  was  from  o t h e r and  to direct  balance.  densities  r  index of r e f r a c t i o n  i n d e x o f r e f r a c t i o n of ethane  t o each  0  2  TT+  i s the  value of  a light-weight c e l l angle  l  i s the Lorentz-Lorenz  <^f  corresponding to The  used  n  function for  1  2  ^  was  precision  to vary  three  £  .  parameters  e x t r a p o l a t i o n f o r the t o be  diameter  32.079 gave a  . 0 0 1 °C  critical  .  (iii)  TABLE OF CONTENTS  1  I.  Introduction  II.  The  L o c a l F i e l d and  III.  The  Lorentz-Lorenz E q u a t i o n  IV.  L i t e r a t u r e Review  V.  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  4 12 17  A.  E x p e r i m e n t a l Research  17  B.  T h e o r e t i c a l Research  20  The Experiment  26  A.  General C o n s i d e r a t i o n s  26  B.  Temperature C o n t r o l  26  C.  Optics  29  D.  The  Cell  35  E.  The  Sample  37  F.  The  Weighing  37  G.  Data C o l l e c t i o n  40  VI.  Data A n a l y s i s  45  VII.  Error Analysis  54  VIII.  Conclusions  57  IX.  Bibliography  58  X.  Appendices  60  (iv)  LIST OF TABLES  Table  I  Page  47  (v)  LIST OF FIGURES FIGURE  PAGE  la  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 S F  51  12  VARIATION OF 5 < WITH  ^ FOR XENON AND ETHANE  13  COEXISTENCE CURVE AND RECTILINEAR DIAMETER OF ETHANE  14  REFRACTION ANALYSIS - WEDGED WINDOW  6  AND ETHANE  52  60  1  I.  INTRODUCTION In  r e c e n t y e a r s 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 l i q u i d - v a p o r  critical  (1)  points, B a l z a r i n i  has s t u d i e d phase t r a n s i t i o n s u s i n g  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  of  temperature,  pressure, and d e n s i t y are s t u d i e d i n an e f f o r t  to  determine  £  ~m—  "vc)  =  i s an example of one  T i s the temperature be determined ^ 2 ~  2  n  parameters  what r e l a t i o n s h i p s e x i s t between them.  T-T  n  point  and  such r e l a t i o n s h i p , where  ^ i s the d e n s i t y , ft and  and the s u b s c r i p t  c  a  are t o  denotes the c r i t i c a l v a l u e .  i s determined by u s i n g the Lorentz-Lorenz e q u a t i o n which i s =  +2  f ^  where  n  i s the index of r e f r a c t i o n ,  Lorentz-Lorenz f u n c t i o n , and  ^  i s the  i s the d e n s i t y of the  substance,  The f i n d i n g s of such o p t i c a l measurements are t h e r e f o r e d i r e c t l y l i n k e d with the v a l i d i t y of the Lorentz-Lorenz e q u a t i o n and a knowledge of the Lorentz-Lorenz  function. (2)  Measurements by Chapman, Finnimore, and Smith Lorentz-Lorenz f u n c t i o n suggest t h a t  ^  v  '. of the  i s constant t o w i t h i n  ± 1%, t h a t no l a r g e anomaly e x i s t s near the c r i t i c a l p o i n t , t h a t the temperature  dependence i s s l i g h t but not n e g l i g i b l e ,  and t h a t a s l i g h t decrease occurs i n a  when xenon becomes  solid. Because of the n e c e s s i t y of having a c c u r a t e v a l u e s of ^  t o support r e c e n t work on s u l f u r - h e x a f l u o r i d e ,  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  SF^ , by  undertook  2  measurements of indicated that d e n s i t y range of  f o r SP^ was 200  T h i s experiment a d e n s i t y range of temperature,  .  Their preliminary findings  constant t o w i t h i n  .5$  over a  amagats. seeks t o measure  250  amagats.  f o r ethane over  Not only i s the  critical  32.08 °C, e x p e r i m e n t a l l y convenient, but more  i m p o r t a n t l y the molecule  l a c k s the s p h e r i c a l symmetry of  xenon and i s t h e r e f o r e a good c h o i c e f o r f u r t h e r  critical  phenomenon s t u d i e s t o t e s t 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 l a c k  s p h e r i c a l symmetry. Studying r e l a t e d r e s e a r c h t h i s author i s concerned t h a t d e n s i t y v a l u e s were a p p a r e n t l y obtained through the use of  c o m p r e s s i b i l i t y data.  To a v o i d the e r r o r p o s s i b l e i n  such d e t e r m i n a t i o n s the present experiment ^  d i r e c t l y by weighing  seeks t o  the contents of the  determine  cell.  The g o a l 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  i n d i c a t e the approach taken by v a r i o u s authors i n attempting to  solve the problem.  The  experimental f i n d i n g s was exposition  degree t o which theory p r e d i c t e d also considered.  A lengthy  of the methods i n v o l v e d i s beyond the scope of  t h i s work, however a d e t a i l e d d i s c u s s i o n of the L o r e n t z Lorenz and C l a u s i u s - M o s s o t t i equations w i l l be The Lorentz-Lorenz by H.A.  e q u a t i o n was  Lorentz and by Lorenz  i n 1880.  conducted.  derived separately I t r e l a t e s the  3  r e f r a c t i v e index of a substance t o the d e n s i t y through r?  n where N  Q  n  2  - 1 + 2  N  '  o 3M  i s the r e f r a c t i v e index and i s frequency dependent,  i s Avagadro's number,  molecular weight, polarizability.  and  ^ i s the d e n s i t y , M  c* i s the atomic  An analogous  i s the  or molecular  equation f o r the l i m i t of low  f r e q u e n c i e s i s the C l a u s i u s - M o s s o t t i equation,  k -1  _  N 0  k + 2 where  k  e°<  3M  i s the d i e l e c t r i c constant and i s frequency  A s o l u t i o n of Maxwell's equations i n a d i e l e c t r i c having /* = 1  gives  of the wave and have  k=n  2  c  v  ph Jt =  ' ^ w  e  r  e  v p  h  ^  i s the speed of l i g h t .  and the two  e x p r e s s i o n s are  s  ^  e  P^  Since  substance a s e  was  velocity  n=—  identical.  The r e l a t i o n s h i p s depend on what l o c a l e l e c t r i c a c t s on a molecule.  dependent.  we V  p  h  field  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  used t o d e r i v e the Lorentz-Lorenz and C l a u s i u s - M o s s o t t i  equations n e g l e c t e d s t a t i s t i c a l f l u c t u a t i o n s and d e n s i t y effects.  The molecular p o l a r i z a b i l i t y  and independent incorrect.  of temperature  was  assumed constant  and d e n s i t y which i s a l s o  Nevertheless i t i s u s e f u l t o f o l l o w  these d e r i v a t i o n s s i n c e they approximate  through  experimental data  to w i t h i n a few percent and serve as a f o u n d a t i o n from which improvements may  be made.  4  II.  THE LOCAL FIELD AND THE CLAUSIUS-MOSSOTTI EQUATION An e s s e n t i a l part o f the d e r i v a t i o n of 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 i s the c a l c u l a t i o n of the l o c a l  field.  T h i s i s the f i e l d t h a t a s i n g l e molecule f e e l s and i s not the space-time average o f the atomic e l e c t r i c f i e l d s . t h i s f i e l d we r e f e r t o F i g u r e l a which isotropic dielectric d i p o l e moments.  To c a l c u l a t e  shows a homogeneous  substance which l a c k s any permanent  Following J a c k s o n ^ a n d  Panofsky and  (6)  Phillips^  ' an imaginary s p h e r i c a l surface of r a d i u s  c o n s t r u c t e d with a s i n g l e molecule a t i t s c e n t e r , l a r g e compared t o m o l e c u l a r dimensions, a l l o w i n g  r  r is is  statistical  averages t o be made over the molecules i n s i d e the sphere, but very small compared to the e x t e r n a l dimensions o f the  —»  dielectric, the  allowing  E  spherical region.  t o be c o n s i d e r e d constant  An e l e c t r i c  field  E  Q  throughout  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 m o l e c u l e s . Charge t h e r e f o r e 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 where  k  along the  i n s i d e the d i e l e c t r i c t o  i s the d i e l e c t r i c constant x  a x i s and so  E = E  x  E =  o ,  of the m a t e r i a l .  E  is  i .  We may c o n s i d e r t h a t a charge d i s t r i b u t i o n e x i s t s on the i n s i d e of the s p h e r i c a l surface due t o p o l a r i z a t i o n of the molecules.  The charge d e n s i t y a t a g i v e n p o i n t on the  surface w i l l depend on the charge movement normal t o the surface.  If  P  i s the p o l a r i z a t i o n per u n i t volume,  amount of charge which c r o s s e s a s u r f a c e of u n i t area  q  the  normal  Figure l a . Imaginary s p h e r i c a l surface i n a d i e l e c t r i c , c o n t e n t s of the sphere are not a l t e r e d .  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  to  E  AX  , and  P = ^ Q2  then  *  1  V  area  i s the d i s t a n c e through which Letting  on t h e s p h e r e , t h e n  dA  If  n  P • n = ^  then  moves,  he a n e l e m e n t o f s u r f a c e  dA • A X 1  i s a volume o f c h a r g e  w h i c h h a s moved a c r o s s t h e s u r f a c e . P • dA = q .  q  So  P =  ax.*1  or AX  i s a u n i t v e c t o r normal t o the s u r f a c e  = cr  cr  , where  i s the surface charge density,  — *  We  want t o f i n d  -*  C  E  r cr- dA» r d9 df>, - (T =  o f t h e s p h e r e where  3  £  = E  a t the center  —•  JI p -1 r sin9 o J  *^cT  E = dA =  E  s i n 9 c o s <)> .  I n spherical coordinates P s i n 9 c o s <j» , and  From t h e symmetry we c a n e l i m i n a t e -4  any  y  component o f  n E  =-TT-^— x  So  T r €  Ex  (8)(  I  2 P  f  £J 0  E x  -2JP TT£  d> s i n 9 ) ( r s i n 9  2  cos  I  ^  2  <J> d$  0  at +  0  we have  cos  &)(r  2  sin 9  d9  2 r  --Wo  =  E  IT  | (P c o s  J  o  E , so i n t e g r a t i n g ' f o r  i 2  1 * 3  Jo „  sin^9  -cos  N R >  Q  d9  which i n t e g r a t e d i s  , cos^9  9 + —  r  —  di)  7  P Evaluation gives  E  = -5-7-  .  C o n t r i b u t i o n s t o the l o c a l  f i e l d from molecules i n s i d e the sphere are obtained  by  summing over a l l the i n d i v i d u a l d i p o l e moments, p , of the m o l e c u l e s . For the x component of the f i e l d a t the c e n t e r of the sphere due t o the surrounding d i p o l e s we have  E  -1 x " 4TT €  G  P  JZ  r  3(p  x  x + p  x y + p  2  x  3  y  r  z  x z)  5  However s i n c e we assumed an i s o t r o p i c homogeneous  dielectric,  the  T h i s means  x ,  y , and  t h a t f o r any  z  d i r e c t i o n s are e q u i v a l e n t .  p„ x z  there i s a  Py x y ,  which means  have  *  y  -p_ x z  ' (p^ x y  + p  and s i m i l a r l y f o r  x z) = 0 .  2  We a l s o  2 YZ  =  y  r  3  XZ  =  ^  ^  "  j  "  3  0  2  o  ^ —  2  =  x  r  5 = 0 .  Thus there i s no c o n t r i b u t i o n  t o the l o c a l f i e l d from the molecules i n s i d e the sphere. The e l e c t r i c f i e l d a c t i n g on a charge a t the c e n t e r —* of the sphere i s then effective field,  ^eff  -*  P  = E + y^-  —» , where  E  e  f  f  i s the  a l s o c a l l e d the l o c a l f i e l d or Lorentz  field.  8  The  relationship  susceptibility  and  £  P = ^"XE  , where %  i s the  i s the p e r m i t t i v i t y of empty space,  Q  i s used i n the d e r i v a t i o n of the C l a u s i u s - M o s s o t t i T h i s r e a l t i o n s h i p has been e x p e r i m e n t a l l y very good approximation t o the data. c o n s i d e r e d a l i n e a r f u n c t i o n of experimentally  electric  E  equation.  v e r i f i e d t o be  That  P  can  i s reasonable  a  be since  a p p l i e d f i e l d s are g e n e r a l l y small compared 11  to the i n t e r a t o m i c f i e l d s which are roughly  10  volts/meter.  -* T h i s means t h a t  E  can be l a r g e and  s t i l l be only a  first  order p e r t u r b a t i o n of the i n t e r a t o m i c f i e l d s , i m p l y i n g It  i s i n s t r u c t i v e t o c a l c u l a t e the f i e l d  spherical cavity i n a d i e l e c t r i c .  linearity.  inside a  Comparison of t h i s  field  with the l o c a l f i e l d h e l p s c l a r i f y the d i f f e r e n c e s between , the two.  F i g u r e l b shows the  s i t u a t i o n where a s p h e r i c a l  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 the c o n t e n t s removed.  As b e f o r e the a p p l i e d f i e l d  -» F  p o l a r i z a t i o n which g i v e s  E = __o k  However u n l i k e b e f o r e  i s along the  E  i n s i d e the x  the c a v i t y and very f a r from the c a v i t y .  E  and causes  °  dielectric.  a x i s only i n s i d e T h i s i s because  m a t e r i a l i s a c t u a l l y removed i n making the c a v i t y which a l t e r s the f i e l d  on the s p h e r i c a l boundary, while the  f i e l d c a l c u l a t i o n i n v o l v e d only an imaginary C a l c u l a t i n g the c a v i t y f i e l d we boundary c o n d i t i o n s .  local  surface.  have the f o l l o w i n g  (1)  39  out  Mini  (2)  2>r  and  r=a  |r=a  k ^out ^ r Jr=a  |r=a  From s o l v i n g L a p l a c e ' s e q u a t i o n i n s p h e r i c a l c o o r d i n a t e s we know t h a t the answer can he expressed i n terms o f Legendre polynomials,  4. = yI2 " 7xA n r • in  P  n  n  P  .  n  i 'out , = V^" £  and  7  4  n=0  n=0  If  E = Zo  T h i s means  From  i s the f i e l d as r-»oo B  Q  = 0 ,  (1) we have  B  = -E ,  1  c  A^ =  o , A  0  and i n g e n e r a l  From k  ±  = k  (2)  A  -E - 2C.  C ,  ,  n 2n+l Q  = 0 A  ?  d  A  = n  ,-(n+l) P + C r n n  B  n  <f ,  out ni  = 0  c = -E + __1 ,  +  — *  for  c A, = _2  ,  n£2 .  which means  A  Q  = 0  2  (n+1) n  2 .  .5  = k ( - C ) 1 _1_ • * 2 5 a^  n 2n+l  -E x .  for n  .3  —k C and i n g e n e r a l  1 1  then  and  1  =  we have  1  B r n  for  ni2  from  (1)  10  Combining these r e s u l t s we see t h a t S  o  c  b  0  0  =  _  i  Since  E a  P  (1-k)  3  ( 1  a  = cos t  1  <b . = " o Tout  n  a  +  E  o a  of  = 0  n  for n>2 .  (l+2k)  where  -» This gives a f i e l d  = A  n  -B ( 3 k )  =  l "  A  we have,  r cost  E  A  D  +2k)  C  =  -y- - e  t tl+^k)(cos"T "  (1-k)  3  2  -»  E e 1-1 - = E0 a  3  rt  —» However, the L o r e n t z f i e l d  P  of  i n s i d e the h o l e .  l+2k  ~* ~* -v. E . = E ( 1 + •^•)  gives  e f f  Q  k since  P = e^E^,  and k = 1 + X g i v e s  E  f  f  k Setting  P=_p_ M  Q  T P o< £ E  .  3  f o r the two cases we get  1  fej = o P ^ M N  = E (2*k)  f o r the Lorentz f i e l d and  K  K  3  (k-l)(2k+l) y  =  N oC p 3 M 0  f  o  r  t  h  e  o t h  er.  Thus we have 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 and the " s i n g l e - h o l e " For  k  very c l o s e t o  where oC = o °^ N  1  we have  = y ~  equation.  f o r both  cases,  and i s c a l l e d the Lorentz-Lorenz f u n c t i o n .  3 M  For k > 1  the two e x p r e s s i o n s d i f f e r however with the  C l a u s i u s - M o s s o t t i e x p r e s s i o n g i v i n g v a l u e s of at from experiment which are n e a r l y constant while the other  11  expression  g i v e s an  ^  which i n c r e a s e s w i t h  I n working on c o r r e c t i o n s t o the  density.  Clausius-Mossetti (7)  e q u a t i o n f o r d i p o l a r l i q u i d s however, Onsager ' v  cavity f i e l d  but  reaction field,  used  supplemented i t with what he termed which was  the f i e l d  due  o f the medium by the molecule i t s e l f . w i t h p o l a r substances we  the the  to the p o l a r i z a t i o n As we  s h a l l not c o n s i d e r  permanent d i p o l e moment c o n t r i b u t i o n s to the  are not theories  concerned concerning  polarizability.  12  III.  THE  LORENTZ-LORENZ EQUATION  A d e r i v a t i o n of the Lorentz-Lorenz equation  involves  c o n s i d e r i n g the charges i n a molecule t o behave l i k e harmonic oscillators.  F o l l o w i n g Feynman  charge  q  and mass  field.  We  choose  E  along the  E  e  .  Using  F = m(  v  = E  l u > t  m  ' we  v  consider a p a r t i c l e  i n a s i n u s o i d a l l y time v a r y i n g x  a x i s so  r +  r* + u>  F = q E r )  2  of  electric i and  where  r  is  the displacement of the p a r t i c l e from i t s e q u i l i b r i u m p o s i t i o n , 2 i s the damping c o e f f i c i e n t , and C O i s a measure of the Q  restoring force,  since  d i f f e r e n t i a t i o n and = m x (-cj  Q  Q  + CO  q/m 5 p O J - to + i c o X  + i ojX)  q  .  e  Q  1 0 3  ^ , upon  have  Solving f o r  x  we  Q  have  E ,  so  q/m p 5 0 J - OJT+ i t o *  x =  and  0  0  letting  x = x  2  E x =  and  s u b s t i t u t i o n we  2 q E  r = x l  p  be the p o l a r i z a t i o n which i s _» /» charge displacement p = q x i or p =  which can be r e w r i t t e n as  -»  p = £ J*E  due to a s i n g l e —* 2 / E q /m 2 - CO 2 + .i c o ov G-> 2  where oc =  O  S7-E= V-B  = 0  f/£  0  (1)  (3)  e  2  J2. ,  5  »  U>  W r i t i n g Maxwell's  q A Q  =  - CO  +  .  1 GO  equations, c V*B 2  V*E  = J / ^ = -  C  - | f  + -|f  ( ) 2  y  a  13  Since the charge d e n s i t y  i s due t o f r e e charge or  p = Pp+ Pf  charge due t o p o l a r i z a t i o n we have Likewise  ? = J  Noting t h a t  p  o  + J  l  f  r  e  .  e  ?  ?pol  a  become upon  n  d  J  =  — » J  ()  = 0  Maxwell's equations  2  (7) (8)  _ ^  i i |  )  g  4 -  _  V  (  V  ^  yxV"E  (5) i n t o (10)  =  0  +  and s u b s t i t u t i n g from  Using the v e c t o r i d e n t i t y and s u b s t i t u t i n g  ^=i | |  V  (6)  i  (8)  VxE* = ~||  y > 7 x g = -^V*B  2g  pol  c  5  T a k i n g the c u r l of  v  and  substitution  *.S--Sf V-B  -V-?=ppol  But  i n a d i e l e c t r i c which l a c k s any f r e e charges —>  =  ree  0 l  . ^  (  g  £  o  (6)  (9)  +  = V ( V-E)  and t h e n i n t o  - V  (9)  +15—^1  o  (10)  2 ] S  gives  (11)  0  Assuming the f i e l d i n the d i e l e c t r i c t o be o f the form  E  x  v  = E  o  e  1  s>  where  k  = v . phase  .  14  Defining  n  through  n = £ ph  , we have  k =  V  C  i u ( t f ) and so  E  x  =  E  o  ^ ^  e  1 2  7 -\T>  But  P = P = 0 y  so  2  V  •p = o .  field  P = P  V  and since the d i e l e c t r i c i s homogenous  e  Q  K - i  2  Assuming  l  w  ^  2  c  2  P  = - % ^  (12) + ^  °  (11)  becomes  .™  (13)  C  2 E  the p o l a r i z a t i o n f o l l o w s the d r i v i n g  and  ot  Substituting - k  i  t  into E  2  and d i f f e r e n t i a t i n g g i v e s  2  = - H— c €-o  n  c  o (13)  0  P^  2  where  P  0  0  b e i n g the macroscopic p o l a r i z a b i l i t y and  Q  of molecules i n v o l v e d .  This y i e l d s  - k  k  =  ( 1 + <* N ) .  N  + ^— c  2 which i s  = e «NE o o o  n  ,  the number  2  2  2  x =0 3>x  d  2 = --^g c  Substituting for  °*  N  k gives  c 2 n  = 1 + O<N  n  2  and s u b s t i t u t i n g f o r oc we have N 5  = 1 +  (  l / § 2  m G  o  •  ^  a s  been assumed  o t h a t the f i e l d field  E*  l o c a l t o a molecule i s j u s t the  applied  , but we know t h a t i n f a c t , c l a s s i c a l l y , we must —V  use the L o r e n t z f i e l d  E  —* ff e  —* =  B  +  p 3~£~  '  15  Using t h i s f i e l d  P = & °<  upon s o l v i n g f o r P  Q  N  0  ,2 GJ -k +  .  c 6.  E e  ff  <>  s  p 0  =  Using t h i s value f o r P  N  £  0  1  C< (  N<* * -1  Ne^E^l-  we have  Q  This gives  —)  2  k  n  1 +  2  2  =  +  1-  , substituting for  N  which  M,*  we have  M  X  1-  N OL  N O N = — ^ —  Since  where  1  N  be  may  .  r e w r i t t e n  k  as  gives  n-1 —  n^+2  =  NCX —~-  .  2 «  3  n +2 2  3  M  i s u s u a l l y c a l l e d the Lorentz-Lorenz  T h i s simple d e r i v a t i o n can be extended  M  function.  t o take i n t o  account  a molecule having s e v e r a l resonant f r e q u e n c i e s by w r i t i n g  V"  04  —5  7  TT—  ^-5  o s c i l l a t o r strength f a c t o r . of i n d i v i d u a l  cx^  where  Considering  f.  tx t o be the sum  contributions i s equivalent to considering  a mixture of two o r more types of molecules, a different and  Ng  o<  .  k  If  N  1  3  2  2 ~  1  n + 2  each type h a v i n g  molecules have a p o l a r i z a b i l i t y  molecules have a p o l a r i z a b i l i t y  n  i s c a l l e d the  = N„1Oc 1 + i  N  o  "2<x 2. o  cx^  , then  oc^ ,  16  (8) To check the a d d i t i v e nature of the p o l a r i z a b i l i t y calculates water,  Feynman  '  v  cXg f o r v a r i o u s c o n c e n t r a t i o n s of sucrose i n  cx^  i s the p o l a r i z a b i l i t y  polarizability  of water,  c*2  i s the  of sucrose, and both are assumed t o be constant  f o r a l l c o n c e n t r a t i o n s of the sucrose s o l u t i o n .  Feynman's  c a l c u l a t i o n s g i v e an cx^ which i s n e a r l y constant and t h e r e f o r e ^ o<^ . k Comparison of t h e o r e t i c a l and experimental v a l u e s of 2  supports the a s s e r t i o n t h a t  where for  n = 1 + 0<N  N cx = y  shows t h a t the harmonic o s c i l l a t o r model  oc e x p l a i n s observed phenomena.  p r e d i c t e d decrease observed.  For  n  in  n  co2*uJ 0 , n  When  t*> < C O  q  the  as t j decreases i s e x p e r i m e n t a l l y has a l a r g e imaginary term which  g i v e s a f a c t o r c o r r e s p o n d i n g t o a decay of E with distance where E = E , e ^ " . T h i s i s c a l l e d a b s o r p t i o n and i s x o e x p e r i m e n t a l l y observed. F o r to>Cu> , n i s l e s s than 1 which c  o  means  v _ r , „ _  pnase Q  > c,  but the group v e l o c i t y  the wave number, remains l e s s than requirements  of s p e c i a l r e l a t i v i t y .  c  ^u> , where ^  k  is  i n agreement w i t h the  17  IV.  LITERATURE REVIEW A. EXPERIMENTAL RESEARCH A l l r e s e a r c h , which was  reviewed by t h i s author, of  the Lorentz-Lorenz and C l a u s i u s - M o s s o t t i equations r e v e a l e d a decrease i n  <5£ as d e n s i t y i n c r e a s e d .  M i c h e l s and  Botzen^^  measured the r e f r a c t i v e index of argon f o r s i x wavelengths 2 5 °C  i n the o p t i c a l range at 2 3 0 0 atm. data.  The d e n s i t y was  and then decrease with i n c r e a s i n g d e n s i t y .  was  of the wavelength  s m a l l however and was  to  calculated using compressibility 2 0 0 amagats  seemed t o have a peak value around  independent  1  f o r p r e s s u r e s from  T h i s was  of l i g h t c o n s i d e r e d . 1%  roughly a  The change  decrease.  L a t e r work by M i c h e l s , t e n 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 a l o n g the  P r e s s u r e s ranged from  2 5 °C k atm.  and  1 2 5 °C  up t o  2700  M o s s o t t i f u n c t i o n , denoted here as a decrease was about was  1%  ^  c  m  »  isotherms. atm.  w  a  The  Clausius-  calculated  s  and  found t o e x i s t as the d e n s i t y i n c r e a s e d and  over the experimental range.  found t o e x i s t with temperature.  No change i n  A p l o t of  a£  5£ vs.  c  was m  o£  c  m  was made which gave a s t r a i g h t l i n e with a slope of one, i n d i c a t i n g t h a t the v a r i a t i o n s i n 5£ E a t w e l l and Jones  (11)  and  ^C  cm  are  similar.  have measured the r e f r a c t i v e  of s o l i d argon i n the o p t i c a l range from  2 0 °K  to  index  8 3 . 8 °K, (12)  the t r i p l e p o i n t .  Regarding e a r l i e r work by Jones and  E a t w e l l and Jones summarized the f i n d i n g s as f o l l o w s :  Smiths  18  " ( i ) The tendency f o r the Lorentz-Lorenz f u n c t i o n t o decrease slowly with i n c r e a s i n g d e n s i t y , a l r e a d y known to occur i n h i g h l y compressed gases and l i q u i d s , was continued i n the sense t h a t v a l u e s o b t a i n e d f o r the s o l i d were g e n e r a l l y s l i g h t l y lower than those f o r the liquid, ( i i ) However, i n the range o f temperature o f the s o l i d phase i t s e l f , the Lorentz-Lorenz f u n c t i o n increased with increasing density. We have r e c e n t l y extended measurements o f the r e f r a c t i v e index of s o l i d argon over a much wider range o f temperature, 20 °K t o 83.8 °K , w i t h a r e s u l t which appears t o r e v e r s e the second of the above c o n c l u s i o n s . " There a r e some problems with s o l i d argon however s i n c e changes occur i n the p o l y c r y s t a l l i n e s t r u c t u r e a f t e r specimen p r e p a r a t i o n .  initial  There i s a l s o a l a c k o f a c c u r a t e  c o m p r e s s i b i l i t y data w i t h which t o c a l c u l a t e the d e n s i t y knowing  the temperature and p r e s s u r e . (11)  Amey and C o l e  v  J  argon, krypton, xenon,  measured the d i e l e c t r i c  constants of  and methane i n the l i q u i d and s o l i d  range near the m e l t i n g p o i n t .  Their results indicate f o r  each substance t h a t 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 l e s s i n the l i q u i d and s o l i d s t a t e s than i n the gaseous by from  .Jfo  to  k.0%  states  depending on the substance. (lk)  Work by A b b i s s , Knobler, Teague, an i n c r e a s e i n boundary  of about  2%  and P i n g s  v  shows  when c r o s s i n g the phase  i s o t h e r m a l l y from the s a t u r a t e d gas t o the s a t u r a t e d  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 t h a n f o r the d i l u t e gaseous forms. i s r e p o r t e d f o r argon which g i v e s a  A temperature dependence 2%  lower value o f 3£  19  at  90  low temperature,  However t h i s was  K  , than at room  a comparison  temperature.  of t h e i r data w i t h data of  Johnston, Oudemans, and C o l e ^ - ^ . Garside, M^lgaard,  and Smitlv  'measured the  refractive  index of xenon l i q u i d and vapor i n c o e x i s t e n c e over a temperature range  of  range .001  change was  .02  to  * 3$  constant t o  -59  of  °C  16.5  to  mol/cm  3  .  X  They r e p o r t  a decrease i n <£f  toward  t o be  ©C  600  f o r argon, perhaps  g r e a t e r change i n  indicated  as d e n s i t y i n c r e a s e s . 20  a f a c t o r of  and Botzen had a range f a c t o r of decrease i n  and a density-  over the range measured, but the  Since t h e i r d e n s i t y range was  Vfo  °C  while Michels  and only d e t e c t e d a xenon would show a  i f the d e n s i t y range were  extended.  From the experimental s i d e there i s l i t t l e  doubt t h a t  the C l a u s i u s - M o s s o t t i and Lorentz-Lorenz equations i n d i c a t e a decrease i n  oC  as d e n s i t y i n c r e a s e s .  Although  experimental v a l u e s of d e n s i t y were determined  the  from  c o m p r e s s i b i l i t y data and the pressure i n the c e l l which would tend t o compound e r r o r s , the tendency is clear. of  However there i s disagreement  identifying  c<  with  Does  cx  toward  a decrease  r e g a r d i n g the  validity  i n f a c t decrease with  d e n s i t y such t h a t the c o r r e c t n e s s of the Lorentz-Lorenz equation i s implied? i n such a way of  the two  that  c<  Does the Lorentz f i e l d i s t r u l y a constant?  vary w i t h d e n s i t y Does a  occur, or i s the molecular approach  fundamentally  inadequate?  combination  to p o l a r i z a b i l i t y  20  A. THEORETICAL RESEARCH C o n s i d e r a b l e t h e o r e t i c a l energy has been expended on the problem of atomic  and molecular  polarizability.  (17)  Ten Seldam and de Groot  '  v  used a s p h e r i c a l box t o c o n t a i n  a helium atom and compressed i t by s h r i n k i n g the They r e p o r t e d a  5i?° decrease  pressure i n c r e a s e d t o 1000  of  r  Q  i n the p o l a r i z a b i l i t y as the  atmospheres.  the wave f u n c t i o n s and energy  sphere.  They had  calculated  l e v e l s of helium as a f u n c t i o n  , the sphere's r a d i u s , and the c o r r e s p o n d i n g p r e s s u r e . (1Q)  Using Kirkwood's^  ' formula i n Hartree u n i t s ,  the  p o l a r i z a b i l i t y of s p h e r i c a l l y symmetric atoms i s CX = ( k/9 k a ) ( y ^ r ) where k = 2 i s the number of 7  2  2  Q  electrons,  a  Q  -  -  2  *  = fi/m e  i s the f i r s t Bohr o r b i t f o r hydrogen,  -  = r„ + r  0  j^U^4lldv = 1  where  r^  and  the n u c l e u s .  '' ' " dv  are the e l e c t r o n c o o r d i n a t e s r e l a t i v e t o The  problem i n comparing t h i s theory w i t h  experimental data i s that the only a v a i l a b l e data a t the only went t o  time  65 atmospheres f o r helium. (19)  I n another work p u b l i s h e d by t e n Seldam and de G r o o t they made s i m i l a r c a l c u l a t i o n s f o r argon and only rough agreement w i t h experimental r e s u l t s was  found f o r the  v  7 1  21  polarizability  of argon  as a f u n c t i o n of pressure.  De B o e r , v a n d e r Maesen, and t e n S e l d a r a ^ ^  used  2 0  (21) Kirkwood's^ ' statistical  m o d e l a n d work by v a n V l e c k  (22) '  v  (23) and B r o w n  v  t o develop  J l  the C l a u s i u s - M o s s o t t i equation  u s i n g both  a H e r z f e l d and a L e n n a r d - J o n e s p o t e n t i a l  They  with  start  p  i  the assumption  «• D - °< y  moment induced i n i t h e  that  ^ -^ik' Pjc  i  expresses the d i p o l e  — molecule.  ocDis the m o l e c u l a r  p o l a r i z a t i o n due t o t h e e x t e r n a l f i e l d molecular  polarizability.  So  - cx \  '  where  «. i s t h e  T ^  k  .p  p o l a r i z a t i o n due t o t h e d i p o l e s o f a l l t h e  "  U  molecules  where  T.,  3  r  tensor  r . r 2 ik i  IK  being the u n i t  k  i i + j j + kk  more t h a n t h e f a m i l i a r  E(f)  field.  =  3  i  .  i s the  surrounding  i*y- ."  k  3  This i s nothing  "  n  which i s  p  |r*-r1  3  the  field  Taking  at  r  due t o a d i p o l e a t  the average  p*. = C D - « ^ T  value  i  k  of  . p *  k  r  ,  p^ , d e n o t e d  +  <* YZ  < ik'Pk T  n =r-r p^  "  , they  T  ik-Pk)  wrote  22  T /  Using to  kTi  y  and  =  7  v  lk  ^ r i j 3  v  ' ( T - p -  p  i k  the a p p l i e d f i e l d  and  which i s a f u n c t i o n of  ) =  fc  S  /where  (£+2)  E  number o f m o l e c u l e s volume o c c u p i e d  S  where  i s a proportionality n'  and  n'"p~ = P = (€-1)  Using  ^  n'=  the  is  constant  temperature  D - ^y-  and  E  T  P = |  .  (€+2)  E  — *  j^p-  gives  (€-1)  or  g = ^y  |^  = |  n'o.(  1  E nV  (€+2) l + s  )  + i  4  TT  n'ot(  1 + B  B  2  n'+  9  Lennard-Jones p o t e n t i a l f i e l d potential field.  While  data w e l l f o r lower o<  are  densities. fact  used  densities,  l a r g e r than the They attempt  cx, d e c r e a s e s  a constant  and  C  C  n'<*  dependence.  and  n* + 0 ( n )  ) using  /3  the  they  in  2  2  9  t h i s theory  t h a t c a l c u l a t i o n s by  show t h a t  temperature  sake o f c o m p a r i s o n w i t h e x p e r i m e n t  e v a l u a t e the c o e f f i c i e n t s  £ - 1 €+ 2  E* 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 w i t h a n added d e n s i t y and F o r the  (6+2)  the  simpler Herzfeld  seems t o r e p r e s e n t  the  the t h e o r e t i c a l v a l u e s  experimental  of  values f o r higher  a n e x p l a n a t i o n o f t h i s t r e n d by Michels,  with  de B o e r , and  increasing density.  c< , a t h e o r y w h i c h l o w e r e d  d e n s i t y w o u l d more n e a r l y r e p r e s e n t t h e  B i j l ^ Since  2  the  ^ they  cx w i t h i n c r e a s i n g  data.  23  However Y a r i s  and K i r t m a n  polarizability increase  calculate  v  due t o many body i n t e r a c t i o n s t o be a n .OkZfo  v a r y i n g from  J a n s e n and M a z u r ^ ^  such a t h e o r y f o r hydrogen  same was  T^  f o rthe i n t e r a c t i o n  k  used  3.5^  f o r helium t o  b e f o r e by de B o e r ,  cx^iO  =OC0(  11+0(2  C  l  potential  due t o d i p o l e s a s  v a n d e r Masen, a n d t e n S e l d a m ^  now  £  T .  cx becomes  k  . T  f  e  i  +  « 2  c<  C  2  ^  , Cg , a r e c o n s t a n t s w h i c h a r e d e n s i t y  So  they write  expanded  dependence  a r g o n t h e y compare a r g o n it  exhibits  0  .  ) -where  o f cx i n c l u d e d Extending t h e i r  rise  R'(n't)  a n d c a n be theory t o  d a t a t o t h e t h e o r y and f i n d  the c h a r a c t e r i s t i c  amagats a n d a d e c r e a s e but  dependent.  o  i n powers o f C*  )  molecule, and  = ^ ~ n'<x ( 1 + R'(n',t)  the functional  t h e same  T ^ T ^ - U  (  of a free  C^  has  that  t o a maximum a r o u n d  thereafter with increasing  200  density,  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 t h e C l a u s i u s - M o s s o t t i (28)  and  L o r e n t z - L o r e n z e q u a t i o n s Mazur and M a n d e l  the t h e o r y developed case  ^.  W i  i s the p o l a r i z a b i l i t y  0  2 0  cx^(rf) where  k/i Where  work o u t  a n d h e l i u m atoms u s i n g t h e  Their modified Clausius-Mossotti equation i s also with the exception that  f o r xenon.  and Mazur and J a n s e n ^ ^  2  just  t h e change i n  o f time  f o rthe s t a t i c  varying fields  electric  v  ' extended  field  t o the  i n o r d e r t o see what t h e  2k  Lorentz-Lorenz density  and t h e L o r e n t z  fluctuations. molecular average  field  i s altered  due t o  polarizability  R and  and  d e n s i t y , temperature, calculations  field,  of the  obtain  represent  and f r e q u e n c y  f o r helium  of the  calculation  they  D  statistical  calculation  and a s t a t i s t i c a l  p o l a r i z a t i o n and l o c a l  Making a c t u a l  OC i s a f u n c t i o n o f  A f t e r a quantum m e c h a n i c a l  polarizability  molecular having  e q u a t i o n becomes when  integrals  dependence.  are quite  difficult  and  a s s u m p t i o n s have t o be made w h i c h l i m i t  the  p o i n t t h a t e f f e c t s a r e g i v e n only t o an o r d e r o f magnitude.  The  important  Lorenz  research are that the Lorentz-  f u n c t i o n i s shown t o be n o t s i m p l y  0< (GJ)  for  0  that  features of their  when  ( X 0 ( 0)  —g- = .1 ,  the accuracy t o  a substitution of  i n t h e C l a u s i u s - M o s s o t t i e q u a t i o n , and where  CJ»  0  i s the s o f t e s t  absorption  ^o frequency, 15%  function correction  i s roughly  l a r g e r than f o rthe corresponding Clausius-Mossotti case. The  the  the Lorentz-Lorenz  next  complication i n theoretical  calculations of  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 J a n s e n a n d  (29)  Solenr  '.  T h e y expand t h e i n t e r a c t i o n H a m i l t o n i a n t o  include  not only dipole-dipole,  quadrapole-quadrapole  b u t d i p o l e - q u a d r a p o l e and  interactions.  They  subsequent paper h i g h e r m u l t i p o l e s w i l l the will  that i n a  be e v a l u a t e d and  Clausius-Mossotti function f o raxially be d e r i v e d .  state  symmetric  molecules  25  When J a n s e n  p u b l i s h e d t h e s u b s e q u e n t p a p e r he s t a t e d ,  w  "A t h e o r y o f t h e s t a t i c d i e l e c t r i c c o n s t a n t i s d e v e l o p e d on a quantum m e c h a n i c a l b a s i s , s t a r t i n g f r o m t h e L o r e n t z 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 t h e c o n c e p t of ' l o c a l f i e l d ' , a m o l e c u l a r v e r s i o n o f the g e n e r a l t h e o r y i s o b t a i n e d and a p p l i e d t o c o m p r e s s e d n o n - ( d i ) p o l a r g a s e s 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 for the d i e l e c t r i c constant. I t i s shown t h a t s u c h a molecular theory i s fundamentally i n e f f e c t i v e i n accounting f o r observed r e s u l t s w i t h i n experimental accuracy." "The q u a n t i t y « i n t h e C l a u s i u s - M o s s o t t i e q u a t i o n d o e s 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 m o l e c u l e ; n e i t h e r may i t be c o n s i d e r e d a s 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 t h e a n a l y s i s o f d i f f e r e n t p h y s i c a l phenomena. I t o c c u r s i n t h i s f o r m o n l y i n c o n n e c t i o n w i t h d i e l e c t r i c p r o p e r t i e s and i s s i m p l y a v a r i a b l e , unknown u n t i l t h e s o l u t i o n h a s b e e n f o u n d , a n d t h e r e f o r e o f no u s e f o r f i n d i n g t h e s o l u t i o n . " This author that  would  with Jansen.  t h e t h e o r e t i c a l a t t e m p t s t o improve  of  the Lorentz-Lorenz  is  c l e a r that while  densities,  machinery,  the simple  5$  •  fail  While  forms fail.  at high improvements  t h e o r y u s e l a r g e amounts o f m a t h e m a t i c a l  the r e s u l t s give  molecules  only q u a l i t a t i v e  a g r e e m e n t and  S i n c e wave f u n c t i o n s f o r  a r e n o t w e l l known,  distribution functions d i f f i c u l t  molecular  t o u s e , and  i n t e r a c t i o n p o t e n t i a l s only approximate, a theoretical  on t h e s i m p l e  expressions  t r u e q u a n t i t a t i v e agreement.  complicated  I t seems  and C l a u s i u s - M o s s o t t i e q u a t i o n s  the e r r o r i s l e s s than  on t h e s i m p l e  not  tend t o concur  s o l u t i o n i s by no means  molecular  i t i s clear  trivial.  that  It  26  V.  THE EXPERIMENT A. The  GENERAL CONSIDERATIONS b a s i c c o n s i d e r a t i o n s i n t h i s experiment were the  maintenance o f a f i x e d temperature t h a t c o u l d be changed e a s i l y , the c o n s t r u c t i o n o f an o p t i c a l system capable o f 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  c o n s t r u c t i o n of a c e l l t h a t would be s t r o n g enough t o withstand the pressure  o f up t o  1000 p . s . i . exerted by the gas, l i g h t  enough t o be weighed on a p r e c i s i o n balance,  l a r g e enough t o  h o l d a u s e f u l sample volume, t r a n s p a r e n t t o a l l o w measurements t o be made, and capable p o r t i o n of i t s contents  B.  light  o f b e i n g emptied of a  d u r i n g the course  of the experiment,  TEMPERATURE CONTROL  As the maintenance o f a constant  temperature i s e s s e n t i a l  c o n s i d e r a b l e a t t e n t i o n was g i v e n t o the c o n t r o l system. i s o l a t i o n was achieved by the c o n s t r u c t i o n of three  Cell  containers,  each i n s u l a t e d from the other with a l a y e r o f styrofoam.  The  i n n e r c o n t a i n e r was an aluminum c y l i n d e r i n which a hole was bored t o house the c e l l .  (See F i g u r e 2)  The outer  surface  of the c y l i n d e r was grooved t o h o l d a h e a t i n g wire which was wrapped about the surface and a s m a l l hole was tapped i n one end  f o r the i n s e r t i o n of a t h e r m i s t o r which was epoxied  a screw. and  into  The aluminum c y l i n d e r was then encased i n styrofoam  i n s e r t e d i n a copper c y l i n d e r about which copper t u b i n g  threaded cap heating wire  I S  copper tubing  hole f o r light  copper cylinder  .aluminum cylinder  Side View  —_hole for l i g h t beam styrofoam insulation  .thermistor i n a screw  plywood cover hole f o r alignment pin hole f o r c e l l "body 1  hole f o r c e l l windows End  View THE CELL CONTAINERS Figure 2  —  external tubing f o r connection to water bath  28  had been wound t o provide f o r f u r t h e r temperature c o n t r o l by r e g u l a t i n g the temperature o f water which flowed i n the t u b i n g . The copper t u b i n g was t h e n encased i n styrofoam and the e n t i r e assembly housed  i n plywood.  Temperature  c o n t r o l was  o b t a i n e d i n two  steps.  Forma S c i e n t i f i c Model 2095 was used t o r e f r i g e r a t e down t o . 5 °C  First  a  water  and pump i t through the copper j a c k e t .  This  was necessary f o r the p o r t i o n of the experiment d u r i n g which the c e l l temperature was below room temperature.  The Forma  was capable of m a i n t a i n i n g c o n s t a n t temperatures t o w i t h i n i  .05  °C  .  Secondly, the h e a t i n g wire around the aluminum  c y l i n d e r and the t h e r m i s t o r embedded i n i t were used t o provide a heating-feedback system w i t h which the temperature c o u l d be maintained t o w i t h i n  .0002 °C . The t h e r m i s t o r was  used as p a r t of a DC b r i d g e w i t h a decade r e s i s t a n c e box as the o t h e r s i d e .  A d i f f e r e n c e between the decade box  and  t h e r m i s t o r r e s i s t a n c e caused a v o l t a g e t o appear which d e t e c t e d w i t h a Hewlett-Packard model 419A  was  DC n u l l v o l t m e t e r .  The output of the n u l l v o l t m e t e r was t h e n f e d i n t o a Kepco model OPS  7-2  a m p l i f i e r which was w i r e d t o f u n c t i o n as an  i n t e g r a t o r i n order t o p r o v i d e a zero average o f f s e t .  The  temperature value used f o r data purposes was t a k e n from an i d e n t i c a l b r i d g e network which was wired t o a c h a r t r e c o r d e r . The t h e r m i s t o r used i n the network was c a l i b r a t e d u s i n g  29  a quartz c r y s t a l thermometer as a standard.  Since the  r e s i s t a n c e of the t h e r m i s t o r obeys the equation  R = R  ,  Q  a knowledge o f R and T f o r v a r i o u s temperatures permitted a least  squares f i t of the data t o f i n d  t h e r m i s t o r i n use  (See F i g u r e 3 )  Q  = .01498583  and  Q  and o< .  F o r the  o< = 3 5 4 4 . 0 6 l  .  The t h e r m i s t o r s t a b i l i t y was not checked  d u r i n g the course G.  R  R  o f the experiment.  OPTICS The  purpose o f the o p t i c a l system was t o provide a  means o f measuring the index of r e f r a c t i o n o f the sample, ethane.  I n order t o do t h i s a plane wave was c o n s t r u c t e d  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 c o l l i m a t i n g telescope.  (See F i g u r e 4 )  F o l l o w i n g the o p t i c s step by step; a monochromatic beam of wavelength 360  6 3 2 8 °. was produced by a M e t r o l o g i c model  helium-neon l a s e r .  attenuate  Crossed  p o l a r i z e r s were used t o  the beam t o a l e v e l safe f o r d i r e c t  observation.  A microscope o b j e c t i v e focused the beam t o a p o i n t and a micron p i n h o l e was p o s i t i o n e d a t the f o c a l p o i n t . was then converted  10  The beam  i n t o a plane wave by p l a c i n g a l e n s such  t h a t the p i n h o l e was a t i t s f o c a l p o i n t . The outer p a r t of the beam was t h e n masked o f f and the c e n t r a l p o r t i o n 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 t o another beam s p l i t t e r which d i r e c t s  HewlettPackard Null Voltmeter Decade r e s i s t a n c e box  Kepco OPS 7-2 DC Amplifi  e r  TEMPER  A T U R E  CONTROL  Figure 3  8  11  9 1. M e t r e - l o g i c L a s e r , 2. C r o s s e d  k. 10 m i c r o n Lens  a  Polarizers  3. M i c r o s c o p e  5.  m o d e l 360  Objective pinhole  10  f=135mm  12  Splitter  e  6. Mask 7. Beam 8. The  Cell  9. The C e l l 10.  Penta  Holder  Prism  11. A d j u s t a b l e 12.  10  Mirror  C o l l i m a t i n g Alignment  Telescope THE OPTICAL SYSTEM Figure  4  32  the i n n e r and outer beams toward the m i r r o r .  10.253  L a n s i n g model  The m i r r o r , a  a d j u s t a b l e m i r r o r with a c a l i b r a t e d  d i f f e r e n t i a l d r i v e , then d i r e c t s the beam i n t o a Davidson O p t r o n i c s alignment t e l e s c o p e model D2?5 which a l l o w s the c o l l i m a t i o n of the beam and v i s u a l alignment with a c r o s s - h a i r . The m i r r o r has a r e s o l u t i o n o f .6  .1 a r c - s e c , r e s e t t a b i l i t y o f  a r c - s e c , and a t o t a l a n g u l a r sweep o f  ~ k degrees.  To  o b t a i n the index of r e f r a c t i o n the f o l l o w i n g a n a l y s i s was used.  \ n.  —»/ / °7  /  REFRACTION ANALYSIS Figure 5 9  i s the angle o f d e v i a t i o n from the o r i g i n a l beam,  n^ s i n <* = rig s i n ^ = n^ s i n "V n S  o  n  =  b  o  n  l  ? sin y sinoc  "1 - » 3 [  oos  or o  r  n  = ^ l sin-ot  n  « ffe|-] . +  r e f r a c t i o n o f the sample, n  where Y =  2  Where  f" [_  8  s i n e * cos 9  + s i n 9 cos  i s the index o f  i s the index o f r e f r a c t i o n o f the  windows, and n^ i s the index o f r e f r a c t i o n o f 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 neglects  the f a c t  t h a t t h e two f a c e s  p a r a l l e l but i n r e a l i t y  wedge t h a t  i t c a n be i g n o r e d .  s e c o n d window c o n t a i n s The  complete  tedious.  The f i n a l  f o l l o w i n g , where  result  4'  of  S  I  N  9  A  A  ct {  sin  f ( e  )  <j) = .08  "  f ( e  A ] )  o  =  "  S  I  N  9  A\  the c e l l  0^ was  l  =  n  3  A  •  0 0 0 0 8 8  73  o f the  where t h e s u b s c r i p t A i s  i n place  beam w h i c h i s  and f i l l e d  .0031°.  Neglecting  with  second o r d e r  t a n ot face  air.  be z e r o .  0 - s i n 0.  1 + c o s 0 - c o s 0A  i s zero.  .J s+i n _ i | f ( e ) - f ( e . ) l A'J  2  reduces t o the e a r l i e r p a r a l l e l ©  .  t h e n c o s <> j i s e s s e n t i a l l y u n i t y and  sin n  49" o r .08034°  surfaces  c a s e o f no wedge a n g l e t h i s w o u l d o b v i o u s l y experiment  The  f o r t h e w o r s t c a s e and s e t <* = 20° ,  used t o denote the d e v i a t i o n o f the l i g h t produced with  .  o f t h e window.  where <x i s t h e a n g l e b e t w e e n t h e two i n n e r  o 0- = 3 » a n d  slight  i n A p p e n d i x A and i s r a t h e r  <J> i s t h e wedge a n g l e  I f we a l l o w  1'  first  i s e a s i l y u n d e r s t o o d and i s t h e  3  windows.  The  beam h a s s u c h a  a wedge a n g l e  r, . ^ 2* i ^cos « n - nr , [ i1+cos 0 - ™«= cos « . + 1 =  wedge s h a p e d .  I t i s l e s s than  analysis i s given  system b u t  o f t h e windows a r e n o t  are s l i g h t l y  window w h i c h i s n o r m a l t o t h e l i g h t  optical  I n the For t h i s  corrections, which  a n a l y s i s i n the case  that  34  The  mirror  i s constructed  such t h a t the micrometer  d i v i s i o n s a r e p r o p o r t i o n a l t o s i n 0. where x i s t h e m i c r o m e t e r r e a d i n g . sin to  0=0  so a = -b x  zero  and  angle,  s i n 0^  50  having  where x  Q  s i n 0^  lines/inch,  ) .  Q  i n the container.  g i v e n by  sin0  t h e medium,  is  t h e number  given  = ~ -  d  When t h e c e l l  i s the reading  Q  To o b t a i n  Edmund S c i e n t i f i c  was p l a c e d  in  i s , s i n 0 = a+bx i s removed corresponding  - s i n 0 = a + b x^ -( a + b  = b ( x^ - x  m  That  The o r d e r s  where  X  b  x ) Q  a Ronchi  stock  ruling  jj05ll,  number  of d i f f r a c t i o n are  i s the wavelength of l i g h t  i s the d i s t a n c e between the l i n e s ,  o f t h e maximum a n d i s a n i n t e g e r .  f o r the l a s e r  i s i n vacuum,  we  and  Since  c o r r e c t t h i s by X  m  the  X  = —~.' n  m X s i n 0„ = - — . m n^ a  Q  3  0  We sin  now 0  have  - sin0  m + 1  (  x  = b ( x  m  m+l " m  *  x  D  b  be v e r y  - x  1  )  m  we have  Solving f o r b  =  3  a  ~"Z *m+l "  • x  D  w  a  the plane  of the m i r r o r ,  wave b u t i n s e n s i t i v e  each c o n t a i n i n g  60  orders,  d  was v e r i f i e d  squares f i t t o f i v e  t o determine  t o be  lense  50  that  ,000064?6 f o r  x  s e t s of data,  - x  m+1 spacing  found t o  t o the v e r t i c a l  b was c a l c u l a t e d t o be  t h i s experiment u s i n g a l e a s t  s  ,  m.  s e n s i t i v e t o the p o s i t i o n o f the f i n a l  constructed plane  +  ri^d ( ^ + 1 - m ) .  =  n  m  Since  . The m  l i n e s / i n c h by u s i n g  m i c r o s c o p e which had a m i c r o m e t e r d r i v e n t a b l e .  a  line  35  D. THE The the  cell  a n aluminum c y l i n d e r t u r n e d  the  The  o t h e r end their  The  windows were s e a t e d  h e l d the take  fit  the  needle cell  thread  hole.  The  was  time,  two  of the  the  cell  cell  1"  The  t o h o l d the  bulk  o f ^20°  wire  valve  used to i n the  The  i n the allowed  cell  end  of the  of the  one  s e a l the  cell  A  f o r simple  i n i t s container.  45°  was  To  cell  and  tapped  cut steel  insure that  and  holder  the  each  matching  A tapped hole  r e m o v a l and  .  made t o  stainless  same p o s i t i o n i n t h e  holder.  and  material  s e a t was  hole.  t o the  other.  side of  the  sapphire  s p e c i a l l y made t o  cell.  of  t o form a s e a l  a brass valve s e a t was  inside  to each  i n d i a m e t e r on  i n indium  p i n s were a t t a c h e d  were d r i l l e d  6  good  )  a t an angle  and  t o t a l weight  replaced  Figure  r i n g s c o v e r i n g paper gasket  a pipe  v a l v e was  out  i n order that  t h e n m i l l e d t o h o l d two  windows i n p l a c e .  down on t h e  ( See  down t o match  o t h e r c r e a t i n g a n edge b e v e l  aluminum r e t a i n i n g  to  was  thick,  d i a m e t e r on t h e  holder  drilled  planes  windows were  cell  made.  l o n g c y l i n d e r was  windows w i t h The  o f the  c o n t a c t w o u l d be  sample.  in  was  inner diameters  thermal of  CELL  i n the  replacement  holes top of  retaining sapphire  ring window  retaining ring screws alignment  cell  pin  body  brass valve  Ok  seat  tapped hole f o r pumping l i n e c o n n e c t i o n  u p p e r end o f needle v a l v e Front  View  THE  CELL  Figure  6  Side  View ON  37  E . THE  Inc.  SAMPLE  Research  grade  Infrared  and mass s p e c t r o m e t e r d e t e r m i n a t i o n s by  supplier  c e n t ethane,  The  ethane  The  i s s h i p p e d under  and  filled  F. THE  mol  the c e l l .  t h i s pressure  ( See  Figure  o f a i r and  any  the d e n s i t y range  The  out w i t h ethane  and t h e  almost  liquid  Room t e m p e r a t u r e  was  entirely  cell at  lowered t o about  to a higher density of the  7 )  pumped on f o r a  have b e e n p r e s e n t i n t h e s y s t e m .  permit f i l l i n g the c e l l  expanding  i t was  l i n e s were f i r s t  t h e volume was  room t e m p e r a t u r e . to  p r e s s u r e and  l i n e were t h e n f l u s h e d  such t h a t  the  99.99  showed i t t o be  hours t o evacuate the system  i m p u r i t i e s t h a t may cell  sample  to flow into  and g a s d e l i v e r y  minimum o f 96  o b t a i n e d f r o m M a t h e s o n Company,  t h e most p r o b a b l e i m p u r i t y b e i n g e t h y l e n e .  caused the ethane cell  was  on a r e p r e s e n t a t i v e  per  that  ethane  and  15  °C  thereby  experiment.  WEIGHING  A S a u t e r Monopan b a l a n c e was  used  T h i s b a l a n c e had minimum d i v i s i o n s c o u l d a c c o m o d a t e w e i g h t s up t o  of  200  t o weigh the .1 m i l l i g r a m  grams.  p e r m i t t e d t h e w e i g h i n g pan t o be  protected  o t h e r f o r e i g n m a t t e r which might  accumulate  between w e i g h i n g s  o f the c e l l  and  cell. and  Sliding  windows  from dust  thus a l t e r  on t h e  and  pan  the t r u e  weights.  Research Grade Ethane  Welch Vacuum Pump and Veeco D i f f u s i o n Pump  K,  valve  valve  Veeco Vacuum Gauge  valve  SYSTEM FOR FILLING THE CELL Figure 7  39  A  plastic  g l o v e was dirty  and  used  t o handle  the  cell  and  replace i t i n i t s container.  a t h r e a d e d r o d was  we  must c o n s i d e r t h e b u o y a n t  to  the  surrounding a i r .  the  inside  the  o u t e r volume  W  P = — g  which the  scale,  m  of  the  sample,  o  acting the we  on the  cell  the  W  = W  connecting  the  density  must be  problem  the  W  o  =  W  =  o  ~  P V  averaging  . ia P  7  W  o  w  a  finding  s  d e  and  X  cell  evacuated  V  i s the  G  on is  Q  have  registered ° density  F i s the  W  force If  '= m - V P o o a i s the weight o f the  and  n  s  to obtain  P  W  since  Q  and  the weight  of the c e l l  i n the  allowed to remain  W  t o be  V. P  was  P  = 0  G  process might  To  P  the  of be  avoid  open a t the  into  = m + V- 9' - V  termined  measurements.  V.  P  evacuated  X  by  set  to permit a i r to d i f f u s e  This gives '  cell  due  t h e n we  i s the weight °  d u r i n g the experiment.  v a l v e was  the experiment  cell  of the  the c e l l ,  weight  s  pumping l i n e  cell  on t h e  cell  a c c e l e r a t i o n of g r a v i t y .  P = 0 so we \s - VL where W  f r o m what i t was  P  W  sample  t o i t , and  of the a i r ,  g i s the  altered  set  exists  o c c u p i e d by  getting  t o remove t h e  o b t a i n the  available  However i t i s d i f f i c u l t the c e l l  to avoid  i s t h e volume  o  means t h a t  of  used  that  ) , where  c e l l , and  s sample.  cell  i s t h e mass o f t h e c e l l ,  i s evacuated  t h e n have  If  , o r volume v  the  To  force  sample h a s  ( m + V. P - V P i V s o a  by  the  this end  cell  and  . T h e n we  1 7 9 . 4 1 0 gm determined  have  ±.001  by  t o be  .016  gm  ^SL  i n d e p e n d e n t l y and  a l s o by w e i g h i n g  the  ^ EL  and  then l e t t i n g  i n a i r and  r e - w e i g h i n g the  cell.  40  This gives  W  13.108 - .003 filled  179.394  =  Q  cc  gm  •  by a v e r a g i n g  with d i s t i l l e d  water  V\  3  was  determined t o  weighings of the  cell. and  and  a l s o f o r the buoyant  W a t e r was  later  j a r which  G.  of water effect  the c e l l  by p l a c i n g t h e c e l l  in a  the c e l l  must c o n t a i n o n l y  o n l y t h e n i s t h e volume o f t h e l i q u i d is first  filled  room t e m p e r a t u r e .  so t h a t ( See  Figure  the c o r r e s p o n d i n g d e n s i t y  is  ^'  .  The  light  index f o r a g i v e n density  liquid  or only vapor  beam w h i c h  8 )  volume  T h i s temperature  of the l i q u i d  and v a p o r  corresponding to  ^  corresponding to  ^  y  combined  of the l i q u i d ,  , the d e n s i t y  of the  i s increased to  T  2  ,  o c c u p i e s more o f t h e c e l l  vapor l e s s .  the l i g h t  visible  since  T^  and  ^  the other  and  ^  volume and  beam t h r o u g h t h e v a p o r  i t s volume i s so s m a l l and  result  vapor.  change a s t h e l i q u i d At  is  one  , the density  As t h e t e m p e r a t u r e  i s T^  i s passed through the c e l l  are observed i n the telescope,  The  i s liquid  r e f r a c t e d by t h e l i q u i d more t h a n by t h e v a p o r and a s a dots of l i g h t  since  o r t h e v a p o r known.  almost the e n t i r e  and  two  bell  c o n n e c t e d t o a vacuum pump.  measurement o f t h e r e f r a c t i v e  means t h a t  at  o f the a i r on t h e  DATA COLLECTION The  cell  a t room  evacuated from the v a l v e b e f o r e weighing  from i n s i d e was  cell  and t h e n f i l l e d w i t h a i r .  C o r r e c t i o n s were made f o r t h e d e n s i t y temperature  be  the bubble  of  y  the  i s not vapor  TEMPERATURE  Figure 8 - DATA COLLECTION PROCEDURE  k2  lies  above t h e  known t h a t  u p p e r edge o f t h e  the  temperature  cell  i s increased  move, i m p l y i n g the  contains to  T'  the  dot  density  i s further increased  l i q u i d at  temperature in  the  cell,  i n d e x and point  to  and  T^  could  and  i s obtained  s i n c e the  Figure  knowledge  near  ^>  c  cell  of the  above t h e  had  to  a  t o be  density gradients  a l t e r the  of the only  experiment.  departure avoid  no  the T-»T  T^  that  sample c a n  longer  a  data  index be  found,  p o i n t s t h a t were r e g i o n was  since point  cell.  obtained  i t i s in this phenomena r e q u i r e  coexistence  In order  to a point  the  restricted  density gradients  existed.  liquid  l i q u i d occupies  from the  .  in  refractive  refractive  The  curve  not  contained  of the  above  curve  i n the  to assure  sample o f homogeneous d e n s i t y increased  is  must have  density  region of data  g r a v i t y as  contained  dot  the  I t i s at  coexistence  The  made  T^  As  further increase  studies of c r i t i c a l  o£. .  of  was  temperature  t o move.  liquid.  i t i s here t h a t the  9 shows t h e  sample c a u s e d by the  not  the  o f the  cell  a l t e r the  volume i s known and  region that recent a  dot  as  course  to l i e just  not  the  t o a known d e n s i t y  cell  the  therefore  hence i t c o u l d  cause the  corresponding  during  T'  i t is  of l i g h t i s seen t o  to  o b s e r v e d t o move w h i c h means t h a t t h e only  Still,  some v a p o r s i n c e when  a change i n t h e  temperature  windows.  that  the T  where  the  TEMPERATURE x * X  X X X  X  X  DENSITY F i g u r e 9 - REGION OF DATA COLLECTION  44  Once p a s t that  the c e l l  temperature  ^  a s i m i l a r p r o c e d u r e was u s e d t o e n s u r e  c  contained only vapor.  was r e d u c e d t o see i f t h e d o t c h a n g e d  If  so t h e n some l i q u i d  to  be i n c r e a s e d  entirely  had formed  slightly  position.  and t h e t e m p e r a t u r e  t o be s u r e t h a t  had  t h e c o n t e n t s were  vapor.  A chart  r e c o r d e r was u s e d t h r o u g h o u t t h e e x p e r i m e n t t o  keep a r e c o r d  o f temperature  variations.  e x p e r i m e n t e r t o judge t h e t i m e come t o e q u i l i b r i u m . the  I n t h i s case t h e  T h i s enabled the  r e q u i r e d f o r the system t o  A l t h o u g h t h i s was n o t i m p o r t a n t i n oC s i n c e  determination of  i t was o n l y n e c e s s a r y t o be  above t h e c o e x i s t e n c e c u r v e , i t was i m p o r t a n t i n d e t e r m i n i n g the  c o e x i s t e n c e c u r v e and t h e r e c t i l i n e a r  d a t a was o b t a i n e d by l e a v i n g t h e c e l l volume and s w e e p i n g o v e r t e m p e r a t u r e . liquid in  known a n d oC-^ calculated  2  -  and<*°  n^ a n d n  experiment  v  K a  this  windows.  n  a t near  '  d  information i s only  1  critical  as the meniscus  While  P,^ and £ a  n  c a n be c a l c u l a t e d  2  This  The d o t s f r o m b o t h t h e  were n o t known,  v  _ =  critical  of the c e l l  since  1  - F — While  filled  and v a p o r p h a s e s were t h e n v i s i b l e  the middle  ru  diameter.  d  P  v  v ^ v  and we  was  were n o t w  e  r  e  have  v„ = ?v*v  '  of secondary importance  i n this  a n d i s o b t a i n e d t o much g r e a t e r p r e c i s i o n n e a r t h e  p o i n t by o t h e r means,  f u r t h e r away  from the c r i t i c a l  i t does p r o v i d e good point.  data  45  VI.  DATA ANALYSIS The  d a t a a n a l y s i s was  "Never t r u s t founded.  made by  a c o m p u t e r " was  found  a r e c o n t a i n e d i n A p p e n d i x B. rig = 1.66  n^.=  index  1.00029  W  v X  o oh = -8.34  A  (J) =  -.080°  by off  ,  gm  cell  used  used  were a s f o l l o w s :  sapphire  cell  weight of the (  sin6  / • a  l  r  i  n  c  e  l  1  evacuated  -, -, "  n  0  c  e  l  l  )  cell  ,, \  i  of the  = b (X  i  o b l i q u e window  - X ) Q  5 n d e d i t i o n o f t h e Handbook o f  the  2  A d i s c u s s i o n of the \ , and  s e c t i o n s o f t h i s work.  p l a c i n g the  computer programs  volume o f t h e  wedge a n g l e  Physics.  constants except  entirely i l l  a n g l e b e t w e e n i n n e r window f a c e s  n^ a r e t a k e n f r o m and  cc  .001  b = .00006476  Chemistry  adage,  by hand f o r t y p i c a l  Constants  micrometer reading  = 20.088°  0<  the  index of r e f r a c t i o n of a i r  = 179.394 *  Q  The  not  of r e f r a c t i o n of  V = 13.108 ± .003  earlier  t o be  A l l c a l c u l a t i o n s were c h e c k e d  data values to insure accuracy.  rig and  c o m p u t e r and  X  A  o r i g i n of  i s given i n detail  ot and  §  were  L i g h t was  in  determined  on a r o t a t i n g t a b l e h a v i n g  on t h e c i r c u m f e r e n c e .  the  degrees  reflected  from  marked  the  s u r f a c e s o f t h e windows and  the a n g u l a r p o s i t i o n  recorded.  The  v a l u e s are the averages  of f i v e  are  to  -  air  i n the  .008° .  n^  X  was  A  r e a d i n g s and  t h e a v e r a g e o f 11  accurate  values taken  with  cell.  i s the  according to  i n d e x o f r e f r a c t i o n o f e t h a n e and  n, 1  = ru 3  (  1  + cos 6 - cos 6 . + A  s  i  n  6  was  calculated  ~ * A t a n &. 3  n  6  ).  46  T h i s a n a l y s i s was u s e d  i n s t e a d o f the f u l l a n a l y s i s which  2 included  t h e c o r r e c t i o n term and a f a c t o r o f c o s § because  .999998 and  cos (j> was 2  .00009  was t y p i c a l l y the  e r r o r introduced  reading  the c o r r e c t i o n term  by t h e u n c e r t a i n t y  )  compared t o  and t h e w e i g h t .  I  is  A  i n the micrometer  to  = « <  of c e l l  fr( f ( e ) - f ( e )  • T h e s e were n e g l i g e a b l e  W*N was d e t e r m i n e d a c c o r d i n g Table  "gin  gives the r e s u l t s  >>  —  + 2  TL  of the a n a l y s i s f o r g i v e n  mass a n d m i c r o m e t e r r e a d i n g .  Since  the major c o n s i d e r a t i o n a value  values  the constancy of  of  was c a l c u l a t e d  2 n e a r ^>  by t a k i n g a l l v a l u e s  c  and be  averaging construed  prevent the  these as  values  at  our s e t t i n g  T=T  t o give£^,. t f=  c  T= T »  r e g i o n of the c r i t i c a l  o f ^ f o r which  plot  vs  <*c to  be a b o u t  w h i c h means  of  «£(T)  in  a s was  b u t r e f u t e d by l a t e r f i n d i n g s , ^  and  Abbiss, should  Knobler, be a g o o d  .  constancy of  o f off- pC  i s not t o  gradients  a value  ) then t h i s average  approximation t o  this  since density  p o i n t and i f  t e n Seldam, and O v e r d i j k  Teague, and P i n g s  The  c  c  While  i t does g i v e  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 , ( Michels,  P  ( ^ - ^ ) < .04  ?  i s shown i n F i g u r e .  10  which i s a  The t o t a l v a r i a t i o n i n o ^ i s  seen  Sc 1% . W = W  between the c e l l  This i s including values Q  .  of 3^for  ^ near 0  I t i s i n this region that differences  w e i g h t when empty a n d when f i l l e d w i t h  small  k? Table  I  DATA AND RESULTS ( * = 6328 X ) Mass (gm) 184.05361 184.00258 183.93675 183.89613 183.84352 183.78652 183.75072 183.70305 I83.65760 I83.57288 183.53306 183.49588 183.45684 183.41906 183.37975 183.31190 I83.25261 183.19706 183.14800 183.11060 183.06472 I83.OI652 182.94860 182.84800 182.78902 182.73998 182.68606 182.63550 182.59896 182.54922 182.51468 182.45106 182.37164 182.33394 182.30102 182.27076 182.23302 182.18898 182.14100 182.10010 182.06508 182.04030 181.98806 181.93390 181.88646  Micrometer 1196.52 1181.79 1161.72 1151.61 •1137.17 1122.05 1112.04 1098.98 IO85.67 1063.49 1053.30 1043.02 1032.58 1022.54 1011.98 993.30 977.91 962.71 947.21 939.03 928.85 915.20 897.50 871.07 854.96 842.00 828.04 814.16 804.49 792.57 782.61 765.38 746.14 735.37 726.67 718.73 709.01 697.60 684.81 674.03 664.64 658.39 644.81 630.77 618.26  Density (gm/cc)  Index  .3555 .3516 .3^65  1.2107 1.2082 1.2047 1.2030 1.2005 1.1979 1.1961 1.1939 1.1916 I.I878 1.1860 1.1842 1.1824 1.180? 1.1789 1.1756 1.1730 1.1703 1.1677 1.1662 1.1645 1.1621 1.1591 1.15^5 1.1517 1.1494 1.1470 1.1446 1.1429, 1.1408 1.1391 1.1361 1.1328 1.1309 1.1294 1.1280 1.1263 1.1243 1.1221 1.1202 1.1186 1.1175 1.1151 1.1127 1.1105  .3394 .3351 .3323 .3287 .3252 .3188 .3157 .3129 .3099 .3070 .3040 .2989 .2943 .2901 .2864 .2835 .2800 .2763 .2712 .2635 .2590 .2552 .2511 .2473 .2445 .2407 .2381 .2332 .2271 .2243 .2218 .2194 .2166 .2132 .2095 .2064 .2038 .2019 .1979 .1937 .1901  Lorentz (cc/ .3781 .3779 .3773 .3777 .3777 .3778 .3778 .3778 .3775 .3778 .3780 .3780 .3780 .3781 .3781 .3780 .3783 .3782 .3773 .3780 .3788 .3786 .3788 .3790 .3788 .3789 .3790 .3788 .3789 .379^ .3791 .3789 .3797 .3793 .3793 .3793 .3794 .3795 .379^ .3793 .3792 .3793 .3793 .379^ .3793  Table I  (cont.)  Mass (gm)  Micrometer  Density (gm/cc)  Index  181.84694 181.78708 181.73338 181.67764 181.63812 181.58498 181.52576 181.48830 181.42338 181.38990 181.34260 181.29532 181.23776 181.17370 181.12364 181.07504 181.02886 180.98216 180.9^216 180.89380 180.85550 180.80800 180.75396 180.70250 180.65644 180.60348 180.55738 180.51108  608.11 592.65 578.71 564.15 55^.35 540.30 524.80 515.18 498.56 490.09 477.76 465.25 450.31 434.66 421.46 408.98 396.71 385.13 37^.92 362.91 353.25 340.68 327.38 313.96 302.60 289.00 277.59 265.74 255.86 243.49 232.83 222.08 209.83 194.91 183.88  .1871 .1825 .1784 .1742 .1712 .1671 .1626 .1598 .15^8 .1522 .1486 .1450 .1406 .1358 .1319 .1282 .1247 .1211 .1181 .1144 .1115 .1079 .1037 .0998 .0963 .0922 .0887 .0852 .0821  1.108? 1.1060 1.1036 1.1010 1.0993 1.0969 1.0942 1.0925 1.0896 1.0881 1.0859 1.0837 1.0811 1.0784 1.0761 1.0739 1.0717 1.0697 1.0679  180.47022  180.42192 180.37790 180.33630 180.28656 180.22740 I8O.I8256  180.14342 180.09006 180.04108 179.98406 179.94180 179.88868 179.81136 179.76842 179.71290 179.66080 179.61404 179.56532 179.50588 179.46234 179.^0892  174.22  161.02 148.47 134.92 124.54 111.38 92.01 81.38 67.5^ 55.05 42.94 31.66 16.87 6.53 -7.21  .0784  .0750 .0719 .0681 .0636 .0601 .0572 .0531 .0493 .0450 .0418 .0377 .0318 .0285 .0243 .0203 .0168  .0130 .0085 .0052 .0011  I.O658  1.0641 I.0619  1.0595 1.0572 1.0552 1.0528 1.0508 1.0487 1.0470  1.0448 1.0429 1.0410 1.0388 1.0362 1.0343 1.0326 1.0302 1.0280 1.0256 1.0238 1.0215 1.0181 1.0162 1.0137 1.0115 1.0094 1.0074  1.0048 1.0029 1.0005  Lorentz (cc/i  .379^ .3794 .379^ .3793 .3796 .379^ .3792 .3792 .3793 .3794 .3792 .3789 .3787 .3793 .3789 .3788 .3784 .3786 .3785 .3788 .3788 .3782 • 3785 .3780 .3782 .3779 .3780 .3777 .3782 .3777 .3782 .3776 .3778 .377^ .3776 .3777 .3777 .3767 .3780 .3782 .3781 .3770 .3766 .3755 .3767 .3720 .3762 .3720 .3753 .2927  .005  0.0 \-  -.005  +  V  r ++  -.01  h  +  +  0.0  F i g u r e 10  + 0.5  -  1.0  VARIATION OF  ^  ETHANE 1.5  WITH  io FOR  2.0  ETHANE  50  amounts o f e t h a n e a r e q u i t e s m a l l a n d p r o d u c e e r r o r s t h a t increase as the  five  W-»W  is  To p r o d u c e a " b e t t e r s c a l e  data p o i n t s nearest  became e x t r e m e . of  .  unpublished The  Q  were o m i t t e d a s t h e v a r i a t i o n  The m a i n body o f t h e d a t a  that i s within .lfo  l e s s than  W  .  f o r t h e graph  shows a  variation  and t h e v a r i a t i o n f o r ^  Figure 11  near  ^  c  shows a c o m p a r i s o n o f r e c e n t  work o n 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 .  two s e t s o f d a t a a r e s i m i l a r f o r h i g h d e n s i t i e s b u t t h e S F ^  d a t a i s more n e a r l y c o n s t a n t shows a c o m p a r i s o n w i t h The  f o r low d e n s i t i e s .  data presented  12  Figure  by S m i t h f o r x e n o n .  e t h a n e d a t a i s c l e a r l y more c o n s t a n t  over the e n t i r e  data  range. The  coexistence curve  was done by Mr. P a l f f y . fit  T c  =  Figure It  The a n a l y s i s  c  = 32.079 -  .3794  ±  . 0 0 1 °C .  .0003  of  1 3 shows t h e c o e x i s t e n c e c u r v e  coexistence curve  diameter  ^C^CC  =  from  this  differed  analysis  PQ  =  .2062  parameter  .0782 - .0001  Using the c a l c u l a t e d  gives a value  was n o t a b l e t o d e t e r m i n e  diameter  involved a three  t o t h e d a t a and e x t r a p o l a t i o n t o g i v e  and  e£s  and r e c t i l i n e a r  value of ±  .0003  and r e c t i l i n e a r  diameter.  data that the  from  a straight  gm/cc .  line.  . A B a l z a r i n i and Palffy unpublished + present investigation  * SF + Ethane i_ 6  0.5  0.0 F i g u r e 11  -  VARIATION  1 OF X  2.0  1.5 WITH  ^ FOR  SF  6  AND  ETHANE  • S m i t h et a l P R L 21.1306.(1968) • + present investigation  .01  •  .005 • 0.0  •a •  • •  • +  + a •  -.005  a  •  + +  +* + a  • • •  +  +  •  -.01 •  0.0  a Xe + Ethane  •  0.5  1.0  1.5  2.0  F i g u r e 12 - VARIATION OF 2£ WITH ^ FOR XENON AND ETHANE  (V)  TEMPERATURE 1—»  CD •  o  cn  ro  00  ro CO cn 4N  ro >)  CO CD o  C CO ro  •  o cn  54  VII.  ERROR ANALYSIS Assuming t h a t  seeks  to explain  i s i n fact  any d e p a r t u r e s  constant  from  this  constancy. o£  t h e known e r r o r  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 .  W  ^  AW_ _ ^ + / o  AX y _ v X - . A  where  and  X^  A^£  X  in ^  and s i m i l a r l y  i n r e a d i n g t h e m i c r o m e t e r was  error  i n the weighing varies is  1  was  10"  and  the corresponding accuracy  For  low d e n s i t i e s  i-  Aj£  is  .5$  F o r d e n s i t i e s near  have  Ao? _  .085$  corresponding  which l i m i t s ^  c  its  actual  lower  in ^ to  For high  of  v a l u e would t e n d  than i t s true value.  The  error  densities  . This i s f o r P~.34gm/cc  is £  ±  .0002 .  06gm/cc  the accuracy  ^  with a i r .  and A X .  The r e s u l t i n g  which corresponds  w h i c h means  A measured v a l u e  .0598  or  filled  d i v i s i o n s and t h e  milligram.  .3  3  f o r AW  .3  over the range o f d a t a .  about  weight.  i s the micrometer reading,  error  n  i s the c e l l  i s the micrometer reading f o r the c e l l  i s the e r r o r  in  due t o e r r o r i n  A  i s t h e empty c e l l w e i g h t ,  Q  W  analysis  To b e g i n t h e  analysis  A o£  i n calculating  error  the error  of o£  to ^^.2  i s known t o  pj w h i c h i s s l i g h t l y t o give a value  -  t o - .002 gm/cc  .0003 .  larger  o f 3*s  we  than  which i s  One e x p l a n a t i o n f o r t h e d e p a r t u r e  .  55  of ^ from side of  p  o£  c  would be t h a t the temperature f o r  i s lower than t h a t a t  c  P  ^ on e i t h e r  T h i s means t h a t i t  c >  i s p o s s i b l e t h a t some moisture from the a i r i s being  weighed  w i t h the c e l l f o r high and low d e n s i t i e s t h a t i s not  being  included at c r i t i c a l density.  T h i s seems reasonable i n  view o f the f a c t t h a t weighings of the c e l l hot and c o l d produced a c o n s i s t e n t i n c r e a s e t h i s i s only  .05$  i n weight of  2 mg.  However  of the sample weight f o r high d e n s i t i e s .5$  and cannot account f o r the observer decrease of The c e l l volume would  i n c r e a s e as the temperature  i n c r e a s e d and since the c e l l volume was measured a t room temperature t h i s would mean t h a t 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 a c t u a l volume.  Increasing  i n c r e a s e the value  the volume would pc  cause the peak around  of  and  t o 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  " o V  V  = 2 x 3 x 15  x 10~5  o r .09$  .  o  T h i s means t h a t the e f f e c t o f thermal expansion i s much l e s s than e r r o r s introduced but  i s comparable  high d e n s i t i e s .  i n measurements a t low d e n s i t i e s  t o measurement e r r o r s at medium and  56  Errors  i n t r o d u c e d by t h e s h i f t i n g  o f the l i g h t  beam  were c o r r e c t e d f o r b y m e a s u r i n g t h e p o s i t i o n o f t h e beam with  the c e l l  removed e a c h t i m e  c o r r e c t i n g the observed reading,  X  Q  .  a d a t a p o i n t was t a k e n a n d  beam d e f l e c t i o n w i t h t h e z e r o beam  The s h i f t i n g  o f t h e z e r o beam d u r i n g t h e  course  o f the experiment  During  t h e measurement o f t h e c o e x i s t e n c e c u r v e  s p l i t t e r was u s e d and  to direct  temperature  The protect  equilibrium  balance  was e q u i p p e d  was h a n d l e d  collecting with  with  sliding  dust while  plastic  of  the data c o l l e c t i o n the c e l l  in  a bell  effect  full  j a r t o remove any m o i s t u r e  surface or inside  weighing  t o be  maintained  g l a s s windows t o n o t i n use and  gloves i n order t h a t the  w e i g h t n o t be a l t e r e d b y c o n t a m i n a t i o n .  the  collection.  ( See F i g u r e 4 )  the pan from  cell  container  t o r e m a i n i n p l a c e f o r t h e sweep o v e r  and p e r m i t t e d thermal  more e a s i l y .  data  .  a beam  t h e beam a r o u n d t h e c e l l  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 t h e  T h i s allowed the c e l l  the  was l e s s t h a n f o u r d i v i s i o n s  Before  o f e t h a n e was p l a c e d t h a t may have b e e n o n  the v a l v e . E f f o r t s t o minimize  e r r o r c a n o n l y be assumed e f f e c t i v e  i s n o t known.  the s t a r t  the  as the a c t u a l  57 VIII.  CONCLUSIONS  1 T h i s work d e m o n s t r a t e s  that  f o r ethane  o£  =  2  - 1  7  " n where for  i s considered constant,  d e n s i t i e s of  variation  200  the Lorentz-Lorenz point  P  of  The  .  optical  entirely  weight  of the c e l l  adequate,  and  hut  c o u l d be  c o n c e r n i n g the weight  s h o u l d be  for  improved.  Future  control  of the c e l l  curve  and  diameter  T was  a  of  critical  ^  on  either the  control  and  the  studies should  of a l l f a c t o r s  i t s d e t e r m i n a t i o n as of u n c e r t a i n t y .  the weight  of the c e l l  Care does  temperature.  P =.2026 ±.0003 gm/cc  line.  t h e use  the 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  straight  showed  t h e method o f d e t e r m i n i n g  taken to insure that  coexistence  data  studies of  found  probably the g r e a t e s t source  not vary w i t h  approximation  s t u d i e s are needed t o v e r i f y  system  +2  index of r e f r a c t i o n to d e n s i t y .  was  p a r t i c u l a r a t t e n t i o n to the  t h i s was  s i n c e the  This justifies  the  in  further  proved  less  excellent  equation f o r o p t i c a l  decrease  hut  c  decrease.  pay  in  phenomena w h i c h r e l a t e A slight  side  amagats and  %fo o r l e s s  of  i s an  ,  n  =32.079 ±.001 not found  data °C  .  gave The  t o vary from  a  58  BIBLIOGRAPHY (1)  Balzarini,  D.A.; Ph.D. T h e s i s ,  Columbia  University,  (1968). (2)  Chapman, J.A., F i n n i m o r e , P.C., a n d S m i t h , B.L.; P h y s . R e v . L e t t . , 21, 1306, (1968).  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Rev.,  112,  434*  ( 1 9 5 8 ) .  (1956)  1291,  60  APPENDIX The  analysis  s l i g h t l y wedge  A.  o f t h e c a s e o f t h e o b l i q u e window b e i n g  shaped  i s due t o D. B a l z a r i n i  1 /n^  and i s a s f o l l o w s .  n.  / V / V  REFRACTION ANALYSIS - WEDGED WINDOW F i g u r e 14 sinp  sinoc =  and  s i n ^ = n ^ s i n (ot + 0 + $ )  where n^ = i n d e x o f r e f r a c t i o n o f e t h a n e , r e f r a c t i o n of sapphire, Since  £ = y-  s i n uv we have Expanding n^  <J>  = index of  and n^ = i n d e x o f r e f r a c t i o n o f a i r .  , by e x p a n d i n g  s i n fi  and s u b s t i t u t i n g f o r  n^ s i n o t = n^ s i n (t*+ 0 + | ) c o s § - n  s i n («+ 6 + $>)  s i n <)| cosH*.  2  gives,  s i n c x = n^ cos <^> sin(c*+0) + s i n <j) f" n ^ c o s $ cos(«x+0)-n 2  2 Expanding cos(«+9)  and l e t t i n g  n^ c o s <|) cos(ot+0) - ix, cos»V .which we  n-cos  (|>(coso(cos e - s i n e * s i n 6)  (1  cosf=  -  nC 2  1  \  sin(ot+ 0 +  a s ((*.+ 0) + §  f  2  f ^ , becomes ,  n Denoting the q u a n t i t y under the r a d i c a l  cos^J  i  - sin y )  call  2  2  and square i t .  sin (c*.+ 6 + 2  2  we  expand  i. 2  61  Using double angle formulas we have, n f = 2  1-  2  2 ( s i n ( « + <j>) + s i n 9 cos 2(tx + 2  + s i n 2(<x+$)sinGcosG  2  "2  2  f a c t o r i n g out  n~ g g s i n (c*+ <J>)  1  2 n  "2  "2 f =(l 2  T  we now have, l e t t i n g —g = a  - a sin ((*+<]))) ( l + x)  where x i s  2  - a ( s i n 6 cos 2(M+<|)) + s i n 2(o<+<t>) s i n 9 cos e) 2  a  n  d  i  s  a  1 - a sin (o<+(J)) 2  small q u a n t i t y .  F o r the case o f o< = 20°, < J > = .08°, and 9 = 3 ° i  x i s only  .003  approximation ( f  • Since we need ( f x ( 1 + x )  a  2  )  = ( l + g )  f  o  we make use of the r  s m a l l x and w r i t e  Since s i n 0 i s always  )== ( l - a sin (*+<j>))^l + |) . 2  2  2  s m a l l , 9 i s about 3° a t i t s maximum, we use the same  2 approximation i n s u b s t i t u t i n g  1 - sin 9  f o r cos 0 . The  2 d i f f e r e n c e between the two f o r 0=3 i s .0000009 and .000003 f o r 0=4° , so the approximation i s q u i t e good. M u l t i p l y i n g out a l l the f a c t o r s we now have f o r f ^ ,  2 n~ c o s <> f c o s o ( - n,, cos <b cosoc s i n 0 - n~ cos § sin<x s i n 0 3 3 2 3 - n ( 1 - a sin (<K+i))2  2  ng( 1 - a sin (oc+$>)) J 2  f ^ c a n be seen t o c o n t a i n two terms which have no 0 dependence and  three terms which have 0 dependence. Thus we c a n w r i t e  f± = P(oc,o) + F(0)  62  S o l v i n g now  f o r n^ we have ,  1 I n^ cos (j) s i n (o<+9) s i n c< ^ J  n., =  s i n <t» F(«,<ji) + F(9) s i n <x L J  +  2  L  which upon expanding s i n (<x+9) becomes = n  3  cosh  When there  (cos 8  + ^  +  [ « « . • ) + P(6)] .  i s a i r i n the c e l l n^ = n^ and the angle made  by the d e v i a t i o n 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 © n  = n  3  , gives ;  A  c o s * (cos 6 2  3  Subtracting  we  A  +  [««.•)  +  n i  = n  3  3  A  have,  • cos 4>(cos 9 - cos 9 2  -n  F(8 >]  +  A  +  s i n 9 - s i n 0.\ )  S o l v i n g f o r n^ g i v e s the f i n a l form which i s , * os 4>(cos 9 - cos 9 2  n  l  =  n  3  1  +  C(  A  +  s i n 9 - s i n 0. v | 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 t o t h a t which i s shown, however t h i s  does not change the a n a l y s i s as  <|>  i s replaced  by  ,  T h i s a n a l y s i s was not needed as i t turned out because c o s $ was 2  .999998  and s i n <> t f F(9) - F ( 0 ) A  s i n «x L  A  J  was  .00009  63  a t the most, which was the w e i g h i n g  and the r e a d i n g o f the micrometer and  therefore neglected.  n  l  =  n  3  which was  1  l e s s t h a n the e r r o r i n t r o d u c e d i n was  This gives  + cos 6 - cos 6.  A  s i n 6 - s i n ©, +  tan  oc  the e q u a t i o n f i n a l l y used i n the d a t a  analysis.  64  APPENDIX  The  f o l l o w i n g program was  B.  used to c a l c u l a t e  n,  65 I M P L I C I T R E A L * & (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 = i 3 , 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 ( 2 X , P 1 0 , 5 , 3 X , F 8 , 2 , 6 X F 7 , « , « X , F 8 , 5 , 7 X , F 7 , 4 ) 1 FORMAT (2X,2F10,S) -rTtmttAi'—t/77V**7^^ PLORENTZ COEFF.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) Fl=l,0sFl — F 5 « »*~AI-R **-A-I-R7-(~i-i-6 6*-l66-)-*-(WWWmmWWS«TH^W+^ 1N(2,0*(AL+PH))*DSIN(TC)*DC0S(TC))/F1 V~ F6= • X A I R * ( D C 0 S ( P H ) * O S I N ( T C ) * D S I N ( A L ) t D C 0 S ( P H ) * D C 0 S ( A L ) * 0 S I N ( T C ) * 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)) F  T  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 - D S I N ( P H ) / D S I N ( A L ) * ( F 3 i ! F 6 ) j D=DC0$(PH)*DC0S(PH) > — - X £ ( I ) eXA IR *~(~lxO^G-0£~(~W • — ~ — — — — ~t~— XESCn = XE(I)*XE(I) 7 ClORZ(I)=(XES(I)«l 0)/((XES(n*2,Q)*RHQ(l)) WRITE (6,3) Y ( n , X ( n , R H 0 ( I ) , X E ( I ) , C L 0 f t Z ( I ) fe XE(I)=XAIR*(l,0+(DCOS(TH)»DCOS(TC)*(X2sXl)/TN)*D)+CTERM V XES(I)=XE(I)*XE(I) VebORZ-(-BM-XE~8-(-I~)M WRITE (6,3) Y ( I ) , X ( I ) , R H O ( I ) , X E ( I ) , C L Q R Z ( I ) i 2 CONTINUE S-T-0-p _ END • •'---*" S  66  T h i s program i s used all  v a l u e s of  <£f  to calculate  f o r which  DIMENSION XC100)r Y ( t 0 0 ) \ FORMAT (2X,2FtO,0) 19 F O R M A T C I X » » C t L = f | F 1 6 , 6 ) Ns90 • — 00 6 I=J#N & READ(5,U X(J),Y(I) RHQC=,20 8 = 0,0 AVE=0 0 0 0 6 0 1 = 1,N D*(X(I)»RH0C)/RH0C JP ( D * D « , 0 4 ) i7 \7,i& 17 AVEsY(I)+AVE BsB*l, 18 C O N T I N U E — (}Q C O N T I N U E CLUAVE/B WRITEC6,1) B WRJTEC6,19) CLU STOP END s  f  £C  ) < 2  c  .04  by  averaging .  T h i s program does a l e a s t line  J 5 3 U  2  f o r the c a l c u l a t i o n o f  b  squares  f i t to a  u s i n g the Ronchi r u l i n g .  I M P L I C I T R £ A L * 8 (A*M>Q*Z) DIMENSION, X(800)iYC200)#Xll(200) FORMAT ( H 2 , < U FORMAT(12) FORMAT ( 6X,'SLOPE•?,7X,»Y INTERCEPT') FORMAT ( I X , a F l 2 4 ) BA=0,0 SLOA=0,0 00 7 J c l , S READ(5,1) A READ(5,5) N SX=G,0 SY=0,0 SXY=0,0 SX2P0.0 C-sN DO 2 I = l # N READ(5 r 1 ) Y d ) SX=SX+A SYaSY+Y(I) SXYsSXY+A*Y(I) SX2=SX2tA*A A=A+1,0 CONTINUE SLOPE?(SX*SY«C*SXY)/(SX*$XsC*SX2) 8  B=(SYBSL0PE*5X)/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 («»•»<•) S L 0 A , B A  STOP END  straight  68  The f o l l o w i n g program was used t o produce the p l o t s of  X  -  £  <5Cc  0  v  s  _ £  ^c  1  DIMENSION X ( 5 0 0 ) , Y ( 5 0 0 ) FQRMAT(2X,2F10,5) 11=90 L2?32 =?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 DO 6 I=L,N X(I)=(X(I)*RHOC)/RHOC Y-(I)aCY(I)»CL3)/Cl8 6 CONTINUE L=L1+I N  s  :  K  RHQC=8,42 CLS=10 527 DO 91 1 = 1 , N X(I)=(X(I)«RHOC)/RHOC Y(I)=(Y(I)*CLS)/CLS 91 CON-T-INUE LBK + 1 8  N= J  CL3?769,8 RHOC=,7357 DO  92  I=L,N  X-H^=-C^m-irRtlE^eiV-R+t«^ Y(I)=(Y(I)*CLS)/CLS 9 2 CONTINUE CALL PL0T(0 Q,0,0,»3) DATA S X , S Y / 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 r S - Y - r T t W r f r ^ r l - 1 CALL AXIS(0,0,0,0,3HRH0,»3,SX,0,0,XMIN,DX) CALL A X I S ( 0 , 0 , 0 . 0 , 2 H L L , 2 , S Y , 9 0 , 0 , Y M I N , O Y ) tr=i NaLi 8  DO  93  I?L,N  e^tL^^M&0^-(^H-^H^TT^T%^'»-H 93 CONTINUE LsLUl N«K  DO  60  I=L,N  C A L L SYMBOL -6-0-eO'N^H-NU'E L=K + 1  (X(I),Y(X),,l,0,0,0,al)  NPJ  DO 80 I°?L,N C A L L SYMBOL 80 C O N T I N U E STOP END  (X ( I ) , Y ( I ) , , 1 , 2 , 0 , 0 , * 1 )  70  curve to  The  f o l l o w i n g p r o g r a m s were u s e d t o f i t t h e  and  the  rectilinear  extrapolate to find  g i v e n which p l o t diameter.  the  T  diameter with c  and  coexistence  £  c  ^  c  curve  three • and  coexistence  parameters  Programs a r e the  and  also  rectilinear  71  IMPLICIT 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 F N V ( 1 O O ) , F N L C 1 0 O ) , Q 1 U Q Q ) DIMENSION C Q 1 ( 1 0 0 ) , C Q 2 ( 2 Q 0 ) D I M E N S I O N R ( 1 0 0 ) , X M V U Q Q ) , X M L U O Q ) , X M Z ( J 00 ) , T (1 00 ) , XO ( 1 00 ) DATA B , X K / , Q 0 0 0 6 « 7 7 7 4 8 D O , » 9 , 9 9 D O / DATA X A I R , A L P , GAM/1 , 0 0 0 2 9 0 0 , 0 , 3 5 0 3 0 0 , 2 , 8 5 0 0 / ' DATA R Q , B E T / 0 , Q H 9 9 D 0 , 3 S 4 4 , 0 6 1 D Q / 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)=BET/DLOG(R(I)/RO) T ( I ) = T U ) * 2 7 3 , 16 XO(I)=XMZO(I)*XK YNL(I)=XAIR+B*(XML(I)*XO(I))/TN lm((B*(XML(I)*XO(I)))**2)/(2,*XAlR) YNV(I)=XAIR+B*(XMV(I)BXO(I))/TN l«((B*(XMV(I)*X0(I)))**2)/(2,frXAlR) FNV(I)=(YNV(!)*YNV(I)*J,0)/(YNV(I)*YNVU)*2,0) FNL(I)=(YNL(I)*YNL(I)*l,0)/(YNL(I)*YNL(I)t2 0) Qi ( I ) = ( F N L ( I ) e f N V ( I ) ) ^ * e A M Q2(I)=(FNL(I)*FNV(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 W R I T E ( 6 , 6 ) U2,V2,W2,ER2 TC2=U /(2.*Wl))*(*VI«D3QRT(Vl*VI«4,*Ui*Wi)) W R I T E ( 6 , « 7 ) TC2 FC=U2*(V2^W2*TC2)*TC2 W R I T E ( 6 , 5 0 ) FC FORMAT(2X,»RCLC ',F15,8) FORMAT ( I X , ITCS-I , F 1 5 , 8 ) F O R M A T d X , I ( N L » N V ) * * B E T , COEFFS ARE',SX, 1tA=l,FI5 8,5X,»B=I,F15 8,5X,»C=«,F15,8,SX,•E=»,F15,8) F O R M A T d X , I ( N L + NV) V S , T C O E F F S , ' 5 X , 1 ! A ? M , F 1 5 , 8 , 5 X , I Be •, F 1 5 , 8 , SX , I C=;' , F 15 , 8 , 5 X , « E= •, F 1 5 , 8 ) FORMAT(iX,9(FiO,5,2X)) DO 8 1=1,N CQ1 ( I ) ? U l * ( V l t W l * T ( I ) ) * T ( I ) CQ2(I)=U2*(V2+W2*T(I))*T(I) WRJTE(6,9) T ( I ) , Y N L ( I ) , Y N V ( I ) , F N V ( I ) F N L ( I ) , Q i ( I ) , C Q t ( I ) f Q 2 ( n , I,CQ2(I? CONTINUE CALL GRAPHi(N,FNL,FNV,T,Q2,FC) CALL GRAPH2(N,T*TC2,FNL,FNV) STOP END t  ?  8  S  (  8  f  72  SUBROUTINE SQPT<M,Y,X,C1,G2 C3#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 0 A«=O O AS=G,0 Bt=0,0 B2=0,0 B3=0,0 DO 10 I 8 1 , M TS=TStY(I)*Y(I) X2=XCI)*XCI) A2=A2+X(I) A3=A3*X2 A4=A«*X(I)*X2 ASoA5+X2*X2 B1=B1*Y(I) B2sB2+Y(I)*X(I) B3*B3*Y(I)»X2. CONTINUE DETaAlA(A3*A5«A4*Aa)eA2*(A2*AStiAa*A3)*A3*(A2*A«eA3*A3) C l = ( l /DET)*(Bl*(A3*A5*Aa*A4)BA2*(B2*A5HB3*A«)fA3*(B2*Aa*B3*A3>-3, C2=( J , / D E T ) * ( A l * ( B 2 * A S » B 3 * A « ) w B i * ( A 2 : M 5 e A 3 * A « > + A 3 * ( A 2 * B 3 - A 3 * B 2 ) ) 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 * C i * B l + T S + A 2 * 2 * C I * C 2 + A3*(C2*e2-t2,*Cl*C3) 1 +A4*2,*C2*C3+AS*C3*C3«2,*C2*B2s2 *G3*e3 RETURN END f  t  e  (  l  l  P  73  S U B R O U T I N E GRAPH 1CN,FNL,FNV,T,Q2,FC) R E A L * 8 FNL,FNV,T,Q2,PC INTEGER S Y M C 3 ) / 3 , « , U / D I M E N S I O N FNL U 0 0 ) * FN V ( H 0 ) T 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 YlCn=FNL(I) r  Y2(I)=PNVm  -Xtl)sT(I) Y3(I)=Q2(I) CONTINUE CALL R L Q T 8 DL3a(Yl(t)»Y2(l))/81 DLT=(XCN)wX(n)/S2 DO- 2 1 = 1, N YS1 ( I ) B - ( Y l ( I > « Y 2 . ( l ) ) / D L S Y82(I)»(Y2(I)»Y2(1))/0L8 Y83(I)s(Y3tI)«Y2.(J))/DL8 XS(I)=tX(I)eX(l))/DUT CONTINUE CAL.L.'-PUOT(0,0,0,Q#«3) CALL AXIS(0,0,0,0,»TEMPERATURE»,+11,82,90,0,X(1)#DLT) C A L L A X I S C O , 0 , 0 , 0 , I R V L V & R L U ', « 9 , S 1 , Q,0, Y2 C 1 ) , D t S ) 0 0 3 1=1,N CALL SYMBOLCYS1(I);XS<I),,H,SYMU),0,Q »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 > 0 , 0 r « * j r  CONTINUE  XM=SJt2,Q  CALL P L 0 T ( X M , 0 , 0 , « 3 ) RETURN END  f  

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