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Hindered rotation of diatomic molecules in the adsorbed phase MacRury, Thomas Bernard 1969

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HINDERED ROTATION OF DIATOMIC MOLECULES IN THE ADSORBED Pi'ASE  by THOMAS BERNARD B.Sc.(Hons0> M.Sc,  MacRURY  U n i v e r s i t y o f B r i t i s h Columbia, 1965  University  o f B r i t i s h Columbia, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n t h e Department of Chemistry  We a c c e p t t h i s t h e s i s as conforming t o t h e required  THE  standard  UNIVERSITY OF BRITISH COLUMBIA May, 1969  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  I further for  shall  the  agree that  his  permission  of  this  written  representatives.  be  available  g r a n t e d by  gain  permission.  Columbia  . tofc^q  for  for extensive  Department  Date  British  the  It i s understood  thes.is f o r f i n a n c i a l  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  f u l f i l m e n t of  U n i v e r s i t y of  make i t f r e e l y  s c h o l a r l y p u r p o s e s may  by  in p a r t i a l  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  Study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  ABSTRACT The  h i n d e r e d r o t a t i o n o f d i a t o m i c molecules  adsorbed phase i s is  investigated  u s i n g two models:  c o n s i d e i e d to be a p l a n a r continuum and the  rotations form V  1) the  = Acos 0; 2  2)  o f the admolecule  h i n d e r e d by a p o t e n t i a l o f the form W = u(l-cos2cb)  the f i r s t  the  l o c a l i s e d h a l f - w a y between two  The i n - p l a n e r o t a t i o n s  plane r o t a t i o n s  of  the adsorbent i s now c o n s i d e r e d t o be a  and the admolecule  sites.  adsorbent  out-of-plane  o f the admolecule are h i n d e r e d by a p o t e n t i a l  square l a t t i c e lattice  i n the  are  and the  out-of-  are h i n d e r e d by a p o t e n t i a l o f the same form as  in  model. U s i n g s p h e r i c a l harmonics as the b a s i s  set  for  the  f r e e r o t a t o r , a v a r i a t i o n a l s o l u t i o n i s p r e s e n t e d f o r f i n d i n g the eigenvalues the  o f the o u t - o f - p l a n e  in-plane hindrance  hindered r o t a t o r .  as a p e r t u r b a t i o n on the  h i n d e r e d r o t a t o r , the e i g e n v a l u e s  ±m degeneracy The  to c a l c u l a t e of  and i n - p l a n e  The o u t - o f - p l a n e  removes a l l the 2 £ + l m degeneracy o f the f r e e  except the ±m degeneracy. this  out-of-plane  o f the o u t - o f - p l a n e  h i n d e r e d r o t a t o r are found u s i n g v a r i a t i o n . hindrance  Then t r e a t i n g  The i n - p l a n e p e r t u r b a t i o n then  leaving a l l eigenstates  eigenvalues  rotator removes  non-degenerate.  found f o r the two models are then used  the h i n d e r e d r o t a t i o n a l heat  capacities  and  entropies  the v a r i o u s n u c l e a r s p i n s p e c i e s o f hydrogen and d e u t e r i u m .  heat  capacities  and e n t r o p i e s  The  o f e q u i l i b r i u m hydrogen ( e q u i l i b r i u m  deuterium) and orthohydrogen  (pai-adeuterium) are found to be quite  s e n s i t i v e to the b a r r i e r heights A .and u.  In contrast, the thermo-  dynamic properties of parahydrogen (orthodeuterium) are rather i n s e n s i t i v e to r o t a t i o n a l hindrance. Theoretical ortho-para separation factors are then compared with those measured experimentally f o r hydrogen adsorbed  on  doped alumina and for hydrogen and deuterium adsorbed on alumina. However, due to large uncertainties i n the experimental data, no d e f i n i t i v e conclusions can be drawn. F i n a l l y , the Henry's law data f o r hydrogen and  deuterium  adsorbed on g r a p h i t i c carbon are analysed using the quantum corrected gas-surface configurational integral and the out-of-plane hindered r o t a t i o n a l model.  The b e s t - f i t parameters are found to be very  s e n s i t i v e to the r o t a t i o n a l b a r r i e r and thus there appears to be good j u s t i f i c a t i o n for incorporating hindered rotations i n the hydrogen (deuterium)-graphite system.  ACKNOWLEDGMENT  I am s i n c e r e l y grateful to Dr. J.R. Sams J r . f o r h i s patience and invaluable guidance during the course of t h i s work. I would l i k e to thank Dr. D.P. Chong for many h e l p f u l discussions i n the s o l v i n g of the eigenvalue  equations.  The f i n a n c i a l a s s i s t -  ance o f the National Research Council of Canada i n the form o f a Postgraduate Fellowship i s also g r a t e f u l l y  acknowledged.  iv  TABLE OF CONTENTS PAGE ABSTRACT  i  ACKNOWLEDGMENT  i i i  TABLE OF CONTENTS  .  iv  LIST OF TABLES  vii  LIST OF FIGURES  ix  CHAPTER I .  INTRODUCTION  . . . . . . . . .  CHAPTER I I .  DEVELOPMENT OF EIGENVALUE EQUATIONS  . . . .  1 11  A ) The Free R o t a t o r  11  B) The O u t - o f - P l a n e H i n d e r e d R o t a t o r . . . .  15  C) The O u t - o f - P l a n e and In-Plane H i n d e r e d Rotator . CHAPTER I I I .  .-  25  QUANTUM STATISTICAL ROTATIONAL THERMODYNAMICS  33  A) The Free R o t a t o r  33  B) The O u t - o f - P l a n e H i n d e r e d R o t a t o r . . . .  35  C) Out-of-Plane and In-Plane H i n d e r e d R o t a t o r .  37  D) R o t a t i o n a l E n t r o p i e s and Heat C a p a c i t i e s o f Hydrogen and Deuterium CHAPTER IV.  .39  HINDERED ROTATIONAL EIGENVALUES AND EIGENFUNCTIONS  . . . . . . . . . .  A) E i g e n v a l u e s o f the O u t - o f - P l a n e  46  Hindered  Rotator B) E i g e n f u n c t i o n s o f the H i n d e r e d R o t a t o r . .  46 57  V  PAGE C) E i g e n v a l u e s o f t h e O u t - o f - P l a n e and In-Plane Hindered R o t a t o r CHAPTER V.  . . . . . . .  66  HEAT CAPACITIES AND ENTROPIES OF HYDROGEN AND DEUTERIUM  . . . . . . . . . . .  82  A) Free R o t a t i o n a l Heat C a p a c i t i e s o f Hydrogen and  Deuterium . . . . . . . . . . . . . . . .  83  B) O u t - o f - P l a n e H i n d e r e d R o t a t i o n a l Heat Capacities  f o r Hydrogen and Deuterium . . .  C) O u t - o f - P l a n e and In-Plane H i n d e r e d Heat C a p a c i t i e s  Rotational  f o r Hydrogen and Deuterium . .  D) H i n d e r e d R o t a t i o n a l E n t r o p i e s  o f Hydrogen  E) H i n d e r e d R o t a t i o n a l E n t r o p i e s  o f Deuterium .  . .  CHAPTER V I .  SEPARATION FACTORS  CHAPTER V I I .  THE EFFECTS OF HINDERED ROTATION ON THE SECOND 179 179  B) Treatment o f t h e Data f o rti^and  on  (2700°)  1  o f the B e s t - F i t R e s u l t s  CHAPTER V I I I . SUMMARY  APPENDIX A . . . . . . . . . . . .  . . . .  . . . . . . . . . .  BIBLIOGRAPHY  APPENDIX B  147 156  .  A) T h e o r e t i c a l Development  C) D i s c u s s i o n  116  164  GAS-SURFACE VIRIAL COEFFICIENT  P33  95  . . . . . .•  8  7  191 197 199  '. .  202 205  vi  PAGE APPENDIX C APPENDIX D . . . . .  210 .  216  vii  LIST OF TABLES PAGE TABLE 1.  Eigenvalues E, *- •" k p,m  f o r a basis set o f 100  )  spherical harmonics . . . . . . . . . . . . . 2.  Comparison of eigenvalues E ^ * +  , m  ^  and  47  *' ^ m  f o r X=10 using d i f f e r e n t sized basis sets . . 3.  Comparison of eigenvalues E j ^ * +  > m  ^  and E ^ ^  , m  51 ^  f o r X=40 using d i f f e r e n t sized basis sets . . 4.  52  Comparison of exact and approximate r o t a t i o n a l energies  5.  58  Number of bound eigenstates f o r various out-ofplane b a r r i e r heights  6.  65  > J f o r a basis set o f 36 spherical  Eigenvalues  m  harmonics . . . . .  .  67  7.  Eigenvalues £.  f o r X=l  69  8.  Eigenvalues £ . -' ' ^ f o r X=l  70  9.  Eigenvalues £ .  ( + 1  (  »  r )  1  ( + 1 > r  r  -  )  f o r X=10 . . . . . . . . . .  72  f o r X=10  73  10.  Eigenvalues Ej '" '  11.  Out-of-plane hindered rotational heat  (  f o r e-H 12.  )  capacities 96  2  Out-of-plane hindered r o t a t i o n a l heat capacities f o r p-H  13.  1,r  101  2  Out-of-plane hindered r o t a t i o n a l heat for 0-H  o  capacities .  106  viii  PAGE TABLE 14.  Out-of-plane hindered r o t a t i o n a l heat f o r e-D  15.  capacities . .  2  113  a) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities  of e-H  2  f o r X=l . . . . . . . . .  118  b) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities o f e-H 16.  2  f o r X=10  119,  a) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities  o f p-H  2  for X=l . . . . . . . .  128  b) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities 17.  of p-H  2  f o r X=10  . 130  a) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities  of o-H  2  f o r X=l  135  b) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities 18.  of o-H  2  f o r X=10 . . . . . . . .  137  a) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities  o f e-D  2  f o r X=l . . . . . . . .  142  b) Out-of-plane and in-plane hindered r o t a t i o n a l heat capacities o f e-D 19.  Separation factors  2  for H  for X=10 2  143  and D o u s i n g the Sandler  l i m i t and the out-of-plane hindered r o t a t i o n a l  20.  model with X=80  169  B e s t - f i t parameters f o r H_ and D  190  ix  LIST OF FIGURES PAGE FIGURE 1 . Energy level diagram for the out-of-plane hindered rotator  . . .  2. The eigenfunctions <j>  C + 1, + 2  - ( 9 ) , <f>  55  (8) and  )  (b *- - (9) for A=40 plotted as a function of 8 , +1,0  )  2  0° <  < 180°  9  62  3. Eigenvalues C^* ' ^ plotted as a function of the 1  1  in-plane barrier height with A=10 4. Eigenvalues  77  * ^ plotted as a function of the ,r  in-plane barrier height with A=10  79  5. Free rotational heat capacities for the different nuclear spin species of H and D plotted as a 2  2  function of temperature  85  6. Contributions to the rotational heat capacity of p-H , o-H , and e-H from the relative populations 2  2  2  91  l;V  7. Contributions to the rotational heat capacity of e-D  2  from the relative populations £;£'.  ......  93  8. Out-of-plane hindered rotational heat capacities of e-H  98  2  9. Out-of-plane hindered rotational heat capacities of p-H 10.  104  2  Out-of-plane hindered rotational heat capacities of o-H  109  2  .. ./continued  X  PAGE FIGURE 1 1 .  Out-of-plane of  12.  e-D  2  hindered  and i n - p l a n e  of e - H . 2  Energy l e v e l  rotator with  heat  o f the  hindered  .  diagram for  out-of-plane  capacities  rotational  diagram for  out-of-plane  115  heat  . . . . . . some o f t h e and i n - p l a n e  lower  121  eigen-  hindered  A=l and 0 < u < 4 .  Energy l e v e l  states  rotational  .  states. o f the  b)  capacities 112  Out-of-plane  a)  heat  |  capacities 14.  rotational  .  2  Out-of-plane of  13.  n-H  hindered  .  some o f t h e and i n - p l a n e  lower  123  eigen-  hindered  r o t a t o r w i t h X=10 a n d 0 < y < 8 15.  Out-of-plane capacities  16.  capacities 17.  heat 133  and i n - p l a n e h i n d e r e d  rotational  heat 140  2  and i n - p l a n e h i n d e r e d  of e-D  rotational  2  of o-H  Out-of-plane capacities  and i n - p l a n e h i n d e r e d  of p-H  Out-of-plane  125  rotational  heat 145  2  18.  Hindered r o t a t i o n a l  molar entropies o f p - H  19.  Hindered r o t a t i o n a l  molar entropies o f e - H  2  . . . .  152  20.  Hindered r o t a t i o n a l  molar entropies  of o-D  2  . . . .  158  21.  Hindered r o t a t i o n a l  molar entropies  o f e-D~  . . . .  160  2  and o - H  .../continued  2 <  149  xi  PAGE  FIGURE 22.  Separation factors e r a t u r e s p l o t t e d as  23.  Separation factors plotted  as  for  and  at  three  temp-  a function of A f o r f-^ and D  2  at  166 20.4°K  a f u n c t i o n o f u f o r A=l and A=10.  .  177  1  CHAPTER I . Until  about  1957,  INTRODUCTION  few i n v e s t i g a t i o n s  o f the heterogeneous  o r t h o - p a r a c o n v e r s i o n i n l i q u i d hydrogen had been c o n d u c t e d . and P o l a n y i ^ i n v e s t i g a t e d s o l i d hydrogen i n the ond  the homogeneous  absence  c o n v e r s i o n i n l i q u i d and  of a c a t a l y s t  and found the  o r d e r i n the c o n c e n t r a t i o n o f o r t h o h y d r o g e n .  s o l i d phase r e a c t i o n s ,  Cremer  r a t e t o be s e c -  In both l i q u i d and  the r a t e c o n s t a n t was found t o be  independent  2 of  the t e m p e r a t u r e .  Swenson  s t u d i e d the heterogeneous o r t h o - p a r a  c o n v e r s i o n i n l i q u i d hydrogen on c h a r c o a l and s i l i c a g e l t h a t the r a t e was f i r s t orthohydrogen.  order with respect  The low temperature,  and r e p o r t e d  t o the  concentration  of  or magnetic,  mechanism f o r  the  o r t h o - p a r a c o n v e r s i o n i n the gas phase has been u n d e r s t o o d f o r some time.  A t h e o r e t i c a l e q u a t i o n f o r the t r a n s i t i o n p r o b a b i l i t y f o r  c o n v e r s i o n by c o l l i s i o n w i t h a paramagnetic substance 3  was  developed  4  by Wigner .  Farkas and Sassche  found t h a t both the homogeneous  gas-  phase c o n v e r s i o n c a t a l y s e d by oxygen and o t h e r paramagnetic g a s e s , and  the  c o n v e r s i o n c a t a l y s e d by paramagnetic i o n s  c o u l d be e x p l a i n e d by the Wigner t h e o r y .  i n aqueous  More r e c e n t l y ,  and McDowell^ m o d i f i e d the Wigner t h e o r y and a p p l i e d i t catalysis  o f the  magnetic s u r f a c e  Harrison t o the  surface  o r t h o - p a r a c o n v e r s i o n i n hydrogen adsorbed on a p a r a at  low p r e s s u r e s  and l i q u i d n i t r o g e n  A f t e r the d i s c o v e r y o f the c a t a l y s e d t o p a r a h y d r o g e n , i t was b e l i e v e d t h a t because ortho-  solution  temperatures.  c o n v e r s i o n of o r t h o the p o t e n t i a l  and parahydrogen were n e a r l y i d e n t i c a l , the s p e c i e s  fields  of  c o u l d not  2  be r e c d i l y s e p a r a t e d .  The f i r s t  i n d i c a t i o n to the c o n t r a r y was  obtainec' by S a n d l e r ^ who showed t h a t  a d s o r p t i o n o f the  p a r a - s p e c i e s on a c t i v a t e d  and T i C ^ at 90°K took  to different  extents.  charcoal  More r e c e n t l y ,  o r t h o - and place  a much more exaggerated  ex7  ample o f t h i s  effect  was demonstrated by Cunningham and J o h n s t o n  i n the i n v e s t i g a t i o n  o f the r a t e  o f o r t h o - t o parahydrogen c o n v e r s i o n  at l i q u i d hydrogen t e m p e r a t u r e s , s a l t impregnated i n f i n e l y n e a r l y independent stantial  d i v i d e d alumina.  a paramagnetic  The c o n v e r s i o n r a t e  o f the c o m p o s i t i o n o f the  l i q u i d phase  f r a c t i o n o f the c o n v e r s i o n , which suggested t h a t  was more s t r o n g l y eratures  u s i n g as a c a t a l y s t  (20.4  was  f o r a suborthohydrogen  adsorbed than parahydrogen at  l i q u i d hydrogen temp-  T h i s view was s u b s e q u e n t l y  confirmed by Cunningham,  K) .  8 Chapin and J o h n s t o n desorptions  , who were able by s u c c e s s i v e a d s o r p t i o n s  to prepare e s s e n t i a l l y  demonstrated t h a t  pure o r t h o h y d r o g e n .  I t was  s t a t e 1=1 were more s t r o n g l y adsorbed In h i s  initial  discussion  a theory of separation f a c t o r s ,  of t h i s  based e s s e n t i a l l y  cases:  (1)  Extreme h i n d r a n c e  c u l e behaved as a t w o - d i m e n s i o n a l  in  the  S a n d l e r ^ proposed  on the i d e a  He c o n s i d e r e d  i n which the  plane r o t a t o r ;  the  (paradeuterium).  subject,  r o t a t i o n was h i n d e r e d f o r adsorbed m o l e c u l e s . limiting  also  o r t h o - and p a r a d e u t e r i u m c o u l d be s e p a r a t e d by  same method and, as i n the case o f hydrogen, the molecules rotational  and  (2)  that two  adsorbed moleA more  extreme  hindrance  o f r o t a t i o n which l e d t o a l o s s o f two r o t a t i o n a l degrees  o f freedom  (vibrator).  The c a l c u l a t e d s e p a r a t i o n f a c t o r f o r o r t h o -  parahydrogen at 9 0 ° K ,  assuming the  first  agreement between t h e o r y and e x p e r i m e n t , the  adsorbent.  theoretical  The second  separation.  separation factor is more s t r i n g e n t  test  l i m i t i n g cases l e d t o  p a r t i c u l a r l y w i t h T i O ^ as  l i m i t i n g case l e d t o a somewhat  Since,  gcod  higher  on such a model, the magnitude  of  determined p r i m a r i l y by a Boltzmann f a c t o r , of the  a p p l i c a b i l i t y of either  be made by comparison o f t h e o r y and experiment  at  the a  l i m i t i n g case can low  temperatures.  7 Cunningham and J o h n s t o n was more than t w i c e as limiting  case was  second was  found t h a t l a r g e as the  assumed,  assumed.  the  observed one when the  and more than t e n times as  Therefore, i t  seemed e v i d e n t ,  a d s o r p t i o n by impregnated a l u m i n a , t h a t was  due t o h i n d e r i n g o f r o t a t i o n ,  sufficiently  great  p r o b a b l y not even  to  calculated separation  l e a d t o the  first  l a r g e when  the  i n the case o f  i f the p r e f e r e n t i a l  the h i n d r a n c e  factor  adsorption  was c e r t a i n l y  not  l o s s o f two degrees o f freedom, and  one.  9 Evett  presented  t i o n ^ o f the d i f f e r e n c e  idealised,  was c o n s i d e r e d , 0 be the  surface  and p a s s i n g  Evett  at low t e m p e r a t u r e s .  model f o r the  r a t h e r than the  Letting  of Sandler's tentative  explana-  i n o r t h o - p a r a c o m p o s i t i o n o f adsorbed hydrogen  compared w i t h the gas phase but s t i l l  a refinement  A more  adsorbed m o l e c u l e  realistic,  on the  l i m i t i n g case o f the p l a n e  angle between the  z axis,  surface rotator.^  drawn p e r p e n d i c u l a r t o  through the c e n t e r o f mass o f the d i a t o m i c  assumed a r o t a t i o n a l h i n d e r i n g p o t e n t i a l  where V ' was the b a r r i e r h e i g h t  to r o t a t i o n .  o f the  form V  Assuming t h a t  the  molecule = V '  this  cos  potential acted on the rotational energies of a rigid rotator and that the rotational wavefunctions could be .written as ( l - £ ) ^ F ( C ) e " ^ 2  m  (when £ = cos8, <b i s the angle betweea some reference plane containing the z axis and a plane containing both the z axis and the molecular axis, and m = 0,1,2 . . . ) ,  (1-S ) ¥L d£ 2  2  arrived at the equation  n e  - 2(m+l) M L d£  [ -n £ 2  +  2  E  -m(m+l)] F = 0,  (1.1)  where A= h /87T I, 2  (1.2)  2  E = (rotational energy) / A, n  2  (1.3)  = V'/A .  (1.4)  Q  This equation i s identical to the spheroidal wave equation considered by Wilson  10  and Stern  opposite sign.  11 2 2 , except that the n £ F term appears with  Wilson's method*^ of expanding F as a power series in  £ was employed and the relationship between n. and E was expressed using continued fractions.  The technique described by S t e r n ^ for  evaluating the roots of these fractions was utilised to obtain the values of E corresponding to various n  2  for the three levels of  interest in the low temperature application.  Separation factors for  hydrogen were calculated and compared with the experimental data of Sandler^ for hydrogen on TiO^, and of Cunningham and Johnston for 7  hydrogen on impregnated alumina.  Evett's results  9  were in much better  5  agreement  with experimental  o f the p l a n e  separation factors  than the  l i m i t i n g case  rotator. 12  White and L a s s e t t r e theoretical  , u s i n g the same method,  separation factors  f o r hydrogen and d e u t e r i u m . was more s t r o n g l y  as a f u n c t i o n o f b a r r i e r h e i g h t , V ^ ,  The t h e o r y p r e d i c t e d t h a t  adsorbed than o r t h o d e u t e r i u m .  the hydrogen s e p a r a t i o n f a c t o r was  l a r g e r at a l l  h e i g h t s than the s e p a r a t i o n f a c t o r f o r d e u t e r i u m .  q u a l i t a t i v e l y w i t h the experiments and observed s e p a r a t i o n f a c t o r s surface  o f Cunningham, et_ al_. .  fact  that  the  agreed  Calculated  f o r hydrogen and d e u t e r i u m (at  might have been due t o the  separation factors  barrier  These r e s u l t s g  coverages) were not i n a c c u r a t e q u a n t i t a t i v e  ever t h i s  orthohydrogen  adsorbed than parahydrogen at a l l b a r r i e r h e i g h t s  and t h a t p a r a d e u t e r i u m was more s t r o n g l y Moreover,  calculated  low  agreement.  How-  experimental  were not determined under comparable  conditions.  13 Evett r e p o r t e d models  l a t e r presented  a refinement  o f the  previously  f o r adsorbed hydrogen by i n c l u d i n g h i g h e r o r d e r terms 14  i n the p o t e n t i a l  .  His results  l e d t o the c o n c l u s i o n t h a t  w i t h d a t a on o r t h o - p a r a h y d r o g e n s e p a r a t i o n c o e f f i c i e n t s  agreement  c o u l d be  o b t a i n e d o n l y by assuming a magnitude f o r the r o t a t i o n a l b a r r i e r which was i n c o n s i s t e n t  w i t h d a t a on a d s o r p t i o n e n e r g i e s .  d i s c r e p a n c y a p e r t u r b i n g term i n v o l v i n g a h i n d r a n c e the p l a n e p a r a l l e l t o the s u r f a c e i n - p l a n e b a r r i e r height) t i o n analysis  showed t h a t  (of the  To e l i m i n a t e to rotation  this in  form Dcos2c{>, where D was  the  was i n t r o d u c e d , and a s e c o n d - o r d e r p e r t u r b a such a term c o u l d l e a d t o agreement  with  6  both separation c o e f f i c i e n t  and a d s o r p t i o n energy d a t a .  I t s h o u l d be noted t h a t the e x p e r i m e n t a l s t u d i e s o f s e p a r a t i o n factors  a l l i n v o l v e systems whicA from the t h e o r i s t ' s  view are e x t r e m e l y q u i t e heterogeneous  complicated.  h i g h so t h a t great  Not o n l y were the s u r f a c e s  ( i n the sense t h a t they e x h i b i t wide  of adsorption energies),  but a l s o the s u r f a c e  i n t e r a c t i o n s between the  importance.  coverages  would be o f  are v e r y d i f f i c u l t  In f a c t ,  to  to  strongly  t h e r e was no a s s u r -  ance t h a t E v e t t ' s h i n d e r e d r o t a t i o n a l model (even though i t give separation factors  involved  i n v o l v e d were  they might be expected  the s e p a r a t i o n c o e f f i c i e n t s .  of  distributions  adsorbate m o l e c u l e s  A l t h o u g h both these e f f e c t s  i n c o r p o r a t e i n t o a workable t h e o r y , influence  point  i n agreement w i t h experiment)  could  provided a  unique e x p l a n a t i o n o f the d a t a , s i n c e  a number o f p o t e n t i a l l y i m p o r t -  ant c o n t r i b u t i o n s were e i t h e r t o t a l l y  i g n o r e d or approximated.  Our i n t e r e s t a c t u a l l y arose out  i n the problem o f h i n d e r e d r o t a t i o n at a s u r f a c e  of quite d i f f e r e n t  have j u s t been d i s c u s s i n g , the type  from those we  and at t h i s p o i n t we s h o u l d d e s c r i b e  o f system we wished t o Constabaris,  considerations  briefly  treat.  Sams and Halsey*^ r e p o r t e d i n 1961 d a t a f o r  the  a d s o r p t i o n o f hydrogen and deuterium on g r a p h i t i c carbon i n the H e n r y ' s law r e g i o n . simplicity  The importance o f these measurements o f t h e system.  v a r i o u s l y as P33 (2700) energetically  Firstly,  is  i n the  great  the p a r t i c u l a r s u r f a c e used  and S t e r l i n g FT (2700)] i s perhaps the  homogeneous  surface  [known most  f o r which a d s o r p t i o n measurements  7  are a v a i l a b l e . actions  Secondly,  since  the isotherms  between adsorbate molecules  d e a l i n g o n l y w i t h the i n t e r a c t i o n s the s u r f a c e . classical  are t o t a l l y n e g l i g i b l e ,  model which c o n t a i n e d the  observed temperature dependence  called capacity factor bent  and Z  Q  the  (where  gas-surface  with  the  two-parameter model w i t h  o f the H e n r y ' s law c o n s t a n t ,  o b t a i n e d e s t i m a t e s o f the g a s - s u r f a c e  i n t e r a c t i o n energy is  is  of a  i m p l i c i t assumption t h a t  By comparing t h i s  inter-  and one  o f i n d i v i d u a l gas m o l e c u l e s  The d a t a were o r i g i n a l l y a n a l y s e d ^ i n terms  adsorbate was monatomic. the  obey H e n r y ' s law,  the s u r f a c e  c o l l i s i o n diameter).  one  and a s o -  a r e a o f the  A quantum  adsor-  statistical  16 17 treatment  o f the  adsorbate)  data  '  ( r e t a i n i n g the  was s u b s e q u e n t l y  centage energy d i f f e r e n c e  c a r r i e d out.  assumption o f a monatomic I t was found t h a t  the  per-  between i s o t o p e s was much l a r g e r than  the  18 corresponding p o l a r i z a b i l i t y difference expected felt  that  o f the  on the b a s i s this  o f the  , and a l s o much l a r g e r than  gas-phase i n t e r a c t i o n e n e r g i e s .  It  d i s c r e p a n c y might be due i n l a r g e p a r t t o the  i n t e r n a l degrees o f freedom o f the  was  neglect  adsorbate molecules  in  the  19 configurational  integral.  Moreover, Meyers and P r a u s n i t z  a n a l y s e d H e n r y ' s law a d s o r p t i o n d a t a f o r s e v e r a l gases on the same s u r f a c e  i n terms  had s u g g e s t e d t h e r e was s i g n i f i c a n t of  simple  had  polyatomic  of a s i m i l a r ( c l a s s i c a l )  model, and  h i n d r a n c e o f the r o t a t i o n a l  degrees  freedom. S i n c e two o f the major c o m p l i c a t i o n s which were i g n o r e d i n  E v e t t ' s model were i n f a c t  absent from the system we wished t o  treat,  8  i t seemed that a s i m i l a r model coulc be used here. 1  calculations were not suitable f o r ou." purposes. was  However, Evett's His main i n t e r e s t  i n low-temperature separation f a c t o r s , and so i t had been  f o r him to c a l c u l a t e only the lowest few states. Constabaris,  et a l , ^ ,  necessary  The data of  on the other hand, were obtained  at f a i r l y high  temperatures (90° - 140°K) and many more states would be required to treat the problem adequately.  It was  therefore necessary  to extend  the c a l c u l a t i o n s to include a larger number of eigenstates, and a v a r i a t i o n a l technique seemed a t t r a c t i v e . It was  also c l e a r that the thermodynamic functions of the  adsorbed phase, i n p a r t i c u l a r the heat capacity and entropy, should quite markedly affected by r o t a t i o n a l hindrance,  be  and i t would be of  i n t e r e s t to examine these. The  following two models f o r hindered  rotation at a surface  have been investigated : 1. The  centre of mass of the admolecule i s placed at some equi-  librium distance from the surface, which i s assumed to be a planar continuum.  We then define an angle 6 as the angle between the z axis  (which i s normal to the surface and passes through the center of mass of the diatomic molecule) and the molecular  axis.  The cb angle  can  be defined as the angle between some reference plane passing through the z axis and a plane passing through the z axis and the axis.  For t h i s model of a planar continuum surface, we  molecular  assume there  i s a r o t a t i o n a l hindering p o t e n t i a l of the form V  = V  1  cos 9, 2  (1.5)  where V  1  is  the b a r r i e r h e i g h t  t o r o t a t i o n and depends  on b o t h the  d i a t o m i c adsorbed molecule and the a d s o r b e n t .  This p o t e n t i a l  t h a t the molecule has  is  a minimum energy when i t  i n the x - y p l a n e ,  which runs t h r o u g h the c e n t e r o f mass o f the m o l e c u l e , energy when the molecule i s 2. The s u r f a c e i s  s t a n d i n g u p - r i g h t on the  affect  spacing i s  r e p r e s e n t e d by a square l a t t i c e  The  admolecule.  The angle  cf) i s now  z axis  l i n e j o i n i n g the two n e a r e s t  When <J> = 0 ° the m o l e c u l e has a minimum h i n d r a n c e a maximum.  i n g p o t e n t i a l o f the  These c o n d i t i o n s  lattice  sites.  and when cf) = 9 0 °  are met f o r a ctp-hinder-  form W  1  W' = — "(1 - cos2<M, 2  (1.6)  where W ' i s  the i n - p l a n e b a r r i e r h e i g h t t o r o t a t i o n .  out-of-plane  is  is  used.  assumed t o be s t i l l p r e s e n t  The h i n d r a n c e  and the p o t e n t i a l i n (1.5)  F u r t h e r j u s t i f i c a t i o n f o r t h i s model w i l l be g i v e n  later.  Chapter II  of  consists  o f the t h e o r e t i c a l development  v a r i a t i o n a l e q u a t i o n s which are used t o f i n d the e i g e n v a l u e s two models. presented  and  and a p l a n e p a s s i n g through the z a x i s and  p e r p e n d i c u l a r t o the  is  the  sites.  angle between a p l a n e p a s s i n g through the  the m o l e c u l a r a x i s ,  the h i n d r a n c e  with  assumed t o be l a r g e enough so t h a t o n l y two s i t e s  the i n - p l a n e r o t a t i o n o f the  d e f i n e d as the  and maximum  surface.  d i a t o m i c molecule s i t t i n g h a l f - w a y between two l a t t i c e lattice  implies  A brief first.  review o f the t h e o r y o f the f r e e  rotator  The o u t - o f - p l a n e h i n d e r i n g p o t e n t i a l i s  the  f o r these is  treated  as  a p e r t u r b a t i o n on the f r e e r o t a t o r e n e r g i e s .  These new  eigenvalues  are then p e r t u r b e d by the i n - p l a n e h i n d e r i n g p o t e n t i a l g i v e n by equation  (J.6). In  equations  Chapter  I I I , the quantum s t a t i s t i c a l  thermodynamic  f o r the r o t a t i o n a l heat c a p a c i t i e s and e n t r o p i e s are p r e -  s e n t e d f o r the f r e e r o t a t o r , the o u t - o f - p l a n e h i n d e r e d r o t a t o r , the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t o r .  and  These e q u a t i o n s  are  then m o d i f i e d f o r the d i f f e r e n t n u c l e a r s p i n s p e c i e s o f hydrogen  and  deuterium.  The  e i g e n v a l u e s and e i g e n f u n c t i o n s f o r the  out-of-plane  h i n d e r e d r o t a t o r , and the e i g e n v a l u e s f o r the o u t - o f - p l a n e and i n p l a n e h i n d e r e d r o t a t o r are d i s c u s s e d i n Chapter heat  The  rotational  c a p a c i t i e s and e n t r o p i e s f o r the d i f f e r e n t n u c l e a r s p i n s p e c i e s  o f hydrogen and deuterium for  IV.  are p r e s e n t e d  and d i s c u s s e d i n Chapter  the f r e e r o t a t o r and t h e two h i n d e r e d r o t a t o r models.  i s devoted  V  Chapter  VI  t o a b r i e f r e - c o n s i d e r a t i o n o f s e p a r a t i o n f a c t o r s f o r the  spin-isomers  o f hydrogen and deuterium.  Henry's law d a t a f o r H^  and D  2  F i n a l l y i n Chapter VII  the  on graphite**^ are a n a l y s e d i n terms o f  a quantum s t a t i s t i c a l model i n c o r p o r a t i n g h i n d e r e d  rotations.  11  .  CHAPTER I I .  DEVELOPMENT OF EIGENVALUE EQUATIONS 20 21  A) The Free R o t a t o r  ' .  The H a m i l t o n i a n o p e r a t o r o f a r i g i d  r o t a t o r can be  w r i t t e n i n s p h e r i c a l c o o r d i n a t e s as h  1 — r  2  8TT M 2  3 3 . — .(r — ) + 3r 3r f  2  1  I  — r  2  2  3 , . 3 ' — (smO — ) 36 30 0  sin9  +  r  where h i s P l a n c k ' s some p o t e n t i a l .  sin 0  2  V(r,0,(j)),  (2.1)  3tJ>'  2  c o n s t a n t , M i s t h e m o l e c u l a r mass and V(r,0,cf>) i s  Assuming no f o r c e s a r e a c t i n g on t h e r o t a t o r ,  V(r,0,cj)) = 0, and p u t t i n g r = 1, M r  = M = I , where I i s t h e moment  2  o f i n e r t i a o f t h e d i a t o m i c m o l e c u l e , we f i n d t h a t S c h r o d i n g e r ' s equation i s  1  3  , .  —  sine  C  0  sin8  39  3¥,  —)  1  3 y  87T Ie  2  2  +  +  39  sin 9 2  where e i s t h e energy  3<j>  h  2  „,  .  ¥ = 0,  (2.2)  2  o f the free r o t a t o r .  The wavefunction  ^(O,^)  can be w r i t t e n as  * (e,<M  (2.3)  *(<M,  = Q(0)  and s u b s t i t u t i o n o f (2.3) f o r ¥(0,<|>) i n t o  3_  9  39  (  s  i  n  Q  3 0  39  )  +  Srrile  h  2  ^  =  _ 1  (2.2) y i e l d s  3 ^  $ d<p  ^ 2  ^  12  Since both sides must be equal to a constant designated as m , 2  (2.4) yields two d i f f e r e n t i a l equations : d  $(cb)  2  d  m $(<J>),  (2.5)  2  cb  2  and —3 sin9 39  . 30— sm9 30 fl  m sin'  0  + £(£ + 1)0  where 8Tr Ie/h has been replaced by £(£ + 1) 2  2  =  0,  (2.6)  Equation (2.5) has the  normalised solutions imcf> $ (cb)  =  (2.7)  m = 0, ±1, ±2,  /2?  m  and equation (2.6) has acceptable solutions only for integral values of 1,1 > |m|.  The normalised solutions for the 0(8) equation are  the normalised associated Legendre polynomials 21, + 1 2  1/2 (£ - m) ! ,m (£ + m) !  PJ  (2.8)  (cos 9).  The associated Legendre polynomials,  (cos0), are  defined as m P? (?) = (1 - S ) '2 2  3„m '"  m  "{P-CO)  (2.9)  13  where E, = c o s 6 . defined  P CC) £  The P^t?) are j u s t the Legendre p o l y n o m i a l s  by the s e r i e s  =  ^  N  |  expansion  C - D  0  W  N  I Si  J  5  w  n  ,  (2.10)  where  (S and that  —— »  =  [— ] means the l a r g e s t i n t e g e r I must be a p o s i t i v e i n t e g e r  e = — —  l{l + 1),  ( 2  which i s < - j . implies  that  -  n )  The r e s t r i c t i o n only the energies  A = 0, 1, 2 ...,  (2.12)  8TT I 2  are  allowed f o r the r i g i d  degenerate s i n c e function  rotator.  f o r any £, m =  These e n e r g i e s are 2£ + 1 -£ + 1, . . . i t .  The t o t a l wave-  i s o f course j u s t  where t h e Y„ (9,cb) f u n c t i o n s £,m v  a r e the s p h e r i c a l harmonics. ^  These  s p h e r i c a l harmonics are o r t h o n o r m a l , w i t h  2u  TT Y  *r,m- ce,«  Y,, te,« s i n e a e d * = 6 , m  U  (2.14)  14  or i n Dirac notation, <Y  V,m'  I£,m  ^  Y  (.M5)  < ' « > - « . mm' e  6  u  The 6  ' s and 6 ,'s are Kronecker 6's. (For convenience Dirac ram' notation w i l l be used throughout the rest of this thesis.) Equation nni ii'  (2.15) implies that the 0. (0) functions and $ (<J>) functions are 36 y m in also orthonormal:  I m  m*  mm!  (2.16)  and < r,m^l £,m^> 0  0  =  ii'  5  (2.17)  The Hamiltonian operator i n (2.1), with V(r, 0,<j)) = 0 and Mr = M = I , can actually be s p l i t up into two operators i n the m 2  subspace: (2.18)  =^n,cpm+--n , 0m' Q  m  where <pm  3 8TT I 2  sin (  Z  2  3<j>  (2.19)  +m  2  and 0m  . 3— —1 9 — sinfl sin0 30 30 0  8TT I 2  irr  sin  (2.20)  15  Operating (2.18) and (2.19) on  $ (40  and 0^ ^(8) respectively,  m  yields  . i i .(bm $m(cb) -' = 0, '  .  v T  v  (2.21) J  and ii  . '(6), 8m 0.£,m(8) = £ 8Jc,nr Q  vJ  J  l{l + 1) 0  *  h  8TT I 2  (8).  O £  '  (2.22)  M  ThusJ'l^operated on $ (<M does not contribute to the energy. For m  the t o t a l wavefunction  '  8ir  I  '  B) The Out-of-Plane Hindered Rotator. We consider a system i n which diatomic molecules are physically adsorbed on an adsorbent which hinders the out-of-plane rotation of the adsorbed molecules.  The adsorption can be either mobile or localised.  It i s assumed that the adsorbed molecules do not interact with each other (Henry's law i s obeyed) so that one can consider the behavior of an individual adsorbed molecule. The adsorbent i s taken to be a planar continuum with the adsorbed diatomic at an equilibrium distance z  from the surface (adsorbent).  This equilibrium distance must be  16  greater than the length of the admolecule no hindrance,  so that at the l i m i t o f  the molecule becomes a free rotator.  The angle 0 i s  defined as the angle between the z axis, which i s perpendicular to the surface and passes through the center of mass of the diatomic, and the molecular axis o f the diatomic.  There exists a molecule-  adsorbent hindering p o t e n t i a l o f the form  V  1  = Xcos 0,  (2.24)  2  where X i s the b a r r i e r height to out-of-plane r o t a t i o n .  This p o t e n t i a l  goes through a maximum when 0 = 0°, i . e . when the molecular axis i s p a r a l l e l to the z axis, and through a minimum when 0 = 90°, i . e . when the molecular axis i s p a r a l l e l to the surface.  The cf) r o t a t i o n i s not  hindered by the continuum and so the cf> angle can be defined as the angle between some reference plane passing through the z axis and a plane passing through the z axis and the molecular axis. In order to obtain the r o t a t i o n a l eigenvalues o f an adsorbed diatomic molecule hindered by the p o t e n t i a l given i n (2.24), i t i s necessary to solve the Schrodinger equation  -X*  = E<j)  (2.25)  where ~K =A  + V  (2.26)  Equation (2.26) can be solved by using Rayleigh-Schrodinger Perturbation Theory  22  or Variation  20 21 ' . Since we are interested  17  in  a l a r g e range o f b a r r i e r h e i g h t s ,  s o l u t i o n i s used.  The v a r i a t i o n a l wavefunction ^ ( 8 A) > where k  d e s i g n a t e s the s t a t e wanted, the  the v a r i a t i o n a l method o f  i s taken t o be a l i n e a r combination o f  f r e e r o t a t o r orthonormal e i g e n f u n c t i o n s  I  =  ml*  Y  *.iu  C f f  »  W  C  m  (8,cb),  l,m,k'  ( 2  where C , are the expansion c o e f f i c i e n t s . x> ,m, K 0  '  2 7 )  E q u a t i o n (2.25) can  then be r e w r i t t e n i n terms o f a m a t r i x e q u a t i o n :  (C  (is  W ) C = C E.  +  (2.28)  a d i a g o n a l m a t r i x w i t h the e i g e n v a l u e s  elements  £ =  :  •  {£„„>= {—  HI + l)}  S i n c e these e n e r g i e s are degenerate  '  •  o f the f r e e r o t a t o r as .-  .  (2.29)  i n the quantum number m, £  can  a l s o be w r i t t e n as  {  %m;*,m  A. i n e q u a t i o n  }  =  { ^  (2.28)  (2.30)  i s the o u t - o f - p l a n e b a r r i e r h e i g h t  V '.  V  is  the p e r t u r b a t i o n m a t r i x g i v e n by  V  =  { ic,m;r,m'> > V  ^  18  wher*  -  m = -£, -l+l, .% ~~  1^  m = ^m-A'.m  1 i s  £,m;i',m'  V  £  =  < Y  g i v e n  i n  t h e  £' ,m' (><M 0  +£,  «•• ^  -£'+1,  1  a n d  2^  ...,  Y  l,m^ '^ Q  I  c o s 2 e  +£',  b  I£  a  s  Y  i  s  (2.32)  a s  C ^) > •  (2.33)  9  ; m  i s a matrix of eigenvectors given by  C and E  = {  W  }  '  (2  -  34)  are the rotational eigenvalues of the out-of-plane hindered  rotator, represented as a diagonal matrix: E  = {E > . kjk  .  (2.35)  Once the perturbation matrix \J i s calculated, the matrix equation (2.28) can be solved to f i n d C , E l and cj)^. To calculate the V„  „, , matrix element given i n (2.33), JC J in y & y m  i t i s convenient to express the perturbation potential cos 0 i n terms 2  of spherical harmonics.  (0,<j>) can be written as  m  19  1/2 Y  (2ft + 1) (&-m)!  (6,<M =  0  4ir (il+m)!  , 1 2 l\ £  e  iracb  , . (-sin )  18, +m (2.36)  (cos 8- 1) 2  d(cos8)  Y  2 o^  ,<P  ^ from this formula i s given by 1/2 (2.37)  or 1/2 (2.38)  cos Also l e t t i n g £ = 0 and m = 0 i n (2.36), one gets 1/2 • o-,o »« Y  (e  =  i  (2.39)  ©  Therefore, substituting (2.39) into (2.38), the perturbation potential can be written completely i n terms of spherical harmonics as  cos 0 = z  2 ,^1/2  £ (TT)  1  2 y  1  /  2  Y ce,« + Y ce,« 2j0  (2.40)  0>0  Using this expression for the potential i n (2.33), the matrix element  20  V„ „, . becomes £,m £* ,m' ;  1/2 V  A,m;r,m.  ! ™ ^ < i - ' ,m'< .«  Y  0 ) 0  (e,cM  ^ I.,m^ '^l &) 1/2<Y  -  I  !^  +  For  1 / 2  e  <Y;  i f n l  2  (e,«|Y  0 > 0  1/2 Y  2.,oC -«|Y e  A-in  Ce,«>  ce,*)|Y . (e « A  f m  (2.41)  f  the e v a l u a t i o n o f the i n t e g r a l o f three s p h e r i c a l harmonics, t h e  f o l l o w i n g f o r m u l a can be used:  < Y  Y^Ce.M •  2 $  6  Y  =  23  * £ m ( ««l o ( ><»|Y » C6,*)> *3* 3 *2, 2 1, 1 6  Y  Q  m  (2£ +1)(2H +1) 1  1/2  2  m  4TT (2JI +1)  C ( £ £ £ ; m m m ) x C (£ i JZ ;000) , 1  2  3  1  2  (2.42)  3  where t h e C(_Z^l l^;ra^m^m^) a r e t h e C l e b s c h - G o r d a n ( o r Wigner) 2  23 24 coefficients.  They can be e x p r e s s e d , u s i n g Wigner's  '  closed  e x p r e s s i o n , as C(£ £ £_;m m m_) = 6 ^ 1 2 3 1 2 3"^ m_,m..+m, J 1 t. 1  0  1  0  1 1/2 (2£ +.l) ( A + i l - i l ) ! (£ -£ +A ) ! ( i ^ + ^ - J l ) ! (^ +m ) ! C^-n^) 3  3  1  2  3  2  3  (£ +£ +£ +l)!(£ -m )!(£ +m )!(A -m 1  2  3  1  1  1  2  3  )! (£ +m )! 2  2  21  x  Z  (-1)  V  V!  (£ +£ +m -v)! 2  3  1  (£ -m +v)!  (2.43)  1  (A - J ^ + J ^ - v ) ! (Aj+m -v)!  (v+£ -£ -m )! 1  2  3  The index v assumes a l l integer values such that none of the f a c t o r i a l arguments are negative.  In fact 1/(-n)! = 0 f o r n a p o s i t i v e  integer.  A l l the arguments of the f a c t o r i a l s are integers. The Clebsch-Gordan following  s e l e c t i o n rules  C(£ £ £ ;m m m ) = 0 1  2  3  1  2  3  c o e f f i c i e n t C{l^L^L^;m.jm m ) has the 2  3  23 :  unless  1. m  = mym^,  3  (2.44a)  • • • »  (2.44b)  The second s e l e c t i o n rule f o r t h i s c o e f f i c i e n t implies £  3  that A ^ , ^ and  must form a triangle (thus the notation ktyyJL^Z )) . The Clebsch-  Gordan c o e f f i c i e n t C(&j£ Jl ;000) i s c a l l e d the p a r i t y c o e f f i c i e n t  and  i t has the s e l e c t i o n rule  (2.45)  Using (2.42) i n (2.41), the perturbation matrix element, V.  „,  ,, can now be written  as  -  22  1/2 V = - f-l £,m;;V ,m' 3 <>5 v  -i  (2£+l)5  1/2 C(£2£' j m o m ' K ^ r ;000)  J  4TT (2£« + l)  1/2  2 , ,1/2 2£+l +  3  ( 7 r )  4TT(2£'+1)  C(£o£';mom') C(£o£';000). (2.46)  For the C(£2£';mom') coefficient, the selection rules given by (2.44a) and (2.44b) give the following allowed values of £,£' ,m and m' (with £ =£,£ =2,£ =£'; 1  2  m^m, m =0, m=m'): from (2.44a)  3  2  3  m' = m + 0 = m,  (2.47a)  £» = £+2, £+!,£, £-1, £-2.  (2.47b)  and from (2.44b)  Using the selection rule of the parity coefficient C(^^^jOOO) given in (2.45), £+2+£' must be even for the coefficient C(£2£*;000) not to vanish.  This implies that £ and £' must be both even integers  or both odd integers (i.e. (-1) = (-1)^'), so that the £' = £ + 1 5,  and £' = £-1 values given in (2.47b) are not allowed.  Therefore,  for C(£2£';mom') C(£2£*;000) we have the following selection rules:  C(£2£';mom*) C(£2£';000) = 0 unless 1. m'=m, 2. £'=£+2,£,£-2, 3.  c-ir = (-ir .  (2.48a) (2.48b) (2.48c)  23  The second term in (2.46) contains the two coefficients C(£o£';mom') and C(JcoJL ;000) . For C(ftoil' ;mom') , the triangle selection rule in 1  (2.44b) implies that l=V and the selection rule in (2.44a) gives m=m'.  For C(£o£';000), the parity coefficient selection rule in  (2.45) implies that l+V must be even or (-1)  = (-1)  . Therefore,  for C(loV ;mom') C(£o£';000) we have that C(loV ;mom') C(£oJo';000) = 0 unless  1.  m'=m,  (2.49a)  2. V=SL,  (2.49b)  3. (-1)* = (-I)"  (2.49c)  1  Using the selection rules (2.48a) to (2.49c), the matrix element VJdjinjx . , jin, can have the following values : 0  0  2&+1  V £,m;£'m'  2V  2ft+l 2£' + l  +  2<L+1 2V  1/2  +1  i f \l-V\  1 3  n  (2.50a)  C(£2£' ;mom') C(Z2V ;000) ,  = 2, m=m'  and (-1) = (-1)^', £  1/2 C(12V 1/2  ;mom') C(£2£ ;000) ,  C(£o£';mom») C(£o£';000) ,  +1  i f l=V , m=m», and (-1) = (-1) ' £  0 a l l other cases, i.e. m/m', and V  f- 1+2,  I,  From the allowed values of V.  1-2.  ..,  (2.50b)  £  (-1)  f (-1)  , (2.50c)  , i t is apparent that m is s t i l l  a good quantum number, for only states with the same m value will be  24  connected by the perturbation matrix V .  However the V matrix can  connect states with t h e i r £ quantum numbers d i f f e r i n g by ±2 (i.e.  £-£ |=2). as well as states with the same £ value. ,  Therefore  £ i s no longer a good quantum number f o r the designation of states. Since £ must equal £' or |£-£'|=2, only states of the same p a r i t y p = (-1)^ w i l l be connected by V •  Thus p i s a good quantum number  and can be used to characterise states. and f o r odd £ values p = -1.  For even £ values p = +1  Both m and p are said to be sharp  because the^V matrix connects only states of the same m and p. Using the sharpness of m and p, the matrix equation given by (2.28) can be rewritten as a set of matrix equations :  (2.51) where  (2.52)  (2.53)  c  with £ e {p},  (2.54)  and  E  (2.55)  25  in diagonal representation.  The r o t a t i o n I e {p} means a l l i v a l u e s  of p a r i t y p .  The a c t u a l o u t - o f - p l a n e h i n d e r e d r o t a t i o n a l  are s p e c i f i e d  as E^'' ^  this  state,  for state {k,'p,m)}.  m  using equation  (2.27)  eigenvalues  The e i g e n f u n c t i o n  and the sharpness  of  o f p and m,  is  given by  -  (P,m)  (2.56)  £,m,k  a e {p} The method o f s o l v i n g the s e t  o f matrix equations  g i v e n by (2.51)  is  o u t l i n e d i n Appendix A .  C) The O u t - o f - P l a n e and I n - P l a n e H i n d e r e d R o t a t o r . The model f o r the a d s o r p t i o n o f a d i a t o m i c m o l e c u l e on a surface  is  now taken as one o f s t r i c t l y l o c a l i s e d  surface is is  r e p r e s e n t e d by a square l a t t i c e  l a r g e compared t o the  two n e a r e s t molecule.  l e n g t h o f the  neighbours w i l l The m o l e c u l e i s  affect  i n the  assumed t o s i t  of the m o l e c u l e ,  z axis,  h a l f - w a y between two  only diatomic  nearest  (with the same r e s t r i c t i o n s from the s u r f a c e .  and the m o l e c u l a r a x i s .  The <J> angle  l i n e j o i n i n g the two n e a r e s t  z axis,  neighbour l a t t i c e  as  The angle 8 i s  which passes through the c e n t e r o f mass  the angle between a p l a n e p a s s i n g through the to the  s p a c i n g which  the i n - p l a n e r o t a t i o n o f the  o u t - o f - p l a n e model)  the angle between the  with a l a t t i c e  The  adsorbed m o l e c u l e so t h a t  n e i g h b o u r s a t an e q u i l i b r i u m d i s t a n c e discussed  adsorption.  is  now taken as  at r i g h t sites,  angles  and a  26  p l a n e p a s s i n g through the  z axis  m d the d i a t o m i c a x i s .  The admolecule  w i l l have a minimum energy when cf) = 0 ° and 1 8 0 ° and a maximum energy when cj> = 9 0 ° and 2 7 0 ° . o f the  These c o n d i t i o n s  are s a t i s f i e d  by a p o t e n t i a l  form W '  W  =  -y-  (l-cos2<f>) = u(l-cos2c{)),  where u(= W ' / 2 ) o  is  the i n - p l a n e b a r r i e r h e i g h t  of-plane hindering potential as t h a t g i v e n by e q u a t i o n not s t r i c t l y a d d i t i v e , o f the p o t e n t i a l s  i.e.  W'  i.e.  changes  and V  q  is  to r o t a t i o n .  Although these two p o t e n t i a l s  the  form o f the o t h e r .  W  and V ' by mean f o r c e p o t e n t i a l s  1  Thus they  o f the W = - ~  neither  are  q  The t h e o r y o f the e i g e n v a l u e a more g e n e r a l p o t e n t i a l  are  and A= f(cp), we assume t h a t  u= f(8)  a r e averaged over a l l s p a t i a l  1  The o u t -  again assumed to be o f the same form  (2.24).  s e p a r a b l e i f one r e p l a c e s V  (2.57)  and  W' O  configurations.  e q u a t i o n s w i l l be developed  for  form  1  W  where a i s  an i n t e g e r  F o r example,  (1-cos acp),  and d e s c r i b e s  the p e r i o d i c i t y o f the h i n d r a n c e  i f the d i a t o m i c m o l e c u l e i s  square l a t t i c e  the i n t e g e r  l o c a l i s e d i n the c e n t e r o f  a w i l l have the v a l u e a = 4.  the a c t u a l s o l u t i o n o f the e i g e n v a l u e a p p l i c a t i o n o f the o u t - o f - p l a n e  (2.58)  m a t r i x equations  However, and i n  exhibits  T h i s w i l l be d i s c u s s e d  in  the  and i n - p l a n e h i n d e r e d r o t a t i o n a l model  t o hydrogen and d e u t e r i u m , the p o t e n t i a l g i v e n i n e q u a t i o n be used because i t  the  the maximum h i n d e r e d r o t a t i o n a l  i n Chapter IV.  (2.57)  will  effects.  27  The H a m i l t o n i a n  f o r the  r o t a t i o n a l model c a n be w r i t t e n  =X+  y^' or s u b s t i t u t i n g  (2.59)  W ,  =X. = A  is  rotational  and  (2.26)  and  y =  W ' o 2  (2.60)  (2.61a)  .  (2.61b)  Schrodinger equation  *(6,<()) = £UQA),  (2.62)  t a k e n t o be a l i n e a r c o m b i n a t i o n o f t h e eigenfunctions  and £ . i s  the  £ 4> (e,4oa w  out-of-plane  out-of-plane  The e i g e n f u n c t i o n s  J  analogous  (2.59),  2  V •,  index j  into  + X c o s 8 + y ( l - c o s a<f>),  lji.ce,*) =  which i s  (2.59)  y ( i - c o s at}>)  +  X =  hindered r o t a t i o n a l energy.  where the  .  (2.24)  where  yj  where,  and i n - p l a n e h i n d e r e d  as  and t h e n  One m u s t now s o l v e t h e  out-of-plane  v  and  hindered  in-plane  can be w r i t t e n  .  as  (2.63)  k  designates the i n form t o the  state.  The v a r i a t i o n a l m a t r i x  out-of-plane matrix equation,  equation, i s given  by  (E  +  uW)<x=  a£ •  <- ) 2 64  28  E art the out-of-plane diagonal  eigenvalues  f o r some X v a l u e .  This matrix i s  and o f t h e form  JL  =  i\  >  k  }  =  {E >-  (2-65)  k  The p e r t u r b a t i o n m a t r i x V V i n t h e <j>, (6,c}>) b a s i s  • W  =  (  w k ) k  .>  i s given by  >  (2.66)  where  W ^ , k  k  = <cb* (0,4.) I  | l - c o s a.<f>| cj> (9,<M> .  (2,67)  k  Cy i s a m a t r i x o f e i g e n v e c t o r s  d  =U  with  } ,  (2.68)  and £ i s a d i a g o n a l m a t r i x w i t h t h e o u t - o f - p l a n e r o t a t i o n a l eigenvalues  8  and i n - p l a n e hindered  as e l e m e n t s ;  = { £ , J >J  J  >•  (2.69)  We c a l c u l a t e t h e p e r t u r b a t i o n m a t r i x V / i n t h e  (6,cf>) b a s i s f o r  m  c o n v e n i e n c e a n d t h e n t r a n s f o r m i t t o t h e <t>(9,<j>) b a s i s b y t h e t r a n s k  formation  i  W  =C W*C • +  . (2.70)  29  The C are the out-of-plane hindered rotational eigenvectors for some i s the transpose of C  X value and are given i n equation (2.34) and  V>/*  i  s  the perturbation matrix i n the  m  (9,cb)  basis and i s given  by  where 5/ — 0 y l ^ 2 j  m = -I, -l+l, ..., I,  m' = -V , -V + l,  (2.72)  and W  tm;r,m- =  O.*) |l-cos a ^ Y ^ C e , * ) > .  (2.73)  Using the property Y  the matrix element W„  l,J >V  = l,JV  Q  „,  Q  *>).,  (2.74)  . can be rewritten as  W„ „, ,m', = A. ., , x m,m'' B ' Jt\,m;£ ft.m^'m l  1  v  (2.75) J  where \,m;£.,m' £ . , m ' ^ V = <0  (6)  > '  .  ^  30  and B  m.m'  = <$*, (cb) I1-cos a<b|$ (cb)> . m' m .  ,  J 1  1  v  J  (2.77)  v  J  Employing the d e f i n i t i o n o f c o s i n e  ix  -ix  cos x =  and  — 2  l e t t i n g x = acb the p o t e n t i a l  1-cos acb = \  S i n c e the f u n c t i o n ^C*)  (2-e  m  m  1-cos acb can be w r i t t e n as  i a < P  -  e" *) .  (2.79)  i a  i s g i v e n by  $ fdo the cb i n t e g r a l B  (2.78)  (2.7)  as  i m ( p e  , , a f t e r s u b s t i t u t i o n o f (2.7) and (2.79) i n t o  (2.77), becomes _ B ..= m,m'  *  1 , im'cb) i l , iacb -iack i imcb •=— < (e hr(2-e - e ) e >. 2TT ^ 2 f  T J  Y  1  V  1  v  ,,, ^ (2.80) •  T h i s i n t e g r a l has the f o l l o w i n g v a l u e s  B  m,m'  , = = = 0  1 ~  i f  m=m'. '  i f |m-m'| = a, a l l o t h e r cases.  (2.81)  31  T h e r e f o r e the p e r t u r b a t i o n m a t r i x element W.  , can be g e n e r a l i s e d  to  Vm;r,m-  Jc.,*«  =  6  • =  < 0  0  ''  f  ffi=m  r,m.^ £,m^ 0  l — ' l => a  > i f  '  a l l other c a s e s .  (2.82)  quantum number m i s no l o n g e r a good one because the V / m a t r i x  The  connects is  4  i  s t a t e s w i t h m=m' and s t a t e s w i t h  |m-m'| = a. However, i f a  an even i n t e g e r , i . e . we w i l l use a = 2, o n l y s t a t e s w i t h t h e same  m p a r i t y w i l l be connected.  H  As i n the o u t - o f - p l a n e case where one  m  d e f i n e s p = (-1) , r = (-1) w i l l be a good quantum number f o r the designation o f states.  Thus f o r the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d  r o t a t o r , % and m b o t h loose t h e i r sharpness and are r e p l a c e d by p and r. Using t h i s p and r p a r i t y the e i g e n v a l u e m a t r i x e q u a t i o n g i v e n i n  (2.64)  can be r e w r i t t e n as f o u r s e t s o f equations,  transformed  (t  C p  '  r ) +  t  pw  by e q u a t i o n  ( p , r )  ) a  ( p  '  r )  afterV/  has  been  (2.70): = a >  r  )  e  ( p  '  r )  ,  (2.83)  with p = +1, r = +1,  f  p = +1, r = -1, p = -1, r = -1, p = -1, r = +1,  (2.84)  32  and where (P,r)  ;  (P>r)  (2.85)  w or  w  (2.86a)  (P>r) _  c ' KS;?, .}c  (p  with l,V  e {r},  +(p r)  ,r)  m  e {p} and m,m'  (P>r)  (2.86b)  (2.87)  and (P,r)  -(P,r)  _  (P,r)  For state [ j ( p , r ) ] with energy £ ^ ' \ T  k or i n the Y  0  (2.88)  the eigenfunction i s given by  'k,j  me{r}  (2.89)  (6,cb) basis as  ^P' (e,(b) r )  e  k  £e{p}  -  c  me{r}  » - ^ k,j s " '£,m,k (  p  (2.90)  The method of solving the general matrix equation (2.64) and the matrix equations using the p and r sharpness given i n (2.83) i s outlined i n Appendix B.  *•  33  CHAPTER I I I .  QUANTUM STATISTICAL ROTATIONAL THERMODYNAMICS  In t h i s the  c h a p t e r the quantum s t a t i s t i c a l  free r o t a t o r ,  plane  the o u t - o f - p l a n e  thermodynamics  h i n d e r e d r o t a t o r and the  and i n - p l a n e h i n d e r e d r o t a t o r w i l l be d e v e l o p e d .  r o t a t i o n a l heat c a p a c i t i e s we .are i n t e r e s t e d quantities.  and e n t r o p i e s  i n the e f f e c t s  out-of-  Only  w i l l be c o n s i d e r e d ,  the since  t h a t h i n d e r e d r o t a t i o n has on these  We assume t h a t the d i a t o m i c molecules  w i t h each o t h e r so t h a t the t o t a l  do not  interact  r o t a t i o n a l p a r t i t i o n f u n c t i o n can  be c o n s i d e r e d as a p r o d u c t o f N m o l e c u l a r p a r t i t i o n f u n c t i o n s , N is  the number o f  of.  where  molecules.  A) The F r e e R o t a t o r . The m o l e c u l a r p a r t i t i o n f u n c t i o n o f a f r e e r o t a t o r can be w r i t t e n  as  l  f  r  =  I' ( 2 U 1 ) £=o  e"  B e  £  ,  (3.1)  the volume o f the system and  degenerate e i g e n v a l u e s f o r the £ t h r o t a t i o n a l l e v e l t h e s e are g i v e n by e q u a t i o n  c h a r a c t e r i s t i c temperature, written  r  25  where 3 =1/^.1, V i s  rotator;  q£ CV,T)  0^ =  n  (2.12).  /8Tr Ik, 2  are the 21+1 o f the  free  Introducing a r o t a t i o n a l  equation  (3.1) can be  as  q  f r  =  V t. (24+1) e  , T  r  .  (3.2)  34  Equation  (3.2) can be shown to have the high temperature l i m i t of  the c l a s s i c a l r o t a t i o n a l p a r t i t i o n function -£(£+1)9 (3.2)  (2&+1) e  1ft  87T IkT 2  = 6  (3.3)  The Helmholtz free energy of the free r o t a t o r , F^ (N,V,T) r  can be written as F  f r  =  (3.4)  - RT£n(q ) fr  The i n t e r n a l r o t a t i o n a l energy, U^ (N,V,T), i s given by r  3£n q L* = RT' fr  8T  fr  (3.5)  -"V  The entropy of the free rotator, S^ (N,V,T), then follows as r  U  fr " fr F  (3.6)  fr  The heat capacity at constant volume, ( C ( N , V , T ) ) i s v  defined as  35  B) Ihe Out-of-Plane Hindered Rotator. The molecular p a r t i t i o n function for the out-of-plane hindered rotational model,  (.V,T), can be written as  _ %  T  -  I e k  +  The E ^  P,m)  _  (+1.0)  3 E  I k  k  I e k  *  I  k  I  2e  I  2e  m>l  (tl.m)  3 £  k  .  m>l  (3.8)  ' s with p = +1 or -1 are the eigenvalues of the out-of-plane  hindered rotator for some value of the barrier height A. The even eigenvalues E^* '^ are non-degenerate and the even eigenvalues 1  ^k ** ^ +  m  a r e  c  * k l y degenerate.  Therefore the factor of two i n the  ou  second term i s for this two-fold degeneracy and the m summation i s over a l l positive m values greater than zero. The same holds true for , the odd p parity states.  The k summation i s over the states that have  the same p and m value. I t i s often convenient to give the lowest rotational eigenstate an energy of zero.  Designating the lowest energy by E°, the  p a r t i t i o n function given-in (3.8) can be rewritten as -3AE,  C + 1  <W hr = I « ^  '  0 )  +1  • •  k  +  l e  k  K  .  -$AE, < ' ) 2e  +  I 'k  I  I  m>l I  k  m>l  m  k  2e  K  ,  (3.9)  36  where  AE CP^) k  ^Cp.m) _  =  E<3<  ( 3 ; 1 0 )  S u b s t i t u t i o n of the hindered r o t a t i o n p a r t i t i o n function, for q ^  i n equations  (3.9),  (3.4)-(3.7) and taking the indicated p a r t i a l  derivatives with respect to T at constant V, the following  express-  ions are found f o r the entropy and the heat capacity at constant volume :  h  Ul  \r  r  +  2 AE.C* '*) e k  I, "  k*l  k  1  _ k=l m>l  "r  e  k  R A F C" ' ) 1 0  k  e  ^(-^m).  *  I  2 A E ^ ' ^ e  I  1  k=l m>l  +  R^n  K  (q ) hr  ,  (3.11)  and RAF f  c  i  -  6  1  / v  /-Ap  (+1,0).  + I I 2(A C ' Ve +1  k=l m>l  K  E]  m  "  2  -RA  E l  "  p  f  C + 1  l  t  '  k  m )  37  ei,o) k=l  +  k  I  2(AE C- ^.)  I  1  k  k=l m>l  c  2  }  k  OAp '1 T  +  ^  (+1,0)  2  Ul  W J hr  .  k  I I 2 AE, ( l k=l m>l +  -BAE, , m  ( + 1  '  m )  ) (-1,0)  -BAE, k=l  (-l,m) z  -BAE, I I 2 AE,/- ^ k=l m>l 1  C) Out-of-Plane  (3.12)  and In-plane Hindered  F o r the o u t - o f - p l a n e  Rotator.  and i n - p l a n e h i n d e r e d  molecular p a r t i t i o n function, R j ( ' J " > ,  V  T  1  S  i r  r o t a t o r , the  g i v e n by  -est* '- ) 1  *'hr -  I  e  + .  1  J  e  I  ^  1  + J  3  e  J  j  -Bfc-.C- '^ 1  + •I e  3  (3.13)  j .(P,r) where t h e £ ^ 's are the e i g e n v a l u e s the lowest  f o r some A and u.  energy s t a t e by £ ° , t h i s p a r t i t i o n  Designating  f u n c t i o n can be r e w r i t t e n  38  as  q' = hr  I  e  j=l  J  +  I e "  J  j=l  + • X j=l  e  + • X e j=l  J  ,  3  (3.14)  where  A£.  Again using equations substituted for  - e°.  (3.15)  1  the e n t r o p y and heat  r  the o u t - o f - p l a n e  g i v e n by (3.9)  c a p a c i t y at c o n s t a n t  volume  and i n - p l a n e h i n d e r e d r o t a t o r are g i v e n by the  f o l l o w i n g two e q u a t i o n s  :  ^ hr  r  £.  ( 3 . 4 ) - ( 3 . 7 ) , but now w i t h q ^  for q£ >  h  =  J  r<  , n  -6A6.( '-^ +1  + • I Ae '-^ e J 5=1  +  +  (+1  .  J  I j=l  Ac^" '" )  I  A £ . <-!.•!)  1  1  J e  J  j =l  + R£n ( q '  e ^  .J  J  h r  ),  (3.16)  39  and  C C  v hr ~T q r  +  I  I  r-{  ^ ^ " / ^  hr <• j = l  3  CAS.^'-V e  3A£.(  + 1  3  '-^  -3A£(-l,-l)  (-1,-1)/  J  e  3  A  ^ " (  1  ,  3 fl  + 1 )  T  -3AS. I  A£.< > ) + 1  + 1  ( + 1  '  +  e ^  UV  ^  hr - 3 A £ (+1,-1)  1 )  +  I  A S . ^ ' - V ^  j= l  BAc-.C-'-) 1  I  j=l  At  (-1, + D  -3A€.  1  (-1, + D .  2  (3.17)  -  D) R o t a t i o n a l E n t r o p i e s and Heat C a p a c i t i e s o f Hydrogen and Deuterium. The  t o t a l wavefunction  o f a homonuclear d i a t o m i c  must be e i t h e r symmetric o r antisymmetric  (depending  molecule  on the n u c l e u s )  26 i n t h e exchange o f the two n u c l e i . antisymmetric  A l s o the w a v e f u n c t i o n must be  i n t h e exchange o f any two e l e c t r o n s .  Thus one can  i n c l u d e i n t h e m o l e c u l a r p a r t i t i o n f u n c t i o n sums o n l y those m o l e c u l a r e i g e n s t a t e s w i t h t h e same symmetry  (e.g., s t a t e s which a r e a n t i s y m m e t r i c  40  in  the  exchange o f the two n u c l e i  the hydrogen n u c l e u s deuterium n u c l e u s wavefunction  contains  can be w r i t t e n  y  r  T \ is t  = r  T o r the  the n u c l e a r symmetry o f Y ^ Y .  there  are a t o t a l  n  just  g  1  Y  0  of  (  (i.e.  Y ^ (7r-9, m  for nuclear  states,  n  has  be the n u c l e a r s p i n and it  can be shown  that  The t o t a l number o f n u c l e a r  + 1  g  r o t a t i o n a l wavefunctions  two n u c l e i  £  ni s i ) n  (6,<j>) as t h e i r b a s i s . the  J  g i v e n by  ^item- )  The  (3.18)  a n t i s y m m e t r i c n u c l e a r s t a t e s and  n  states is  n  exchange o f n u c l e i so t h a t Y  g ^ ( g ^ + l ) / 2 symmetric n u c l e a r s t a t e s . n  and f  states  Letting s  of g i ( S i ~ l ) / 2 n  total  v  the n u c l e a r ground s t a t e degeneracy,  n  r o t a t o r the  for translations, Y for vibrations, ' v  and Y ^ are symmetric i n the  n  and the  V Y Y ¥ , v r e n '  t  the w a v e f u n c t i o n  g j (=2s +l)  free  since  as  for rotations, ¥ for electronic ' e  Y , Y  and symmetric f o r  an odd number o f nucleons  an even number.  T where  for  •  n l  have the s p h e r i c a l harmonics  F o r any s p h e r i c a l h a r m o n i c , an  9 •*• ir-0 and cb -*• cb+iT)  cb+ir) = $ (cb Tr) Q m  +  £ j m  (3.19)  (Tr-9)  interchange  yields  .  (3.20)  41  For the <J) dependent .  wavefunction  * C*+ir) = e m  imfcb+TTi  ^  = (-1)  m  imcb  e  y  ,  (3.21)  and f o r t h e n o r m a l i s e d a s s o c i a t e d Legendre p o l y n o m i a l s , w i t h £ = c o s 8 so t h a t the i n t e r c h a n g e o f n u c l e i amounts t o l e t t i n g £  1/2 0  i?x,,< m> ^  =  m  Thus  2&+1  (&-m)!  4ir  (£+m)!  2 -m  (-1)*  m  m  P ( 0 ..  (3.22)  £  ( - 9 > <j>+ir) becomes Tr  m  Y^ (7r-e,(}) +TT) m  = (-1)  £  Y^^te,*).  T h e r e f o r e the r o t a t i o n a l wavefunctions  (3.23)  a r e seen t o be  i f % i s odd and symmetric i f % i s even.  This w i l l  antisymmetric  a l s o be t r u e f o r  the o u t - o f - p l a n e h i n d e r e d r o t a t i o n e i g e n f u n c t i o n s and the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t i o n e i g e n f u n c t i o n s s i n c e , as has been shown p r e v i o u s l y , the p e r t u r b a t i o n m a t r i c e s V* the same p ( = ( - l ) ) p a r i t y . neucleons, ¥ even £, ^  r  ¥  andV^/only  connect  For n u c l e i w i t h an odd (even) number o f  must be antisymmetric  (symmetric).  For s t a t e s o f  i s symmetric and so one must use antisymmetric  nuclear states.  F o r s t a t e s o f odd I, V  must use symmetric  states o f  (antisymmetric)  i s antisymmetric  nuclear states.  (symmetric) and so one  For n u c l e i  with  odd mass number, the f r e e r o t a t o r p a r t i t i o n f u n c t i o n i s w r i t t e n as  42  q  =  £ r  +  (2£+l)  Z  n  l  n  e  £ even  2  £ ( £ l) £ odd  l  2  2  +  e  ,  1  (3.24)  and for nuclei with even mass number as gnl gnl (  q  =  £ r  v " Z (2£+l) e £ even  + 1 )  o  2  p  B £  n  £  p f „  l  .  (3.25)  £ odd  2  S i n c e the hydrogen nucleus has an odd mass number and a n u c l e a r degeneracy of  of g ^ 2 , =  n  the f r e e  e q u i l i b r i u m hydrogen i s  q  f r  (e-H )  q  f r  ( p - H ) =1  2  = q  f  r  r o t a t o r molecular p a r t i t i o n  function  g i v e n by  (p-lh,) .+ q  f  r  (o-H )  ,  2  (3.26a)  where 2  (2£+l)  *  e  ,  *  .  (3.26b)  £ even and  _ q  f r  (o-H ) 2  = 3 I  (2£ l) +  e  g  £  £  .  (3.26c)  £odd Hydrogen w i t h even r o t a t i o n a l s t a t e s i s w i t h odd r o t a t i o n a l s t a t e s i s  c a l l e d parahydrogen and hydrogen  c a l l e d orthohydrogen.  The deuterium  nucleus has an even mass number with n u c l e a r degeneracy g ^ 3 and so =  R  43  q^  r  f o r equilibrium deuterium i s given by q  f  r  (e-D ) = q 2  (o-D ) + q  £ r  2  (p-D ),  f r  (3.27a)  2  where q  (o-D ) = 6 I  f r  (2£+l) e  2  \  (3.27b)  £ even  and  _ q  (P-D ) = 3 I  £ r  g e  (2£+l) e  2  .  1  (3.27c)  £ odd  Deuterium with even rotational states i s c a l l e d orthodeuterium and deuterium with odd r o t a t i o n a l states i s c a l l e d paradeuterium. Using the appropriate nuclear spin degeneracies f o r hydrogen and deuterium  and equation (3.9), the out-of-plane hindered r o t a t i o n a l  p a r t i t i o n functions are given by the following equations :  q  hr  q  ^  ( e _ H  2  )  =  q  hr (P'V  +  q  -BAE, CP"H ) = I e k  h r °" 2 '  (+1  h r  h r  hr  C  _  H  3  0)  + I  2  - B A E .  q  '  k  •  q  C  1  '  0  D  )  =  q  hr " 2 ( 0  D  ; )  +  q  hr ^ ~  ( + 1 v  ^  D  ) 2  '  m )  ,  k  - B A E . ^  + 6 I I e k m>l  2  " 2  -BAE  )  Co-H ) = 3 I e k  ( e  I 2 e k m>l  C 3 > 2 8 a )  k  1  (3.28b)  ' ^  ,  (3.28c)  (3.29a)  44  -3AE, q  (o-D ) = 6 I e k  h r  C + 1  '°). + 12  k  2  and  J - J e k m>l  , , -BAE,^ '^  1  (P"D ) = 3 I e k  h r  1  •  2  1  k  ,, -BAE,^ '^  q  C* ^ , (3.29b)  -BAE  + 6 I . J e k m>l  .  (3.29c)  The lowest energy E ° w i l l be an even p p a r i t y energy f o r e q u i l i b r i u m hydrogen ( e q u i l i b r i u m deuterium) and parahydrogen ( o r t h o d e u t e r i u m ) . For  orthohydrogen (paradeuterium), E ° w i l l be the lowest odd p p a r i t y  energy. For  the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t o r , the  p a r t i t i o n f u n c t i o n s f o r the n u c l e a r s p i n s p e c i e s o f hydrogen and deuterium,using  (3.14) and the c o r r e c t n u c l e a r s p i n d e g e n e r a c i e s , can  be w r i t t e n as  q  ' h r  (  e  -  H  2  )  !  q  S  'hr  ( P _ H  2  )  +  q  hr  C  °" 2 ' H  )  (  3  '  3  0  a  where -BA£.C q  'hr  ( p  - 2 H  }  ^  =  6  + 1  >  3  + 1  )  - B A ^  l  +  j  e  J  'hr  q  'hr  =  ( e  - 2 D  }  =  3  q  E  e  'hr °- 2 (  D  3 I e  +  )  +  q  )  (3.30b)  1  1  3  1  -BA.e.c- ^ ^  -BA.O-.C- >- > q  ' -  '  3  and 1  1  'hr  1  ,  J  (3.30c)  '  ( 3  '  3 1 a )  where -BAt\< > > +1  q  'hr  C  °- 2 D  )  =  6  I  e  j  3  +  -BA£.  1  +  6  I  e  3  j  -  C + 1  '  '-  i y  (3.31b)  )  45  and -SAc . C" ' ) 7  q'  (P-D ^  =  I  3  2  h r  1  3  e  -6A£\ (" ».  - 1  +  1  I  3  j £°  c a n be e i t h e r  hydrogen  e  o f even p p a r i t y o r c f odd p p a r i t y f o r  rotator,  the  will  be t h e  entropy  For orthohydrogen  The  (3:31c)  equilibrium  (paradeuterium)£ ° "  lowest even p p a r i t y e n e r g y .  and h e a t  found by u s i n g equations  )  •  J  will be t h e l o w e s t odd p p a r i t y e n e r g y and f o r p a r a h y d r o g e n it  1  j  (equilibrium deuterium).  deuterium)  +  For the  f o r the  i n equations  e n t r o p y and h e a t c a p a c i t y o f  (3.28a)-(3.31c)  in  capacities Classical rotator,  free  and o u t - o f - p l a n e and e n t r o p i e s  expressions the  found  g i v e n by  ( 3 . 4 ) - ( 3 . 7 ) .•  In C h a p t e r V , t h e rotational,  p a r t i t i o n functions  out-  out-of-plane  and i n - p l a n e h i n d e r e d r o t a t i o n a l h y d r o g e n and d e u t e r i u m c a n b e appropriate  are  (3.4)-(3.7).  o f - p l a n e h i n d e r e d r o t a t i o n a l h y d r o g e n and d e u t e r i u m and  s i m p l y by employing the  free  c a p a c i t y o f h y d r o g e n and d e u t e r i u m  (3.26a)-(3.27c)  corresponding equations  (ortho-  plus  rotational,  out-of-plane  in-plane hindered r o t a t i o n a l  o f h y d r o g e n and d e u t e r i u m a r e  f o r t h e h e a t c a p a c i t y and e n t r o p y  out-of-plane  in-plane hindered rotator  hindered rotator are  hindered  and t h e  given i n Appendix C.  heat  discussed. o f the  out-of-plane  free and  46  CHAPTER IV. HINDERED ROTATIONAL EIGENVALUES AND EIGENFUNCTIONS A) Eigenvalues of the Out-of-Plane Hindered Rotator • For a basis set of lowest 1 5 even eigenvalues values E ^ * heights.  , m  spherical harmonics  100  E ^ ^ +  , m 3  k  the  and the lowest 2 1 odd eigen-  ^ are given i n Table 1 for various out-of-plane b a r r i e r  The eigenvalues and the b a r r i e r heights have been reduced  to u n i t l e s s quantities by 8 i T I / h . 2  is  JJJCQJ*)  l i s t e d f o r two reasons.  2  None of the higher energy states  F i r s t l y , they are not required i n the  c a l c u l a t i o n o f the entropies and heat capacities o f hydrogen and deuterium due to the low temperature  range that we are interested i n .  Secondly, a basis set of only 36 out-of-plane hindered rotational eigenfunctions (5^(9,<b) i s subsequently used f o r the out-of-plane and in-plane hindered rotator.  Thus only the 36 E ^ ' k  m 3  eigenvalues given  i n Table 1 are required f o r the c a l c u l a t i o n of The even eigenvalues i n Table 1 are for states of even p p a r i t y (p=+l) and the odd eigenvalues are f o r states o f odd p p a r i t y (p=-l) .  The k index i s used to designate the ordering o f states within  each (p,m) symmetry group.  Since m i s s t i l l  a good quantum number,  the eigenvalues are l i s t e d i n groups with the same m value and within each m group they are ordered by k with k=l being the lowest  energy.  I t should be stressed that only the p p a r i t y and the m quantum number characterise the eigenstates {k,(p,m)'} and that k i s a dummy index f o r ordering the states.  The ±m degeneracy has not been removed by the  hindering p o t e n t i a l , as one would expect, since the perturbation  47  TABLE 1 a) Eigenvalues E^  for a basis s?t of100 spherical harmonics •  #  k  m  X=l  1  0  .31900  2  0  3  0  6.5335 20.508  X=5 1.3574 8.8107 22.578  X=10  X=20  X=40  X=80  2.3050  3.6567  5.5358  8.170:  11.790  17.541  27.051  40.568  25.251  30.913  43.253  67.365  13.193  18.440  26.199  29.921  39.732  57.034  1  ±1 •  2  ±1  1  ±2  2  ±2  20.402  22.004  23.979  27.797  34.803  46.609  1  ±3  20.275  21.330  22.596  24.958  29.131  35.994  1  ±4  20.090  20.441  20.857  21.625  22.972  25.182  6.4247 20.482 6.1409  8.0478 22.430 6.6699  9.9162 24.906 7.2650  8.2935  9.9455  12.438  TABLE 1 f 1 ml b) Eigenvalues E^ for a basis set of 100 spherical harmonics. t  k  m  1  0  2  0  12.514  14.643  3  0  30.505  32.550  1  ±1  2  ±1  12.468  14.354  16.706  21.247  29.328  42.069  3  ±1  30.488  32.458  34.957  40.064  50.515  70.788  1  ±2  12.331  13.612  15.122  17.875  22.554  29.887  2  ±2  30.436  32.185  34.374  38.735  47.211  62.587  1  +3  12.110  12.532  13.023  13.907  15.400  17.756  2  ±3  30.350  31.739  33.447  36.761  42.943  53.715  1  ±4  30.230  31.132  32.221  34.285  38.024  44.382  1  ±5  30.077  30.376  30.735  31.412  32.629  34.689  X=l 2.5931  2.1955  X=5_  X=1Q  X=20  X=40  X=80  4.8228  7.2853  11.270  16.995  24.949  17.467  23.503  35.502  54.681  35.158  40.558  52.196  77.770  2.8996  3.6439  4.8441  6.6438  9.2377  49  matrixV' connects only states with the same m quantum number. I f one uses the property * Y^ce.cM = c-i) Y ^ c e . w . t h e n the m a t r i x element V„  V  ' oi  £,-m;Jc',-m  0  = i3 C^) v  J  1/2 v  . ,  2  J  1  .i)  i s g i v e n by  0  (-I)" "  ( 4  <Y ,  £',m' 0  |2 \_5J  \  1  /  2  Y. _ +0,0' Y J£,m Y * >.  2,0  n  (4.2)  Since  2m i s always even  V  and  £ , - m ; £ ' ,-m "  V  £,m;£ ,m ,  ,  ( 4 , 3 ; )  so t h e ±m degeneracy remains w i t h E ^ ^ ' ^ m  The  eigenvalues  = Ej ^  , + r a  t  ^ .  c a l c u l a t e d f o r a b a s i s s e t o f 100 s p h e r i c a l 13  harmonics a r e i n e x c e l l e n t agreement w i t h those c a l c u l a t e d by E v e t t using  the continued  f r a c t i o n method t h a t W i l s o n ^ and S t e r n * ^  for  s o l v i n g t h e s p h e r o i d a l wave e q u a t i o n . „ However, E v e t t  the  lowest e i g h t non-degenerate e i g e n v a l u e s  hindered  rotator.  f o r the  ,  developed  o n l y gave  out-of-plane  Each energy was c a l c u l a t e d i n d i v i d u a l l y from the 27  tables of S t r a t t o n et a l .  .. Our v a r i a t i o n a l method has the advantage  t h a t any d e s i r e d number o f e i g e n v a l u e s  and e i g e n v e c t o r s  t e d , depending on the s i z e o f the b a s i s s e t .  can be c a l c u l a -  F o r example, u s i n g a  b a s i s s e t o f 100 s p h e r i c a l harmonics 100 s t a t e s are immediately  avail-  50  a f t e r s o l v i n g the a p p r o p r i a t e m a t r i x equations g i v e n by However, one must be sure t h a t the b a s i s  set  is  (2.51).  l a r g e enough so  that the v a r i a t i o n a l method has converged f o r l a r g e o u t - o f - p l a n e barriers.  To check t h i s  convergence o f the v a r i a t i o n a l method,  eigenvalues  f o r X = 10 and X = 40 have been c a l c u l a t e d f o r d i f f e r e n t  sized basis  sets.  Table 2 gives E j ^  p , m  >  values  sets o f 16,36,64 and 100 s p h e r i c a l h a r m o n i c s . a r e indexed the same way as i n T a b l e 1. a basis  set  The {k,(p,m)} s t a t e s  I t s h o u l d be noted t h a t  a v a i l a b l e f o r comparison.  For t h i s  lowest 10 e i g e n s t a t e s are l i s t e d i n T a b l e 2. set  is  i n c r e a s e d from 16 t o 100,  become c o n s t a n t , set o f 36.  for  o f 16 s p h e r i c a l h a r m o n i c s , t h e r e are o n l y 10 unique  eigenvalues  basis  f o r X = 10 and b a s i s  as e x p e c t e d .  F o r X = 40, b a s i s  r e a s o n , o n l y the  As the s i z e  the e n e r g i e s  The e i g e n v a l u e s  o f the  decrease  and f i n a l l y  have converged a t a b a s i s  s e t s o f 36,64,100,194 and 256 are used  f o r comparison i n T a b l e 3 f o r the same e i g e n s t a t e s as those g i v e n i n T a b l e 2. basis  o f 100,  o f 100i least  At a basis  set  o f 194 the e n e r g i e s  have converged and f o r a  they are w i t h i n about one p a r t i n 20,000.  Below a b a s i s  the d i s c r e p a n c i e s i n c r e a s e f a i r l y r a p i d l y so t h a t a b a s i s  of at  100 s h o u l d be used at these l a r g e b a r r i e r s . 22 P e r t u r b a t i o n methods  can a l s o be used t o f i n d the  v a l u e s o f the o u t - o f - p l a n e h i n d e r e d r o t a t o r i f X < 1. convergence i s v e r y p o o r .  F o r X > 1 the  E ^ ^ through t o E ^ ^ are l i s t e d  Appendix D f o r the same s t a t e s g i v e n i n T a b l e 1. s t a t e <jv.(0,cJO can be c a l c u l a t e d from  eigen-  in  The energy o f the  c  51  TABLE 2 Comparison of eigenvalues Ej^ ' > and E^ f o r A=10 k k using different sized basis sets. +1  E  m  1  k  k  m  basis = 16  basis = 36  basis = 64  basis = 100  1  0  2.3349  2.3052  2.3050  2.3050  2  0  12.237  1  ±1  10.286  1  ±2  7.4286  11.796  11.790  11.790  9.9201  9.9162  9.9162  7.2662  7.2650  7.2650  (-lim)  F  -  k  k  m  basis = 16  basis = 36  basis = 64  basis = 100  1  0  7.3010  7.2853  7.2853  7.2853  2  0  1  ±1  2  ±1  17.018  16.707  16.706  16.706  1  ±2  15.333  15.123  15.122  15.122  1  ±3  13.111  13.023  13.023  13.023  17.810 3.6488  17.470 3.6439  17.467 3.6439  17.467 3.6439  52  TABLE 3 Comparison of eigenvalues E^* *" 1  0  and E^' '" 1  0  for X=40 -  using different sized basis sets. E  C +  l,m)  k  m basis = 36 basis = 64 basis = 100 basis = 194 basis = 256  1  0  2  0  28.054  27.098  27.051  27.050  27.050  1 ±1  18.865  18.454  18.440  18.439  18.439  1 ±2  10.044  5.5846  5.5369  5.5358  9.9477  F  9.9455  5.5358  9.9454  5.5358  9.9454  C-l fin)  k  m basis = 36 basis = 64 basis = 100 basis = 194 basis = 256  1  0  17.065  16.997  16.995  16.995  16.995  2  0  36.067  35.523  35.502  35.502  35.502  1  ±1  2  ±1  29.709  29.338  29.328  29.328  29.328  1  ±2  22.718  22.558  22.554  22.554  22.554  1  ±3  15.439  15.401  15.400  15.400  15.400  6.6536  6.6440  6.6438  6.6438  6.6438  53  E =  f . A  K  N  n=o  E  M  k  (4.4)  f  where E ^ ^ i s the unperturbed energy of the free rotator. 0  The energies of nine even {k, +l,m)} eigenstates and six r  odd {k,(-l,m)} eigenstates are plotted against the potential barrier X i n Figure 1. As discussed above, the 2H+1 degeneracy of the free rotator i s partly removed, only the ±m degeneracy remaining.  At  X = 0, one just has the 2£+l degenerate energies of the free rotator. With the introduction of A, the levels s p l i t and the sharpness of % i s removed along with the 21+1 m degeneracy.  Within any free rotator  % group of degenerate states, l+l singly or doubly degenerate states are produced with the state of lowest m value lying highest. In Figure 1, no two states i f m=m'.  < J >  k  /  +  1  ,  m  (9,c}>)  >  and  < J >  k  ^  +  1  ,  m  can ever cross  >(6,<f>)  However states of different m can cross; for example  <J > > °Vc» and c^ V > <». + 14 +  1+C  2  The same of course holds true for the odd states.  Consider-  ing the even and odd levels together, any state of even p parity can cross any state of odd p parity, regardless of the m values.  For i n -  -stance, the energy of the (1,(+1,±2)} state i s greater than that for i  ( l , ( - l , 0 ) } below A = 10 but less above 10. Thus, at barriers greater than A = 10, there i s a reordering of the even and odd p parity energies, i . e . below A = 10, and above A = 10, E ^ *  1  ^  <  E  /  +  1  E ^ '  '  0  1  )  '  ' ^  <  E ^ " <  E  ^  1  "  ' ^ ^  1  i s due to the fact that the perturbation V = V  <  1  E ^ "  1  ^  <  E/" '^. 1  <  C E  L  +  1  '  This  cos 6 has a greater 2  ±  2  >  FIGURE 1 Energy l e v e l  diagram f o r the o u t - o f - p l a n e  E n e r g i e s are p l o t t e d as a f u n c t i o n o f A, solid lines lines  hindered r o t a t o r 0 < A < 40.  The  are f o r {k,(+l,m)} e i g e n s t a t e s and the dashed  for {k,(-l,m)} eigenstates.  The l i n e s  are  as f o l l o w s :  1.  U,(+1,0)}  10.  {2,(-1,0)}  2.  {1,(-1,±1)>  11.  {!,(+!,±4)}  3.  U,(-1,0)}  12.  U,(+l,±3)}  4.  {!,(+!,±2)}  13.  {2,(+l,±2)}  5.  {1,(+1,±1)}  14.  (2,(+l,±l)}  6.  {2,(+1,0)}  15.  {3,(+1,0)}  7.  {1,(-1,±3)}  8.  {1,(-1,±2)}  9.  (2,(-l,±l)}  labelled  55  56  e f f e c t on the lower m states than the higher ones.  It i s felt  that t h i s reordering of states may be responsible, i n part, for the enrichment of orthohydrogen (odd states) r e l a t i v e to parahydrogen (even states) that occurs i n the adsorbed phase compared to the gas 6 7  phase at the same temperature. '  This p o s s i b i l i t y w i l l be discussed  i n Chapter VI. Here the r o t a t i o n a l energy levels o f the out-of-plane hindered r o t a t o r have been computed by using v a r i a t i o n to solve the exact quantum mechanical wave equation.  Other discussions'''  4,28  of  e s s e n t i a l l y the same model have used r o t a t i o n a l energy levels which are the solutions to the approximate wave equation. d G(g) 2  dC  2  +  8Trfl_ h  2  (4.5)  n,m  8TT I 2  where £ = cos9 and Q  (0) = G(£).  g  (The term dG(Q/d?. has been dropped)  This equation amounts to assuming that at i n f i n i t e l y large b a r r i e r s , the hindered rotator behaves like an out-of-plane in-plane rotator with the following expression  n,m  n +  2  X / 1  2 +  m  *  vibrator and an  f o r i t s eigenvalues:  (4.6)  with  n = 0,1,2,  (4.7)  m = 0, ±1, ±2,  (4.8)  and  I f X i s u n i t l e s s , then the energy E w i l l be u n i t l e s s . ' n,m &  The s t a t e s  /  f o r the approximate wave e q u a t i o n w i l l be d e s i g n a t e d by [ n , m ] . {k,(p,m)} s t a t e s ,  {l,(+1,0)},  t o the lowest v i b r a t i o n a l s e t , The e n e r g i e s  {l,(-l,+l)} [0,0],  differences  and { l , ( + l , + 2 ) } c o r r e s p o n d  [0,1],  and [0,2]  o f these s t a t e s a r e g i v e n i n T a b l e 4.  o f the approximate s o l u t i o n t o the exact i n the e n e r g i e s 6  The •  respectively.  The convergence  one i s extremely s l o w .  The  6 .= E - E . ^ P ^ a r e a l s o g i v e n i n T a b l e 4. n,m k , m  &  The s t a t e  [ 0 , 0 ] i s the o n l y one t h a t seems t o be c o n v e r g i n g towards  t h e exact  energy o f ( 1 , ( + 1 , 0 ) } . The others, at l e a s t up to X = 80,  appear t o be d i v e r g i n g . imate s o l u t i o n s al  T h i s would seem t o imply t h a t the approx-  s h o u l d never be used f o r o u t - o f - p l a n e h i n d e r e d r o t a t i o n  energy l e v e l s i f one wants the a c t u a l magnitude o f the  eigenvalues.  A l s o from T a b l e 4, i t can be seen t h a t the energy d i f f e r e n c e between t h lowest two s t a t e s l a r g e X.  found from the exact  This i s just  the energy d i f f e r e n c e  found from t h e approximate s o l u t i o n ^ ' the S a n d l e r l i m i t . ^  e r a t u r e s where few l e v e l s X > 40.  2 8  f o r the two lowest  states  and i s o f t e n r e f e r r e d t o as  T h e r e f o r e i f o n l y energy d i f f e r e n c e s  the approximate r o t a t i o n a l e n e r g i e s  height,  s o l u t i o n approaches u n i t y at  are r e q u i r e d ,  can p r o b a b l y be used at low temp-  a r e r e q u i r e d and at v a l u e s o f the b a r r i e r •  However, anyone wanting t o c a l c u l a t e r o t a t i o n a l  p r o p e r t i e s a c c u r a t e l y would be a d v i s e d t o use the exact  eigenvalues.  B) E i g e n f u n c t i o n s o f the Hindered R o t a t o r . I n Chapter II the v a r i a t i o n a l w a v e f u n c t i o n was w r i t t e n as  TABLE 4 Comparison of exact and approximate rotational energies ( A l l energies have been reduced by h /8ir I and are given as unitless quantities ). 2  2  E l  X  fc  1  ( + 1  >  0 )  .3190  [0,0] E  0,0  {1,(4-1,+2)}  U.C-i [0, +1]  U,(4-1,0)}  •6  E  l  E  o,i  6  El K  C + 1  '  ± 2 )  [0,+2] E 0,2  6  1.000  0.681  2.196  2.000  -0.196  6.141  5.000  -1.141  5  1.357  2.236  0.879  2.900  3.236  0.336  6.670  6.236  -0.434  10  2.305  3.162  0.857  3.644  4.162  0.518  7.265  7.162  -0.103  20  3.657  4.472  0.815  4.844  5.472  0.628  8.294  .8.472  0.178  40  5.536  6.325  0.789  6.644  7.325  0.681  9.946  10.325  0.379  60  6.967  7.746  0.779  8.048  8.746  0.698  11.285  11.746  0.461  80  8.170  8.944  0.774  9.238  9.944  0.706  12.438  12.944  0.506  59  S i n c e the $ (40 energy,  p a r t o f the s p h e r i c a l harmonics has no e f f e c t  o n l y the  0 p a r t o f the w a v e f u n c t i o n i s  on the  i m p o r t a n t f o r the  p r e s e n t d i s c u s s i o n and we can w r i t e (p,m)  «k  (p,m)  te) - J  (p,m)  e .ce) c  p}  t>  (p,m) where 0. (0) are d e f i n e d by equations J6,m  ,  perturbation V  chosen f o r i n v e s t i g a t i o n .  (2.8)-(2.10).  To see how the  the r o t a t i o n a l wavefunctions ( l , ( + l , + l ) } and { l , (+1 ,+2)}  The r e a s o n f o r t h i s  l a r g e v a l u e s o f the b a r r i e r h e i g h t , c o r r e l a t e d with a d i f f e r e n t  choice i s  {2,(+1,0)} ^ - ^ >  cb,  have been  that,  :  (4.10a)  [1, + 1 ] ,  U , ( + l , + 2)} ^ — - >  at  [ n , m ] , o f the approx-  [2,0] ,  : {l,(+l, +l ) } ^ - ^ >  (0),  each o f these s t a t e s can be  vibrational state,  imate wave e q u a t i o n g i v e n by (4.5)  '  (4.10b)  [0,+2] .  (4.10c)  i f the o u t - o f - p l a n e h i n d e r e d r o t a t o r i s becoming s i m i l a r  t o an o u t - o f - p l a n e functions,  (4.9) Cp,m)  = Xcos 6 a f f e c t s  the'{k,(p,m)} states'{2,(+1,0)},  Therefore,  ,  1 > B > k  <J>  t o e x h i b i t two,  2  C+1,0)  v i b r a t o r and an i n - p l a n e r o t a t o r , the t h r e e  (e),  <j>  C+1, + 1  1  - (e) )  one and zero nodes,  and  cb  +2) ( 1  :  +  l  j  (e),  respectively.  wave-  should s t a r t  60  Using the eigenvectors found from solving the matrix equations (2.51) f o r X = 40, these three wavefunctions can be written approximately as (+1,0) ,(+1,0) 4>£ ' '(6) - .5142 0 y e ) + .5914  (+1,0) 0  0  (6) - .5942  (+1,0) © ^ (9) 4  + .1788 ^(+1, + !)  ^  c e )  > 8 8 g 0  Q  (+l, + l )  (4.11)  _  ( 0 ; )  0| J' ^(9) +1  +  > 4 4 7 6  + .1042  G^ j' +  + 1 )  (9),  (4.12)  (+l,+2) cb (9) 1  = .9649 0  (+l,+2) 2  2  (9)  -.2587 ©  (+l,+2) (9)  4  + .0458  2  (+l,+2) 0  A l l terms with C  (p,m) , x» ,m, K  (9).  6 ) 2  -  (4.13)  < .01 have been neglected.  0  In Figure 2 these  wavefunctions have been plotted for 0° < 0 < 180°, at t h e i r appropriate energies.  Curve l a i s f o r <j> ^ ^ ' ^ (9), curve 2a f o r c f > ^ ' ^ (9) +  +  +  and curve 3a f o r <f> *'^(8). The hindering p o t e n t i a l V +  2  = X cos 9, 2  with X = 40, i s also shown (curve 4). The dashed curves represent the free r o t a t o r wavefunctions  f o r each o f the three states :  (+l,+2) curve lb :  ^  ,_ (0)  (+1,+1) curve 2b : curve 3b :  ^ $  2  2  2  (9)  —> 0  2  1  (6)  ,  (4.14a)  x  (9) (+1,0)  > Q  u  x  (0) -  _  J  (4,14b)  n  —> 0  2  Q  (9).  (4.14c)  FIGURE  The functions  cj> (  + 1  >  + 2  )  (0),  <J>  2  (9) and ^> ^ '°\d) +l  +  2  p l o t t e d as a function of 9 f o r 0° < 9<  180° and A=40.  dashed curves are the free rotator wavefunctions. potential V  = (40)cos 9 i s also plotted. 2  +  m  The energy scale  scale i s given on the r i g h t hand side.  curves are l a b e l l e d as follows:  la. V>> + 1 + 2  lb.  0  2 > 2  2a.  i f ' * "  2b.  Q  1  2  3a. cf> 2  3 b  -  G  2,0 V* = Xcos ( 2  The  The  i s given on the l e f t hand side of the figure and the fy</ ^' ^^  are  The  62  63  These f r e e r o t a t o r w a v e f u n c t i o n s be p l o t t e d a t an  a r e d e g e n e r a t e and t h e y s h o u l d  energy £^ = 6.0 ( u n i t l e s s ) .  actually  However t h e y have been  drav.-n a t t h e same e n e r g i e s as t h e h i n d e r e d r o t a t i o n a l e i g e n s t a t e s f o r direct  comparison. The  <j> C  +1,+1  > (6) e i g e n f u n c t i o n has been drawn as a symmetric  f u n c t i o n f o r convenience  o f p l o t t i n g , although i n f a c t i t i s a n t i -  s y m m e t r i c about 0 = 90° because o f t h e symmetry p r o p e r t i e s o f t h e n o r m a l i z e d a s s o c i a t e d Legendre p o l y n o m i a l s w h i c h a r e t h e b a s i s f u n c t i o n s . F o r an i n t e r c h a n g e o f n u c l e i , 0 -*• 0-TT, and  Using  (4.15) i n (4.9) w i t h m = +1, p = +1 and £ £ { + l } , t h e e i g e n -  f u n c t i o n f o r t h e s t a t e {1,(+1,+1)}, a f t e r an i n t e r c h a n g e o f n u c l e i , can be w r i t t e n as  4!' ;  •^'^Ce-iO -  I  (-I)*"  1  +1 )  (0)  eft'!"  (4.16)  S i n c e £ e { + l } , i . e . £ must be o f even p a r i t y , £-1 must be odd s o t h a t  <j>J ^(6-Tr) +1 +1,  (0) .  = -  <j>  (+1, + 1  with respect t o p a r t i c l e interchange. <f>j^  +1, +  1 )  (0) i s t h e r e f o r e a n t i s y m m e t r i c  However t h e complete  ^ (6,$) must be symmetric s i n c e t h e b a s i s f u n c t i o n s  wavefunction j(8,<f>) a r e  symmetric w i t h r e s p e c t t o p a r t i c l e i n t e r c h a n g e f o r even £, i . e . <j)  ( + l j+ 1 ) 1  (0-TT,  cf) 7T) = c f ) +  C + 1 1  ' (6,c) ). + 1 )  )  I n g e n e r a l , cf>  C+1 k  ' (e,<f,) m)  functions  are symmetric, but <j>  '  k  antisymmetric,  (9) f u n c t i o n s  depending on the p a r i t y o f m. 1,m  are a n t i s y m m e t r i c , b u t §^  >(9)  <}>,'•  can be symmetric; o r (9,<f>)  1  functions  can be e i t h e r symmetric o r a n t i -  symmetric. I t s h o u l d be n o t i c e d i n F i g u r e 2 t h a t the o u t - o f - p l a n e e r e d r o t a t i o n a l wavefunctions character.  They e x h i b i t  f o r A = 40 have become v i b r a t i o n a l i n  the number o f nodes expected  harmonic o s c i l l a t o r on a c i r c u l a r p l a t e .  $^  + l > +  ^ (6)  functions  are zero at  ty^* ' ^ ( 0 ° ) = ^  However  1  0  +  1  '  0  )  9=  (180°)  b a r r i e r o f A = 40 i s n o t l a r g e enough. A = 80, < J > ^ ' ° > ( 0 ° ) = . 0 2 . +1  2  c h a r a c t e r o f <J> ^ 180°.  +  1  Thus,  The cj) ^  + 1 , +  for a linear  ' > (9) and 1  0 ° and 1 8 0 ° as they s h o u l d b e .  t 0.0 due t o the f a c t t h a t a I f the b a r r i e r i s i n c r e a s e d t o  f o r A = 40, i t i s t h e r o t a t i o n a l  ' ^ (6) which i s c a u s i n g t h e n o n - z e r o v a l u e at 0 ° and  Increasing the b a r r i e r r e s u l t s  v i b r a t i o n a l i n character u n t i l function exhibits  i n §2*^  t h e p r o p e r boundary c o n d i t i o n s .  the molecule v i b r a t i n g i n s i d e  becoming more  ^  at b a r r i e r s g r e a t e r than A = 80, t h e  as the b a r r i e r i s i n c r e a s e d , the p r o b a b i l i t y ,  I t i s apparent p,m  2  diatomic.  i n Figure 2 represent  When the molecule  and the  the w e l l d e c r e a s e s .  chance o f the m o l e c u l e t u n n e l i n g through the b a r r i e r w a l l shaded areas  that  |<J>j^ > (9) | , o f f i n d i n g  the p o t e n t i a l w e l l i n c r e a s e s ,  p r o b a b i l i t y o f t h e molecule b e i n g o u t s i d e  The  hind-  the degree  Thus the  decreases.  o f t u n n e l i n g o f the  i s i n s i d e the b a r r i e r ,  Ej^  P , m  >  > V , and  fD m*) when  J  < V  tunneling through.  ( r e p r e s e n t e d by t h e shaded areas) As one moves  from s t a t e  the molecule  {l,(+l,+2)} to state  is  65  {2,(+1,0)}, since  the amount o f t u n n e l i n g i n c r e a s e s ,  states  o f h i g h e r energy w i l l be l e s s s t r o n g l y h i n d e r e d by the  potential barrier. for  as one would e x p e c t ,  different  In table  5 are g i v e n the number o f bound s t a t e s  b a r r i e r heights.  These bound s t a t e s have  energies  s nil Ej.  ^  a n <  ^  t  n  u  s  they  are not p u r e l y r o t a t i o n a l i n c h a r a c t e r .  X = 1, o n l y t h e ( l , ( + l , 0 ) } s t a t e i s bound.  As t h e b a r r i e r i s  For  increased  t o X = 80 a l l the 36 s t a t e s g i v e n i n T a b l e l a and l b a r e bound.  This  b e h a v i o u r agrees q u a l i t a t i v e l y w i t h what one expects from the approx14 imate s o l u t i o n p r o p o s e d by H i l l  .  However, i t s h o u l d be p o i n t e d out  t h a t h i g h e r e i g e n s t a t e s w i l l n o t be b o u n d , even f o r a b a r r i e r of  X = 80.  rotational  It is  o n l y when t h e b a r r i e r approaches  i n f i n i t y that a l l  e i g e n s t a t e s w i l l be b o u n d , i . e . a l l o u t - o f - p l a n e  w i l l be c o m p l e t e l y h i n d e r e d and t h e molecule w i l l become of-plane  v i b r a t o r and i n - p l a n e  rotator.  TABLE 5 Number o f bound e i g e n s t a t e s f o r v a r i o u s out-of-plane b a r r i e r heights.  Barrier  (unitless)  Bound E i g e n s t a t e s  1.0  1  5.0  4  10.0  8  20.0  13  40.0  28  80.0  36  height  rotations  a true out-  66  C) E i g e n v a l u e s o f the O u t - o f - P l a n e and I n - P l a n e H i n d e r e d R o t a t o r . I n . s o l v i n g the m a t r i x e q u a t i o n I  (E P' (  for  a basis  set  the e i g e n v a l u e s  r ) +  pW  cb^^' ^ 1  o f 36 E^P' ^ 1  A=10.  £^(p> )^ r  ( P , r )  =a  ( P , r )  [j,(p,r)]  +  and r = (-1)  ,  (2.83)  set  o f 36 s p h e r i c a l  eigenvalues,  .  F o r an o u t - o f - p l a n e  ^ , ^  are l i s t e d  and  ^' ^  u,  in  are g i v e n  +  t  F o r an o u t - o f - p l a n e b a r r i e r  A=10, the even p p a r i t y e n e r g i e s  height  are g i v e n i n T a b l e 9 and the odd  are l i s t e d i n T a b l e 10 f o r s i x v a l u e s  o f the i n - p l a n e .  u. m degeneracy i s now c o m p l e t e l y removed by the  b a r r i e r and 36 nondegenerate the f o u r ( p , r )  ness o f p and r . lies  and  The odd p p a r i t y e n e r g i e s , £S  The  of  )  o f the i n - p l a n e b a r r i e r h e i g h t  +  i n T a b l e 8 f o r the same u v a l u e s .  barrier,  r  m  t h e even p p a r i t y e n e r g i e s , $L^ ^~' ^  p p a r i t y energies  '  are grouped a c c o r d i n g t o the evenness o r  b a r r i e r o f A=l and d i f f e r e n t v a l u e s  of  P  and i n - p l a n e h i n d e r e d r o t a t i o n a l  o f p and r where p = (-1)  T a b l e 7.  (  are g i v e n i n T a b l e 6 f o r A=l  Z oddness  £  (6,cb) f u n c t i o n s , as o u t l i n e d i n Appendix B ,  These e n e r g i e s  The o u t - o f - p l a n e for states  )((  c a l c u l a t e d from a b a s i s  harmonics must be known. and  ( P , r )  g r o u p s , none o f the  are p r o d u c e d .  levels  W i t h i n any one  can c r o s s due t o the s h a r p -  F o r an o u t - o f - p l a n e b a r r i e r A<10, the  lowest at a l l v a l u e s  energy l i e s next  eigenstates  in-plane  lowest.  ^  , +  energy  o f u = W ' / 2 employed, and the c T ^ " " " ^ 1 ,  Q  However from T a b l e s 9a and 10a i t  is  apparent  t h a t the energy d i f f e r e n c e between these two l e v e l s decreases  as u  increases.  the  In f a c t ,  the  [ 1 , ( - 1 , - 1 ) ] s t a t e may even l i e below  TABLE 6 a) Eigenvalues E, ^  +1,m  > for a basis set of 36 spherical harmonics.  k  m  X=l  1  0  .31900  2  0  3  0  1  ±1  2  ±1  1  ±2  2  ±2  20,405  24.188  1  ±3  20.273  22.727  ±4  20.091  20.909  1  "  6.5335 20.511 6.4247 20.484 6.1409  X=10 2.3052 11.796 25.535 9.9201 25.171 7.2662  TABLE 6 Eigenvalues E ^  i > m  ^ for a basis set of 36 spherical harmonics.  m  X=l  X=10  0  2.5931  7.2853  0  12.514  17.470  0  30.508  35.398  ±1  2.1955  3.6439  ±1  12.468  16.709  ±1  30.490  35.186  ±2  12.331  15.123  ±2  30.438  34.569  ±3  12.110  13.023  ±3  30.351  33.592  ±4  30.231  32.308  ±5  30.077  30.769  69  TABLE 7 for X=l  a) Eigenvalues y=0.25  i  i  .56425  y=0.50 .80002  y=2  y=3  y=4  1.2436  2.0254  2.6843  3.2447  2  6.3836  6.6134  7.0490  7.8850  8.7041  9.5049  3  6.3899  6.6366  7.1236  8.0719  8.9864  9.8684  4  6.7931  7.0703  7.6622  8.9285  10.258  11.626  5  20.340  20.586  20.991  21.598  22.213  22.852  6  20.340  20.589  21.084  22.067  23.048  24.031  7  20.588  20.744  21.142  22.177  23.259  24.377  8  20.656  20.911  21.429  22.498  23.602  24.736  9  20.830  21.187  21.913  23.386  24.882  26.395  b) Eigenvalues  ^ for \=1  j_  y=Q.25  y=o.so  y=i  y=2  y=3  y=4  1  6.5487  6.6708  6.9096  7.3664  7.7982  8.2079  2  6.7987  7.1707  7.9084  9.3574  10.768  12.138  3  20.504  20.662  20.900  21.351  .21.801  22.253  4  20.518  20.759  21.245  22.235  23.260  24.322  5  20.629  20.849  21.372  22.465  23.583  24.721  6  20.865  21.252  22.029  23.589  25.154  26.723  70  TABLE 8 a) Eigenvalues £\ ^ ' ^ for X=l 1  j_  y=0.25  u=0.50  y=l  y=2  y=3  y=4  1  2.3191  2.4397  2.6726  3.1082  3.5084  3.8783  2  2.5690  2.9394  3.6702  5.0891  6.4449  7.7309  3  12.357  12.584  12.892  13.353  13.785  14.200  4  12.359  12.607  13.104.  14.115  15.156  16.235  5  12.596  12.744  13.185  . 14.217  15.268  16.322  6  12.844  13.221  13.975  15.469  16.943  18.396  7  30.327  30.574  30.879  31.300  31.720  32.143  8  30.327  30.575  31.069  32.039  33.001  33.975  9  30.553  30.674  31.084  32.099  33.135  34.193  10  30.592  30.828  31.311  32.326  33.398  34.509  11  30.666  30.926  31.479  32.615  33.775  34.956  12  30.877  31.271  32.064  33.654  35.248  36.846  71  TABLE 8 1  b) Eigenva lues p . ^ > V 1  1  +  for A=l  i  u=0.25  y=0.50  y=i  y=2  y=3  y=4  i  2.8406  3.0832  3.5536  4.4364  5.2443  5.981!  2  12.543  12.717  13.046  13.699  14.359  15.028  3  12.580  12.828  13.318  14.280  15.216  16.127  4  12.804  13.132  13.813  15.198  16.598  18.007  5  30.474  30.678  30.960  31.487  32.026  32.580  6  30.476  30.713  31.171  32.072  32.974  33.883  7  30.604  30.783  31.268  32.300  33.360  34.443  8  30.694  30.960  31.510  32.648  33.810  34.989  9  30.850  31.222 .  31.975  33.494  35.028  36.575  72  TABLE 9 a) Eigenvalues £\ ^  for X=10  + 1 ,+ 1 3  2  u=0.25  y=0.50  y=2  1  2.5490  2.7807  3.2083  3.9346  2  7.5151  7.7617  8.2480  9.1939  10.981  14.188  3  7.5210  7.7853  8.3413  9.5490  12.194  17.114  y=i  y=4 5.0106  y=8 6.4328  4  12.045  12.289  12.769  13.690  15.396  18.858  5  21.160  21.413  21.927  22.979  25.183  29.931  6  21.160  21.413  21.927  22.979  25.191  30.036  7  24.435  24.674  25.134  26.002  27.678  31.191  8  24.438  24.688  25.189  26.191  28.199  32.246  9  25.790  26.057  26.620  27.844  30.529  36.367  bO  Eigenvalues  8j^  +1>  "  13  for  X- 10  2  u=0.25  u=0.50  y=i  y=2  y=4  y=8  1  10.044  10.166  10.402  10.852  11.668  13.060  2  10.294  10.665  11.401  12.840  15.580  20.440  3  22.978  23.230  23.738  24.758  26.609  28.875  4  22.978  23.231  23.742  24.791  27.018  31.883  5  25.296  25.422  25.678  26.208  27.541  32.057  6  25.546  25.922  26.675  28.186  31.221  37.321  TABLE 10 c- f-1 a) Eigenvalues c•  for  X=  y=2  u=0.25  U=0.50  3.7674  3.8875  4.1191  4.5497  5.3042  4.0172  4.3872  5.1163  6.5277  9.1358  y=l  y=4  u=8 6.5191 13.397  13.274  13.526  14.032  15.052  17.073  19.811  13.274  13.526  14.035  15.078  17.287  21.518  16.833  16.956  17.198  17.670  18.593  22.307  17.083  17.456  18.197  19.660  22.499  27.779  31.020  31.273  31.783  32.823  34.981  38.651  31.020  31.273  31.783  32.823  34.982  39.613  33.842  34.090  34.578  35.475  36.735  41.412  33.842  34.091  34.592  35.603  37.697  42.272  35.312  35.442  35.713  36.354  38.237  42.793  35.562  35.940  36.700  38.232 .  41.322  47.555  74  TABLE 10 I  b) Eigenvalues  1  y=0.25  u=0.50  u=l  1  7.5318  7.7713  8.2294  ,+  *  3  for A=10  y=2 9.0650  u=4  y=8  10.451  12.431  2  15.372  15.620  16.109  17.067  18.901  22.266  3  15.374  15.626  16.132  17.153  19.164  22.681  4  17.721  17.974  18.485  19.533  21.756  26.744  5  32.558  32.810  33.318  34.347  36.437  40.012  6  32.558  32.810  33.319  34.348  36.463  40.851  7  34.807  35.023  35.411  36.109  37.509  41.412  8  34.819  35.070  35.573 . 36.585  38.635  42.883  9 ' 35.662  35.949  36.579  40.838  46.874  37.947 .  75  [ 1 , ( + 1 , + 1)] levels,  s t a t e f o r some b a r r i e r u>8.  i.e.  an odd p p a r i t y s t a t e  w i l l have a most n o t i c e a b l e  effect  l y i n g below an even p p a r i t y  f  lower e i g e n v a l u e s ,  and w i t h X=10.  unperturbed eigenstates on the  ^> )  for states  r  lowest  at u=0 levels  The l e t t e r s  t  at u=0  [j,(p,r)]  , r  >.  states.  o f c o u r s e , due t o the  are then l e f t  nondegenerate  levels  diagram f o r the  {k,(p,m)}  out-of-plane perturbation.  f o r {k,(p,m)} s t a t e s  levels These  (m^O) as the r o t a t i o n s  of  are p e r t u r b e d f u r t h e r by the i n - p l a n e h i n d e r i n g  F o r example,  {1>C+1,0)} -  {!,(+!,±2)}  states  [j,(p,r)]  u n s p l i t f o r { k , ( p , 0 ) } s t a t e s o r s p l i t i n t o two  the d i a t o m i c m o l e c u l e s potential.  for states  The s p l i t t i n g o f the  the  and the numbers  F i g u r e 3a shows the energy l e v e l  [ j , ( + l , + l)]  is,  r e f e r to  {k,(p,m)} w i t h energy E ^ P ' " ^  curves r e f e r t o the i n d i v i d u a l j v a l u e s  w i t h energy £ j ^ P  The  discussed  p l o t t e d i n F i g u r e s 3 and 4 as a f u n c t i o n o f the i n - p l a n e b a r r i e r  h e i g h t u(=W / 2 )  six  These p r o p e r t i e s w i l l be  Chapter V . Some o f the  are  state,  on the p r o p e r t i e s o f e q u i l i b r i u m  hydrogen and e q u i l i b r i u m d e u t e r i u m . in  Such a r e o r d e r i n g o f energy  v  Perturbation^  :  >  [  l  (  +  1  >  +  1  )  ]  f  (  4  .  1  7  )  [2,01,-H)],  (4.18a)  [3,C+1,+1)]•  (4.18b)  [ 5 , ( + l , + l ) ] and [ 6 , ( + l , + l ) ]  up t o an i n - p l a n e b a r r i e r h e i g h t  f  o f u=7.  are g r a p h i c a l l y i n d i s t i n g u i s h a b l e  FIGURE 3 Out-of-plane and in-plane hindered rotational eigenvalues  ~ (~ r^1 r") -  £j  '  plotted as a function of the in-plane barrier  height with A=10. unitless.  A l l energies and barrier heights are  The labels on the curves are the j values. In  Figure 3a, r=+l and the letters a to d at u=0 refer to th following out-of-plane {k,(p,m)} eigenstates: a.  U,(+1,0)} ,  b.  U,(+l,±2)},  c.  (2, (+1,0)} ,  d.  U,(+l,±4)}.  In Figure 3b, r=-l and the letters a and b at y=0 refer t the following eigenstates: a.  (1,(+1,±1)},  b.  {!,(+!,±3)}.  77  28  24  20  + 00  28  FIGURE 4 Out-of-plane and in-plane hindered rotational eigenvalues £j^  1 , r 3  X=10.  plotted as a function of the in-plane barrier with The labels on the curves refer to the j values. In  Figure 4a, r=-l and the letters a to c at u=0 refer to the following out-of-plane (k,(p,m)} eigenstates: a.  U,(-l,±l)} ,  b.  (l,(-l,±3)} ,  c.  {2,(-l,±l)} .  In Figure 4b, r=+l and the letters a to c at u=0 refer to the following eigenstates: a.  U,(-1,0)} ,  b. {1,(-1,±2)}, c.  {2,(-1,0)} .  The  energy  3b and 4a r e s p e c t i v e l y  arise  from the s p l i t t i n g o f the u n p e r t u r b e d  s t a t e s { k , ( + l ,m) } and { k , ( - l , m ) } , where m i s tive  integer.  The  ( ^* +  3a and 4b r e s p e c t i v e l y  +  D  previously, group.  none o f the  This is  [4,(-lyl)]  and £ j ^ ^> ^ +  result  { k , ( p , m ) } s t a t e s , where m i s  levels given i n Figures  an odd p o s i t i v e  energies  or nega-  given i n Figures  from the i n - p l a n e p e r t u r b a t i o n even  [j,(p,r)]  (including zero).  on  As mentioned  l e v e l s may c r o s s w i t h i n any  apparent i n F i g u r e 4a, where below u=6 the  (p,r)  levels  and [5, ( - 1 , - 1 ) ] are a p p r o a c h i n g each o t h e r and a f t e r  y=7,  t h e y are s e p a r a t i n g . The complete energy l e v e l diagram f o r the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t o r would be a s u p e r p o s i t i o n o f F i g u r e s 3 and 4. The r e s u l t i n g diagram would give the energy l e v e l s  of  equilibrium  hydrogen or e q u i l i b r i u m dueterium (and any other d i a t o m i c m o l e c u l e ) . The c o m b i n a t i o n o f F i g u r e s 3a and 3b, which have even p p a r i t y , g i v e the energy  levels  f o r parahydrogen o r o r t h o d e u t e r i u m , w h i l e  odd p p a r i t y e i g e n v a l u e s levels  i n F i g u r e s 4a and 4b would g i v e the  f o r a p o t e n t i a l o f the  W  energy  g i v e n by (2.83) have a l s o been  = y (l-cos4c|>),  t o v e r i f y the  solved  form  adsorbed m o l e c u l e i s  sites),  the  f o r orthohydrogen and p a r a d e u t e r i u m . The m a t r i x equations  (the  would  (4.19)  assumed t o s i t  assumption t h a t  i n the c e n t e r o f f o u r  a potential  o f the  form  lattice  = u (1-cos 2<b),  W  w i l l e x h i b i t the  largest  (2.57)  in-plane hindered r o t a t i o n a l e f f e c t s .  found t h a t the i n - p l a n e p e r t u r b a t i o n i n (4.19) degeneracy Thus the  of the  {k,(p,m)} s t a t e s i f m i s  lowest two odd p p a r i t y degenerate  not s p l i t by e q u a t i o n  (4.19).  o f the  as w i l l be seen i n Chapter V .  Therefore  s t a t e s i s not removed, the  adsorbed d i a t o m i c molecule w i l l not be g r e a t l y  by the i n - p l a n e p e r t u r b a t i o n .  = -1.  m  the  o f the { l , ( - l , ± l ) }  states  r = (-l)  These two s t a t e s g r e a t l y a f f e c t  if  erties  odd, i . e .  are  admolecule,  is  does not remove the ±m  (l,(-l,±l)}  p r o p e r t i e s o f the the degeneracy  It  prop-  altered  CHAPTER V. |  HEAT CAPACITIES AND ENTROPIES OF HYDROGEN AND DEUTERIUM  I n t h i s c h a p t e r t h e heat c a p a c i t i e s and e n t r o p i e s o f t h e  d i f f e r e n t s p i n s p e c i e s o f hydrogen and d e u t e r i u m w i l l be d i s c u s s e d . The h e a t c a p a c i t i e s o f t h e f r e e r o t a t o r a r e p r e s e n t e d f i r s t .  Although  29 these are well-known  , i t i s f e l t t h a t t h e subsequent d i s c u s s i o n o f  t h e e f f e c t s o f h i n d e r e d r o t a t i o n on t h e heat c a p a c i t y w i l l be more l u c i d i f the behaviour o f the C  v e r s u s T curves f o r t h e f r e e r o t a t o r  v are f i r s t examined i n some d e t a i l .  I n p a r t i c u l a r , a t t e n t i o n w i l l be  p a i d t o t h e r o l e s o f d e g e n e r a c i e s and e i g e n v a l u e s p a c i n g s i n d e t e r m i n i n g t h e o v e r a l l shape and magnitude o f heat c a p a c i t y c u r v e s . In  c o n s i d e r i n g t h e e f f e c t s o f h i n d e r e d r o t a t i o n , i t i s con-  v e n i e n t , and i n d e e d , v i r t u a l l y e s s e n t i a l f o r t h e sake o f c l a r i t y , t o t r e a t t h e o u t - o f - p l a n e h i n d r a n c e and t h e o u t - o f - p l a n e p l u s i n - p l a n e h i n d r a n c e as s e p a r a t e c a s e s .  T h i s a l l o w s us t o c o n s i d e r t h e former as  a p e r t u r b a t i o n o f t h e f r e e r o t a t o r , and t h e i n - p l a n e h i n d r a n c e as a perturbation of the out-of-plane hindered rotator. F i n a l l y t h e e n t r o p i e s o f hydrogen and d e u t e r i u m a r e d i s c u s s e d for  t h e o u t - o f - p l a n e and t h e o u t - o f - p l a n e p l u s i n - p l a n e h i n d e r e d  r o t a t i o n a l models. comparison.  The f r e e r o t a t i o n a l e n t r o p i e s a r e i n c l u d e d f o r  S p e c i a l a t t e n t i o n i s p a i d t o t h e e f f e c t o f t h e X and u  r o t a t i o n a l b a r r i e r s on t h e e n t r o p y curves at low t e m p e r a t u r e s  where  t h e n u c l e a r and r o t a t i o n a l d e g e n e r a c i e s o f t h e d i f f e r e n t n u c l e a r s p i n s p e c i e s are very important.  A) Free Rotational Heat Capacities of Hydrogen and Deuterium. I  The free r o t a t i o n a l heat capacity curves f o r hydrogen and  deuterium are given i n Figure 5.  The curves are plots of C /R as y  a function of temperature over the range 0°K < T < 190°K.  The heat 29  capacities are not i n exact agreement with those given by Farkas due to the fact that s l i g h t l y d i f f e r e n t c h a r a c t e r i s t i c r o t a t i o n a l temperatures are used:  © ( H ) = 87.49°K r  and 0 ( D ) = 43.75°K.  2  r  2  These are calculated using an equilibrium interatomic distance of o 30 .7415 A f o r hydrogen and deuterium At  low temperatures a l l the nuclear- spin species of hydrogen  (Figure 5a) behave as monatomic gases, and t h e i r r o t a t i o n a l heat capacities are zero.  The equilibrium hydrogen curve (curve 1) begins  to r i s e rapidly as the temperature i s increased from 15°K.  I t goes  through a maximum at 50°K with C /R = 2.06 and then approaches the v  high temperature c l a s s i c a l l i m i t  (C /R = 1.0). v  Curve 2 f o r para-  hydrogen reaches a maximum and then s t a r t s t o slowly decrease, approach ing the c l a s s i c a l  l i m i t from above.  The orthohydrogen curve (curve 4)  demonstrates no maximum and never exceeds the c l a s s i c a l value of C^/R. The heat capacity of normal hydrogen, given by  C (n-H ) = y  2  j.C (p-H ) + v  2  |  i s also shown i n Figure 5a by curve 3.  C (o-H ), y  2  (5.1)  As expected, curve 3 l i e s  between the parahydrogen and orthohydrogen curves, resembling the ortho hydrogen curve i n shape.  At high temperatures, i t w i l l coincide with  84  FIGURE 5  Free r o t a t i o n a l heat c a p a c i t i e s p l o t t e d temperature.  as a f u n c t i o n  Curves 1 t o 4 i n F i g u r e 5a are f o r the  of foll-  owing s p e c i e s o f hydrogen:  1.  e-H  2  2.  p-H  2  3.  n-H  2  4.  o-H  2  Curves 1 t o 4 i n F i g u r e 5b are f o r the of  deuterium: 1.  e-D  2  2.  o-D  2  3.  n-D  2  4.  p-D_  following  species  85  T(°K)  86  the curve for the heat capacity of equilibrium hydrogen, j  In Figure 5b, the equilibrium deuterium heat capacity curve  (curve 1) begins t o rise rapidly after 5°K and exceeds the c l a s s i c a l limit.  At high temperatures i t returns to the c l a s s i c a l limit as  expected.  Curve 2 for orthodeuterium also goes through a maximum,  after which i t approaches the c l a s s i c a l l i m i t from above.  For para-  deuterium, no maximum i s seen i n the heat capacity curve (curve 4) and i t never exceeds the c l a s s i c a l l i m i t .  The heat capacity of  normal deuterium, found from C (n-D ) = | C (o-D ) +. | C (p-D ), y  2  y  2  y  (5.2)  2  i s given by curve 3. This curve l i e s between the heat capacity curves for orthodeuterium and paradeuterium.  I t exhibits a maximum as does  the orthodeuterium curve and then coincides with the equilibrium deuterium curve. It should be noted that i f one were to plot C /R as a function v  of T* (=T/0 ), the parahydrogen and orthodeuterium curves would coincide as would the orthohydrogen and paradeuterium curves.  In other  words, if.the nuclear spin species i s pure, the heat capacity of the two gases at relative reduced temperature i s independent of the d i f f erences i n nuclear spin degeneracies between them. At this point a comment should be made as to the reason for the maxima that appear i n the equilibrium hydrogen, parahydrogen, e q u i l i brium deuterium, and orthodeuterium heat capacity curves. Since the  e n e r g i e s o f m o l e c u l e s are q u a n t i s e d i n t o d i s c r e t e l e v e l s , t h e h e a t c a p a c i t y w i l l depend upon t h e r e l a t i v e p r o b a b i l i t i e s o f a m o l e c u l e b e i n g i n s t a t e H o r ft' and t h e energy d i f f e r e n c e a s s o c i a t e d these s t a t e s .  with  L e t t i n g g^ denote t h e p r o d u c t o f t h e 21+1 degeneracy  o f t h e f r e e r o t a t o r e n e r g i e s and n u c l e a r s p i n degeneracy, t h e h e a t c a p a c i t y can be w r i t t e n as  -Be  ~  C£ H  H  e  3  (5.3a)  -Be,  R6'  • where B=(kT)  -1  and  K  H  ~  2 >  < e  i  > 2  (5.3b)  >  = £(£+1) ( h V 8 i r I ) •  Defining the p r o b a b i l i t y o f  f i n d i n g t h e m o l e c u l e i n t h e Ith s t a t e as  h  =  *i  e  f  \ h  e  '  (5.4)  and s u b s t i t u t i n g (5.4) i n t o (5.3a) y i e l d s  (5.5a) RB' (5.5b)  There i s nothing special about H' as compared with I  and restoring  the symmetry by averaging with the exact equivalent terms i n which  I  % and  are exchanged, equation (5.5b) becomes  C  v  ... (5.6 a)  v  7X  I  p  z  p v  ^1  ^ -  -  )2  (5  v  6b)  The double summation i n (5.6b) consists of three kinds of terms; those for which 1>V , those for which l=V , and those for which 1<V . The second kind make no contribution to the heat capacity since (e^-e^) = 0, and the other two are exactly equivalent.  2  Thus equation (5.6b) can  be rewritten as  (5.7) kT Each term i n (5.7) represents the contribution to C /R from the relative y  populations of states I and I .. That this depends on the probability 1  of occupancy of the I'th state as well as the Jlth state may seem surprising at f i r s t , but this i s required by the principle of microscopic r e v e r s i b i l i t y . Heat capacity curves for some of the important terms i n equation (5.7) have been calculated for parahydrogen, orthohydrogen,  e q u i l i b r i u m h y d r o g e n , and e q u i l i b r i u m d e u t e r i u m .  They are  as a f u n c t i o n o f temperature i n F i g u r e s 6 and 7.  Also included in  I  each f i g u r e  is  contributions for  plotted  the t o t a l heat c a p a c i t y , which o f course i n c l u d e s and not merely those i n d i v i d u a l ones p l o t t e d .  o r t h o - and p a r a d e u t e r i u m are o m i t t e d ,  t o those f o r p a r a -  since  function of r e l a t i v e  Curves  they w i l l be  and orthohydrogen r e s p e c t i v e l y  identical  i f plotted  as a  reduced t e m p e r a t u r e .  For parahydrogen ( F i g u r e 6a)  almost the e n t i r e  t o the heat c a p a c i t y between 40°K and 190°K i s  contribution  due t o the  relative  populations  o f the £ = 0 and 1=2 s t a t e s which w i l l be d e s i g n a t e d  £=0; £ ' = 2 .  The £ = 0 l e v e l  state  ( £ = 2 ) has  is  non-degenerate,  a degeneracy  o f g^ = 5.  w h i l e the  This  i n C ^ / R at a temperature  molecules  t o . b e e x c i t e d t o the £ = 2 s t a t e .  rush,  there i s  states  (while  2  =  5  Q  '  g  4/g  =  9  2  and the heat c a p a c i t y d e c l i n e s . from £ = 0 ; £ ' = 4 become  /  5  ^  After this  (  - 4" 0£  £  ) / ( e  2 ~ (P e  =  C 3~ i  3  = 14/5,  excited  state).  £  e  plotted is levels,  just  of  sudden  to s t i l l 1  0  higher /  3  )  At around 140°K a s m a l l c o n t r i b u t i o n  apparent.  The heat c a p a c i t y f o r orthohydrogen ( F i g u r e 6b) peculiar behavior.  excited -  at which l a r g e numbers  no g r e a t tendency t o e x c i t e molecules S /g  first  as  leads t o a r e l a t i v e l y  sudden i n c r e a s e start  all  (Note t h a t i n t h i s  so no s p e c i a l  case g j / g  =  7/3  shows no such and (e^-e^)/  c o n s i d e r a t i o n a t t a c h e s t o the  first  The e n t i r e heat c a p a c i t y over the temperature due t o the r e l a t i v e p o p u l a t i o n s  the p r o b a b i l i t y o f occupancy o f the  range  o f the £ = 1 and £ ' = 3  l a t t e r slowly  increasing  FIGURE 6  C o n t r i b u t i o n s to the r o t a t i o n a l heat c a p a c i t i e s from the r e l a t i v e w i t h the  curves  populations  l a b e l l e d as 1.  F i g u r e 6b i s  £;£'.  o f hydrogen  F i g u r e 6a i s  for p-H^  follows:  2.  Total C /R v £=0;£'=2  3.  £=0;£'=4  f o r o-H^.  The s i n g l e  curve i s  f o r the  h e a t c a p a c i t y and the heat c a p a c i t y f o r £ = 1 ; £ ' = 3 . is  for e-H  2  and the  curves  are l a b e l l e d  1.  Total C y R  2.  £=0;£'=1  3.  £=0;£'=2  4.  £=1;£'=2  as  follows:  total F i g u r e 6c  91  92  FIGURE 7  C o n t r i b u t i o n s t o C^/R  f o r e q u i l i b r i u m deuterium.  are l a b e l l e d as f o l l o w s : 1.  T o t a l heat  2.  SL=0;V=1  3.  £=0;£'=2  4. •  =2  capacity  The  curves  93  with temperature. The h e a t  c a p a c i t y o f e q u i l i l r i u m hydrogen ( F i g u r e 6c)  v e r y l a r g e maximum at about 5 0 ° K . i n t o the £ ' = 1 l e v e l considerable  from the ground s t a t e .  increase  in C / R . v  b u t i o n from £ = 0 ; £ ' = 1 d e c r e a s e s . now c o n t r i b u t e and the heat The interesting  Thii' i s  shows a  caused by the sudden rush S i n c e g^/g  At temperatures  9 there i s  =  a  above 55°K the c o n t r i -  Promotions t o h i g h e r e x c i t e d  states  c a p a c i t y approaches C / R = 1 . v  e q u i l i b r i u m deuterium curves  ( F i g u r e 7)  are r a t h e r  i n t h a t the r e l a t i v e p r o b a b i l i t i e s £ = 0 ; £ ' = 1 and £ = 0 ; £ ' = 2  b o t h show d i s t i n c t maxima w i t h the c o n t r i b u t i o n from £ = 0 ; £ ' = 2 b e i n g c o n s i d e r a b l y l a r g e r and at h i g h e r temperatures contribution.  T h i s can be e x p l a i n e d by the s t a t e d e g e n e r a c i e s .  £ = 0 l e v e l has a t o t a l state  (£'=1)  increase start  degeneracy  had a degeneracy  o f g^ = 6 w h i l e the f i r s t  o f g^ = 9.  t o be e x c i t e d  to the £ ' = 1 s t a t e .  to decrease.  ing  p o p u l a t i o n o f the £ ' = 2 l e v e l begins  for  the v a l u e  = 3.  t o the £ ' = 2 l e v e l because &2/g  (.e2~s )/(.ZJ-SQ)  The heat  = 9/6  However, u n l i k e e q u i l i b r i u m hydrogen, t h e r e i s  and  Q  S i n c e g, ,  molecules of  Above 35°K the c o n t r i b u t i o n from £ = 0 ; £ ' = 1  s t r o n g tendency t o e x c i t e molecules  v  a sudden  y  v  C /R.  Thus t h e r e i s  The  excited  i n C / R at a temperature at which l a r g e numbers o f  C / R i s not v e r y h i g h . starts  than the £ = 0 ; £ ' = 1  Thus at temperatures  above 20°K the  a =  3  increas-  to contribute s i g n i f i c a n t l y  to  c a p a c i t y curve f o r £=0;" £ ' = 2 has a l a r g e r maximum t h a n  £ = 0 ; £ ' = 1 and so i t  is  maximum o f the t o t a l heat  r e a l l y the former which p o s i t i o n s capacity curve.  the  The £ = 0 ; £ ' = 1 c o n t r i b u t i o n  causes a s l i g h t heat  s h o u l d e r at about 3C°K and o f course causes the  total  c a p a c i t y t o be n o n - z e r o around 1 0 ° K .  B) O u t - o f - P l a n e H i n d e r e d R o t a t i o n a l Heat C a p a c i t i e s f o r Hydrogen and Deuterium. V a l u e s o f C / R f o r e q u i l i b r i u m hydrogen are found i n T a b l e  11  y  and h e a t  c a p a c i t y curves f o r X = 0 , 5 , 1 0 ,  f u n c t i o n o f temperature i n F i g u r e 8a. rotation increases,  c a p a c i t y curves w i l l  h i n d e r e d r o t a t o r o r d e r i n g (see  the free  curves  1-3  above a l l t h e s e .  c a p a c i t y above the f r e e  f i n a l l y decrease  The approach t o c l a s s i c a l  o f F i g u r e 8a,  classical  rotator  capacity of  and approach the  ordering i s  and at h i g h e r temperatures  (e.g.  d e u t e r i u m ) , the heat  b a r r i e r s w i l l behave more c l a s s i c a l l y  evident  curve 4 w i l l  in lie  capacity for large  at temperatures  around 1 9 0 ° K .  F i g u r e s 8b and.8c are shown c o n t r i b u t i o n s to the  c a p a c i t y from some o f the r e l a t i v e p o p u l a t i o n s { k ' , ( p , m ' ) } ) between the ,  At  F o r d i a t o m i c gases w i t h s m a l l e r c h a r a c t e r i s t i c  r o t a t i o n a l temperatures  In  left.  approach the  At v e r y h i g h temperatures the heat  c l a s s i c a l hindered rotator w i l l rotator limit.  t o the  Appendix C ) , i n which the l a r g e r the  the g r e a t e r the heat  v  to  c a p a c i t y become enhanced  and the maximum a l s o s h i f t s  h i g h temperatures the heat  v a l u e o f C / R = 1.0.  As the b a r r i e r h e i g h t  c o n t r i b u t i o n s t o the heat  at lower t e m p e r a t u r e s ,  b a r r i e r height,  and 40 are p l o t t e d as a  (designated  lower energy l e v e l s o f an  h i n d e r e d r o t a t o r f o r X=l and X=20.  The r e a d e r w i l l  what f o l l o w s  level  t o r e f e r t o the energy  as  heat {k,(p,m)};  out-of-plane find i t helpful i n  diagram g i v e n i n F i g u r e  1.  96  TABLE 11 ) Out-of •plane hindered rotational heat capacities for e-L^. T( K)  A=0  A=l  A=5  A=10  A=20  A=40  A=80  10.0  .0.0001  0.0001  0.0015  0.0067  0.0199  0.0347  0.0459  15.0  0.0105  0.0136  0.0601  0.1477  0.2792  0.3836  0.4486  20.0  0.1089  0.1238  0.3164  0.5686  0.8407  1.0089  1.0985  25.0  0.3956  0.4220  0.7521  1.0907  1.3564  1.4769  1.5277  30.0  0.8512  0.8764  1.1979  1.4666  1.5971  1.6171  1.6112  . 35.0  1.3475  1.3582  1.5164  1.6139  1.5879  1.5252  1.4801  45.0  1.9900  1.9641  1.6931  1.4595  1.2567  1.1352  1.0704  55.0  2.0226  1.9846  1.5309  1.1541  0.9175  0.8086  0.7557  65.0  1.7510  1.7202  1.3045  0.9166  0.6867  0.6007  0.5626  75.0  1.4368  1.4175  1.1122  0.7681  0.5467  0.4759  0.4492  85.0  1.1774  1.1674  0.9695  0.6848  0.4682  0.4024  0.3831  90.0  1.0742  1.0677  0.9147  0.6600  0.4451  0.3784  0.3617  100.0  0.9155  0.9140  0.8318  0.6332  0.4221  0.3485  0.3345  120.0  0.7417  0.7448  0.7449  C.6352  0.4370  0.3374  0.3202  140.0  0.6860  0.6905  0.7261  0.6757  0.4973  0.3668  0.3355  160.0  0.6930  0.6976  0.7469  0.7341  0.5785  0.4199  0.3656  190.0  0.7528  0.7570  0.8108  0.8308  0.7094  0.5205  0.4227  C  1.0 5.0  FIGURE 8  Out-of-plane hindered r o t a t i o n a l heai capacities f o r equilibrium hydrogen. on C /R. v  Figure 8a shows the effect of X  The curves are labelled as follows: 1.  X=0  2.  X=5  3.  X=10  4. X=40 Contributions to the heat capacity f o r X=l are given i n Figure 8b. Relative populations are designated by {k,(p,m)};{k',(p ,m')}. 1  The curves are labelled as follows  1.  Total C /R v  2.  {l.C+1,0)}; {1,(-1,±1)}  3.  {1,01,0)}; {1,(-1,0) }  4.  (1,C 1,0)}; {1,(+1,±2)} +  Contributions f o r X=20 are shown i n Figure 8c. The curves labelled as follows: 1. 2.  Total C /R v {1,01,0)}; {1,(-1,±1)}  98  T ( ° K )  99  For  a b a r r i e r height  from { l , ( + l , 0 ) } ; pared  to nine  o f A = l , the f i r s t  {l,(-l,±l)}.  f o r free  in  curve 2 reaches  ever,  there  two l e v e l s f o r the f r e e  and the { l , ( - l , 0 ) }  the { l , ( - l , ± l ) }  Now t h e degeneracy  b u t i o n from t h e s e r e l a t i v e  The heat  l e v e l s i s not v e r y l a r g e so t h a t  i s considerably  l e s s than  that  capacity  i s dependent on  whereas f o r A=0 o n l y the lowest  contri-  ( F i g u r e 8 c ) , o n l y the c o n t r i b u t i o n from the two  controls  the h i g h e r l e v e l s s t a r t C / R now occurs  i s important at low t e m p e r a t u r e s .  adding i n t h e i r c o n t r i b u t i o n s .  The maximum  T h i s i s due t o the f a c t  A i n c r e a s e s the energy d i f f e r e n c e  between these two s t a t e s  F i g u r e 1) s o t h a t  This  the heat c a p a c i t y up t o 55°K at which p o i n t  at about 3 2 ° K .  (see  (see  the c o n t r i -  Thus the b e h a v i o r o f the heat  {1,(+1,0)};{1, ( - 1 , ± 1 ) }  term c o m p l e t e l y  v  How-  important. For A=20  states  capacity  f o r A=l the s p l i t t i n g  e q u i l i b r i u m hydrogen f o r A=l at low temperatures  two i m p o r t a n t c o n t r i b u t i o n s  in  that  r a t i o i s only three  populations  from ( l , ( + 1 , 0 ) } ; { 1 , ( - 1 , ± 1 ) } .  bution i s  popula-  i s another important c o n t r i b u t i o n due t o p o p u l a t i n g the  of  of  the c o n t r i b u t i o n  a maximum o f C / R = 1.48 and then d e c r e a s e s .  T h i s i s owing t o the f a c t  1).  so t h a t  rotator.  state {1,(-1,0)}.  Figure  r a t i o i s s i x (com- .  as the c o n t r i b u t i o n from the r e l a t i v e  v  o f the f i r s t  Here the degeneracy  e q u i l i b r i u m hydrogen]  t o C / R i s n o t as g r e a t tions  important c o n t r i b u t i o n comes  lower temperatures  that  as  decreases  are r e q u i r e d f o r the promo-  t i o n o f molecules from the lowest s t a t e t o the f i r s t  excited  I n c r e a s i n g A a l s o i n c r e a s e s t h e energy s e p a r a t i o n between  state.  (l,(+l,0)}  100  and  { l , ( - l > . 0 ) } so t h a t the c o n t r i b u t i o n from p o p u l a t i n g the { l , ( - l , 0 ) }  s t a t e d e c r e a s e s i n importance at low t e m p e r a t u r e s . why o n l y t h e { l , ( + l , ± l ) } temperature heat  This  explains  s t a t e i s o f major importance f o r the low  c a p a c i t y o f e q u i l i b r i u m hydrogen f o r A=20.  The parahydrogen heat  capacities  are g i v e n i n T a b l e 12 and  are p l o t t e d f o r A=0, 10, 20 and 40 i n F i g u r e 9a over the temperature range 0 ° K < T < 3 8 0 ° K .  The l a r g e r the b a r r i e r ,  c a p a c i t y o f parahydrogen at temperatures below r o t a t i o n a l temperature.  about the c h a r a c t e r i s t i c  There i s then a r e - o r d e r i n g o f the c u r v e s , and  at i n t e r m e d i a t e temperatures lowest.  the g r e a t e r the heat  the curve f o r the l a r g e s t b a r r i e r  As the b a r r i e r h e i g h t  increases,  t h e maximum i n the p a r a -  hydrogen curve d e c r e a s e s and e v e n t u a l l y d i s a p p e a r s It  is interesting  curve o f l a r g e s t  A l y i n g highest  o r d e r e d at 3 8 0 ° K ) .  at  (only the A=40 curve i s n o t c o r r e c t l y  It is for this  reason t h a t  the heat  since  this  capacities  190°K.  some o f t h e c o n t r i b u t i o n s t o the heat the r e l a t i v e populations  which have a degeneracy {l,(+l,0)}  of  r e - o r d e r i n g i s not  In F i g u r e 9b and 9 c , f o r A=l and A=20 r e s p e c t i v e l y ,  A=l  the f r e e  and b e g i n t o r e - o r d e r a g a i n w i t h the  parahydrogen were c a l c u l a t e d up t o 3 8 0 ° K , evident  as shown by curve 4.  t o note t h a t the h i n d e r e d curves r e - c r o s s  r o t a t o r curve at h i g h temperatures  lies  ; {l,(+l,±l)}  one b e c a u s e ,  are shown  capacity o f parahydrogen.  of states  {l,(+l,0)}  For  and { l , ( + 1 , ± 2 ) } ,  r a t i o o f two, c o n t r i b u t e the most t o C ^ / R . contributes  as seen i n F i g u r e  slightly  1, E ( ]  + 1 , ± 1  )  i  l e s s than the p r e v i o u s s  s l i g h t l y greater  than  101  TABLE 12 Out-of- •plane h i n d e r e d T(°K)  A=0  r o t a t i o n a l hea" c a p a c i t i e s  A=l  X=S  A=10  A=20  for  p-H^*, A=40  A=80  1.0 5.0 10.0 15.0 20.0 25.0  0.0001  30.0  0.0002  0.0005  0.0009  0.0012  35.0  0.0003  0.0004  0.0006  0.0013  0.0025  0.0040  0.0053  45.0  0.0058  0.0059  0.0078  0.0122  0.0197  0.0278  0.0342  55.0  0.0326  0.0327  0.0369  0.0482  0.0680  0.0881  0.1032  65.0  0.1010  0.1009  0.1041  0.1196  0.1513  0.1842  0.2083  75.0  0.2214  0.2207  0.2164  0.2264  0.2607  0.3016  0.3318  85.0  0.3882  0.3866  0.3683  0.3607  0.3826  0.4224  0.4542  90.0  0.4859  0.4818  0.4550  0.4345  0.4443  0.4795  0.5103  100.0  0.6866  0.6836  0.6390  0.5878  0.5633  0.5813  0.6068  120.0  1.0667  1.0627.  0.9913  0.8792  0.7655  0.7244  0.7283  140.0  1.3277  1.3239  1.2480  1.1103  0.9095  0.7966  0.7685  160.0  1.4474  1.4447  1.3833  1.2399  1.0024  0.8253  0.7604  190.0  1.4400  1.4395  1.4131  1.3137  1.0766  0.8378  0.7179  continued  102  TABLE 12 (continued)  T(°K)  X=Q  X=l  X=5  X=1Q  A=20  X=40  X=80  200.0  1.4113  1.4113  1.3962  1.3155  1.0910  0.8413  0.7046  240.0  1.2622  1.2638  1.2819  1.2692  1.1212  0.8648  0.6745  280.0  1.1401  1.1423  1.1758  1.2067  1.1367  0.9083  0.6865  320.0  1.0630  1.0653  1.1037  1.1575  1.1486  0.9612  0.7260  380.0  1.0000  1.0015  1.0357  1.1027  1.1549  1.0343  0.8031  f o r o-D  at o n e - h a l f the temperature.  * These v a l u e s are a l s o a p p l i c a b l e  FIGURE 9 O u t - o f - p l a n e h i n d e r e d r o t a t i o n a l heat c a p a c i t i e s f o r p a r a hydrogen p l o t t e d o v e r t h e temperature range 0°K < T < 380°K. F i g u r e 9a shows t h e e f f e c t o f X on C /R. The c u r v e s a r e l a b e l l e d as f o l l o w s :  Contributions F i g u r e 9b.  Contributions  1.  X=0  2.  X=10  3.  X=20  4.  X=40  t o t h e heat c a p a c i t y  f o r X=l a r e g i v e n i n  The c u r v e s a r e l a b e l l e d as f o l l o w s : 1.  T o t a l C /R v'  2.  {1,(+1,0D};{1,(+1,±2)}  3.  {1,(+1,0)};{1,(+1,±1)}  4.  {1,(+1,0)};{2,(+1,0)}  f o r X=20 a r e shown i n F i g u r e 9c. The  c u r v e s a r e l a b e l l e d as f o l l o w s : 1.  T o t a l C /R v  2.  {1,(+1,0)};{1,(+1,±2)}  3.  {1,(+1,0)}.;{1,(+1,±1)}.  4.  {1,(+1 0)};{2,C+1,0)}  5.  {1,(+1,±2.)};{1,(+1,±1)}  J  104  105  '*  +  .  Curve 4 i n F i g u r e 9b shows the c o n t r i b u t i o n t o the heat  c a p a c i t y from t h e r e l a t i v e p o p u l a t i o n s {2,(+1,0)} s t a t e .  o f the ground s t a t e and the  Here the degeneracy r a t i o i s u n i t y and so the  heat c a p a c i t y i s much s m a l l e r than f o r the f i r s t The  reason f o r these t h r e e  c o n t r i b u t i o n s b e i n g o f importance at  a p p r o x i m a t e l y the same temperature i s t h a t of and  the f r e e  is  level.  are n o t w i d e l y s p l i t  For X=20  f o r X=l and so a l l t h r e e  ( F i g u r e 9c) o n l y the c o n t r i b u t i o n  t i o n o f the m o l e c u l e s  degeneracy  are i m p o r t a n t .  {1,(+1,0)};{1,(+1,±2)}  Then the c o n t r i b u t i o n from the promo-  from the ground s t a t e t o the second  { 1 , ( + 1 , ± 1 ) } becomes i m p o r t a n t .  the e f f e c t  fold  However as seen i n F i g u r e 1, the energy  i m p o r t a n t up t o about 8 0 ° K .  state  the f i v e  r o t a t o r i s p a r t l y removed t o two s e t s o f ±m degeneracy  one nondegenerate  levels  two c o n t r i b u t i o n s .  o f the b a r r i e r h e i g h t  excited  T h i s can a g a i n be e x p l a i n e d by  on the e n e r g i e s o f the t h r e e  lowest  eigenstates. The heat c a p a c i t i e s for of  several  different  temperature  o f orthohydrogen are g i v e n i n T a b l e 13  barrier heights,  and are p l o t t e d  f o r . X = 0 , l , 5, 10 and 40 i n F i g u r e 10a.  as a f u n c t i o n A few o f the  i m p o r t a n t c o n t r i b u t i o n s to C a r e shown f o r X=l and 20 i n F i g u r e s 10b • y  and  10c, r e s p e c t i v e l y .  Note t h a t the i n t r o d u c t i o n o f a s m a l l b a r r i e r  l e a d s t o a pronounced maximum at low t e m p e r a t u r e s ,  owing t o the v e r y  small difference  and { 1 , ( - 1 , 0 ) }  states  i n energy between the { l , ( - l , ± l ) }  (see F i g u r e 1 ) , and the f a c t  {l,(-l,±3)}  t h a t the next lowest  state  i s n o t s i g n i f i c a n t l y p o p u l a t e d u n t i l much h i g h e r tempera-  106  TABLE 13  O u t - o f - p l a n e h i n d e r e d r o t a t i o n a l heav c a p a c i t i e s f o r o-H *. T(°K)  A =0  X=l  X= 5  X=10  A=20  A =40  A=80  1.0 5.0  0.0230  10.0  0.1809  15.0  0.2402  0. 0008  20.0  0.2244  0. 0078  25.0  0.1904  0. 0270  0 .0002  30.0  0.1575  0. 0574  0 .0014  35.0  0.1301  0. 0936  0 .0046  45.0  0.0910  0. 1622  0 .0211  0 .0003  55.0  0.0664  0. 2096  0 .0511  0 .0020  0. 0002  0 .0002  65.0  0. 0006  0.0507  0. 2342  0 .0892  0 .0073  0. 0011  0 .0014  75.0.  0. 0027  0.0418  0. 2431  0 .1295  0 .0185  0. 0042  0 .0048  85.0  0. 0084  0.0396  0. 2442  0 .1688  0 .0373  0. 0112  0 .0120  90.0  0. 0132  0.0414  0. 2441  0 .1877  0 .0498  0. 0168  0 .0174  100.0  0. 0283  0.0515  0. 2460  0 .2249  0 .0810  0. 0325  0 .0322  120.0  0. 0842  0.1005  0. 2699  0 .3014  0 .1660  0. 0847  0 .0778  140.0  0. 1744  0.1863  0. 3282  0 .3880  0 .2732  0. 1615  0 .1398  160.0  0. 2885  0.2975  0. 4149  0 .4860  0 .3920  0. 2546  0 .2101  190.0  0. 4724  0.4786  0. 5678  0 .6406*  0 .5709  0. 4051  0 .3171  continued  107  TABLE 13 (continued) f  T(°K)  A=0  200.0  0.5332  240.0  A=l  A=5  A=10  A=20  A=40  0.5388  0.6208  0.6924  0.6291  0.4563  0 .3531  0.7320  0.7359  0.7981  0.8659  0.8276  0.6399  0.4803  280.0  0.8610  0.8641  0.9156  0.9829  0.9721  0.7892  0 .5891  320.0  0.9331  0.9356  0.9809  1.0496  1.0679  0.9051  0.6829  380.0  0.9782  0.9803  1.0190  1.0889  1.1462  1.0302  0.8021  *These values are also applicable for p-D  A=80  at one-half the temperature.  FIGURE 10 Out-of-plane hindered hydrogen.  r o t a t i o n heat c a p a c i t i e s f o r o r t h o -  F i g u r e 10a shows t h e e f f e c t o f A on C /R. v  The  c u r v e s a r e l a b e l l e d as f o l l o w s : 1.  A=0  2.  A=l  3.  A=5  4.  A=10  5.  A=40  C o n t r i b u t i o n s t o t h e heat c a p a c i t y f o r A=l a r e g i v e n i n F i g u r e 10b.  The c u r v e s a r e l a b e l l e d as f o l l o w s : 1. 2.  T o t a l C /R v {1,(-1,±1)};{1,(-1,0)}  3.  {l,(-l,±lj};{l,(-l,±3)}  C o n t r i b u t i o n s t o C^/R f o r A=20 a r e shown i n F i g u r e 10c where the c u r v e s have t h e same l a b e l l i n g as t h o s e i n F i g u r e 10b.  109  T(°K )  110  tures are reached.  As X i s increased, the energy difference  -  increases quite r a p i d l y , while E ^  decreases.  Furthermore, the degeneracy r a t i o s for these two pairs  v  - E^  of states are -j and l , respectively, so that population of the {1,(-1,0)} state i s not so strongly favoured r e l a t i v e to the {l,(-l,±3)} state.  This then leads to the disappearance of the maximum i n C as y  the b a r r i e r to r o t a t i o n i s increased. In Figure 11 are given the heat capacity curves of normal hydrogen calculated from equation (5.1) and the out-of-plane  hindered  r o t a t i o n a l heat capacities of parahydrogen and orthohydrogen l i s t e d i n Tables 12 and 13 respectively, f o r X =0, 1, 4 and 40.  At temperatures  below 40°K there i s no contribution to C /R from parahydrogen so that v  i n t h i s region  C  y  (n-H ) = 2  C (o-H ) . v  2  For X = 5, the normal hydrogen  heat capacity does not show the plateau that orthohydrogen does because of the contribution from parahydrogen above 40°K. noted that at high a monotonically  I t should also be  ( c l a s s i c a l ) temperatures, the heat capacity w i l l be  increasing function of b a r r i e r height.  Values o f the out-of-plane  hindered  r o t a t i o n a l heat capacities  of equilibrium deuterium are given i n Table 14. Heat capacity curves for X=0, 5, 10 and 40 are p l o t t e d i n Figure 12a as a function of temperature and some of the important contributions are indicated i n Figures  12b and 12c f o r X=l and X=20, respectively.  It i s interesting  to note the development of a shoulder and f i n a l l y a plateau between 20°-40°K with increasing X.  This i s caused by several f a c t o r s .  Firstly,  FIGURE  11  O u t - o f - p l a n e h i n d e r e d r o t a t i o n a l heat c a p a c i t y c u r v e s f o r normal hydrogen.  The c u r v e s a r e l a b e l l e d as  1.  A=0  2.  A=l  3.  A=5  4.  A=40  follows:  112  113  TABLE 14 Out-of-•plane hindered rotational heat capacities for e-D^. T(°K)  A-0  A=l  A=5  A=10  A=20  A=40  A=80  0.0003  0.0011  0.0033  0.0058  0.0077  1.0 5.0 10.0  0.0182  0.0207  0.0534  0.0975  0.1480  0.1816  0.2006  15.0  0.1483  0.1533  0.2225  0.2954  0.3538  0.3825  0.3961  20.0  0.3499  0.3511  0.3808  0.4185  0.4424  0.4497  0.4521  25.0  0.5219  0.5182  0.4841  0.4669  0.4599  0.4565  0.4554  30.0  0.6436  0.6371  0.5576  0.4943  0.4662  0.4618  0.4627  35.0  0.7368  0.7293  0.6276  0.5312  0.4842  0.4805  0.4844  45.0  0.9013  0.8947  0,7878  0.6517  0.5550  0.5383  0.5431  55.0  1.0432  1.0385  0.9481  0.7970  0.6433  0.5886  0.5847  65.0  1.1369  1.1342  1.0690  0.9254  0.7283  0.6227  0.6007  75.0  1.1784  1.1774  1.1390  1.0199  0.8036  0.6486  0.6004  85.0  1.1811  1.1815  1.1672  1.0807  0.8681  0.6740  0.5948  90.0  1.1734  1.1742  1.1702  1.1007  0.8966  0.6878  0.5924  100.0  1.1485  1.1501  1.1627  1.1254  0.9468  0.7187  0.5914  120.0  1.0915  1.0938  1.1259  1.1387  1.0255  0.7910  0.6105  140.0  1.0481  1.0506  1.0900  1.1329  1.0823  0.8688  0.6541  160.0  1.0199  1.0222  1.0629  1.1217  1.1218  0.9426  0.7116  190.0  1.0000  1.0000  1.0302  1.0981  1.1520  1.0329  0.8028  FIGURE 12 O u t - o f - p l a n e h i n d e r e d r o t a t i o n a l heat c a p a c i t i e s briura d e u t e r i u m .  f o r equil]  F i g u r e 12a shows t h e e f f e c t o f X 6 n C /R  The c u r v e s a r e l a b e l l e d as f o l l o w s :  1.  X=0  2.  X=S  3.  X=10  4.  X=40  C o n t r i b u t i o n s t o t h e heat c a p a c i t y  f o r X=l a r e g i v e n i n  F i g u r e 12b and t h e c u r v e s a r e l a b e l l e d as f o l l o w s :  1.  Total  C /R v  2. 3.  {1,(+1,0)};{1,(-1,±1)} ' {1,(+1,0)};{1,(-1,0)}  4.  {1,( 1,0)};{1 (+1,±2)}  5.  {l,(+l,0)}j{l,-(+l,±l)}  +  In F i g u r e 12c t h e c o n t r i b u t i o n s  V  f o r A=20 a r e shovim.  c u r v e s a r e l a b e l l e d t h e same as t h o s e i n F i g u r e 12b.  The  115  T ( ° K )  116  *i.  u  •  •  4-u  A-ce  '  c ("I.*!)  E (+1,0)  as t h e b a r r i e r i n c r e a s e s t h e energy d i f f e r e n c e  - E^  d e c r e a s e s , so t h a t t h e {1,(-1,±1)} s t a t e can be p o p u l a t e d a t l o w e r i  temperatures.  S e c o n d l y , t h e p r o b a b i l i t y o f p o p u l a t i n g t h e {l,(+l,±2)}  s t a t e i s enhanced r e l a t i v e t o t h a t o f t h e s t a t e { 1 , ( - 1 , 0 ) } , t h e l a t t e r becoming q u i t e u n i m p o r t a n t a t l a r g e b a r r i e r h e i g h t s .  T h i s i s due b o t h  t o t h e f a c t t h a t t h e s e l e v e l s c r o s s at about X=10, w i t h  {1,(-1,0)}  now l y i n g h i g h e r ( s e e F i g u r e 1 ) , and t o a f o u r f o l d d i f f e r e n c e i n degeneracies. by comparing  The importance o f degeneracy  r a t i o s can a l s o be seen  c u r v e s 3 and 5 o f F i g u r e 12c. Thus, a l t h o u g h t h e  {1,(+1,±1)> s t a t e s t i l l  l i e s above t h e { l , ( - l , 0 ) } s t a t e , t h e p r o b a b i l i t y  o f p o p u l a t i n g t h e f o r m e r l e v e l i s g r e a t e r at a l l i m p o r t a n t t e m p e r a t u r e s because o f i t s degeneracy  and i t t h e r e f o r e g i v e s a l a r g e r  contribution  to C . y  H i n d e r e d r o t a t i o n a l h e a t c a p a c i t i e s f o r o r t h o d e u t e r i u m and p a r a d e u t e r i u m may be found i n T a b l e s 12 and 13 r e s p e c t i v e l y .  The  b e h a v i o r o f t h e h e a t c a p a c i t y need n o t be d i s c u s s e d s i n c e as we have a l r e a d y mentioned,  o r t h o d e u t e r i u m (paradeuterium) and parahydrogen  ( o r t h o h y d r o g e n ) have t h e same h e a t c a p a c i t y i f C /R i s p l o t t e d as a v  f u n c t i o n o f r e l a t i v e reduced t e m p e r a t u r e , T* (=T/0 ) .  F o r example,  t h e maximum t h a t curve 2 i n F i g u r e 10a d i s p l a y s at 15°K f o r o r t h o hydrogen w i l l o c c u r a t 7.5°K f o r p a r a d e u t e r i u m . C) O u t - o f - P l a n e and I n - P l a n e H i n d e r e d R o t a t i o n a l Heat C a p a c i t i e s f o r Hydrogen and Deuterium. V a l u e s o f t h e heat c a p a c i t y o f e q u i l i b r i u m hydrogen a r e  given i n Table  15a f o r X=l and T a b l e 15b f o r X=10 at v a r i o u s  p f the i n - p l a n e b a r r i e r .  Figure  13a shows the heat  as a f u n c t i o n o f temperature f o r X = l ; y=0, plane b a r r i e r i s  2 and 4.  value.s  capacity  (C /R) v  When the i n -  i n c r e a s e d the maximum i n C / R decreases and moves  v to  lower t e m p e r a t u r e s .  curves  lie  above t h e  At h i g h ( c l a s s i c a l )  curve f o r X = l ; y=0.  out-of-plane hindered rotator, must l i e  above the  C (X,u=4) v  classical  < C (X,y=8), etc.  This  r o t a t i o n a l heat  capacities  As was the case w i t h  a l l X and y h i n d e r e d heat  value of C / R = 1 . 0 v  largest  can be seen from the  out-of-plane  v  as the i n - p l a n e b a r r i e r h e i g h t  is  capacity  states  level  o f the o u t - o f - p l a n e  letters [j(p> )l r  and the o u t - o f - p l a n e  from the  and i n F i g u r e  lowest energy  o f the  eigenstates  lower  eigen-  14b f o r X=10, 0 < y < 8. ( k , ( p , m ) } are l a b e l l e d by  and i n - p l a n e h i n d e r e d r o t a t i o n a l  are l a b e l l e d by numbers. v  (curve 1 i n F i g u r e  diagrams f o r some o f the  hindered r o t a t i o n a l states  c o n t r i b u t i o n t o C / R at  lower  and i n - p l a n e h i n d e r e d r o t a t o r are g i v e n i n  F i g u r e 14a f o r X = l , 0 < y < 4 The o u t - o f - p l a n e  to  i n c r e a s e d can be e x p l a i n e d by  y  energy  capacity  and i n - p l a n e h i n d e r e d  c o n t r i b u t i o n s t o C / R from the r e l a t i v e p o p u l a t i o n s reason,  heat  g i v e n i n Appendix C .  temperature r e l a t i v e t o the r e f e r e n c e heat  For t h i s  capacities  y v a l u e has the g r e a t e s t heat  classical  the  and C ( X , y = 0 ) <  The l o w e r i n g o f the maxima and t h e i r s h i f t i n g  13a)  the h i n d e r e d  Thus f o r any g i v e n X v a l u e the  v  c a p a c i t y curve w i t h the  temperatures  For y=2 i n F i g u r e  low temperatures  results  13a,  states  the main  from the sudden rush  [1,(+1,+1)] i n t o the next h i g h e s t  state  TABLE 15a Out-of-plane and in-plane hindered rotational heat capacities of e-H^ for A=l. T(°K)  ii=0.25  y=0.50  y=i  y=2  y=3  y=4  0.0003  0.0056  1.0 5.0 10.0  0.0002  0.0004  0.0017  0.0207  0.1147  0.3522  15.0  0.0156  0.0219  0.0511  0.2143  0.5397  0.8808  20.0  0.1313  0.1539  0.2438  0.5690  0.9102  1.0237  25.0  0.4330  0.4655  0.5842  0.9007  1.0471  0.9248  30.0  0. 8849  0.9091  0.9902  1.1270  1.0372  0.7819  35.0  1.3594  1.3619  1.3617  1.2560  0.9814  0.6701  45.0  1.9514  1.9138  1.7728  1.3343  0.8919  0.5680  55.0  1.9695  1.9251  1.7626  1.2924  0.8620  0.5682  65.0  1.7093  1.6773  1.5586  1.2036  0.8594  0.6099  75.0  1.4115  1.3935  1.3254  1.1044  0.8608  0.6613  85.0  1.1649  1.1573  1.1269  1.0125  0.8583  0.7085  90.0  1.0664  1.0624  1.0456  0.9719  0.8554  0.7290  100.0  0.9144  0.9153  0.9172  0.9031  0.8477  0.7635  120.0  0.7466  0.7517  0.7699  0.8154  0.8347  0.8138  140.0  0.6926  0.6984  0.7202  0.7842  0.8372  0.8545  160.0  0.6996  0.7052  0.7261  0.7923  0.8578  0.8959  190.0  0. 7597  0.7644  0.7823  0.8433  0.9103  0.9620  .TABLE 15b Out-of-plane and in-plane hindered rotational heat capacities of e-H for A=10. 2  T(°K)  -0.25  y=0.50  y=l  U=2  y=4  y=8  1.0  0000  0.0000  0.0000  0.0000  0.0000  0.0896  5.0  0000  0.0000  0.0000  0.0074  0.4488  0.5464  10.0  0093  0.0180  0.0658  0.3887  1.0023  0.1383  15.0  1651  0.2167  0.4104  0.9104  0.6679  0.0580  20.0  5905  0.6517  0.8384  1.0201  0.4087  0.0314  25.0  0968  1.1113  1.1258  0.9059  0.2636  0.0195  30.0  4524  1.4098  1.2467  0.7640  0.1816  0.0133  35.0  5875  1.5113  1.2519  0.6585  0.1345  0.0097  45.0  4335  1.3592  1.1107  0.5613  0.1010  0.0070  55.0  1390  1.0953  0.9420  0.5500  0.1229  0.0116  65.0  9100  0.8906  0.8161  0.5734  0.1850  0.0286  75,0  7665  0.7617  0.7374  0.6090  0.2734  0.0625  85.0  6859  0.6891  0.6942  0.6468  0.3749  0.1142  90.0  6619  0.6675  0.6821  0.6651  0.4271  0.1462  100.0  6361  0.6446  0.6713  0.6997  0.5294  0.2199  120.0  6385  0.6488  0.6843  0.7609  0.7107  0.3914  140.0  6788  0.6886  0.7241  0.8174  0.8537  0.5702  160.0  7369  0.7457  0.7781  0.8743  0.9646  0.7381  190.0  8335  0.8407  0.8678  0.9603  1.0885  0.9549  120  FIGURE 13 Out-of-plane of  and i n - p l a n e h i n d e r e d r o t a t i o n a l h e a t  e q u i l i b r i u m hydrogen.  X=l and t h r e e v a l u e s  capacities  F i g u r e 13a g i v e s C ^ / R c u r v e s  of u.  The c u r v e s  for  a r e l a b e l l e d as  follows:  Figure  1.  X = l , y=0  2.  X = l , y=2  3.  X = l , y=4  13b shows C ^ / R c u r v e s  The c u r v e s  are  l a b e l l e d as  f o r X=10 a n d t h r e e v a l u e s o f y .  follows:  1.  X = 1 0 , y=0  2.  X=10, y=2  3.  X=10,  y=8  FIGURE 14a Energy l e v e l  diagram f o r some o f the lower e i g e n s t a t e s o f  the o u t - o f - p l a n e  and i n - p l a n e h i n d e r e d r o t a t o r w i t h X=l  and u=0 t o 4. The energy of  X , 0 < X < 1.  are p l o t t e d  The o u t - o f - p l a n e  s t a t e s {k,(p,m)} a r e l a b e l l e d by l e t t e r s and i n - p l a n e h i n d e r e d r o t a t i o n a l  a r e l a b e l l e d by numbers.  (k. (p,m)} a.  as a f u n c t i o n  as a f u n c t i o n o f u , 0 < u < 4.  the o u t - o f - p l a n e  [j,(p,r)]  plotted  Then X i s h e l d c o n s t a n t and the e n e r g i e s  hindered r o t a t i o n a l and  is first  {l  a  Ol,0)}  [j, (P,r)] 1.  [1, (+1,+1  b.  '{1, ( - 1 , ± 1 ) }  2.  [1, ( - 1 , - 1  c.  {1, (-1,0)}  3.  [1, C-1,+1  d.  {1, ( + 1 , ± 2 ) }  4.  [2, ( - 1 , - 1  e.  '{1, ( + 1 , ± 1 ) }  5.  [1, (+1,-1  f.  ( 2 (+1,0)}  6.  [2,  7.  [3, (+1.+1  8.  [4, (+1, 1  9.  [2, (+1,-1  S  ( i, i +  +  +  states  Energy o  ro  (J)  oo  o  FIGURE 14b Energy l e v e l  diagram f o r some o f the  out-of-plane  and i n - p l a n e h i n d e r e d r o t a t o r w i t h X=10 and u=0  to 8.  The energy  0 < X < 10. plotted the  is  first  plotted  same as those i n F i g u r e  and the e n e r g i e s  0 < u < 8. 14a.  the  as a f u n c t i o n o f X,  Then X i s h e l d c o n s t a n t  as a f u n c t i o n o f u ,  lower e i g e n s t a t e s o f  are  The e i g e n s t a t e s  are  126  [ 1 , ( - 1 , - 1 ) ] ( s e e F i g u r e 14a). o f s i x f o r t h e y=0 c u r v e .  The degeneracy r a t i o i s now t h r e e i n s t e a d  T h i s i s because t h e i n - p l a n e p e r t u r b a t i o n  removes a l i t h e ±m degeneracy so t h a t o n l y t h e n u c l e a r s p i n degeneracy o f t h e odd hydrogen s t a t e s a r e l e f t . t h e maximum.  T h i s t h e n e x p l a i n s t h e drop i n  As t h e h e a t c a p a c i t y from t h i s c o n t r i b u t i o n b e g i n s t o  decrease, the molecules  can t h e n p o p u l a t e t h e s t a t e [ 1 , ( - 1 , + 1 ) ] , which  i s t h e t h i r d lowest s t a t e at u=2.  F o r u=4, t h e two lowest s t a t e s a r e  t h e same as b e f o r e b u t t h e energy d i f f e r e n c e £ . ^ ' + 1  has decreased  ^  +1 3  from 1.08 t o 0.634. Thus t h e temperature  required to  p o p u l a t e t h e [1, (-1,-1)] s t a t e i s l e s s and t h e maximum o c c u r s at a lower t e m p e r a t u r e . promotion  The peak i s n a r r o w e r t h a n t h e one f o r u=2 because  t o t h e next h i g h e s t s t a t e s t i l l  temperature.  o c c u r s at about t h e same  T h e r e f o r e t h e maximum i s v i r t u a l l y due t o o n l y t h e r e l a t i v e  p o p u l a t i o n s o f t h e ground s t a t e and t h e [ l , ( - l , - l ) j  state.  The same e x p l a n a t i o n s h o l d t r u e f o r t h e s h i f t i n g and l o w e r i n g o f t h e maximum i n t h e heat c a p a c i t y curves i n F i g u r e 13b. the p r o m o t i o n  Here however  from [1,(+1,+1)] t o [ 1 , ( - 1 , - 1 ) ] c o n t r i b u t e s up t o temp-  e r a t u r e s around 50°K, at w h i c h p o i n t h i g h e r c o n t r i b u t i o n s w i l l become important.  T h i s i s due t o t h e l a r g e energy s e p a r a t i o n between t h e ground  s t a t e and t h e second e x c i t e d s t a t e , i . e . £ ^  _ 1 ,+ 1  1  >  +  - 6.  The peak a t 5°K f o r u=8 i s v e r y sharp due t o t h e f a c t t h a t t h e energy difference  S^" '" ^ 1  1  -£ ^ 1  + 1 ,+ 1  ^ i s o n l y 0.09.  Since the [1,(-1,-1)]  l e v e l may c r o s s t h e [1,(+1,+1)] l e v e l , t h e energy d i f f e r e n c e w i l l be z e r o a t some v a l u e o f y and t h e peak w i l l d i s a p p e a r  momentarily,  127  r e a p p e a r i n g when [ 1 , ( - 1 , - 1 ) ] Heat c a p a c i t i e s are g i v e n i n T a b l e the heat  [1,(+1, + 1 ) ] .  f o r parahydrogen at d i f f e r e n t  slightly  as y i n c r e a s e s  t e m p e r a t u r e , the heat  in-plane b a r r i e r height.  F o r A=10 (Table 16b),  y is  the heat  170°K.  and f o r A=10 t h i s  increased. T h i s explains  below 4 5 ° K , for  increased,  C / R increases v  energy  decreases  energy i n c r e a s e s  as  170°K.  4 between 0°K and 3 8 0 ° K .  curves  4 and 8.  capacity  is  F i g u r e 15b g i v e s  In g e n e r a l the C ^ / R curves  (curve 1 i n F i g u r e s 15a and 15b)  because  degeneracy has been removed and, o t h e r than the d i s c r e p a n c y mentioned  above f o r the A=l c a s e , the energy d i f f e r e n c e l a r g e r so t h a t h i g h e r temperatures i n t o these h i g h e r l e v e l s . curves  lie  eratures  above the r e f e r e n c e  reference  The  between l e v e l s becomes  are r e q u i r e d to promote the  At c l a s s i c a l  the h i n d e r e d curves  rotational  ies  every-  w i t h i n c r e a s i n g y and f o r A=10 C- / R decreases  s i m i l a r curves f o r A=10 and y=0,  all  capacity  F o r A = l , the  In F i g u r e 15a the parahydrogen r o t a t i o n a l heat  l i e below the r e f e r e n c e  How-  why f o r A = l , at any g i v e n temperature  a l l temperatures up u n t i l  p l o t t e d f o r A=l and y=0,  16a,  below 4 5 ° K .  d i f f e r e n c e between the two lowest even p p a r i t y s t a t e s as y i s  In T a b l e  c a p a c i t y decreases w i t h i n c r e a s i n g  where d e c r e a s e s f o r i n c r e a s i n g u up u n t i l  slightly  in-plane barriers  16a f o r A=l and T a b l e 16b f o r A=10.  capacity increases  e v e r above t h i s  l i e s bslow  curves.  temperatures  molecules  a l l y-hindered  However at i n t e r m e d i a t e  l i e below the o u t - o f - p l a n e  temp-  hindered  curves.  out-of-plane  and i n - p l a n e h i n d e r e d r o t a t i o n a l heat  o f orthohydrogen are g i v e n f o r v a r i o u s y v a l u e s  capacit-  i n T a b l e 17a f o r  TABLE 16a jOut-of-plane and in-plane hindered rotational heat capacities of p-H for X=l*. 2  y=0.25  y=0.50  y=l  y=2  y=3  y=4  35.0  0 0004  0.0004  0.0004  0.0005  0.0006  0.0007  45.0  0 0059  0.0059  0.0060  0.0063  0.0067  0.0072  55.0  0 0326  0.0326  0.0324  0.0319  0.0314  0.0312  65.0  0 1008  0.1004  0.0990  0.0944  0.0891  0.0844  75.0  0 2203  0.2193  0.2155  0.2025  0.1867  0.1720  85.0  0 3860  0.3841  0.3770  0.3523  0.3214  0.2915  90.0  0 4811  0.4788  0.4699  0.4389  0.3995  0.3607  100.0  0 6826  0.6795  0.6674  0.6245  0.5683  0.5113  120.0  1 0613  1.0574  1.0419  0.9852  0.9063  0.8208  140.0  1 3227  1.3192  1.3052  1.2520  1.1728  1.0802  160.0  1 4440  1.4417  1.4324  1.3949  1.3335  1.2541  190.0  1 4393  1.4391  1.4377  1.4290  1.4060  1.3648  T(°K) 1.0  5.0 10.0 15.0 20.0 25.0 30.0  continued  TABLE 16a (continued) T(°K)  y==0.25  V- :0.50  y=i  y=2  y=3  y=4  200.0  l . 4114  1.4117  1 .4126  1 .4122  1.4012  1.3734  240.0  l . 2643  1.2660  1.2726  1.2945  1.3184  1.3331  280.0  l . 1429  1.1451  1.1535  1.1836  1.2230  1.2604  320.0  l . 0659  1.0680  1.0766  1.1079  1.1517  1 .1983  380.0  0.9984  1.0004  1.0080  1.0369  1.0793  1.1283  *These values are also applicable for 0-D2 at one-half the temperature.  130  . TABLE 16b Out-of-Dlane and in-plane hindered rotational heat capacities of p-H for X=10*. 2  T(°K)  y=0.25  u=0.50  y=i  30.0  0.0002  0.0002  0.0002  35.0  0.0012  0.0012  45.0  0.0121  55.0  y=8  y=2  y=4  0.0010  0.0005  0.0001  0.0116  0.0104  0.0062  0.0018  0.0005  0.0477  0.0465  0.0419  0.0290  0.0110  0.0036  65.0  0.1187  0.1163  0.1075  0.0808  0.0373  0.0139  75.0  0.2250  0.2215  0.2084  0.1667  0.0886  0.0365  85.0  0.3588  0.3545  0.3380  0.2831  0.1676  0.0748  90.0  0.4325  0.4279  0.4102  0.3501  0.2169  0.1002  100.0  0.5856  0.5808  0.5619  0.4952  0.3314  0.1631  120.0  0.8771  0.8729  0.8560  0.7913  0.5972  0.3286  140.0  1.1010  1.0984  1.0874  1.0404  0.8607  0.5245  160.0  1.2386  1.2379  1.2340  1.2112  1.0784  0.7236  190.0  1.3122  1.3137  1.3189  1.3296  1.2872  0.9859  1.0 5.0 10.0 15.0 20.0 25.0  ../continued  131  TABLE 16b (continued) T(°K)  y=0.25  y=0.50  200.0  1.3137  1.3157  1.3231  1.3425  1.3285  1.0591  240.0  1.2654  1.2686  1.2809  1.3214  1.3923  1.2713  280.0  1.2008  1.2041  1.2172  1.2634  1.3742  1.3820  320.0  1.1503  1.1535  1.1657  1.2103  1.3332  1.4293  380.0  1.0957  1.0983  1.1086  1.1472  1.2662  1.4360  *  y=2  y=i  These values are also applicable for  0-D2  y=4  y=8  at one- half the temperature  132  FIGURE 15 O u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t i o n a l heat of  parahydrogen p l o t t e d over the temperature range 0°K < T  < 380°K. of  capacities  y.  F i g u r e 15a g i v e s C / R curves f o r A=l and two  The curves are l a b e l l e d as 1.  X = l , y=0  2.  X = l , y=4  follows:  In F i g u r e 15b are g i v e n C / R curves f o r A=10 and t h r e e v  of  y.  The curves are l a b e l l e d as  1.  X=10,  y=0  2.  X=10,  y=4  3.  X=10,  y=8  values  follows:  values  X=l  and T a b l e 17b f o r A=10.  C / R curves f o r u=0,  0.25,  v  temperature i n F i g u r e {l,(-l,±l)} the  is  F o r an o u t - o f - p l a n e  2 and 4 ar^ p l o t t e d as a f u n c t i o n o f  16a.  F o r u=0 the  d o u b l y degenerate  o f the  lowest s t a t e i s  between the two new s t a t e s , 0.25 for  However f o r u=0.25,  removed.  The energy  ratio is  the  is  about  one i n s t e a d o f o n e - h a l f  o f the o u t - o f - p l a n e  causes the h i g h maximum at 1 2 ° K .  into  difference  [ 1 , ( - 1 , - 1 ) ] and [ 2 , ( - 1 , - 1 ) ] ,  at u=0.25 and the degeneracy the two lowest s t a t e s  lowest odd p p a r i t y s t a t e  and so the f i r s t promotion i s  { 1 , 0 1 , 0 ) } s t a t e . ( s e e F i g u r e 14a).  degeneracy  b a r r i e r of A = l ,  hindered r o t a t o r .  As the i n - p l a n e b a r r i e r i s  This  increased  t h e s e p a r a t i o n o f these two lowest l e v e l s i n c r e a s e s .  In f a c t ,  u=2 and above,  [1,(-1,-1)]  [ 1 , ( - 1 , + 1)]  the r e l a t i v e p o p u l a t i o n s  with£  ( 1  " '  h i g h e r temperatures  1  +1  level.  the r e l a t i v e populations ^  -1  1  1.33  become i m p o r t a n t .  are r e q u i r e d f o r promotion from the  t o the second lowest  with  -£ C '" ) =  )  o f the s t a t e s  "^1^  The next  ~ 1*98'  bute t o the maximum o f curve 3,  [1,(-1,-1)]  thus making the peak r a t h e r b r o a d .  at  As the maximum f o r promotion t o t h i s  [1,(-1,+1)] state i s  reached,  the c o n t r i b u t i o n from the next  the heat  c a p a c i t y remains r e l a t i v e l y constant  states.  C / R then r i s e s  from  Both o f these promotions c o n t r i -  p o p u l a t i o n o f the  After 130°K,  Thus  and [ 2 , ( - 1 , - 1 ) ]  F o r a b a r r i e r o f u=4, low t e m p e r a t u r e s .  and  ground s t a t e  important c o n t r i b u t i o n i s  o f the s t a t e s ^  at  important  level  is  l e v e l becomes important and instead of  decreasing.  again due t o c o n t r i b u t i o n s from h i g h e r  135  TABLE 17a Out-of-plane and in-plane hindered rotational heat capacities *  of o - l ^ for A=l .  T(°K)  u=0.25  u=0.50  5.0  0.2439  0.0138  10.0  0.6032  15.0  u=l  y=2  0.3437  0.0390  0.0012  0.5878  0.6814  0.2509  20.0  0.4744  0.7377  25.0  0.3678  30.0  y=3  y=4  0.0272  0.0041  0.0007  0.5210  0.1133  0.0294  0.0085  0.6537  0.6963  0.2485  0.0880  0.0346  0.2860  0.5411  0.7558  0.3961  0.1739  0.0828  35.0  0.2257  0.4398  0.7376  0.5258  0.2730  0.1483  45.0  0.1483  0.2935  0.6107  0.6828  0.4628  0.3008  55.0  0.1038  0.2044  0.4752  0.7162  0.5981  0.4428  65.0  0.0768  0.1492  0.3680  0.6786  0.6687  0.5513  75.0  0.0609  0.1150  0.2903  0.6132  0.6882  0.6219  85.0  0.0541  0.0957  0.2371  0.5449  0.6761  0.6604  90.0  0.0542  0.0910  0.2184  0.5138  0.6636  0.6705  100.0  0.0616  0.0900  0.1954  0.4623  0.6343  0.6789  120.0  0.1073  0.1270  0.1995  0.4092  0.5875  0.6771  140.0  0.1910  0.2050  0.2572  0.4197  0.5808  0.6853  160.0  0.3011  0.3114  0.3502  0.4769  0.6149  0.7183  190.0  0.4824  0.4893  0.5156  0.6056  0.7127  0.8038  1.0  ./continued  136  TABLE 17a (continued)  T(°K)  y=0.25  y==0.50  200.0  0.5409  0 .5470  240.0  0.7373  280.0  y=2  y=3  y=4  0.5705  0 .6517  0.7506  0 .8372  0 .7415  0.7576  0 .8151  0.8899  0 .9619  0.8651  0 .8682  0.8806  0 .9254  0.9866  1 .0499  320.0  0.9365  0 .9391  0.9491  0 .9865  1.0394  1 .0974  380.0  0.9809  0 .9830  0.9910  1 .0212  1.0657  1 .1177  *  These values are also applicable to p-D~ at one-half the temperature.  137  TABLE 17b Out-of-plane and in-plane hindered rotational heat capacities of of o-H for A=10*. 2  T(°K)  u=0.25  y=.0.50  5.0  0.2353  0.0122  10.0  0.4337  0.2352  0.0124  15.0  0.3251  0.4143  0.1001  0.0013  20.0  0.2245  0.4337  0.2364  0.0131  25.0  0.1589  0.3861  0.3498  0.0471  0.0003  30.0  0.1178  0.3264  0.4156  0.1036  0.0018  35.0  0.0930  0.2743  0.4413  0.1732  0.0068  0.0001  45.0  0.0764  0.2080  0.4307  0.3142  0.0367  0.0016  55.0  0.0887  0.1853  0.4014  0.4271  0.1018  0.0094  65.0  0.1165  0.1895  0.3813  0.5057  0.1975  0.0303  75.0  0.1502  0.2072  0.3729  0.5563  0.3095  0.0690  85.0  0.1850  0.2307  0.3731  0.5867  0.4228  0.1260  .90.0  0.2023  0.2436  0.3755  0.5966  0.4765  0.1604  100.0  0.2368  0.2709  0.3846  0.6100  0.5739  0.2380  120.0  0.3097  0.3340  0.4194  0.6290  0.7242  0.4114  140.0  0.3941  0.4121  0.4774  0.6594  0.8282  0.5841  160.0  0.4907  0.5044  0.5551  0.7102  0.9083  0:7414  190.0  0.6448  0.6544  0.6905  0.8158  1.0124  0.9432  y=i  y=2  y=4  y=8  1.0  ./continued  138  TABLE 17b (continued) T(°K)  =0.25  y==0.50  y=i  U=2  y=4  y=8  200.0  0 .6951  0,.7037  0 .7363  0.8549  1 .0453  1 .0024  240.0  0 .8674  0,.8733  0 .8963  1.0001  1 .1569  1.1977  280.0  0 .9834  0,,9879  1 .0054  1.1051  1 .2302  1 .3322  320.0  1 .0489  l , .0525  1 .0666  1.1635  1 .2648  1 .4123  380.0  1 .0868  i . ,0895  1 .1003  1.1869  1 .2672  1 .4559  *  These values are also applicable to p-D„ at one-half the temperature.  FIGURE 16 O u t - o f - p . ane a n d i n - p l a n e h i n d e r e d r o t a t i o n a l h e a t 1  of  orthohydrogen.  four values  Figure  F i g u r e 1 6 a shows C / R c u r v e s v  of y.  The c u r v e s  A = l , y=0  2.  A = l , y=0.25  3.  A = l , y=2  4.  A = l , y=4  y  are  l a b e l l e d as  1.  16b g i v e s C / R c u r v e s  The c u r v e s  are  l a b e l l e d as  capacities  f o r A=l and  follows:  f o r A=10 a n d f o u r v a l u e s  follows:  1.  A = 1 0 , y=0  2.  A=10, y = 0 . 2 5  3.  A = 1 0 , y=2  4.  A = 1 0 , y=8  of y.  Heat c a p a c i t y curves in  F i g u r e 16b.  F o r y=0,  f o r X=10 and y=0,  with eigenstates  0.25,  {k,(p,m)},  2 and 8 are the heat  r i s e s v e r y s l o w l y because o f the  l a r g e s e p a r a t i o n of the  and  - E ^ "  {1, C-l, 0)}  states  h i n d e r a n c e removes  (E^" '* ^ 1  difference  ,  0  3  - 3.7).  [2,(-1,-1)]  (see  F i g u r e 14b).  1  13  -E^" '" 1  1 3  r a p d i l y at 2 0 ° K ,  [1,(-1,+1)] is  Thus low  temperature the  energy  important.  but shows no maximum.  energy s e p a r a t i o n between the two lowest s t a t e s i s  For y=8  Thus the  even g r e a t e r  the s e p a r a t i o n between the two lowest {k,(p,m)} s t a t e s at y=0. curve 4 l i e s below  curve 1 at  y=0,  in  populations  F i g u r e 12b.  that  the  capacities  i n T a b l e 18a  F o r X = l , heat c a p a c i t y curves  2 and 4 are p l o t t e d i n F i g u r e 17a as a f u n c t i o n o f relative  Thus  contribute.  and i n - p l a n e h i n d e r e d r o t a t i o n a l heat  X=l and i n T a b l e 18b f o r X=10.  than  I t then c r o s s e s  e q u i l i b r i u m deuterium are g i v e n f o r v a r i o u s y v a l u e s  for  The  low t e m p e r a t u r e s .  curve as h i g h e r l e v e l s b e g i n t o Out-of-plane  of  energy  has i n c r e a s e d c o n s i d e r a b l y and a l s o  p o p u l a t i o n o f the s t a t e  curve 3 r i s e s  y=0  the  F o r y=0.25, the  F o r y=2,  for  temperature.  c o n t r i b u t e to the y=0 curve were shown  As was d i s c u s s e d p r e v i o u s l y ,  '  one gets  [1,(-1,-1)] to  The heat c a p a c i t y then decreases u n t i l the  cE/" '-  relative  However as the y  approximately 0.25.  i s h i g h enough f o r promotions t o h i g h e r l e v e l s .  the  {l,(-l,±l)}  are r e q u i r e d f o r t h i s promotion and curve 2 shows a sharp  peak at 8 ° K .  difference  capacity  o f the ground s t a t e ,  lowest o f these two s t a t e s  between these two s t a t e s i s  temperatures  1  the double degeneracy  a sudden r u s h from the next lowest s t a t e  1  shown  the maximum i n curve 1 at  142  TABLE 18a Out-of-plane and in-plane hindered rotational heat capacities of e-®2 f° T(°K)  y=0.25  y=0.50  1.0  0.0000  0.0000  5.0  .0.0000  10.0  r  • y=2  P=3  y=4  0.0000  0.0000  0.0000  0.0000  0.0001  0.0003  0.0034  0.0192  0.0599  0.0220  0.0258  0.0411  0.0991  0.1728  0.2264  15.0  0.1551  0.1604  0.1795  0.2293  0.2548  0.2411  20.0  0.3511  0.3507  0.3487  0.3312  0.2891  0.2319  25.0  0.5163  0.5107  0.4891  0.4160  0.3282  0.2481  30.0  0.6343  0.6262  0.5957  0.4967  0.3870  0.2951  35.0  0.7264  0.7180  0.6863  0.5828  0.4668  0.3688  45.0  0.8925  0.8860  0.8608  0.7741  0.6668  0.5660  55.0  1.0371  1.0329  1.0165  0.9553  0.8691  0.7757  65.0  1.1336  1.1315  1.1231  1.0877  1.0283  0.9526  75.0  1.1773  1.1769  1.1751  1.1625  1.1310  1.0792  85.0  1.1818  1.1827  1.1856  1.1911  1.1845  1.1578  90.0  1.1747  1.1760  1.1808  1.1933  1.1972-  1,1822  100.0  1.1507  1.1527  1.1601  1.1831  1.2039  1.2094  120.0  1.0946  1.0971  1.1067  1.1398  1.1799  1.2128  140.0  1.0514  1.0539  1.0636  1.0985  1.1450  1.1909  160.0  1.0229  1.0252  1.0343  1.0676  1.1144  1.1647  190.0  0.9926  1.0023'  1.0316  1.0748  1.1248  0.9946  143  TABLE 18b Out-of-plane and in-plane hindered rotational heat capacities of e-D for X=10. 2  T(°K)  y=0.25  y=0.50  1.0  0.0000  0.0000  0.0000  0.0000  0.0002  0.1597  5.0  0.0016  0.0030  0.0110  0.0663  0.2345  0.0878  10.0  0.1017  0.1134  0.1531  0.2304  0.1762  0.0271  15.0  0.2951  0.2938  0.2862  0.2414  0.1063  0.0128  20.0  0.4142  0.4015  0.3561  0.2330  0.0725  0.0076  25.0  0.4617  0.4465  0.3922  0.2465  0.0672  0.0071  30.0  0.4896  0.4760  0.4266  0.2865  0.0901  0.0141  35.0  0.5274  0.5165  0.4760  0.3515  0.1417  0.0335  45.0  0.6494  0.6434  0.6199  0.5333  0.3169  0.1213  55.0  0.7958  0.7935  0.7834  0.7339  0.5433  0.2688  65.0  0.9250  0.9253  0.9251  0.9086  0.7683  0.4507  75.0  1.0200  1.0220  1.0289  1.0390  0.9593  0.6401  85.0  1.0809  1.0841  1.0955  1.1257  1.1043  0.8179  90.0  1.1008  1.1044  1.1173  1.1552  1.1597  0.8988  100.0  1.1250  1.1290  1.1439  1.1933  1.2407  1.0412  120.0  1.1366  1.1407  1.1564  1.2176  1.3157  1.2472  140.0  1.1291  1.1328  1.1473  1.2113  1.3267  1.3655  160.0  1.1167  1.1199  1.1328  1.1949  1.3105  1.4236  190.0  1.0927  1.0954  1.1058  1.1607  1.2665  1.4427  y=2  y=4_  y=8  144  FIGURE 17 Out-of-plane of  equilibrium deuterium.  curves as  and i n - p l a n e h i n d e r e d r o t a t i o n a l h e a t  capacities  I n F i g u r e 1 7 a a r e shown t h e  f o r A - l and t h r e e v a l u e s  of y.  The c u r v e s  are  C^/R labelled  follows:  Figure  1.  A = l , y=0  2.  A = l , y=2  3.  A = l , y=4  17b g i v e s C / R c u r v e s  The c u r v e s a r e  y  l a b e l l e d as  f o r A. =10 a n d t h r e e v a l u e s  follows:  1.  A=10, y=0  2.  A = 1 0 , y=2  3.  A=10,  y=8  of y.  145  146  80°K was caused by the contributions of not a single level but promotions to four states as given below : 1.  (l,(+l,0)}  ->  {l,(-l,±l)},  • 2.  U,(+1,0)}  -  (l,(-l,0)} ,  3.  {1, (+1,0)}  -  (1,(-1,±2)},  4.  {1,(+1,0)}  ->  {l,(+l,±l)}.  Although the u hinderance removes the ±m degeneracy, none of the new [j,(p,r)J states are close enough to the [1,(+1,+1)] ground state to cause any peculiar behavior i n C /R u n t i l u reaches a value of 4. v  Thus the u=2 curve i s very similar i n shape to curve 1.  However when  u i s increased to 4, the energy difference between these two levels, i s small enough to enable population of the state [1,(-1,-1)] at low temperatures. C /R v  Curve 3 therefore shows a low temperature maximum of  = .24 at 16°K.  The heat capacity then decreases u n t i l promotions  to other levels begin to contribute to the heat capacity.  The maximum  at 16°K i s not very high because the removal of a l l ±m degeneracy results i n a degeneracy ratio between the two lowest states of onehalf.  At high temperatures the u hindered curves have the correct  ordering, the curves of largest u lying highest. For A=10, C /R curves for u=0, 2 and 8 are plotted i n Figure 17b. v  As u i s increased the energy difference between the two lowest ^ and u hindered rotational eigenstates, [l,(+l,+l)j and [1,(-1,-1)], decreases (see Figure 14b) and for u=2 a low temperature maximum appears.  147  F o r y=8, t h e energy d i f f e r e n c e £ ^  ^  - <£j^ ^ ^ +  ,+  very s m a l l .  1 S  The peak becomes v e r y sharp and now o c c u r s a t about 1°K. c a p a c i t y thsn decreases  almost t o z e r o .  The heat  A t 20°K t h e heat capacity-  b e g i n s t o i n c r e a s e a g a i n as o t h e r l e v e l s c o n t r i b u t e t o C . mentioned f o r e q u i l i b r i u m h y d r o g e n ,  As  whencE^ ^  approaches z e r o , t h e peak w i l l g e t n a r r o w e r and approach 0°K. Cl^^  + 1 , +  =  ^ i ^  ^  1» . 1) < £^(-''> -'-) +  P ^  t n e  +  f  w i l l d i s a p p e a r e n t i r e l y and when  ea  i  <  e  t h e two l e v e l s have c r o s s e d , t h e peak  >  A move' away from * and 0o K as £  w i•lili r e a p p e a r  n  For o r t h o d e u t e r i u m  v  (+1,+1) ~ C 1(-1,-1) •i n c r e a s e s .  t h e o u t - o f - p l a n e and i n - p l a n e h i n d e r e d  r o t a t i o n a l heat c a p a c i t i e s a r e g i v e n f o r v a r i o u s y i n T a b l e s 16b.  When  Those f o r p a r a d e u t e r i u m  are given i n Tables  the f e a t u r e s o f t h e two d e u t e r i u m  16a and  17a and 17b. A l l  n u c l e a r s p i n s p e c i e s w i l l be t h e  same as f o r t h e c o r r e s p o n d i n g hydrogen s p e c i e s b u t t h e y w i l l  occur at  o n e - h a l f t h e temperature.  D) H i n d e r e d  R o t a t i o n a l E n t r o p i e s o f Hydrogen.  The  e n t r o p i e s o f parahydrogen and orthohydrogen  f o r different  v a l u e s o f o u t - o f - p l a n e and i n - p l a n e b a r r i e r h e i g h t s a r e p l o t t e d as a f u n c t i o n o f temperature  i n F i g u r e 18.  As seen from F i g u r e s 18a and  18b, t h e e n t r o p y o f parahydrogen i s q u i t e i n s e n s i t i v e t o X, b u t e x h i b i t s a s t r o n g e r dependence on y.  As one would e x p e c t , t h e r e i s a l o w e r i n g  of the entropy w i t h i n c r e a s i n g r o t a t i o n a l hindrance.  At 0°K a l l p a r a -  hydrogen e n t r o p i e s a r e z e r o , s i n c e t h e p a r t i t i o n f u n c t i o n s q ^ ( p - ^ ) , ' c  lh (p-^2 r  3  a n c  * hr' q  a  PP  r o a c n  unity.  T h i s i s n o t so f o r t h e e n t r o p y  148  FIGURE Hindered r o t a t i o n a l orthohydrogen and the labelled  molar entropies  for various values  in-plane barrier as  Figure  18  y.  o f parahydrogen  o f the  Figures  out-of-plane  Figure  18b  1. . A=0; A = l , y=0  1.  A=0  2.  2.  A = 1 0 , y=0  3.  A=10,  y=4  4.  A=10,  y=8  Figures  A = l , y=4  18c a n d 18d f o r  Figure  18c  o-H  2  are  l a b e l l e d as  follows  F i g u r e 18d  1.  A=0  1.  A=0  2.  A = l , y=0  2.  A=10, y=0  3.  A = l , y=0.25  4.  A = l , y=4  .3. 4.  barrier A  1 8 a a n d 18b f o r p - H  follows:  18a  and  A=10, y=0.25 A=10,  y=8  2  are  150  of  orthohydrogen,  shown i n F i g u r e s 18c and 18d, because o f  and n u c l e a r s p i n d e g e n e r a c i e s . total just  the  groui-d-sta.te  degeneracy  R£n9 = 4.366  e.u.  When the  free  For free is  rotator is  2&+1 r o t a t i o n a l degeneracy  T h e r e f o r e the  lowest  has a degeneracy  nine,  is  of s i x ,  and the  degeneracy  zero p o i n t  the  rotational eigenstate s p i n degeneracy). (curves  This i s  entropy i s an entropy  lowest o u t - o f - p l a n e  lowest o u t - o f - p l a n e  [1,(-1,-1)].  It  Thus the  loss of  apparent  .806  entropy curves  at the  f o r the  This entropy  removes  the ±m  hindered  of three  (the  l o s s o f 1.377  o r t h o molecules t o the  from F i g u r e s  curves  largest  18c  barriers lie  and 18d,  that  as  e.u.  can be  lowest  state  the orthoA l s o the  lowest.  A and u h i n d e r e d r o t a t i o n a l  as a f u n c t i o n o f temperature  show a v a n i s h i n g  nuclear  e.u.  0°K v a l u e up t o h i g h e r t e m p e r a t u r e s .  For e q u i l i b r i u m hydrogen, are p l o t t e d  e.u.  transition  u o r A. b a r r i e r h e i g h t s are i n c r e a s e d the r o t a t i o n a l e n t r o p y o f hydrogen remains  e.u.  zero p o i n t e n t r o p y becomes R£n3 = 2.183  t r a n s i t i o n of the is  R£n6 = 3.560  and i n - p l a n e  [ 1 , ( - 1 , - 1 ) ] has a degeneracy  3 and 4 o f F i g u r e 1 8 c ) .  a t t r i b u t e d t o the  {l,(-1,±1)},  hindered r o t a t i o n a l  Introducing a u hinderance completely so t h a t  barrier  removed except f o r the ±m degeneracy.  o f as b e i n g g r a d u a l l y g i v e n up by the  the o r t h o m o l e c u l e s t o the  the  S^^Co-H^) at 0 ° K i s  (k(-l,m)} s t a t e f o r orthohydrogen,  which can be thought  eigenstate.  so t h a t  orthohydrogen  h i n d e r e d by an o u t - o f - p l a n e  as seen i n curve 2 o f F i g u r e 18c.  of  rotational  rotational  i n F i g u r e 19a and 19b.  zero p o i n t entropy s i n c e the  entropies All  lowest s t a t e i s  a  151  FIGURE 19 H i n d e r e d r o t a t i o n a l molar e n t r o p i e s for various values plane b a r r i e r u.  o f e q u i l i b r i u m hydrogen  o f the o u t - o f - p l a n e  b a r r i e r A and the  In F i g u r e 19a are shown the f o l l o w i n g  curves:  In F i g u r e 19b the  1.  A=0; A = l , y=0  2.  A=l,  y=2  3.  A=l,  y=4  curves are l a b e l l e d as  1.  A=0  2.  A=10,  y=0  3.  A=10,  y=4  4.  A=10,  y=8  .  follows:  inentropy  152  153  nondegenerate  even p p a r i t y s t a t e .  above t h e f r e e r o t a t o r temperatures.  The h i n d e r e d r o t a t i o n a l curves l i e  curve at low t e m p e r a t u r e s and below at h i g h  Curve 4 f o r A=10, u=8 i n F i g u r e 19b i s i n t e r e s t i n g i n  that the entropy increases very r a p i d l y  as t h e t e m p e r a t u r e i s  i n c r e a s e d from zero and t h e n l e v e l s o f f at a v a l u e o f S = 2.75 e.u.. Not u n t i l about 75°K does t h e e n t r o p y b e g i n t o i n c r e a s e a g a i n .  This  at f i r s t seems r e m a r k a b l e .  However, i f one l o o k s a t t h e s p a c i n g o f t h e  two l o w e s t [ j ( p , r ) ] s t a t e s ,  [1,(+1,+1)] and [ 1 , ( - 1 , - 1 ) ] , (see F i g u r e  14b) t h e r e a s o n f o r t h i s sudden jump i n e n t r o p y becomes a p p a r e n t .  For  an o u t - o f - p l a n e b a r r i e r o f A=10 and an i n - p l a n e b a r r i e r o f u=8, £^(+l, + l )  =  6 > 4  3 2 8 and(£  ( : 1  " " 1 ,  1 )  = 6.5191.  A t 0°K, t h e e n t r o p y i s  c o n t r o l l e d by t h e [1,(+1,+1)] s t a t e which i s nondegenerate.  The two  l e v e l s a r e s o c l o s e l y spaced t h a t a t 20°K, t h e y behave as two degenerate levels giving  an e n t r o p y o f R£n(l+3) = 2.755 e.u.  I f t h e s e two l e v e l s  have e x a c t l y t h e same energy t h e e n t r o p y a t 0°K would be R&n4.  Further  i n c r e a s i n g o f t h e b a r r i e r h e i g h t may r e o r d e r t h e l e v e l s so t h a t [1,(-1,-1)]  lies  lowest.  R£n3 = 2.183 e.u.  The z e r o p o i n t e n t r o p y w i l l t h e n become  I n g e n e r a l , the reason f o r the i n c r e a s e i n entropy  o f h i n d e r e d r o t a t i o n a l e q u i l i b r i u m hydrogen  at l o w e r t e m p e r a t u r e s  (compared t o t h e f r e e r o t a t i o n a l e n t r o p y ) can be a t t r i b u t e d t o t h e d e c r e a s e i n t h e energy s e p a r a t i o n o f t h e two l o w e s t e i g e n s t a t e s , one o f w h i c h has even p p a r i t y and t h e o t h e r odd p p a r i t y . F o r any o r t h o - p a r a h y d r o g e n m i x t u r e , c o n t a i n i n g a f r a c t i o n x o f parahydrogen,  t h e e n t r o p y at any g i v e n temperature i s g i v e n by  154  S  x-H  the  xS(p-H )  =  2  last  + (l-x)S(o-H )  term r e p r e s e n t i n g  (below 2 0 ° K )  - R jx£nx + ( 1 - x ) £ n ( l - x ) j ,  2  the  free  the entropy o f m i x i n g .  At low  (5.8)  temperatures  r o t a t i o n a l entropy o f e q u i l i b r i u m hydrogen  is  g i v e n by  S (e-H ) f r  and  for  2  = S (p-H ) f r  2  = 0.0,  (5.9)  orthohydrogen, S (o-H ) f r  2  = R£n9 = 4.366 e . u .  Then a c c o r d i n g t o e q u a t i o n (x = j) i s  (5.8),  .  (5.10)  the e n t r o p y o f normal hydrogen  g i v e n by  = 4.39  e.u.  •  (5.11)  3 Of t h e s e 4.393 e n t r o p y u n i t s , up by the t r a n s i t i o n rotational level  levels,  ^R£n3 -  1.637  e.u.  are g r a d u a l l y  o f the o r t h o molecules t o the  which r e s u l t s  from the s p l i t t i n g  given  lowest o f the o f the  three  1=1 o r t h o  i n the s o l i d .  The r e m a i n d e r , 2.756 e . u . , r e p r e s e n t s the entropy 3 1 due t o the n u c l e a r s p i n (^R£n3) and the entropy o f m i x i n g (^R£n4 + 3 4 •^R£n •—)-..  The n u c l e a r s p i n entropy w i l l p r o b a b l y be d i m i n i s h e d  extremely  low temperatures  lowest o f the of  mixing  three  by the t r a n s i t i o n  o f the molecules t o  l e v e l s caused by the n u c l e a r s p i n , w h i l e  (1.118 e . u . )  may p e r s i s t  u n t i l the  at  the  the entropy  o r t h o - and parahydrogen  155  phases are s e p a r a t e d . entropy curves  However, i t  the  all  i n t h i s c h a p t e r are b a t e d on the assumption t h a t  n u c l e a r s p i n degeneracy that  s h o u l d be p o i n t e d out t h a t  rotational  is  not  the  removed at v e r y low temperatures  degeneracy  is  not  the  and  removed u n l e s s p e r t u r b e d by a  rotational hindrance. For large 3.560 e . u .  over the  out-of-plane  b a r r i e r s t o r o t a t i o n where ^ ^ ( 0 - ^ )  temperature  range 0°K t o 2 0 ° K ,  the e n t r o p y  =  of  normal hydrogen w i l l be g i v e n by  S  h r  (n-H ) 2  = | R £ n 6 + 1.118  e.u.,  = 3.788 e . u . .  However f o r s m a l l A b a r r i e r s , S ^ f o - H ^ low temperatures  S  h r  (n-H )  in-plane b a r r i e r is  2  = jS (o-H ) hr  large,  S '(n-H ) h r  '  2  2  + 1.118  e.u. .  f o r A and u b a r r i e r s t o r o t a t i o n . regardless  normal hydrogen below 20°K  ••  does not remain constant  at  and so  S i m i l a r arguments h o l d t r u e  of  (5.12)  o f whether A. = 1 or 10,  (5.13)  If the  the entropy  is  = | R £ n 3 + 1.118  = 2.755 e . u . .  e.u.,  (5.14)  156  For s m a l l u b a r r i e r s  S  High temperature  h r  '(n-H ) 2  = |s  classical  h r  '(o-H ) 2  entropies  + 1.118  e.u..  (5.15)  are g i v e n i n Appendix C .  E) H i n d e r e d R o t a t i o n a l E n t r o p i e s o f Deuterium. The e n t r o p i e s temperature is  o f o r t h o d e u t e r i u m are p l o t t e d  i n F i g u r e 20a and 20b.  g i v e n by R£n6 = 3.560 e . u .  (i.e.  o r t h o d e u t e r i u m has  of s i x ) .  the  corresponding to greatest hindrance l i e  curves  three. curves  fact  discussion  lowest.  The entropy  as a f u n c t i o n o f T * ( = T / 9 ) . a n u c l e a r s p i n degeneracy  o f the b e h a v i o u r o f the  to  This  r  t h a t b o t h s p e c i e s have  Thus the  temperature  i n c l u d e d because they are i d e n t i c a l  those f o r orthohydrogen i f p l o t t e d due t o the  curves  a nuclear spin  Note once again t h a t w i t h i n c r e a s i n g  f o r p a r a d e u t e r i u m are not  of  The zero p o i n t entropy f o r a l l  degeneracy curves  as a f u n c t i o n  is  of  orthohydrogen  can be a p p l i e d t o those of p a r a d e u t e r i u m at o n e - h a l f the  temp-  erature. The h i n d e r e d r o t a t i o n a l e n t r o p i e s  o f e q u i l i b r i u m deuterium are  shown i n F i g u r e 21a and 21b f o r v a r i o u s v a l u e s o f X and at 0°K i s  just  that  o f orthodeuterium  (3.560 e . u . ) .  y.  As b a r r i e r s  r o t a t i o n are i n t r o d u c e d , the h i n d e r e d r o t a t i o n a l entropy i s the  free  r o t a t i o n a l entropy at  h i g h temperatures. that  low temperatures  The e x p l a n a t i o n  is  to  greater  and l e s s than S  o f these e f f e c t s  a l r e a d y g i v e n f o r e q u i l i b r i u m hydrogen.  The entropy  £  analogous  at to  than  157  FIGURE 2C H i n d e r e d r o t a t i o n a l molar e n t r o p i e s various values barrier U.  o f the o u t - o f - p l a n e  c f orthodeuterium f o r b a r r i e r X and the  In F i g u r e 20a are shown the f o l l o w i n g  curves:  The curves  1.  X=0; X = l ,  2.  X=l,  y=0  y=4  i n Figure- 20b are l a b e l l e d as 1.  X=0  2.  X=10,  y=0  3.  X=10,  y=8  follows:  in-plane  entropy-  159  FIGURE 21 H i n d e r e d r o t a t i o n a l molar e n t r o p i e s for various values  o f the o u t - o f - p l a n e  in-plane barrier y. as  o f e q u i l i b r i u m deuterium  The curves  b a r r i e r A and the  i n F i g u r e 21a are  labelled  follows:  "1.  A=0;  A=l,  2.  A=l,  y=4  y=0  In F i g u r e 21b are shown the f o l l o w i n g e n t r o p y c u r v e s :  .  1.  A=0  2.  A=10,  y=0  3.  A=10,  y=4  4.  A=10,  y=8  161  For any p a r a of  orthodeuterium,  orthodeuterium mixture c o n t a i n i n g  the entropy at any g i v e n temperature  a fraction x can be found  from  S  x-D  =  x S  (o-D )  + (l-x)-S(p-D )  2  -  2  At low temperatures  (below 1 0 ° K ) ,  RJxAn;- +  the f r e e  £  n  j  .  (5.16)  r o t a t i o n a l entropies  of  d e u t e r i u m are g i v e n by  S  fr  ( e _ D  2  ; )  =  S  fr^°~ 2 D  )  =  3  S (p-D )  = 4.366 e . u . ,  S (n-D )  = |s (o-D )  f r  2  f r  2  '  5  6  2  R  e  '  u  >  J  (  '  1  7  )  |s (p-D )  +  £ r  2  +  =5.093 e.u. ,  (5.19)  2.373 e . u . o f which are due t o the n u c l e a r s p i n degeneracy deuterium and 1.455 e . u . o f which are due t o the r o t a t i o n a l s p i n degeneracy  5  (5.18)  - {i4 m f r  0  of paradeuterium.  of orthoand n u c l e a r  The remaining 1.265 e . u . are due t o  the e n t r o p y o f m i x i n g . For l a r g e  out-of-plane  b a r r i e r s t o r o t a t i o n , where S ^ ( p - D ) r  3.560 e . u . , the e n t r o p y o f normal deuterium at temperatures  2  =  below 10°K  is S (n-D ) h r  2  = |  (3.560)  = 4.825 e . u .  + | .  (3.560) + 1.265 e . u . (5.20)  162  However, f o r s m a l l X b a r r i e r s , the entropy o f paradeuterium does not remain constant  between 0°K and 1 0 ° K i.o t h a t the normal deuterium  e n t r o p y must be c a l c u l a t e d from the  S  hr  ( n  ~ 2) D  I  =  S  hr  C p  following equation :  " 2 D  )  +  3  "  6  3  8  e  " '  '  u  (  S i m i l a r arguments h o l d t r u e f o r X and u b a r r i e r s t o r o t a t i o n plane  and i n - p l a n e h i n d e r e d r o t a t o r ) .  Sj '(p-D2)  = 2.183  i r  and  e.u.,  5  >  2  1  )  (out-of-  For large u b a r r i e r s with  the entropy o f normal deuterium between 0°K  10°K i s  S  h r  '(n-D ) 2  =4.366 e.u.  .  (5.22)  For s m a l l u b a r r i e r s ,  .. hr't - 2 S  The d i f f e r e n c e s (5.20)  n  i n t o the In  capacities  )  =  between the  and (5.22)  molecules  D  l hr' S  (  p  "  D  2  )  +  3  ,  6  3  8  e  u  '  *  low temperature e n t r o p i e s  (  g i v e n by  lowest r o t a t i o n a l  and e n t r o p i e s  3  )  (5.19),  c h a p t e r on the h i n d e r e d r o t a t i o n a l heat  o f hydrogen and deuterium a few  seem t o be q u i t e above 6 0 ° K .  the g r e a t e s t e f f e c t s the p e r t u r b a t i o n s .  2  states.  qualitative  comments s h o u l d be made on the h i n d e r e d r o t a t i o n a l e f f e c t s  temperatures  ,  are j u s t due t o the t r a n s i t i o n o f the p a r a d e u t e r i u m  concluding t h i s  The e n t r o p i e s  5  insensitive  discussed.  to small b a r r i e r heights  Orthohydrogen (paradeuterium)  entropies  show  due to the removal o f the r o t a t i o n a l degeneracy However the e n t r o p i e s  r a p i d l y r e t u r n t o the  at  free  by  rotator value as the temperature is increased from 0°K.  Therefore  unless tho system exhibits very large barriers to rotation, entropy measurements would not be of interest except at very low temperatures Cprobably below 20°K). I n contrast the rotational heat capacities of equilibrium hydrogen (equilibrium deuterium) and orthohydrogen (paradeuterium) show very marked effects at small barrier heights and at temperatures ranging from 0°K to 100°K.  T h e parahydrogen (orthodeuterium) hindered  rotational heat capacities however are very insensitive to A or y. T h u s i t would be of interest to make calorimetric measurements on a  well defined system at a l l accessible temperatures below 100°K. Ideally  one would want to take pure orthohydrogen (paradeuterium)  adsorbed on a uniform surface and then measure i t s hindered rotational heat capacity by some rapid calorimetric technique.  (A f a i r l y rapid  measurement would be required to ensure the minimal amount of ortho to para conversion.)  I n fact one could in principle study the ortho  to para conversion by measuring changes in C /R.  164  CHAPTER VI. SEPARATION FACTORS FOR HYDROGEN AND DEUTERIUM If translations, vibrations and rotations are independent and if  i n the adsorbed phase only the rotational motion i s perturbed  by the adsorbent, a straightforward application of s t a t i s t i c a l mechanics to the rotational populations of the ortho and para states of hydrogen or deuterium i n the gas phase (free rotator) and i n the adsorbed phase (hindered rotator) leads to the following expressions:  ^  * h r  (H ) 2  =  [q (p-H )/q (o-H )] i tl - , fr  2  fr  (6.1)  l  tV<P- 2>/«hrCo-H )] H  2  and A  ^ (D ) h r  2  =  [q (o-D )/q (p-D )] 1 *I i  (6.2)  t  where the q's are the appropriate molecular rotational p a r t i t i o n functions given by equations (3.26a)-(3.29c).  The quanties ^ ^  are called separation factors or separation coefficients and are a measure of the difference i n parahydrogen and orthodeuterium concentrations resulting from rotational hindrance i n the adsorbed • phase. Separation factors have been calculated for H and D at 2  2  20.4°K, 25°K and 30°K. The results are given as a function of the unitless barrier height A i n Figure 22 for H and D 2  2  The 20.4°K  FIGURE 22 Separation factors  f o r hydrogen and deuterium at t h r e e temp-  e r a t u r e s p l o t t e d as a f u n c t i o n o f the o u t - o f - p l a n e height  (0 < X < 8 0 ) .  The curves are l a b e l l e d as  1.  H  2  at  20.4°K  2.  H  2  at  25.0°K  3.  H  2  at  30.0°K  4.  D  2  at  20.4°K  5.  D  2  at  25.0°K  6.  D  2  at  30.0°K  barrier follows:  166  167  curves f o r  a  n  ^  C  c u r  ves  1 and 4 r e s p e c t i v e l y )  are i n  excellent  12 agreement  w i t h those c a l c u l a t e d by White and L a s s e t t r e  separation factors the  factors  .  The  are seen to be v e r y s e n s i t i v e to temperature  d e c r e a s i n g w i t h i n c r e a s i n g temperature.  temperature c o e f f i c i e n t hydrogen c o n v e r s i o n at  is  This  negative  a l s o c h a r a c t e r i s t i c o f the o r t h o t o  low temperatures  with  on v a r i o u s c a t a l y t i c  para-  adsorb-  29 ents . F o r the c a t a l y t i c c o n v e r s i o n o f orthohydrogen t o p a r a h y d r o g e n , the c o n v e r s i o n r a t e was f e l t to be o n l y dependent on the c o n c e n t r a t i o n 29 i n the adsorbed phase.  Thus Farkas  temperature c o e f f i c i e n t  was s i m p l y due to the f a c t  e r a t u r e was i n c r e a s e d dependence  l e s s gas was a d s o r b e d .  of t h e o r e t i c a l  just  are independent  At v e r y l a r g e v a l u e s  the  that  negative as the  However the  separation coefficients  a h i n d e r e d r o t a t i o n a l model i s separation factors  concluded t h a t  temp-  temperature  calculated  using  due to a Boltzmann f a c t o r . o f the t o t a l  o f A, the  theoretical  approach the S a n d l e r l i m i t , where o u t - o f - p l a n e 6  amount o f gas  These adsorbed.  separation  rotation is  factors  completely  hindered : A  f^^lW ,  ~  (6.3)  where  Ae = e x  1  - e  Q  ,  (6.4)  o r i n terms o f the o u t - o f - p l a n e  = Jexp(AE ~ ' C  1  1  ±1)  hindered r o t a t i o n a l eigenvalues  /kT)  ,  E^P' ^ 11  (6.5)  168  wher? A E ^  ^  1  and a t i n f i n i t e  AE ^  - E ^ -  ^  - E^  +  '  1  1  A  e /  2  f  = 1  l  F o r hydrogen,  h2/8TT I  86.9 c a l / m o l e .  Q  \  C6.6)  barriers  f-1>-+11 ) A-*» * > l  1  2  h  2  ^  (6.7)  -87T F 2  .= 173.8 c a l / m o l e and f o r d e u t e r i u m h / 8 T r I = 2  L i s t e d i n T a b l e 19 a r e v a l u e s o f  f o r both  hydrogen and d e u t e r i u m a t 20.4°K, 25°K and 30°K.  Included f o r  2  comparison a r e t h e c o r r e s p o n d i n g e x a c t s e p a r a t i o n f a c t o r s f o r A=80. I t can be seen t h a t t h e convergence  o f t h e exact s e p a r a t i o n f a c t o r s  and t h e ones c a l c u l a t e d u s i n g (6.6) i s v e r y poor. negative temperature c o e f f i c i e n t o f d Boltzmann  f a c t o r i s now v e r y a p p a r e n t .  c o n s t a n t , so t h a t ^ ^ ( T )  The f a c t t h a t t h e  i s dependent o n l y upon t h e F o r hydrogen Ae^/^ i s a  = -^exp(const/T), and  ^ ^ ( T ) w i l l be an  e x p o n e n t i a l l y decreasing f u n c t i o n o f temperature. The t r e n d s shown i n F i g u r e 22 a r e i n q u a l i t a t i v e agreement 6 8 w i t h experiment  i n two i m p o r t a n t a s p e c t s .  Firstly,  orthohydrogen  i s more s t r o n g l y adsorbed t h a n parahydrogen, n o t o n l y i n t h e S a n d l e r l i m i t but at a l l intermediate stages. s t r o n g l y adsorbed than o r t h o d e u t e r i u m .  Moreover p a r a d e u t e r i u m i s more Secondly, the s e p a r a t i o n  f a c t o r f o r deuterium at a given temperature i s c o n s i s t e n t l y s m a l l e r t h a n t h a t f o r hydrogen a t t h e same t e m p e r a t u r e .  T h i s i s j u s t due t o  the d i f f e r e n c e i n t h e c h a r a c t e r i s t i c r o t a t i o n a l t e m p e r a t u r e s o f  169  TABLE 19 Separation factors for  and  using the Sandler l i m i t ^ and  the out-of-plane hindered rotational model with X=80. Note that  T(°K)  =  ~  A  */  hr  hr  W  W  A*?  A-  Ar  20.4  48.6  36.4  .251  5.70  4.92  .137  25.0  22.0  17.4  .209  3.84  3.40  .115  30.0  12.4  10.1  .185  2.82  2.59  .082  hydrogen and d e u t e r i u m , °  i.e. 0  J  f o r H, = 87.49°K and 0 f o r D» = 2 r 2  r  43.75°K.  I f s e p a r a t i o n f a c t o r s are c a l c u l a t e d f o r hydrogen and  deuterium  a t t h e same b a r r i e r h e i g h t and the same reduced  e r a t u r e , T*(=T/0 ) , then the s e p a r a t i o n f a c t o r s are  temp-  identical.  B e f o r e a q u a n t i t a t i v e comparison between t h e t h e o r e t i c a l s e p a r a t i o n f a c t o r s f o r the out-of-plane hindered r o t a t o r experiment  and  can be made, i t i s n e c e s s a r y t o b r i e f l y o u t l i n e t h e  temperature determined  experimental r e s u l t s .  low  S e p a r a t i o n f a c t o r s have been  i n two ways a t 20.4°K.  F i r s t , i n the r a t e s t u d i e s o f  7 Cunningham and J o h n s t o n  u s i n g doped a l u m i n a , i t was  assumed t h a t  t h e o r t h o t o parahydrogen c o n v e r s i o n took p l a c e o n l y i n t h e adsorbed was  first  l a y e r and t h a t homogeneous c o n v e r s i o n i n l i q u i d hydrogen  s l o w and c o u l d be n e g l e c t e d .  except i n the f i r s t  l a y e r , was  para concentration.  The  w i t h a paramagnetic s a l t . on t h i s s u r f a c e was  16.0  Thus a l l the l i q u i d hydrogen,  assumed t o be a t t h e same o r t h o -  absorbent The  used was  alumina  impregnated  s e p a r a t i o n f a c t o r f o r hydrogen found  ± 3.0.  The mechanism o f c o n v e r s i o n i s one  i n w h i c h s p i n r e v e r s a l i s caused by a c t i o n o f a magnetic i o n on hydrogen n u c l e u s . molecule  the  T h i s o c c u r s o n l y when t h e i o n and t h e hydrogen  are i n c l o s e p r o x i m i t y .  I t seems r e a s o n a b l e t o assume  t h e r e f o r e t h a t o r t h o - p a r a c o n v e r s i o n i s o c c u r r i n g i n the p h y s i c a l l y adsorbed  first  monolayer and t h e s e p a r a t i o n r e f e r s t o the  con-  c e n t r a t i o n o f t h e hydrogens i n t h i s l a y e r r e l a t i v e t o the remainder o f the m i x t u r e .  I f t h e o u t - o f - p l a n e h i n d e r e d model i s used, i t can  be seen from F i g u r e 22 t h a t a s e p a r a t i o n f a c t o r o f 16.0 i s o b t a i n e d f o r a b a r r i e r h e i g h t o f A=13 (= 2.25 K c a l / m o l e ] .  A p p l y i n g the  e x p e r i m e n t a l l i m i t s o f ±3.0 f o r t h e s e p a r a t i o n f a c t o r , t h e b a r r i e r found has t h e l i m i t s 2.25 ± .40 K c a l / m o l e . However t h e measured s e p a r a t i o n f a c t o r i s due not o n l y t o h i n d e r e d r o t a t i o n b u t a l s o t o t h e c a t a l y t i c mechanism o f t h e paramagnetic  ion.  Thus i t i s not  r e a s o n a b l e t o c o n s i d e r a s t r i c t l y h i n d e r e d r o t a t i o n a l model i n such a system. The s e p a r a t i o n o f o r t h o h y d r o g e n from parahydrogen  and o f  p a r a d e u t e r i u m from o r t h o d e u t e r i u m , by p r e f e r e n t i a l a d s o r p t i o n on a l u m i n a (where l i t t l e o r no c a t a l y t i c mechanism i s i n v o l v e d ) , have g  been i n v e s t i g a t e d by Cunningham, Chapin and J o h n s t o n . I n t h e s e e x p e r i m e n t s , normal hydrogen and d e u t e r i u m were adsorbed on a l u m i n a t o t h e e x t e n t o f a monolayer and t h e c o n c e n t r a t i o n s o f o r t h o h y d r o g e n and p a r a d e u t e r i u m were determined a t v a r i o u s s t a g e s of the desorption process.  The maximum c o n c e n t r a t i o n o f o r t h o -  hydrogen and p a r a d e u t e r i u m i n t h e adsorbed phase were g i v e n by g  Cunningham et_ a_l.  as 93.5% and 54% r e s p e c t i v e l y .  Using the  formulae  yt (H ) = 2  ([P-H ]/[Q-H ]) 2  2  ([p-H ]/[o-H ]) 2  2  ^  gas p h a s e  a d s o r b e d  p  h  a  s  e  ^  172  A n, y& CP )  ^  °- 2^^- 2^\ =. a  D  a5  2  «.L  separation factors and  2.4  2  J  £_gas L  2  r  phase pnase  ^  adsorbed phase  1J  f o r hydrogen and deuterium are found t o be  respectively.  869 c a l / m o l e  D  From F i g u r e 22,  one f i n d s  f o r hydrogen and 673 c a l / m o l e  a barrier  for deuterium.  4.8  of White  12 and  Lassettre  assumed t h a t the b a r r i e r h e i g h t s f o r hydrogen on impregn-  a t e d a l u m i n a and. f o r h y d r o g e n and d e u t e r i u m on neat alumina were a l l equal.  From the v a l u e o f /$  they found a b a r r i e r h e i g h t b a r r i e r to calculate alumina o f 4 . 1 .  f o r hydrogen on impregnated alumina o f 2.35  K c a l / m o l e and then used  this  a s e p a r a t i o n f a c t o r f o r d e u t e r i u m on impregnated  However t h e i r assumptions  are u n r e a s o n a b l e  on s e v e r a l  counts. The on  the m o l e c u l e  erent on  e f f e c t i v e b a r r i e r height adsorbed and the  to r o t a t i o n i s  adsorbent.  both  s h o u l d be  diff-  f o r hydrogen adsorbed on impregnated a l u m i n a , hydrogen adsorbed  a l u m i n a and deuterium adsorbed on a l u m i n a .  impregnated a l u m i n a system might be expected l a r g e r b a r r i e r to o u t - o f - p l a n e system due,  The hydrogento e x h i b i t  a slightly  r o t a t i o n than the h y d r o g e n - a l u m i n a  i n p a r t , t o the presence  However the b a r r i e r h e i g h t of  Thus i t  dependent  o f the  l a r g e paramagnetic  ion.  o b t a i n e d from the s e p a r a t i o n  16 f o r hydrogen on impregnated alumina cannot be compared w i t h  factor that  for  hydrogen on alumina because o f the c a t a l y t i c mechanism i n v o l v e d i n  the  former.  F o r hydrogen  and d e u t e r i u m adsorbed on a l u m i n a ,  hydrogen  e x h i b i t s a l a r g e r b a r r i e r than d e u t e r i u m , i . e . f o r hydrogen A = 869 c a l / m o l e and f o r d e u t e r i u m A-673 c a l / m o l e .  This ordering 31  can be j u s t i f i e d as f o l l o w s . are  As l o n g as o n l y d i s p e r s i o n f o r c e s  i n v o l v e d , the i n t e r a c t i o n between a m o l e c u l e i n the  adsorbed  phase and t h e s u r f a c e ( c o n s i d e r e d as an i n s u l a t o r ) can be e v a l u a t e d by summing ( o r i n t e g r a t i n g ) the c o n t r i b u t i o n s from each m o l e c u l e or s e c t i o n t h a t makes up t h e s o l i d , c o n s i d e r i n g t h e s e as dipoles. directly  fluctuating  Then the o v e r - a l l van der Waals a t t r a c t i o n can be o b t a i n e d 16 17 from one o f t h e formulae  d i s p e r s i o n energy  '  which g i v e the i n t e r m o l e c u l a r  (e.g., Kirkwood-Muller) using e m p i r i c a l q u a n t i t i e s  such as p o l a r i z a b i l i t i e s and d i a m a g n e t i c s u s c e p t i b i l i t i e s .  Thus,  an e s t i m a t e o f t h e t o t a l p o t e n t i a l between one m o l e c u l e and t h e s u r f a c e (and t h e heat o f a d s o r p t i o n ) can be o b t a i n e d by t h e u s u a l Lennard-Jones  (12,6) p o t e n t i a l t o h o l d between a p o r t i o n 32  o f t h e s o l i d and the m o l e c u l e and i n t e g r a t i n g Jones  (9,3) p o t e n t i a l .  t o get a Lennard-  When the Henry's law d a t a f o r ti^ and  g r a p h i t e were a n a l y s e d u s i n g the quantum c o r r e c t e d second 17 surface v i r i a l  assuming  coefficient  w i t h a Lennard-Jones  on  gas-  (9,3) p o t e n t i a l  and t h e K i r k w o o d - M u l l e r f o r m u l a f o r e v a l u a t i n g t h e g a s - s u r f a c e c o l l i s i o n diameter z was  t h e hydrogen-adsorbent  interaction  found t o be g r e a t e r t h a n t h e d e u t e r i u m - a d s o r b e n t  energy.  energy  interaction  T h i s r e s u l t e d i n d e u t e r i u m h a v i n g a g r e a t e r apparent Z  v a l u e than'hydrogen.  Q  The same b e h a v i o r would be expected on a l l  174  coiitinuum s u r f a c e s .  Thus, s i n c e d e u t e r i u m  i s f u r t h e r away from the  s u r f t e e than hydrogen and i n t e r a c t s l e s s s t r o n g l y w i t h the s u r f a c e , i t wouid be expected t o e x h i b i t t h e s m a l l e r b a r r i e r h e i g h t t o outof-plane r o t a t i o n . I t s h o u l d be m e n t i o n e d , however, t h a t t h e s e p a r a t i o n f a c t o r s g  o b t a i n e d from the a d s o r p t i o n experiments  o f Cunningham e t al_.  not be t o t a l l y r e l i a b l e .  T h e o r e t i c a l l y the s e p a r a t i o n f a c t o r s  c a l c u l a t e d from e q u a t i o n s  (6.1) and  may are  (6.2) which assume e q u i l i b r i u m  between t h e gas phase and t h e adsorbed  phase.  Since the  adsorbent  had t o be heated on d e s o r p t i o n , i t i s not c e r t a i n t h a t t h e  desorbed  sample has t h e c o m p o s i t i o n o f t h e adsorbed  initially  i n e q u i l i b r i u m w i t h t h e f e e d gas.  l a y e r which was  C o n s i d e r a t i o n of the  experiments  on hydrogen i s f u r t h e r c o m p l i c a t e d by the f a c t t h a t a slow  but  n o n - n e g l i g i b l e c o n v e r s i o n o f o r t h o - t o parahydrogen accompanied the d e s o r p t i o n .  These e x p e r i m e n t a l p r o b l e m s , c o u p l e d w i t h the  fact  t h a t c a l c u l a t e d s e p a r a t i o n f a c t o r s are e x t r e m e l y s e n s i t i v e t o changes i n t h e c o n c e n t r a t i o n o f o r t h o h y d r o g e n ,  ( i . e . a concentration  range o f 98% t o 96% l e a d s t o s e p a r a t i o n f a c t o r s r a n g i n g from 16 t o 8 ) , make i t i m p o s s i b l e t o p i n down any d e f i n i t i v e v a l u e f o r the hydrogen rotational b a r r i e r height.  T h e r e f o r e , not u n t i l v e r y  accurate  e x p e r i m e n t a l s e p a r a t i o n f a c t o r s are a v a i l a b l e w i l l one be a b l e t o determine  j u s t how  s u c c e s s f u l the o u t - o f - p l a n e h i n d e r e d r o t a t o r model  i s f o r the p r e d i c t i o n o f r o t a t i o n a l  barriers.  S e p a r a t i o n f a c t o r s f o r the o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t o r can be c a l c u l a t e d from e q u a t i o n s  (6.1) and  (6.2)  by r e p l a c i n g q ^ (3.30a -  r  3.31c).  by q ^ ' r  For X= l  which are  and X = 1 0 , r e p a r a t i o n  are  g i v e n i n F i g u r e 23 as  the  in-plane b a r r i e r height  factors  rise  (curves  3 a n d 4)  for  out-of-plane  the  Let  us  experimentally. will  be  a f u n c t i o n o f the increases,  very r a p i d l y , but are  not  as  longer unique. solutions  F o r any g i v e n A v a l u e ,  separation factors  o f 16 i s  measured  < 16.,  factor the  case  above.  s u c h that%j  Therefore  As  in  there agreement  solution is  no  1 and 2 i n F i g u r e 23 g i v e t w o  :  X = 1 0 . 0 = 1.74 y = 0.45  2.  X = 1.0 y = 1.98  unless  X and y ,  factor  a separation  For example, curves  1.  Thus,  in-plane barrier.  discussed  a separation  experimental value of 16..  20.4°K  T h i s was a l s o t h e  factors  yield  equations  at  the deuterium s e p a r a t i o n  separation  assume t h a t  3 by  factors  the hydrogen  sensitive.  a y value which w i l l  with the  given i n Chapter  one h a s  or the  ment b e t w e e n However t h e  = 174 c a l . = 344 c a l .  o f one o f t h e m ,  to rotation  theoretical existence  = 78.2 c a l .  some i d e a as t o t h e  magnitude  what b a r r i e r h e i g h t s  Kcal.  are  separation  r e l a t i v e magnitudes it  is  impossible to  required to y i e l d factors  of  good  and e x p e r i m e n t a l  of in-plane hindered rotation i s  state  agreeones.  certainly  FIGURE 23 Separation factors  f o r hydrogen and deuterium at  as a f u n c t i o n o f the l a b e l l e d as  in-plane b a r r i e r .  follows:  Curve  1.  H , A=10  Curve  2.  H , A=l  Curve  3.  D , A=10  Curve  4.  D , A=l  2  2  2  9  20.4°K,  The f o u r curves  plotted are  i m p o r t a n t b e c a u s e o f the extreme s e n s i t i v i t y o f the t h e o r e t i c a l s e p a r a t i o n f a c t o r s t o y.  U n t i l very accurate  experimental  s e p a r a t i o n f a c t o r s are a v a i l a b l e and u n t i l something i s known o f t h e r e l a t i v e magnitudes o f A and y f o r a g i v e n  molecule-adsorbent  s y s t e m , b a r r i e r s t o r o t a t i o n can not be d e t e r m i n e d ,  n o r can  one  make any d e f i n i t i v e c h o i c e as t o w h i c h o f the two m o d e l s , t h e outo f - p l a n e h i n d e r e d r o t a t o r o r t h e o u t - o f - p l a n e and i n - p l a n e h i n d e r e d r o t a t o r , r e p r e s e n t s the true p h y s i c a l hindered best.  rotation  179  THE EFFECT OF HINDERED ROTATION ON THE SECOND GAS-SURFACE VIRIAL COEFFICIENT.  CHAPTER V I I .  A) T h e o r e t i c a l Development. F o l l o w i n g the development Helmholtz f i ? e  33 o f Barker and E v e r e t t * ^ ,  the  energy o f an i d e a l d i a t o m i c gas can be w r i t t e n  as  -kTAnQ,  where the p a r t i t i o n f u n c t i o n Q i s  (7.1)  g i v e n by  N  3  1  2iTMkT  (7.2)  N!  M is  the m o l e c u l a r mass, N the number o f molecules  volume o f the q^  r>  system.  The m o l e c u l a r p a r t i t i o n f u n c t i o n s ,  are the v i b r a t i o n a l and f r e e r o t a t i o n a l p a r t i t i o n  L e t t i n g nN=N and Nk=R, the f r e e  F =  -nRT-i 1 + y £ n  -nRT£n(q  of  chemical p o t e n t i a l the f r e e  v i b  and V the q ^  g =  3F — 9n  and  D  functions.  2TrMkT  )  -nRT  in \~ nN'  2  -nRT£n(q )  (7.3)  f r  g can be found by t a k i n g the p a r t i a l  energy w i t h r e s p e c t  v  energy can be r e w r i t t e n as  h  The  present  derivative  t o the number o f moles o f gas  :  180  -RTj&n  ,2  J  -RT£n(q ) -RT£n(q- ) ^vib' fr^ v n  kT  -RT£n  (7.4)  where V/nN i n (7.3) has been replaced by kT/p, i.e. the gas phase i s treated as ideal. We now consider a system of admolecules whose out-of-plane rotation i s hindered. The p a r t i t i o n function of such a system can be written as follows:  Q  =  a d s  KitP  fr r>f  fa  h  (/) d r ,  (7.5)  N  N!  where q^ i s the out-of-plane hindered rotational p a r t i t i o n function r  and the Slater sum ^' 1*> 3  35  ( r ) i s the exact quantum mechanical N  analogue of the Boltzmann factor \i ( r ) . At high temperatures where N  quantum deviations are small,-»• W^.  The series expansion for  'l'^(r^) i s well known^ ^ and i s given by  «  (r ) N  H  N  1 * X  2  I w  2  ^  +  '...}  (7.6) i n which  W (r ) - X-™ e-WcA N  N  (7.7)  181  X =  1 (2irMkT) 2  ,  N  4>(r ) = I i=l N  w.  (j) _  3  V/4>  <Kr..) ,  I  cpfr ) +  (7.8)  - |  (V cp)  (7.9)  (7.10)  2  24TT  0)  w  __§_ i „ ^  =  9607T  4  5V. cj)(V.cp) 2  2  3 6  2 ( 3  2V. (V.cp) 2  3  ~J  + 3V.cp.V. (V.cb)  2  + 8V.cp.V. cp + 5(V. .<j>) 3  ~J ~J  2  J  :  (7.11) 24  "3  and  ^  N  ( l  )  =  The t o t a l  Z (±1)  X  e x  (7.12)  P  intermolecular potential  [equation  (7.9j]is  assumed t o be  a sum o f N a d m o l e c u l e - s u r f a c e  i n t e r a c t i o n s p l u s another sum o f p a i r -  wise i n t e r a c t i o n s between the  admolecules.  refers  to B o s e - E i n s t e i n s t a t i s t i c s  The upper s i g n i n  and the lower s i g n to  (7.12)  Fermi-Dirac  statistics. S i n c e the i d e n t i t y p e r m u t a t i o n o p e r a t o r causes the o f the exponent  to v a n i s h , 10  = X  i f the symmetry o f the  argument eigen-  182  functions  has no i n f l u e n c e .  This value A ^  is  the v a l u e o f W f o r xt  N N an i d e a l gas (<j>(r ) 0 ) The d i s t a n c e over which the molecules i n f l u e n c e each o t h e r due to the symmetry o f the wave f u n c t i o n s i s =  of  1/2 the o r d e r o f magnitude h/(2MkT) factor,  is  w h i c h , except f o r a n u m e r i c a l  the de B r o g l i e wavelength  temperature T .  o f the m o l e c u l a r motion at a  As t h i s wavelength becomes much s m a l l e r than  molecular diameter at high temperatures, statistical  e f f e c t s become v e r y s m a l l .  temperatures ^ j j ( r )  W ^ (r  these d e v i a t i o n s  potential  V by  are c o n s i d e r e d to behave  cb(r ) i n  (7.7)  is  to at h i g h  ). obeyed so t h a t the  t h a t o f an adsorbed molecule w i t h the s u r f a c e .  the admolecules  N  due  Thus f o r an i d e a l gas  We now assume t h a t H e n r y ' s law i s interaction is  the  only All  i d e n t i c a l l y so t h a t the .  ~\ \ N  r e p l a c e d by Nc£ (r) and ^ ( r s  ) is  replaced  N  C*^(r)) .  The p a r t i t i o n f u n c t i o n g i v e n by (7.5)  now becomes  N Q  ads  =  ( C  J-W'cr)  W  dr}  A"  3 N  ,  .  • (7.13)  where  ^'(r)  = W '(r) j l + A I w ^ 2  x  +  n J 1  +  °'(r)  A  - £ ^Cj)  .  +  _}  (7.14)  183  In  equation  (7.14)  -3<t> (r) s  W  (r)  I  =  e  (7.IS)  and ° i  V 1  (r) ~  = ± exp  (- — r ) . X  (7.16)  2  2  (upper s i g n B . E . ) Using  (7.3),  written  the H e l m h o l t z f r e e  energy o f the  adsorbed gas  can be  as  F  a  d  s  =  -kT£n(Q  a d s  )  2TrMkT  = -nRljl +  :  -nRT£n(q . ) v  ^ nRT£n(q ) h r  h  2  b  •  -nRT£n  ^1  (r)  dr  (7.17)  nN  The  chemical p o t e n t i a l  g . ads 6  = -RT|tn 2  '  2 T r M k T  g  a d s  -RT£n  o f the adsorbed gas t h e n f o l l o w s  (q . ) v  b  -RT£n(q ) h r  -  RT£n  ^  as  (jJ dr nN (7.18)  E q u a t i n g the phase, is  chemical p o t e n t i a l s  one f i n d s  g i v e n by  that  the  o f the  gas phase and the  t o t a l number o f moles o f gas  adsorbed  i n the  system  184  n =  111  q  (7.19)  fr  Assuming t h e gas phase t o be i d e a l , t h e number o f moles i n the gas phase i s j u s t  n° =  where V  V  (7.20)  <RTj  geo  i s t h e geometric volume o f the system. °  geo  The number o f moles  J  o f gas adsorbed then f o l l o w s as  n j = ads  n-n  %T  ^'(r)  hr  + V geo  fr  -V  geoj  E RT  -1  V.1. (~r ) d ~r  The term  dr - V  geo  (7.21)  or  id. 1  quantum c o r r e c t e d second g a s - s u r f a c e v i r i a l quantum B^g, t a k e n t o o r d e r fi  h  ( r ) -1 dr  JI  coefficient  i s j u s t the B  A C  .  The t o t a l  i n t h e Wigner-Kirkwood expansion, i s  g i v e n by Cl B  AS  =  B  AS  . +  B  AS  „ +  B  AS  *  perf B  AS  ' ^ PP u  e r  s i  S  n  B  - -^ E  (7.22)  185  where  -e<f>  -i  dV,  (7.23)  geo  -3<f>  2 2 Q  AS  12M  dV,  (7.24)  V geo  K ho3 3 A  240M  S  e  S  V>  s  -  |  2V  Vcp  2  geo  + 8Vtp .V cp  +5  3  s  '  1  s  '  *  + 3Vcp .V. •Vcb XT s j  N] *r 1  53J  s  24  Vcp  5V cp 2  •av,  (7.25)  and perf 3  -s*-A/1/2 ,  AS  where ^  (7.26)  i s the surface area o f the adsorbent,  cp i s t h e i n t e r a c t i o n s  p o t e n t i a l between an i s o l a t e d gas m o l e c u l e and t h e s o l i d i n t h e d i f f e r e n t i a l volume element dV, and we s h a l l use t h e f o l l o w i n g tion f o r this quantity.  approxima-  I f one assumes t h a t t h e gas m o l e c u l e s i n t e r -  act w i t h i n d i v i d u a l atoms o f t h e s o l i d t h r o u g h a Lennard-Jones potential  and t h a t t h e s o l i d can be c o n s i d e r e d t o be  (12,6)  semi-infinite  ( i . e . , i n f i n i t e i n x and y and bounded by t h e p l a n e z = 0) and t o  186  obey t h e continuum model (12,6) p o t e n t i a l solid  (i.e.  can be i n t e g r a t e d over the t h r e e - d i m e n s i o n s  to y i e l d a L . J . (9,3)  potential  'g  s  s  s  o f the form  (7.27)  L  i s t h e maximum energy o f g a s - s u r f a c e  distance action  the  o f the  3 /3  •* (z)  £g  uniform d i s t r i b u t i o n o f m a t t e r ) ,  i n t e r a c t i o n and Z  between a gas atom and the p l a n e s u r f a c e  q  i s the  at zero net i n t e r -  energy. For convenience  virial  coefficient  is  the quantum c o r r e c t e d second  o f t e n w r i t t e n i n reduced form as  u it* B  AS  =  B  AS  gas-surface  + A  s AS  +  B  s  B.„  AS  ±  p e r f ¥A B s AS  (7.28)  where -1 AS  B  (7.29)  AS  and A'  B  AS  '  B  AS  a n c  h/z  *  B  AS  v  adsorption isotherm,  " a ads  -  I  a  (Me ) gs  ° l  u  e  s  1/2  (7.30)  have been p u b l i s h e d p r e v i o u s l y ,  The  ( 7 . 2 1 ) , can now be w r i t t e n as  r hi q  (j f r L  geo  fr  -l  RT"  (7.31)  187  B) Treatment of the Data for H and D on P33(2700°). 2  2  Experimentally the adsorption isotherm i s written as  exp  n ads = BAS  P•RT-'  P- + AAS c  eX1>  R T  2  + . .. ,  (7.32)  and when terms higher than the quadratic are neglected, plots of exp n*ads^'^' , /p versus p are linear with gradient C..„/(RT) and intercept '- - ° y j-j-n^cij. w-i-uii a u i c u t ^AAS vt  L  2  3w  B^gP/RT. Therefore these so-called gas-surface v i r i a l coefficients can be obtained directly from adsorption isotherms measured at low surface densities. Henry's law d a t a for H and D on P33(2700°) have been exp by f i t t i n g the B^g values to the quantum corrected 15  2  analysed  1 7  2  theoretical expression for the gas-surface v i r i a l coefficient. In 15 g XT) the work of Constabaris, et a l . , data for B^g as a function of temperature were f i t t e d to B^g by adjusting the two parameters F  e gs /k and5$zo, thus obtaining • b e s t - f i t values for these parameters, r  From the classical f i t , Z was calculated from the well known Kirkwood39 40 41 Muller ' , Slater-Kirkwood or London formulae and a c l a s s i c a l q  • made. MacRury 17 f i t t e d the B P values to the quantum estimate of A* e x  corrected second gas-surface v i r i a l coefficient, where B  AS  = f(T*, A* ) , s  (7.33)  with T* = (3 e ) a-  5  _ 1  •  (7.34)  Best-fit values of £ /k and were then found and a new A gs' o s value calculated. This process was repeated u n t i l self-consistent 17 *" egs'/k, ' ^ z o and A s values were obtained. The results found for H  2  and  are given i n Table 20. To investigate the effect of out-of-plane hindered  rotations on these data for H and D^, the following assumption 2  w i l l be made. Since the measurements were made at f a i r l y high temperatures  (90° - 140°K), and since the system was allowed to 42  equilibrate at each temperature for six to eight hours  , i t seems  reasonable to assume that the experimental results pertain to equilibrium hydrogen and deuterium rather than to normal hydrogen and deuterium.  Thus the equilibrium hydrogen and equilibrium deuterium  hindered rotational p a r t i t i o n functions w i l l be used. Two different f i t t i n g procedures were employed, the f i r s t being similar to that given above.  From (7.31) and (7.32) one can  equate the coefficients of the p/RT terms : exp q BAS = q ^  *  h  (&z  A C  f r  B  )  o AS A C  +  V geo  rq  h  fr  > '•1  (7.35)  which can be rearranged as exp t AS " geo i  B  ^  z  o  =  V  .  q  hr  ^fr  •  .  ( 7 > 3 6 )  exp B  AS  and £  is first  f i t t e d t o the c l a s s i c a l B 'AS  /k e s t i m a t e s .  to  obtain classical$Az  Then the K i r k w o o d - M i i l l e r f o r m u l a i s used t o  * c a l c u l a t e z , which i i i t u r n g i v e s one a f i r s t e s t i m a t e o f A . o' s 6  v a l u e f o r the o u t - o f - p l a n e b a r r i e r o f X=l q, /q~  temperatures.  i s continued u n t i l s e l f - c o n s i s t e n t values f o r these  parameters are found. i n Table  A  i s used and the r a t i o  i s c a l c u l a t e d f o r each o f the e x p e r i m e n t a l  procedure  o  The b e s t f i t v a l u e s f o r  and  are g i v e n  20.  that a plot of ^ t z  o  v e r s u s the cube r o o t o f the c r i t i c a l  volume ( o r  the gas phase c o l l i s i o n d i a m e t e r a) s h o u l d be l i n e a r * ' ^ . 9  o t h e r words, the s u r f a c e area o f the adsorbent independent  o f the a d s o r b a t e used.  3  In  s h o u l d be a c o n s t a n t ,  For t h e v e r y homogeneous non-  porous s o l i d c o n s i d e r e d h e r e , t h i s i s a v e r y r e a s o n a b l e  assumption.  T h i s i s found t o be the case f o r the i n t e r a c t i o n o f the r a r e gas atoms Ne, A r , Kr and Xe w i t h t h i s s u r f a c e .  However the b e s t f i t  17 (9,3) r e s u l t s  from  and  ( i g n o r i n g the i n t e r n a l modes)  i n d i c a t e d t h a t the hydrogen and d e u t e r i u m r e s u l t s f a l l w e l l o f f the l i n e d e f i n e d by the r a r e gas d a t a , as do r e s u l t s f o r s i m p l e p o l y 19 43 atomic m o l e c u l e s . The model d e s c r i b e d by Sams, et_ al_. neglected r o t a t i o n a l degrees o f freedom o f the admolecules and i t i s l i k e l y  190  TABLE 20 Best-fit parameters for H and D . 2  2  Note 5 = [e(H)-e(D)]/e(H) g  17 1. Quantum F i t  , no hindered rotation.  e /k(°IQ H  2  z (A)  ^(mV ).  2.004  9.53  ,74^(10^  gs  674  o  19.10  1  s.d.(io ) 8  6 cio ) 2  s  0.197 9.50  D  610  2  22.88  2.043  11.2  Z free).  2. Quantum F i t with hindered rotation X  H  ^  e  gs  173.8  2  / k C  °  K )  M  z  o ^ °  647  k  ^  o^  Z  18.68  0.314  q  '^CmV ) 1  2.004  s.d.flO ) 5 (10 )  9.32  8  2  s  0.202 7.11  D  86.9  2  601  24.90  2.054  12.1  3. Quantum f i t with hindered rotation a n d $ z A  H  2  Se) 173.8  V  k  r  K  621  )  ^ V  1  0  ^  22.57  Z  o ^ 2.032  Q  0.217  fixed.  ^V ) 1  11.1  s.d.flO ) 6 U0 ) 8  2  s  2.80 1.13  D  2  86.9  614  22.50  2.038  11.0  1.10  191  that the failure of the model for the hydrogens and simple polyatomic molecules i s due to hindered rotation.  Thus (9,3)O4Z values for the q  rare gas were plotted against O and a least squares straight line was 44 constructed.  Using the o values found by Michels, e_t al_.  and D.~, the following ^z  for  values are required to put the results  for these gases on the line : H  20  D. 2  ^z o = 22.55 x 10~" —gm  a = 2.959 A,  , ' 3  a = 2.952 A,  = 22.51 x l(f" — . o gm  exp ^ One now f i t s B^g to the classical B^g to get f i r s t estimates ofzQz  o and egs/k from which A*s i s calculated.  Then quantum corrections  and hindered rotational effects are included and a new estimate o f ^ z i s found i n the usual way.  However the usual 'errors squared' method  i s not used as the c r i t e r i o n for best f i t .  o  Instead the new ^z  i s compared with the predetermined value from the' iz ?  o  value. o versus a plot  and the difference between these two values i s minimised, i . e . H and 2  D2 are forced to l i e on the line. 0  minimises this difference,  z o  When an egs'/k value i s found which  and A  s  are recalculated.  This procer  dure i s continued u n t i l a self-consistent e /k value i s obtained. The results from this f i t for A=l are also given i n Table 20. C) Discussion of the Best-fit Results. It i s apparent from Table 20 that the b e s t - f i t gas-surface interaction energy for hydrogen i s very sensitive to hindered rotation  whereas  that  f o r deuterium i s not as s e n s i t i v e .  rotation with»$z  o  free,  e  / k f o r H_ d e c r e a s e s by about 4.0% and gs' 2 J  £ ^ / k f o r D„ d e c r e a s e s by o n l y 1.5%. two r e a s o n s .  Firstly,  Including hindeied  This is  to be expected  for (A=l)  a reduced r o t a t i o n a l b a r r i e r o f u n i t y  is  a b a r r i e r of  If  one compares t h e s e two b a r r i e r s w i t h the r e s p e c t i v e quantum  £  / k values  gs  173.8  cal/mole for  for free r o t a t i o n  expect l a r g e r e f f e c t s effects  in  for H  2 <  17  and 86.9  cal/mole  g i v e n i n T a b l e 20, 5  one would  S e c o n d l y , the h i n d e r e d r o t a t i o n a l  s h o u l d be s m a l l e r than those i n H  e r a t u r e because  f o r D^.  o f the d i f f e r e n c e  2  at the same temp-  in their characteristic rotational  temperatures. When one now compares the second f i t • ( q u a n t u m f i t hindered r o t a t i o n a n d ^ z rotations,  Q  fixed)  one n o t i c e s t h a t  w i t h t h e quantum f i t  for  with  free  fixing ^z  i n c r e a s e s the £ / k v a l u e f o r o gs' D_ by 0.8% and d e c r e a s e s the £ / k v a l u e f o r ti by 7.9%. This is 2 gs 2 6  n  due to the  fact  that  the $ z  o  (from 22.88 x 10"** cm /gm to 3  J  v a l u e f o r D„ must be d e c r e a s e d by 2  1.6%  }  22.51  x 10"  cm /gm)  4  3  must be i n c r e a s e d by 18% (from 19.10 x 10"  11  and &Z  Q  for H  2  cm /gm t o 22.55 x 10""* 3  cm /gm). 3  Another i m p o r t a n t f e a t u r e 6" = [ £ ( H ) - e ( D ) ] / £ ( H ) . s  v a l u e o f 9.50%  f o r the  is  the decrease  I t has been p o i n t e d o u t (9,3)  quantum f i t  1 7  i n 6 ', where t h a t the  (with f r e e r o t a t i o n s ) 18  not i n good agreement w i t h the p o l a r i z a b i l i t y d i f f e r e n c e between H„ and D .  M o r e o v e r , the energy d i f f e r e n c e  of  6  g  was  1.32%  was much l a r g e r  193  t h a n one would expect on t h e b a s i s o f the c o r r e s p o n d i n g gas-phase I values.  I  I f the combining 5  rule e gs 44  measured b u l k gas parameters e(H) s h o u l d be =2.0% r u l e may  - fe e ) g  1/2  i s assumed and  the  used, i t f o l l o w s t h a t [e(H) - e ( D ) ] /  f o r the hydrogen.  Although t h i s  combining  n o t be e s p e c i a l l y a c c u r a t e , i t does not seem p l a u s i b l e  t h a t t h e d i f f e r e n c e s h o u l d be l a r g e r f o r the g a s - s u r f a c e case  than  44 f o r t h e b u l k gas (5 = 4.1%). Even u s i n g t h e more s o p h i s t i c a t e d 45 46 c o m b i n i n g r u l e s o f Hudson and McCoubrey and Fender and H a l s e y does n o t improve t h e s i t u a t i o n .  The  f i r s t method o f f i t t i n g w i t h  h i n d e r e d r o t a t i o n lowers t h e 5  v a l u e t o 7.11%, which i s s t i l l s ' The second method ( f i x i n g ^ z ) y i e l d s a 6 v a l u e o f  rather high. 1.13%  o  w h i c h seems much more r e a s o n a b l e .  On t h i s b a s i s , t h e method  of  fixing<s4z  of  c o u r s e , depend on the A. v a l u e choosen f o r c a l c u l a t i n g  o  seems t o be the b e s t .  s  partition function q^.  H^ and B^.  s  values  will,  the  A l s o as p o i n t e d out i n C h a p t e r V I , i t i s  n o t p o s s i b l e a t p r e s e n t t o determine for  However t h e s e 6  the r e l a t i v e magnitude o f A  In t h e s e f i t s a u n i t l e s s b a r r i e r o f A=1.0  imples  t h a t t h e d e u t e r i u m r o t a t i o n a l b a r r i e r i s o n e - h a l f t h a t f o r hydrogen. -This p r o b a b l y does n o t r e p r e s e n t the t r u e p i c t u r e , a l t h o u g h on the b a s i s o f t h e g a s - s u r f a c e i n t e r a c t i o n e n e r g i e s o f the two s p e c i e s , it  appears t h a t u s i n g a s m a l l e r b a r r i e r f o r B^ i s j u s t i f i e d . The  agreement i n the s u r f a c e areas f o r hydrogen and  d e u t e r i u m i s not improved by the i n c o r p o r a t i o n o f h i n d e r e d i f t h e f i r s t method o f f i t t i n g i s employed.  rotation  In f a c t the d i s c r e p a n c i e s  194  in  &  have i n c r e a s e d s l i g h t l y .  When t h e second method o f f i t t i n g  i s used, t h e s u r f a c e areas a r e , c f c o u r s e , f o r c e d t o e s s e n t i a l l y t h e same v a l u e .  I t i s i n t e r e s t i n g t o note t h a t t h e i n c l u s i o n o f  hindered r o t a t i o n s with for  deuterium  and l e a v e s t h e hydrogen f i t s t a n d a r d d e v i a t i o n s a l -  most t h e same. ered r o t a t o r procedure  f r e e improves t h e s t a n d a r d d e v i a t i o n s  I f enough e i g e n v a l u e s f o r t h e o u t - o f - p l a n e h i n d -  were a v a i l a b l e one might be a b l e t o r e p e a t t h e f i t t i n g  u n t i l t h e s t a n d a r d d e v i a t i o n s were m i n i m i s e d .  However i t  i s f e l t t h a t t h e d a t a a r e not s u f f i c i e n t l y a c c u r a t e t o j u s t i f y an attempt.  The s t a n d a r d d e v i a t i o n s a r e n o t as good f o r t h e second  f i t t i n g procedure, for  such  b u t t h i s i s t o be expected  since the c r i t e r i o n  b e s t f i t i n t h i s case i s t h e \$-Z v a l u e o b t a i n e d . q  However,  a g a i n i t may be p o s s i b l e , f o r some v a l u e o f A, t o produce s t a n d a r d d e v i a t i o n s which a r e b e t t e r than those f o r t h e quantum f i t w i t h no hindered  rotations. A b a r r i e r h e i g h t o f X=5 was a l s o employed i n t h e f i t t i n g  of the v i r i a l data.  The b e s t - f i t g a s - s u r f a c e i n t e r a c t i o n  a r e g i v e n below:  1.  $ z H D  o  free:  2  e /k = 731°K gs'  2  e /k = 484°K gs'  0  0  energies  2. <&z fixed: o H„ £ /k = 556°K 2 gs' D  2 0  e /k = 645°K gs  In the f i r s t f i t the energy difference i s unreasonably large (6^ = 34%). The second f i t gives deuterium the larger interaction energy.  This i s contrary to the fact that the lighter of two  isotopes has always been found to exhibit the larger interaction energy^' '^>^. 17  The reason why the b e s t - f i t results for A=5  are not satisfactory i s probably due to the use of a barrier height which i s physically unreasonable for the hydrogen-graphite and deuterium-graphite system. It i s satisfying that the v i r i a l data for hydrogen and deuterium are extremely sensitive to hindered rotation so that there appears to be good j u s t i f i c a t i o n for incorporating hindered rotation i n the ^ f ^ ) - P33(2700°) system.  On the basis of reason-  able agreement between the 6 value obtained from the f i t and one s  obtained using the combining rule e = fe e ) gs g s J  1/2  or the polariza^  b i l i t y difference, the second method of f i t t i n g seems to be superior.  However i t i s s t i l l impossible to actually determine  the value, of A for H^ and from such f i t t i n g procedures. A CcL 1 value of A=173.8 —=- for H_ on graphite seems quite reasonable mole 2 \ &  but there i s no assurance that  n  w i l l have a barrier which i s  o n e - h a l f t h a t f o r H^-  Assuming the p r e s e n t b a r r i e r f o r  then  the b a r r i e r f o r d e u t e r i u m c o u l d be found i f an a c c u r a t e v a l u e for  6  s  c o u l d be d e t e r m i n e d .  parameters  One  could set 6  and  s  o4- z  o  as f i x e d  and i t e r a t e t h e d e u t e r i u m f i t t i n g p r o c e d u r e f o r d i f f -  erent X values u n t i l  the f i t t e d  6  and *A z s  were i n good agreeo •° 6  ment w i t h t h e f i x e d v a l u e s . I t would a l s o be i n t e r e s t i n g t o i n v e s t i g a t e t h e h i n d e r e d r o t a t i o n s o f the n u c l e a r s p i n s p e c i e s o f hydrogen adsorbed on g r a p h i t e .  and  deuterium  However one would have t o d e v e l o p a v e r y  r a p i d and a c c u r a t e method o f d e t e r m i n g a d s o r p t i o n i s o t h e r m s t o ensure the minimum amount o f o r t h o t o p a r a c o n v e r s i o n .  Until  measurements have been attempted s u c c e s s f u l l y , i t does not  appear  w o r t h w h i l e i n t r y i n g t o improve the h i n d e r e d r o t a t i o n a l model e x c e p t as a p u r e l y t h e o r e t i c a l  exercise.  such  197 1  CHAPTER VIII. S'JMMARY As discussed i n Chapter IV, the out-of-plane hindered rotational eigenvalue equations and the out-of-plane plus in-plane hindered rotational eigenvalue equations could be solved for a large range of barrier heights very accurately using a variational method. The eigenstates of the out-of-plane hindered rotator were characterl  ised by the quantum numbers p(=(-l) ) and m.  A l l the 2£+l m degen-  eracy of the free rotator was removed except for the ±m degeneracy. The eigenstates of the out-of-plane and in-plane hindered rotator could be characterised by the quantum numbers p and r ( = (-l) ) . m  Treating the in-plane hindered rotational potential as a perturbation on the out-of-plane hindered rotator completely removed the +m degeneracy of the out-of-plane hindered rotational eigenstates. In Chapter V the hindered rotational heat capacities and entropies of the nuclear spin species of hydrogen and deuterium were studied by using the hindered rotational eigenvalues given i n Chapter IV.  The A and u barriers had a very marked effect on the thermodynamic  properties of equilibrium hydrogen (equilibrium deuterium) and orthohydrogen (paradeuterium).  The heat capacities and entropies of para-  hydrogen (orthodeuterium) were not as sensitive due, i n part, to the fact that the lowest rotational eigenstate was nondegenerate so that the A and u perturbations only altered the energy of this state, not i t s rotational degeneracy.  198  The comparison of t h e o r e t i c a l separation factors with the experimental ones i n Chapter VI was uncertainties.  clouded by the large experimental  The doped alumina -H  system could not be analysed  2  using a hindered r o t a t i o n a l model because of the c a t a l y t i c mechanism present.  The adsorption experiments  of hydrogen and deuterium  on  alumina were complicated by the desorption process required to analyse the concentration of orthohydrogen  i n the adsorbed phase.  Thus no  d e f i n i t i v e conclusions could be drawn. In Chapter VII, the Henry's law d a t a  f o r H^ and D  1 5  2  on  P33(2700°) were reanalysed by incorporating, an out-of-plane r o t a t i o n a l hindrance  The b e s t - f i t results were found to be very s e n s i t i v e to \  and so i t appears that hindered rotation may be very important i n the H (D )-graphite 2  2  system.  However i t was not feasible to determine  magnitudes of the out-of-plane hindered r o t a t i o n a l b a r r i e r f o r H T>2 adsorbed on graphite.  2  the and  U n t i l very accurate Henry's law data are  available f o r hydrogen and deuterium adsorbed on a well-defined adsorbent, i t does not appear to be worthwhile  to r e f i t the present  data to a more sophisticated hindered r o t a t i o n a l model. however, be of i n t e r e s t to attempt  It would,  to calculate, possibly on t h e o r e t i c a l  grounds, an accurate a p r i o r i estimate of 6 (=(e(H)-e(D))/e(H)). Then g  i t would be possible to f i t the v i r i a l data to 6  andS^z s  sequently determine hydrogen and  and to cono  the r e l a t i v e magnitudes of the b a r r i e r heights f o r  deuterium.  199  BIBLIOGRAPHY  1.  E . C.-emer and M. P o l a n y i ,  2.  C . A . Swenson, J . Chem. Phys.  3.  E . Wigner, Z . p h y s i k .  4.  L . Farkas and H . S a s s c h e , Z . p h y s i k .  5.  L . G . H a r r i s o n and C . A . McDowell, P r o c . Roy. S o c . A220,  Z. physik.  chem.  18_, 520  7.  C M . Cunningham and H . L . J o h n s t o n ,  (1950).  chem. B23, 19 (1933).  J . Am. Chem. S o c . 80,  (1958). J . Am. Chem.  80i, 2382 (1958) .  9.  A . A . Evett,  10.  A . H . W i l s o n , P r o c . Roy. S o c . (London)  11.  T . E . Stern,  12.  D. White and E . N . L a s s e t t r e ,  13.  A . A . Evett,  14.  T.L. Hill,  15.  G. Constabaris, 367  (London)  Chem. 58, 58 (1954).  C M . Cunningham, D . S . C h a p i n and H . L . J o h n s t o n , Soc.  (1933).  77 (1953).  Y . 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Chem., Vol. 1, Acad. Press, 1964.  23. M.E. Rose, "Elementary Theory of Angular Momentum", John Wiley and Sons, Inc., New York, 1957. 24.  E.P. Wigner, "Gruppentheorie", Friedrich Vieweg und Sohn, Braunschweig, 1931.  25.  T.L. H i l l , "An Introduction to S t a t i s t i c a l Thermodynamics", Addison-Wesley Publishing Company, Inc., 1960, Chapter 4.  26.  T..L. H i l l , i b i d , Chapter 22.  27. J.A. Stratton, P.M. Morse, L.J. Chu, J.D.C. L i t t l e and F.J. Corbato, "Spheroidal Wave Functions", Technology Press of Massachusetts Institute of Technology, John Wiley and Sons, Inc., New York, 1956. 28.  D. White and J . Haubach, J . Chem. Phys. 30, 1368 (1959).  29.  A. Farkas, "Light and Heavy Hydrogen", Cambridge Press, 1935.  30.  "Interatomic Distances", The Chemistry Society, Burlington House, London, W.1., 1958.  31.  0. Sinanoglu and K.S. Pitzer, J . Chem. Phys. 32, 1279 (1960).  201  32.  T.L. H i l l , J , Phys. Chem. 63, 456 (1959).  33.  J.A. B a r k e r and D.H. E v e r e t t , T r a n s . Faraday Soc. _58, 1608 (1962).  34.  J.C. S l a t e r , Phys. Rev. 38, 237 (1931).  35.  J.C. S l a t e r , J . Chem. Phys. 1_, 687 (1933).  36.  G. Uhlenbeck  37.  J.O. H i r s c h f e l d e r , C F . C u r t i s s and R.B. B i r d , " M o l e c u l a r  and E. B e t h , P h y s i c a 3, 729 (1936).  Theory o f Gases and L i q u i d s " , John W i l e y and Sons, I n c . , New Y o r k , 1959, C h a p t e r 6. 38.  J.G. K i r k w o o d , Phys. Rev. 44, 31 (1933).  39.  J.G. K i r k w o o d , Z. P h y s i k 33, 57 (1939).  40.  A. M i i l l e r , P r o c . Roy. Soc. (London) A154, 624 (1939).  41.  H. Margenau, Rev. Modern Phys.  42.  J.R. Sams, p r i v a t e  43.  J.R. Sams, G. C o n s t a b a r i s and G.D. H a l s e y , J . Phys. Chem.  1_1, 1 (1939).  communication.  64, 1684 (1960). 44.  A. M i c h e l s , W. de G r a a f f and C.A. Ten Seldam, P h y s i c a 26_, 393  45.  (1960).  •  G.H. Hudson and J.C. McCoubrey, T r a n s . Faraday Soc. 56, 761  (1960).  46.  B.E.F. Fender and G.D. H a l s e y , J . Chem. Phys. 36, 1881 (1962).  47.  J . P . O l i v i e r and S. Ross, P r o c . Roy. Soc. (London) A265, 477 (1962).  4.8.  G.T. A r m s t r o n g ,  F.G. Brickwedde  N a t l . B u r . Standards  and R.B. S c o t t , J . Research  55, 39 (1955).  202  APPENDIX A.  SOLUTION OF THE OUT-OF-PLANE HINDERED ROTATIONAL MATRIX EQUATION.  From Chapter  I I , we found t h a t the o u t - o f - p l a n e m a t r i x  e q u a t i o n can be w r i t t e n as a s e t o f m a t r i x  equations:  S i n c e t h e o u t - o f - p l a n e h i n d e r e d r o t a t o r r e t a i n s i t s ±m degeneracy, i.e. E ^ ^ ' * ^  E ^ ^ ' ^ , the above m a t r i x equations need o n l y be 3  =  solved f o r p o s i t i v e values o f m s p h e r i c a l harmonics values 0 < reduced  SL< 9.  m  (0 < m <&).  F o r a b a s i s s e t o f 100  (9,<M> t h e I quantum number takes on the  F o r convenience  equation  (A.l) i s w r i t t e n i n  ( u n i t l e s s ) form by m u l t i p l y i n g both s i d e s by 8 i T I / h . A l l 2  e i g e n v a l u e s and b a r r i e r h e i g h t s a r e t h e r e f o r e expressed  2  i n unitless  quantities. In s o l v i n g t h e m a t r i x e q u a t i o n s , one s t a r t s by c h o o s i n g some v a l u e o f A and l e t t i n g m = 0.  The m a t r i x V  z e r o and then one c a l c u l a t e s the  Q . ^ I Q m a t r i x elements by  V  £,0;r,0  = I 21+1 3  A  i s s e t equal t o  1/2  21'+1  [2C {.121 •; 000) C {121'; 000)  +C(£0£';000) C(J£0£>;000)].  The Clebsch-Gordan  (A.2)  c o e f f i c i e n t s are c a l c u l a t e d by e q u a t i o n  u s i n g double p r e c i s i o n l o g a r i t h m s f o r the f a c t o r i a l s . 1 0 x 10 m a t r i x , w i t h d i a g o n a l elements  (2.43)  (A.l) gives a  (£=£') and elements two spaces  203  o f f t h e d i a g o n a l (|£-£'|=2)  V  C P , 0 )  =  1 £,0;£',oj' V  v d t h  V  = U2  > > ~l  l  2  ( A  -  3 )  T h i s m a t r i x i s then s p l i t up i n t o two 5 x 5 m a t r i c e s , depending upon the p p a r i t y :  Ol o)  f  c + 1  '  1  0 )  V and (-1,0)  (-1,0) V  V  £,0;£',o|-  <- ) A  4b  The same p a r t i t i o n i n g p r o c e d u r e i s used f o r t h e reduced ^ ( P ^ m a t r i x . (P,0) The f r e e r o t a t o r e i g e n v a l u e s a r e c a l c u l a t e d from £„ = £(£+1) and , m  (p,or then converted  t o d i a g o n a l m a t r i x elements e. .  r (p,0h ^  r m a t r i x i s then p a r t i t i o n e d  (P,O)  _  . The C  i n t o two 5 x 5 m a t r i c e s :  and  Equation  ( A . l ) f o r m = 0 c a n now be w r i t t e n as two 5 x 5 m a t r i x  equations :  C£  C + 1  '  0 )  • AV  C + 1  ' ) 0 )  C ' =C ' t C+1  0)  (+1  0)  ( +  l j 0 )  «  (A.6a)  204  and  C  + xv C- - ) C " C  1  { C  '  l  t  6  )  lj0)  05  -C" (  lj0)  E " C  ^  ll0)  A J a c o b i m a t r i x d i a g o n l i z a t i o n s u b r o u t i n e , w r i t t e n by D. Kennedy  of  t h i s department f o r t h e I.B.M. 7044, i s used t o s o l v e t h e s e m a t r i x e q u a t i o n s , thus o b t a i n i n g t h e e i g e n v a l u e s the corresponding eigenvectors used f o r convergence  £~ ( * ' ^ and £  £ ^ * ' ^ and £ +  +  ^  C-1,0)  a  m  j  The c r i t e r i o n  i s t h a t a l l o f f - d i a g o n a l elements a f t e r d i a g o n a l _7  i s a t i o n must be l e s s than 1. x 10 t h e i n d e x k Cfor f-~ . . . , E^ ^'^ +  . The e i g e n v a l u e s a r e o r d e r e d by  one g e t s t h e e i g e n v a l u e s  E ^  +  ^ B ^ ^ ^ '  i f (A.5a) i s a 5 x 5 m a t r i x e q u a t i o n ) . The same p r o c e d u r e  i s then r e p e a t e d f o r m = 1, 2, ... 9,  each t i m e c o n s t r u c t i n g t h e a p p r o p r i a t e V andC m a t r i c e s and p a r t i t i o n i n g them a c c o r d i n g t o t h e i r p p a r i t y .  A f t e r t h e m a t r i x e q u a t i o n s have  been s o l v e d f o r a l l p o s s i b l e m v a l u e s [0 < m < I), chosen and t h e e n t i r e p r o c e s s r e p e a t e d .  a new X v a l u e i s  205  APPENDIX B. SOLUTION OF THE OUT-OF-PLANE AND IN-PLANE HINDERED ROTATIONAL MATRIX EQUATION. The matrix equation can be written as  CE  + yV) a  =c £  (B.i)  ,  where  r  t -_  L  k.kf  (B.2)  r  '  for some value of \, and  V /  =  {w  k ) k  ,}  CB.4)  ,  i n the ^(.Q A) basis or W * i n the original  £  =  (w*  A  ,  (B.5)  £,m;£',m'J ' 0 I  (0,<(O basis.  m  m  Recall that  are the eigenvalues of the out-of-plane and in-plane hindered  rotator.  As shown i n Chapter II i t i s convenient to calculate the  in-plane perturbation i n the Y s p l i t into two parts, A„ '  0  (0,<{O basis. , and B  ,, with  W  0  „,  , can be  206  1  i f m = m'  • \  if  0  a l l other  Xi r  (B.7)  |m-m'|=2,  T h e r e f o r e the o n l y p a r t o f W the A„ n i l part. £ , m ; £ ,m'  ,  cases,  , t h a t needs t o be c a l c u l a t e d  y IH y X/ y m  These elements can be w r i t t e n  is  as  IT  \  V,m'W  Q  £ , m ; £ ' ,m'  L e t t i n g £ = cos6 and d £ = s i n 6 d9, Q 2£+l  (?)  g  1/2  (£-m)!  ™  m  is  sine  (B.8)  g i v e n by  m  (B.9)  £  (£+m)!  d9,  where m  m P  CO  o  Cl -z ) 2  7  m 3  3£  £  CO ,  (B.10)  £ 2  and P  £ CO  f£] n  2* n=0  i.  T a k i n g the m ^ d e r i v a t i v e o f "£' t  (B.ll)  1  2  *  P„£  m  2  *~ ] 2n  <  '2£-2n  I  -£-2n  * J  (B.ll)  yields  (£-2n)C^-2n-l)  n=0  CB.12)  207  Substituting (B.12) into (B.8) for £,m and £',m', the matrix element A  0  •  -fM  , can be found by integrating the following equation using  a sixteen point gaussian quadrature :  i  £,m;£',m'  '21+1 (A-m).O 2 (£+m)!  1/2 r  L  \h 2 J  1 r 2 n=0  - l  n  (TI n  ,r i=o  C-i)  '2V-21  V.  *J  (£-2n)  1/2  2  (r+m') !  (£'-2i) (£'-2i-l) ... (r-2i-(m»-l))  V \  I  2£'+l (£'-m')  x (£-2n-l) ... (£-2n-(m-l)) ££-2n-m  i  '2£-2n  J  J  .£ -2i-m' ,  (B.13)  A basis set of t h r i t y - s i x spherical harmonics i s used. From the out-of-plane hindered rotational eigenvalue equations, this basis set was found too large enough for convergence i f A<10.(this i s discussed in Chapter I I I ) . The I.B.M. 7044 computer can only work with two-dimensional arrays, i.e. two indices, and therefore a l l matrix elements W„ „. , f= A„£,m;£*,m' „. . B m,m" ' ) were converted from the four £,m;£'m' indices £,m;£',m' to two indices I and J (W „, , -+ W ) by means 0  of the following tabulation :  At yTH y As  T  yXTl  T  J. J tJ  208  l,V values  m,m' values  -5  0 1 2 3 4 5  -4  -3  -2  -1  0  1  2  3  4  5  1 3 4 5 6 7 8 12 13 14 20 21 22 30 31 32  9 15 23 33  16 24 34  25 35  36  2  10 18 28  17 27  26  Thus the m a t r i x element  11 19 29  W. _ „ ., becomes W,„ , . 4,-3;2,-l 18,6  hindered r o t a t i o n a l eigenvalues  are now w r i t t e n as d i a g o n a l  1,2,  F o r a g i v e n A v a l u e the a p p r o p r i a t e E the computer. t o the  The o u t - o f - p l a n e ^  . . . 36.  is  (B.14)  matrix i s  The c o r r e s p o n d i n g e i g e n v e c t o r s  I , J index andV-/= jw^. j j  elements  C  read i n t o storage are a l s o  found from ( B . 6 ) .  converted One now p i c k s  a v a l u e o f y and s o l v e s the m a t r i x e q u a t i o n g i v e n by ( B . l ) .  Again  the J a c o b i a n method o f m a t r i x d i a g o n a l i s a t i o n i s u s e d , w i t h an  off-  _7 d i a g o n a l c r i t e r i o n o f 1.  x 10  .  Once the program has been  entered  i n t o the computer, the time r e q u i r e d f o r s o l v i n g the 36 x 36 m a t r i x  J-|'  (=  E + y V * 0 i s about one minute f o r each u v a l u e chosen.  A=l and 10,  (B.l)  A=l;  has been s o l v e d f o r the f o l l o w i n g v a l u e s  y= 0 . 2 5 ,  A=10; y= Due t o the  0.5,  1,  2,  For  of y :  3, 4  0 . 2 5 , 0 . 5 , 1 , 2 , 4 , 8 . fact  that  in  the m a t r i x c o u l d be s o l v e d  one m i n u t e , the m a t r i x e q u a t i o n was not broken up i n t o the  i n about set  of  209  matrix equations  with P= +1, r = +1, p= +1, r = - 1 , p=  - 1 , r= - 1 ,  p=  - 1 , r= +1.  (B.16)  However t h i s procedure would be necessary i f larger basis sets were required, i . e . X > 1 0 .  For instance i f one hundred spherical harmonics  were required f o r convergence of the out-of-plane eigenvalues, \-] would be a ( 1 0 0 x 100) matrix.  1  A matrix t h i s large would r a p i d l y use  up the storage space of the I.B.M. 7044 computer.  Using the p a r i t y  of p and r , one could solve the following four smaller matrices.  t-l  H  ,U,i)  (25 x 25) ,  ,(i,-D  (20  X  20) ,  ,(-i,-D  (30  X  30) ,  ,(-i,D  (25  X  25) .  For a given value of p and r , the appropriate YL (p,r) to be converted to W  T  T  .5,  , would have  by the method discussed previously and then  the equation ( B . 1 5 ) could be solved.  210  APPENDIX C. CLASSICAL ROTATIONAi HEAT CAPACITIES AND ENTROPIES. In this Appendix the classical rotational heat capacities and entropies for the free rotator, the out-of-plane hindered rotator and the out-of-plane and in-plane hindered rotator are given. ' A) The Free Rotator. The c l a s s i c a l Hamiltonian for a freely rotating diatomic molecule can be written as 1_  fr  P  2  21  P +  »  CCD  sin e 2  The free rotational molecular partition function then follows as oo  -oo  oo  -oo  2 IT  o  TT  -Bh  fr  dOd^dpgdp^  (C2)  O  where $= (kT) . Substituting (C.2) into C d ) and integrating over angular momentum coordinates and the two polar angles yields 87T IkT 2  <*fr  (C.3)  The c l a s s i c a l free rotational heat capacity and entropy, assuming that the diatomic molecules do not interact and using equation (C.3) i n equations  (3.4)-(3.7),  are then given as follows :  (C ) . = R, v fr ' 1  (C4)  211  8TT IkT S  fr  =  "'  R  1  +  ^  (C.5)  n  B) The O u t - o f - P l a n e H i n d e r e d R o t a t o r . The c l a s s i c a l H a m i l t o n i a n now becomes  h, = h- + X cos 9 hr fr 2  where X i s  the o u t - o f - p l a n e  ,  (C6)  hindered r o t a t i o n a l b a r r i e r .  plane hindered r o t a t i o n a l p a r t i t i o n function follows  oo  ^hr  1  oo 2TT  ( (  o  TT  -3h  hr  The  out-of-  as  dGdcJidpgdp^  CC.7)  o  I n t e g r a t i n g over momentum space and the cf> angle  yields  -BAcos e 2  47T IkT 2  r T F  sinG e  d0.  (C8)  I t can be seen t h a t the i n t e g r a l i n (C.8) has the v a l u e two f o r X=0 or i n f i n i t e  temperatures,  and thus  (C.9)  or  X  -*•  o  T  •+  0 0  212  Using of  (C.8) i n equations  the  out-of-plane  equations  (3.4)-(l'.7)  t h e h e a t c a p a c i t y and  hindered rotator  :  are  g i v e n by the  entropy  following  TT  -BAcos G 2  sin9 cos B e  d9  2  RT + A •  v  h  0  3T  r  /•TT  -BAcos 6 2  sin9 e  d9  TT = R + RT  2  2  cos o s m 9  2 •  -6Acos 9  -6Acos 9 d9  e  z  2  cos 9  sin0 e  2  d0  -3Acos 6  -6Ac6s 9  r 7 r  sin9 e  d9  sine e  x  d6  (cio)  TT  -3Acos 9 2  S  h r  =  R  cos  ;o  +  v sine  d9  e  -BAcos 9  r T T  2  sine  e  d9  /•TT  -6Acos 6 2  +R£n  At  A -»- o o r T ->  free  rotator  0 0  4TT I k T  the heat  limits.  Using  sine e  d9  c a p a c i t y and e n t r o p y h a v e , (C.4),  (C.5),  ( C I O ) and  (C.H)  of course, (Cll),  the  the  213  differences between the heat capacity and entropy of the free rotator and the out-of-plane hindered  AC  =  (C v  ) .  v'fr  -  rotator are given by  (C ). v  v^hr  ' -6Acos e  -3Acos 6  2  2  dB'  cos 8 sin9 e  sine e  d6 -2  -3Acos e  TT  -3Xcos 0 2  de  cos D sine e  2  d6  sin0 e  . .  AS  S  (C.12)  f r " hr' S  -3Acos 0 2  cos 6 sin6 e TT  de  + R£n /•TT  -3Xcos e 2  sine e  d0  -8Xcos 0 2  sin6 e ...  It can be shown by expanding AC  V  (C.12)  and  C(13)  d6 (C.13)  i n power series that  < 0 and A S > 0 f o r 3A < 1 . Thus the hindered  rotator has greater  heat capacity and less entropy than the free rotator under these conditions.  C ) The Out-of-Plane and In-Plane Hindered  Rotator.  The c l a s s i c a l Hamiltonian of the out-of-plane hindered  rotator can be written as  and in-plane  214  h  hr'  \ r KO  =  +  = h  where y' =  2y  c o s 2 (  M  + A c o s 0 + y' sin cp , 2  f r  (C.14)  2  and y i s the i n - p l a n e b a r r i e r .  c l a s s i c a l p a r t i t i o n function  The c o r r e s p o n d i n g  ', a f t e r i n t e g r a t i n g over the momentum,  i s g i v e n by  8Acos e 2  2TTlkT  sm  2 T T  -6y'sin cp 2  d6  Oe  dcp  (C.15)  A g a i n t h i s p a r t i t i o n f u n c t i o n has the p r o p e r t i e s  £imq ' - q hr  (C.16)  fr  o  x,y o r T -> °°  The heat c a p a c i t y and entropy, u s i n g  (C.15) i n ( 3 . 4 ) - ( 3 . 7 ) ,  then  f o l l o w s as 2TT  2TT  -Ry'sin <p  By's i n <p  2  C V h r  CVhr  =  2TT  2  sin cp e  2  dcp  dcp J  ^ o  kT'  -gy»sin cp 2  s i n cp e  ^  +  > -  2TT  "HI  By'sin cb 2  e  2  -dcp!  e  dcp  2  (C.17)  215  2TT  and  -By'sin cf> 2  $  sin  R  +  e  d(J)  -By'sincj> 2  /•27T e  ^ X +  T  -6Xcos 0 2  cos% s i n 9 e  —  TT  0  d<J>  d6  -3Xcos 6 2  sinG e  d9  2TT  -6Xcos 9  -By'sin 4>  2  + R£n  |2TTlkT I  A s ( i ' + 0, i t out-of-plane  £im  y'+0  h  sinO e  (C.18)  2  d8  e  d<|)  2  can be seen t h a t  the heat  c a p a c i t y and e n t r o p y o f  and i n - p l a n e h i n d e r e d r o t a t o r  (C ) »  v  have the  limits  + (C ) .  (C.21)  v hr  hr  the  and M m S, ' hr  --»•  (C22)  S, hr  y'->0  As was the case with the out-of-plane hindered rotator, CC ") ' > ' ^ v hr ^v-'hr - ^SPfr hr' - hr - fr ° § out-ofplane barrier and By' and BX < 1. r  a n dS  S  J  S  f  ra  ivenv a l u e o ft h e  216  APPENDIX D a) Perturbation energies E ^ ^ to E^ "' for even p parity states. 1  7  Note that 0.0 1 = 0.0001. 3  k K  [1] •k . E  E  [2] k  E  [ 3 ] E  h  k  [ 4 ]  k  h  F ^ k  F W k  F ^ k  E  E  E  '{k,(+l,0)} 1  .333333  -.014815  .0 47031  2  .523810  .010150  -.0 47608  3  .506493  .0 17751 2  3  3  .0 53147 5  .0"l3587 -.0 24281 -.0 56788 5  -.0^14109  {k,01,±l)} 1  .428571  2  .480519  -.0 38873 2  ,0 14424 4  .0 56819 6  .0 11902 -.O^ISO' -.0 56685 2  6  {k,( l,±2)} +  1  .142857  -.0 19436  2  .402597  -.0 21399  2  3  .0 36060 4  -.O^OgeS  -.0 58224 e  .0 60527 6  '(k,(+l,±3)} 1  .272727  -.0 13870 2  .0 76421 5  {k,( l,±4)> +  1  .0*90909 -.0 57793 3  .0 57316 5  .0 24412 -.0 56614 5  .0 50404 -.0 13219 6  ,0 57753 7  7  7  7  .0 10507 7  .0 10508 7  APPENDIX D (continued)  [1] b) P e r t u i b a t i o n e n e r g i e s E^'"' "' t o E ^ ^ 1  f o r odd p p a r i t y  [1]  states  [4]  [5]  {k,(-l,0)} 1  .600000  ,(r6857i  .0*60952  ,0 25896  ,0 87280  2  .511111  ,0 32942  .0*59599  .0*26542  ,0 88776  3  .504273  ,0 11299  .0 11653  ,0 79733  2  2  5  5  /  7  /  {k,(-l,±l)} 1  .200000  .0 45714  -.0 12190  -.0*21034  •.(T20571  2  .466667  ,0 13648  .0 11825  .0 21357  .0 22972  3  .487179  .0 88946  -.0 31166  -.0 33726  2  2  3  3  3  5  5  7  {k,(-l,±2)> .333333  ,0 22447  .0^12790  ,0 60682  .435897  ,0 25849  ,0*10486  ,0 53192  Z  J  7  7  {k,(-l,±3)} .11111.1  .0 99763  .0*13264  -.0 16984  .350427  .0 49200  -.0 92311  .0 17498  3  3  5  b  6  {k,(-l,±4)} ,230769  .0 91033 3  .0 44360 5  -.0 11848 7  (k,(-l,±5)} .076923  ,0 36413 3  .0 27883 5  -.0 22441 7  7  

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