UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Conservation laws in recombination kinetic theory Sze, Pui King Ivy 1986

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1986_A6_7 S97.pdf [ 8.51MB ]
Metadata
JSON: 831-1.0059475.json
JSON-LD: 831-1.0059475-ld.json
RDF/XML (Pretty): 831-1.0059475-rdf.xml
RDF/JSON: 831-1.0059475-rdf.json
Turtle: 831-1.0059475-turtle.txt
N-Triples: 831-1.0059475-rdf-ntriples.txt
Original Record: 831-1.0059475-source.json
Full Text
831-1.0059475-fulltext.txt
Citation
831-1.0059475.ris

Full Text

CONSERVATION LAWS IN RECOMBINATION KINETIC THEORY  By  PUT KING IVY SZE B.Sc.  (Hons.), The U n i v e r s i t y of Reading, 1982  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE STUDIES (Department of Chemistry)  We accept t h i s t h e s i s as conforming to the r e q u i r e d  THE  standard  UNIVERSITY OF BRITISH COLUMBIA August 1986  ® P u i King Ivy Sze, 1986  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  Department  V6T  DE-6(3/81)  1Y3  Columbia  I further  purposes  gain  the  requirements  I agree  shall  that  agree  may  representatives.  financial  permission.  T h e U n i v e r s i t y o f British 1956 M a i n M a l l Vancouver, Canada  study.  of  It not  be  that  the  Library  permission  granted  is  by  understood be  for  allowed  an  advanced  shall for  the that  without  head  make  it  extensive of  my  copying  or  my  written  ABSTRACT  The  hydrodynamic equations  monomers and dimers are s t u d i e d .  of change f o r a r e a c t i n g gas mixture of The gas i s c o n s i d e r e d  to be d i l u t e and )  d e s c r i b e d by the k i n e t i c theory of Lowry and S n i d e r ( J . Chem. Phys. 61, 2320 (1974)). representing  From the k i n e t i c equations  f o r the d e n s i t y  the monomer and dimer, the equations  one-molecule o b s e r v a b l e s  are o b t a i n e d .  operators  of change f o r  Since the energy  operator  i n v o l v e s the i n t e r m o l e c u l a r p o t e n t i a l energy, i t i s necessary the energy balance  equation  i n c l u d e s molecule-molecule  to d e r i v e  from the von Neumann e q u a t i o n , s i n c e t h i s correlations.  As w e l l , the k i n e t i c  theory  formulated  by Lowry and S n i d e r i s r e w r i t t e n so that rearrangement  collisions  are emphasized.  A collisional  sum r u l e i s d e r i v e d i n v o l v i n g the commutation  p r o p e r t i e s of channel  p r o j e c t o r s and t h e i r r e s p e c t i v e p o t e n t i a l s .  known p r o p e r t y of the o p t i c a l theorem i s t h a t i t i d e n t i f i e s  the r e a c t i v e  l o s s terms as p a r t of the n o n - r e a c t i v e t r a n s i t i o n s u p e r o p e r a t o r s . sum  r u l e i s a p p l i e d to r e w r i t e the n o n - r e a c t i v e  so as t o d i s p l a y the r e a c t i v e l o s s terms.  superoperators  of mass, l i n e a r  momentum,  A form o f the o p t i c a l theorem i n which  k i n e t i c energy o f f - d i a g o n a l i t y i s allowed optical  The  This aids i n e s t a b l i s h i n g  c o n s e r v a t i o n laws f o r the p h y s i c a l o b s e r v a b l e s a n g u l a r momentum and energy.  transition  A  f o r i s also derived.  Both the  theorem and the sum r u l e a r e based on the s t r o n g o r t h o g o n a l i t y  h y p o t h e s i s , which p l a y s a fundamental r o l e i n the Lowry-Snider  theory.  - iii -  On l o c a l i s i n g the physical attributes at the centres of mass of the molecules, the contributions to the equations of change from c o l l i s i o n a l transfer (due to the forces and torques between the c o l l i s i o n partners) and from the transfer of the physical attributes from the reactants to the products are i d e n t i f i e d .  The transformation of dimer internal  degrees of freedom into monomer translational degrees of freedom or vice versa when a dimer Is dissociated or formed i s found to contribute to the equations of change by virtue of the d i f f e r i n g l o c a l i t y of the c o l l i s i o n partners.  The decomposition of the kinetic energy operator into i t s  components for r a d i a l and rotational motions allows the k i n e t i c energy flux contributions associated with the pressure tensor and the molecular angular momentum flux to be i d e n t i f i e d .  - iv -  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  ..'  iv  ACKNOWLEDGEMENT  vi  1.  INTRODUCTION  1  2.  A KINETIC THEORY FOR A MONOMER-DIMER REACTING GAS  9  A. B. C. D.  3.  4.  5.  9 13 17 26  THE FORM OF THE EQUATIONS OF CHANGE  34  A. B.  34 39  D e s c r i p t i o n of M o l e c u l a r Observables Formal E q u a t i o n s of Change  A SUM RULE  49  A. B. C.  49 56 66  Channel C o u p l i n g and the G e n e r a l i s e d O p t i c a l Theorem Gain and Loss of Monomer Observables G a i n and Loss of Dimer Observables  HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE NON-REACTIVE COLLISIONS A. B. C. D. E.  6.  D e s c r i p t i o n of S t a t e s Using the D e n s i t y Operator Formalism............... P r o j e c t o r s and Channel S t a t e s D e s c r i p t i o n of C o l l i s i o n s Time E v o l u t i o n of the System  OBSERVABLES I : 69  C o l l i s i o n s I n v o l v i n g 2 Monomers C o l l i s i o n s I n v o l v i n g a Monomer and a Dimer C o l l i s i o n s I n v o l v i n g 3 Monomers C o l l i s i o n s I n v o l v i n g 2 Dimers C o l l i s i o n s I n v o l v i n g 2 Monomers and 1 Dimer  HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE REACTIVE COLLISIONS  69 76 79 82 85  OBSERVABLES I I : 92  v  A. B. C. D.  C o l l i s i o n s I n v o l v i n g a Monomer and a Dimer C o l l i s i o n s I n v o l v i n g 3 Monomers Dimer-Dimer C o l l i s i o n s C o l l i s o n s I n v o l v i n g 2 Monomers and 1 Dimer  94 114 121 136  - v -  Page 7.  THE ENERGY BALANCE EQUATION A.  B. C.  8.  158  The P r o d u c t i o n and Loss of Energy A s s o c i a t e d w i t h the Monomers and w i t h the Dimers  158  A.l  The Energy Operators  158  A.2  Energy Balance f o r the Monomer  161  A.3  Energy Balance f o r the Dimer  168  Non-Reactive C o l l i s i o n s Reactive C o l l i s i o n s  SUMMARY  BIBLIOGRAPHY  172 190  240 242  - vi -  ACKNOWLEDGEMENT  I would l i k e t o express my h e a r t f e l t g r a t i t u d e to Dr Robert my s u p e r v i s o r , who has not only been g i v i n g me i n t e l l i g e n t guidance,  Snider,  a d v i c e and  but a l s o has been most courteous and p a t i e n t w i t h me when I  show s i g n s of s l a c k n e s s on my work.  H i s d e v o t i o n to h i s r e s e a r c h and  f a s c i n a t i o n w i t h i t a r e i n s p i r i n g and deserve my utmost r e s p e c t .  My thanks  are a l s o due t o Miss A n i s s a Yeung, who p r o o f - r e a d my  t h e s i s , and t o Mr S t a n l e y Luck f o r h i s i n t e r e s t  i n my work as w e l l as  g i v i n g me encouragement and a p p r e c i a t i o n f o r what I have  The  company of Miss L u c i l l e Fung, Messrs Kachung Hui and Marcus  K a r o l e w s k i have been most h e a r t e n i n g . enjoy the moments I share w i t h them. has  accomplished.  I a p p r e c i a t e t h e i r c h a r a c t e r s and I t i s my f r i e n d s h i p w i t h them t h a t  p r o v i d e d me a d e l i g h t f u l environment  i n which I can work on.  Moreover, I owe my perseverance with my work through  the d a r k e s t  hours  d u r i n g my r e s e a r c h to Miss Susanna Tse, f o r every time when I l o s e h e a r t , I will  remember her d e d i c a t e d n e s s and s i n g l e - m i n d e d n e s s .  Jeeva Jonahs, Her  who typed up my t h e s i s s u r e l y has my warmest  d e v o t i o n and m e t i c u l o u s n e s s  thanks.  i n s p i r e not o n l y me, b u t , as I am p r e t t y  s u r e , other people who work with h e r t o o .  Thanks a l s o go t o those who were most h e l p f u l and u n d e r s t a n d i n g as I was wandering a i m l e s s l y d u r i n g my e a r l y s c i e n t i f i c Drs M a r j o r i e Jeacock s h a l l never  endeavour.  and David R i c e have my r e s p e c t and g r a t i t u d e :  f o r g e t t h e i r c o r d i a l i t y and t h o u g h t f u l n e s s .  I  - 1 -  CHAPTER 1 INTRODUCTION  Fundamental to the laws of physics are the conservation of mass, l i n e a r momentum, angular momentum and energy.  In f l u i d dynamics, these  are expressed by the nature of the equations of change for the densities 1  of these physical a t t r i b u t e s .  While i t i s well established that these  hydrodynamic equations are consistent with (derivable from) s t a t i s t i c a l 2  mechanics  and from the k i n e t i c theory of gases for bimolecular reactive 1  and non-reactive c o l l i s i o n s , how the recombination and breakup reactions' affect the flux expressions has not been studied.  It i s the object of  this thesis not only to obtain conservation equations but also to investigate the effect of chemical reactions on the various fluxes. S p e c i f i c a l l y , the equations of change are to be derived from the k i n e t i c theory formulated by Lowry and Snider  for a dimer-monomer reacting gas.  In their c l a s s i c paper, Irving and Kirkwood  derived the hydrodynamic  equations for mass, linear momentum and energy d i r e c t l y from the c l a s s i c a l L i o u v i l l e equation for the entire system.  The fluxes are  expressed i n terms of expectations of molecular variables.  Dirac delta  functions are introduced for the purpose of l o c a l i s a t i o n of the molecules which are considered to be point masses interacting v i a pair potentials. The quantum mechanical equivalent can be found i n a paper by Snider and Lewchuk  on the i r r e v e r s i b l e thermodynamics of a system of molecules  possessing internal angular momentum i n which the equation of change for angular momentum i s also considered.  - 2 -  These f o r m u l a t i o n s d e a l w i t h the time e v o l u t i o n o f a system composed of a s i n g l e s p e c i e s .  For a c h e m i c a l l y r e a c t i v e system i t i s more  a p p r o p r i a t e to express  the e q u a t i o n s o f change f o r each s p e c i e s and to  e s t a b l i s h the c o n s e r v a t i o n laws f o r the t o t a l mass, momentum, a n g u l a r momentum and energy procedure,  f o r the f l u i d mixture  i t i s necessary  the chemical  as a whole.  To c a r r y out t h i s  to d e r i v e the k i n e t i c equations  s p e c i e s present i n the system.  f o r each o f  In p a r t i c u l a r , when no  r e a c t i o n s take p l a c e , and o n l y one s p e c i e s i s p r e s e n t , the e q u a t i o n s o f change reduce  Kinetic hierarchy assumption  5  to those o f S n i d e r and Lewchuk.  e q u a t i o n s a r e g e n e r a l l y obtained by t r u n c a t i n g the BBGKY a t the two- or t h r e e - p a r t i c l e l e v e l ,  t o g e t h e r with the  t h a t the two- and/or t h r e e - p a r t i c l e d e n s i t y o p e r a t o r s o r  distribution  f u n c t i o n s are f u n c t i o n a l s o f the o n e - p a r t i c l e d e n s i t y  operator or d i s t r i b u t i o n function. r e p r e s e n t a d e n s i t y expansion  In t h i s way the k i n e t i c  o f the L i o u v i l l e e q u a t i o n .  equations  For a  one-component d i l u t e g a s , assuming the v a l i d i t y o f Boltzmann and  t h a t the H a m i l t o n i a n  Hamiltonians  statistics,  takes the form o f a sum o f o n e - p a r t i c l e  and t h e i r p a i r p o t e n t i a l s , we have, f o r the f i r s t  BBGKY e q u a t i o n  6  i f i 3t A-p (l) (1)  quantum  P  ( 1 )  (l)]  -  = Tr[V(l,2),  p (12)J (2)  .  2  (1.1)  Here p ^ \ l ) and p ^ \ l 2 ) are s i n g l e t particle  and d o u b l e t d e n s i t y o p e r a t o r s f o r  1 and f o r the p a i r o f p a r t i c l e s l a b e l l e d  1 and 2.  ^5^(1)  - 3 -  i s the H a m i l t o n i a n  for particle  between p a r t i c l e s 1 and 2.  1, and V ( l , 2 ) i s the p o t e n t i a l  operator  I f the h i e r a r c h y i s t r u n c a t e d at the  t w o - p a r t i c l e l e v e l , by s e t t i n g  P  ( 2 )  ( l , 2 ) = 8(1,2) P  ( 1 )  (l) P  0 )  ( 2 ) ^(1,2)  (1.2)  w i t h ft, the M p l l e r o p e r a t o r a s s o c i a t e d w i t h b i n a r y c o l l i s i o n s between particles  1 and 2 and  being i t s h e r m i t i a n a d j o i n t , then t h i s i s  c o n s i s t e n t w i t h the p i c t u r e that the molecules  a r e independent b e f o r e the  collision.  On s u b s t i t u t i n g e q u a t i o n (1.2) i n t o e q u a t i o n ( 1 . 1 ) , a  generalised  Boltzmann e q u a t i o n i s o b t a i n e d , namely  i*kp  ( 1 )  <i> = L ^ d ) , P  ( 1 )  5  (DL  + T r [ V ( l , 2 ) , 0(1,2) p 2  ( 1 )  (l)  p  ( 1 )  (2)  T  fi (l,2)]_.  (1.3)  Inherently, t h i s equation describes nonlocal c o l l i s i o n s , that i s , the incoming p a r t i c l e s a r e not a t the same p o s i t i o n as the p a r t i c l e 1) t h a t i s being observed.  (labelled  T h i s n o n l o c a l i t y and i t s r e l a t i o n to the 7  corresponding  localised  form  (known as the Waldmann-Snider  equation)  p  has been d i s c u s s e d by Thomas and Snider .  Because o f the n o n l o c a l i t y o f  the c o l l i s i o n o p e r a t o r i n e q u a t i o n ( 1 . 3 ) , the macroscopic f l u x e s ( f o r example, the p r e s s u r e tensor) g e n e r a l l y have c o l l i s i o n a l c o n t r i b u t i o n s . T h i s i s to be c o n t r a s t e d w i t h the usual Boltzmann e q u a t i o n , i n which  - 4 -  c o l l i s i o n s are l o c a l and the fluxes of momentum and energy are associated solely with the k i n e t i c motion of the individual molecules.  If two p a r t i c l e s are bound, then i t i s physically obvious that the fluxes ( i n particular of the energy) w i l l depend on the i n t e r a c t i o n of the p a r t i c l e s .  Besides, i n the c o l l i s i o n of the bound pair with other  molecules, there i s no separation of the bound pair before c o l l i s i o n so the closure r e l a t i o n (1.2) cannot be v a l i d . 3  9  The proper closure r e l a t i o n s 10  have been considered by various workers > »  •  This thesis w i l l  a  follow the work of Lowry and Snider , which i s reviewed  i n Chapter 2.  The chemical description of bound pairs requires one to recognise these dimers as d i s t i n c t species.  As such, the d i l u t e gas k i n e t i c theory  of dimers depends on binary c o l l i s i o n s of dimers, which are equivalent to a particular type of f o u r - p a r t i c l e c o l l i s i o n .  If only dimers are  present, the resulting Boltzmann equation has the same structure, equation (1.3) as for the case of free p a r t i c l e s but now the single p a r t i c l e density operator p ( D i s replaced by the bound "pair" density operator p (12) with analogous changes f o r the free motion Hamiltonian D  and c o l l i s i o n operators.  The description of the free (dimer) motion i n  terms of equivalent atomic (monomer) properties has been given by Olmsted and  11  Snider .  For the more general situation i n which recombination  and decay of  dimers can occur, the fluxes and equations of change involve properties of both monomers and dimers.  While the chemical description of these  - 5 -  properties i s n a t u r a l l y given and  i n terms of the  i n d i v i d u a l f r e e monomers  d i m e r s , the e q u a t i o n s of change need a u n i f y i n g d e s c r i p t i o n i n  to prove c o n s e r v a t i o n terms of the has  s e t the  out  and  principles.  The  unifying description i s clearly in  atomic (monomer) p i c t u r e and stage f o r how  t h i s can be  order  the work by Olmsted and  accomplished.  v e r i f i c a t i o n of the c o n s e r v a t i o n  I t i s the  e q u a t i o n s that  Snider  working  form the work o f  this thesis.  As mentioned p r e v i o u s l y , s e v e r a l groups of workers have endeavoured  q to d e s c r i b e studied only  a r e a c t i n g diatomic-monatomic system.  a system c l o s e  slightly  truncating  to e q u i l i b r i u m .  from e q u i l i b r i u m .  the  BBGKY h i e r a r c h y  Then the Wigner f u n c t i o n s  at the  on  transport  the  fails  the bound s t a t e s  The  i n p a r t i c u l a r , the  o f dimers.  course c o n s i s t e n t with t h e i r l i m i t i n g  also studied  a k i n e t i c e q u a t i o n f o r the  as  treatment of bound s t a t e s as a  Moreover, t h e i r  incomplete as dimer-dimer c o l l i s i o n s are not  10  are  f r e e s t a t e s whereas the chemists view i s  to emphasize t h i s chemical p i c t u r e and  s i g n i f i c a n t concentration  McLennan  a  c o n s t i t u e n t atoms i n a dimer molecule i s viewed  a c h a r a c t e r i s t i c of the dimer. perturbation  and  c o e f f i c i e n t s were  to t r e a t a m o l e c u l e as a d i s t i n c t chemical s p e c i e s and boundedness of the two  differ  three-atom l e v e l , they obtained  However, the d e s c r i p t i o n i s atomic and  t r e a t e d as a p e r t u r b a t i o n  Curtiss  Using the m o l e c u l a r chaos assumption  k i n e t i c e q u a t i o n from which v a r i o u s calculated.  Olmsted and  a  treatment i s  considered,  low dimer  disallows  but  which i s of  concentration.  a monomer-dimer r e a c t i v e m i x t u r e and s i n g l e t density operator,  again  using  obtained the  -  molecular  chaos assumption  three-atom  level.  6 -  and t r u n c a t i n g the BBGKY h i e r a r c h y a t the  P a r t o f the d o u b l e t d e n s i t y o p e r a t o r i s to r e p r e s e n t a  p a i r o f f r e e atoms and i s thus equal operators. channels  of s i n g l e t density  The t r i p l e t d e n s i t y o p e r a t o r i s w r i t t e n as a sum over  o f products o f d e n s i t y o p e r a t o r s f o r the m o l e c u l e s  the r e s p e c t i v e channels. expansion  to a product  present i n  As i t i s known t h a t a d i v e r g e n c e o f the d e n s i t y  o f the BBGKY h i e r a r c h y o c c u r s a t the l e v e l of four-atom  c o l l i s i o n s , McLennan t r u n c a t e s h i s expansion  at the three-atom  level.  In  n e g l e c t i n g dimer-dimer c o l l i s i o n s w h i l e three-monomer c o l l i s i o n s a r e i n c l u d e d , t h e d e s c r i p t i o n can o n l y be v a l i d concentrations.  f o r low dimer  As w e l l , McLennan's expansion  i s i n terms o f the d e n s i t y  of atoms and the m o l e c u l a r nature o f the r e a c t i v e system i s not emphasized.  3 A more formal approach was g i v e n by Lowry and Snider . being  s t u d i e d i s an i d e a l gas i n which m o l e c u l e s  collisions. isolated  The system  i n t e r a c t o n l y by  The atoms t h a t p a r t i c i p a t e i n a p a r t i c u l a r c o l l i s i o n  subsystem, so t h a t the von Neumann e q u a t i o n d e s c r i b i n g t h i s  colliding  s e t o f atoms can be i n d e p e n d e n t l y s o l v e d .  equations  f o r the d e n s i t y o p e r a t o r s r e p r e s e n t i n g the dimer and the  monomer a r e o b t a i n e d . free f l i g h t  and v a r i o u s c o l l i s i o n  processes.  a n g u l a r momentum a r e obtained from  Snider.  Coupled  kinetic  The e q u a t i o n s c o n t a i n terms which correspond to  e q u a t i o n s o f change f o r the m o l e c u l a r and  form an  a t t r i b u t e s o f mass, l i n e a r momentum the k i n e t i c e q u a t i o n s o f Lowry and  C o n s e r v a t i o n laws f o r each i n d i v i d u a l  or n o t , a r e e s t a b l i s h e d .  In t h i s t h e s i s , t h e  collision  type, r e a c t i v e  The c o n t r i b u t i o n s to momentum and angular  - 7 -  momentum f l u x e s from c o l l i s l o n a l  t r a n s f e r are a l s o i d e n t i f i e d .  As f o r  the e q u a t i o n o f energy b a l a n c e , a more e l a b o r a t e f o r m u l a t i o n i s r e q u i r e d s i n c e energy I s not a s i n g l e - m o l e c u l e o b s e r v a b l e and i t s time e v o l u t i o n cannot be o b t a i n e d from the k i n e t i c e q u a t i o n s f o r the i n d i v i d u a l however, the u n d e r l y i n g p r i n c i p l e o f Lowry and and m u t u a l l y e x c l u s i v e c o l l i s i o n s i s used.  species,  S n i d e r ' s work o f i s o l a t e d  Within t h i s  formulation,  c o n s e r v a t i o n o f energy i s v e r i f i e d .  The crux o f e s t a b l i s h i n g  these c o n s e r v a t i o n laws f o r each  type i s to r e c o g n i s e that the v a r i o u s l a b e l l e d  channels (how  collision the atoms  arrange themselves as d i s t i n c t groups of m o l e c u l e s ) are coupled by the c o l l i s i o n operator, i n p a r t i c u l a r  through the channel p o t e n t i a l .  completeness r e l a t i o n s h i p (sum  r u l e ) expressed i n terms of the  commutators of the arrangement  channels and t h e i r r e s p e c t i v e  is  A  potentials  derived.  Using  t h i s sum  r u l e , the c o n t r i b u t i o n s to the hydrodynamic  of change from c o l l i s l o n a l  equations  t r a n s f e r ( t h e t r a n s f e r o f the p h y s i c a l  a t t r i b u t e s d u r i n g a c o l l i s i o n from t h i s molecule to o t h e r m o l e c u l e s participating physical  i n the same c o l l i s i o n event) and the t r a n s f e r o f the  a t t r i b u t e s from the r e a c t a n t s to the p r o d u c t s due  rearrangement  are i d e n t i f i e d .  to atom  On u s i n g the u n i f y i n g p i c t u r e o f atomic  1 1  m o t i o n , the t r a n s f o r m a t i o n o f the r e l a t i v e motion between two monomers i n t o dimer i n t e r n a l motion when r e c o m b i n a t i o n takes p l a c e i s i d e n t i f i e d as c o n t r i b u t i n g reactions result  to the e q u a t i o n s o f change.  S i m i l a r l y , decomposition  i n a l o s s i n dimer i n t e r n a l degrees of freedom and the  - 8 -  creation  of monomer t r a n s l a t i o n a l  reactions,  we  see  a c o u p l i n g of  the  m o l e c u l e s b e f o r e and  The  k i n e t i c energy due  kinetic and  the  freedom.  a p a r t due  after c o l l i s i o n  destruction  behaviour of the  to the  T h e r e f o r e , not  dimer-monomer r e a c t i n g  of dimer i n t e r n a l  states  the  derived  channel s t a t e s . the  approximation. o b t a i n the  gas  are  affects  a p a r t due  motion.  to  the  rotation  only conservation derived the  but  the  creation  macroscopic  e q u a t i o n s of change i s t h a t  o r t h o g o n a l i t y approximation p r e c l u d e s any  r i g o r o u s and  dimer i n t e r n a l  system.  A l i m i t a t i o n of  different  exchange  to these motions i s i d e n t i f i e d by decomposing  to r a d i a l motion.  f o r the  For  r e l a t i v e t r a n s l a t i o n a l motion between  energy o p e r a t o r f o r k i n e t i c motion i n t o  relations and  degrees of  The  coherences between  kinetic description  Lowry-Snider theory i s o b t a i n e d as Thus, i t may  be  that  conservation relations  coherence e f f e c t s are  allowed.  the  the  by  strong  the  Evans e t a l  the  first  1  1  is  order  Evans t h e o r y should be  used  to  f o r the more g e n e r a l s i t u a t i o n where  - 9 -  CHAPTER 2 A KINETIC THEORY FOR A MONOMER-DIMER REACTING GAS  The s t a r t i n g  p o i n t of the present i n v e s t i g a t i o n i s the k i n e t i c  3 e q u a t i o n s o b t a i n e d by Lowry and S n i d e r  f o r pf and p^,, the d e n s i t y  o p e r a t o r s r e p r e s e n t i n g the monomeric and d i m e r i c m o l e c u l e s w i t h a s i n g l e atomic s p e c i e s .  associated  Below i s a review of the Lowry-Snider  formulation.  A.  Description of States Using the Density Operator Formalism  The system  i s a dilute  i d e a l gas w i t h no i n t e r a c t i o n between t h e  m o l e c u l e s except v i a c o l l i s i o n s .  T h i s i s r e p r e s e n t e d by the d e n s i t y  N  operator  p ^ ^ , N b e i n g the number of atoms of the system.  isolated  and independent  operator  p ( ) i s used  n  taking a p a r t i a l statistics  collision  F o r an  i n v o l v i n g n atoms, a reduced  to d e s c r i b e the event;  density  n  p ( ) i s o b t a i n e d by N  t r a c e of the N-atora d e n s i t y o p e r a t o r p ( ) . Boltzmann  i s assumed  throughout.  As the gas under c o n s i d e r a t i o n i s a d i l u t e and i d e a l one, p(N)  c  be thought  o f as b e i n g f a c t o r i s e d  i n d i v i d u a l molecules.  ( N )  Taking i n t o account  1  M  = g(N, M, D ) " M~ D~  D  n  i n t o a product of d e n s i t y o p e r a t o r s f o r  a s s o c i a t i o n s o f atoms, t h i s becomes  p  a  E (a,3)  the p o s s i b i l i t y of d i f f e r e n t  the sum of p r o d u c t s  n p (i) n i*a (jk)t(3 1  p (jk).  (2.1.1)  - 10 -  Here M and  D a r e numbers o f monomers and dimers r e s p e c t i v e l y , w i t h  N - 2D + M.  (2.1.2)  g i s the number of arrangements  g(N, M,  D) = M!  and  P f ( i ) and  ^ — D! 2  f o r M monomers and D dimers  from N atoms,  = (*) ^ j D! 2  (2.1.3)  P ( j k ) are the reduced d e n s i t y o p e r a t o r s f o r the D  monomer i and dimer ( j k ) r e s p e c t i v e l y .  N o r m a l i s a t i o n s are such that the t r a c e over a l l atoms o f e q u a l s u n i t y and  D  (N)  the t r a c e s over the reduced d e n s i t y o p e r a t o r s f o r the  m o l e c u l e s g i v e the number of m o l e c u l e s  (N) Tr 1...N  p  w  = 1  (2.1.4)  Tr p ( i ) = M i  (2.1.5)  r  Tr P , ( j k ) = D. •  i  The  p  ( n )  (2.1.6)  D  n  reduced d e n s i t y o p e r a t o r p ( ) f o r atoms 1 t o n i s d e f i n e d  ( l . . . n ) B (N)  T  r  n+l...N  so  p  (N)  (  by  2  >  1  .  7  )  -  Tr p 1.. .n  Any  (  n  )  11 -  ( l . . . n ) - (*).  (2.1.8)  g i v e n atom can be e i t h e r a monomer o r p a r t o f a dimer, so a  s i n g l e t density operator p(D  w i l l d e s c r i b e the combination  of a  monomer and part o f a dimer molecule whose p a r t n e r i s a t r a c e d - o v e r  atom  ( g h o s t ) , namely  P  The  ( 1 )  (D  =  P ( D + p£° f  quantity p j ^ ( l )  (1).  (2.1.9)  i s the 1-atom bound-state d e n s i t y o p e r a t o r , d e f i n e d  by  1 }  p £ ( l ) - 2 Tr P ( 1 2 )  (2.1.10)  b  w i t h norm  Tr P ^ U )  = 2D.  (2.1.11)  For the p a i r d e n s i t y o p e r a t o r  p ( 2 ) Q 2 ) , on using  the assumption  t h a t a s m a l l i n t e g e r i s n e g l i g i b l y small when compared with N, M and D, one  obtains  p  ( 2 )  (12) = { p  (  1  )  (l)  P  ( 1 )  ( 2 ) + P (12). b  A p p l i c a t i o n o f the l a r g e N, M, D approximation results  (2.1.12)  a l s o l e a d s to the  - 12 -  P  ( 3 )  (123) -  I P  ( 1 )  + P  U )  P (1234) - ^ P (4)  (  (l)  (2)  1  >  P  (2)  ( 1 ) P  P (13) + p  (3)  ( 1 )  b  ( 1 )  P  + j  ( 1 )  P  ( 0  p  12  p  ,  )  ( 1 )  (D  P  ( 1 )  (2)  p  ( 1 )  ( 1 ) [ P  (1)  P w  (23)  +  ( 3 ) P (12)]  (2.1.13)  fe  P ><3) p ><4>  « >  p  (l)  ( 1 )  ( 1 )  (  I  +  (1  (1  ( 1 )  ( 2 ) p (34) + p b  (4)  p (23) + p  ( 1 )  (4)  p (13) + p  ( 1 )  b  b  p  ( 1 )  (2) p  ( 1 )  (3)  ( 1 )  (l)  p  ( 3 ) p (24) + b  ( 3 ) p (14) + b  (4)  [ P ( 1 2 ) p ( 3 4 ) + p ( 1 3 ) P (24) + p ( 1 4 ) b  b  b  b  b  p (12) b  p (23)]. fe  (2.1.14)  n  The g e n e r a l form f o r p ( ) i s a sum over a l l p o s s i b l e o f pb and p ( D . the  The n u m e r i c a l c o e f f i c i e n t s f o r each term depends on  number o f fragments f :  N  ( ) g(N-n, M+n-2f, D+f-n) n  for  '  g(N, M, D) M  M  (n)  =  z (a,B)  s m a l l n.  2 ^ f n n!  w  2  i*>a  n-f (2.1.15)  n!  D°  n  l a r g e M, D and s m a l l n.  >  for  factorisations  So the g e n e r a l form f o r p ( ) i s  ( 1 ) P  (i)  n  P  (jk^B  ( D  i  k  )  (2.1.16)  - 13 -  B.  P r o j e c t o r s and Channel S t a t e s  T h i s s e c t i o n i s devoted  to the c l a s s i f i c a t i o n o f f r e e and bound  states.  C o n s i d e r 2 atoms 1 and 2 i s o l a t e d Hamiltonian  a l l other atoms, the  assumes the form  (2) tf. "Kj  w h e r e i s potential  from  + V  (2.2.1)  1 2  the 1-atom H a m i l t o n i a n  i  f o r atom i , and V j 2  energy o p e r a t o r between 1 and 2.  can be s p l i t  i n t o a centre-of-mass  s  the 2  The p a i r H a m i l t o n i a n  )  p a r t 3*CM and a r e l a t i v e motion p a r t  1<rel  ^  and  (  2  )  =*CM  +  K  2  rel  i f 1 and 2 a r e bound t o g e t h e r , ^  operator with eigenkets  K  rel  |  b  ±  >  =  E  bi |  M  and  r e  l will  Jb-^ > b e l o n g i n g  be the i n t e r n a l  2  D  ( 2  2  * *  3 )  the i t h bound s t a t e i s d e f i n e d by  b i > <bi| J  the p r o j e c t i o n o p e r a t o r onto  energy  to n e g a t i v e e i g e n v a l u e s e £  >*  The p r o j e c t i o n o p e r a t o r onto  P.. = DI  2  ( - - >  (2.2.4)  the f u l l  space  o f the bound s t a t e s i s  -  the sum  P  14  over a l l P b i ' s  = J P .  b  (2.2.5)  M  With the d e f i n i t i o n o f P^, one may i n t o a bound  K  -  <  2  • P iC  "  )  *< > 2  Hamiltonian  part  - X.  ( 2 )  b  and a f r e e  s p l i t up the p a i r  P  < 2 )  (2.2.6)  b  part  (i -  ,X< >  f  2  P ) D  5<  1  +J<  + Vj  2  (2.2.7)  2  SO  (2)  X  i-  V  s  =  K  th  12  =  V  e  (2)f  +  R  (2)  b >  (  modified p o t e n t i a l  12  ~^CM  P  £  between f r e e  P  ( 2  For the N-atom system, the H a m i l t o n i a n i s assumed  (K^  1fc ' a  N  = EK i  >  2  >  g  )  molecules.  b ~ I bi bi  1 - p a r t i c l e H a m i l t o n i a n s and p a i r  2  2  ' '  9 )  to be a sum of  potentials  N  +  Z Kj  V  .; 3  (2.2.10)  - 15 -  n-body p o t e n t i a l s w i t h n ^ 3 a r e being i g n o r e d . partition  i n t o Harailtonians f o r i n d i v i d u a l m o l e c u l e s and  intermolecular potential channel-dependent not  One would l i k e to  terms.  However, such a s p l i t t i n g i s  w h i l e the d e f i n i t i o n of independent  channel s t a t e s i s  unique.  The boundedness o f any p a i r of atoms ( i j ) B  projection Pbij• free.  t  t h e r e i s d i f f i c u l t y i n d e f i n i n g when an atom i s  The obvious c h o i c e i s the o p e r a t o r  N  (N) P  u  i s c l e a r l y d e f i n e d by the  E  fl  1  -  P  ^  2  bli  but t h i s i s not idempotent chemically i n t u i t i v e  2  t ' '  s i n c e the P b l i '  8  a  s o l u t i o n to the problem  r  e  n  o  t  a  H  orthogonal.  1 1  )  A  i s to assume t h a t no atom i s  s i m u l t a n e o u s l y bound to two o t h e r atoms, namely  P  bij  P  6  P  2  bik = j k b i j '  t h i s assumption  2  12  < ' - >  being c o n s i s t e n t w i t h the n o t i o n that no t r i m e r s  From the a p p r o x i m a t i o n (2.2.12), idempotency  exist.  immediately f o l l o w s .  It i s  (N)  a l s o assumed t h a t P ^  P  f i  }  * i  =  K  T h i s assumption for  and  commute:  i ff-  (2.2.13)  F  i s c h e m i c a l l y r e a s o n a b l e s i n c e -Kj i s the H a m i l t o n i a n  monomer i and P ^ ^  Is the p r o j e c t i o n  i n t o the space f o r t h i s  (free)  -  molecule.  16 -  The assumption as formulated  r e q u i r e s t h a t the molecules  by e q u a t i o n  (2.2.12)  under c o n s i d e r a t i o n are not  tacitly  undergoing  (N) collisions.  In other words,  d e f i n e s a molecule  free  from  interactions.  The f o r e g o i n g d i s c u s s i o n i s on p r o j e c t i o n o p e r a t o r s a c t i n g on k e t s which o n l y r e p r e s e n t pure s t a t e s . represented for  But the n o n - e q u i l i b r i u m gas i s  by the d e n s i t y o p e r a t o r  the gas to be i n any p a r t i c u l a r  projection  A  P  -= b i j  A P  o p e r a t o r space i s by  onto the bound i j s t a t e i s d e f i n e d by  Atom i i s assumed  l j ; t i  f o r the f r e e - i  s t a t e i n the N-atom  analogy  (2.2.H)  to be e i t h e r  F bij  2  ' '  operator.  A > P < » AP<f.  +  t requires  ( 2  The p r o j e c t i o n s u p e r o p e r a t o r  fi  at any i n s t a n t  bij  where A i s an a r b i t r a r y  f | f  channel  xo i d e n t i f y the p r o b a b i l i t y  superoperators.  The p r o j e c t i o n s u p e r o p e r a t o r  ^bij  p(N).  = 5  (  N  )  f r e e o r bound:  (2.2.16)  -  where  i  s  the i d e n t i t y  The  assumptions  P  AP  f i  b i  17 -  superoperator  i n the N-atom space.  . = 0  (2.2.17)  and  " b i j ^ b i k "  are i m p e r a t i v e  0  '  ( 2  ^  2  ' -  1 8 )  f o r consistency i n presentation.  (N) These p r o j e c t o r s < P j j  a  n  b  respectively,  i n particular  ^ ^ f±  identify  the dimer ( i j ) and monomer i  the c o r r e s p o n d i n g  density operators are given  by P (12) = ( ) b  2  p (l) = N f  C.  Tr < f 3.. .N  Tr f < 2...N  N  )  p  M 2  p  ( N )  ( N )  (2.2.19)  .  (2.2.20)  Description of Collisions  The  system i s c o n s i d e r e d  collisions. individual  The i n i t i a l  to be an i d e a l gas i n t e r a c t i n g  and f i n a l  o n l y through  s t a t e s are assumed to be products o f  monomer and dimer d e n s i t y o p e r a t o r s .  The p h y s i c a l r e a l i s a t i o n  - 18 -  of  such a system  short-range  The  i s a d i l u t e gas o f molecules  o n l y through a  potential.  collisional  types one envisage;  o f monomers and dimers a r e n o n - r e a c t i v collisions,  interacting  f o r a r e a c t i v e system collisions,  consisting  rearrangement  as w e l l as decomposition a d recombination c o l l i s i o n s i n  which e i t h e r a monomer o r a dimer a c t s as the t h i r d  body.  These a r e a l l  encompassed by ( 2 . 3 . 1 ) :  M + M + M +M  (2.3.1a)  M + D -*•. M + D  (2.3.1b)  D + D + D +D  (2.3.1c)  M+M+M  (2.3.Id)  + M+ D  (2.3.Ie)  M + M + D + D + D.  Here M and D denote monomer and  Collisions and  i n v o l v i n g more than t h r e e m o l e c u l e s  a r e thus i g n o r e d .  are r a r e o c c u r r e n c e s  3-fragraent to 3-fragment c o l l i s i o n s  are a l s o  c o n s i d e r e d r a r e events, however as Lowry and Snider p o i n t o u t , p a r t o f the 3-fragraent-to-3-fragment 3-fragment-to-2-fragment collision  non-reactive t r a n s i t i o n operator consists of  l o s s terms.  I t i s thus proposed  that the  types  M + M + M + M + M + M  (2.3.If)  M + M + D + M + M + D  (2.3.lg)  - 19  s h o u l d be i n c l u d e d . equivalent  The  collision  -  types  ( 2 . 3 . I f ) and  ( 2 . 3 . l g ) are  to the l o s s t r a n s i t i o n s i n the paper by Lowry and  Snider.  t e r m o l e c u l a r to t e r m o l e c u l a r t r a n s i t i o n s are r a r e , n o n e t h e l e s s not c l o s e d and and  t h e r e f o r e should be r e t a i n e d .  ( 2 . 3 . l g ) a l s o has  The  allowance  the advantage t h a t a t e r m o l e c u l a r i n i t i a l  t r e a t e d i n the same manner as 3-fragment to or from  are  for (2.3.If)  does not p r e c l u d e a t e r m o l e c u l a r f i n a l s t a t e nor v i c e v e r s a and be  they  The  state so should  2-fragment  collisions.  The  collision  types l i s t e d  i n equation  (2.3.1) can be c l a s s i f i e d  the t r i p l e  (n, f * , f ) , where n i s the number of atoms i n v o l v e d i n the  collision,  f  the number of fragments b e f o r e the c o l l i s i o n ,  number of fragments In the f i n a l s t a t e . (n,f) final  The  initial  c o n s i s t s of d = n - f* dimers and m = 2 f channel  ( n , f ) has n - f dimers and  1  and f the  channel g i v e n by  - n monomers, and  2f - n monomers.  3-fragment to 3-fragment p r o c e s s e s , then the c o l l i s i o n the ones  by  the  Including  types we  study  are  satisfying  2 ^ n _< 4, n <_ f + f» < 6,  and  f _< 3, f ' _< 3.  For each f r a g m e n t a t i o n of  partitioning  labelled format  (n,f),  there may  be more than one  the i n d i v i d u a l atoms over the set of molecules  channels  way  so  ( a , 3 ) are r e q u i r e d to c a t a l o g u e the c l a s s i n the same  as i n (2.1.1).  involving  channel  For the c o l l i s i o n  atoms 1, 2 and  3, one  1  type (n, f , f ) = (3, 2,  possibility  of the l a b e l l e d  2)  channel i s  -  (a,3)=l(23)  20 -  where atom 1 i s f r e e and atoms 2 and 3 are bound.  2 possible collision  types  1(23)  + 1(23), a n o n - r e a c t i v e c o l l i s i o n ; and  1(23)  -»• 2 ( 1 3 ) , an exchange c o l l i s i o n .  There are g ( n , 2 f - n , n - f ) , g as d e f i n e d by (2.1.3), l a b e l l e d a given fragmentation  The is  channel  f o r m u l a t i o n of P Q-Q» the d e n s i t y o p e r a t o r f o r n c o l l i d i n g C  t r a c e d back i n time  and p ' s . D  to a time b e f o r e the c o l l i s i o n began and so to a time  With the assumption  i n t o a product of  t h a t the n-atom c o l l i s i o n  i s o l a t e d event, the d e n s i t y o p e r a t o r  i  s  embedded  i s an  i n t o the e n t i r e  space.  The  collision  types g i v e n by (2.3.1a-g) r u l e out the e x i s t e n c e of  " g h o s t s " , t h a t i s , where t h e r e are atoms h e l p i n g to determine of  atoms  t h a t the c o l l i s i o n dynamics can be  when the n-atom d e n s i t y o p e r a t o r c o u l d be f a c t o r i s e d  N-atom  channels f o r  (n,f).  achieved by u s i n g the assumption  Pf's  There are  a collision  accounted  event, but t h e i r dynamics i s not being  the outcome  explicitly  f o r , and " s p e c t a t o r s " , where one or more atoms are i n c l u d e d i n n  the d e s c r i p t i o n p ( ) but where these atoms p l a y no r o l e i n d e t e r m i n i n g the c o l l i s i o n outcome.  Consider  representing equation  an i s o l a t e d n-atom c o l l i s i o n ,  the c o l l i d i n g  n  the time dependence of p ^ , , coll  atoms i s d e s c r i b e d by the n-atom von Neumann  - 21 -  ,K_9 l f i  (n) 3F c o l l rt  P  =  I  p(n) (n) coll p  v  (n)  K  where  P  (n) (n) (n) coll " coll *  i s the L i o u v i l l e  Tl  (  1  i=l n E I i=l  £ A ±  while  ±i  . 3  *  2  )  i<j n E  + 1  (2.3.3)  Kj Hamiltonian  that i s ,  = J ^ A - AK  (2.3.4)  i s the i n t e r a t o m i c p o t e n t i a l  * A  ,  superoperator  A c t i n g on an a r b i t r a r y o p e r a t o r A, i. ± i s the atomic commutator,  2  n I  j.(n)  =  .  P  = V .A ±  Equation  commutator, a c t i n g  - A Vy  to g i v e  (2.3.5)  ±  (2.3.2) i s v a l i d  o n l y when the n-atom  system i s i s o l a t e d and  does n o t i n t e r a c t with other atoms i n the e n t i r e N-atom g a s . The formal s o l u t i o n o f (2.3.2) i s  P ^ t )  It  i s assumed  assumes  = exp[- ± £  ( n )  (t  - t )] p o  ^  (t ).  t h a t long b e f o r e the c o l l i s i o n  Q  (2.3.6)  ( t •+• -<*»), p^°?, ( t ) o ' coll o  the form o f a product o f monomer and dimer d e n s i t y o p e r a t o r s ,  - 22 -  There a r e d i f f e r e n t all  the p o s s i b i l i t i e s i s taken.  l i m i t i n g value  t  possible factorisations  lim + — o  o  1  In analogy w i t h e q u a t i o n  (2.1.5), t h i s  o f tp^",,] ( t ) i s g i v e n by coll o  [p;"},] ( t ) = I («. .) c  and a s t a t i s t i c a l average o f  1  2  -  0  n  !  B  lim n — 16a' o  — — f  P (i,t ) H ° (jk)eg'  P,(jk,t ).  f  f  b  J  (2.3.7)  The  time e v o l u t i o n of the dimer and monomer d e n s i t y o p e r a t o r s i s  ascribed  to t h a t o f f r e e monomers and d i m e r s :  p (i,t ) f  P (jk, b  The P  D  the  c  1  o  f  t ) = exp[-±£( Q  ^ V  ik)  d e n s i t y operator  P^^l  a t time t by t a k i n g channel M 0 l l e r  i s defined  c  ^  a  n  t  n  0  1  k  P (J »t)-  (2.3.9)  b  u  s  b  e  expressed  the mathematical l i m i t t  s u p e r o p e r a t o r f o r the i n i t i a l  i n terms o f p 0 0  Q  •»• - .  This  f  and  involves  1  channel ( a , 8 ' ) ,  which  by  l i r a  ^(a',3')=  t  «Plir*  (n)  t] Q  e x  5  E  1  S '  (2.3.10)  — °°  a  3')  [  P -^(a'e-)  Here £ ( ' , 3 ' ) i s the L i o u v i l l e o p e r a t o r  '(a'  (2.3.8)  = exp[- • | f ( t - t ) ] p ( i , t )  Q  1  i  +  1  T h i s development p a r a l l e l s  f o r the channel  (a',3')  * (1k)'  t h a t of the channel M i l l e r  (2.3.11)  operator  - 23 -  V.0')  "  having  channel  * V  ^)~=  l  l  m  e  x  t •»• o  1  i  t  +  ±  n(a',6')  arbitrary i n i t i a l  channel  In  *  ]  e  x  p  [  s  -^(a'$')  0  " V , 0 ' )  r  ]  (2.3.12)  ( 2  e x i s t s as a strong l i m i t  2  . <£•>  P  3  ' -  ( 2  n  a  «  1 3  3  * *  1 4 )  f o r n-atom c o l l i s i o n s i s g i v e n by  n - f (n,)-!^ (  1 3 )  f o r any  (a*3') i n v o l v i n g n atoms, i t f o l l o w s t h a t  3  }  e  (n)  i*a'  the d e n s i t y o p e r a t o r f o r n c o l l i d i n g  1  o  ^(a'.B-/-  (a',3«)  P<;0  t  **MkV  t h i s way, the Mriller s u p e r o p e r a t o r  (n)  and  )  1  that  A  n  Hamiltonian  On the assumption  ^cx'S')  (  P l ^  f  n  ( . .15)  f  2  (jk)€3«  i  b  j  3  k  atoms i s to be i d e n t i f i e d as  c>  n  f  I 2 " ' (n!)-\pj (a- ,3*) ^  8  »  M p  ;  H i€a»  p (i) r  n (jk)eg'  P,(jk). b  (2.3.16)  For a t r a n s i t i o n from the i n i t i a l channel  channel ( a ' , 3 ' )  to the f i n a l  ( , ^ ) , the t r a n s i t i o n s u p e r o p e r a t o r 3"( 6)j(a',3' ) i s d e f i n e d , a  a>  - 24 -  viz:  *(..»,(.•.»•)  5  »(«.»»(...»•) ^  /  f  i  (  j  l  < 2  £ , " V 6  3  - -  1 7 )  where v ~ ( g ) i s the s u p e r o p e r a t o r r e p r e s e n t i n g the i n t e r m o l e c u l a r a>  potential  ^  « (a,3)  f o r the channel  5  ^  (  n  Hence the t o t a l  It  )  £  - ,  (a,3):  <n(a,3)  (2.3.18)  t r a n s i t i o n s u p e r o p e r a t o r i n t o the channel  has been assumed that c o l l i s i o n s are i s o l a t e d  (a,3) i s  events i n v o l v i n g a  d e f i n i t e number o f atoms, so the N-atom d e n s i t y o p e r a t o r d e s c r i b i n g a c o l l i s i o n i n v o l v i n g n atoms w i t h l a b e l s i , j , k . . . l i s g i v e n by  r„W [ p  i ijk...l coll ]  =  - / ^ " l n> (  [ p  r (N) i ijk...l coll 1  p  (N-n) •  ,„ „ _ (2.3.20)  t h a t i s , the d e n s i t y f o r the e n t i r e gas i s f a c t o r e d i n t o a product o f the n-atom d e n s i t y o p e r a t o r f o r the c o l l i s i o n  [p^?? ,] ,., and i t s ijk...l coll  complementary p a r t .  C o l l i s i o n s i n v o l v i n g d i f f e r e n t s e t s o f atoms a r e c o n s i d e r e d exclusive.  T h e r e f o r e an n-atom c o l l i s i o n i s r e p r e s e n t e d by  mutually  x  - 25  [P  ( N )  1  ]  «. = n-atom c o l l  -  S [pi?£ Jcoll. . .. ., i j k . ,.1 i<j<k...l  (2.3.21)  J  n i  n  J  The  density operator  f o r c o l l i s i o n i s a sum  i n v o l v i n g v a r i o u s numbers of c o l l i d i n g  of d e n s i t y  operators  atoms  ' " " ' ' . l i • n>2 I, " ' " V . W . coll.  Lowry and  Snider  have s t u d i e d  the  t r a n s i t i o n s u p e r o p e r a t o r J ( a, 3) ; ( a, 3)  3 r  (a,3);(a,3)  A  =  ^(a,  =  t  3)^(a, 3)  (a,3);(a,3)  +  t  A  s t r u c t u r e of the  A  A  "  (ct,3);(a,3)  A  t  a ?  g ) . ( ^g) a  ±  s  t  he  transition  [  t  [ (  1  ( a , 3); ( a , 3 ) ^  ^  t )  A  [  (a,3);(a,3)  and t j (  The  a )  f i r s t terms on the r i g h t  t , are c o n s i d e r e d  ]  ( 2  3  ' '  2 3 )  operator  (2.3.24)  O N  g ) i s the channel Green's  l t  '< a,3) ; (a,3) ' '  o\ / O N = V, fi, (<*,3);(a,3) (a,3) (a,3) o x  non-reactive  :  ~ ^ )(a,3);(a,3)  Here t (  2  < ^">  superoperator  s i d e of (2.3.23), those which are l i n e a r i n  to be l o s s terms.  This a s s o c i a t i o n i s formally  verified  D.  i n Chapter 4 when a g e n e r a l i s e d o p t i c a l  theorem i s d e r i v e d .  Time E v o l u t i o n o f the System  The  system evolves with r e s p e c t to time a c c o r d i n g to the von Neumann  equation  =  <Tt  *  (2.4.1)  O p e r a t i n g on e q u a t i o n (2.4.1) by of  the dimer d e n s i t y o p e r a t o r P ( 1 2 ) and D  Pf(l)  the monomer d e n s i t y o p e r a t o r  the present work the f o r m u l a t i o n of Lowry and  emphasize the f a c t  channel  t h a t the molecules  i n s t e a d of the o p p o s i t e l o g i c  aforementioned. treatment  These two  Hamiltonian potential  Snider i s r e w r i t t e n  are bound a c c o r d i n g to the  t h a t has been g i v e n i n the  schemes are m a t h e m a t i c a l l y e q u i v a l e n t but  i n which the arrangement channels  i s more convenient  We  we get the time e v o l u t i o n  respectively.  In to  or  are regarded  as  the  fundamentals  to use because i t Is e a s i e r to decompose the  total  i n t o the channel H a m i l t o n i a n  and  the c o r r e s p o n d i n g  than to d e a l w i t h a l l the one-  and  two-particle Hamiltonians.  have the commutation r e l a t i o n s  ^ ( i j ^ b i j  =  6  >  £  bij (ij)  channel  - 27 -  and  =  V f i  f  2  fi<i  which are c o n s i s t e n t w i t h (2.2.12) and (2.2.13). require  l  and  the  i %  3  Moreover, we  also  approximation  P  "  4  <-->  f j  £  2  i  < '^>  a c c o r d i n g l y the r e l a t i o n  P  * i f j  =  P  fj^ i *  (2.4.5)  For a l a b e l l e d  channel  (a,3), a p r o j e c t i o n operator i s defined  a c c o r d i n g to  P, ( a  = P,.  Q N  '  6 )  f  = P  w i t h channel  ^ ( a ^ ' ( a  3 )  Henceforth channel.  =  t  l  n  p  (jk) 3  b  ^  k  (2.4.6)  c  Hamiltonian  1  r = C  i*a  1  (jk)^3  the l e t t e r c i s used  (2.4.7) (  j  k  )  i n p l a c e of (a,3) as the l a b e l  W i t h i n the approximations  for this  (2.2.13) and ( 2 . 4 . 5 ) , the commutation  relation  H  P = P c c c c  (2.4.8)  - 28 -  is valid,  and c o r r e s p o n d i n g l y , the p r o j e c t i o n  = n c i«-a  <P  superoperator  n tp... (jk)*3  commutes w i t h the channel  Llouville  (2.4.9)  s u p e r o p e r a t o r d e f i n e d by (2.4.10)  C  iea  1  (jk)fr3  U  ;  where  (2.4.11)  the commutation being w r i t t e n as  L 9 " f l ee c c  and  (2.4.12)  which h o l d s w i t h i n the assumptions  (2.4.12) i s then used  (2.4.2) - (2.4.4).  t o o b t a i n k i n e t i c equations f o r the monomer d e n s i t y  o p e r a t o r P f ( i ) and the dimer d e n s i t y o p e r a t o r P ( j k ) . D  label  c to r e p l a c e (a,B) i s i n accordance  channels.  The r e l a t i o n  The use o f the  w i t h the emphasis on l a b e l l e d  The import of such a change i n symbolism i s r e v e a l e d i n  Chapter 4 where a sum r u l e i s d e r i v e d on the assumption  that the  p r o j e c t i o n o p e r a t o r / s u p e r o p e r a t o r commutes with the H a m i l t o n i a n / L i o u v i l l e o p e r a t o r f o r t h i s channel r e g a r d l e s s o f how many s p e c i e s a r e i n v o l v e d . So with the assumptions Lowry-Snider  that have been contemplated  f o r m u l a t i o n can be extended  o t h e r m o l e c u l a r s p e c i e s may be p r e s e n t . n o t a t i o n a l l y than ( a , 3,  ...).  so f a r , the  to a more g e n e r a l case i n which Then c i s more convenient  - 29 -  Kinetic Equation for the Monomer (N) Acting  on (2.A.l)  on the l e f t  by the p r o j e c t o r  3  tf ^  and  taking  t r a c e s over atoms 2 to N g i v e s  i n  1  gt" Pf^ ^  N  P  f l  *  (2.4.13)  2.. .N  The d e f i n i t i o n o f the p r o j e c t o r i n t o a g i v e n c h a n n e l , (2.4.6) and (2.4.9) t a c i t l y assumes the mutual e x c l u s i o n o f a l l other the  s e t o f atoms i n v o l v e d .  l a b e l l e d channels f o r  From the approximations l a i d  B, and the d e f i n i t i o n ( 2 . 4 . 9 ) , i t f o l l o w s  out i n S e c t i o n  that f o r a system of n atoms  ^  (2.4.14) cl  where c l i s the l a b e l f o r a channel i n which atom 1 i s f r e e . rigorous  formulations,  Instead  of  i t i s assumed that the channels and t h e i r  c o r r e s p o n d i n g p r o j e c t i o n s can be d e f i n e d  i n the same manner i n the e n t i r e  N-atom space,  f  ( N ) _ £ g> (N) _ £ f i ( n ) j ( N - n ) f l  cl  C l  where J(N-n) ± space.  s  cl  the p r o j e c t i o n i n t o the complementary (N-n) atom  Corresponding to the channel c l i s the channel L i o u v i l l e and the  potential  superoperators  the l a t t e r g i v e n by  y.(n) cl  C l  = < t  (n)  (n) cl  and  t^  e  former g i v e n  by (2.4.10) and  - 30 -  or i n the e n t i r e N-atom  Substituting  (2.4.15) i n t o  ifilr P < ° - N 9 t  space  f  (2.4.13) g i v e s  Tr E (P 2...NC1 C  -  N  Tr £ 2...N  1  o  C  l  C  p  C  ( N )  + N  l  Tr  l  ( N ) +  C  T r E <P 2...N c l  N  l  C  (  N  )  P  T r E «P  + N  M  XT  O  2...N  C  E *  2...NC1  p  E cl  + N Tr E <P < » V < » > 2...NC1  >  l  C  pj«  Tr L  = N  C  I t <?> <P<?>  Tr  2...NC1  - N  ( N P l  ( N  (  N  )  N  (  ^ C  l  N  )  N  )  (  N  )  P  l  W  V C  (  N  )  P  l  N  (  V< >  Cl  1  (  l  ( P  l  N  )  P  .  Cl  2...N c l (2.4.18)  S i n c e i t has been assumed simultaneously,  < ™  P  (  N  )  -  that an atom cannot be f r e e and bound  then  P f  (D  0  P  ^ (2...N),  (2.4.19)  t h e r e f o r e the e q u a t i o n  e ±  4t"  p  f  (  1  )=  < t  l  p  f  (  1  )+  N  T  r  E  el^S  p  (  N  >  (2.4.20)  2...N c l  i s obtained. second  term  " C ^ P f O ) d e s c r i b e s the f r e e e v o l u t i o n o f p ^ ( l ) , and so the i s attributed  to be a c o l l i s i o n a l  term.  A c c o r d i n g l y p(N)  (N) i s r e p l a c e d by P ^ QQ  an  <* the k i n e t i c e q u a t i o n  l a r g e N, M and D) becomes  for  P  f  ( l ) ( f o r small n and  -  i f ^ P ^ l )  +  =i  P  7 5  (<f  2  2  +  •  (l)  f  J  ^ 1 2 3  +  +  l  Tr(f  +  +  123  1 2  J  J  +  123 123  1  2  (  + . . . +  3  o  4  )  :  f  P  1 2  ^1(23) l ( 2 3 )  ^1(23)  l  2  Tr  (  2  3  4  n _ f  )  '  3  ( p  ^12(34) ^ 1 2 ( 3 4 )  234^  T  31 -  1 2  ) ( p  1(23)  l(23)  )  P  (34)  +  [(n-1)!]  2...T1  p  - 1  2 p  (13)2  )  123  (12)(34)  ( P l 2  +  +  2 p  (13)(24)  (12)34  +  2 p  )  13(24)  +  2 p  (13)24  )  I <P J ?> I p , + . . . . cl cl . c (  n  )  (  cl  c'  (2.4.21)  where  c' denotes the i n i t i a l  i s d e f i n e d from P f ^ / ^ .  i  6  channel.  J u s t l i k e the p r o j e c t o r  a, P ^ j ^ / ^ j ^ *  k  ( J ) «=" &>  t l l e  P/ c  c  c  arrangement  channel d e n s i t y o p e r a t o r i s d e f i n e d as  p  =  n iea  p (i)  n P (jk). (jk)*B  (2.4.22)  K  Thus the m o l e c u l a r and arrangement channel d e s c r i p t i o n s can be t r a n s l a t e d i n t o one another i n a s t r a i g h t f o r w a r d manner.  K i n e t i c E q u a t i o n f o r the Dimer The treatment i s the same as that f o r the monomer. label  L e t c(12) be the  f o r a channel i n which atoms 1 and 2 are bound t o g e t h e r ,  ^  = 1 1 2  _  n  f <> c(12) '<»« j  =  then  y f W e(12)  «  1  2  C2 4 231 > "  (  "  - 32 -  With we  a c t i n g on the l e f t  d 1 2  on (2.4.1) and t r a c i n g over atoms 3 to N,  obtain  «St  %  < 1 2  >  "  <2> *  < M2 *  *(12> "b(12)  +  (  >™  < W  2>  *  3  H  c  (  V  1  ° (2.4.24)  J . . .N  l^P (12) b  =X  ( 1 2 )  +  ^  b  3  ^(12)3 ( 1 2 ) 3  (  ^(12)34  ( p  2  T r 34  .  J  (<?  ( p  +  2 p  ( 1 2 ) 3  (12)3  (12)34  (12)(34)  ( P  +  +{Tr<P  p (12)  + Tr 34  +  cC12;  +  (?  T  +  +  p  +  +  2 p  (13)2  (12)(34)  <P  p  ( 1 2 ) 3  (13)(24)  (12)34 (12)34  12(34)  T  1  1  c'  P  1  )  J  )  +  2 p  13(24)  1  +  2 p  (13)24  I c(12)  c  (  1  2  )  )  c  (  1  2  )  c' c  + . . . where n i s s m a l l compared w i t h N, M and  (12)(34)  )  Tr 2 - '- [(n-2)!]34..,n f  )  (12)(34) ( 1 2 ) ( 3 4 )  (12)(34)  n  123  (2.4.25) D.  - 33  The of  kinetic  intermediates  theory d e s c r i b e d  -  above does not a l l o w f o r the  ( s p e c i e s i n metastable  s t a t e s ) which may  subsequent c o l l i s i o n s or decay spontaneously energy.  I n t u i t i v e l y , one  significant equations  expects  r o l e i n the f l u i d  i n the present  situations.  e m i t t i n g a quantum of  that the i n t e r m e d i a t e s can  i n the f o l l o w i n g chapter may  good d e s c r i p t i o n of the behaviour  scope of  undergo  play a  dynamics of a r e a c t i v e system, so  of change as obtained  r e s u l t s obtained  existence  of the system.  research w i l l  the  not g i v e a  I t i s hoped t h a t  be g e n e r a l i s e d to a wider  - 34 -  CHAPTER 3 THE  The  FORM OF THE EQUATIONS OF CHANGE  e q u a t i o n s o f change f o r monomer and dimer o b s e r v a b l e s a r e  o b t a i n e d by using the k i n e t i c  equations  f o r the r e s p e c t i v e s p e c i e s as  d e r i v e d by Lowry and S n i d e r .  A.  D e s c r i p t i o n o f Molecular  Observables  The hydrodynamic d e n s i t i e s o f mass, l i n e a r momentum, angular momentum and  energy  a r e s t u d i e d i n t h i s paper.  Their corresponding  molecular  o b s e r v a b l e s can be a s c r i b e d to be a t t r i b u t e s o f a s i n g l e molecule f o r the i n t e r m o l e c u l a r p o t e n t i a l . treatment  T h i s c h a p t e r thus focuses on the  o f the s i n g l e - m o l e c u l e a t t r i b u t e s , namely mass, l i n e a r and  a n g u l a r momenta and k i n e t i c  Let  except  energy.  be an o b s e r v a b l e a s s o c i a t e d w i t h the monomer i , and <l>(jk)  an o b s e r v a b l e f o r the dimer ( j k ) .  The c o r r e s p o n d i n g  o b s e r v a b l e s f o r the  whole N-atom system a r e g i v e n by  z  bij  ) 4>.  (3.1.1)  f o r the monomer,and  (3.1.2)  -  f o r the dimer.  35 -  The s u b s c r i p t s M and D a r e used to i n d i c a t e monomeric and  dimeric properties  respectively.  The e x p e c t a t i o n v a l u e s o f the p h y s i c a l o b s e r v a b l e s f o r the s p e c i e s a r e t h e r e f o r e equal to  « 0  =  Tr  Z  ( 5 - Z(P  1 . . . N 1  M  b  j  ) * l  j  p  ( N )  1  (3.1.3)  = T r <j> p (1)  1 f o r the monomer, and  <<i>> = n  D  "  Tr E f 1...N j<k  £  *(12)  P  (  N  )  P b  i  b °  j  2  )  ( 3  1  ' -  4 )  f o r the dimer.  To d e s c r i b e a r e a c t i v e system i n which a dimer m o l e c u l e may be broken up i n t o or formed from two monomer m o l e c u l e s a t c o l l i s i o n s , i t i s n e c e s s a r y to be a b l e to t r e a t d i m e r i c and monomeric a t t r i b u t e s on a unified  basis.  The p h y s i c a l o b s e r v a b l e s (whether f o r a monomer or a  dimer) can always be t r e a t e d  as atomic a t t r i b u t e s s i n c e the b a s i c u n i t o f  matter i s the atom f o r the purpose o f t h i s t h e s i s . these atomic o b s e r v a b l e s a r e u n e q u i v o c a l l y l o c a l i s e d i . e . , atomic c e n t r e o f mass.  For the monomer, at the m o l e c u l a r ,  When i t comes to the dimer m o l e c u l e , one  - 36 -  may l o c a l i s e  these a t t r i b u t e s of the c o n s t i t u e n t atoms at the c e n t r e of  mass of the m o l e c u l e , or at the c e n t r e s of mass of the r e s p e c t i v e atoms. The  r e l a t i o n s h i p between these two schemes was d i s c u s s e d by Olmsted and 1 1  Snider .  In the present work, the l o c a l i s a t i o n Is always  c e n t r e of mass of the r e s p e c t i v e m o l e c u l e . a r b i t r a r y , but i t emphasizes  to be at the  T h i s c h o i c e may be c o n s i d e r e d  the m o l e c u l a r aspect of the system and i s  t h e r e f o r e the n a t u r a l d e s c r i p t i o n used i n c h e m i s t r y .  The  l o c a l mass d e n s i t i e s  f o r the monomer M^(r_,t) and f o r the dimer  Mrj(r.,t) a t macroscopic p o s i t i o n _r and time t are d e f i n e d by  M^r.t) "  =  1 1  Tr 1...N  Z (<f - Z <?,.,) m6 i j b  = Tr m 6. p 1  6±  J  p  ( N )  (t)  1  (l,t),  (3.1.5)  = &(r_ - _£i), _r_i being the p o s i t i o n o p e r a t o r f o r atom i , and  V I ' "  s  ,  ^ / b  - T r 2m 6  J  f  l  x = <5(r (jk) ° ^ " =  —- i  2  j  p  ( 1 2 )  +  < »>  k  b  S  W  W  "  M  (12,t),  (3.1.6)  +  -^k ^k — ii i —-ik ), ^ 2 =L ( 2 >» 2 i(jk)  b  =  e  i  n  g  t  h  e  p o s i t i o n operator f o r  the c e n t r e of mass of the p a i r ( j k ) . The dependence of macroscopic d e n s i t i e s on _r and t w i l l indicated  be understood and w i l l not be e x p l i c i t l y  i n the formulae from here onwards.  - 37 -  For an a r b i t r a r y m o l e c u l a r o b s e r v a b l e <$>± f o r the monomer i and <j>(jk) f o r the dimer ( j k ) , the l o c a l d e n s i t i e s respectively  < v  per u n i t mass a r e  d e f i n e d by  l  T  ~=\ ,  M^  r  „ j 1...N i  1  - . j  Tr (•jfi^g  z  <  s  (  V  *± i>s 6  p  (  N  )  P (D  (3.1.7)  f  and  <$ > n  Tr I C... 1...N j<k ^  = M ^  b  = V  £  k  k  )  <*<12) ( 1 2 ) s b  (  6  (  )  ^  P  (  1  2  6,...) J k  )  )  p  (  N  )  8  ( 3  '  1  ' -  8 )  When < J > and 6 do not commute, one has t o p i c k the a p p r o p r i a t e symmetrisation  to ensure h e r m i t i c i t y .  F o r the hydrodynamic d e n s i t i e s o f  mass, l i n e a r momentum and angular momentum, the symmetrised  operator i s  g i v e n by  (  =  (  +  (3a  Ws i Vi W  < Wak)^ • r  (  W  ( j  k)  +  6 ( j  k)*  ( j  9  - >  k)>-  For mass (<{>£ = m), l i n e a r momentum ( ^ - P±) and a n g u l a r momentum (<(>£= _r_£ x  +  where s± i s the atomic  the c o n n e c t i o n between dimer and atomic  angular momentum),  attributes i s  3  1  1  ( - - °)  -  •<jk)  *J  =  3 8  -  V  +  For k i n e t i c energy,  (  the atomic  3  -  1  -  1  1  >  o b s e r v a b l e i s the o n e - p a r t i c l e Hamitonian  <fri = ^ i w h i l e the d i m e r i c o b s e r v a b l e i s the sum of the H a m i l t o n i a n s for  the two c o n s t i t u t e n t atoms p l u s the i n t r a m o l e c u l a r p o t e n t i a l  =J  +Jf  V) S k The symmetrised  +  ( 3  V  k i n e t i c energy  operator ( ^ i 6 i )  s  K 1 2 )  -  f o r the monomer i  is  <*i  V  s  =  t  6  IPi i  +  2  6  £i • i-Ei  therefore for the dimer (jk),  +  ( 3 -  the corresponding  symmetrised  1  -  1  3  )  kinetic  energy operator i s  (K  uk) W s  =^  6  +  +  + (  W s  -k  V  [ p  (jk)  6  k  (  6 ( j  j  k)  +  k )  ^ W s 6  +  2-P-j ' ( j k ) £ j  +  2  i \ •  6  6  V a*) +  (jk)£k  6  +  (jk) Pjl  6  (  p2 j  k  )  k  ]  3  1  14  < - - >  s i n c e the p o t e n t i a l depends o n l y on p o s i t i o n s and so V and 6 commute,  V6  =  6V.  (3.1.15)  - 39 -  B.  Formal E q u a t i o n s o f Change  C o n s i s t e n t w i t h the k i n e t i c e q u a t i o n s of Chapter a typical  monomer and a t y p i c a l  2 f o r the s t a t e s o f  dimer i s the S c h r o d i n g e r p i c t u r e i n which  the p h y s i c a l o b s e r v a b l e s a r e e x p l i c i t l y  time-independent.  For the  monomer, the change w i t h r e s p e c t to time o f the d e n s i t y o f each o f the physical  !t-  p r o p e r t i e s i s thus  ( M  M<*M»  =  T  '  (  p  W s i n r  f  (  1  >  (3.2.1)  The  first  during term  term  involving  free f l i g h t  the s u p e r o p e r a t o r <£ ± g i v e s r i s e  (between one c o l l i s i o n  i s a s s o c i a t e d w i t h the c o l l i s i o n a l  analogy,  IF  ( M  to the change  and the n e x t ) , w h i l e the l a s t  change o f  <*M>.  By  the dimer r a t e o f change i s  D  p  <V>  12  < W < 1 2 ) > s l t b< >  " -^ *(12) (12 8 (12) (  +  6  )  t  ^ * ( 1 2 ) ( 1 2 ) s 4F (  6  )  [  p  f  P  (  b  ( 1 2  1  2  The  f r e e motion terms f o r dimer m o l e c u l e s  and  S n i d e r and have the same s t r u c t u r e as t h e i r  )  >  W  *  ( 3  2  ' '  2 )  have been o b t a i n e d by Olmsted counterparts f o r  - 40 -  monomers.  In t h i s chapter we w r i t e down the e x p l i c i t  motion terms while the c o l l i s l o n a l chapters. first.  The e q u a t i o n s  forms o f the f r e e  terms w i l l be d i s c u s s e d  i n subsequent  o f change f o r the monomer a r e to be d e s c r i b e d  I n d i v i d u a l properties are discussed  separately.  The monomer mass d e n s i t y changes because o f the mass f l u x a s s o c i a t e d with  the stream v e l o c i t y w h i l e c o l l i s i o n s can l e a d to a p r o d u c t i o n o r  l o s s o f monomers and o f monomer mass, e x p l i c i t l y  (3.2.3)  Here u i s the m a c r o s c o p i c (stream) v e l o c i t y of the monomer d e f i n e d by  (3.2.4)  The  r a t e o f change o f monomer momentum d e n s i t y i s g i v e n by  (3.2.5)  where  (3.2.6)  (_p. <5 J 2 b e i n g  the tensor transpose  I t i s u s u a l l y convenient  of p 6 p  t o break the k i n e t i c  f l u x of l i n e a r momentum  -  into  4 1  -  c o n v e c t i v e and c o n d u c t i v e c o n t r i b u t i o n s to o b t a i n the more f a m i l i a r  e q u a t i o n of motion  l ^ u + u -  (M) Here P  p£  M)  V „ - - l £  V  i ^ u ]  c  o  n  .  (3.2.7)  i s the monomer k i n e t i c p r e s s u r e tensor d e f i n e d by  1  = m"  1v[(  ~  Rl  mu) (£j - mu) 6 ^  The a n g u l a r momentum o f any molecule momentum and the t r a n s l a t i o n a l  p (l).  (3.2.8)  f  c o n s i s t s o f the i n t e r n a l  angular momentum.  angular  For the monomer i , the  a n g u l a r momentum o b s e r v a b l e Jj_ i s a c c o r d i n g l y made up o f a translational  i i  =  ±i  +  p a r t l± = rj_ x  Ij. = ±  + ±  1  x  and an i n t e r n a l  £-  ±  part s ,  (3.2.9)  ±  By d e f i n i t i o n , the monomer angular momentum d e n s i t y per u n i t mass i s  <±H>  ^  T  r  (  i l V s  P  f  ( 1 )  »  (3.2.10)  whose r a t e o f change i s hence g i v e n by  S i n c e [ J ^ , ^ ] . . = 0, the f r e e motion term can be r e w r i t t e n so t h a t  - 42 -  |F < V 4 M »  = -  V • Tr U 6 p ) 1  1  1  p (l)  s  f  (VlM>  +  ) ]  coll  (3.2.12)  where  (ViVs = { [ V A ^ i i +  6  +  +  1  (3  Wi -  2  13)  --  J u s t as the angular momentum d e n s i t y c o n s i s t s of t r a n s l a t i o n a l and internal  p a r t s , that i s ,  where  the k i n e t i c and  internal  1  m"  p a r t o f the angular momentum f l u x i s the sum o f t r a n s l a t i o n a l angular c o n t r i b u t i o n s  Tr(J 6 p ) 1  1  1  M )  s  p ( l ) = r x(g£ + J ^ u u ) * f  1  m"  TrCSj 6j p ^ P U ) . f  (3.2.16)  On combining the e q u a t i o n of motion (3.2.7) and the e q u a t i o n o f change o f a n g u l a r momentum, the e q u a t i o n s f o r the changes o f i n t e r n a l momentum d e n s i t y and the t r a n s l a t i o n a l o b t a i n e d , namely  angular  angular momentum d e n s i t y a r e  - 43 -  TF < V i M m t » f  "  m _ 1 v  -  '^ i i W s  [ p  f  (  1  )+  h  ^ ^ . m ^ ' c o n (3.2.17)  and  - ^ r  x ^  u) = V • [ ( P <  M )  + M  M  u u ) x r ]  +  [ ^ ( r x M  H  i^coll* (3.2.18)  The  l a t t e r can be r e w r i t t e n as  | ^ ( r x u) + (u • V) r x u = M^  1  M )  V • (p£ x r) + | ^ (r x  u)l  c  o  l  l  (3.2.19)  which has a form and  analogous  (3.2.18) a r e indeed  to the e q u a t i o n o f motion.  coupled  equations as i n t e r n a l  Equations  (3.2.17)  and t r a n s l a t i o n a l  angular momenta evolve i n t e r - d e p e n d e n t l y .  L i k e w i s e , k i n e t i c energy independent  quantities.  and p o t e n t i a l  energy o f a system are not  Since k i n e t i c energy  i s a molecular  observable,  we may w r i t e down the e q u a t i o n o f change f o r i t .  We d e f i n e the k i n e t i c energy  T  M =  Y  (  J  Vl s }  p  f  (  1  )  per u n i t mass o f the monomer as  (3.2.20)  - 44 -  whose rate of change i s given by It  ( M  MV  " -  m _ 1  V  *^  i  V  P  s  f  ( 1 )  l  +  T*  (  ]  W  (3.2.21)  coll  where  m" T r ( ^ j f i ^ g P ( D = m Tr 1  1  8  f  (  +  W  l  +  K  +  55  (  4 ^1  +  l V l  6  +  l ^ l -P-l V P  55  *1>  P (D f  (3.2.22) is  the energy f l u x due to f r e e motion of the monomer molecules which can  be broken i n t o a c o n v e c t i v e term, a c o n d u c t i v e heat f l u x and a term due to the energy c a r r i e d  1  m"  Tr(K  l £ l  6  l ) s  by the k i n e t i c momentum f l u x  p (l) = ^ f  » T  M  +  c j f > £<*>.„  (3.2.23)  +  Here  ^  CM)  -TS.  is  m  =H  T r [ (  z ^  £1 _L_ m  -P-l  u  )  —  the k i n e t i c c o n t r i b u t i o n  The e q u a t i o n s of change f o r the monomer.  -P-l  . r _ l _u ) ( - i m — m  u) 6 ] p (1) — l s f  (3.2.24)  t o the heat f l u x from the monomers.  f o r the dimer are e x a c t l y  of the same form as  These equations and the d e f i n i t i o n s of the  corresponding q u a n t i t i t e s  are l i s t e d  f o r further  reference.  - 45 -  The dimer mass d e n s i t y changes a c c o r d i n g to  (3.2.25)  where _v i s macroscopic (stream) v e l o c i t y o f the dimers  H' W c V s  -1  ^  E  M  D  p  b  (3.2.26)  ( 1 2 )  The e q u a t i o n o f motion f o r the dimers i s  (3.2.27) where  p£  is  D )  = (2m)"  1  Tr[(£  the dimer k i n e t i c  (p,.,v P/.,\ ~ ( j k ) -Hjk)  (jk)'s  - 2mv) ( P  ( 1 2 )  ( 1 2 )  - 2mv) «  (  1  2  )  ]  8  P (12)  (3.2.28)  b  p r e s s u r e t e n s o r , w i t h the symmetrised  operator  d e f i n e d by  =  ^jk^jkAjk)^ 4  ••ECjkAjlO^jk)  +  (  +  £  6  6  (jk) (jk)-E(jk)  ^(jk) (jk)£(jk)  ) , : +  6  ( j k ) £ ( j k ) -E(jk)  (3.2.29)  As w i t h the monomer, the a n g u l a r momentum  •^Kjk)  -3  o p e r a t o r f o r the dimer ( j k )  (3.2.30)  -k  can  be expressed as the sum o f a t r a n s l a t i o n a l  for  the dimer molecule as a whole  a n g u l a r momentum  operator  - 46 -  ^jk)  and  X  ( 3  - ^ j k ) ^ j k )  2  ' '  3 1  >  the i n t e r n a l momentum o p e r a t o r a s s o c i a t e d w i t h the sum of the  i n t e r n a l momenta o f i n d i v i d u a l  atoms _Sj + ey_ and that  due to the  r e l a t i v e motion between the c o n s t i t u e n t atoms j and k, jr-ji^ x _pjk where  r.  k  = r. - r  i s the p o s i t i o n  P  j k  (3.2.32)  k  o p e r a t o r o f k r e l a t i v e to j , whereas  " j(Pj " P>  (3.2.33)  k  i s the r e l a t i v e momentum o p e r a t o r of k from j .  The angular momentum  d e n s i t y per u n i t mass  <V  can  the  «  (  1  2  )  )  s  P (12)  E  "D  1  ll  +  ^2  +  (3.2.34)  b  thus be broken i n t o a t r a n s l a t i o n a l  <^,int>  By  Tr ( J ^ ,  =  Ll2  X  part r x v and an i n t e r n a l  6  ]  *12> ( 1 2 ) s  P  b  ( 1 2 )  «  ( 3  2  ' '  part  3 5 )  d i r e c t analogy to the equations o f change f o r the monomer d e n s i t i e s , r a t e s o f change of the t r a n s l a t i o n a l  densities  jriv  and i n t e r n a l angular momentum  are respectively  x v ) + (v • V) r x v = M^  1  V • (p£  D )  x r ) + [|^ <r x v > ]  c o l l  (3.2.36)  - 47 -  and  1 * %  <-iD,int  - -(2*)-  +  It  ( M  1  > )  V • Tt[(.  D  1 +  s  r  2 +  1  x p  2  ) p  1 2  (  1  2  )  6  (  1  2  )  ]  s  %  {  l  2  )  (3.2.37)  ^D.int^coll  where t(£j  + \  +r.  x p. ) ^  k  k  j  k  )  6  (  j  k  £  j  j  k  )  ]  s  4  +  +  +  <£,  J l  +  6  ^jk)  6  (jk) £  + k  £  k  P  %  +  k) %  +  (jk)  ( j  x j  j k  >  JL  P  5 (  +  k  +  -ik  )  j  P  x k  ^ k  (  x  k  j k  )  >  As i n the case o f the monomer, w h i l e the f r e e translational collislonal  and i n t e r n a l  (3.2.38)  ^ k >  flight  contribution  to  angular momenta are s e p a r a t e l y conserved, the  parts are not.  F i n a l l y , analogous  w i t h the monomer, the dimer k i n e t i c energy d e n s i t y  per u n i t mass i s  l  h=~\  H  (  *(12) W s  whose change i s g i v e n by  P  b  (  1  2  )  (3.2.39)  - 48 -  ^  (M T ) = - (2m)" D  1  V • Tr 12  D  +  lit  ( M  T  D D  ) ]  ( K  (  1  2  )£ ( 1 2 )  «  (  1  2  )  )  ^(12)  8  coll  " V ' [M, v T  D  D  +  +  P< > • v]  (M T )]  +  D  D  c o l l  . (3.2.40)  Here 2^P^ i s the k i n e t i c contribution to the heat flux vector from the dimers  (3.2.41)  and  the symmetrised operator ( ^ j ^ ) £(jk) ^(jk)^s *  ( H  (jk) ^ j k ) ^ j k ) ^  =  [(J  ? V)  6  ak)>s  +  -p-(jk)  S  ^  e t i n e c  £<jk)  ^ by  ^ak)  ^ j k ) ^  1  (3.2.42)  The changes i n these densities due to free motion a l l have the same structures as i n the c l a s s i c f l u i d dynamic description.  However, since  reactions are necessarily quantal i n nature, the f l u i d dynamics of a reactive system, here the dimer/monomer mixture, requires a quantum k i n e t i c theory.  - 49 -  CHAPTER 4 A SUM RULE  The  kinetic  equations  (2.4.21) f o r the monomer d e n s i t y o p e r a t o r  p f ( l ) and (2.4.25) f o r the dimer d e n s i t y o p e r a t o r f o r m a l and c o n t a i n no e x p l i c i t of the c o l l i s i o n  processes.  a s s o c i a t i o n w i t h the g a i n and l o s s  F o r example, f o r c o l l i s i o n s  monomer and a dimer, the l o s s of a dimer molecule not e x p l i c i t l y  0^(12) are very  shown i n the k i n e t i c  equation  aspects  involving a  due to decomposition i s  f o r the dimer d e n s i t y  operator:  i f J [  lt  p  b  ( 1 2 ) ]  M+D  =  T  3  ^(12)3 (12)3  T h i s i s because the channels  ( P  1(23)  +  are c o l l i s i o n a l l y  e v o l u t i o n of the system of c o l l i d i n g molecules Hamiltonian  of the system, and the f i n a l  p r o j e c t i o n s onto  this  channel,  A.  onto  coupled.  The time  i s generated  channel  by the  i s r e p r e s e n t e d by the  of the c o l l i d i n g  the v a r i o u s p o s s i b l e f i n a l  system and the  channels.  Channel C o u p l i n g and t h e G e n e r a l i s e d O p t i c a l Theorem  C o n s i d e r an i s o l a t e d Hamiltonian onto  )  (12)3 *  t h e r e f o r e i t i s i m p e r a t i v e to study the  r e l a t i o n s h i p between the H a m i l t o n i a n projections  2 p  for this  a g i v e n channel  system of n p a r t i c l e s .  isolated  system, and P  c with  and V  c  c  c  Let  be the t o t a l  be a p r o j e c t i o n  b e i n g the channel  operator  Hamiltonian  - 50 -  and  p o t e n t i a l energy f o r t h i s channel r e s p e c t i v e l y .  p r o j e c t o r s over a l l channels i s the i d e n t i t y so that partitioned  R  ( n )  The sum of C  a n be  a c c o r d i n g to  = E P H c  (  n  )  c  =!P c  c  <K  + V ).  (4.1.1)  T h i s i s a l s o e q u a l to  &  For  ( n )  ( E P ) = l(i< + V ) P . c c c c c c  a typical  (4.1.2)  channel c, i t s p r o j e c t o r P  c  and H a m i l t o n i a n Ji\.  commute  ** P = P H • c c c c  As a consequence, respective  (4.1.3)  a sum r u l e r e l a t i n g  the channel p r o j e c t o r s and t h e i r  p o t e n t i a l s i s o b t a i n e d , namely  E P V = E V P . c c c c c c  (4.1.4)  With the approximations o u t l i n e d i n Chapter 2, the analogous superoperators  sum r u l e f o r  follows:  E e KT = E V e . c c c c c c  This  (4.1.5)  sum r u l e can be expressed i n another form.  V i n t , c °f  t n  e molecules associated  The i n t e r n a l energy  w i t h the channel c i s g i v e n by  - 51  -  n V, . int.c  c  so the t o t a l H a m i l t o n i a n  E P ( zV c 1=1 C  EH i  ±  E P c  V  +  Since E P = 1 , c c  C  (4.1.6) 7  i '  C  +  one  = E P c  which immediately  can be broken  V  has  ) - E( E K ' c i=l  the  into  ±  + V  C  +  V  l n  ) P .  (4.1.7)  C  relation  E}* i  c  (4.1.8)  leads t o  E P V. . = EV. . P c int.c int.c c c c *  whose c o r r e s p o n d i n g  (4.1.9) 7  form f o r s u p e r o p e r a t o r s i s  E<P vT = E\r.. ^ c int.c Int.c c c '  v  c  (4.1.10) '  with  V r  int,c  Although  A  =  t V  int,c'  A ]  rule,  4  '  i  a  i  )  whether r e a c t i o n takes p l a c e or not depends on the e n e r g i e s of  the c o l l i d i n g m o l e c u l e s , molecular  (  «  chemists  are used  to t h i n k i n g i n terms of  e n e r g i e s of the s p e c i e s i n v o l v e d and  (4.1.9) and  (4.1.10),  i s i n accordance  the l a t t e r  with t h i s  the  form of the  view.  sum  -  To  see  (4.1.5)  is  collision  E P c  or  physical  allowed when  c  c  to  the  <W  V  e  s i g n i f i c a n c e of  act  on<fV i,  e  c  <P  c  the  c  i n i t i a l  = T, V~  '  -  labelled  ^  the  sum  Mailer  channel  rule,  the  identity  superoperator is  describing  c' :  (4.1.12)  , ,  c  a  '  equivalently  c'  , , e'e'  this  may  f ,2  and  the  52  Acting  on  <c' =  E  (  =  be  p i c  c'c'  t  {v1  c  c  f  P  c  E <P  EIT f , e c c ' c  seen  as  it  gives  c«  -  c'  p  c'  c c*c'  the  JT , cc'  c  (4.1.13)  superoperator  form  of  the  optical  theorem.  c '  p  -  f  c  [  t  t  p  cc'  c'  dec'  ,t  ,) cc'  r  - (A  tfee'  ,t  ,)  cc'  P  ,t /]}. c' cc' 1  (4.1.14)  To  obtain  (4.1.14),  the  structure  of  the  Miller  operator  has  been  considered:  c'  c' =  P  c'  c'  ,  c'  cc +  (-/  (  _  <  f  (~/, c  c'  , c*  c'  +  +  :  c'  +  i E ) -  K  c ' ~ iO"  1  V  1  H  c  , c'  +  SI , c*  i  e  V ,ft , c  c  )  _  1  (  " ^ c '  +  ^ c '  "  J  ^ c  +  i  E  )  - 53 -  c'  0  +  e  c c ' c' c'  vcc'  c'  c  l , t , + ft , (tf - * ,) <)cc' cc' |jcc' c c'  ^ c c ' (c'  V  p , c' (4.1.15a)  or  U = Z P « , = P , + Z P (A , t , c' c c c c l cc cc c c  (4.1.15b)  where P Tl , = 6 ,P , + P (. , t ,. c c' c c ' c' c ^ c c ' cc'  Taking the h e r m i t i a n a d j o i n t , we  E  •  P  +  C  (  t  lc' cc'  ) t  P  c  (4.1.16) '  v  have  ( 4  c  1  - -  1 7  >  and  fi  If tfo,  c'  P  c  =  6  P  cc' C  +  (  ^ c c ^ c c ^V  4  -  the H a m i l t o n i a n s f o r a l l the channels are equal and are denoted and  ifp « c  1  '  by  and e q u a t i o n (4.1.14) are r e s t r i c t e d to be d i a g o n a l i n  energy, then t h i s e q u a t i o n i s reduced to  t  f , ( t , , p , - p , t , , ) c' c'c' c' c' c'c'  = Z{v-< ^,p,-2Tri(Pt , P , c c c' c' c cc' c' c 0  1  =  Z{V P c C  [ 6 U ) t  o  f  , ] } cc' '  exp[ijtt] p , exp[-|itt] c  - 2irif t p [6(/ ) t / ] } c cc c o cc  (4.1.19)  1  8  >  - 54 -  since  p  ^ f o ' c c ^ c^cc^  - 'cc'Pc'^fo'cc'^ - ~  2 i r i  t  P  cc' c'  [ 6 ( /  'o  )  'cc'*  1  (4.1.20) for the r e s t r i c t e d case when CK ,P ,]_ = 0 and [ f t . t ^ ^ ^ t 0  For pure states, p  c  i  replaced by P  s  P  Wc'  c  c  =  0  \  and (4.1.14) becomes  = 1  " 'c'c^ c'  c c c' c' c  (4.1.21) This i s to be compared with the o p t i c a l theorem i s f a m i l i a r to us:  P , t , , - t*, ,P , c' c'c' c'c c' = P ,V , Z P (L , t ,) + P ,V ,P , c' c' c ecc cc c' c' c' c - {Z (A L  , t , ) * P V ,P , + P ,V ,P ,} c c c' c' c' r '  Occ cc'  Z c  (P ,V P c  c  c  +  P  c  " ^cc-'cc'^  J  CK -* ,)P c  ( P  V  c  P  c c c'  +  P  c f )  (J .t cc  (  c ^ c -«- '> c  ccf  P  )  C>  expressed i n a form which  - 55 -  Z cc"  jc  c' c c'  ^cc'cc'*  *  [tt  c  1  c c jcc cc'  V c W c - V c ^  P  c c ' c ^cc-'cc'* - ^ c c - ' c c ' ^  Vcc'l (4.1.22)  Equations (4.1.21) and (4.1.22) are equivalent since  t  Z c  ,P (U , t ,) - pc (k«cc' ,t ,) t ,P cc' c Jcc' cc cc' cc' c t  t  r  ,t tc ' cc'  z  c  p t , - p t ,(6 ,t ,) p] c cc' c cc' 7cc' cc' c  t* ,p a , - p Q , t* ,p cc' c c c c' cc' c t  ,  -  c' c cc'  z  c  p t ,n\-p ) c cc' c c  ( t  P  c'c' c'  + (p ,t t , c' c'c'  p it t 1 ? I ) c' c'c' c' p I t I ,p ,) c' c'c' c'  p n , - p n .^.v p c' c c c' c c' c c c  J^.V  - n^.p v n , + p v . S ^ . P  c' c c c  c c c' c' c  1-  = z<P ir-fi .n , = z i r (P n  c  c c c  c  c  c c c c  (4.1.23)  -  E q u a t i o n (4.1.22) i s concerned initial  channel  w i t h the t r a n s i t i o n o f a s t a t e i n the  c' to a l l a c c e s s i b l e f i n a l  w i t h e q u a t i o n (4.1.23) i s found superoperator l / "  to be e f f e c t e d by the p o t e n t i a l together.  theorem, e q u a t i o n s (4.1.21) and (4.1.22) are  to the o p e r a t o r form o f t h i s theorem.  Gain and Loss of Monomer Observables  I t i s found labelled and  comparison  c o n d i t i o n t o o b t a i n (4.1.19), the  on-the-energy-shell o p t i c a l  B.  channels which on  which couples a l l the channels  c  Under the s p e c i f i e d  reduced  56 -  that the a p p l i c a t i o n o f (4.1.5) r e g a r d l e s s o f the i n i t i a l  channel  r e s u l t s i n a r e l a t i o n s h i p between ( 3 / 3 t [ p f ( l ) ] } n  {9/<*t[pb(12)] }  c o  c o  n  which immediately  mass and by c a r e f u l examination, momenta.  l e a d s to the c o n s e r v a t i o n o f  c o n s e r v a t i o n o f l i n e a r and angular  However, to see how the s p e c i e s are c r e a t e d o r d e s t r o y e d by the  i n d i v i d u a l r e a c t i o n s , i t i s more a p p r o p r i a t e to use the o p t i c a l (4.1.13).  For the sake o f c l a r i t y , the implementation  equations i n v o l v i n g d i f f e r e n t i n i t i a l considered  (i)  theorem  o f these two  fragmentation channels a r e  i n separate s e c t i o n s .  2-Monomer C o l l i s i o n s The  result  2M  (4.2.1)  - 57 -  i s immediately o b t a i n e d from the r e l a t i o n  f  12^2  V  tf  ( 4  " 12 12-  2  * '  2 )  The s u b s c r i p t 2M i n d i c a t e s that two monomers are i n v o l v e d i n the collision.  S i n c e V\2 and 6j commute, the mass d e n s i t y f o r t h i s type  o f c o l l i s i o n i s conserved  [  =  kVm  T  \  m  - £  s  m  ih  [ p  f  ( 1 ) ]  2M  V12 W 1 2  = 0.  (4.2.3)  For these c o l l i s i o n s , the change i n the d e n s i t y o f a monomer o b s e r v a b l e i s due to the i n t e r a c t i o n between the c o l l i d i n g  [  l F *M < V 2 M = " ]  2rT  H^Ws  +  (  *2 V s  This p o t e n t i a l r e s u l t s i n an i n t e r m o l e c u l a r  ]  pair,  V n ^ W  ( 4  2  ' '  4 )  t r a n s f e r o f l i n e a r and  a n g u l a r momenta when c o l l i s i o n s a r e n o n - l o c a l .  (ii)  Monomer - Dimer C o l l i s i o n s For  t h i s type of c o l l i s i o n s , s u b s t i t u t i o n o f  1(23)  1(23)  123 123  c c  v  (12)3 (12)3  (13)2 ( 1 3 ) 2  ;  (4.2.5)  - 58 -  into  the a p p r o p r i a t e term i n the k i n e t i c  e q u a t i o n (2.4.21) f o r the  monomer d e n s i t y o p e r a t o r g i v e s  %  P  f  ( 1 ) ]  M D +  = - sr T r * 23  P  ( A  Tr 23  (  T  ^c c- ^(12)3 (12)3  <  ( J  2 Tr  P  L(23) 1(23)  Z i r  c  c*c-  2 f  P  b  c  J  )  (13)2 (13)2 ^  P  '(12)3 (12)3  +  v f V  p  (13)2 (13)2  )  7  H(23) 1(23)  €  >A  ( P  (12)3 (12)3}  P d2)]  £ T r IV li c  +  +  (^  +  JV/  P  (13)2 (13)2  +  P  ^(12)3 ( 1 2 ) 3  )  M + D  ( 2 3 )  P  1 ( 2 3 )  +  2^  1 2 ) 3  p  ( 1 2 ) 3  )  (4.2.6)  where the dimer-monomer term i n the k i n e t i c e q u a t i o n (2.4.25) f o r the dimers has been used collislonal  t o i d e n t i f y the  r a t e of change.  d e n s i t y changes are r e l a t e d  (12)3 terms as equal to the dimer  So f o r monomer-dimer c o l l i s i o n s , the mass by  - 59 -  9  9  [  "9T VM+D  =  [  - 9 r VM+D  2n 2i  1 £i  m Tr 12  2  f  -1  "  Z  ^  6  ±\2  J  M+D  1  2n  9  2n  Z  n=2  [ p  (12) (12)3 (12)3 l(23)  2i  £  +  P  (13)2  1 -12 -12 m VV : Tr 2 2 2 123 [ P  1(23)  +  +  P  (12)3  fig,  ]  T  J  ( 1 2 f (12)3 (12)3  P  (12)3  +  P  (13)2  ]  (4.2.7)  An e x p r e s s i o n i d e n t i f y i n g  the change a s s o c i a t e d with  individual  r e a c t i o n types may be d e r i v e d by r e c o g n i s i n g t h a t the n o n - r e a c t i v e t r a n s i t i o n s u p e r o p e r a t o r c o n t a i n s the r e a c t i v e l o s s terms  >  :r  ^ l(23) i(23),l(23)  l w  J  ^ 0 ^ 1 ( 2 3 )  U2)3 (12)3,l(23)  (13)2°(13)2,1(23)  123 123,1(23)  J  (4.2.8)  which g i v e s  [  U  P  f  (1)  WD  - 60 -  1  ^c ^c ( 2 3 )  * 23  +  ^(12)3^(12)3 +  Tr 23  ( E  1  V r c  +  +tf5  ^m ^  < ?  c c^l(23)  2 ( < ?  7  123 123  ]  T  f >  2 < P  P  { 13)2\ 13)2  l(23) l(23)  "  +  e  )  (P  7  (12)3  (12)3 (12)3  T  l(23) l(23)  )  l(23)  )  P  p  +  P  (13)2  }  l(23)  (12)3  (A.2.9) Thus, e x p l i c i t l y monomer 1 i s l o s t w h i l e monomer 1 i s gained The  (12)3  i n (12)3 •»• 123 and (12)3 •»• 1(23) c o l l i s i o n s .  r a t e o f change o f the d e n s i t y  o b t a i n e d by m u l t i p l y i n g  due to the r e a c t i o n 1(23)  o f an a r b i t r a r y monomer o b s e r v a b l e i s  t h i s e q u a t i o n by (<Pi ^1)s  a n c  * t a k i n g the  t r a c e over 1.  [  Z  V  r  c ^c( 2 3 )  + 2(tf  Tr 123  (  1  Ws  c  s  +  +  «Ws  3  1 2 3  3  +  1 2 3  i e  f >  <( 1 2 ) 3 " ( 1 2 ) 3 l ( 2 3 ) 3  : r  1  ]p  (23) i(23)  )  P  (12)3  Wl(23)  + 2[(* 6 ) 3  >  "  -  (  W  s  )  ^  ( 1 2 ) 3  2  ( 1 2 ) 3  1(23)  ^ V s ^ m ^ (4.2.10)  - 61  The  first  term  -  i n (4.2.10) i s analogous  to the c o n t r i b u t i o n to the  e q u a t i o n of change f o r 2-monomer c o l l i s i o n s , appears  and a term of t h i s  form  i n each c o l l i s i o n process t h e r e f o r e the t r a n s f e r of the  observables  other than the t r a n s f e r from the r e a c t a n t s to the  can be p i n p o i n t e d .  The  second  term g i v e s the g a i n and  o b s e r v a b l e s f o r exchange r e a c t i o n s :  physical  products  l o s s of monomer  the l o s s of a monomer molecule  the r e a c t a n t monomer (atom 1) becomes p a r t of the dimer (12) and of a product monomer molecule  as atom 3 emerges as a monomer.  g a i n of two monomers on decomposition  i s g i v e n i n the l a s t  channel  any  at _ r j ;  c o n s i s t e n t w i t h the p i c t u r e t h a t a r e a c t a n t molecule  the g a i n  The  net  term.  Moreover, when atom 1 i s p a r t of the dimer (12) ( f i n a l p h y s i c a l a t t r i b u t e associated with i t i s l o c a l i s e d  as  (12)3), this i s  i s annihilated  at  the p o s i t i o n of the r e a c t a n t i t s e l f , not at the p r o d u c t ' s , so t h a t  k  +  l  T  ( < 13  6  )  ( f  : r  P  *'l l s (12)3 (12)3 l(23)  d e s c r i b e s the l o s s of the r e a c t a n t monomer.  C o r r e s p o n d i n g l y , the  product  monomer should emerge at the v e r y s i t e i t i s produced; t h i s i s a l s o formally e s t a b l i s h e d i n equation  (4.2.10).  With the l o s s or p r o d u c t i o n of the monomer and a t t r i b u t e s associated with i t i d e n t i f i e d second  way  collision  of a p p l y i n g the sum  f o r each c o l l i s i o n  (i.e.,  However, the equations  type,  of r e a c t i o n s on the f l u i d  the  a c t i n g on the n o n - r e a c t i v e  of change f o r M^  e q u a t i o n (4.2.6) i s more convenient  effect  physical  s u p e r o p e r a t o r ) p r o v i d e s a more complete d e s c r i p t i o n of  collisions. from  rule  the  obtained  to use f o r s t u d y i n g the  dynamics of the system.  the  overall  - 62 -  (iii)  3-Monomer C o l l i s i o n s  The same kind o f m a n i p u l a t i o n s as i n the p r e v i o u s s e c t i o n the  leads to  results  2 T r [ | - p,(12)], 9  1  [  V  7t  3M  [  M  t  ]  " at D 3 M  +  b  - ^—  M  3M  2i ^ t  mV  V  2Ti  Tr  I V f «^  o  I23 123, P  1  :  2  3  v  c  2!  2  2  T  (4.2.11)  • • • -J  123  (12)3  <f>  T  P  (12)3 (12)3 123  (4.2.12)  and  the e q u a t i o n o f change  [  7t M < V l M - - - K M  T r  3  123  123  -!<•,«,). - (•JV.I ( , 2 > 3 ^ F  IK  Tr j [ ( * 6 ) 123 1  1  s  +  (* 6 ) 2  2  s  1 2  )3  +(* « ) ]i:u- P ^ c 3  3  8  c  c  123  (4.2.13)  63  -  -  T h e r e f o r e , the a p p l i c a t i o n o f formal k i n e t i c t h e o r y a g a i n s u c c e s s f u l l y shows that the r e c o m b i n a t i o n r e a c t i o n , the monomer a t t r i b u t e w i t h the r e a c t a n t monomers, h e r e , the atoms w i t h l a b e l s  associated  1 and 2, would be  d e s t r o y e d as a r e s u l t .  (iv)  Dimer-Dimer C o l l i s i o n s There cannot be any l o s s o f monomers when two dimer  participate  lt  [  molecules  in a collision:  21 P  f  ( 1 ) ]  2D  1 2  ^ 23  7  ( p  (34)' 12(34) (12)(34)  +  2 p  (13)(24)  )  (4.2.14)  g i v i n g the c o r r e s p o n d i n g e q u a t i o n o f change  hTF *M < V J D 2  =  " If  +  " Ws  ( P  (12)(34)  (  *2 W  +  2 P  ^12(34) ^12(34)  (13)(24)  )  or to express i n a form c o n s i s t e n t w i t h the e q u a t i o n s i n the p r e v i o u s sections,  [4t^<VJ2D  =  -* ' T  [  (  W s  +  ( < t >  )  2 2 s 12(34) i2(34) 6  ] f  : r  1234  P  +  2  t(<P <5 ) 1  1  + s  (12)(34).  <Ws^l4(23) 14(23) 7  (4.2.15)  I t i s a l s o p o s s i b l e to express the time e v o l u t i o n o f P f ( l ) due to 2-dimer c o l l i s i o n s i n terms o f t h a t o f Pb(12)  - 64 -  lt  [  P  f  ( 1 ) ]  21 3 h . ! , 1 2 ( 3 4 ) ^12(34) 1234  2D  T  t<P  (P 2i 3* ^  [ 4  ( J  (12X34)  +  3  ^ / c  2 P  +  ^13(24)^13(24)  (13)(24)  f  P  -2 T r [ ^ P d 2 ) ] b  "^^34 r ^  1  +  J  1  ^14(23) 1 4 ( 2 3 )  )  " < (12)34<>(12)34  \ l 2 ) ( 3 4 ) (12)(34)  +  2>A  +  f  ( 12)(34) ^ 1 2 ) ( 3 4 ) > J  P  (13)(24) ( 1 3 ) ( 2 4 )  )  2 D  2  ^  3  ^  P ( 1 2 ) ( 3 4 )  +2  p  )  ^13)(24) (13)(24) ' (4.2.16)  (v)  Monomer-Monomer-Dimer C o l l i s i o n s The c o n t r i b u t i o n  the  from c o l l i s i o n s  i n v o l v i n g a dimer and 2 monomers t o  k i n e t i c e q u a t i o n f o r the monomers can be w r i t t e n i n the a l t e r n a t i v e  forms  [  7F  P  f  ( 1 ) ]  2M+D  ^ Tr 3n 2  3  4  Z c  \T <P  c c  SV  + 2 , n  "  2  r [ 2  lt"  P  b  ( 1 2 ) 1  P  12(34) 1 2 ( 3 4 )  P  +  ( 1 3 ) 2 4 (13)24  2M+D  p  ^(12)34 (12)34  +  2v/  13(24)  P  13(24)  - 65 -  =  * Tr 234 c  c c 12(34)  " (12)34  3  +  J  +  (12)34  +  ^(12)(34)  2(J  13(24) 13(24)  +  2<P  2IP  (13)(24) (13)(24)  [ P  ft 234 ^ 1 2 ( 3 4 )  7  (12)(34)  7  (13)24  P  12(34)  (13)24  J  3  [ P  12(34) (12)34  +  2 p  +  13(24)  2 p  ]  (13)24 * (4.2.17)  From the second  form, the g a i n and l o s s o f a monomer o b s e r v a b l e a r e  e x p l i c i t l y g i v e n by  lt  [  >  " M <*M 1 M D 2  +  i Tr W 1234  S V  ^ • i V . ^ W . ' +  2[(* VB "  (  W s  4  [  (  i  ^  W s V  s  ]  +  W*2<34)  ^(13)24^(13)24  <* W> 2  ^(12)34^(12)34 '12(34)'  (* 6 ) ] 2  +  2  S  ^(12X34) ( 1 2 ) ( 3 4 ) 3  +  2  :r  ^(13)(24) (13)(24)  (4.2.18)  - 66 -  The exchange r e a c t i o n 12(34) + (13)24 b r i n g s about the g a i n of the monomer 4 which i s compensated by the l o s s of monomer 1 so t h e r e Is no net  change i n the number of monomers.  break-up and recombination number of monomers. monomer molecules  L i k e w i s e , the  simultaneous  12(34) + (12)34 r e s u l t s i n no change i n the  The l a s t b l o c k of terms r e f l e c t s the l o s s of the  when recombination  ( w i t h or without  exchange) takes  place.  C.  Gain and Loss of Dimer Observables  A f t e r the a p p l i c a t i o n of the o p t i c a l theorem (4.1.13) the expected g a i n and l o s s terms f o r d i f f e r e n t k i n d s of c o l l i s i o n s a r e o b t a i n e d . Since the m a n i p u l a t i o n s  and p h y s i c a l meanings are the same as f o r t h e  monomers, the equations  of change a r e l i s t e d w i t h no d e r i v a t i o n shown,  and  except  for section  (i),  no account  on the meanings of the terms  d e r i v e d w i l l be g i v e n .  (i)  Monomer-Dimer C o l l i s i o n s  6  )  Z  (12) (12) s  U  c  c c (12)3  6  )  (12) (12) s p J  1(23) l(23)  "  (  6  )  *(12) (12) s  (12)3  123^123  (4.3.1)  - 67 -  A f t e r a chemical  r e a c t i o n has  taken p l a c e , the i n i t i a l dimer (12)  l o n g e r e x i s t s , thus t h e r e i s the d e s t r u c t i o n of the r e a c t a n t dimer  +  h  T  r  )  ^12) (12) s  (  5  3  ( P  cV(12)3'  C  ( 1 2 ) 3  *  (12)  *  For the exchange r e a c t i o n (12)3 + 1(23), the g a i n of the dimer (23) the m o l e c u l a r i s observed  (ii)  no  a t t r i b u t e a s s o c i a t e d w i t h t h i s dimer i s expected  and  and this  i n (4.3.1).  3-Monomer C o l l i s i o n s [  h  (iii)  [  W 3M ]  =  ~  2S 5 * ( 1 2 ) ( 1 2 ) s T  (  6  )  ( 4  ^(12)3^(12)3^23  3  3  ' '  2 )  2-Dimer C o l l i s i o n s  3t  V V  r Tr *1234  T  +  I  ]  (  2 D  W<12>>s  ( [ (  6  2 ( (  6  (  )  +  (12) (12) s  6  )  *(13) (13) s  P(13)24  "  )  *(13) (13) s  [ ( , f )  6  J  (  +  +  (  ( < t  '  ]  *(34)Ws  6  1  )  *(24) (24) s 6  ]  '(34) (34) s  ] )  [ ( <  )  >  )  6  *c*A (12)(34)  ' (12) (12) s  " f>  +  !1  (13)(24) (13)(24)  (  6  )  *(34) (34) s  P  (12)(34)  1 )  7 (13)24  )  *(12) (12) s  6?  7  12(34) 12(34)  (4.3.3)  - 68 -  (iv)  Monomer-Monomer-Dimer  [  WW  9F  TfT  Collisions  Tr 1234  ( t  +  (12) (12) 8 6  ( < | )  ^0^(12)34 c  )  6  )  f  (34) (34) s  2 ( l (  6  )  *(13) (13) s  *(13)(24)  ( <  J  (12)(34) (12)(34)  )  6  J  +  (  5  )  *(24) (24) s  4 [ (  6  "  ( < f >  5  )  (12) (12) s  p  )  ( 12)34  (13)(24)  )  (  6  )  + [ f (34) (34) s ~ * ( 1 2 ) ( 1 2 ) s  +  ]  )  *(13) (13) s  _  (  t >  6  )  ' (12) (12) s  ]  3  P  12(34) 12(34)  7  P  (13)24J(13)24  (4.3.4)  On comparing for is  the c o l l i s l o n a l  c o n t r i b u t i o n s to the e q u a t i o n s  o f change  the p h y s i c a l a t t r i b u t e s a s s o c i a t e d w i t h the monomers and dimers, i t found  loss  t h a t f o r a g a i n o f a dimer m o l e c u l e ,  term f o r the monomers, and a l o s s  g a i n f o r the monomers. processes.  t h e r e i s the c o r r e s p o n d i n g  f o r the dimers i s coupled  Thus mass i s conserved  for individual  with a  collision  -  69  -  CHAPTER 5 HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES I: NON-REACTIVE COLLISIONS  In the preceding chapter, e x p l i c i t expressions for the c o l l i s l o n a l gain and loss of monomer and dimer molecules and the physical observables associated with them have been derived.  In this chapter, for the  p a r t i c u l a r attributes of mass, l i n e a r momentum, and angular momentum, the c o l l i s i o n a l contributions to the appropriate hydrodynamic equations are obtained.  The energy balance equation w i l l be discussed i n Chapter 7  since i t requires a more elaborate formulation.  This i s because the  intermolecular potentials are not single-molecule observables so their time evolution cannot be described by the kinetic equations.  For the equations of change for mass, l i n e a r and angular momentum d e n s i t i e s , the treatments of non-reactive c o l l i s i o n s i s f a i r l y straightforward so they are presented f i r s t .  An immediate result i s that  for non-reactive c o l l i s i o n s , the mass densities of the dimer and the monomer do not change.  As i n the previous chapter, d i f f e r e n t c o l l i s i o n  types are treated i n separate sections.  A.  C o l l i s i o n s Involving 2 Monomers  The rate of change of the density of a single-molecule property f o r t h i s c o l l i s i o n type i s given by  - 70 -  (5.1.1)  Since d e l t a f u n c t i o n s commute with the p o t e n t i a l energy o p e r a t o r , the mass d e n s i t y change i s zero.  For the d e n s i t i e s o f l i n e a r and angular  momenta, the c o l l i s o n a l change i s due to the t r a n s f e r o f the r e s p e c t i v e a t t r i b u t e from one monomer to the o t h e r as (<J>i + <j>2) i n (5.1.1) commutes w i t h the p o t e n t i a l  f°  r  =  a n  $i  _Pi  =  ^ $1 Jj.•  For momentum d e n s i t y change, t h i s i s g i v e n i n terms o f the momentum f  l  u  x  £ M 2  6 -6  riL l  ul = - Tr 9t ^1- 2M M.  —  J  •  "  v  * £  where P_„, the pressure =2M  —  -  2  2  (V 1  2  V 1  ) <f  2  1  2  1  vA/ 2  1  p  2  (  M  5  *  1  *  2  )  tensor due to 2-monomer c o l l i s i o n s , i s d e f i n e d by  2n+l S M=2  T  r  12  <  6  (V v  ( 1 2 ) ¥  V  )  12 12'  +  tf  V n=l P  (  12 12^12  M  )  !  r  l  T  2  '  v  2  n  6  (12)  )  .  (5.1.3)  - 71 -  This  pressure tensor gives r i s e  to an angular momentum d e n s i t y  when the p o t e n t i a l has s p h e r i c a l it  i s the s o l e  f l u x , and  symmetry o r depends o n l y on p o s i t i o n s ,  f a c t o r f o r the change i n angular momentum  A s p h e r i c a l l y symmetric p o t e n t i a l  density.  commutes with the atomic angular  momentum  [  - i '  likewise  V  (  r  i j  )  ]  -  =  °»  f o r a p o t e n t i a l V i j that  depends o n l y on the r e l a t i v e p o s i t i o n  between I and j , so that  = ? • (P  In g e n e r a l the p o t e n t i a l  2 M  x r).  i s not s p h e r i c a l l y symmetric and depends on  f a c t o r s such as s p i n o r the c o u p l i n g translational  (5.1.4)  between j s ^ ' s , so i n t e r n a l and  angular momenta a r e not i n d e p e n d e n t l y conserved.  of change o f t r a n s l a t i o n a l a n g u l a r momentum d e n s i t y  consists  The r a t e  o f a term  which corresponds to a n e t t r a n s l a t i o n a l angular momentum change and that  which i s due to the torque between the c o l l i d i n g  [|t ( £  x  \ii)] M = 2  V  * <£ M 2  X  i> - H ^ T ^ 1 2  X  pair  V  V  )  < 12 12 1 2 ^ 1 2 1 2 f  P  (5.1.5)  - 72 -  and  t h i s i s compensated by the change i n i n t e r n a l  =  2S  V  '  H^ 6  (  "'> ^ r l  +  \l2)  ) 2  angular momentum  P  p  ^12 1 2 ^ 1 2 1 2  + 6  i l 2 - ± T r ( — - — - — — - ) IT * 12 2 2  1  2  f 1  2  Jl/ p 1  2  1  2  (5.1.6) to g i v e a t o t a l a n g u l a r momentum c o n s e r v a t i o n law  =  V  (P  • =2M  X  +  Ifi  V  *  T  r  (  2  ¥  6  (12)  +  ->  i  ) 2  ^12 ^12^12^2-  (5.1.7)  Equations  (5.1.5-7) show t h a t i t i s the s p a t i a l inhomogeneity  c o l l i d i n g molecules  t h a t causes the i n t e r c h a n g e o f i n t e r n a l  momentum and t r a n s l a t i o n a l their  free  angular momentum.  f o r example, Chapter  w i t h time are independent  the i n t e r n a l  and t r a n s l a t i o n a l o f each o t h e r ( s e e ,  Another p o i n t to note i s  angular momentum f l u x i n (5.1.6) has the form  the d i s t r i b u t i o n o f the atomic molecules.  are i n  3 o f t h i s t h e s i s ) , so i t i s v i a c o l l i s i o n s t h a t the  angular momenta t r a n s f o r m i n t o one another. that  angular  When the m o l e c u l e s  f l i g h t , however, the changes o f i n t e r n a l  angular momentum d e n s i t i e s  o f the  a k i n to  a n g u l a r momentum over the two c o l l i d i n g  - 73 -  F o r n o n - r e a c t i v e c o l l i s i o n s i n v o l v i n g a h i g h e r number o f atoms, the contribution like  to the e q u a t i o n o f change  f o r l i n e a r o r angular momentum,  the monomer-monomer c o l l i s i o n s , i s due to the t r a n s f e r o f the  attribute  from one p a r t n e r to the o t h e r s .  the i n i t i a l  Consider an n-atom  collision,  channel g i v e n by the l a b e l c', the change i n the monomer  hydrodynamic d e n s i t y o f l i n e a r momentum (<pj = jy±) o r angular momentum (<Pi = Jj.) i s , on s u p p r e s s i n g the numerical  c o e f f i c i e n t due to  c o u n t i n g , as f o l l o w s :  n-atom,c',no r x t  •h  Tr 1.. .n  +...+<P ) 6 j ] n  -  g  [(<p +...+<p ) fij] 2  n  (5.1.8)  (here c' i s a channel contribution (<|>,  i n which 1 i s f r e e ) and f o r the dimer, t h e  to the hydrodynamic e q u a t i o n f o r l i n e a r momentum  > = p, + p.) or angular momentum  = J_i + J_.) i s  - 74 -  l  3t  D  J  F n  atom, c", no r x t  = - :=• Tr [(<().+(()-) 6 , J V- ..P..«^..p . n . 1 2 (12) s c c c c i • • •n 1 0  Tr [(*3 i.. .n  (here c" c o n t a i n s the dimer  ...  +  +  (12)).  *) n  «  (12)  ] vr ..P ..J...P „ s  c  c  c  The commutativity of t o t a l  a n g u l a r momentum of the system w i th the p o t e n t i a l has been used (5.1.8) and ( 5 . 1 . 9 ) . p o t e n t i a l s , and  Let V  t  o  t  a  i  be the sum  (5.1.9)  l i n e a r or to o b t a i n  of a l l i n t e r - a t o m i c  l e t c' be a channel i n which ( i j ) , ( i ' j ' )  . . . are bound  as d i m e r s , then  I*!  +  +  V  V  J _  C  = [<fr + ... + 4, 1  v  n  Uj  +  + ...  ]  total —  + 4>, v  [<j> + . . . 1  ]_  N  + <|> v n >  l t  ,]_  +  = 0  where  (5.1.10)  = j ) ^ , or  localisation  = J_^.  From e q u a t i o n s (5.1.8-9), we  see that  the  scheme t h a t has been adopted here c o r r o b o r a t e s with the  p i c t u r e that c o l l i s l o n a l t r a n s f e r i s the t r a n s f e r of the o b s e r v a b l e one i s concerned w i t h from the c e n t r e of mass of one molecule to the c e n t r e s of mass of i t s c o l l i s i o n p a r t n e r s .  - 75 -  B.  C o l l i s i o n s Involving a Monomer and a Dimer  The  r a t e o f change i n the d e n s i t y o f a p h y s i c a l observable o f t h e  molecules  from a n o n - r e a c t i v e monomer-dimer c o l l i s i o n i s  +  VV^M+D.no  [ [  (  3  123  LK  *1  +  rxt  )]  *(23) s  <P  +  [ ( 6  6  1 ~ (23)  )  ]  3  s'  P  1(23) 1( 2 3 ) ^ ( 2 3 ) l ( 2 3 ) (5.2.1)  There a r e d i f f e r e n t ways o f d i v i d i n g has  the terms up, but the one used above  the advantage t h a t the term on the l e f t may be i n t e r p r e t e d as the  c o n t r i b u t i o n to the e q u a t i o n due to the c e n t r e o f mass motion, and as t h e 6 - f u n c t i o n s on the r i g h t g i v e s the r e l a t i v e p o s i t i o n between the molecules, colliding  t h i s i s the change due to the r e l a t i v e movement between t h e molecules.  For the monomer, the change o f  J  hTt **M — M+D,no r x t  6  ~ 123 l  ( V l  V  <23) l(23)  ) < P  t /  P  l(23) l(23) l(23) (5.2.2)  (with J ^ j k )  =  +  - ^ i ~ "2 ^ — j  —k^ being the p o s i t i o n o p e r a t o r  f o r the p a i r  -  (jk)  relative  76 -  t o i ) i s compensated by the change i n the dimer momentum  density  a [  lt  ]  =  **D - M+D,no r x t  ^3  6  (V  V  ( 2 3 ) 1(23) 1(23)  )  f  1(23)^1(23) l(23) * P  (5.2.3)  The r a t e of change of the d e n s i t y of t o t a l momentum f o r t h i s c o l l i s i o n can be expressed  £M+D no r x t ' collision  [-5F  ( M  3  t  e  r  m  a r i s i n  &  type o f  as the divergence o f the p r e s s u r e t e n s o r from the i n t e r m o l e c u l a r f o r c e between t h e  partners  V  M H. D ^ M D , n o r x t " " +  M  )]  +  * S D,no r x t '  f- )  n  2  Izj  ( 5  M+  -1  n  n  —1 f23">  _  1  2  ' '  4 )  1 1 - 1  n=Z (V  V  )  p  1 ( 2 3 ) 1 ( 2 3 ) ^1(23) ^ 1 ( 2 3 ) l ( 2 3 ) (5.2.5)  where R i s the c e n t r e of mass of the c o l l i d i n g m o l e c u l e s , and f o r collisions  involving  atoms i , j and k, R -  knowledge i n i r r e v e r s i b l e a n g u l a r momentum f l u x  identified  from  (r_^ + r_j + r^.)/3.  thermodynamics, we expect a  (-P„._. ^ x r ) and t h i s =M+D,no r x t —  translational  term i s a c t u a l l y  the e q u a t i o n showing the c o n s e r v a t i o n o f t o t a l  momentum f o r t h i s  From our  J  angular  c o l l i s i o n type.  From the equations f o r the changes In the angular momentum  densities  f o r the i n d i v i d u a l  species  3 [  ~o~t V^I^M+D.no r x t  v 1  TT J  2  r 2  u  -ri's  3  3 Vl [  1(23)^1(23)1(23)^1(23)  <£i  +  x  J>lVs  U  5  ^l(23) l(23) l(23) l(23)  ]  P  2  6  < ' - >  p  ^W'wD.no r x t P  -4^ (V23) (23)^^l(23)n(23rt(23) l(23) 5  3  {  m "  2 +  V r  -  3  K23)  +  < P  "  2  3  X  ^  v /  2  3  >  6  (  2 3  >  +  K £  (23)  *  W (23)U 6  p  l(23) l(23) l(23) (5.2.7)  with  the changes i n i n t e r n a l and t r a n s l a t i o n a l  being  angular momentum  e x p l i c i t l y g i v e n , the c o n s e r v a t i o n r e l a t i o n f o r t o t a l  angular  momentum  fit % ^ + V ^ V D . H O rxt 2  i =  T  ~* U3  L  l i ~ (£2+13) 3  (  6  V (23)^s^l(23/l(23)^(23) l(23) p  densities  - 78 -  (—p x r ) + L. =M+D,no r x t — ^M+D,no r x t  = -V  (  " ^ 3 il(23)"°  6  RlfT  (  2  2  ^r « - ^3  )  V  2  P  1(23)^1(23)^(23) 1(23)  (5.2.8)  f o l l o w s immediately.  As (-P„ _ x r ) i s the t r a n s l a t i o n a l =M+D,no r x t —  momentum f l u x , the r e s t must be the i n t e r n a l  =M+D,no r x t  , the m o l e c u l a r  angular momentum  angular momentum  angular  flux.  flux  =M+D,no r x t  3  7  1  123  (£i(23) K  +  , }  }  6  R-23  X  ( V  V  23 1(23)  )  v A  P  ^l(23) l(23) l(23) (5.2.9)  w i l l be the o n l y c o n t r i b u t i o n  t o the i n t e r n a l angular momentum  flux  a s s o c i a t e d w i t h monomer-dimer n o n - r e a c t i v e c o l l i s i o n s when the p o t e n t i a l depends e n t i r e l y on p o s i t i o n s .  To see how t r a n s l a t i o n a l  and i n t e r n a l  a n g u l a r momenta transform i n t o one o t h e r , the changes i n the d e n s i t i e s are c o n s i d e r e d s e p a r a t e l y .  i T F * * ( " H » N D ^ W H O rxt +  = -V •  (-p  2 6  ~ 123  w  =M+D,no r x t 1  +  6  3  x r)  —  (23)  -K23)  X  ( V  V  1(23) 1(23)  )f  v/  P  l(23) V(23) l(23) (5.2.10)  - 79 -  [  8T  'V^int*  +  V^D,int  (26 +6 * 123  t  > ) ]  M f D . n o rxt  ) +  (l 2  3  1  + 2  l  + 3  X  i23 -£23  ) ]  P  ^(23)^1(23)^1(23) l(23) ( 6  6  r ( )  )  2 3  - V •L =M+D,no r x t  (25  1 +  6  ( 2 3 )  )  Tr 123  (  }  -1(23)  X  ( V  V  1(23) 1(23)  )  - V i + £ t  6  ( 6  1  _ 6  (23) 3  (  < a  2  3  ' =M+D,no r x t  )  )  (B 2s -2s )]Ur  2  3  i ( 2 3 )  P  l(23) i(23) l(23) (5.2.11)  T h e r e f o r e the i n t e r c h a n g e of the i n t e r n a l and t r a n s l a t i o n a l momenta o f the c o l l i d i n g  pair  i s effected  angular  by the torque between the  molecules.  C.  C o l l i s i o n s Involving 3 Monomers  The  form o f the e q u a t i o n o f change f o r t h i s c o l l i s i o n type i s  -  80 -  - ( 6 . + 6„ + 6 , ) 3M+3M  6Ti  Tr 123  (  W*3 s  + [(«  +  [(6  )]  2  1  ^(6  2 +  63))  (2* -# -* )/3] 1  2  3  8  123 123 123^123  (5.3.1)  Since t o t a l l i n e a r  and a n g u l a r momenta commute w i t h the p o t e n t i a l o f t h e  system, c o n s e r v a t i o n o f these q u a n t i t i e s f o l l o w s  immediately.  Same as b e f o r e , t h e l i n e a r momentum d e n s i t y change can be expressed i n terms o f a p r e s s u r e t e n s o r due t o t h e f o r c e s between t h e c o l l i s i o n partners  3t  —^3M+3M  V  * S3M+3M  (5.3.2)  where P i s t h e c o n t r i b u t i o n t o t h e p r e s s u r e t e n s o r from t h i s type o f =3M>3M collisions  E3M+3M  -  \- Tr 123 6  6  ( V  (23^23  I  +  1  (  2  V  23 123  n  +(|)  -  1  n  )  n  )  +  6  -  81  ( V  2n-l . _£__ - l - | 3  (n!)"  1  V  R^1(23) 1(23) 123  2n-2 2n-2 . 6 v  [I ( £  n  + r  ( 1 2 ) 3  )  ( 2 3 )  n ( n ) 2  (V  2 3  V  1 2 3  ) - (-  )  r  n  1 ( 2 3 )  ) ]  n=2 n-1  n-1 V  9  123  (r 6  V  V  V 1(23) 123  )  v/y p 123 123  1 2  M  ...) 6  +  ( 1 2 )  (V  1 2  V  1 2  ) + (r  1  3+  ...) 6  ( J  3  )  ( V ^ )  123 +  (  ^23  >  %/  +  —  6  > (23)  ( V  V  23 23  )  P  ^ 123 H23 123* (5.3.3)  The second e x p r e s s i o n f o r P_.. ... i n d i c a t e s that i f the m o l e c u l e s i n t e r a c t =3M->"3M o n l y v i a p a i r p o t e n t i a l s and i f there i s no r e a c t i o n t a k i n g p l a c e , the system o f t h r e e m o l e c u l e s c o l l i d i n g  t o g e t h e r behaves as t h r e e s e t s o f  d o u b l e t s whereby each p a i r i s independent from the o t h e r s .  T h e r e f o r e the  e q u a t i o n s o f change  collision  type have  f o r angular momentum d e n s i t i e s f o r t h i s  s t r u c t u r e s which are analogous to those f o r monomer-monomer  collisions:  - 82 -  lat  X  ^  "M-^SIMM  -  (  * 2 M+3M  V  3  5+6  —  x  I)  6+6  £  1  2  x (V  1 2  V  1 2  ) + -  T  -  r  n  x (V  1 3  V  1 3  )  123  6  6 +6 2  - T - ^ 2 3  +  123  X  (  V  23 23 V  )  123^123 (5.3.4)  C  (  >:)1  9 t V4l,int 3M*3M 6 -6  -1 2r7 ^ T  3  "V  ^ " V  2 ,  ^12^123^123^23  6.+6, *  +  m  ^ "  £  1 2  X  ( V  V  12 12  )  e  123^23 123 P  (5.3.5)  and  the e x p r e s s i o n s can be i d e n t i f i e d i n the same manner as p r e v i o u s l y .  D.  Collisions Involving 2 Dimers  For  an a r b i t r a r y dimer p h y s i c a l  collisions results  l  M  < $  > ]  o b s e r v a b l e <<pn> t h i s type o f  i n a change i n <$n> a c c o r d i n g to  9t D D 2D,no rxt  - 83 -  6 r  (12)  [ 2  (  (34) (  2  *1234  L  + 6  *(12)  f  + < ( ,  ) ]  (34) s  /  +  , ^(12)~ [ (12)- (34) 2 ( 6  6  < ) )  )  (34)  )  ]  1  s  P  ^12)(34) (12)(34)' Vl2)(34) (12)(34) (5.4.1)  This c o l l i s i o n  type can be l i k e n e d  to monomer-raonomer c o l l i s o n s :  a r e n o n - r e a c t i v e processes i n v o l v i n g c o n t r i b u t i o n s to the e q u a t i o n s  two e q u i v a l e n t m o l e c u l e s .  of change f o r l i n e a r  both Their  and t r a n s l a t i o n a l  a n g u l a r momentum d e n s i t i e s a r e analogous to each o t h e r .  For the dimers,  they a r e  [  3t  M  ]  D - 2D,no r x t  V • Tr 1234  -(12X34) 2  ( V  V  V (12)(34) (12)(34)  ;  2n+l +  Z n-1  < V  [(2n l)!]"  1  £  +  <  1 2  >(  3 4  >  2  V  (12)(34) (12)(34)  2  "  6  D R  )  P  *( 12) ( 3 4 ) ^ 12) (34) ( 12) (34) V •P =2D,no r x t (where r . = r, - r,, i s the o p e r a t o r f o r the r e l a t i v e —(ij)(kl) -nij) —(kl)  (5.4.2)  -op-  position  o f ( k l ) to ( i j ) , and P i s the c o n t r i b u t i o n to the —z IJ j no rx t  pressure  tensor from t h i s kind o f c o l l i s i o n s ) and  x  [|t < £  "  V  %1^2D,no  (P  ' =2D,no r x t  6  _ X  rxt  I>  X  + 6  (12) (3A) ^12)(34) 234 2 2  X  Q V  V  (12)(34) (12)(34)  ;  P. vrt> . . . (12)(34) (12)(34) (12)(34)* P  H  (5.4.3)  The  change i n the i n t e r n a l  counterpart  angular momentum d e n s i t y i s s i m i l a r  f o r monomer-monomer c o l l i s i o n s although  additional  f l u x term g i v e n r i s e by the m o l e c u l a r  to i t s  now there i s an  angular momentum of the  molecules  =2D,no r x t  = - I  (  ^  £  (  1  2  2 °  4  )  +  -"  ) 6  R^12  X  V  V  X  12 (12)(34)-^34  V  V  3 4 ( 12) (34) >  P  ^(12)(34)^12)(34) (12)(34)(5.4.4)  a M  hit D ^ D . i n t ^ D . n o r x t  _ i - " TT  6 T  t  r  , 1234 T  r  6  (i2)- 34) 2 (  < ?  y y y - g * 2  P  ^ _ (12)(34) V  (12)( 4rtl2)(34) (12)(34) 3  v  .  L  ^2D,no r x t  6 +  Tr 123A  +6 r  (12)  (34) ( 1 2 ) ( 3 A ) X  2  ( V  V  (12)(34) (12)(34)  )  M  (12)(34)  (12)(34) (12)(34) (5.4.5)  K.  Collisions Involving 2 Monomers and 1 Dimer  For t h i s c o l l i s i o n p r o c e s s , we have  [  3T  ( M  -i 2-fi  M<V  +  VV  6  Tr 1234  K  l  +  6  ) ]  2 M D , no r x t +  2  2  +  + [(—2—  «r«  4>i+4>  +  +  2  5  - «  ( 3 4 )  •(34), 2 ! J  °(34)'  )  2  ]  s  2  p  12(34) 12(34) 1 2 ( 3 4 ) 1 2 ( 3 4 ) (5.5.1)  The to t r e a t  c o n s e r v a t i o n laws f o l l o w d i r e c t l y from (5.5.1) but we might wish the system as three d o u b l e t s as b e f o r e .  motion, f o r i n s t a n c e , can be expressed  The e q u a t i o n s o f  i n a manner which  parallels  - 86 -  ( 5 . 5 . 1 ) , o r In forms which show how  the molecules  interact  pairwise,  i.e.,  no r x t  y  V«2  Tr 1234  [  6  ( V  2  V  (12)(34) (12)(34)  )  6  r 2  " V ^  +  ( V  V  12 (12)(34)  ) ]  12(34) 12(34)^12(34)  — 2  Tr 1234  6  V  1  1(34)  ( V  13  + V  14  )  +  6  V  2  2(34)  ( V  23  + V  23  )  6 -6 1 2 + — — - (V V ) 2 12 12 v v  v  ;  (P P  12(34)^2(34) 12(34) (5.5.2) and  M  3  f 9 t D — 2M+D,no r x t  = T 2  =  1 2  Tr -  2 ^  3 4  U  6,„.,(V (34) (12)(34) (12)(34) V V  6  (34)  V  [ V  1(34)  J1  ( V  13  + V  14  ) +  p  ^12(34)^2(34) 12(34)  V 2  (34)  ( V  23  + V  24  ) ]  P  ^12(34) 12(34) 12(34)  giving  )  (5.5.3)  - 87 -  [  9tT  »D2)]2M+D,no  Accordingly,  - 4  T r  the presure t e n s o r P  6  1234  (V  V  rxt  V  (12)^12 12 12(34)  )+  6  2  M  +  D  £2M+D,no  >  n  ^  o  (5.5.4)  rxt*  can be w r i t t e n  (V  i n two forms  V  R^(12)(34) (12)(34) 12(34)  )  2n+l oo  + 2  2n .  ^. 1  Z  2  [(2n+l)!]~ (Zl -)  2n 6,  V  *(V V  V  )  V  (12r 12 12(34)  2  n-l  ;  2n Z n=l  [(2n-l)!]  1  2n-l •  =ii  2n-l V .6 (12)  1  2n+l +  Z  [(2n+l)!]  2  -d K34)  - 1  ( V  .  v  fi)  z  n=l V  (12)(34) (12)(34)  V/1  R  )  P  'l2(34) 12(34) 12(34)  6  1234  V  '^134) ( 1 3 4 ) 1 ( 3 4 )  +  6  (V  V  (12) ^12 12 12  ( V  13  + V  14  )  +  6  V  (234)^2(34) 2 ( 3 4 )  )  2n+l 00  +  E n=l  r [(2n+l)!]  V/  P  - 1  (=|1)  12(34) 12(34) 12(34)  2n  2n  • V  5 (  x  2  )  (  V  l  V 2  12  )  ( V  23  + V  24  )  88  -  1234 n=2  l-(-2) n!  n  n  -1(34)  _  -V  -  1  n  -  1  V  *  6  (134)  6  (234)  V  V  +V  1(34)< 13 14>  n r_„^._,v +  "V  n-l  L  *  n-l V  V  2(34)  ( V  23  + V  24  )  P  12(34)^2(34) 12(34) (5.5.5)  where r . . . , . = ( r . + r , + r, )/3 i s the p o s i t i o n o p e r a t o r f o r the c e n t r e —(ijk) — i - j —k of mass o f atoms i ,  j and  k.  To break up the c o l l i d i n g  system i n t o a s e t of i n d i v i d u a l  i n t e r a c t i o n terms has the advantage collisions  bimolecular  that as f a r as n o n - r e a c t i v e  are concerned, when the s t r u c t u r e s of the f l u x e s due to  b i m o l e c u l a r c o l l i s i o n s a r e known, t h e i r c o u n t e r p a r t s f o r more  complex  c o l l i s i o n s can be o b t a i n e d immediately because the l a t t e r can be d i v i d e d into  terms, each f o r a p a i r o f m o l e c u l e s , which a r e comparable  f l u x e s f o r analogous b i m o l e c u l a r c o l l i s i o n s . and  to the  So i t f o l l o w s from  (5.1.5)  (5.2.10) t h a t the t r a n s l a t i o n a l angular momentum change f o r t h i s  collision  [f^r  type i s  x (Mj, u + M,, v ) ]  2  M  = V • (P x r) =2M+D,no r x t —  +  D  >  n  o  t  x  t  - 89 -  j  Tr 1234  6  ^ l  + 6  (34)  3  ^1(34)  26 +6 2  +  6  l  V  ^2(34)  + 6  1(34)  ( V  13  + V  14  )  ( 3 4  —  X  V  2(34)  ( V  23  + V  24  )  2  -T-^12  +  X  X  ( V  V  )  12 12  P  12(34) 1 2 ( 3 4 ) 1 2 ( 3 4 )  (5.5.6)  whereas f o r i n t e r n a l  a n g u l a r momentum d e n s i t y , o t h e r than the term t h a t  negates the second term i n (5.5.6) i s  - V • L„ =2M+D,no r x t  + V •— ^  All  Tr 1234  (12) ^-1 i  v  2  ' 12  £  , 1(34) +  (  -  ^  . . . )  6  (  1  3  (  2  3  4  4  )  [^-2(83^)]  (  r  1 3 +  ir ) 4  £  , 2(34) +  k  ...)  6  )  [s - (s 2  2  3 +  s )] 4  P  12(34)"l2(34) 12(34)  where L , i s the m o l e c u l a r momentum g i v e n by =2M+D,no r x t &  J  cr u ) 23+  24  9 0  -  -  =2M+D,no r x t  =  Tr 1234  v  g  • • • •) 6 (134) -34  • • *  ) <5 (234) 3 4 r  X  V  34  ( V  x  V  34  ( V  + V  13 14>  23  + V  24  )  12(34)^2(34)^2(34) (5.5.7)  Another kinetic  m o l e c u l a r o b s e r v a b l e under study i n t h i s r e s e a r c h , the  energy  of the m o l e c u l e s , u n l i k e l i n e a r momentum and a n g u l a r  momentum, i s not conserved  f o r non-reactive c o l l i s i o n s .  monomer-monomer c o l l i s i o n , the change i n k i n e t i c  energy  C o n s i d e r the due to t h i s  c o l l i s i o n type i s g i v e n by  Tr  =- 2 ^  0f  1 +  ^ > 2  ° V V  « >  1  2  ^  2  P  1  2  W l 2  To o b t a i n the v e r y l a s t e x p r e s s i o n above r e q u i r e s the i n t e r t w i n i n g relation  c' c'  c' c'  ^c'  » c  - 91  -  I f p\2  i s d i a g o n a l i n *<i , <&12Pl2 v a n i s h e s so that k i n e t i c  itself  i s conserved.  2  However, i f the c o l l i d i n g  inhomogenous, i . e . , when  P  j  2  at  collisions.  spatial  a n c  * kinetic  energy  inhomogeneities.  For c o l l i s i o n s i n v o l v i n g  f o r n o n - r e a c t i v e c o l l i s i o n s even i f p » c  homogenous because now One  the p r o j e c t o r ^ » c  energy  ( o r p ") c  ( o r & ")  does change  c  formally v e r i f i e d  i s not  conserved  is spatially i s no l o n g e r the  i s conserved.  i n Chapter 7 where the energy  q u a n t i t y owing  a h i g h e r number of  would a n t i c i p a t e an i n t e r c h a n g e of k i n e t i c  e n e r g i e s a t c o l l i s i o n s so t h a t t o t a l energy  studied.  energy  i s i n g e n e r a l not a conserved  atoms In which r e a c t i o n s are p o s s i b l e , k i n e t i c  identity.  ( b e s i d e s being  T h e r e f o r e , even f o r a simple case l i k e 2-monomer  collisions, kinetic to  system i s s p a t i a l l y  i s position-dependent  moraentum-depenent) , [**i2» P12J — * 0  energy  and  potential  Indeed t h i s i s  balance r e l a t i o n  is  CHAPTER 6 HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES I I : REACTIVE COLLISIONS  The structure of the equations of change for the monomers and dimers due to any given reaction suggests that there are two contributing factors e f f e c t i n g the change, therefore each of the equations can be divided into two parts.  One i s analogous  to the c o l l i s i o n a l transfer  between the molecules for non-reactive c o l l i s i o n s , and on neglecting the numerical factor due to counting, t h i s i s  -5-  Tr  (<f>,<5.) V/ tf JV.p  T l ,  , (where c * c' and c' involves  1 1 S C C C C  l...n  ,  ,  x  the monomer 1)  for the monomers, and  ~k ,  T  r  (  5  )  u  f  p  *(i2) (i2) s 'c c^c" c"  l...n f o r the dimers.  i *?>  {  e:  involves  the dimer (12)) Same as with non-reactive events, f o r the observable  mass, these terms vanish, and f o r l i n e a r and angular momenta, these correspond to the transfer of the physical attributes between the c o l l i d i n g molecules.  This kind of transfer i s designated as the  " c o l l i s i o n a l transfer" to be distinguished  from the r e d i s t r i b u t i o n of the  physical attributes from the reactant molecules to the product molecules as the atoms regroup into a d i f f e r e n t set of molecules.  The contribution  to the hydrodynamic equations for the monomers from t h i s second kind of transfer i s  i 1  when there i s no monomer i n the i n i t i a l  i  Tr  n 1.. .n  t V W s "  channel,  < V l W c c '  p  ct  Here i i s a product monomer, and f o r the dimers,  i  Tr E (<p (jk) ^ j k ) ^ c c c l...n (jk) |  i  Tr l...n  [ E  (jk)  3  (jkAjk^s  ( j k ) being a product dimer. term  ( < t ,  "  l P  ci  6  >  otherwise  this i s  or  )  (12) (12) s  ] < f  : r  p  c cc" c"'  The l a t t e r c o n t r i b u t i n g  term  i s the o n l y  t h a t g i v e s the change i n the number or mass d e n s i t y of e i t h e r  species.  F o r the observables of l i n e a r and angular momenta, t h e i r  evolution i s explicitly  time  shown not o n l y i n the d e n s i t y o p e r a t o r d e s c r i b i n g  the r e a c t i o n , but a l s o i n the g a i n and l o s s of the observables a s s o c i a t e d r e s p e c t i v e l y with the r e a c t a n t s and p r o d u c t s , thus i t can be r e f e r r e d to as the " r e a c t i v e " part of the hydrodynamic e q u a t i o n s . may not be a p p r o p r i a t e i n the sense  The nomenclature  t h a t both " c o l l i s i o n a l  t r a n s f e r " and  " r e a c t i v e " p a r t s are t r a n s f e r terms at r e a c t i v e c o l l i s i o n s nonetheless i t differentiates  the k i n d of t r a n s f e r common t o a l l c o l l i s i o n types  the k i n d a t t r i b u t a b l e o n l y to r e a c t i o n b e s i d e s being convenient  from  to use.  -  A.  94 -  C o l l i s i o n s I n v o l v i n g a Monomer and a Dimer  The r e a c t i o n s t h a t may take p l a c e are e i t h e r decomposition whose c o n t r i b u t i o n s to the e q u a t i o n s o f change a r e  a M  <  >  h i t M *M ^M+D-»-3M  n ^ 123  (  W s  ^123^123^1(23) 1(23)  +  +  t(* Vs 2  <*3 3>s 6  lf  123^123  (6.1.1) and  l  8t  D  J  D M+D^3M  =r mT r 1  *  123  (  •( 23 ) ( 23 )  )  6  s ^ 1 2 3 **12 3°H( 2 3 ) 1(23) '  [ ( < { ,  6  )  (23) (23) s  < ?  J  123 123  (6.1.2)  or exchange i n which case the changes i n the hydrodynamic d e n s i t i e s are g i v e n by  -  [  9t  95  -  "M^M^M+D.exch 21  Tr 123  ^ l V s  ^(12)3*02)3^(23) 1(23)  +  (  " W s  6  )  " *l l s^(12)3^(12)3  (6.1.3)  and  ^ 9t W^MfD.exch  U 1 1  123  J V  (23) (23) s '(12)3^(12)3 K23) 6  )  V  1(23), +  [ ( < ,  6  )  (  6  )  '(12) (12) s ~ * ( 2 3 ) ( 2 3 ) s  ] f  T  (12)3 (12)3  (6.1.4)  F i r s t , c o n s i d e r the decomposition r e a c t i o n which i n v o l v e s a n e t g a i n o f 2 monomers and the n e t l o s s o f a dimer. e q u a t i o n s f o r both s p e c i e s . unambiguously j^j +K-k  +  Vjk>  These  a r e indeed found i n the  As f o r the t h i r d body, i t s r o l e i s  shown i n Chapter 7.  On p u t t i n g  <$>± = ^ i , ^ ( j k )  =  i t p r o v i d e s the energy f o r or c a r r i e s away the  energy generated i n the r e a c t i o n when the r e a c t a n t dimer breaks up. In f a c t , t h e r e i s an exchange o f the k i n e t i c energy a s s o c i a t e d w i t h the c o l l i s i o n p a r t n e r s and the p o t e n t i a l energy a s s o c i a t e d w i t h the products and  the t h i r d  body.  - 96 -  For the  the e q u a t i o n o f change f o r mass d e n s i t y , the c o n t r i b u t i o n  from  break-up r e a c t i o n i s  [  TE  (  M  M  +  V W  M  3  1  1 2Lo* lo-i  _ i loo £ [(2n)!] (-fi)  Z  123 6  =  _ i  V  V :  <  Z  Z  2  0 0  - - 4 V V : T r (2m) [ 1 - - £ i +  n 2  n  _  1  2 2  n  • V  _  2  ]  l  n=2 P  (23)  123 123 1(23)  j r m^ 1= § 1 6 ^ ^ 2 3 J  1 2 3  p  i ( 2 3 )  .  (6.1.5)  N e g l e c t i n g terms o f h i g h e r o r d e r s i n VV, a r a t e dispersion  The  o f mass over the product monomers i s o b t a i n e d .  mass d i s p e r s i o n  X=M^  T r i m  = ff}2)  r  1  2  tensor % i s d e f i n e d by  r  1  2  «  (  1  2  )  p (12) b  + j <r> U .  (6.1.6) (2)  U i s the u n i t the  o f change due to the  second rank tensor and Tft  symmetric t r a c e l e s s  1 and - j <T> U are r e s p e c t i v e l y  and the zeroth-weight t e n s o r s of'ft.  M ^ O , the  l o c a l d e n s i t y o f the s c a l a r moment o f i n e r t i a o f the dimer molecules has (2) been a s s o c i a t e d  w i t h pure v i b r a t i o n a l  t r a n s i t i o n s whereas M^1"A-  both pure r o t a t i o n a l and r o t a t i o n - v i b r a t i o n a l  1 1  transitions .  to  - 97 -  The r a t e o f change o f l i n e a r momentum d e n s i t y i s g i v e n by  lit % i i C  = -V •  +  M  Z>W3M  D  t  P  5M+D+3M  - £  W  :  T  r (  1 2  i  3  -2r¥¥  +  - ) [ %  +  P > 3  6  1  P  P  (23) s« 123^123 l(23)  *2 " ^3 [  £  ( 6  2  2"V  ]  P  s fWl23 l(23) (6.1.7)  ct where ?j^rj-»-3M +  *  St b e c  s  n a  °m i° l  t r a n s f e r p a r t of the p r e s s u r e  tensor  a s s o c i a t e d w i t h the r e a c t i o n M+D+3M and i s d e f i n e d by  C t  P =M+D+3M =  =  T V ((rr - . - ^ v  -Tr  123  _  1  (  6  R  2  3  (V  )  +  7  +  L  ,n r , , „_ ° l~(-2) - K 2»3—) — : N  n  n=2 V  1(23) 123  )  !  n-1 n-1 . .) • V v  3  lA  ^123 i(23) l(23)* P  (  6  1  ' '  8  )  The second term i n (6.1.7) corresponds to the change due to the t r a n s f e r o f momentum third  from the r e a c t a n t (23) to the product monomers whereas the  term i s the change due to the r e l a t i v e motion o f the product  monomers.  T h i s r e l a t i v e movement i n f a c t r e p r e s e n t s a t r a n s f o r m a t i o n o f  v i b r a t i o n a l and r o t a t i o n a l degrees o f freedom i n t o t r a n s l a t i o n a l degrees o f freedom.  Neglecting  third  the c o n s e r v a t i o n r e l a t i o n i s  and h i g h e r order terms i n the g r a d i e n t s ,  - 98 -  3M = -V  • (p ) =M+D+3M'  V  "- ' +  4  £SD*3M  V  6  '  2  V  " I  J  :  V  3  IT  R  [<P  f  ;  + 2  P3  ) 6  (23)  ]  6? S  P  123^23 1(23)  p  (23)^2 E23U 123 i23 l(23) 3  V  X  123  6  (  2  3  )  1  2  3  X  1  2  3  P  ^123^123 1(23)  (6.1.9)  Here {n^jP^jlg *  s  t  n  e  t e n s o r i a l l y symmetrical  part o f (_£.Py)  : s  (6.1.10)  T h i s i s connected dimers.  with the r o t a t i o n a l  The zeroth-weight  and v i b r a t i o n a l  tensor gives r i s e  t r a n s i t i o n s of the  to pure v i b r a t i o n a l  transitions  £  «  ( 1 2 )  (2L  i n  1 2  '£  1  (v + l )  2  )  l  /  2  s  P (12) - | u b  (v + 2 )  1  /  2  : ^TVv  <v + 2, j X r | P |v j X r> b  jXv  - v  1 / 2  (v - 1 )  1 / 2  <v - 2, j X r I p  fe  Iv j X r>  (6.1.11)  - 99 -  ^corresponds  t o the p r o d u c t i o n of mass d i s p e r s i o n due to i n t e r n a l  motions of the dimer m o l e c u l e s  2  M  ~= H  6  D ^  = i  T  6  r  2  (12) (  £ l 2 t l 2  Js  p  b  ( 1 2 )  P  (12) 7 ^ ( 1 2 ) ' i l 2 ^ -  ( 1 2 )  b  (6.1.12)  '  to the H e i s e n b e r g p i c t u r e , V ^ i s the change of 9^1 w i t h r e s p e c t t o  According  time, and hence the n o t a t i o n . d  (^jJllj)^  o  e  s  n  o  t  F o r the second weight t e n s o r ^  commute w i t h  x 2.±y  z •  n  o  r  w  i  t  n  Ojjj"  , as  aa* + k a^a  (a and a^ a r e the d e s t r u c t i o n and c r e a t i o n o p e r a t o r s a s s o c i a t e d w i t h a harmonic o s c i l l a t o r and k i s c o n s t a n t ) , t h i s i s r e l a t e d vibrational  transitions  of the dimers.  components of r-t j and 2±j application  [q .  P  w  it  L =  p + qp + pq + 4 ^x y ^y x *x^y  i  T6n"  r  U  , q rp  +  as f o r pure v i b r a t i o n a l  *L  H  2  2  + q ] y M  2>  is  p q ) y x  y y  [p p , q x*y' x K  K  +pq  yy  z  be the  motions are not the ± 2  the x y component of  r  w  i*.  quadrupole-induced t r a n s i t i o n s  ~(q  and p  relation  i s found t h a t such v i b r a t i o n a l  In p a r t i c u l a r ,  w  a l o n g the w-axis (w = x, y, z ) , on the  of the commutation  W  Let q  to r o t a t i o n -  n  ~(qp  +pq)] M  x x  x x  2  2  •  + [p + p , q q 1 x y' x y • l l  F  4  4  transitions.  -  1  Jr  ~7T* 1 U P P .M q  =  16Ti  2  2. + q ] y -  'x'y' x  100 -  . 2 2 . + IP + p , q q x *y' x y M  M  i  J  (6.1.13)  and  the o t h e r components  Ji  =  M  i n a s i m i l a r manner.  Comparing  Sv^due to the MrD->-3M r e a c t i o n  the r a t e of change o f  [  can be obtained  D ^M D*3M +  "fi  J23  6 ( 2 3  7  >  - Tr [ r -23 123  0  0 v  P  ^-23^23^ ^ 123 123 1 (23) (V V ) + ( V „ v J r „ J (P ^V 23 123 23 123 - 2 3 123 1(23) 1(23) 0 0  1 0 0  with equation ( 6 . 1 . 9 ) ,  y  v  y  100  J  K  (6.1.14)  the c o n t r i b u t i o n to the p r e s s u r e tensor f o r t h i s  r e a c t i o n from the l o s s i n v i b r a t i o n a l motions can be i d e n t i f i e d .  By  analogy to the harmonic o s c i l l a t o r , the product monomers a p p a r e n t l y e x h i b i t o s c i l l a t o r y behaviour when they are formed. o f the p r e s s u r e  2h  T  6 (  *  ["ft  S i m i l a r l y the part  tensor  2 3 ) -23  V  ^  X  W  f  - ^ 2 3 123^123*1(23)  s  x  U  M x U  +  6  ( 2 3 ) ^23  X  ( V  V  23 123  ) < ?  123  v A  p  'l(23) l(23)  (6.1.15)  -  represents  101  -  the t r a n s f o r m a t i o n of pure r o t a t i o n a l motions of the dimer  molecule to the r o t a t i o n  between the product  a n g u l a r v e l o c i t y of the dimer i s g i v e n  (Mp  As  Tr 6  r  ( 1 2 )  f o r non-reactive  translational other  1  <r»  1  x p  2  M + D  +3  S)  x M  between the product  l  s  monomers and  d  u  momentum d e n s i t y due  (6.1.16)  b  The  tensor g i v e s r i s e to a  However, there are a l s o f l u x  e  t o  t h e i r t o t a l l i n e a r momentum the study the energy  -23 -23 T—  [ (  +  ) 6  ]  -£2 ^3 (23) :  p  i23-123 l(23)  i (23)^23£2 Js 3  1  V  X  \H  angular  to t h i s r e a c t i o n i s :  r  6  conservation  r a t e of change i n t r a n s l a t i o n a l  123  V  6  ( 2 3 ) ^23  X  terms  the c o u p l i n g of the d i s t a n c e  r e p e r c u s s i o n of which i s r e v e a l e d when we r e l a t i o n i n Chapter 7.  molecular  by  c o l l i s i o n s , the p r e s s u r e  which  The  P <12).  n  angular momentum f l u x .  than ( ~ £  monomers.  <P  I  P  123 123 1(23)  < P  : I  P  X  ^  ^23 123 123 1(23)  X  ^  x r)  -  +  i  •  V  1 2  ^-T  2 ^23  6  i *  o  123 ~  2  6  i  2  3 X  +  6  = - V  ^  2  3  +  6  V  -  ]  (23) s  ^123^23*1(23)  +  2  3  2  3s  f l 2  7  P  3 123 l(23)  C 9 ^ v  - Tr — J23 3  ' r x (V V " > P -1(23) 1 ( 2 3 ) 1 2 3 °123 l ( 2 3 ) l ( 2 3 ) X  ~ SM+D*3M  k^  X  [ (  1  V  D p  ;  -  T f - TT P  ~¥  0  +  2^23  X  [(  +  -E -B 2  ) 3  6  ]  (23) s  0  1  123  Tr 123  %  [  X  102  123^1(23)  V 3 S  i •F £ 2 3 [  x  -^23  ]  2 s  f  123^123 1(23)  + l—k 3  (23) -1(23)  v X  V  V  1(23) 123  ;  p 123 1(23)  (6.1.17)  Compensating  the n e t change  changes i n i n t e r n a l  i n translational  angular momentum.  momentum o f the monomers  angular momentum a r e the  The g a i n i n the i n t e r n a l  angular  -  [  <  103 -  >J  3 t \ lM,int M+D->-3M  Tr 123  ^ i V  ^123^123^(23) p  +  [(s « )f 2  (s «3)]^  2  3  1 2 3  3  (6.1.18)  l(23)  3  1 2  i s due to the break-up o f the dimers which transforms the i n t e r n a l momentum o f the c o n s t i t u e n t atoms o f the decayed translational internal  dimer molecule  i n t o the  angular momentum between the product monomers and the  (atomic) angular momenta o f the products  Ut ViD.int^M+D-^M  Tr 123  <»  +  ±3  2  +  ^23  X  -£23  )  6  (23) ^123^123^(23) 1(23),  " <I  X  23  ^23  +  -2  +  -3  )  6  (23)  C  123*123 (6.1.19)  The net change i n atomic  angular momentum i s s o l e l y c o l l i s i o n a l  in origin.  p a r t o f the r a t e  The r e a c t i v e  to the t r a n s f e r  o f the p h y s i c a l  attribute  transfer  o f change i n i t s d e n s i t y i s due from the r e a c t a n t dimer to the  product monomers as w e l l as the d i s t r i b u t i o n over the product monomers.  [  4t"  ( M  < J  M M,int>  +  M  D<4,int  >)]  M D>3M +  = "  V  * SI£D*3M  -  104 -  6,-6 Tr 123  t-S^  (  2  ^ 1 - l 2 " l 2 > l ^123 123°i(23)  6 +6 + s ) ( — 2  +  2 6  + Tr 123  l  + 6  [ ( 8  2  2  2  f  6 -6  3  2  «  3  ( 2 3 )  '1(23) 3  +(s -s ) 2  )] ^  3  2  3  J  1  2  3  (23) -1(23)  x  ( V  V  1(23) 123  )  ^123^(23) }  +  1  6  Ti ( 2 3 ) ^ 2 3  X  1(23)'  -223 ^123^123  (6.1.20) The  relation  been used  +  [s^ + ±  —3  2  +  —23  to o b t a i n (6.1.20).  =M+D+3M  (  3 ^  x  =  £.23'  ~^-l(23)  X  h  -^1(23)'  a  s  The term  r  - -l(23)-"  )  6  X  R^23  ( V  V  23 123  )  C ?  p  123^(23) l(23)  (6.1.21)  which a r i s e s  from  be a m o l e c u l a r  the i n t e r c h a n g e o f i n t e r n a l  angular momentum f l u x , having  and a n g u l a r momentum must the t r a n s l a t i o n a l  and atomic  a n g u l a r momentum f l u x e s a s s o c i a t e d w i t h t h i s r e a c t i o n i d e n t i f i e d . Comparing  (6.1.17) w i t h (6.1.20),  molecular  a n g u l a r momenta are found  second  the net changes i n t r a n s l a t i o n a l and to c a n c e l each o t h e r .  Again a term  o r d e r i n the g r a d i e n t s i s o b t a i n e d  2fi  Tr 123  L_r23 ^ 3 . 2! 2 2 '  v  v  (23) 2 3 -*-23 £  9 + r -23  1  X  x D -P-23 2!  .E.23 —23 —  2  2  1 J  D  P  123 123 1(23)  • VV 6 * °(23) V  V  - 105 -  V  V  TO * H  r  3  16ti  VV  S  1  p  \23) 'l23 l23 im)  X  2 3 ^23 *23  (2)  : Tr 123  -23  X  ^-23/  :  (  £ 2 3 -23 ^ (23)  P  123 123 1(23)  (2) (£ 3 W  +  x  ^23 *23  2  = _ v • CT  -  T  V  =M+D*3M  ~i SM+D+3M'* C  T  (6.1.22) We denote t h i s f l u x , which a r i s e s  from the t r a n s f o r m a t i o n of m o l e c u l a r  angular momentum i n t o the monomer t r a n s l a t i o n a l angular momentum by ct (L - L )„, . „ because i t i s the most convenient = = M+D+3M riJ  use  0  f o r f l u x terms of such o r i g i n ,  t a k i n g the t e n s o r product  c h o i c e of n o t a t i o n t o  i s a f o u r t h - r a n k t e n s o r obtained by  o f two second rank u n i t  t e n s o r s i n the manner  as below  ^D=  x\x) + y\%) + z\zj .  As f o r the exchange r e a c t i o n ,  there i s no net change i n the mass o f  the monomer o r dimer:  *3t *VM+D,exch  =  "^  f  123 ^ " " V  : r  p  (12)3 (12)3 l(23)  00  2i  jr. " n-l n - l  1  V • Tr mr 123  6  + 2  J  I (n!)' n=2  P  (12)3 ( 1 2 ) 3 1 ( 2 3 )  1  f Z  •  V  « J  -  106 -  2 2i V • Tr m T ^(12)3 " rT 123 (  -1(23)  )  n +  Z  2 ) » (n..)"  (  1  (r  (  1  2  )  3  ^1(23)  n=2  R (12)3  )  n-1 •  n-1 V  (12)3*1(23) (6.1.23)  Ut  =  =  ^M+D.exch  " n  1  T  2 m ( 6 2  3  ~ IT £  Ut  21 n~  2  m  6  (12)~ (23)  [  |  )  J  P  ^(12)3 ( 1 2 ) 3 1 ( 2 3 )  ^(12)3 " ^ 1 ( 2 3 )  )  +  '••  ]  6  3  P  R?12)3 U2)3 1(23>  ^M+D.exch  Tr 2m 123  ,1 -12 -12 "2T"2~"T"  v  C  .  °(12) ^12)3^(12)3*1(23)  I  ^23 -23 *2! 2 2  Ut  ^MfD.exch  -  V  V  2  :  .) 6  (23)  M  U t D^M+D,exch* (6.1.24)  While  t h e r e i s no n e t change In the masses o f the monomers and  dimers, the exchange r e a c t i o n molecules  r e s u l t s i n a change i n the momenta of t h e  - 107 -  [  9t %  ]  - M+D,exch  (  *  123  £ l V s ^(12)3^(12)3^1(23) 1(23) (  6  )  + [<J> Vs - £ l l s  ] t f ,  3  (12)3^(12)3  (6.1.25)  ^3t \  2i  -^M+D,exch  Tr 123  (  6  )  -£(23) (23) s  V r  P  v /  (12)3 (12)3 l(23) 1(23)  +  I (  )  -E(12) (12) s " - -(23) (23) s ( 1 2 ) 3 ( 1 2 ) 3 6  (  P  6  )  1e  T  (6.1.26)  Combining the two e q u a t i o n s , c o n s e r v a t i o n o f t o t a l momentum i s a g a i n established  tit  ( M  Mii  f o r the exchange  +  M  reaction,  ]  D^ M D,exch +  = -V • P =M+D+3M  c t  -V • P - l i =M+D,exch "h  Tr  ^<  3  6  1  + 2 < S  (  J  ( 6  6  ]  ^(12)3 (12)- 3> s  (23)  _ P _L_il3) [  +  " s  P  )  ]  s  _  (12)3 (12)3 1(23)  tP  1 ( 2 3 )  (6 -6 1  ( 2 3 )  )]  s  -  ct -V • p. =M+D,exch  2i VV ti  Tr 123  108 -  A  1  (  +  2  12)3 - ( 1 2 ) 3 _ - 1 ( 2 3 ) - 1 ( 2 3 ) , -i T 5 )  v  -Vs  f  3  P  (12)3 (12)3 1(23)  1 V  ' £  ¥^(12)3^(12)3^ 7  -  ill(23>El(23)J  ] S  P  ^(12)3 (12)3 1(23)  " 2  V  £  V (12)3  X  (P  X  ^(12)3 " -Il(23)  T  X  ^1(23)  )  P  (12)3 (12)3 1(23) (6.1.27)  where L E ^ J ^ — ( i j )k^ (  )  i(i )k^(ij)k s'  t  h  a  i  t  S  t  1  *  ie  t e n s o r  a  i -'-iy  symmetric p a r t o f  8  j  ^ij)k-£(ij)k^s = 2 ^  t £  .ct =M+D exch' * t  i e  c  s  °m i  o  n  a  ij)fcE(ij)k 8 )  3  (ij)k  l  +  [(  -^(ij)k-£(ij)k  )  1 S  ^  (6.1.28)  >s *  transfer contribution  to the pressure  tensor  due t o the exchange r e a c t i o n as i n d i c a t e d i n the s u b s c r i p t s i s d e f i n e d by  SM+D.exch  S  "  2 T  (  ^ il(23)  +  —>  V  V  V  1(23) (12)3  }  P  *(12)3*1 (23) l ( 2 3 ) ' (6.1.29)  - 109 -  We can e a s i l y i d e n t i f y the change i n the l o c a l i s a t i o n o f momentum of the molecules  from being at those o f the r e a c t a n t s to being at those o f the  p r o d u c t s and the changes a s s o c i a t e d with the change i n the r e l a t i v e linear  and t r a n s l a t i o n a l angular momenta.  e x p r e s s i o n i s t h a t the i n t e r n a l  s t a t e v a r i a b l e s do not appear here.  a t t r i b u t e s can be d i v i d e d i n another  P  6 3  - P  +  " Vl  3  6 3  +  -£(23) (23)  6  V 3 ^ ( 2 3 ) —2—  way  +  6  -E(12) (12) " V l  ~  V  {  2  V 2 6  6  =  The  6  Al2)\l2)  *2*2 " £ ( 2 3 ) ( 2 3 )  3  The problem with t h i s s o r t o f  6  ~ (23)  }  +  (  -P-23 W  +  *( 1 2 ) \ 12) (  2  '  - j»cvv 2 1  (6.1.30)  so that the c o n s e r v a t i o n r e l a t i o n can be w r i t t e n i n a form which i s analogous  (M  to the p r e v i o u s r e a c t i o n  +  [ < E M -52-  M  ) ]  D ^ M D,exch +  c t  - V • P =M+D,exch  1 -12 -12 Tr 2! 2 2 123  <  6  )  £(12) (12) 8 <?  1 —23 —23 2 \ — — ^2 [  +  ^  6  ]  (23) s  :r  p  (12)3 (12)3 l(23)  -  2i  v  *  T  r  IIE12I  123  V x Tr 123  6  •12^s (12)  (6  110 -  ^23^23^8  (12) £l2 -El2 X  6  6  (23)  " (23)-23  ]  x  ^(12)3^12)3*1(23)  ^23  )  p  ?12)3 3l2)3 l(23) —  —  •  (6.1.31)  The  c o u p l i n g between the i n t e r n a l motions of the dimer and the r e l a t i v e  m o t i o n between the dimer and the monomer i s e v i d e n t comparing and  (6.1.31).  for  t h a t m a t t e r ) , one sees an i n t e r c h a n g e of i n t e r n a l  degrees the  Indeed, j u s t  (6.1.27)  l o o k i n g at the p r o d u c t s alone ( o r r e a c t a n t s  o f freedom which a r i s e s  from  the d i f f e r e n c e  and t r a n s l a t i o n a l  i n positions  between  molecules.  As i n the p r e v i o u s c a s e , t o t a l reaction,  but t r a n s l a t i o n a l  s e p a r a t e l y conserved.  =  V  (  * =M D,exch  X  +  angular momentum i s conserved  and i n t e r n a l  For t r a n s l a t i o n a l  for this  a n g u l a r momenta a r e not angular momentum, we have  £>  2i  + ± ± v • Tr 123  (  ^(12)3  + , , , )  -(12)3 x  (  -l(23)  + , , , )  -1(23)  (  l V s (12)3 (12)3 1<23) P  7  P  - Ill -  2i _  6  (12)  + 2 6  3  3  * 123  -(12)3  X  -2(12)3 J  P  ^(12)3 (12)3 1(23) 2 6  +  1  6  (23) -1(23)  2 6  -2  Tr — 123  1  +  £1(23)  6  (23) — 3 — r , -1(23) x (V, , , ) <(12)3'Y(23) P 1(23)V(12)3 1(23) X  (  V  -iv A l D l 3 123  C t  (P x r ) + — VV £M+D,exch -> H V  +  x  V  (  21 "fi  V  ' ^  V^(12)3£(12)3^s  7  o N o  P  ^12)3 (12)3 1(23)  )  ^R s  < F  ),P  P  -(12)3 _ -1(23) - 1 ( 2 3 ) , 3 3 3" ;  J  J  " ^l(23)£l(23)^  X  [  1  -  \ V x( T r R -(12)3 £ ( 1 2 ) 3 " - K 2 3 ) ^ 123 6  P  (12)3 (12)3 l(23)  X  X  £l(23)  ]  x r)  P  ^(12)3 ^ 1 2 ) 3 1 ( 2 3 )  41 V • Tr ^ ( 1 2 ) 3 £(12)3 9H 123 x  £l(23) —1(23)  tf  (Z R>s ( 1 2 ) 3 ( 1 2 ) 3 1 ( 2 3 ) 6  7  P  X  £  -  2 6  -  Tr  2  1  +  6  21  V  -1(23)  X  ( V  6  Tr  (£(12)3  123  x  V  1(23) (12)3  -^(12)3  V  (  -l(23)  C  * £ M D,exch  X  VV  2 6  P  (12)3^(23) l(23)  }  +  1  e  3,  3  -^1(23)  )  6  < P  (12)3 (12)3 1(23)  J  P  < r  (12)3  (23), }  3  ±>  +  21  X  +  (12)  2 6  (  "  -  (23) 3  123  112  (Tr  2!  2  2  P  ;  ^- -(12)°(12) s  : r  p  X  (12)3 l(23)  123  1 -23 -23 TT 2 2  2i  -12 -12 ~T~~2~*  Tr  , <• \ ^(23)°(23) s ;  , f. , ^ ( 1 2 ) (12) s ;  P  *U2)3*(12)3 1(23)  X  -  123  -23 -23 , t\ — — x ^P 3) (23) s 0  ;  (2  2±  V -(Tr [ { £  1 2  P  1 2  } « S  " {£ 3E23^s (23) 6  ( 1 2 )  1 ( S  2  : r  p  X  '(12)3 (12)3 l(23)  123  1  V  x ( T  ^  [ £  X  6  £  X  6  1 2 - £ l 2 ( 1 2 ) - 23 -£23 (23)  l f f  (12)3  : r  p  (12)3 l( 3) 2  X  }  - 113 -  6+6 +  (£  ^f  2 3  12 ^ 1 2 - V - ^ 2 3 X  \  2  i  +6  2  "  6+6  (  2 3  £23-\  1  )  s  ^(12)3^(12)3*1(23)  )  3  [23  X  —K23)  x  (V  V  )<?  V  1(23) (12)3 (12)3 ^1(23)*1(23) (6.1.32)  A l l the terras can be i d e n t i f i e d i n the same way as with the decomposition reaction.  Also, just as before, we can i d e n t i f y the flux term other than  the translational and atomic angular momentum fluxes as the molecular angular momentum flux associated with this reaction.  This i s obtained  when equation (6.1.32) i s combined with the rate of change i n the i n t e r n a l angular momentum density due to this reaction  [  Tt" V i M , i n t > (  +  V^.int^M+D.exch  = - V •L =M+D,exch + 2i  17.  T  r  (  *  123  £l(23)--*  + [-(r  1 2  )  6  (  2  2  R 3 £r ± - ^3  )  "(12)3^(12)3*1(23)  2  ...) 6  ( 1 2 )  -l"-2 5  +  (  £ 3-° 2  6  1(23)  (23)  ^(12)3^(12)3.  + % VV n  Tr 123  ,1-12-12 2\——  r "  x . , (12) ^1 ^ 2 6  {  (  +  \ )  6  r -*  0  6  (  (23) ^2  +  V  :r  P  °(12)3 (12)3 l(23)  -  2i  1  "  123 ~  2  1  2  6  l  1  +  3  6  2  X  £  l  2  2  ~  2  114 -  £  2  3  ~  X  2  3  ~  ~  1  3 )  g  (12)3  < f  (12)3 l(23)  : r  P  ( 2 3 ) -H23)  3  X  V V  1(23) (12)3 V  (12)3 1(23)  ;  J  P  l(23)* (6.1.33)  Thus  again  atomic  thereactive  angular  molecular  part  momentum.  angular  does  notgive  As w e l l ,  momentum  flux  rise  t o a n yn e t change  thecollisional  of this  reaction  transfer  part  i n o ft h e  i s given by  C t  L =M+D,exch  =  1  ^  (  £  l ( 2 3 )  +  " *  )  6  R^23  X  (  V  23 (12)3 V  )  <  r  (12)3  :  i  l(23) l(23)P  (6.1.34)  B.  Collisions Involving 3 Monomers  Recombination However,  whereas  partitioning  may  be thought  the part  of physical  has  a structure  analogous  the  collisional  transfer  of  molecules  contributions respectively  of as thereverse  of theequation attributes to that part  participating  from  o f change  f o r decomposition  hasa different  o f change  due to  thereactants  the  to the products  but of opposite  structure  i n t h e c o l l i s i o n event  to theequations  of decomposition.  because  i s different.  f o r t h emonomers  the  sign, set  The  anddimers a r e  - 115 -  Tr 123  2fT  t  (  *lVs  -KWB  +  +  (  W s  +  (  *3Vs  ]f  (  }U  *3Vs I(23)^l(23n23 123  l(23)Tl(23) (6.2.1)  and  l  h  W3M+MH-D - " 2ff ^ ]  Except to  t ( 4 , 3  6  ]  :T  P  ( 2 3 ) ( 2 3 ) s ^1(23) 1 ( 2 3 ) 1 2 3 '  ( 6  2  * *  2 )  f o r the c o l l i s i o n a l t r a n s f e r c o n t r i b u t i o n s which are analogous  the n o n - r e a c t i v e 3-monomer c o l l i s i o n s , m a n i p u l a t i o n s are the same as  w i t h monomer-dimer c o l l i s i o n s , l i k e w i s e f o r the i d e n t i f i c a t i o n The  hydrodynamic equations a r e l i s t e d  o f terms.  below.  Mass  [  3T M ( M  2-h  +  VV  ]  V 3M+M D +  1  : Tr 123  2 3  £ 3 2  °°  1  -23  2 n  2  n  ~  2  2  n  "  2  n=z 6  1,3  L  f  T  P  (23) l(23) l(23) 123  (6.2.3)  - 116 -  L i n e a r Momentum With monomers being the o n l y s p e c i e s i n the i n i t i a l collisional  c h a n n e l , the  t r a n s f e r c o n t r i b u t i o n to the e q u a t i o n o f motion comes  e n t i r e l y from the e q u a t i o n f o r the monomers:  r— I9  "M-^M+M+D  t  n  „Ct  = - V * P S3M+M+D 2 2  T  T  (  123  2  3  )  3  2  K 2 3 )  S  1(23)  J  ^123  6 -<5 +  "  ^  V  123  [  2  " "  £ 3 )  ct * FoM+M+D  i 2fi  V  X  i 2tT  +  l s  <?  K23) i(23) 123 :r  ^  6 3  P  2 3 ^ ( 2 3 ) ~T~~ l ( 2 3 ) ' i ( 2 3 ) 123 6  + < 5  ]  T 2  3  '(23)^23^23^  123  2  J  ~^ "  (23)^23  X  E  u  f  :r  :T  P  P  1(23) 1(23) 123  < P  : T  P  - 23 1(23) 1( 3) 1 3 2  2  (6.2.4)  ct Here P „  w  _ i s the c o l l i s i o n a l  tensor associated  C t  P =3M-»-MfD  t r a n s f e r c o n t r i b u t i o n to the p r e s s u r e  w i t h the r e a c t i o n 3M+M+D and i s g i v e n by  Tr 123  ( 6  (23)£ 3  +  ( V  2  V  23 1(23)  )  1(23) + ( 6  l£l(23)  +  •  ,  °  ( V  V  1(23) 1(23)  }  123  - 117 -  Tr ( r 123  + ...) 6  1 2  (V  ( 1 2 )  1 2  V  1 2  ) p  1(23) 123 123' ( r  +  1  3  +  ...) 6  (V  ( 1 3 )  1 3  V  1 3  )  (6.2.5)  Again f o r t h i s r e a c t i o n , the change i n momentum d e n s i t y can be expressed in  terras o f a p r e s s u r e  tensor  3M+M+D o  „Ct  =3M+M+D  +  i 2f\  W  !  T  (  2  i 2r7  3  123  " ^  1 ^23 *23 2T"2~"1~ ^(23) (23) s  V  X  6  (  2 3  6  123  )^-  (  2  3  )  2 3 £  ~  6  23^  2  3X  s  P  ~  )  <P  7  P  1(23) 1(23) 123  T 7  P  1(23) 1(23) 123  2  3  |P  :I  P  H23) 1(23) 123  (6.2.6)  Whereas f o r the break-up r e a c t i o n  the i n t e r n a l motions o f the r e a c t a n t  dimer i s transformed i n t o the r e l a t i v e  t r a n s l a t i o n a l motion between the  product monomers, here the r e v e r s e i s observed. v i b r a t i o n s and r o t a t i o n s when i t i s formed.  And the dimer  undergoes  118 -  A n g u l a r Momentum  >  ,lnt ^3M->-M+D  Tr 123  2ti  [  ^ l > s S  +  -t<£ V 2  ( l 2  S  2 >  + (s  s  + S  3 6 3  <l3Vs'^(23) l(23)^23 P  ) ] P s  1 ( 2  „3  ,  123  (6.2.7) r "8t l  =  <  M ^D,int  [  " ** 123 -  lt  [  •  (M  +  ( M  2 +  '  •  3M^M+D  1  3  +  X  £  6  ]  23>' ( 2 3 ) (s2 3 ^ ) s2 3 l^( , 2 3 ) l ( 2„ 3 ) , 1 2 3 (6.2.8) J  y  J  P  i  r  V^D.lnt»l  +  M<^,int>  2 W  >)]  ^ , < £  1  2  -  •  »  S  )  So  U 2 ) < i r V 1 2 l ( 2 3 / W 123 V  P  - s„ 2  123  3M+M+D  J  P  (23) 1 ( 2 3 ) 1 ( 2 3 ) 1 2 3  *?23 123' +  6  (23)l23  X  ^23 ^ 1 ( 2 3 ) ^ 1 ( 2 3 )  (6.2.9)  - 119 -  J u s t as f o r the decomposition the net change i n atomic term.  r e a c t i o n , the o n l y term  that g i v e s r i s e to  angular momentum i s the c o l l i s i o n a l  transfer  But now s i n c e o n l y monomers a r e i n v o l v e d i n the i n i t i a l  channel,  no m o l e c u l a r  angular momentum f l u x due t o c o l l i s i o n a l  transfer i s  id entified.  Combining with the e q u a t i o n o f change f o r t r a n s l a t i o n a l  angular momentum  [-^r  = V  x  M  (  U+ M  M  =3M>MfD  IK  L  t  +  1  3 M >  M+D  -  23  1  6  7  X  v)]  D  (  6  T r  =  ^  +  ^23 '•° ^  X  £l2  X  ( V  -^T-Ul3  +  t (  V-B3  )  6  ]  (23) s  d 3  T  p  l(23) l(23) 123]  2  2  123  1  V  12 12  x (V  6  i  " ti -23 l  + 6 2  X£  23  ~2~  1 3  ^1(23)^23  )  V  1 3  )  123  3 1  3  s^l(23) l(23)  ct  V •P =3M*M+D  " 2 ^  W  :  H ll  l  3  T  T  -T- -f-  (  -H(23)  f  P  X  Ws l(23)%23) 123 £  ]  - 120 -  +  4  V  6  *  T  V  X  (  T  P  J  P  (23)^23£23^s 1(23) 1(23) 123  6  ^  3  X  P  J  X  *>  P  ( 2 3 ) ^23 - £ 2 3 1 ( 2 3 ) 1 ( 2 3 ) 1 2 3  X  ±>  r r —23 —23  i V  "  ' 123 ~  «  X  ~  ~  (  2  3  )  6  (  2  3  )  )  s  + 6  y Tr 123  P  ^K23)^1(23) 123  6  2  -12  X  ( V  V  12 12  )  +  —  + 6 —  ^13  X  7  V  < 13 13>  P ,  ^ p 1(23) 123^123  +  i 2rT ! L -23 123 [r  T  X  6„ + 6 2 3 2 s ^1(23) l(23) 123' 1  ^23  J  P  (6.2.10) one f i n d s t h a t the m o l e c u l a r angular momentum f l u x a r i s e s from the f o r m a t i o n o f a m o l e c u l a r a n g u l a r momentum and the d e s t r u c t i o n o f a translational  angular momentum  5  +  2rT 3* ^ 2 3 - E 2 3 X  T  12  T6n"  -  V  Tr -23 123  VV  X  ( 6  r  2 ^ 3 ^ 2 3 / /  +  2 -23 -23 -23  =3M+M+D  2  o  3  J  's  u  1(23T1(23)^123  / 3  +  • T  6  +  2  (23)  -^23 2 3  „  X  :  ^-23^23  6  P  ( 2 3 ) ^ 1 ( 2 3 ) ^ 1 ( 2 3 ) 123  ^23 (6.2.11)  -"121  C.  -  Dimer-Dimer C o l l i s i o n s  With two dimers, the r e a c t i o n s that may take p l a c e a r e decomposition o f a dimer without exchange, d e c o m p o s i t i o n w i t h exchange, and exchange without d e c o m p o s i t i o n .  T h e i r r e s p e c t i v e c o n t r i b u t i o n s to the e q u a t i o n s  o f change f o r the m o l e c u l e s a r e as f o l l o w s :  Decomposition  Without  Exchange  hTt W ^ D - ^ M + D . n o  "  r— l  [  *  1234  D  *lVs  (  +  <  ]f  W s  p  'l (34)3i2(34) (l )(34)' 2  ( 6  2  3  - -  1 }  >i  M <$  9t  (  exch  J  D 2D->2M+D,no exch  ^hll^in^s  * 1234  +  U  6  )  (34) (34) s  1  U  <:>  v/  12(34) 12(34) Vl2)(34)  " *(12) (12) s ^12(34)^2(34) (  P  6  (12)(34)  )  5  (6.3.2)  Exchange  Ut  Decomposition  W^D^M+D.exch  I(  1234  6  >  *1 1 B  +  (  Ws^l3(24)^13(24) (12)(34)' p  ( 6  3  ' -  3 )  - 122 -  Ut  M  <$  >  I  D D - 2D+2M+D,exch „  2 1  zr- T r  [  i  \ \ 2 ) \ \ 2 , \ +  ( < ( >  < S  )  (34) (34) s  1  ^3(24)^13(24) ^12)(34)  1234 + [  <*(24) Ws  - « * < 1 2 > Ws  +  6  <*(34) ( 3 4 ) > s>  3  ^13(24)^13(24)  (12)(34)' (6.3.4)  Simple Exchange  {{  ^1234  \\2)\\2)\  +  ^(W^n^B  ( < | )  (34) (34) s ri3)( 4)Xl3)(24)^12)(34) 6  )  1 U  2  +  ( < t >  6  )  (24) (24) s f  ( < t >  6  J  (13)(24)' (13)(24)  )  (34) (34) s  (12)(34). (6.3.5)  For the simple decomposition  r e a c t i o n , the changes i n the d e n s i t i e s  of mass, l i n e a r momentum and angular momentum other than t h a t due to collisional  t r a n s f e r are found  to be e s s e n t i a l l y the same as t h a t f o r the  - 123 -  M+D+3M r e a c t i o n except t h a t now i t i s a dimer m o l e c u l e which a c t s as the third  body.  Mass D e n s i t y  L  8t  2D->-M+D,no exch 2  i li  V  V  2  :  ,15, 1234  l  2  ' 8t  m  (  r r 1 —12 —12 2 T " 2 ~ ~  +  ~'  )  6  r  p  (12) i2(3A)^2(34) (12)(34)  D = 2D*2M+D,no exch (6.3.6)  L i n e a r Momentum  ["at  ( M  M ±  +  % £ 2D>2M D,no ) ]  exch  +  = -V • P =2D+2M+D,no exch = - V •  c  t  P  =2D->-2M+D,no exch  1234 <i  :r  P  12(34) i (34) 12(34) 2  '  "  " I  6  1234  V  X  + (third  }  <12)^122l2 s  < r  ?5, ( 1 2 ) £ l 2 £ l 2 1234 6  X  J  p  i2(34) 12(34) (12)(34)  < P  12(34)l2(34) (12)(34) p  and h i g h e r order terms i n V)  (6.3.7)  -  £2D>2M+D,no for  exch'  t  h  e  c o l l i s i  n  ° al  124 -  t r a n s f e r part o f the p r e s s u r e t e n s o r  the r e a c t i o n i s g i v e n by  C t  P =2D+2M+D,no exch  - Tr ^12X34) 1234  6  R  2n+l (2n+l)!  n=l  (V  2  V  )f  X/  P  (12)(34) 12(34) 12(34) Vl2)(34) (12)(34) (6.3.8)  A n g u l a r Momentum  r x  u + ^  v ) ]  2  D  +  2  M  +  D  j  n  o  e  x  c  h  C t  V • (P x =2D+2M+D,no exch ±-  }  1 +  if "  r r -12 -12 • < , 2 ) > 1 . ' I ( 2 3 ) ^ l ( 2 3 ) ' l 2 3 * I>  ' < !?, 2 ! — —  I f l  ' 1 2 M (12)^12^1 Js  V  (  6  2  f  J  p  12(34) 12(34) (12)(34)  X  ±>  -1 2  V  X  (  T  ^  6 3  4  X  f  p  (12) £l2 ^ 1 2 12(34)^12(34) (12)(34)  x  I>  -  < r  .  S T  3  P  i2(34) 12(34) (12)(34)  + 5  (12)  r  1 2  T  125 -  (34) 2  4  x  r  (  -(12)(34)  X  v  ) ^ (12)(34) (12)(34) V  V  V  ;  P  ^ 1 2 ( 3 4 ) ^ 12)(34) (12)(34)  6 +6 i  1 2 1  *T J234 "  2 X  ]  ' 1 2 ~~2~ s £  f  P  1 2 ( 3 4 ) ^ 2 ( 3 4 ) ( 12) (34)  (6.3.9)  U t  ( M  M  k  v v  ^ M . i n ^  :  12  T  r 3 4  +  %  ^D.int^UD-^M+D.no  TT ~T~ ~1T  <W  V  (£j + Tr 'h 1234  [ ( 6  6  (12)~ (34)  ±) 2  )  ^  -  exch  s ^12(34)^: 12(34) 12(34)^(12)(34) J  (£3 + J4)  2  ^12(34)^2(34)^12X34)  -l~-2  + [C6 -« ) — 2 — 1  -  2  ct V • L =2D+2M+D,no e x c h  ] s  ^12(34) ^12(34)  ]  s  P  (12)(34)  -  6  +  Tr 1234  (12)  + 6  (34)  2  +  126 -  £(12)(34)  6  -fi ( 1 2 ) ^12  X  X  V  V  ^ (12)(34) 12(34)  ;  r  l2(34rU2)(34)  £ l 2 ^12(34)^12(34)  (12)(34) (6.3.10)  The flux  collisional  transfer  contribution  to the m o l e c u l a r a n g u l a r momentum  I s d e f i n e d by  =2D+2M+D,exch  +  -  1  ^  ( 4  <?  The  6  12(34)  v/2  V  12 12(34)  )  ~ £  3  4  X  P  ( 7  V  34 12(34)  )  angular momentum between the products  (  Tr 1234  +  (  (6.3.11)  o f the m o l e c u l a r angular momentum o f the  to r e l a t i v e t r a n s l a t i o n a l  2rT vv  ( 7  d2)(34) (12)(34)*  term due to the t r a n s f e r  reactant  X  ^12  £(12)(34)  J_rl2 2! 2  r r 1 —12—12 2 f T " ~  zl2 2 '  .  X  -12 - £ l 2  P  ^12(34) ^12(34) (12)(34) = - V • (L - L  C t  (6.3.12)  ) 2D->-2M+D,no exch  c l e a r l y p a r a l l e l s the f l u x  term a r i s i n g  from such t r a n s f e r  o f angular  - 127 -  momentum from the r e a c t a n t dimer t o the product monomers f o r the M+D+3M reaction.  For  the exchange-decomposition  r e a c t i o n , the a p p r o p r i a t e e q u a t i o n s o f  change a r e :  Mass d l  (M  3t  +  M  ]  V 2D->2M+D,exch  VV  2i tr  :  r r -12 + Tr 2m ( 1— -12 — ^ ^5 2l 2 2 ~'>\l2) 1234 +  K  _  r  l 2\  K  r 24 2  r 24 2  J  6  l  +  + ( -34-34 2 T ~ ~ (  .) 6  (34)  (24)  P  ^13(24) 1 3 ( 2 4 ) ( 1 2 ) ( 3 4 )  1  VV : [|- M  2  9t  Oil] J  D = 2D*2M+D,exch'  L i n e a r Momentum  U + M,, v ) j  Mr  = -V • P =2D+2M+D,exch a  -  V  •  C  t  P  =2D-»-2M+D,exch  2D+2MfD(exch  (6.3.13)  -  21  r r 1 .£12 £ l 2 : Tr 2 T — — 1234  VV  (  +  1 2  (  +  128 -  ••'  ^3 A "~3 A l — —  S  TT~  (  P  + 3  )  6  2  -E4  )  (12) s ]  6  (34) s ]  ,  w  0  2 "  :J  +  "Pi £  [  .1-24-24 (  ~  )  [ (  +  ?2 V  6  ]  (24) s  P  °13(24) 1 (24) (12)(34) 3  21 V  '  m 4  t  6  (  1  2  ^ 1 2 ^  )  +  6  ( 3 4 ) ^ 3 4 H 3 4 ^ s " ( 2 4 ) ^24*24 >s 6  ]  ^13(24) i3(24) (12)(34) P  : r  4-  V x Tr ( 12)^12 ^ 1 2 1234 6  -  X  6  (24) -24 3  X  +  6  X  ( 3 4 ) ^34 -P-34  -^24 P  ^13(24) 1 3 ( 2 4 ) ( 1 2 ) ( 3 4 )  (6.3.  ct where P„„ _„ „ - i s the c o l l i s i o n a l t r a n s f e r =2D-»-2M+D,exch tensor a r i s i n g molecules,  p a r t o f the p r e s s u r e  from the exchange d e c o m p o s i t i o n i n v o l v i n g  namely  two dimer  - 129 -  =2D+2M+D,exch  (r 6 + •-.) - ( 1 2 ) ( 3 4 ) °R  2 Tr 1234  (V (12)(34) 13(24)  D  P  v/  V  )  P  ' 13(24) ]fl2)(34) (12)(34)  (6.3.15)  A n g u l a r Momentum  [  "D <4,int>^  1>F M <iM,mt> ( M  +  6  -2 T r 1234  + 6  (12) (34) 2  £(12)(34)  13(24)  k  +  2D-»-2M+D,exch  X  ( V  V  (12)(34) 13(24)  P  (12)(34) (12)(34)  X  6  X  6  ^ 2 4 ^ 2 4 ( 2 4 ) ~ £ l 2 £ l 2 ( 1 2 ) " -^34 ^13(24) ^ 3 ( 2 4 ) ^ 1 2 X 3 4 )  C  t  V •L =2D>2M+D,exch  21 •h  Tr 1234  r(  + s  8  +  ^±l ±2  }  -  s  -  -3  8  ^  V  (  1  2  )  ( 3 4  2  >i ]  s  ^3(24) ^13(24)^12X34)^12X34)  x  6  P 4 (34) 3  ]  - 130 -  2i  Tr  6  -l"-2  V ^ V V  +  V 2 2 < - V V ("V --  6  <12>>  1234  +  - i z  ±  (  ^  (  V  3  6  v v 7  )  +  (  S  3  +  S  a  )  (  (  _ i _ J L _  6  (  3  A  )  )  - y y -V-- w (  P  ^13(24) 13(24) (12)(34)  ;  (6.3.  ,ct the s t r u c t u r e o f L«" . , . i s referred =2D- 2M+D, exch  to the m o l e c u l a r  >-o v l  angular  momentum f l u x due to n o n - r e a c t i v e dimer-dimer c o l l i s i o n s , and  [-37 *  X  (M  M 2  +  )]  ^ 2D+2M+D,exch  = V • (P x r) =2D+2M+D,exch  Tr 1234  (  +  -  f  (  )+  )  ~T T r ~ l ! "- ~ r (  ¥ TT-TT> (  <^T  (  T T " IT*  7  P  +  +  '"  - >  ) : 3  r  ^  13(24) 13(24) (12)(34)  x (  x (  6  }  £(12) (12) s  6  )  £(34) (34)  (  6  !  )  ^(24) (24) s  - 131 -  2  i r  6  + 6  d2)  (34)  f  2  1234  £(12)(34)  X  ( V  V  (12)(34) 13(24)  )  P  ^13(24)^12)(34) (12)(34)  2i *  <?  V •  1  1234 ~  -  (L  2  X J £  6  +  12 (12)  £34  X  6  r  £ 4 ( 3 4 ) " 24  X  3  -E 4 (24) 1 6  2  P  13(24)Jl3(24) (12)(34)  L  C  t  )  2D-*-2M+D,exch' (6.3.17)  For the simple exchange, c o n s e r v a t i o n laws a r e again  established.  Mass  [  3t  ^2D,exch  ^ 1 4  ( i - r i ! £12  VV : Tr 2m 1234  v  21  2  •••z  2  0  (12)  - 4(1,— -13 -13 Z i £...) }6 * 2  ( 1 3 )  2  ^2!  2  2  - ( /I _ _ -24 _ . ^24 . . )  6  (34)  N  ,5.  (24)  6  I"  [  3V  "  ^  f  R  D  .  1  P  (13)(24) (13)(24) (12)(34)  e  x  c  h  (6.3.18)  - 132 -  L i n e a r Momentum  Jt  - 2D,exch = ~  %  [  2  c t  V • - i  * S D,exch  V  ]  P =2D,exch  VV  *  • Tr r i - ( £  • , : o , 1234  l  1 3  2  >( *>  2 !  ( P 6  Tr  )  1  (13)(24)  2  2  f  £(12)(34) £(12)(34) 2T  J  2  2  A +  *  ..]  p  R s ( 1 3 ) ( 2 4 ) (13)(24) (12)(34)  5  1234  £  ^<13)(24)  1  -E(13)(24)U P  "  ^{12)(34)- -(12)(34) 3  p  *(13)(24) (13)(24) (12)(34)  1 - 4 V x 2  Tr 6 1234  X  R  £<13)(24) - E ( 1 3 ) ( 2 4 )  - £(12)(34) r  X  £(12)(34)  J  P  U3)(24) (13X24) (12)(34)  - V •  +  * VV  c t  P =2D,exch  1 £ l 3 -13 Tr 2\ ~T 2 1234 r  ,  K  \ ;  ^(13)°(13) s  1 If  r  £ 2 4 £24 . . . 2 2 - -(24)°(24) s l  1  £-12 £ 1 2 (P .) 2! 2 2 ( 1 2 ) (12) s 2! 1  VJi  P  X 1 3 ) (24)^(13) (24) ( 1 2 X 3 4 )  P  ;  £34 £34 2  2  x  .)  ( 3 4 ) (34) s  -  i  V  •  Tr 1234  UlsPjA  "  {r  1  2  P  1  6  2  }  133 -  +  (13)  «  s  (  6  ll24£24^s ( 2 4 )  1  2  -  )  T  [r  3  4  P  3  A  }  6  s  (  3  4  )  P  ^(13)(24) (13)(24) (12)(34)  -~ V x  Tr X  ^13  1234  -111  6  ^ 1 3  x  (13)  6  -2l2  Ql3)(24)  : r  +  £  X  2 4  6  ^ 2 4 (24)  (12) "^34  X  6  ^ 3 4 (34)  p  (13)(24) (12)(34)  (6.3.19) Here  ct £2D,exch  5  "  1  2  ^  (6 4  R^(12)(34)  " ^  +  (  v/  V  (  12) ( 3 4 )  V  ( 12) ( 3 4 )  }  P  ^(13)(24) Vl2)(34) (12)(34) (6.3.20)  is  the part o f the pressure  transfer symmetric  f o r t h e exchange part of ( r  (  i  j  )  (  tensor  associated with  reaction  k  l  )  P  (  i  j  )  involving  (  k  l  )  >  s  the c o l l i s i o n a l  two d i m e r s .  1« g i v e n b y  The t e n s o r i a l l y  -  -i  -E(ij)(ki>U  te<ij)<ki>  (  134 -  )  - ( i j ) ( k l ) -£(ij)(kl) s  +  ( £  (ij)(ki) -BcijXkl)^  +  - i =  - j ~ - k ~ -1 2  ^ij)(kl)  (6-3.21)  Angular Momentum The r a t e o f change i n t r a n s l a t i o n a l  angular momentum d e n s i t y  a s s o c i a t e d w i t h the 2-dimer exchange r e a c t i o n i s  ljt±*  Kj^D.exch  = V • (p x r) =2D,exch —  ti  £  1 \ -13  Tr 1234  [ ( 1  0  " f i  ~T-"  + K i -jr)  ]  13  =  1 [(l  < F  6 Tr  1234  - 2f>  (  -H(13)  -IA...] - f A x (  r [ ( 1  x  ~T  \\3)h  6  £ ( 2 4 )  (  2  4  l —  1  ^3 A  Z =  (  T  X  ^(12)  6  )  (12) s  "~"3 A  x  1  V*-*  -(34)  T -  u  (34)'s V-u^.  U3)(24) (13)(24) (12)(34) : r  P  (12) (34) ( 1 2 ) ( 3 4 ) < ?  )  r i  +<5 2  )  £  P  ( X  (  7 V  v V  ' (12)(34) (13)(24)  (13)(24)^12)(34) (12)(34)  ) ;  -  i „ + zr T r * 123A  6  £l3  X  +  1  ^13  "£l2  < P  3  6  +  2  6  X  6  •  +  1  135 -  6  ^24  X  +  2  4  ^24  2  6  ^12  5  3  +  6  4  X  " £34 £34  J  P  (13)(24) (13)(24) (12)(34)' (6.3.22)  Comparing internal  the terms i n (6.3.22) t h a t corresponds  to a net change i n  a n g u l a r momentum w i t h the r a t e o f change i n i n t e r n a l  angular  momentum d e n s i t y , one f i n d s that the m o l e c u l a r angular momentum  flux  Lor, • i s g i v e n by =2D,exch 6  -  3  V • L  =2D,exch  =  V  +  ' * 234 - ( 1 2 ) ( 3 4 ) 1  ) [  * 12 <13)<24>>- ^34 ( 7  £l2  V  x ( V  V  3  4 (13)(24)  ) ]  P  ^ 1 3 X 2 4 ) ^ 1 2 X 3 4 ) ( 12) (34)  6  "4T  T r  1234  X  £-13 ^ 1 3  ( 6  1 +  e  x 1  2  P  ( 6 i 2  3  2  (13)  6  " £  6  (12)  l  6  •>  + 6  2  2  +  X  -E 4 - E 4 2  ) - r  2  3  4  x p  ( 6  2  + 6  4.  (24)  3 4  (6  (  3  4  )  p  { 1 3 ) ( 2 4 ) ^ 13)(24) ( 12)(34)* (6.3.23)  - 136 -  D.  2 Monomers and 1 Dimer  C o l l i s i o n s Involving  There are four p o s s i b l e kinds o f r e a c t i o n s :  simultaneous  d e c o m p o s i t i o n and r e c o m b i n a t i o n , r e c o m b i n a t i o n without exchange, exchange-recombination, to  and exchange without r e c o m b i n a t i o n .  By analogy  the r e a c t i o n types d e a l t w i t h e a r l i e r i n t h i s c h a p t e r , the  hydrodynamic equations f o r mass, l i n e a r momentum and a n g u l a r momentum d e n s i t i e s can be obtained  Simultaneous  immediately.  Decomposition-Recombination  The g e n e r a l e q u a t i o n s o f change a r e :  ^9t ^Si ^M^M+D.recomb, d ecomp  [ 2  (  * 1234  +  W s  + [(• 8 ) 3  3  (  W s  ]  ^(12)34^(12)34^2(34)  + ( W s  s  " <*lVs  )  " *2 2 s^l2)34 (  6  : 3  (12)34  12(34) ' (6.4.1) and f — M <$ >1 3t D D 2M+D,recomb,decomp  2fT  Tr 1234  (<|)  )  ( 3 4 ) ( 3 4 ) s (12)34 °(12)34 12(34) 6  Vr  4  v/  P  +  t ( < f >  )  (12) (12) s " 6  ( < t >  }  (34) (34) s 6  ]  12(34)'  ^(12)34^12)34  (6.4.2)  - 137 -  Substituting performing  i n t o the above equations the a p p r o p r i a t e o b s e r v a b l e s , and  c a l c u l a t i o n s analogous t o the M+D+3M and 3M+M+D r e a c t i o n s ,  c o n s e r v a t i o n laws a r e e s t a b l i s h e d  for this  reaction.  Mass  hnr  ( M  M  V  +  3  2M+D,recomb,decomp  1 £34 £34 ,  2f7 vv  Tr 2m U 1234  ~2  1~  +  °(34)  " J  (12)34  ::r  p  (12)34 12(34)  1 £l2 £l2 , , ~ 2T ~2 2~ °(12) +  « _ J. 2  rl— M ^ 1  VV :  J  9t D = 2M+D,recomb,decomp (6.4.3)  L i n e a r Momentum  ["aT =  (  M  M i i  - V •  +  ^ ^ ^ M + D . r e c o m b . d ecomp  P  =2M+D,recomb, decomp r?  cr _  V •  „ct P  =2M+D,recomb,decomp  2rT  VV  : Tr 1234  £-34 £ 3 4 , — — t<P  ,  r  2!  v  24)  + 3  r 6  , ]  (34) s  £l2 £l2 , . , , - — — £ l ^ 2 (12) r  l (  +  P  N  ) 6  ^12)34^(12)34 12(34)  ]  !  - 138 -  T  5 [ (34) ^ M^34^34 I I ^ Js J  u  1234  P  l l l ? P ls? } j ( 12) ^12-^12 ;  ? J  p  (12)34 (12)34 12(34)  2  V  X 1  23  [ 6 4  ( 3 4 ) £34  e>  X  6  P34 " (12)£l2  X  Pl2  ]  1 d  p  (12)34 (12)34 12(34)  (6.4.4)  =2M+D,recomb,decomp  ( 6  1234  +  (12)£l  ( 6  +  ( V  2  R£(12)(34)  +  V  12 (12)34  -*  0  ( V  )  V  (12)(34) (12)(34)  n  P  P  (12)34 1 2 ( 3 4 ) 1 2 ( 3 4 )  — 2 z  Tr (£ 1234  +  (  1 ( 3  4)  +  £ (34)  6  '••> ( 1 3 4 )  +  2  <r  '••  vfi-' (12)34 12(34)  ) 6  ( V  (234)  1(34)  ( V  ( V  2(34)  13  ( V  +  23  V  +  14»  V  2A»  p  12(34) (6.4.5)  is  the c o n t r i b u t i o n  from c o l l i s i o n a l  to the p r e s s u r e tensor a s s o c i a t e d w i t h t h i s  transfer.  reaction  - 139 -  A n g u l a r Momentum  ht  -  X  (M  M -  +  -^2M+D,recomb,decomp  = V • (p x r) =2M+D,recomb,decomp —  + i -V 2n  1 1 TT - IT  N  Tr 1234  (  5  1  -34 -34 — —  1  J z  26  2  4  6  ]  (34) s  —  x  (  +  )  t -Ei £  6  2  (12)  ]  P  .  3— - - l(34)  2^  +  2  S  £  V  x  l(34)  ( V  13  +  V  14>  ( 3 A )  3  -2(34)  /  x  V  2(34)  ( V  23  +  V  24  )  P  (12)34 *i2(34) 12(34)  6 +6 3  Tr 2h" 1234  —  :i  L  +  +  3  (12)34 (12)34 12(34)  + 6.  Tr 1234  (  f P PA>  -12 -12  N  (  ~ T T ~ TT> <P  X  (r  _  .34 -E 4 — r ~ > s 3  J  6+6  4  x  (  x  ~ £l2 -El2  —2~\  P  (12)34 (12)34 12(34) (6.4.6)  As w i t h t h e c o l l i s i o n two  terms c o n t r i b u t e  total  types t h a t have been s t u d i e d to the change  a n g u l a r momentum  i n internal  i s conserved:  earlier  on, the l a s t  a n g u l a r momentum  and the  - 140 -  Ut  +  ^M^iM.int/*  M  <  >  D ^D,int ^2M+D,recomb,decomp  6+6  Tr 2fi 1234 6  + 6  l  2  - K s ^ ) ^ -  I  6  (  1  2  )  ))  P  ^12)34" (12)34 12(34)  C  t  V • L =2M+D,recomb,decomp  s 2  6  <V  1234  +  2  + j Z  (  3  (  ^3  +  U  24  )  P  5  x  6  (12>^12 -£l2  2 6  X  P  )  T  P  ( 3 4 ) -34 - -34 ^(12)34 ( 1 2 ) 3 4 1 2 ( 3 4 )  $ri/\  +  i  3  +  '  J  Tr 1234  +  'I ^  S  W  ^13+^4*  (12)34 12(34) 12(34)  <6  ~ 2*  W  <V  P  - 2s. - 2s.  o  -1(34)  X  V  l(34) ^13  +  V  14  ;  +  1  5  3  ^ *>  T  £  x  2(34)  X  V  CV + V ) 2(34) ^ 2 3 24 V  V  ;  (6.4.7) y  /  P  ^(12)34 i 2(34) 12(34)*  -  141 -  The m o l e c u l a r angular momentum f l u x L , „ , , for this =2M+D,recomb,decomp r e a c t i o n i s g i v e n by the c o l l i s i o n a l  transfer  part  =2M+D,recomb,decomp  \  Tr 1234  (£  1 ( 3 A  - <r  )  +  <S  — >  2 ( 3 4 )  '(12)34  £  X  (134) 3 4  + ..O \  2  3  A  )  t V  £  34  3  ( V  13  x [V  4  +  V  3 4  14  (V  ) ]  2 3  + V )] 2 4  12(34)^12(34) (6.4.8)  and the f l u x  term obtained by summing the p r o d u c t i o n l o s s terms i n  (6.4.6-7) i . e . , (L - L  - V • (L - L  C t  ct, ) 2M+D,recomb,decomp'  ) 2M+D,recomb,decomp  6  [ £ 2 f l  1234  X  12 ^ 1 2  ( 6  (12)  1  +  6  2 ) ]  2  s (12)34  6  ~ t£  3 4  x p  3 4  (6  ( 3 4 )  3  +  6  ^  P  (12)34 12(34)  4 ) ]  g  (6.4.9)  Recombination Without  Exchange  The d i f f e r e n c e between r e c o m b i n a t i o n - d e c o m p o s i t i o n and simple r e c o m b i n a t i o n i s t h a t i n the l a t t e r  case, the dimer molecule that  - 142 -  p a r t i c i p a t e s i n the c o l l i s i o n does not d i s i n t e g r a t e but a c t s as a t h i r d body.  C l e a r l y the c o n s e r v a t i o n f o r the p h y s c i a l a t t r i b u t e s o f mass,  l i n e a r momentum and angular momentum h o l d s .  The r a t e s o f change o f the  d e n s i t y o f an a r b i t r a r y s i n g l e - m o l e c u l e o b s e r v a b l e due to t h i s  reaction  are  2M+D+2D,no exch  i  2b  Tr 1234 l * l V s (  +  (  W s  ]  Vr  /  (12)(34)^(12)(34)i 2(34) P  -  K ^ V s  +  12(34)*  7  <*(12)(34) (12)(34)  (6.4.10) and  at  J  D  i  D 2M+D>2D,no exch  Tr 1234  " " 2rT  6  )  V/  f  ( 3 4 ) ( 3 4 ) s '(12)(34) (12)(34)  12(34) P  +  (<t>  6  )  (12) (12) s  tf  12(34)*  :I  '(12)(34) (12)(34) (6.4.11)  The hydrodynamic equations o f change f o r mass, l i n e a r momentum and a n g u l a r momentum are a l s o  listed.  - 143 -  Mass  Ut  ( M  M  +  M  )]  D 2M+D+2D,no  2*  1  r  1  2  V  V  3  2!  4  exch  r  r  2  2  J  (12)  M  (12)(34) (12)(34) 12(34)  3  ~ 2 Ut  1  (M  D^UM+D>2D,no  exch (6.4.12)  L i n e a r Momentum  UrT  ( M  +  M  ) ]  *D £ 2 M D + 2 D , n o  exch  +  = -V • P =2M+D-»-2D,no exch C t  - V • P =2M+D*2D,no exch  +  VV :  r r 1 ,-12 -12 Tr ^ - ( 2! 2 2 1234 v  +  <  7  £(12) (12) 8 )  6  P  ^12)(34)" (12)(34) 12(34)  i 2ti  I  1234  "  2  V  V  :I  P  '( )li-i2-iil2 s (12)(34) (12)(34) 12(34) 1 2  X  6  1234  (12)-12  X  -El2 ^ ( 1 2 ) ( 3 4 ) ( 1 2 ) ( 3 4 ) 1 2 ( 3 4 ) J  p  (6.4.13)  ^ _ , i s d e f i n e d by =2M+D-»-2D,no exch  -  P  144 -  C t  =2M+D+2D,no exch  y  Tr 1234  ( 6  (12)£l2  +  5  (<S  +  ( V  R£(12)(34)  +  V  12 (12)(34)  }  ' "  v0  ( V  )  V  (12)(34) (12)(34)  )  P  (12)(34) f2(34) 12(34)* (6.4.14)  A n g u l a r Momentum  [•§7 £  x  ( M  +  M i  ^  )]  ^ 2M D-2D,no  V • (P  2M+D-»-2D,no exch  -•=rV. 2n  2 6  + 6  1  2  +  e  i 2h" J ' / 1234  6  o  (p  x  2  T  /10  ,6, ,). i0  ^ 1 2 ) (12)'s P  (12)(34) (12)(34) 12(34)  (34) £-1(34)  +  +  I i i I i i  2 f!  y Tr * 1234  x r)  C^-^)  Tr 1234  exch  +  X  V  l(34)  ( V  13  +  V  14>  +  1 r  3  x  £2(34)  J  X  V  (V + V ) 2(34) ^ 2 3 24  ;  p  (12)(34) (12)(34) 12(34)  V ! i  ^^^n T ^ s  J  P  ^(12)(34) (12)(34) 12(34) (6.4.15)  - 145 -  TF ^ M ^ M . i n ^  +  M  <  D J-D,int ^2M D-2D, no >)  exch  +  6+6  ^  1234 (P, (12)(34) (12)(34) 12(34) 7  6-6  1  J, Tr 2ti 1234  ~ 2s  < V '»*)>  - 2s.  £„ - 2s +  P  «(34>>  <V  (12)(34)  2n  £  3  V/2  ^ u + ' V - 2s,  1  ^23  +  V  P  12(34) 12(34)  12  X  ]  P  ; T  P  ^12 s (12)(34) (12)(34) 12(34)  V •L =2M+D-»-2D,no exch  +f  2 6  Tr 1234  6  1  + (34) 3 1(34) £  2 6  +  2  +  6  3  p  x  V  l(34)  ( V  + V  1 3 14>  (34) ^2(34)  P  x  V  2(34)  (12)(34) 1 2 ( 3 4 ) 1 2 ( 3 4 )  ( V  23  +  V  24  )  - 146 -  Simple Exchange R e a c t i o n The c o n t r i b u t i o n s to the e q u a t i o n s o f change from t h i s r e a c t i o n a r e :  2M+D->-2M+D,exch  (  * 1234  »lVs  +  (  W s  ^(13)24X13)24^12(34)  ]  12(34) +  " W s  "  (  6  )  *l l s^(13)24J(13)24  (6.4.17)  and  L  8t  D  2i  D  Tr 1234  2M+D+ 2M+D, ex ch  (  X 3 4 ) ( 3 4 ) 8 ^(13)24^(13)24^2(34) 6  )  12(34)' +  I (  6  )  *(13) (13) s"  ( < | ,  6  )  ]  (34) (34) s Xl3)24J(13)24  (6.4.18)  Besides the c o l l i s i o n a l  t r a n s f e r p a r t , t h i s i s e s s e n t i a l l y the same as  the exchange r e a c t i o n i n v o l v i n g  a dimer and a monomer, o n l y now there i s  an e x t r a monomer molecule which p a r t i c i p a t e s i n the c o l l i s i o n event and does not r e a c t i t s e l f .  E x p l i c i t l y , the r a t e s o f change o f the  hydrodynamic d e n s i t i e s o f mass, l i n e a r momentum and angular momentum associated  w i t h t h i s r e a c t i o n a r e as below.  - 147 -  Mass  [  37  (  M  M  +  V W D •»-2M+D,exch  2i  4r-  VV : Tr * 1234  2  m  [  ( £  ?  P  £  £  £  (13)4 (13)4 " 1(34) 1(34)  ) +  6  —1  (134)  T J  P  '(13)24 (13)24 12(34)  21  £  Tr 2m (1 2! 1234  VV  V  1 3 -13 2 2  6  (13)  tf  7 (13)24 (13)24 12(34) J  - (1_ 2!  - I 2  P  =34 =34 2 2 • • • ^ o (34)  VV • f— M % * 3 t D = 2M+D>2M+D,exch l  ll  M  (6.4.19)  L i n e a r Momentum The  contribution  [•§£ =  ~  ~ ~  R V  V  2i  +  % -  to the e q u a t i o n o f motion from t h i s r e a c t i o n i s  )  ]  2M+D+2M+D, exch  S2M+D*2M+D,exch ct * S2M+D->-2M+D,exch  £  VV  : Tr  (  £  3  1234  (  £  (13)4 (13)4 _ 1(34)  E  3  6  £U34)  3  )  r  3  T  }  P  - (134) (134) s (13)24 (13)24 12(34)  -  2i  V • Tr 1234  6  148 -  (  ( 1 3 4 ) 11(13)42(13)4 Js " (ll(34)-El(34)  U  )  p  *( 13)24 ^(13)24 12(34)  V x Tr 1234  6  (  R £(13)4  2(13)4 " £l(34)  x  r  x  -El(34)  )  p  ^(13)24- (13)24 12(34)  c t  - V •P =2M+D+2M+D, exch  +  2i w  Tr 1234  1 —13 —13 U ~ ~ T  {  )  (  6  £(13)  )  (l3) s J  P  ?13)24 (13)24 12(34) X ^3 A ^3 A ~ 2 T ~T~ ~T~ 2(34) ( 3 4 ) s )  (  2i V  ' m4  t  6  (  1  3  (  6  ^^13^8  )  J  -  6  )  (34)tl34£34U  P  (13)24 (13)24 12(34)  V  *  X  1234  f  6  (  1  3  )  £  l  J  3  X  £  l  3  "  6  (  3  4  )L  U X  "  3  4  1  P  (13)24 (13)24 12(34)  (6.4.20)  where _P(ijk) ^ by  s  the t o t a l momentum of atoms i , j , and k, and i s defined  - 149 -  Jkijio  =  £  i  +  £  j  +  £  (6.4.21)  v  -Ct  a n c  * S M+D-»-2M+D exch' ** t  e  c o  2  associated  :  s  -'-l '- i  o n a  -'-  t r a n s f e r p a r t o f the p r e s s u r e  tensor  with the monomer-monomer-dimer simple exchange r e a c t i o n i s  d e f i n e d by  C t  P =2M+D*2M+D,exch  2  Tr 1234  (a  +  f  (12>^12  ( 6  +  —  }  R-(12)(34)  ( V  V  12 (13)24  +  ( V  )  V  (12)(34) (13)24  P  ( 13)24^12(34) 1 2 ( 3 4 ) * (6.4.22)  Angular Momentum  £  lit  V  X  %  +  )]  *D ^- 2M D+2M+D,exch +  *^22M+D->-2M+D,exch  x  2i  -13 -13 1 —) ~T x 2! T  1 Tr (1234 1!  -34 -34 — —  ,  ,  .  ^(lsAlS^s  x  , . . <P(34) (34) s 6  '(13)24^13)24^2(34)  )  - 150 -  +  ^ 1S4  t 6 ( 1 3 )  6 2  1 +  Tr 1234  6  £  1  3 X £  l  3  2  w  +  +  +  6  3  4 X £ 3 4 1  p  *a3)243(l )24 12(34) 3  6  v  (34) -1(34)  x  ( V  V  1(34) 1 4  }  (34) £  3  <?  6 ( 3 4 )1  £ 1 0 X" 1 ( 72, n1V2, J -12  3  2 6  "  2  2  2 6  l  2(34)  v/  x  V  2(34)  ( V  23  +  V  24  )  P  (13)24 T.2(34) 12(34) (6.4.23)  r ° [  "at  ( M  <  M 4l,int  21 -h  >  +  M  <  D -D,int  >)]  2M+D*2M+D,exch  6.-6„ Tr 1234  s. - 2s_ - 2s.  _s„ - 2s. - 2 ^ +  <VW  ~  1"^ ^23  ^(13)24^2(34)^2(34)  (  +  V  151 -  -  21  6  Tr 1234  1  +  6  - 1 ~ -1  3  6, •+ 6, ( 6  "  P  C  js - s,  ( 3 4 ) " - V"^  :r  (  ^3  +  +  V  ( 6  "V*  3 " V  P  (13)24 (13)24 12(34)  t  - V •L =2M+D->-2M+D,exch 21 T~  1  2  ^  v 4  "(13)-13 " A 3  6  1 +  + 2 Tr 1234  6  1  26 +  A  y  J  -^34 " ( 1 3 ) 2 4 ( 13)24"l2( 34)  2  L , x (V V,„) -12 12 12 1 0  x  2  2 6  "(34)-34  + 6  (34) 3  + 6.  -1(34)  X  V  X  V  V  l(34) 14  .  3 — ^ ^ 2 ( 3 4 )  2(34)  ( V  23  +  V  24>  M  (13)24 1 2 ( 3 4 ) 1 2 ( 3 4 ) (6.4.24)  The p h y s i a l reaction (L - L  C t  associations  o f the terms can be r e f e r r e d  to the exchange  i n v o l v i n g a dimer and a monomer, and the f l u x can be obtained as b e f o r e . ) 2M+D-»-2M+D,exch  Recombination w i t h Exchange Corresponding monomers  to t h i s r e a c t i o n a r e the equations o f change f o r the  - 152 -  M  <$  >  Ut M M UM+D->-2D,exch  -1  Tr 1234  [  (  +  W s  (  *  W  2  V /  e  (13)(24) (13)(24A (34) 2  P  [<4>lVs  +  (*  6 2  2  )  ]  P  S  12(34)'  J  (13)(24) (13)(24)  (6.4.25) and f o r the dimers  T — l  < $  M  3t  D  >1  D  2M+D+2D, exch  ( < t >  1234  +  6  )  U  < S  (34) (34) s U3)(24) U3)(24)°12(34)  [ (  6  )  *(13) (13) s  +  ( < t >  5  )  (24) (34) s "  J  (<f,  (34)  p  6  12(34)  )  (34) s  ]  P  *(13)(24) (13)(24) 12(34)  (6.4.26)  T h i s may be compared dimers.  w i t h the exchange decomposition  Since c o l l i s i o n a l  i n v o l v i n g two  t r a n s f e r o f l i n e a r momentum and angular  momentum obeys the c o n s e r v a t i o n laws f o r these a t t r i b u t e s , the c o n s e r v a t i o n o f mass, l i n e a r momentum and angular momentum  follows.  Mass  UF  (  M  " " 2  M  W  V  +  :  [  2M+D-»-2D,exch  ]  (M  )1  ¥ t D^' 2M+D->-2D,exch (6.4.27)  - 153 -  L i n e a r Momentum  ["37  ( M  M^  V  +  M  )]  D£ 2M D+2D,exch +  * S2M+D-»-2D,exch C t  « _ y • P =2M+D+2D,exch  r  r  r- VV : Tr 2! * 1234  6  ]  (l2) s  £(13)(24) £ ( 1 3 X 2 4 )  £(12)(34) £(12)(34)  + (  (!  +4  3  P  (13)(24) (13)(24) 12(34)  " (12) ^ l z W s  V • Tr 1234  6  +  6  (  R  tf  ^£(13)(24)£(13)(24)^(12)(34)£(12)(34)K  :I  )  P  (13)(24) (13)(24) 12(34)  V x  Tr 1234  =  6  (12) £l2 £ l 2  X  +  6  R £ ( 1 3 ) ( 2 4 ) £ ( 1 3 ) ( 2 4 ) ~ £ ( 12)(34) £( 12)(34)  (  '(13)(24)  X  X  p  (13)(24) 12(34)  }  - 154 -  c t  -V  • P =2M+D+2D, exch  + -=y- VV : Tr ^ 1234  i l ^ n ^ O s A l S ^ s  (  P  6  + £  £  24 24  ( £  6  )  (24) (24) s  )  -34-34 - -(34) (34) s  <r  i  Hr  :I  P  (13)(24) (13)(24) 12(34)  1 3  P  1 3  } 6  + {r^}  ( 1 3 )  6  ( 2 4 )  - { r ^ } «  ( 3 4 )  ]  1234 J  P  (13)(24) (13)(24) 12(34)  4-  V x Tr 13 1234 £  X  6  -  £ 3 4  +  ^ 1 3 (13)  x  5  £ 3 4  P.  (  3  £  X  24  £  6  2 4 (24)  4 )  ;I  P  (13)(24)' (13)(24) 12(34)  (6.4.28)  The c o l l i s i o n a l  t r a n s f e r part o f the p r e s s u r e tensor a s s o c i a t e d w i t h the  exchange-decomposition  involving  two dimer m o l e c u l e s i s d e f i n e d by  ct =2M+D-»-2D,exch  p  - Tr 1234  ( 6  +  T.  (12)-12  ( 6  +  •"•  R-(12)(34)  (13X24)  ^  ) ( V  +  V  12 (13)(24)  -*  0  ( V  }  V  (12)(34) (13)(24)  o  12(34)^12(34)* (6.4.29)  -  155 -  A n g u l a r Momentum  2M+D>2D,exch  = V • (p x r) =2M+D-»-2D,exch  —13  V • Tr 1234  1  1  —13 +  (— (yr - Tf) ...)  . ,-24 .1 1 . . ^24 + ( — (TT " Yf) + •••) — J  _ ,-34(1  1  2  2!  v  1!  J  ) + ...)  fi l  [(^+£3)  x  —  (13) 8  r / x  6  [(£ +E4> 2  £34x  r /  [  x ,  A  (p  3  ]  (24) s  + p  )  4  6  ,  (34)  ]  s  P  (13)(24) (13)(24) 12(34)  6 Tr 1234  1 +  6  2  I n -12  2  2 6  1  +  +  6  x  (V V ) 12 12  v  1 0  v  1 0  (34)  -  + —2  £1(34)  1341 3  x  V  V  l(34) 14  x V  r  -2(34)  X  V  + H 1234 " ^ I S  " T ^ s  3  V  2(34) 23  6  6  V A +  (£  P  V  V -*  1  v/  '(13)(24) l2(34) 12(34)  X  24 -B24  "V^s  6 +6 (£  X  34 -%4  "V^s  1  P  ^(13)(24) (13)(24) 12(34) (6.4.30)  - 156 -  [  3t  +  ^lAl.int^  V-D.int^WD^D.exch  6-5  *  1234 ±1 - 2s  - 2s^ /  p  ^(13)(24)' 12(34) 12(34)  JLo ~ 2jjn ~ 2s, +  (  V  W  ,  1  ~  6-6.  =^^23  6-6.  6-6  P  X l 3 ) ( 2 4 ) ^13)(24) 12(34)  6  +  Tr 1234  V 2 -2—^12  X  ( V  V  12 12  26. + 6 , . . . 1 (34) 3 £-1(34)  2 6  +  2  6  i  f  X  V  1(34) 1 4  r  p  (13)(24M2(34) 12(34)  (34)  3  V  )  £2(34)  X  V  V  2(34) 23  * =2M+D>2D,exch  6  ^  Tr  f [  +  6  l 3 — r,. —2— ^13 t —  0  X  x  p,.  +  V +  *13 "  6  —  ~1  4 —  6  V 4 ^24 * -^24 " ~"~2 ^34 r_  OA  x _p_„  X  ^34  J  P  '(13)(24r ( 1 3 ) ( 2 4 ) 1 2 ( 3 4 ) (6.4.31)  - 157 -  I t has been the c o n j e c t u r e of c h e m i c a l k i n e t i c i s t s dissociation the  fact  These  internal  states.  unimportant: determine  motions o f the dimer m o l e c u l e s have been  f o r both r e a c t i v e and n o n - r e a c t i v e c o l l i s i o n s .  a necessary feature  internal  the  r e a c t i o n i s preceded by v i g o r o u s r o t a t i o n s and v i b r a t i o n s o  reactants.  identified  that  These  are i n  f o r c o l l i s i o n s i n v o l v i n g molecules possessing  Whether these i n t e r n a l  i t i s the k i n e t i c  the a c c e s s i b i l i t y o f  motions  are " v i g o r o u s " or not i  energy of the c o l l i d i n g m o l e c u l e s reactions.  that  - 158 -  CHAPTER 7 THE ENERGY BALANCE EQUATION  The  k i n e t i c equations f o r P f ( i ) ,  the d e n s i t y o p e r a t o r r e p r e s e n t i n g  a t y p i c a l monomer i , and P ( j k ) , the d e n s i t y o p e r a t o r f o r the dimer D  molecule  ( j k ) have been used  to o b t a i n equations o f change f o r the  p h y s i c a l o b s e r v a b l e s mass, l i n e a r momentum and angular momentum. i s how the e n t i r e system p(N) i n d i v i d u a l molecules  e v o l v e s w i t h time i n s t e a d o f how  change t h a t d e s c r i b e s the change i n the e n e r g i e s  a s s o c i a t e d w i t h the s p e c i e s because the i n t e r m o l e c u l a r p o t e n t i a l the m o l e c u l e s  A.  But i t  renders  interdependent.  The Production and Loss of Energy Associated with the Monomers and with the Dimers  In  this  defined.  s e c t i o n , the energy  The e q u a t i o n s  o b s e r v a b l e s f o r the two s p e c i e s w i l l be  showing the g a i n and l o s s a s p e c t s o f the energy  a s s o c i a t e d w i t h the dimers and monomers expressed those  i n Chapter 4 w i l l  be o b t a i n e d .  i n an analogous  From these e q u a t i o n s ,  form t o  energy  c o n s e r v a t i o n i s e s t a b l i s h e d f o r each k i n d o f c o l l i s i o n .  1. The Energy Operators  The  o b s e r v a b l e s a s s o c i a t e d with the monomers and the dimers o f t h e  system have been d e f i n e d i n Chapter  3:  -  159 -  1*±  6 D  =  9  Z tf* A, . bjk (jk) 9  <  k  These e q u a t i o n s have to be m o d i f i e d intermolecular  The  f o r the energy o p e r a t o r s to i n c l u d e  potentials.  o p e r a t o r d e s c r i b i n g the energy o f the monomer i n the system i s  g i v e n by  tf-j'a  <*i*T  V  1  i  <7  -'- > 1  J  so the e x p e c t a t i o n v a l u e o f the monomer energy i s  1 • • «n  i  Tr ( 1. . .N  = N  = TrK  1  f  1  +  +  N  I Z V ) 1=Z u  3),  <f  p  ( N )  T r j J v ^ S f 1...N  Having i d e n t i f i e d  value  K  p (l)  1  v a l u e (Chapter  j  t P ^ W '  (7-1.2)  i=z  the o b s e r v a b l e o f k i n e t i c  energy and i t s e x p e c t a t i o n  the p o t e n t i a l energy o p e r a t o r and i t s e x p e c t a t i o n  can be d e f i n e d .  For the dimer, the o p e r a t o r d e s c r i b i n g i t s energy and the expected value  are r e s p e c t i v e l y  - 160 -  (7.1.3) l*j*k and  (N)  p  (N)  bl2  coll* 1... N  The  k i n e t i c energy  i=3  (7.1.4)  and p o t e n t i a l energy  same way as the monomer. the H a m i l t o n i a n  can be s e p a r a t e l y a s s i g n e d i n the  The sum of (7.1.1) and (7.1.3) indeed  f o r the system.  gives  The k i n e t i c energy a s s o c i a t e d  w i t h the monomer o r dimer i s a d e q u a t e l y d e f i n e d r e s p e c t i v e l y by the d e n s i t y o p e r a t o r s d e s c r i b i n g a t y p i c a l monomer P f ( i ) and a t y p i c a l dimer P ( j k ) .  On the o t h e r hand, the p o t e n t i a l energy has to be  D  defined  N  by [ p ( > ]  c o  ll.  For the time b e i n g , focus i s put on the energy of the s p e c i e s . reduced  The energy  The energy  either  balance e q u a t i o n f o r e i t h e r s p e c i e s should be  to the e q u a t i o n s f o r k i n e t i c and p o t e n t i a l  this species.  associated with  energy  densities for  a s s o c i a t e d with the monomer or w i t h the dimer  has been r e c o g n i s e d a l t h o u g h t h i s i s not a one-molecule p r o p e r t y . t h i s d e f i n e d , we may l o c a l i s e  i t at where the molecule  is.  Such  Having  -  161  -  l o c a l i s a t i o n i s c o n s i s t e n t w i t h the l o c a l i s a t i o n scheme adopted I n t h i s t h e s i s and w i t h the f i n d i n g e a r l i e r on t h a t the p h y s i c a l a t t r i b u t e s a r e localised  a t the p o s i t i o n s o f the m o l e c u l e s p a r t i c i p a t i n g i n the  c o l l i s i o n which I s i n t u r n a d i r e c t r e s u l t from the l o c a l i s a t i o n we have used t h r o u g h o u t .  2.  Energy Balance f o r the Monomer  The l o c a l energy d e n s i t y per u n i t mass o f t h e monomer i s d e f i n e d by  - t f  N  Tr [(^ 1. .. N  +  I  I 1=2  V l i  )  6^9™  p  (7.1.5)  The time e v o l u t i o n o f the energy d e n s i t y i s t h e r e f o r e  1. .. N  1=2  N (7.1.6)  On a p p l y i n g  the von Neumann e q u a t i o n  ifi  and u s i n g the r e l a t i o n s  <P^  ;  = E ^ ^ ,  <PjX  cl  = ^  c l  the energy b a l a n c e e q u a t i o n f o r t h e monomers, namely  ^  > «"1  ^  =  £  cl  + U  cl'  - 162 -  I'M  4rr 3F  (  W  N  Tr 1...N  p  ^ l V s ^ / f l  +  (  ?,",  V  1=2  +  t (  Z  l l V .  +  Vi>s  / c l cl  A  i  v  £  c l  uV  E  1=Z  = T r (JI 6,) JC, ^ 1 1 s 1  + N  p  (  N  >  p  / c i ^ c i  (  N  )  Cl  p_(l) i  T r (K.6.) I <P ,\T ,p , „ 1 l's . c l c r 1.. .N cl  (N)  N +  J * 1... N 1=2  (  V  6  }  li l s  V cl  ^  +  p  l ^cl^l>  W  (7.1.7)  i s obtained.  As Tr  d  P  1  l^i^s ^i f( ^  d e s c r i b e s the f r e e motion change i n  energy d e n s i t y of the monomers, the remaining terms must be due t o collisions:  the f i r s t  contribution  to the k i n e t i c energy d e n s i t y change,  the  of these i s a t t r i b u t e d  change i n p o t e n t i a l energy.  to be the c o l l i s i o n a l and the second one t o  Using the same arguments as i n Chapter  2, the e q u a t i o n s o f change f o r k i n e t i c energy and p o t e n t i a l  energy  d e n s i t i e s o f the monomers a r e o b t a i n e d .  The  change i n k i n e t i c energy d e n s i t y I s g i v e n by  l f i  l t < W "f  < W s  V f  (  1  ) +  f  <*lVs ^ 1 2 ^ 1 2  +  - 163 -  + I c'  (n-1)!  (7.1.8)  Tr (J*. 6,) I f . i r . p ,+ 1 I s , c l c l c' 1...n cl  t h i s has the same s t r u c t u r e as the e q u a t i o n s of change f o r any single-molecule equation  observables  f o r the monomers, and the form o f the  showing the g a i n and l o s s a s p e c t s  collisional  processes  of various  kinds of  i s r e f e r r e d to Chapter 4, p u t t i n g  <j)^=i^.  For the p o t e n t i a l energy, the p o t e n t i a l energy d e n s i t y per u n i t mass of the monomers i s d e f i n e d as  (7.1.9)  coll*  N e c e s s a r i l y monomer 1 must be i n the process terms v a n i s h .  Now the sum r u l e d e r i v e d  to d e r i v e the e q u a t i o n  o f c o l l i s i o n , otherwise a l l i n Chapter 4 may be a p p l i e d  o f change f o r monomer p o t e n t i a l energy d e n s i t y  i n a form t h a t g i v e s e x p l i c i t g a i n - l o s s terms, which i s  ifi  IF  (  M  M V -H  V  6 V 112 2  ^12^12^2  +V 12 13 2  + f  J  12 12>  p  12  p  J  123 123  + <P  1(23)  1(23)  123  - 164 -  V  + Tr 6 1 123  12  + V  13 +  2  +  ( P  v +  ( P  H  < F  12(34)  <1-T  p  2 P  (P  3  )  J  1(23) 1(23)  }  (12)3  14  12(34)^  +  +  +  (  +v  +v V  ^2(34)  7  123 123  1(23)  12 1 3  Tr 6 1234  (P  ^l(23fl(23^  4 )  +  (12)34  +  2 P  ^ 1 2 ^ 2 ^ 2  J  *T.2(34) 12(34)}  13(24)  +  2 p  (13)24  }  +  +  V +V 2  1  1  123  2  123 123 123  V  +  Tr 123  6  12  1  + V  [  [(?  ^123^123  V +V V  12^ 23 2  (<S  +  J  123 123  +  <  , . 3 6  +  t e  + 2(6  12 3  2  6  3  ^123  ] v /  l(23)  V  13^ 23, 2 }  <P  l  *123 123" l(23)  1(23)  1  V +v 23  ^  < P  i(23) l(23)  1  x 5  2  v  2  l  13, }  +7  +  ;  V +v  V  V +V of A,  4.  (12)3 ( 1 2 ) 3  13  2  + fx 2 +  ^ 1 ( 2 3 ) 1(23) 123  f < P  ^13 23 2 ^ (12)3 (12)3 i(23) <f  yL  7  (12)3 (12)3  -  V  Tr 6  +  V  165 -  V  12 13 14 2  ^12(34)^12(34)^(12) (34)  1234 +  A[£  (H  n  +  13(34) 13(24)' '(12)(34)  4J  7  12(34) 12(34)  +63  ]  13(24)^13(24)  1  (12)(34)  v +  Tr  (fi V  1234  2  2  3  +y 3  3  14  V  v +v +v 12 13 1 4 ^ 1 2 V  „ (12)34 (12)34  ;  24 2  K  - 6  fi  4  2  rfi 4  9 1  +y  1  +V  34  12  13 2  " °1  J  14, ;  T  (13)24 (13)24  +V  12 13 14 1 2 [6  V  J  "(12)(34) (12)(34)  +V  +  2  f  T  (13)(24) (13)(24)  ]  +V  12 13 14 If  12(34)*12(34)  +  2  <  *13(24) l3(24)  v.,+v +v,. 12 13 34 p fp 3 2 (12)34 (12)34  ]  n  .  r  V  + +  26 ^ ^°4  4  +  V  2  4  2  +  V  3  4  12(34)  / ^(13)24 (13)24  12(34)* (7.1.10)  - 166 -  For  n o n - r e a c t i v e c o l l i s i o n s , the term  n  i 1.. .n  is  V  l i  1=2  the change o f the d e n s i t y o f the p o t e n t i a l energy a s s o c i a t e d w i t h a  typical  monomer when t h i s molecule undergoes c o l l i s i o n s .  compared w i t h the change i n k i n e t i c  T h i s may be  energy d e n s i t y a t n o n - r e a c t i v e  collisions  n , 1.. .n  1 1 s  c' V  c' c'  What these two terms correspond  to can be i n f e r r e d  by the commutation  relation  [  j 7 l i l ' n : ' ] - = 2 i=2 }j^U l>V-\ V  6  [8  " V ^ . L  +  2  Vs>' (7.1.11)  Here the symmetrised o p e r a t o r s a r e d e f i n e d by  (V  _ 1  [6 ,* ]J ±  =^  g  T V « ) • V + V . • (V 6) i  £ ±  + 3£  =  I  [  £  i  .  (  V  i  ±  £  "  ±  • V ( V 6 ) + 3 ( V 6 ) • yp^ ±  ±  6)V] ,  (7.1.12)  s  and  (tV,^]_  6 ) = {{[V,« ]_ s  ±  6 + 6[V,^]_ + 2[V,  £ i  • 6 ]_} £ i  - 167 -  +  1_ 8m  1_ 8m  (V V)  • 26  + 3  • 6 ( 7 ^ ) + 3(V V) • 6 ^  ±  P  ±  ±  +  6  • (7 V)  P i  ±  ±  (7.1.13)  Add  (7.1.11) t o [ 0 * 6 ) 1  1  s  , V , ] , one g e t s c  (7.1.14)  T h e r e f o r e , the change In energy d e n s i t y owing involving  to n o n - r e a c t i v e  a monomer i s due to the i n t e r c h a n g e o f k i n e t i c and p o t e n t i a l  e n e r g i e s o f t h i s molecule as w e l l as the s p a t i a l gas.  inhomogeneity o f the  Due to such inhomogeneity, the p o t e n t i a l energy d e n s i t y  u n l i k e the hydrodynamic  For r e a c t i v e c o l l i s i o n s , the  monomer p o t e n t i a l energy d e n s i t y change i s  Tr 1.. .n  w. + * 1 ] c' c c  -  n E 6 i=2  change,  d e n s i t y o f l i n e a r o r a n g u l a r momentum, i s  non-zero even i f c o l l i s i o n s a r e l o c a l .  i  collisions  -  The  gain  168 -  i n monomer p o t e n t i a l energy when the product monomers j a r e  formed and the l o s s i n monomer p o t e n t i a l energy when the r e a c t a n t (1)  i s destroyed  first  pretty straightforwardly.  However, the  term, which p a r a l l e l s the change i n p o t e n t i a l energy d e n s i t y f o r  non-reactive for  can be assigned  monomer  c o l l i s i o n s , i s associated  single-molecule  w i t h the product monomers whereas  o b s e r v a b l e s , the a s s o c i a t i o n i s with the r e a c t a n t s :  Thus the r i g h t hand s i d e o f e q u a t i o n (7.1.14) becomes  The  s i g n i f i c a n c e o f t h i s c h a r a c t e r i s t i c o f p o t e n t i a l energy i s apparent  i n Section studied  3.  C when the energy balance e q u a t i o n f o r r e a c t i v e p r o c e s s e s i s  in detail.  Energy Balance f o r the Dimer  The  E  local  - 1  D  = M D  energy d e n s i t y  Tr Z <T bjk 1...N j<k  per u n i t mass o f the dimer I s d e f i n e d  by  (N)  (N)  (7.1.15)  -  which can be broken i n t o  V  D  5  M  (  2>  1  D  = "D  and  (  1  the p o t e n t i a l  (  ,  ,  T  r N  1...N  V  1  (  *  +  l i  1= j  i . .. N  2>  169 -  V  6  6  2i>  the k i n e t i c energy d e n s i t y T .  (12Al2  p )  ( 7  ( 12)^12^011  1  ' '  and t h a t o f the p o t e n t i a l  (  - It  W  On p u t t i n g  4*^^  =  ( M  +  +  V  energy,  energy  +  DV  ^j ^k  )  out to o b t a i n the energy b a l a n c e e q u a t i o n f o r the  dimer which can be separated i n t o two p a r t s : t h a t o f the k i n e t i c  h  1 6  The same procedure as f o r the  D  monomer can be c a r r i e d  )  2 i  V  +  li  1=3  V  energy d e n s i t y  IF  t  jk'  ( M  *  (  DV-  i e  e  c  l  u  a  t  i  '-  o  n  or  7  1  - -  1  7  )  M  " change f o r < $ > , the D  D  d e n s i t y o f an a r b i t r a r y dimer p r o p e r t y i n Chapter 4 g i v e s the e q u a t i o n o f change o f the k i n e t i c energy d e n s i t y  ^  ( M T ) = ft [ ( ^ D  D  +  V  1 2  ) \  n  )  ]  s  irP (l2).  (7.1.18)  h  Coupled w i t h t h i s e q u a t i o n i s the e q u a t i o n o f change f o r the d e n s i t y o f the  potential  a s s o c i a t e d w i t h the dimers  h  - <2>  (  W  Tr  , J<\ U 2i> 1. . . N 1=3 (  V  W  6  (12)^bl2 I F  ^ c o l l ' (7.1.19)  The changes  i n the p o t e n t i a l  energy d e n s i t y o f the dimers f o r d i f f e r e n t  collisional  types are e x p l i c i t l y g i v e n by  - 170 -  i  3t " D V  n  v  l  3 V  2 ^  °(12)  (1  V 6  Tr 123  V  +  2 6  + V  12  + 1 3  (12)  13  7  '(12)3 (12)3  )  6  (23)  . (12)  V  13  + V  23 — '  P  123  J  1(23) 1(23)  (12)3  2  C  <P  • (12)3 (12)3  vA/  (12)3  + 7  12 13 2 ^1(23)^1(23)^(12)3  V  - 6 13 (12)  + V  + V  14 23 2  v +v 19  + offi (13) +  <r  V 3  2  7  Tr 1234  +  J  2  V 6  fr  +V 13 23 . 2 i23 123  2 ( 6  +  ?3  2  (12)3 (12)3^123  < r  f 6 (23)  2  +  + v  f  (12)  +  n  12  2 ( 6  + V  24  f J  12(34) 12(34)  +V„,+V„. 14 23 34 2  V  fi (12)  13  + V  14  + V  23  + V  24.  6  (12)(34)  J  \ l 3 ) ( 2 4 ) (13)(24)  V  12 + 2 (6,, , (13) +  2  (  6  0  + V  14  + V  23 -  (13)24^(13)24  + V  V  34 2  6  (  1  2  )  13  + V  14  + V  23 -  + V  24,  )  171 -  V  +  Tr  13  + V  14  + V  23  + V  24  (12)  1234  + <  ^(12)34X12)34  d2)(34)Xl2)(34)  }  p  V + 26  (  +v  V  V  12 1 4 2 3 34  <P  ^(13)24 (13)24  V  + V  +  2  + V  u  ^13)(24) (13)(24)  V  13 14 23+ 24 2  V  + (6  (12)(34) (12)(34)  +v  (13)  6 °(34)  + y Tr 1234  +v  13  + V  14  + V  23  ff V  + V  J  (12)(34) (12)(34)  24  (34)  V  - 6  13  + V  14  + V  23  + V  24.  (12)  J  12(34) 12(34)  V  + 4(6  12  + V  14  + V  23  + V  34  V  - 6 (12)  (13)  13  + V  14  + V  23  + V  24.  X13)24 (13)24 J  +  2((6,_ 6, , ) '(13)^(24)' +  0  J  N  V  l2  (13)(24) (13)(24)  P  (12)34  + V  V  l4^ 23  + V  34  _  - 13 (12)  -  V  + 2  Tr  + V  13  14  + V  23  + V  172  -  24  (12)  1234  ( £  (12)(34)  + 4 6  +  (12)(34)  12  14  °'(13) "**  4  23  24 £  0  13  + V  14  + V  23  + V  f  )  p  (13)24°  (13)24  24  (12)34  (34)  P  (<f (34) 12(34)  +  £  P  (12)(34) (12)(34)  1 2  V  +  < ?  £(12)34 (12)34  %  V  + 6  6 >  2 ( 6  (13)  +  12  + V  14  + V  23  + V  p  (12)34'  )  24  W  3)(24)^(13)(24)  (7.1.20)  B.  Non-Reactive C o l l i s i o n s  The earlier  change due  potential in  origin.  i n energy d e n s i t y  to the t r a n s f e r  energy  flux  f o r non-reactive  o f e n e r g y between  o f the molecules which  The e q u a t i o n s f o r k i n e t i c  energy  c o l l i s i o n s i s , a s shown  the m o l e c u l e s plus the i s not c o l l i s i o n a l and p o t e n t i a l  transfer  energy  will  be  - 173 -  dealt with separately and one finds that the net gain of one i s compensated by the loss of the other.  (i)  Monomer-Monomer C o l l i s i o n s The  contribution from this type of c o l l i s i o n to the energy balance  equation i s as follows: For the k i n e t i c energy density, we have  [  h  W  M -k  H ^ i V s ^ 1 2 ^ 1 2  =  2  6 2*  1  *T*  6  2  '12 u  = -V  P  1 2  ;  2  J  s  12*12 12  •u  =2M  -  r P - T r ( = i i + ...) [ ( = -  12  2  2  (  m  7  12 12 V  )  12 12^12  6+6 2ra  Tr ^12-V^B 12  V  V  ' < 12 12> P  6+6  P  12^2 12'  (  7  2  - '  1  )  -  174 -  P^^ • ju i s the c o n t r i b u t i o n to the energy f l u x due to the mechanical work done to the system.  -  The term  2  Tr ( - i - + ...)  P^ being  [ ( - - u) 6  to the heat f l u x  between the two monomers.  r — 3t  1  2  )  ]  • (V  s  1 2  V  1 2  )  1  2  1 2  p  1 2  ,  the t o t a l momentum of the c o l l i d i n g m o l e c u l e s , i s the c o n d u c t i v e  contribution  L  (  M V 1  M M 2M  arising  from the i n t e r m o l e c u l a r  force  For p o t e n t i a l energy,we have  ^ ^ ^ 1 2 ^ 1 2 = - v  [ M  V  M M^2M  + Tr[(^--u) j m — 2  + y - Tr 2m 1  5 V. ] (P A P 1 12 s 12 12 12 1  9  o  1 9  9  V V (£  2  1 2  2  ;  ( V  s  V  12 12  )  12  12^12  V V +  (V  V  12 12>  (  ' *12  2  ;  s  (7.2.2) where [p.6.V..1 i s d e f i n e d by — i i ij s  -  [  and  6  v  ~?  =  ]  £i i ij s  A  [M^M^^M  N  D  T r  ^ i V i j  i2  ^Z^  m  +  175 -  Wlj  ~ ^  6  +  V  i ^ i  V  6  i  3v  +  <f>  u\Zi >  P  ]  i 12^ s *^ 12 12 12  a  r  e  r  ( 7  e  s  P  e  c  t  i  v  e  2  - -  l  3 )  y  the c o n v e c t i v e and c o n d u c t i v e p a r t s o f the p o t e n t i a l energy f l u x due t o monomer-monomer  collisions.  The l a s t  term on the r i g h t  (7.2.2) i s equal and o p p o s i t e to the l a s t  side of equation  term o f e q u a t i o n (7.2.1).  the k i n e t i c energy f l u x a r i s e s from c o l l i s i o n a l  Thus  t r a n s f e r o f energy  between the c o l l i s i o n p a r t n e r s ) whereas the p o t e n t i a l energy f l u x not i n v o l v e any i n t e r m o l e c u l a r energy t r a n s f e r .  (i.e., does  The i n t e r c h a n g e o f  k i n e t i c energy and p o t e n t i a l energy a s s o c i a t e d w i t h e i t h e r m o l e c u l e i s due  to the c o u p l i n g between the r e l a t i v e momentum between the c o l l i s i o n  p a r t n e r s and the f o r c e between them.  I t i s nonzero even when t h e  c o l l i s i o n p a r t n e r s occupy the same l o c a l i t y because o p e r a t o r i s not d i a g o n a l i n £12-  the c o l l i s i o n  density  Summing (7.2.1) and (7.2.2), the r a t e  o f change i n energy d e n s i t y due to monomer-monomer  c o l l i s i o n s can be  expressed i n terms o f a f l u x  [  h  V  ]  ¥M 2M  where _q^ *  s t  M  n  e  •  " • t =2M * JI  +  + T M  V  ]  M M^ 2M  (7  W  2  ''  4)  p a r t o f the c o n d u c t i v e heat f l u x which i s a s s o c i a t e d  w i t h the c o l l i s i o n s i n v o l v i n g  two monomers, and i s g i v e n by  Tr 2M  12  m  —  1 12 s  e (V  i2  ^ P 2  12  .  V  12 12>  (7.2.5)  - 176 -  I t i s s a t i s f y i n g that  f o r monomer-monomer  non-reactive c o l l i s i o n s , identified  the c o n v e c t i v e p o t e n t i a l energy f l u x i s  so t h a t we can i d e n t i f y the complete c o n v e c t i v e energy f l u x  (when no r e a c t i o n s  take p l a c e ) and e x p l i c i t l y and f o r m a l l y  k i n e t i c energy c o n t r i b u t i o n potential  (ii)  c o l l i s i o n s , and i n f a c t ,  connect the  to f r e e motion o f the m o l e c u l e s and the  to c o l l i s i o n s .  Monomer-Dimer C o l l i s i o n s The e q u a t i o n s o f change f o r k i n e t i c and p o t e n t i a l energy d e n s i t i e s  associated  [  "3T  ( M  with t h i s c o l l i s i o n  T  M M  +  M  " -i f  )]  D D M+D,no r x t  I C  23  T  are r e s p e c t i v e l y  Vl>s  +  (  6  )  ^(23) ( 2 3 ) s  ]  V /  P  l(23)^l(23ft(23) l(23)  y- T r 123 5  r i  p  ^1(23) l ( 2 3 ) i ( 2 3 ) l ( 2 3 )  V  '  1 2  *  (  £l(23)  +  — >  - (P/3m 6 ) • ( V R  +  s  ( V  ^t%3  V  1 ( 2 3 )  ' 23 1(23)  n  P  ^(23) i( 3) l(23) 2  )  V  +  )  1 ( 2 3 )  ( V  V  23 1(23)  )  '^23  - 177 -  1 32m  [ (  ( V  ^1(23)"-  V  R  1(23) 1(23)  [3((r  f  ) 6  1 ( 2 3 )  )  + 3  £l(23)  +  ...)6  ( V  (  )  ^l(23) ^l(23)-" V  V  1(23) 1(23)  )  *  2^2^)*  R  t ]  +  (  P  l(23A(23) l(23)  i  6  [  1  ** <*1  *OV  +  (  ,/  2 ( 2 3 )\M* 13() 2H3()2 3 «) 1(23) 1 (2  ^23)>  123  P  (P • vM =M+D, no r x t V  ;  £  )  ^l(23)-  I  (  3lT-^>  V s  ( V  '  V  1(23) 1(23)  :  ^1(23)^1(23)^(23)  Tr 123  (  -l(23)*  , , ; )  6-1(23)  X  ( V  V  1(23) 1(23)  )  v/l/  1(23) 1(23)^1(23)  +  ^  (  ^l(23)"-  )  —^23  X  ( V  V  )  ) 6  Pi(23) ^l(23)--- R  p  23 l(23) *'l(23fi(23) l(23)  1  - 178 -  £  • Tr 123  (  6  -l(23)"° R  23 ra  £-23 ,  r  '  IT  9_  Ur^  2 3  1(23)  )  X p  + ("23  -23 _  y  2  .  )  V  x  2 3  V  1 ( 2 3 )  )  mr23  <  +  3 £l(23) . 4m£l(23) * r 3r  3  v  l  )  -l(23) *** ~  1  3 £  + (  1 (23) 4mr  X  (  2  3  )  -El(23)  V 1  (  2  3  )  l(23)  ;  -VD>  1(23)  ' £l(23)  X  ( V  1(23)  V  l(23)  )  —I s  1(23)  1(23X1(23)  *I * £ 1- <lV-* (( 2 2 33))> ^ F  1  1  1  3  ( M  V  M M  + M  V  D D  ) 3  +  ^123  V (  6  l+ 6  <  2 3  >  ) _  2  + V 1 2  ^1(23)^1(23)^(23)^(23)  rxt  M D,no  i  ^  13 f  P  *l(23) l(23)*l(23) l(23)  ( 7  2  ' '  6  - 179 -  =  _ V • Try!!  kf  [ 2 3  6 l +  ^ g i  V  ? U23)  ( (  '  6  ( 2 3 )  6  )^ l ]  )  P  g  )  1 ( 2 3 )  ^  ( 2 3 )  P  1 ( 2 3 )  ]  V (23) ^l(23) s -^1(23)^1(23)^(23)  = - V  + Tr 123  t ^ " ^  6  1  J  2  s P  V  , ,,£(23) , , + K - S T - " ^ (23)  +v 12 13  1(23) 1 ( 2 3 ) 1 ( 2 3 ) n  6  i " * 123  6  t V l ( 2 3 ) >  Here the symmetrised  (  6  *(ij) (ij)  V )  s  2  )  1  (7.2.7) < f  3  *(ij)W  +  V  6  +  ^<ij) (ij)  We d e f i n e the symmetrised k i n e t i c  P  1 ( 2 3 ) ^ ( 2 3 ) l (23)'  o p e r a t o r (£,( i j )) "5(ij )V) 1 8  =  V (23) ^K23) s -  (  s  i s g i v e n by  W(ij) +  3  V  6  V  (ijAij)  (7.2.8)  energy o p e r a t o r s r e s p e c t i v e l y f o r  r a d i a l and r o t a t i o n a l motions v i z .  _i 4  I ' P ' - ^ v f p ^  +  + 3 l ' (  ' l + 3 p  ¥  J  ?  . f p J ^ . p r '  r  (7.2.9)  -  180 -  and r  x p r - x (4-.. V ) l — or s  r  1 T  x p  ± r" x  x _p  V) + r " x (|p. V)  r_  jc  x _p  —  + 3 r« r " x (|p. V)  +  x _p  —  3  r " x (-|p. V) r '  (7.2.10) As w e l l  to •=M-D  S  r  %  i(23) V f  (  1  )  p  b  (  2  3  )  ]  _  1  H  3  £-1(23)  X  *1(23> W  1  )  p  b<  2  3  >  (7.2.11)  is  the r e l a t i v e  angular v e l o c i t y between the monomer and the dimer.  The  velocity  u + 2v (7.2.12)  is  the average  v a l u e o f the stream v e l o c i t i e s o f the monomer and the  dimer a s s o c i a t e d w i t h monomer-dimer c o l l i s i o n s . (g  M + D  n  Q  r  x  t  • _v') i s the c o n t r i b u t i o n to the energy  w i t h the p r e s s u r e tensor f o r t h i s c o l l i s i o n t y p e . energy  flux  The term  carried  flux associated The c o n d u c t i v e  along by L „ . _ i s i d e n t i f i e d by n o t i n g t h a t =M+D,no r x t J  &  -  181 -  a x (b x c) • d = a • (b x c) x d  = a • (b c)  and  • d - ( a • b) c • d,  by comparing (7.2.6) with (5.2.10-11).  c o n t r i b u t i o n s from v i b r a t i o n a l components p ^ • r ^ 3 / 3 ^ V e q u a t i o n (6.1.11), motion o f the gas. energy  The k i n e t i c  1  (  2  3  )  + (3/3^  V  As f o r dimer r o t a t i o n ,  D  v e l o c i t y between a monomer and a dimer.  •  and  k i n d of k i n e t i c  flux  )  • p^),c f .  f l u x due to b u l k  c l e a r l y i t i s conducive  a s s o c i a t e d not o n l y w i t h _u>, but a l s o w i t h jo^  L  1 ( 2 3 )  does not c o n t r i b u t e to the heat  the r o t a t i o n  energy  motion ( t h e term w i t h the r a d i a l  f l u x a s s o c i a t e d w i t h the angular momentum  collision,  (7.2.13)  >  to t h e  f l u x L, which i n t u r n i s  the r e l a t i v e  angular  As a monomer i s i n v o l v e d i n the  o f t h i s monomer i s n a t u r a l l y i n c l u d e d i n the energy  flux  ( 7 . 2 . 7 ) , one o b t a i n s the energy  term.  Combining e q u a t i o n s  (7.2.6)  conservation r e l a t i o n f o r this  collision  [  1TF ^ M + W W n o  = - V  rxt  (P • v') + a =M+D,no r x t — ' -^M+D,no r x t v  + (M V u + M V v) MM D D -'M+D.no r x t  (7.2.14)  3w + to -M-D no r x t  Here q„, ^ i s the c o n d u c t i v e heat •^M+D no r x t collision.  flux associated with  this  - 182 -  (iii)  C o l l i s i o n s I n v o l v i n g 3 Monomers  F i r s t l o o k a t the change i n the k i n e t i c energy d e n s i t y  [  M  T  ]  T t M M 3M+3M  V 3 6  Tr 6fi 123  3 [ ( 2 ^ - ^ ) )  +  ( 6  - ^ ) ]  1  s v  (  + j t o v v ^ v  V • (P  +  i  6  +6  2 3  123 123 123  i  ) ]  • u)  =3M+3M  - V  6  p  123  Tr 123  -  - (£i(23)**'  L  )  [  ~  /3m  6  J  pl R s , ( V  r (  r :  - JT^- f  "  (  £23-"  )  L  '  +  Ws  K ^ r n ^ - ^  P  A  V . . . ) [ ( P / 3 m - u) 6  '  (  < 7  r  )  ]  s  V  23 123  )  N  [  +  (  £  )  16  ( V  2  - -l(23)* * r  2\ / l  3  P  1 2 3 _,_ , 2 3 t, . 4 £ 3 — — 2 3 (23)  l (  2  V  ,  LL23)_ . / 4m  R R  6  )  V V  I  +  (v  —23 —23  ~! ~ 2 2  — 2  '  v  V  1(23) 1 2 3  - - - )  )  J  (23) '3  N _ ^23  V V  ^ 1(23) 123  ;  m  ( V  V  23 123  )  1  —is 123 123 123  )  1(23) 123 '  - 183 -  " m f -  V  —  [  23  ' %M*3M  + j V • Tr  (  123  6  +  p  <«i + 2  V J s Wl23*f23 123  • »>  )  ^12  + ( r  ••' ^ 2 m - - « >  1 2  V  1 2  )  V-P-3  1  3  ...) 0  V  2m + ( r  ' (V  2  3  ...)  2 (.  3 2 m  6  1  6  +  V  - £ > ' < 13 13>  f  p  123^i23 123  " «> * ( v v ) 23  23  9  ^123  6  +  [ (  +  [  C  +  1  6  3  ^ ^ )-J- _2 +  V  2  3  +  *  2  3  )  - ^  T  ^  ] s  - ^ ]  s  i r  2  123  P  123 123'  3  (7.2.15)  The symmetrised o p e r a t o r s i n the second e q u a l i t y i s d e f i n e d  i n equation  (7.1.13).  As f o r the change i n the p o t e n t i a l energy d e n s i t y , i t i s g i v e n by  3M+3M  P  123  1  Z  n  23 123 123 123  - 184 -  V 1  V . Tr ( 6 ,„  V  5  1 3  1-1  i 1  +V 2  l P l  IU  [  12  + V  ) s> ;  2  o P  s  123 123 123  13  ~~ ~~'  ( y  2  i23 l>sJ-^123^i23 123 6  p  = -V (M  V  M M ^3M+3M  + Tr 123  6fi  Tr 123  [(-I - ) & —  + [ V  +  (  12' ^1  [ V  1  2  13  U  m  ( (  13' *1  1  2  )  ]  *>  ^  p  123 123 1 2 3 l123^123  8  p  i  ^2 -V > ls  +  ^ 3  )  - T ^ >  L  s  s  p  123 123 123  -  (7.2.16)  As b e f o r e , by i n t r o d u c i n g relation  [  ii  the heat f l u x  q3M>3M» the energy  balance  can be expressed as  (M  E  )  M M 1 M-3M 3  (7.2.17)  - 185 -  (iv)  C o l l i s i o n s I n v o l v i n g 2 Dimers Except  f o r the change a s s o c i a t e d with the i n t e r n a l  e q u a t i o n o f change f o r the k i n e t i c  energy f l u x , the  energy d e n s i t y due to t h i s kind of  c o l l i s i o n has the same form as that f o r monomer-monomer c o l l i s i o n s .  In  b o t h c a s e s , two e q u i v a l e n t m o l e c u l e s a r e i n v o l v e d , but now, the m o l e c u l e s possess m o l e c u l a r energy which g i v e s r i s e to an i n t e r n a l Using the same method as i n ( i i ) ,  energy  flux.  the energy f l u x a s s o c i a t e d w i t h the  m o l e c u l a r a n g u l a r momentum f l u x can a l s o be i d e n t i f i e d .  M  T  ^9t D D^2D,no r x t  = -7  (P • v ) =2D,no r x t —  Tr 1234  Al2)(34) V s  ,  , • • •)  [  +  (  ( V  4 ^ " ^  V  Vs* (12)(34) (12)(34)  1 -El 7 T ViT •  ( V  V  12 (12)(34)  v  12 (12)(34)  ~l Jlr' 6  ( V  -12, m  ) /  V  34 (12)(34)  £  +  ^(12)(34)  s/L  p  ( V  V  34 (12)(34)  tl2)(34) (12)(34)  )  )  34x * IT>  )  -  186 -  1 Zh  V  <P  v/1  '(12)(34) (12)(34)  P  (12)(34) (12)(34) (7.2.18)  For p o t e n t i a l energy, we have  no r x t  V  13  + V  14  + V  23  + V  24 ]  V  i  T  r  r  13  + V  14  + V  23  + V  24  „  2  t T j1234 f.t  >  s  P  SJ  P  (12)(34) Yl2)(34) (12)(34)  ^ (6  (12Al2)(34) s )  ]  M  (12)(34)  (12)(34) (12)(34)  = - V ( M  +  V  D D ^ 2 D , no r x t  Tr 1234  r A i 2 )  .  .  V  v/1  13  + V  14  + V  23  + V  24.  p  (12)(34) '(12)(34) (12)(34)  Tr [V 2n 2 (12)(34) L V  1  3 4  r ( 1 2 ) ( 3 4 ) ,, 2 (12)(34) s 6  + 6  0  >  1  3  ]  ]  (P (12)(34)^(12)(34) (12)(34)' P  (7.2.19)  -  187 -  A g a i n , t o t a l energy i s c o n s e r v e d , and the energy f l u x  can be broken  into  a p a r t due t o P-V work, a c o n v e c t i v e ( p o t e n t i a l ) energy f l u x , a c o n d u c t i v e energy f l u x , and a f l u x  carried  along by the i n t e r n a l  angular  momentum f l u x L__ =2D, no r x t  l  ¥ D 2 D ,no r x t ]  7t  = - V  (P •v =2D,no r x t — v  +  ( M  +  (v)  (7.2.20)  V  D D^2D,no rxt  =2D,no r x t * —  +  -^2D,no r x t  C o l l i s i o n s I n v o l v i n g 2 Monomers and 1 Dimer The  [  "§T  change i n the k i n e t i c  (M  T  +  M M  V •  -51  M  D D 2M+D,no T  )]  Tr 1234  (  -12  energy d e n s i t y f o r t h i s c o l l i s i o n i s  rxt  , , , ) ( _  2m~  6  /  v  ( 1 2 ) s • '12*12'  , w (134) . ^-l(34)*** 3m °(134)'s £  n  1  v  ' 1 ( 3 4 ) " 13 '' 14'  •±-(131*} 2  ~  ^2(34)'"^-Sir 12(34)  M  6  12(34) 12(34)  )  (234) s *  t V  2(34)  ( V  23  + V  24  ) ]  - 188 -  +V« i Tr  6  K£ 34)---) 1 (  J  (i34)  1234  r l  3£  i  K34),  2  l ( 3 4 >(v,,+v ) 1 3 1 41A)  n  v v  4m  v  ;  3^34 m  -^ 2  V  34  ( V  13  + V  14  ) ]  —is  (  £2(34)*  3  [  , , : )  6  (234)  P  1 - -2(34) 2" 4m"  V  v  , ,  2(34) '23 24  /  3 £34 2 m  v,  v  '34 23' 24 —Is  Vl  P  ^12(34) 12(34) 12(34)  6+6 Tr 2fi 1234  6+6  12(34) 12(34)^12(34).  The second b l o c k o f the f l u x i M+D 2  a n c  no r x t '  * ^  (7.2.21)  terms g i v e s the energy f l u x a s s o c i a t e d  with  i n t r o d u c i n g v " , the average stream v e l o c i t y  a s s o c i a t e d w i t h t h i s c o l l i s i o n g i v e n by  v" =  the  I  (u + v) = { (u + 3 v ' ) ,  (7.2.22)  energy f l u x a s s o c i a t e d w i t h P can a l s o be i d e n t i f i e d . =2M+D, no r x t o w l T k  - 189 -  The n e t change In k i n e t i c  energy i s compensated by the n e t change i n  p o t e n t i a l energy whose d e n s i t y changes a c c o r d i n g to  [  9t  (M  V  M M  +  M  V  D D^2M+D,no r x t  ( V  = - V  u  M  +  M V D  v )  D  2  M  •P-l  +  V  D  j  n  12  o  r  + V  Tr K ^ - u ) \ 1234  +  4.  1  +  x  13  t  + V  14 1 *12<34)<12<34> 12<34)  2  P  8  T T - r/-(34) x 2 ^ 2m- " ^ (34) [ ( _  13  14  6  2  4  tf>  [V  Tr 1234  Zh"  +  V  = - V  2 )  V  + V  ^ ^ 1 2 ( 3 4 ) ^ -  (  24' -M  l i l K  (=2M+D,no r x t ^ " —  +  1  ^i2(34) s -  + V  t 23  P  5  t 13 14«  +  ]  12(34)" 12(34) 12(34)  «1 + , ( 2  1 2  l  23 24. s  =2M+D,no r x t  +  f  v/  p  12(34) Y2(34) 12(34)  )  12(34) sl-  -^M+D.no r x t  4  + (M V u + M V v)„ V M D D -2Mf-D.no r x t V  TE  [  ( M  T M  M  + M  D V 2 M D , n o rxt' ]  +  (7.2.23)  -  190 -  Because of the r e l a t i o n  (P  =2M  +  =3M,no r x t  }  *-  +  =2D,no r x t * -  ±  "M  + P • v' + P • v" =M+D,no r x t — =2M+D,no r x t —  =P •— =no r x t  +  M  D  +  £  —  (P + P + — P + — P )M =2M =3M,no r x t 3 =M+D,no r x t 2 =2M+D,no r x t D 7  (u - v) -  ?  1  CP + — P + — P ) =2D,no r x t 3 =M+D,no r x t 2 =2M+D,no r x t  v  the k i n e t i c energy f l u x  carried  along by the p r e s s u r e  n o n - r e a c t i v e c o l l i s i o n s can be i d e n t i f i e d .  f l u x which i s connected  carried  "M*  V  t e n s o r due to  As they a r e two d i f f e r e n t  s p e c i e s i n the system, t h e r e i s an a d d i t i o n a l  species.  M  y  c o n t r i b u t i o n t o the heat  to the d i f f e r e n c e i n the v e l o c i t i e s of the two  L i k e w i s e , owing t o the presence along by the m o l e c u l a r  of monomers, the energy  angular momentum f l u x  case) i s not a s s o c i a t e d w i t h u>, the angular v e l o c i t y  flux  ( i n the n o n - r e a c t i n g o f the dimer.  O b v i o u s l y , when dimers a r e the only s p e c i e s i n the system, and when no r e a c t i o n s take p l a c e , the k i n e t i c energy f l u x a s s o c i a t e d with ,  molecular  ,  angular momentum f l u x i s L _. . w=L co. ° =no r x t — = — 4  C.  Reactive  Collisions  For the hydrodynamic d e n s i t i e s  of mass, l i n e a r momentum and a n g u l a r  momentum, the c o n t r i b u t i o n s t o the equations transfer  a r e comparable to those  of change from  for non-reactive processes.  collisional However,  - 191 -  the c o l l i s i o n a l  t r a n s f e r p a r t s o f the energy balance equations f o r  r e a c t i v e and n o n r e a c t i v e c o l l i s i o n s as the i n t e r c h a n g e o f k i n e t i c  are d i f f e r e n t  from each o t h e r ,  and p o t e n t i a l e n e r g i e s g i v e r i s e  term o n l y i n the former c a s e .  to a f l u x  T h i s f l u x a r i s e s by way o f t h e d i f f e r e n c e  i n l o c a l i t i e s o f the products and r e a c t a n t s .  (i)  C o l l i s i o n s I n v o l v i n g a Monomer and a Dimer F i r s t , c o n s i d e r the decomposition  and  reaction.  From e q u a t i o n s  (6.1.1)  ( 6 . 1 . 2 ) , we have, f o r k i n e t i c energy d e n s i t y , the r a t e o f change  [  i  (M  T  M M  = -V  VD  +  ) ]  M D+3M +  C t  P =M+D+3M *  +  ,ct, W  [ (  +  T 7 1  3  ^  ? ki-  <E - S  ~*123  ct [+D+3M  +  +  ^ D +  -SM+D+3M  M ' £  3  +  -23 *"  )  6  (23)  1 -^23 ^23 Tr ("2T ~~1 T~ 123 +  J  (  f  a  # - - i i  +  p  )  1  ,  : r  P  •i23 a* 123 i23 l(23)  5  •••)t(£2 - -3^ ( 2 3 ) ^  (V u )  r  P  123 123 1(23)  - VV :  "  (  123  TT 2 4 =  1  3  1 ..2  + 6  (23)  2  m  ^123^123^(23)  T  u  - 192 -  VV  1  : Tr 123  £  2  3  2\——  {  .  ,l  r  +  K  +  £ 3  2 m  .£2 P3- £ 4m  2  ) [  -23 £23  v  2~ ' "  2 T ~1  . , £  -  . £3  n  (  + 2  2mu)  6  (  2  3  )  ]  g  £23  ~  }  ]  £23 s  6  (23)  ^ 3 P; x 123 123 1(23)  k^ *  6  [  V (23) 2 123 s K  ]  1(23)  ^123^123^1(23)  123 +  V  23  6  7  (23)^123 123  (7.3.1)  To a r r i v e at ( 7 . 3 . 1 ) , the r e l a t i o n  V • (r»  £ ' a • b) = (V • £ ' r ' a) • b + Or' £ ' a) : (V b)  has been used. symmetrised  The s t r u c t u r e o f J^J+D^JJ *  o p e r a t o r s are d e f i n e d  t(£ ...)  ^  ± j  =7  t(r  =- i  l£  ± j  + 2,) ' £  s  referred  The  by  1 : j  5  s  ...) p . +  (jByUij...))'] • ( 6  £  ± j  ±  to (7.2.6).  (7.3.2)  ( l j )  (£  ±  + Pj)) , s  (7.3.3)  and  ± j  ± j  ^  *£  +£  ± j  '  £  £ j  £  ± j  + 2(  2 i j  £  t  ± j  ) . (r^  P ^ ) ^ (7.3.4)  -  The  c o l l i s i o n a l transfer  structure channel  193 -  contribution  to the energy f l u x has the same  as f o r n o n - r e a c t i v e monomer-dimer c o l l i s i o n s except  potential  non-reacting  collision  partners, that  2M+D-3M  a n d  As w i t h the  from the r a d i a l and r o t a t i o n a l  by the f o r c e s and torques between the c o l l i s i o n p a r t n e r s  i n the i n i t i a l  ^M+D^3M  superoperators.  case, t h i s flux a r i s e s  motions caused (molecules  and p r o j e c t i o n  f o r the  m U S t  b e  channel).  Because of the a s s o c i a t i o n  with the  p a r t o f the energy f l u x which i s r e l a t e d to  (3j£ + J V D  a s s o c i a t e d  ) / 4  A N D  angular v e l o c i t y and v e l o c i t y monomer-dimer c o l l i s i o n s .  - '  T H E  The term  VV *  123  2  f  ~  (  ~  2  3  )  J  P  123 123 1(23) 2  U  (7.3.5)  i s the r a t e  o f change i n k i n e t i c  i n mass d i s p e r s i o n  energy d e n s i t y r e s u l t i n g  due to t h i s r e a c t i o n  (M+D+3M).  from the change  The f l u x  term i n which  the r e l a t i v e and t o t a l l i n e a r momenta o f the product monomers a r e coupled together  n ^3-23  Tr 123  6  (23)  •£(23) 2m * -&23 s ]  (  1*23  6  ' £(23)  P  P  123^23 1(23)  )  (23) s P  ^123^123 1(23: 2 £23 -H23 (  X  )  X  (  P  6  )  - -(23) (23) s  (7.3.6)  -  194 -  I s a s s o c i a t e d with the t r a n s f o r m a t i o n o f i n t e r n a l motions o f the r e a c t a n t dimer i n t o r e l a t i v e t r a n s l a t i o n a l motion between the product monomers. T h i s c o n t r i b u t e s to the energy  f l u x connected  w i t h the p r e s s u r e t e n s o r  compare t h i s with e q u a t i o n ( 6 . 1 . 9 ) .  for t h i s r e a c t i o n :  t r a n s f o r m a t i o n o f energy,  the heat  Besides  such  f l u x a l s o c o n t a i n s the term a s s o c i a t e d  w i t h the t r a n s f e r of t o t a l momentum from the r e a c t a n t to the products  _  i  r  * 123  J_z23 M = l 2 5  f  2  +  - 3  " <£  4m  2  +  P  2  2m 3  ^  Ws  P  123~123 l(23)*  With regard to the f l u x  term a s s o c i a t e d with - ( V  , i t i s connected  w i t h the v i s c o s i t y o f the monomers and to the monomer angular 1/2 V x _u.  When o n l y the second  order term  comparing w i t h e q u a t i o n (6.1.17), to the t r a n s l a t i o n a l relative position  velocity  i n rj>3 i s kept, on  t h i s energy  f l u x i s found  to be r e l a t e d  angular momentum f l u x due to the c o u p l i n g o f the  between the product monomers w i t h t h e i r t o t a l  linear  momentum  r I  H  3  r (  TT^f-^-*  6  )  J  P  ^(23) ( 2 3 ) s ^ l 2 3 1 2 3 l ( 2 3 )  *  I JE.23 JI-23 —23 Tr 4 ~T~ ~ 2 ~ - -(23) (23) s -£(23) (23) s ~ T ~ 123 [  (  P  6  )  +(  6  )  (  _  T>  V  X  -(Vu + (Vu) )  ]  P  ^123~123 1(23)  The  energy  flux carried  along by the angular momentum f l u x due to the  t r a n s f e r of the m o l e c u l a r  angular momentum o f the r e a c t a n t dimer to the  -  195  -  r e l a t i v e a n g u l a r momentum between the product monomers i s a l s o i d e n t i f i e d using  the r e l a t i o n  £ x _£ • _q_ x £  _p x  ' 5. x p  2  2  q  _g_ £L " £ • £ " £  2~  q  _g_ being the p o s i t i o n  *£  '  (7.3.7)  q  operator.  In Chapter 6, (p-j j  connected w i t h ± 2 pure v i b r a t i o n a l t r a n s i t i o n s  • S i j ) s has  f o r the dimer  been ( i j ) . From 2  e q u a t i o n (7.3.7), one can deduce  that  the i n t e r n a l k i n e t i c energy p^  can be broken i n t o the v i b r a t i o n a l k i n e t i c  JJij  and  * <  m  ^  rotational  £  ± j  )  _ 1  £ij £ij  /m  energy  ' £ii  energy  x j ^ j • ^  x  2  £ i  ./m r . .  The d e c o m p o s i t i o n of j> • j> i n t o a r o t a t i o n a l  term and a r a d i a l motion  14  term has a c t u a l l y been accomplished b e f o r e . However, the s t r u c t u r e of 2 the r a d i a l motion term p does not resemble what i s o b t a i n e d here and q no p h y s i c a l relation  association  (7.3.7) can be s u b s t i t u t e d  contributions excitations  V  :  i23  [ < 2 T  1-22 —23 ~2~ ~  of f a c t , the  i n t o e q u a t i o n (7.2.6) to o b t a i n the  i n the same manner as i t i s a p p l i e d  I V  In point  to the energy f l u x due to r o t a t i o n a l and  I " h  of i t has been g i v e n .  vibrational  to  —23 '"^  ~  ' 23 s £  ]  6  ( 2 3 ) 123  P  123 1(23)  - 196 -  k  VV:  , J _ ^23 ^23 2 2  Tr 123  . . (23)  4.'  £ 3 2  (  ,  1 . ~ 2 ^  X  -P-23 — m r 23  * (£ 3  X  m 2 2 23  _  £23  2  r  . ^  r r £23 £23 +  1"  £23  m r  2  * £23  3  —I s J  P  123 123 1(23)  T J23  +j  3  f i - Z i i l i * 2 1  Tr 123  2  >> fi < >  2  £23  X  23  r  6  2,  m 1  £23  2  <S  3  2  3  I  2  3  1(23) 2  '  , J _ £ 2 3 £23 V 2! 2 2 ***'  £23/  .1 £ 2 3 £ 2 3 + ( , 2 2  2  2  :,  X  £23  p  (23) 'l23 123 l(23)  (7.3.8)  so  that  one can i d e n t i f y the c o n t r i b u t i o n  to the energy f l u x from the  t r a n s f o r m a t i o n of i n t e r n a l , i . e . , r o t a t i o n a l the  dimer i n t o  monomers. the  energy of  the energy due to the r e l a t i v e motion between the product  Whereas the v i b r a t i o n a l  rotational  and v i b r a t i o n a l  part,  of which s t r u c t u r e  by a s s o c i a t i n g  term shows up only i n the heat i t w i t h _w, g i v e s r i s e a l s o  i s analogous to the d i s p e r s i o n  of  <D  flux,  to a term  co /2 as w e l l  as a term where the m o l e c u l a r angular momentum of the r e a c t a n t dimer i s dotted  into  OJ. As one i s d e a l i n g w i t h the t r a n s f o r m a t i o n of i n t e r n a l and  translational  states,  the a s s o c i a t i o n  w i t h u) i s i n f a c t a r b i t r a r y .  One  -  may i n t r o d u c e u^, such that  the energy f l u x due to L i s L •  be an a p p r o p r i a t e c h o i c e i n t h i s  As  f o r the p a r t  6  2 3  197 -  , which may  case.  that g i v e s a net change i n k i n e t i c  energy, the term  V  •fi ^ 3 < > 2 3 ^123^123^(23)  corresponds to the l o s s i n i n t e r n a l broken up t o g i v e two monomers. energy between the p r o d u c t s .  energy of the dimer  (23) when i t i s  T h i s i s transformed i n t o the p o t e n t i a l  T h i s g a i n term i n the p o t e n t i a l  d e n s i t y a s s o c i a t e d w i t h the monomers i s e x p l i c i t l y  energy  shown i n the e q u a t i o n  *~3t VM'M+D+3M V  (K 1 ( 6  123  V  +V  12 2  V  13 . * 2 +  6  V  +V  12 2  +V  23 +  6  . 3  V  +v  13 2  + V  23 p p jv ^123^123^(23) N  1(23)  V +v 23  (7.3.9)  For the dimers, the r a t e of change i n the p o t e n t i a l energy d e n s i t y i s  [.£_ M V L  =  These  8t  1  D D M+D+3M  £  -  ^  (  123  give  12 2  3  )  2  13^ A?  (7.3.10) J  P  123 123 1(23)*  -  - V  ( M  198 -  V  M M-^M+D+3M  Tr 123  V  Pi  [(-£ - u) m —  6  12 1  +  V  2  13 ] 1 3  L  1  s  +  -P-2 2 [ ( - |  m  6  ^  1" s ]  2  P  123  1(23) 1(23)  -23 12  123 n r .  r  9  +  nT<-T>  123  n-l n-l '  7  5  V  > (23) 13  p  123 l(23)  6+6  6+6  6„+6  >-  * 123  +  (  ^23 -V  *123>8>  '123VH(23)  + «2 3 +6  P  V  <P T 23 123 123 J  (7.3.11)  l(23)*  As e x p e c t e d , the d i s p l a c e m e n t o f energy from the r e a c t a n t s t o the products r e s u l t s i n a p o t e n t i a l energy f l u x (which i s p a r t o f the heat flux).  For c o l l i s i o n a l  t r a n s f e r the p o t e n t i a l  i s a s s o c i a t e d w i t h the  products whereas the m o l e c u l a r a t t r i b u t e s l i k e mass, l i n e a r and a n g u l a r momenta, and k i n e t i c energy a r e a s s o c i a t e d  w i t h the r e a c t a n t s .  This i s  -  not  surprising really.  kinetic  For n o n - r e a c t i v e p r o c e s s e s , the i n t e r c h a n g e of  and p o t e n t i a l e n e r g i e s has a c o n t r i b u t i n g  to n o n l o c a l i t y of the c o l l i s i o n . kind  199 -  p a r t which i s not due  When the c o l l i s i o n s  of i n t e r c h a n g e p r o v i d e s the energy f o r as w e l l  energy generated d u r i n g reactant this  energies.  the r e a c t i o n .  Evidently,  are r e a c t i v e ,  this  as d i s s i p a t e s the  So the energy terms cannot be  conservation  only  of energy i s e s t a b l i s h e d f o r  reaction  [  1F ( V M  +  ct =M+D-»-3M  +  W  :  M  3  4l-D  T  '<£ - 2  W3M  C t  + %  V  }  + *  £ U~T--T-  M  +  3(0  , . ct %+D+3M  n  « _v  ~k  W  W  ii M+D>3M  * ii  [(  ^2  +  ill?  ^23 ^23 ^23  +  lln"^  V^23  V123z T T  1  ^  (  r  +  X  f  r  23  6  *3>  6  ^23  I  ^23 -23 23>  [  W3M  = " ^  * i2  -3M+D^3M  +  3  X  +  «  2/2  +  ]  (23) s  6  V  <?123 123 l ( 2 3 ) p  P  :  ( 2 3 ) ^123 ^ 2 3 l ( 2 3 )  6  23 (23)  : ( V  2  3  (23)  /  P  n 3Jl23 l(23)  K  (  :(  V  V  -  -  ii  }  ) 1  t  }  123^123^(23)  (7.3.12)  - 200 -  The  flux  i s a s s o c i a t e d w i t h V _OJ and (V ut)t can be reduced to a  that  term which i s connected w i t h the c u r l of the dimer angular v e l o c i t y ( t h e c o u p l i n g between the r o t a t i o n of the r e a c t a n t dimer and the r o t a t i o n of the gas) and another term connected w i t h the symmetric  t r a c e l e s s p a r t of  the t e n s o r V u) (energy f l u x due t o r e s i s t a n c e to r o t a t i o n of the dimer)  Tr 123  16h  r  2  r  3  2  x ( r  3  2  3  x £  )  2 3  6  " \<r  ^Tr  [r  6  2  2 3  3  XJ> ) x r  ( r  tf  2 3  2  x £  3  7  2  2  3  3  )  /• r  (  2  )  2  r  P  (23) i23^23 l(23)  V  * 2~  + ^  x £  2  3  r  2  3  )  (  2  )  p  j V • V u U) .  In other words, t h i s i s the a n g u l a r momentum analogue of the energy  which  (V  a  )  flux  from the r o t a t i o n a l motion and v i s c o s i t y of the product monomers  i s i n t u r n connected w i t h the l i n e a r momentum of the m o l e c u l e s .  As a monomer i s s p h e r i c a l , t h e r e i s no r o t a t i o n a l terms  -  (23) 123' 123 l(23)  V oi + (V w) ( 2  arising  X  3  above which i s a s s o c i a t e d w i t h t h i s  species.  term analogous  to the  I f we use V I U Q and  t  Q  ) i n s t e a d , the same argument goes although now one i s concerned  w i t h the r e s i s t a n c e  to the r o t a t i o n between the product monomers.  In a d d i t i o n , t h e r e i s a term which  corresponds to the d i s p e r s i o n of the  p o t e n t i a l energy between the atoms w i t h l a b e l s 2 and 3. the l a s t  T h i s term and  term o f e q u a t i o n (7.3.12) c o n s t i t u t e the r a t e of change i n  energy d e n s i t y f o r t h i s  reaction resulting  from the bulk flow of the gas.  - 201 -  F o r the exchange r e a c t i o n , to express the energy f l u x due to mechanical work. In terms o f _v' , the v e l o c i t y a s s o c i a t e d w i t h t h i s b o t h b e f o r e and a f t e r c o l l i s i o n , we note  3 =  (  +  £ i  3 <£i+  = (  £ i  +  JLj -P-k  £j  +  P  )  +  event  that  £dj)k  - 3m v') + p ^ ^ + 2m v«  k  + £ . - 2m _v') + 2m v' ,  (7.3.13)  t h e r e f o r e the r a t e o f change i n k i n e t i c energy d e n s i t y i s  {  1£  (M  T  W Wexch  +  M M  ,  P • v =M+D,exch —  3  O t  ^  + L. =M+D,exch  03  +  • 4  ct +  (  i~ h  ^M+D.exch * -  [(r *  1 2  ...)  5  +  ( 1 2 )  -^M+D.exch  v—gjj- - ^  123  "  K£ 3*" 2  )  6  -H23) (23) ^m— " - ^ (  <P 7 P ( 1 2 ) 3 ( 1 2 ) 3 1(23) J  J  12 s  ]  *£ 3 s 2  - 202  2i  ,1 "2T  Tr 123  (  £ l 2 -12 ^~-2~  -  x  r l  . P  , 6  J  - -(12) (12) s  (V v ' ) 1 - (T  —23  —23  r- • • •)£ ( 2f_P 3)  6  ]  (23) s  P "7 (12)3 ( 1 2 ) 3 1 ( 2 3 ) P  Tr £ l 2 £-12 ! l 2 123  X  (12)  :(V to)*  ~ £.23 £ 2 3 £ 2 3 :f  6  *12 x  6  -^23  (23)  P  ^(12)3 (12)3 l(23)  Tr 123  : (V oi)  P  :I  - VV : { [ ^ M ^ J D  +  |*VV  Tr 123  P  (12)3 (12)3 1(23)  M + D ) e x c h +  j _ ^12 2! 2  ...}  -12  T~  6  (12)  r  12 m 2/ "2 w /2 n  ,1  -12 -12 2"  x  " U~2 k  ?(12)3^(12)3 7 P1(23)  ,  2 (12) 1 2 r  C  - 203 -  + 2i  v v  ,1 -12 -12 2T~2~~2~  .  s  .  l  123  £l2 (  °(12)  -El2  x  m r  -  jo) • ( r  7  1  2  x p  -  1 2  T  m _2 r <£) 1  2  12  -12 +  1  Pl2  -12  2 m r 12  *£l2  -23 -^23 . . ~2T ~2 2~ ' " ' °(23)  ,1 (  (-23  X  £23 _ 1  £  .  )  ( £ z 3  X  £  2  3  m 2 2 2 3 •£> N  r  m r 23 £23 £ 3 +  £ 3  *  2  '1  m m  r  2  ' £23  23 —I s  f  2i  Tr 123  ( V  12  6  : i  P  (12)3 (12)3 l(23)  V  6  (12) " 2 3 ( 2 3 )  )  ^(12)3^(12)3 1(23)  6  +  [  ^(12)3  l  + 6  (23) 2 s^(12)3 (12)3 ]  f  V / )  l(23)  (7.3.14)  - 204 -  The  term  +  6  Tr 123  [  — m r  '£  1  2  x  £  l  2  (6  V  1 -—j->1  ( 1 2 )  12 (12)3  -  represent  -P-23 2 m r 23  -23  i  6 + 6 2 2  X  X  * £23  P  (6  - -23 (23)  P  (12)3 1(23)  3, s J  the g a i n and l o s s i n r o t a t i o n a l energy as the product dimer i s  formed and when the r e a c t a n t dimer i s d e s t r o y e d .  But the r e a c t i v e  part  of the k i n e t i c energy e q u a t i o n can be broken i n such a way t h a t the change i n r e l a t i v e  t r a n s l a t i o n a l degrees of freedom ( c f . e q u a t i o n  (6.1.27) and the second l a s t  e q u a l i t y of e q u a t i o n (6.1.32)) i s r e v e a l e d ,  and hence expressed i n terms of r e l a t i v e a n g u l a r v e l o c i t y between t h e molecules.  As f o r the p o t e n t i a l energy e q u a t i o n , we have  [  !£  (M  V  M M  = - V  +  WWexch +  <VM^ V D V  13 + 2 Tr [ 123  +  r  ^MfD.exch  23 A 2  , . ,  V  6  ]  3 s P  (12)3 1 ( 2 3 ) l ( 2 3 ) +  r  V l 3  +  V  " r < > £  1 2  ^  A  i  - 205 -  _ • i Tr r V  *  123  - (- lEli !^ + ) 2  3  ~  21 ( (12) Tr 2 H 123 6  (12)3  +  6  6  3  v  v v  + 6  1  R  ?  6 v  10 J  13  P  (12)3 (12)3 1(23)  (23)  23  V  12  }  P  3  (12)3 (12)3 P  1  i .y  S  6  _ r (12)  a  ^  l(23)*  ^ ( 1 2 ) 3 ^ ^ ( 1 2 ) 3 X 1 2 ) 3 1(23)  2  (7.3.15)  Clearly,  total  energy i s conserved f o r t h i s r e a c t i o n .  e x p r e s s the p o t e n t i a l  We may a l s o  energy e q u a t i o n i n an analogous form to (7.3.11)  which the t r a n s f o r m a t i o n o f p o t e n t i a l  energy from k i n e t i c  energy i s  explicit.  (ii)  C o l l i s i o n s I n v o l v i n g Three Monomers The o n l y r e a c t i o n that may take p l a c e i s the r e c o m b i n a t i o n r e a c t i o n  w i t h a monomer m o l e c u l e as the t h i r d  body.  The energy balance f o r t h i s  r e a c t i o n i s g i v e n by  hnr  ( M  T  M M  = -V  +  M  DV 3M+M+D ]  ct ?X~ =3M->-M+D  • u + (P -  —  + L  2%  c  t  )  „  '3M+M+D  • v  • a) + a '  =3M+M+D  +  P  123  -  -±3M+M+D  6  ^ 2 3 — >  P  ( 2 3 ) ^23 '  1(23)*1(23) 123  0  (23) 2m  v')]  8  in  -  Tr 123  206 -  1 £-23 -23 >. , ZT~T~ •* °(23) ,  K  (  £23J1^3  ;  _  .  - -  x  r  2  3  m r23 -23 -23 7— * £ 2 3 m r23  ^23  p  7  1(23)  P  1(23) 123  1 -23 -23 + Tr ( 2T 2 2 123  £  r (23)'  6  (23)  1(23)  (  m  v  '  2  +  ( 2 3 )  T  -2m  r  v')6  23  P  1 ~*23 ""23 ) l ,( 2 3 )p l ( 2 3 ) ("5T 5 - • • 0 ( £(/2, 3 o )^ ( 2 3 . )) ' st Pl ( 2 3. T u  r ? 3  + — - Tr -23 -23 -23 123  X  6  -223 (23)  1 6  3  0  K  u ? 3  ( V u))'  P  *1(23)- 1(23) 123  vr„ 1 6  3  x p  2 3  _r  2 3  r  123  *1(23)  J  P  1(23) 123  ( 2 3 )  1(23) 123  u  123  .(p  4m  J  T r  -2m v'  (23)  :  ( V a))  : ( V  -  - 207 -  r 2rT T123  +  +  V  +  *2  6  6  *3>  6  +  l  V  \N(23)^1(23/I23  123  7  23 (23) 'l(23) l(23)  (7.3.16)  •3T  - V  ( M  V  M M  + M  V  D D  ) ]  3M^D  ( V M - ^ V D ^ W D  2  h  (23)  123  2  /  2m  J  s f  V  F6 1 2 + 1 3 1 2  + +  4fT  Tr 123  V  P  -£ 3 ~~2~~  l  r  [ 6  (  U O  „ i-"£  )  ]  s  2  (  +  6  (23) V  12  D -T 1(23) 1(23) 123 J  <-|l  +  L  P  l(23) l23 123|  ...)  +  6  ( 2 3 )  V  P  1 3  J  2  ^ 1 2 3 s ^ 1 ( 2 3 ) ^ 1 ( 2 3 ) 123 123  +  2  V  23  J  1(23) l(23)  (7.3.17)  - 208 -  All  the terms can be I d e n t i f i e d  for c o l l i s i o n s  as f o r monomer-dimer c o l l i s i o n s ,  that i n v o l v e four atoms whose c o n t r i b u t i o n to the  e q u a t i o n s o f change f o r k i n e t i c and p o t e n t i a l  (iii)  Dimer-Dimer  e n e r g i e s a r e g i v e n below.  Collisions  Decomposition w i t h no Exchange  [  lt  ( M  « _ V  T  M M  likewise  + M  T  D D)]  2D+2M+D,no exch  ,ct ,ct, • v + (p - p )_ •v =2D->-2M+D,no exch 2D>2M+D,no exch — P::  2  =2D*2M+D,no exch  + 4- T r 1234 n  1  (  P  6  )  : (V u)' 3  f  p  12(34) 1 2 ( 3 4 ) ( 1 2 ) ( 3 4 )  i^l2  ^  2D->-2M+D,no exch  £-12 £-12 2~ - - ( 1 2 ) ( 1 2 ) s  2T 2 f  +  —  <%ST-'2> ^ 1 2 J  Ws  p  12(34) 12(34) (12)(34)  1234  L n L  1  2  L  x 1  2  2n  6  (12)  :  (  v  u  •>  t  p  ^12(34)^12(34) (12)(34)  m  1234 x l l 2 f  X  £-12 -12  :7  -12/'  p  : 6  (12)  12(34) 12(34) (12)(34)  ( V cu)  - 209 -  TV  1  - vv  *  1  2  3  ^12 -El2  1 2!  4  f  2  2m  A l l ) ~ Am  2 3  ^"  (P  ( 1 2 )  - 2m v") 6  (  1  2  )  ]  g  P  12(3A) 12(3A) (12)(3A)  2  "1  " D ^ J 2D->-2M+D,no exch  &  i  A -23 -23 . . 2 m (.-^—, ^— ...; 6 (23) 2 3 2 2! 2 2  • Tr  1  1  2  v" /2  r  3  4  U  2 ,„ 7  2  (P T 12(3A) 12(3A) (12)(3A) J  P  £  - ^ VV:  ,1 -12 12 (2T"T~~  Tr 123A  2« ^ _P| 2  r  =  + P  m r  1 2  *  —  v ,  °(12) 1 - 7 JH>  •  (£12  x  m 2 -P-12 ~ T 1 2 ^ r  12 -12 -12 m r  12 —is P  ^12(3A)^2(3A) (12)(3A)  Tr 123A  [CK  V  _  6  +«  r,«  (12)  + 1  t3A)  . (12)  + 6  (3A)  2  }  f  J  s  p  12(3A) 'l2(3A)^12)(3A) (12)(34)  V  6  J  P  12 (12) ^12(34) 12(34) (12)(3A) (7.3.18)  - 210 -  As w i t h the monomer-dimer exchange r e a c t i o n , we can a s s o c i a t e the energy flux  ct along by the p r e s s u r e tensor (P - P , with = = 2D->-2M+D,no exch  carried  b  v" by u s i n g  J  r  the i d e n t i t y  £i + 2 j =J £i (  +  +  +  +  2  = (j>^ -Ej ~  P  +  Pj -Ek -El>  + k  m  .Y.")  4 <Pi  +  +  2  +  Pj  -E(ij)(kl) m  .Y."  P  +  (7.3.19)  +  Pl> £ i ( j k l )  k  With t h i s r e l a t i o n s h i p , . t h e i d e n t i f i c a t i o n o f P _ „ • v" from ' =2M+D,no r x t — w  p  the term  T  1<  W.  £12  •  ( V  V  12 1 2  )  1234 2  - r -1(34)  v  (-He- fi„„.o„ • 3m  -£ (34) < T T 2  (134)'s  I  *(234) 8 )  1  [v  . xv v  1(34) 1 / 0 /  [ V  * 2(34)  (v  13  ( V  23  1 0  +  V  +14v .)] y  24  w  ) ]  12(34) 12(34)^12(34)  in  (7.2.19) can be made i n a more d i r e c t way.  The r a t e o f change i n the p o t e n t i a l energy d e n s i t y due to t h i s reaction i s :  - 211 -  *3t  (M  V  V  M M  '  +  M  V  D D^2D+2M+D,no exch  t™M M^ V  +  +  M  V  D D ^ 2 D > 2 M D , n o exch +  V  -El  Tr 1234  2  "iT"^  6  i  12  +  V  13  —  V  2m  4 +  V  14,  +  13  ]  ~  V  14  s  +  V  23  + V  24.  "(34) J  P  12(34) (12)(34) (12)(34)  + -i- T r 1234  ( :  r Tr¥¥-i  +  r  r 2  2!  )  6  ( v  (  2  (12)  2  6  + 6  1  P  1234  >. ^ 1 3  6  + (^  r  6  ;  6  ^12(34)  +  2 + (34)  12(34)  4 +  V  (34)  12(34)  1  +  6  s  ^ 23  24  ;  2 f ^ s ^12 p  i2(34Al2)(34) (12)(34)  6  t  + 6 2  2  V  V  r  ~P T n 12(34) 12(34) (12)(34) J  +  i2) i3  P  12 ^12(34)^12(34) (12)(34)  v  23  2  - 212 -  Compared w i t h the e q u a t i o n o f change f o r k i n e t i c  energy d e n s i t y  c o n s e r v a t i o n o f energy i s a g a i n v e r i f i e d .  Exchange Decomposition  [  !Tt  (M  T  +  M M  - V  M  T  )]  D D 2D>2M+D,exch  C  •v + L  P ^ =2D+2M+D,exch  —  •  =2D+2M+D,exch  »  ct +  (  )  I " £  +  1234  2D>2M+D,exch ' -  +  t  £ D->-2M+D,exch 2  1^24  "  0)  —  'P  Ws  2 4  r A12) \ , , £ l 2 — S S T " ^ ' £ l 2 (12) s  [  (  6  [£34 < ^ ~ 1 >  J  ]  W*  *£34  P  13(24) 13(24) (12)(34)  +  , 2i ~fi 1  T  1 £ 4 £24 , 2T — — £(24)  .  2  2  *  (  4  J  P  6  ,„ „,t  v )  (24) s  13(24) 1 3 ( 2 4 ) ( 1 2 ) ( 3 4 )  :  (  V  £ >  - 213 -  +  1 £l2 £l2 , 2J 21 2 2 A12) * 1234 T  v  R  r  ;  (12) s (V v ) *  , 1 £34 £34 , . , 2l 2 2 A 3 4 ) °(34) s ;  <P  21  :f  +  £12 £12  Tr  p  13(24) 13(24) (12)(34)  (  £i2 £ i 2  } t  6(12)  1234 + £34 £34  <f  8n  +  (  £34 £ 3 4  :I  } t  u - v  6(34)  p  13(24) 13(24) (12)34)  Tr  1234  £ 4 £ 4 £ 4 2  2  X  2  6  £ 4 (24) 2  ~ £l2 £l2 £l2  X  £ l 2 (12)  ~ £34 £34 £34  X  £34 (34)  3  6  (V a))'  6  P  13(24) 13(24) (12)34)  8h  Tr  1234  ^24  X  £ 4£24£247*  6  2  (24)  : (V co)  r  \ 34  x  6  £34 £34 £347* (34)  p  13(24) 13(24) (12)34)  - 214  2i  Tr 1234  1  -  2  ^24 ^24 -£(24) " 2 2 4m  TC  1 2!  -12 -12 2  m  £ "  2 4 )  - 2m v £(12) • (P( 4m  2  2  1 £ 4 £34 £(34) ~ 2T 2 2 4m  m  - 2m v")  6  ( 2 4 )  - 2m v)  5  ( 1 2 )  1 2 )  ^  3  o  • (Fv  2  -1 J  p  13(24) 1 3 ( 2 4 ) ( 1 2 ) ( 3 4 )  T 4t [  V^W^D.exch  + 21 ft 3 T  1 2  r  i _ ^12 £l2 2! 2 2 (  4  P  + ^ T r 1234 1 1  J  £34 £ 3 4 2 2  } ;  (  v" ^  2  2  - v ) 2 ;  P  13(24) 13(24) (12)(34)  ,1  £24 £24  . . 2 (13) i 3  ^TTT" T ~  r  1_-12 £l2 2\ 2 2 "'  ( K  }  p  2  w  ,1 £ 3 4 £ 3 4 V "21 2 2— * * *  6  (P(34) - * v) ( 3 4 )  r  °(12) 12  . r  (34) 3 4  13(24) 1 3 ( 2 4 ) ( 1 2 ) ( 3 4 )  S*/2 2  - 215  + 2 i VV:  Tr 1234  1 £24 £ 2 4 T F T T  (  £24 (  +  V m  }  "  x  £24 2 m r;24  (24)  1 v . , 7 JS> • (£24 v  M  2  1  . 6  „ m 2 v £24 " 7 2 4 M  x  r  £24 £24 ~2 * £24 m r  £24  l  -  4  £12 £l2  v  . 6  2 T T T  X  (-12  f  12  m r  1  -|.)  (12)  • ( r  1  2  x p  -  1 2  f  2  r , «)  2  £l2 £l2 + £12  '  ' £12 m r  1  2  —Is  f  {  1 -34 -34  2T~  ~  N  . °(34)  i ^ l ^ L - ^ ^ m r  +  £34  3  . (£  3 4  x p  3 4  - f r  2 4  »)  4  -34 -34 — 34 m  „ ' -£34  r  —Is  J  P  13(24) 13(24) (12)(34)  - 216 -  [(  1234 +  ^ (12 )"^"^(34) ^13(24) ~ 2 s ^12(24) ^ 3 ( 2 4 ) ^ 1 2 ) ( 3 4 ) ( 1 2 ) (34) )  [ V  6  V  1  6  P  V  24 (24) " 12 (12) " 34 (?  :J  6  (34)  ]  P  13(24) 13(24) (12)(34) (7.3.21)  Ut  (M  V  M M  ^V^D-^M+D.exch  +  (  = - V  M  V  M M£  +  M  V  D D ^D^M+D.exch  V  2[(=i - ) 6  + 2 Tr 1234  U  m  —  f&ZSl  +  +  12  V  13  V  - v) 6  + l ( — 5 —  + V  14,  l  +  V  1 2 14  1> °(24)  >  Tr 1234  A  6  .  +  +  P  +  *1 ^ 2  /  3  6  6  (12)  2 ~  v  (i  6  + 3  K  6  (24)  (34)  2  _  >  +  6  (12)  '  13  (34), _ 2 14 ;  6  6  +  * (24) (12) 2 ~ ::I  P  13(24) 13(24) (12)(34)  V  6  (34). 2  V  23  + V  2  ^13(24) (12)(34) (12)(34)  2i  +  ; v  23  34  ]  J  s  - 217 -  2i  6  (i<  1234  13(24)  6 l  +  S  ^13(24)  i3(24)  6 [ 1  (12)  v/l/  6  3 }  2  +<*13(24)  r  +  +  1  (  V  +  s ^13  2  2  4  ( 2 4 )  \  +  ^12  )s  ^23  ^14>  V  +  P  (12)(34) (12)(34)  + 5  5  (34) 2  v V  24  J  _ V (24) 2  6  v V  12  V (24) 2  y V  . 34  ]  P  13(24) 13(24) (12)(34)  (7.3  Exchange w i t h no Decomposition  [  W2D,exch  9t  * "  V  [  • =2D,exch * i + i  2  D  ,  e  x  c  h  ' » +  q  2 D > e x c h  ]  218 -  4 n  1 -13 -13 , T r 2T ~1 2~ A l 3 ) 1234 1.24  1  i  — ^  1  "2T ~~2  -2h  *  ;  p  t  —£H  x  °(13) s  2~~ £ ( 2 4 ) C  6  \ ;  (24) s (V v ) '  1 £-12 £ l 2 , "2T ~T ~T A l 2 )  1  ,  ;  £34 £34 ,  ,  " 2 T " T ~ ~ 2 ~ A34)  f>  II  N ;  °(34) s  P  (13)(24) (13)(24) (12)(34)  T r £13 1234  '£l3  £ 4 <HST--2>  +  x  °(12) s  2  6  f  Ws  '-£24  £12 <=TSp--2> * £ l 2  £34  ]  (13) s  6  (12) s  *£34  3  P  (13)(24) (13)(24) (12)(34)  ]  W-  - 219 -  Tr x 6 1234 i l 3 -13 -13 £l3 (13)  TF  +  x  £ 2 4 ^24 -24  6  £24 (24) (V  X  6  X  6  to)  ~ £12 £l2 -12 £l2 (12) " £34 £34 -34 £34 (34) P  (13)(24)J(13)(24) (12)(34)  r 1  6  x p  2 3  2 3  £  2  r 5  3  (  2  3  +  x  \£ 4  £24^24  £2  (24)  v£l2 £12 £12 £1  '(12)  2  x  " \^4*.£34 i34£34/  (13)(24)  v  J  -  * (£(  2  ,1 ^24 £ 2 4 , £ ( 2 4 ) " ) ^2T 4m  m  £  r  vf  r,„ r  1  9 )  ~  1  ~  2  2! 2 (i1  1  2  2  2!  1  r  J  1 3 )  ~ 2m v) 6  , J (13) s n  J  , * (P" (24)  Am  r±Q2) ) [4m  -^34 ^34, -£(34) ~ =4m  2  (34)  2 r a  6  £>  ]  (24) s  - 2m v  ^  }  v  <o)  p  1  ~2~ ~2~  6  (V  (13)(24) (12)(34)  j_£l3_£l3 ^ 1 3 ) 2! 2 2 ' 4m  .  )  1234  (P(  2  m  £  [  2  P  (13)(24) (13)(24) (12)(34)  1 2 )  - 2m v) o , J "(12) s 1  , „ ^(34) £>  (  2  m  x  x 6  0  J  , (34) s ]  - 220  + £ VV:  Tr 1234  l  -13  f  v^r —  (  £ l 3  +  -13  -  . .  —  o  1  " f m r 12  3  ( 1 3 )  co) • ( r  1  x p  3  1 3  - | r  2 3  i  £  )  £-13 £-13 2— * £13 m r 13  -El3  —I s  ,1  £ 2 4 £24  (~  24  X  2  f m r 24  4  . . (24)  - j jo) • ( £  £ 4  £24 • - hr-'  1T ~T~  ,1  {  -12  ("12  r  -12  ~T~  **12  2 4  - ?  2  r , co!  2  J  m  x p  £ 4  2  +  2 4  2  £24  A  "  —I  s  —I  s  , m)  _ 1  °(12)  £ )  .  ( £ i 2  x  _m ^  ^  m r 12 £l2  £l2 12 ' £l2 m r  + -£12 P  1  i  2  (13)(24)- (13)(24)Xl2)(34)  - 221  VV:  Tr 1234  1 £34 £34 . . 7 T T T ••• (34)  (  )  x  .£34 (—  +  ( F  |VV  - £ vv  £34  6  1  2— m r 34  m _2 (  " 7 ^  * £34  x  £34 ~ T 34 r  £ 3 4 £34 2 — * £34 m r 34  £34 *  U3)(24)3(13)(24) (12)(34) p  : { [ f e ^ ] ,  Tr 1234  -  „  "2T ~~2 1  2  c  -13 -13  + 1  TT  h  v /2}  * 2 2~~ °(13) 1 3 r  ~2 2~ (24)  £24 £24  .  r  2 24  f  1 £l2 £l2 2! 2 2  . (12)  1 £ 3 4 -34 2! 2 2  . 2 (13) 1 3  r  2 12  r  P  (13)(24)°(13)(24) (12)(34)  a, /2 2  2$  - 222 -  Tr 1234  [V  f  6  13 (13)  + V  6  V  6  V  6  2 4 (24) " 12 (12) " 34 (34)^  3  P  (13)(24) (13)(24) (12)(34)  +  + K  +y  2  3  6  +# )  (  1  2  )  +  4  v/U  5  (  3  4  )  2  ]  \^ w  s  S  (  (13)(24)  p  (13)(24) (12)(34) (12)(34)  (7.3.23)  Ut « _ V  2D, exch  (M  V  D D  2D,exch  V 3  12  2  + 2 Tr [(-^l - - - v) 6 1234 ~ 2 m  (  1  3  +  V  14  +  V  23  +  V  )  J  P  ^(13)(243 (13)(24) (12)(34)  ,£(13X24) £(13)(24) 2 2  v  1234  z <  (V  14  £(12)(34) £(12)(34) 2 2  P  1  *(13)  + — Tr * 1234  (13)(24) V  P  (13)(24) (13)(24) (12)(34)  +  6  (24)  2 %/2  J  s p  (13)(24)X13)(24) tl2)(34) (12)(34)  +  V  23>  24,  - 223 -  5  "K  (13)  1  +  6  (24)  ( V  2  6  .  f  12  +  V  34  +  §  (12)  )  (34) , ( V  2  , 13  +  V  v  24  ) ]  1234 V  J  P  C13)(24) (13)(24) (12)(34) (7.3.24)  (iv)  Monomer-Monomer-Dimer C o l l i s i o n s  Recombination-Decomposition  M  T  +  *3t ( M M  m - V  M  T  D D^2M+D,recomb,decomp  M  P =2M+D,recomb,decomp  + L  ct 2M+D,recomb,decomp  TfT  ^ CL  —  3  ^  +  "  + ' q  —2M+D,recomb,decomp  * £12  ^ 3 4 ^ - ^ )  (12)34  Tr  ^M-D 4  t£l2 < ^ - r )  Tr 1234  L ^ ^) = 2M+D,recomb,decomp  * P 4 3  W :  6  ]  (34) s  p  (12)34 12(34)  6  \ ^ 2 J ^ E l 2 ^12 £12/ (12)  "33S 1234  (V w) (34)  f  p  (12)34^(12)34 12(34)  * w —  - 224 -  - V  "3"2TT  Tr £l2 £l2 £l2 1234  x  6  £l2 (12) : (V co)'  - r. .34 £34 £34  J  x  6  £34 (34)  P  *(12)34 (12)34 12(34)  + V  2h  Tr 1234  1_ £ l 2 £ l 2 2! 2 2  (  £(12)  6  )  (12) s (V v")  1 TT  £34 £34 , ~T ~2~ A 3 4 )  (12)34  , . °(34) s ;  P  (12)34 12(34)  2  + VV : ( r — M ^ l ' 3 t D - 2M+D,recomb,decomp  v" / } ' ' 2  1 1  r,„ r,„ 1 -12 -12 + -=£ VV : Tr 2! 2 2 1234  p,,„ Ml2) 1  x  - 2m v" (P(12) "  4m  z  i _ £ 3 4 £34, £ 2! 2 2 -) [r  K  w  ( 1 2 ) 3(12)34 ^ — ^ 12(34)  " 4m  2  m  v  2 m  £">  6  ]  (i2) s  " (P(  3 4  ) "  2  m  6  £ " ) (34)  ]  - 225  + ±=- VV Z  n  ,1 -12 -12 2 f l " T  : Tr 1234  . . ••• (12)  (  (  £ l 2  X  1  f m r 12  }  2  P  6  - 2 ^  -12 +  -  (  * ^12  X  m 2 2 12  v ^  r  ^12  -12  ' r2  i 2  m  £l2  12  , 1 -34 -34 v . ("2T ~ 2 ~ ~ 2 ~ *••-' °(34)  m  r  34 -34  +  £34 m  r  -34 — * £34 34 -4s  J  P  ^(12)34 (12)34 12(34)  -  VV  2n  1 ,-12 ^12_ Tr I T 2 2 1234 1  l$  :j  2 12  P  . _ -34 fv34 -34 ^34 2 — (12) 2 — 2 34  (12)34 (12)34 12(34)  r  v m  . 6  (34)  )  2  w  ,2, '  - 226 -  IT  Tr 1234  F<*l+W  +  +  6?  +  ^ 2  (jf  6  l  +  % 4 ) >  6  (  2  "  3  2  4  )  ]  6  %  +  s  (  3  ^13 ^14>  4  )  ]  ( ^ 3 + ^ )  s  }  (34) ( 3 4 ) s ^ 3 4  ,rt  p  (12)34 12(34) 12(34)  +  ( 6  V  6  V  ) P  : r  p  (12) 12 ~ (34) 34 (12)34 (12)34 12(34)  (7.3.25)  +  ["9t  - V  %  ^VVUM+D.recomb,decomp  V  M -  +  + -y T r 1234  M  V  D D —^2M+D,recomb, decomp  , [r (^ 3^ 2  V  , 63 x 13 - u)  +  V  + V  23j 34,  ],  z  + ttyp-'  v) «  V/  «  ( 1 2 )  ' 1 3 ^ 1 4 ^ 2 4 , ( 1 2 )  p  (12)34 i'2(34) l (34) 2  6  2fi  Tr 1234  6  (12) " (34) ( V  6_-6, + ( 3 +  +  13  V  14  +  V  23  +  V  6.-6.  1 „ 13  L  . _4  V  P  14  )  6_-6  6,-6_  0  +  V  V  24  +  (12)34 ( 1 2 ) 3 4 1 2 ( 3 4 )  3  2  2 V  v 23  +  + — 2  2 V  V 24  v }  - 227 -  2  1  ^(12)34 ^ P  * 1234  6  (12)34  6  ^ 3  +  ^ 2 3 >  + 6  (12) 4 2 's  P  v  14  24  ^ ( 1 2 ) 3 4 ^ 2 ( 3 4 ) 12(34)  + 6  + —  —V V  2  +  ^  (  V  2  34  12  V  2  )  J  P  3 4 ^(12)34 ( 1 2 ) 3 4 1 2 ( 3 4 )  (7.3.26)  Recombination Without Exchange  [  ~9t  (M  T  M M  » _ V  +  M  P  +  +  +  T  )J  D D 2M+D^2D,no exch  C t  C t  v  3uo +  4l-D 4  ct =2M+D+2D,no exch  CT -  V  =  C  L *'') ;  =  2M+D>2D,no exch  • a)  -  -3-2M+D>2D,no exch  r +  • v" + (P - P ) — = = '2M+D->-2D,no exch  2M+D+2D,no exch  2i?  1234  (12) x , , ^ T - - ^ * 1 2 (12) s  t  "12 f  (  £  T  6  P  (12)(34) (12)(34) 12(34)  ]  •u -  - 228 -  - V  1 £j_2 £ l £ 2T 2 2 £(12)  Tr 2TT 1234  (V v )  (  1  e. (12)(34) (12)(34) 12(34) 3  1  Tr  37R" 1234  p  S£l2 -El2 £l2 £l2 (12J x  : V to  6  p  (12)(34r(12)(34) 12(34)  + Tr 1234  -\2 £l2 (12) £l2 6  J  x  -El 2  (V co)*  p  X l 2 ) ( 3 4 ) (12)(34) 12(34)  VV  Tr  •ar  1 £12^12 £(12) " 4m 2 2  ^34 <P  T  r  1234  m  £ (  2m v) 6  £(12)  ( 1 2 )  2 1  :I  P  (12)(34) (12)(34) 12(34)  " 1 lit IfT  2  2  2M+D->-2D,exch  6  r  v /2  <P  P  ^12 —12 ( 1 2 ) 2 1 2 ( 1 2 ) ( 3 4 ) ^ ( 1 2 ) ( 3 4 ) 1 2 ( 3 4 )  0 ) 2 / :  ]  s  - 229 -  1 -12 -12 2! 2 2 (12)  - VV 1234  Al  x  -^i2  m r 12  +  £l2 '  (r  " 2$  ' -12-P-12 " X  m 2  -12 -12 2 ' *12 m r 12  P  ^(12)(34) 7 J 2 ) ( 3 4 ) 1 2 ( 3 4 )  2T  Tr 1234  +  ^  U  %  4  2  )  6 l  +  2  (  ^(34)  3  4  \  ^13  J  2  +  ^4>  W  V  s 23  24  ;  P  ^(12)(34)^12(34) 12(34) +  [  3t  (M  V  M M  +  6  V  (12) 12  P  ^(12)(34) ^12)(34) 12(34)  W * 2 M + D - » - 2 D , n o exch  (M  V  D D-^M+D^D.no  exch V  +  Tr [ ( - ^ 1234 2  m  ~ v) 5 " (  1  2  13  + V  )  P  ^(12)(34)°T2(34) 12(34)  14  + V + V 23 24.  230  2¥  J  [^(12)  r 3  4  -  < 13  V  V  +  V  -  14>  < (12)  +  6  -  < 23  V  V  +  V  24>  3  (12)(34) 12(34) 12(34) J  + i*  P  6  Tr (12)(34)  1234  V  +  (12)  6  (34)  ^12)(34)^(12)(34)*°12(34) 12(34) p  \  + 6  2  2  +  V  12  f  (12)(34) (12)(34) 12(34) 7  p  (7.3.28)  Exchange  [  8t  -  ( M  Recombination  M M T  +  - V  M  D D 2M+D-»-2D,exch T  )]  =2M+D>2D,exch  +  +  , v" +  L =2M+D-»-2D,exch C  t  (L -  L  1234  C  t  )  [  .  3  ~  ~  +  t£  9 A  t£  3 4  i-L24  ct, P" ) u  2M+D+2D,no  exch  ^ D  i  4  2M+D->-2D,exch  £l3 £l3  +  (£ -  * -  * '"Tm  £2M+D-»-2D,exch  +  1  -Z>  (M 1 33 ) ^ s  5  J  p . . • (^2£> -^24 " " T T " "- ^v ) (24) s V  p  34 *  (13)(24)  u  ^  " v")  (13)(24) 12(34) p  ]  «  (  3  4  )  ]  8  —  - 231 -  1 - V  -  Tr 1234  £34 £34  (  -P-34 ^34  :I  P  1 -13 -13 (2/ 2! 2 2 -H13) (13)'s V  (V v ) , 1 -24 -24 , . . " 2 T ~ 2 ~ ~ ^(24)°(24) s ;  p  (13)(24)  Tr  (34)  (13)(24) (13)(24) 12(34)  +  + 7  u - v  )  2  6>  Tr 1234  +  1  (13)(24) 12(34)  £ 3 4 -34  1234 U~2  2~  ,  -  J  .  (13)^8  ^(13)  „,t :  (  V  ^ >  P  ^13)(24) (13)(24) 12(34)  T6n"  Tr -13 -13 -13 1234  +  X  -El3 (13) 6  £24 £ 2 4 -24  X  -224 (24)  -34 -34 -34  x  - -34 (34)  J  6  p  6  P  ^(13)(24) (13)(24) 12(34)  (V w)'  f c  - 232 -  - V  Tr 1234  S2Li3  +  x  5  P13 £13 Ln/'  (i3)  \ I 4 £ 2 4 £24j2>/' X  2  -VI.34 £34 £34 JL-.  6  V to  (24)  x  J  (34)  P  (13)(24) (13)(24) 12(34)  + VV: i  Tr 1234  ,1  ^13 -^13  ( T T I  •  £  l  X  75—  *  -~ m r 13 3  1  P  .  .  ,  0 (13)2  )  " 7 ^  3  ^13 +  .  m  m 2 x 2 1 3 »> r  -£l3  ^13  T~ r  i 3  x  " ^13  ' £i3  13  —Js  . 1 -^24 -2k 7 T ~ 2 ~ T "  v  C  r  .24  .  (24)  x  £j 4 - L - 1 to) • ( r m r 24 2  T  -£24  2  4  x p  2 4  -  7  m _2 r to) 2  4  •^24 ^24 2 ^ * £24 24 m  r  —J s  P  ^(13)(24)^(13)(24) 12(34)  - 233 -  + VV :  Tr 1234  1 1-34 -34 " TT — ~T  . 6  {  X  —34  . (34)  1  -H34  2 m r 34  +  £34 '  •  + VV :  L  2 8t  P  V  2  2  -34 -34 2 2  ^ 2 (24) r "(24) / N  £«o  2  /2  r,,„x2 (13) "(13)  7  p  (13)(24) (13)(24) 12(34)  2  2  -34 -34 m(v - v " ) , 2 2 2 (13)(24) (13)(24) 12(34) p  J  1234  r  r r £l3 £l3 r,,„.2 2 2 (13) (13)  1 2!  P  ^  m _2 P 3 4 - 7 3 4 ^  R  J  1 2!  T V  x  (13)(24) (13)(24) 12(34)  2!  1  4  n  D = 2M+D+2D,exch  1234  3  £34 £ 3 4 1 ' £34 34 M  P  ( £  r  J  P  - 234 -  Tr 1234  l ^ V ^ T - V l Z *  +  ^ 2  +  ^(34)>  +  ^  /  (  2  3  4  )  ]  2  +  ^l *(34)>  +  s^23  (  *(34) W  >s^4  3^34  P  (13)(24)' 12(34) 12(34)  +  ( 6  V  (13) 13  +  5  (24)24 - (34) 3 4 > V  6  V  p  (13)(24r(13)(24) 12(34)  (7.3.29)  2M+D->-2D,exch  V  ^  D —^2M+D+2D,exch V  +  2  6  1234  + V  14  + V  23  + V  34-  <»>  !  >  12  P  ^ (13)(24T 12(34) 12(34)  Al3)  - Tr ^ 1234  6  (24)  V _ V _  2  l  .  +  6 f  v  2  6  6  +  ' 12  6  (13) + (24) 1 (34), 2 " 2 ' 14 v  v  6  (13) ^ 2 r  +  6  (24)  6  2  +  (34) , 2 ' 23  6  y  P  3  p  (13)(24) (13)(24) 12(34)  - 235 -  6  + i Tr 1234  [ t t (12)(34) f  +  5  (34)  V  1 J  2  V  s (13)(24)  P  (13)(24r 12(34) 12(34)  6  . +  (12)  l  r l  6  * (34) _ 2 13  6  . 2  +  6  6  (34)  V  f  V 2  :I  (13)  +  24  6  (24) .2  V  . 34  J  p  '(13)(24) (13)(24) 12(34)  (7.3.30)  Exchange Without Recombination  Ut  m  -  ^  T  M  +  T  ^ D^2M+D-*- M+D,exch 2  V =2M+D+2M+D,exch  —  =2M+D+2M+D,exch  ct + (L - L ) M+D>2M+D,exch * —  +  2  [  1234  £l3  r  ^ " Z " )  -^2M+D->-2M+D,exch  *£l3  E  /  - (34)  6  ...  T (13)24 (13)24 12(34) J  0 P  ]  (13) s  . 34  f  4  . (34) s J  - 236 -  - V  2i  Tr 1234  1 -13 -13 , (13)  hn^s : (V v")'  1  -3h  £34 T~ A 3 4 ) ° ( 3 4 ) s ;  " T T ~2  Xl3)24 (13)24 12(34) :r  P  Tr -13 -13 -13 1234  x  6  : (V co)  - £34 r £34 £34 6>  -Bl3 (13)  :J  x  £34°(34) P  (13)24 (13)24 12(34)  Tr £l3 1234 -  x  6  £13 £13 £13 * (13) : (V co) x  £34 £34 £34 £34 " "(34)  e  1  D P  (13)24 (13)24 12(34)  1 £i £i3, £ ( i 3 ) " 2i -) [ 4m + VV : 4F- Tr <7T 2! 2 2 * 1234 2 m  3  .  r  r_. r_.  2 m  ) 6  ]  £" (13) s  p, -2m v 0/N  1  f^fl ' J  '(£(13)-  ^ P  (13)24 (13)24 12(34)  "  2  )5  -<l(34)- "^ (3 )' 4  S  -  1 J , . 7T 1234  2i  - VV  1T  l  ,1-13 - 1 3 . 2 ~ 2 ~ " ~2~ ° ( 1 3 ) 1 3 r  ?  + VV : 2 i  1  fit  Tr 1234  . 2 . °(34) 34 r  ;  f  p  V  ~2  ,1  -13 -13 2~~  ^7T  ,-13  <-  x  v  ^13  m r  2 / 2  . 6  (13)  1  " 7  (  »>  * ^13  x  Pl3 " 7  m -2 1 3 »> r  13  J— * r  -13 Pl3  -13  m  . „  -P-13  13  1 £34 £34  (  2T-2"T"  X  m r  f 3  3  4  5  (34)  - \  »>  ' (£34  x  P34 " 7  r  3 4 »>  4  £34 £34 . „ +  £34 m  r  2 — 34  * -234  —is  (13)24  P  (13)24 12(34)  2  . /2  (13)24 12(34)  V^2D*2MfD.exch "  +  -34 -34 ~ ~T~T  J  (13)23  +  237 -  - 238 -  6 , + 6 /  Tr 1234  +  +  X) 2  +  ^ 2  -V l V i  8  ^34)  1  ^ T  )  ,n;  1  2  [ ( ^ ^  +  1  ^  (tf  (  3  4  )  23 24>  +  +U  -M^'B  )  (  " l 4  6  ^(34) (34) > s ^34  p  (13)24 12(34) l (34) 2  +  (6  V  6  V  ( 1 3 ) 13 " (34) ( 3 4 )  }  ^(13)24^(13)24^2(34)  (7.3.31)  (M  V  MM  +  M  V  )1  D D 2M+D-2M+D,exch  (  +  ¥M  M  + 2 Tr 1234  V  )  D D ^ 2M+D^2M+D,exch  .A_ [  V  , 6 , 14 _ u)  (  +  4  V  2i  Tr 1234  2-  +  ( 6  V  J1  ]  2  2rm  P  + V  24 - 34.  12  +  V  14  +  g  V  + V  23 34-  p  (13)24 l2(34) 12(34)  6  (13)- l  ( 6  6  ) V  } V  12 l  2 4 ~ ( 3 4 ) 24  J  P  +  +  (  (13)24 (13)24 12(34)  ( 6  6  (13)- (34)  2  ) V  23  2  }  V  14  - 239  -  1234  p  6  +  [Of  2 +  ^  ( 1 3 )  8  1 *  1  S  2 <P  +  ('n )  6  ) 6,  +  2  ^(13)24^12(34) 12(34)  ,  ( 1 3 ) 2  i t r 23 s  2  6,  +  6  (34)  4  +  6  (13) V  13  34  ]  P  (13)24^(13)24 12(34)  (7.3.32)  Thus, when chemical r e a c t i o n s take p l a c e , owing  to atom  rearrangement, i n a d d i t i o n to the heat f l u x and the  and  L =  • cu —o  types of terms, the energy f l u x a l s o c o n t a i n s terms which r e s u l t  from  d e l o c a l i s a t i o n of the p h y s i c a l a t t r i b u t e from the r e a c t a n t s to the products.  - 240  -  CHAPTER 8 SUMMARY  In this work, conservation laws for the physical observables of mass, l i n e a r and angular momenta, and energy are attained for a dimer-monomer reacting gas as described by the Lowry-Snider k i n e t i c theory.  Following  Olmsted and Snider, the physical observables are v i s u a l i s e d as molecular attributes and are l o c a l i s e d at the centres of mass of the molecules concerned.  This "composite" picture allows the effects of reactions on  the equations of change to be taken into account.  A sum rule  C  v  C  c  C*C  c  i s derived, assuming the v a l i d i t y of the completeness r e l a t i o n  which i s related to the strong orthogonality assumption, which i s i n turn a c r u c i a l feature of the Lowry-Snider k i n e t i c theory.  From the sum rule,  the contribution to the equations of change from each c o l l i s i o n type i s identified.  For a more general situation i n which coherences between the  (different channels play a s i g n i f i c a n t role, the completeness r e l a t i o n i s no longer v a l i d and the sum rule w i l l thus also need modification to allow for coherence e f f e c t s .  - 241 -  The  equations o f change obtained i n t h i s work can be compared  those obtained using a macroscopic the mass f l u x treatment,  i s due e n t i r e l y  1 5  p o i n t of v i e w .  In the l a t t e r  to m o l e c u l a r v e l o c i t i e s .  with case,  In the p r e s e n t  when chemical r e a c t i o n s o c c u r , there i s a l s o a term a s s o c i a t e d  w i t h the change i n mass l o c a l i s a t i o n .  This additional  term v a n i s h e s when  the system i s homogeneous.  Another f e a t u r e of the Lowry theory i s that c o l l i s i o n s are c o n s i d e r e d to be i s o l a t e d isolated  and independent  events.  When c o l l i s i o n s are no l o n g e r  or when i n t e r m e d i a t e s are i n v o l v e d , the k i n e t i c  be g e n e r a l i s e d and the Evans theory i s a r e a s o n a b l e this generalisation.  I t i s surmised  m o d i f i c a t i o n , and the l o c a l i s a t i o n  starting  that the sum r u l e ,  scheme used  theory needs to point f o r  with  i n this thesis w i l l  form a b a s i s on which the g e n e r a l theory can be b u i l t .  still  - 242 -  BIBLIOGRAPHY  1.  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 Theory of Gases and L i q u i d s ( W i l e y , New York, 1954).  2.  J.H. I r v i n g and J.G. Kirkwood,  3.  J.T. Lowry and R.F. S n i d e r , J . Chem. Phys. 6J_, 2320 (1974).  4.  R.F. S n i d e r and K.S. Lewchuk, J . Chem. Phys.,  5.  N.N. Bogoliubov, Problems of a Dynamic Theory i n S t a t i s t i c a l P h y s i c s , i n S t u d i e s i n S t a t i s t i c a l Mechanics, J . de Boer and G.E. Uhlenbeck ( e d s . ) , V o l . I ( N o r t h - H o l l a n d P u b l i s h i n g Co., Amsterdam, 1962); M. Born and H.S. Green, P r o c . R. s o c . A 188, 10 (1946); 189, 103 (1947); 190, 455 (1947); J.G. Kirkwood, J . Chem. Phys. J_4, 180 (1946); _15_, 72 (1947); J . Yvon, La T h e o r i e des f l u i d e s et de l ' e q u a t i o n d ' e t a t ( P a r i s : Hermann et C i e , 1935).  6.  R.F. S n i d e r and B.C. Sanctuary, J . Chem. Phys. J55, 1555 (1971).  7.  L. Waldmann, Z. N a t u r f o r s c h . Chem. Phys. 32, 1051 (1960).  8.  M.W.  9.  R.D. Olmsted and C F . C u r t i s s , J . Chem. Phys. 62, 903 (1975); j>2_, 3979 (1975); 63, 1966 (1975).  J . Chem. Phys.  18_, 817 (1950).  46, 3163 (1967).  A 12, 660 (1957); R.F. S n i d e r , J .  Thomas and R.F. S n i d e r , J . S t a t s . Phys. 2_> 6  1  (1970).  10.  J.A. McLennan, J . S t a t s . Phys. 28, 257; 28, 521 (1982).  11.  R.D. Olmsted  12.  D.K. Hoffmann, D.J. K o u r i and Z.H. Top, J . Chem. Phys. 70_, 4640 (1979); J.W. Evans, D.K. Hoffman and D.J. K o u r i , J . Math. Phys. 24, 576 (1983); J . Chem. Phys. 78, 2665 (1983).  13.  J.M. Jauch, B. M i s r a and A.G. Gibson, Helv. Phys. A c t a 41_, 513 (1968).  14.  A. Messiah, Quantum Mechanics, V o l . I , E n g l i s h t r a n s l a t i o n by G.M. Temmer, ( W i l e y , New Y o r k ) .  15.  S.R. de Groot and P. Mazur, N o n - E q u i l i b r i u m Thermodynamics ( N o r t h - H o l l a n d P u b l i s h i n g Co., Amsterdam, 1962).  and R.F. S n i d e r , J . Chem. Phys.  (1976).  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0059475/manifest

Comment

Related Items