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

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CONSERVATION LAWS IN RECOMBINATION KINETIC THEORY By PUT KING IVY SZE B.Sc. (Hons.), The University of Reading, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Chemistry) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1986 ® Pui King Ivy Sze, 1986 In p r e s e n t i n g this thesis in partial fu l f i lment of t h e requ i rements for an a d v a n c e d d e g r e e at the Univers i ty o f Brit ish C o l u m b i a , I agree that t h e Library shall m a k e it f reely avai lable for re fe rence a n d s tudy . I further agree that p e r m i s s i o n fo r ex tens ive c o p y i n g o f this thesis for scho la r l y p u r p o s e s may be g ranted by the h e a d o f m y d e p a r t m e n t o r by his o r her representat ives . It is u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f this thesis fo r f inancia l ga in shall no t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t T h e Un ivers i ty o f Brit ish C o l u m b i a 1956 M a i n M a l l V a n c o u v e r , C a n a d a V 6 T 1Y3 DE-6 (3 /81) ABSTRACT The hydrodynamic equations of change for a reacting gas mixture of monomers and dimers are studied. The gas i s considered to be d i l u t e and ) described by the k i n e t i c theory of Lowry and Snider ( J . Chem. Phys. 61, 2320 (1974)). From the k i n e t i c equations for the density operators representing the monomer and dimer, the equations of change for one-molecule observables are obtained. Since the energy operator involves the intermolecular p o t e n t i a l energy, i t i s necessary to derive the energy balance equation from the von Neumann equation, since this includes molecule-molecule c o r r e l a t i o n s . As well, the k i n e t i c theory formulated by Lowry and Snider i s rewritten so that rearrangement c o l l i s i o n s are emphasized. A c o l l i s i o n a l sum rule i s derived involving the commutation properties of channel projectors and t h e i r respective p o t e n t i a l s . A known property of the o p t i c a l theorem i s that i t i d e n t i f i e s the reactive loss terms as part of the non-reactive t r a n s i t i o n superoperators. The sum rule i s applied to rewrite the non-reactive t r a n s i t i o n superoperators so as to display the reactive loss terms. This aids i n esta b l i s h i n g conservation laws for the physical observables of mass, l i n e a r momentum, angular momentum and energy. A form of 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 for i s also derived. Both the o p t i c a l theorem and the sum rule are based on the strong orthogonality hypothesis, which plays a fundamental role i n the Lowry-Snider theory. - i i i -On localising the physical attributes at the centres of mass of the molecules, the contributions to the equations of change from collisional transfer (due to the forces and torques between the coll i s i o n partners) and from the transfer of the physical attributes from the reactants to the products are identified. 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 differing l o c a l i t y of the collision partners. The decomposition of the kinetic energy operator into i t s components for radial and rotational motions allows the kinetic energy flux contributions associated with the pressure tensor and the molecular angular momentum flux to be identified. - i v -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS ..' i v ACKNOWLEDGEMENT v i 1. INTRODUCTION 1 2. A KINETIC THEORY FOR A MONOMER-DIMER REACTING GAS 9 A. Description of States Using the Density Operator Formalism............... 9 B. Projectors and Channel States 13 C. Description of C o l l i s i o n s 17 D. Time Evolution of the System 26 3. THE FORM OF THE EQUATIONS OF CHANGE 34 A. Description of Molecular Observables 34 B. Formal Equations of Change 39 4. A SUM RULE 49 A. Channel Coupling and the Generalised Optical Theorem 49 B. Gain and Loss of Monomer Observables 56 C. Gain and Loss of Dimer Observables 66 5. HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES I: NON-REACTIVE COLLISIONS 69 A. C o l l i s i o n s Involving 2 Monomers 69 B. C o l l i s i o n s Involving a Monomer and a Dimer 76 C. C o l l i s i o n s Involving 3 Monomers 79 D. C o l l i s i o n s Involving 2 Dimers 82 E. C o l l i s i o n s Involving 2 Monomers and 1 Dimer 85 6. HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES I I : REACTIVE COLLISIONS 92 v A. C o l l i s i o n s Involving a Monomer and a Dimer 94 B. C o l l i s i o n s Involving 3 Monomers 114 C. Dimer-Dimer C o l l i s i o n s 121 D. Col l i s o n s Involving 2 Monomers and 1 Dimer 136 - v -Page 7. THE ENERGY BALANCE EQUATION 158 A. The Production and Loss of Energy Associated with the Monomers and with the Dimers 158 A . l The Energy Operators 158 A.2 Energy Balance for the Monomer 161 A.3 Energy Balance for the Dimer 168 B. Non-Reactive C o l l i s i o n s 172 C. Reactive C o l l i s i o n s 190 8. SUMMARY 240 BIBLIOGRAPHY 242 - v i -ACKNOWLEDGEMENT I would l i k e to express my h e a r t f e l t gratitude to Dr Robert Snider, my supervisor, who has not only been giving me i n t e l l i g e n t advice and guidance, but also has been most courteous and patient with me when I show signs of slackness on my work. His devotion to his research and f a s c i n a t i o n with i t are i n s p i r i n g and deserve my utmost respect. My thanks are also due to Miss Anissa Yeung, who proof-read my t h e s i s , and to Mr Stanley Luck for h i s i n t e r e s t i n my work as well as giving me encouragement and appreciation for what I have accomplished. The company of Miss L u c i l l e Fung, Messrs Kachung Hui and Marcus Karolewski have been most heartening. I appreciate t h e i r characters and enjoy the moments I share with them. It i s my friendship with them that has provided 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 darkest hours during my research to Miss Susanna Tse, for every time when I lose heart, I w i l l remember her dedicatedness and single-mindedness. Jeeva Jonahs, who typed up my thesis surely has my warmest thanks. Her devotion and meticulousness i n s p i r e not only me, but, as I am pretty sure, other people who work with her too. Thanks also go to those who were most h e l p f u l and understanding as I was wandering aimlessly during my early s c i e n t i f i c endeavour. Drs Marjorie Jeacock and David Rice have my respect and gratitude: I s h a l l never forget t h e i r c o r d i a l i t y and thoughtfulness. - 1 -CHAPTER 1 INTRODUCTION Fundamental to the laws of physics are the conservation of mass, linear momentum, angular momentum and energy. In fluid dynamics, these are expressed by the nature of the equations of change for the densities of these physical attributes 1. While i t is 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 kinetic theory of gases for bimolecular reactive and non-reactive c o l l i s i o n s 1 , how the recombination and breakup reactions' affect the flux expressions has not been studied. It is the object of this thesis not only to obtain conservation equations but also to investigate the effect of chemical reactions on the various fluxes. Specifically, the equations of change are to be derived from the kinetic theory formulated by Lowry and Snider for a dimer-monomer reacting gas. In their classic paper, Irving and Kirkwood derived the hydrodynamic equations for mass, linear momentum and energy directly from the classical Liouville equation for the entire system. The fluxes are expressed in terms of expectations of molecular variables. Dirac delta functions are introduced for the purpose of localisation of the molecules which are considered to be point masses interacting via pair potentials. The quantum mechanical equivalent can be found in a paper by Snider and Lewchuk on the irreversible thermodynamics of a system of molecules possessing internal angular momentum in which the equation of change for angular momentum is also considered. - 2 -These formulations deal with the time evolution of a system composed of a single species. For a chemically reactive system i t i s more appropriate to express the equations of change for each species and to e s t a b l i s h the conservation laws for the t o t a l mass, momentum, angular momentum and energy for the f l u i d mixture as a whole. To carry out t h i s procedure, i t i s necessary to derive the k i n e t i c equations for each of the chemical species present i n the system. In p a r t i c u l a r , when no reactions take place, and only one species i s present, the equations of change reduce to those of Snider and Lewchuk. K i n e t i c equations are generally obtained by truncating the BBGKY hi e r a r c h y 5 at the two- or t h r e e - p a r t i c l e l e v e l , together with the assumption that the two- and/or t h r e e - p a r t i c l e density operators or d i s t r i b u t i o n functions are functionals of the one-particle density operator or d i s t r i b u t i o n function. In th i s way the k i n e t i c equations represent a density expansion of the L i o u v i l l e equation. For a one-component d i l u t e gas, assuming the v a l i d i t y of Boltzmann s t a t i s t i c s , and that the Hamiltonian takes the form of a sum of one-particle Hamiltonians and their pair p o t e n t i a l s , we have, f o r the f i r s t quantum BBGKY equation 6 i f iA -p ( 1 ) ( l ) - P ( 1 ) ( l ) ] = T r [ V ( l , 2 ) , p ( 2 )(12)J . 3t - 2 (1.1) Here p ^ \ l ) and p ^ \ l 2 ) are s i n g l e t and doublet density operators for p a r t i c l e 1 and for the pair of 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 Hamiltonian for p a r t i c l e 1, and V(l,2) i s the potential operator between p a r t i c l e s 1 and 2. If the hierarchy i s truncated at the two-particle l e v e l , by setting P ( 2 ) ( l , 2 ) = 8(1,2) P ( 1 ) ( l ) P 0 ) ( 2 ) ^ ( 1 , 2 ) (1.2) with ft, the Mpller operator associated with binary c o l l i s i o n s between p a r t i c l e s 1 and 2 and being i t s hermitian adjoint, then t h i s i s consistent with the picture that the molecules are independent before the c o l l i s i o n . On su b s t i t u t i n g equation (1.2) into equation (1.1), a generalised Boltzmann equation i s obtained, namely 5 i * k p ( 1 ) < i > = L ^ d ) , P ( 1 ) ( D L + T r [ V ( l , 2 ) , 0(1,2) p ( 1 ) ( l ) p ( 1 ) ( 2 ) fiT(l,2)]_. 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 are not at the same p o s i t i o n as the p a r t i c l e ( l a b e l l e d 1) that i s being observed. This 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 l o c a l i s e d form (known as the Waldmann-Snider equation) p has been discussed by Thomas and Snider . Because of the n o n l o c a l i t y of the c o l l i s i o n operator i n equation (1.3), the macroscopic fluxes ( f o r example, the pressure tensor) generally have c o l l i s i o n a l c o ntributions. This i s to be contrasted with the usual Boltzmann equation, i n which - 4 -collisions are local and the fluxes of momentum and energy are associated solely with the kinetic motion of the individual molecules. If two particles are bound, then i t is physically obvious that the fluxes (in particular of the energy) wi l l depend on the interaction of the particles. Besides, in the co 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 relation (1.2) cannot be valid. The proper closure relations 3 9 10 have been considered by various workers > » • This thesis w i l l a follow the work of Lowry and Snider , which is reviewed in Chapter 2. The chemical description of bound pairs requires one to recognise these dimers as distinct species. As such, the dilute gas kinetic theory of dimers depends on binary collisions of dimers, which are equivalent to a particular type of four-particle 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 particles but now the single particle density operator p ( D i s replaced by the bound "pair" density operator p D(12) with analogous changes for the free motion Hamiltonian and c o l l i s i o n operators. The description of the free (dimer) motion in terms of equivalent atomic (monomer) properties has been given by Olmsted and Snider 1 1. For the more general situation in 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 i n terms of the i n d i v i d u a l free monomers and dimers, the equations of change need a unifying d e s c r i p t i o n i n order to prove conservation p r i n c i p l e s . The unifying d e s c r i p t i o n i s c l e a r l y in terms of the atomic (monomer) picture and the work by Olmsted and Snider has set the stage for how t h i s can be accomplished. It i s the working out and v e r i f i c a t i o n of the conservation equations that form the work of t h i s t h e s i s . As mentioned previously, several groups of workers have endeavoured q to describe a reacting diatomic-monatomic system. Olmsted and Curtiss studied a system close to equilibrium. Then the Wigner functions d i f f e r only s l i g h t l y from equilibrium. Using the molecular chaos assumption and truncating the BBGKY hierarchy at the three-atom l e v e l , they obtained a k i n e t i c equation from which various transport c o e f f i c i e n t s were ca l c u l a t e d . However, the d e s c r i p t i o n i s atomic and the bound states are treated as a perturbation on the free states whereas the chemists view i s to treat a molecule as a d i s t i n c t chemical species and in p a r t i c u l a r , the boundedness of the two constituent atoms i n a dimer molecule i s viewed as a c h a r a c t e r i s t i c of the dimer. The treatment of bound states as a perturbation f a i l s to emphasize t h i s chemical picture and disallows a s i g n i f i c a n t concentration of dimers. Moreover, t h e i r treatment i s incomplete as dimer-dimer c o l l i s i o n s are not considered, but which i s of course consistent with th e i r l i m i t i n g low dimer concentration. McLennan 1 0 also studied a monomer-dimer reactive mixture and obtained a k i n e t i c equation for the s i n g l e t density operator, again using the - 6 -molecular chaos assumption and truncating the BBGKY hierarchy at the three-atom l e v e l . Part of the doublet density operator i s to represent a pair of free atoms and i s thus equal to a product of s i n g l e t density operators. The t r i p l e t density operator i s written as a sum over channels of products of density operators for the molecules present i n the respective channels. As i t i s known that a divergence of the density expansion of the BBGKY hierarchy occurs at the l e v e l of four-atom c o l l i s i o n s , McLennan truncates his expansion at the three-atom l e v e l . In neglecting dimer-dimer c o l l i s i o n s while three-monomer c o l l i s i o n s are included, the de s c r i p t i o n can only be v a l i d for low dimer concentrations. As well, McLennan's expansion i s i n terms of the density of atoms and the molecular nature of the reactive system i s not emphasized. 3 A more formal approach was given by Lowry and Snider . The system being studied i s an i d e a l gas i n which molecules i n t e r a c t only by c o l l i s i o n s . The atoms that 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 form an is o l a t e d subsystem, so that the von Neumann equation describing t h i s c o l l i d i n g set of atoms can be independently solved. Coupled k i n e t i c equations for the density operators representing the dimer and the monomer are obtained. The equations contain terms which correspond to free f l i g h t and various c o l l i s i o n processes. In t h i s t h e s i s , the equations of change for the molecular a t t r i b u t e s of mass, l i n e a r momentum and angular momentum are obtained from the k i n e t i c equations of Lowry and Snider. Conservation laws for each i n d i v i d u a l c o l l i s i o n type, reactive or not, are established. The contributions to momentum and angular - 7 -momentum fluxes from c o l l i s l o n a l transfer are also i d e n t i f i e d . As for the equation of energy balance, a more elaborate formulation i s required since energy Is not a single-molecule observable and i t s time evolution cannot be obtained from the k i n e t i c equations for the i n d i v i d u a l species, however, the underlying p r i n c i p l e of Lowry and Snider's work of is o l a t e d and mutually exclusive c o l l i s i o n s i s used. Within t h i s formulation, conservation of energy i s v e r i f i e d . The crux of est a b l i s h i n g these conservation laws for each c o l l i s i o n type i s to recognise that the various l a b e l l e d channels (how the atoms arrange themselves as d i s t i n c t groups of molecules) 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 . A completeness r e l a t i o n s h i p (sum rule) expressed i n terms of the commutators of the arrangement channels and t h e i r respective potentials i s derived. Using t h i s sum r u l e , the contributions to the hydrodynamic equations of change from c o l l i s l o n a l transfer (the transfer of the physical a t t r i b u t e s during a c o l l i s i o n from t h i s molecule to other molecules p a r t i c i p a t i n g i n the same c o l l i s i o n event) and the transfer of the physical a t t r i b u t e s from the reactants to the products due to atom rearrangement are i d e n t i f i e d . On using the unifying picture of atomic m o t i o n 1 1 , the transformation of the r e l a t i v e motion between two monomers into dimer i n t e r n a l motion when recombination takes place i s i d e n t i f i e d as contributing to the equations of change. S i m i l a r l y , decomposition reactions r e s u l t i n a loss i n dimer i n t e r n a l degrees of freedom and the - 8 -c r e a t i o n of monomer t r a n s l a t i o n a l degrees of freedom. For exchange reactions, we see a coupling of 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 molecules before and a f t e r c o l l i s i o n to the dimer i n t e r n a l motion. The k i n e t i c energy due to these motions i s i d e n t i f i e d by decomposing the k i n e t i c energy operator for k i n e t i c motion into a part due to r o t a t i o n and a part due to r a d i a l motion. Therefore, not only conservation r e l a t i o n s f o r the dimer-monomer reacting gas are derived but the creation and destruction of dimer i n t e r n a l states a f f e c t s the macroscopic behaviour of the system. A l i m i t a t i o n of the derived equations of change i s that the strong orthogonality approximation precludes any coherences between the d i f f e r e n t channel states. The k i n e t i c d e s c r i p t i o n by Evans et a l 1 1 i s rigorous and the Lowry-Snider theory i s obtained as the f i r s t order approximation. Thus, i t may be that the Evans theory should be used to obtain the conservation r e l a t i o n s for the more general s i t u a t i o n where coherence e f f e c t s are allowed. - 9 -CHAPTER 2 A KINETIC THEORY FOR A MONOMER-DIMER REACTING GAS The s t a r t i n g point 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 equations obtained by Lowry and Snider f or pf and p^ ,, the density operators representing the monomeric and dimeric molecules associated with a single atomic species. Below i s a review of the Lowry-Snider formulation. A. Description of States Using the Density Operator Formalism The system i s a d i l u t e i d e a l gas with no i n t e r a c t i o n between the molecules except v i a c o l l i s i o n s . This i s represented by the density operator p^ N^, N being the number of atoms of the system. For an i s o l a t e d and independent c o l l i s i o n involving n atoms, a reduced density operator p( n) i s used to describe the event; p( n) i s obtained by taking a p a r t i a l trace of the N-atora density operator p ( N ) . Boltzmann s t a t i s t i c s i s assumed throughout. As the gas under consideration i s a d i l u t e and i d e a l one, p(N) c a n be thought of as being f a c t o r i s e d into a product of density operators for i n d i v i d u a l molecules. Taking into account the p o s s i b i l i t y of d i f f e r e n t associations of atoms, t h i s becomes the sum of products p ( N ) = g(N, M, D)" 1 M~MD~D E n p ( i ) n p ( j k ) . (2 .1.1) ( a , 3 ) i * a 1 (jk)t(3 - 10 -Here M and D are numbers of monomers and dimers r e s p e c t i v e l y , with N - 2D + M. (2.1.2) g i s the number of arrangements for M monomers and D dimers from N atoms, g(N, M, D) = ^ — = (*) ^ j - (2.1.3) M! D! 2 D! 2 and P f ( i ) and P D ( j k ) are the reduced density operators for the monomer i and dimer (jk) r e s p e c t i v e l y . Normalisations are such that the trace over a l l atoms of D(N) equals unity and the traces over the reduced density operators for the molecules give the number of molecules (N) Tr p w = 1 (2.1.4) 1...N Tr p ( i ) = M (2.1.5) i r Tr P,(jk) = D. (2.1.6) • i D The reduced density operator p( n) for atoms 1 to n i s defined by p ( n ) (l...n) B (N) T r p(N) ( 2 > 1 . 7 ) n+l...N so - 11 -Tr p ( n ) (l...n) - (*). (2.1.8) 1.. .n Any given atom can be either a monomer or part of a dimer, so a s i n g l e t density operator p (D w i l l describe the combination of a monomer and part of a dimer molecule whose partner i s a traced-over atom (ghost), namely P ( 1 ) ( D = P f ( D + p£° (1). (2.1.9) The quantity p j ^ ( l ) i s the 1-atom bound-state density operator, defined by p £ 1 }(l) - 2 Tr P b(12) (2.1.10) with norm Tr P ^ U ) = 2D. (2.1.11) For the pair density operator p ( 2 )Q2), on using the assumption that a small integer i s n e g l i g i b l y small when compared with N, M and D, one obtains p ( 2 ) ( 1 2 ) = { p ( 1 ) ( l ) P ( 1 ) ( 2 ) + P b(12). (2.1.12) Ap p l i c a t i o n of the large N, M, D approximation also leads to the re s u l t s - 12 -P ( 3 )(123) - I P ( 1 ) ( l ) p ( 1 ) ( 2 ) P ( 1 ) ( 3 ) + I [ P ( 1 ) ( 1 ) P w ( 2 3 ) + + P U ) ( 2 ) P b(13) + p ( 1 ) ( 3 ) P f e(12)] (2.1.13) P ( 4 )(1234) - ^ P ( 1 > ( 0 P ( , ) « > P(1><3) p(1><4> 12 p( 1 ) ( l ) p ( 1 ) ( 2 ) p b(34) + p ( 1 ) ( l ) p ( 1 ) ( 3 ) p b(24) + P ( 1 ) ( D P ( 1 ) ( 4 ) p b(23) + p ( 1 ) ( 2 ) p ( 1 ) ( 3 ) p b(14) + P ( 1 ) ( 2 ) p ( 1 ) ( 4 ) p b ( 1 3 ) + p ( 1 ) ( 3 ) p ( 1 ) ( 4 ) p b ( 1 2 ) + j [P b(12) p b(34) + p b(13) P b(24) + p b(14) p f e(23)]. (2.1.14) The general form for p( n) i s a sum over a l l possible f a c t o r i s a t i o n s of pb and p ( D . The numerical c o e f f i c i e n t s for each term depends on the number of fragments f : ( N) g(N-n, M+n-2f, D+f-n) w 2 n-f n' g(N, M, D) M M D° n! (2.1.15) fo r large M, D and small n. So the general form for p( n) i s > ( n ) = z 2 ^ f n P ( 1 ) ( i ) n P ( i k ) (a,B) n! i*>a ( j k ^ B D (2.1.16) for small n. - 13 -B . Projectors and Channel States This section i s devoted to the c l a s s i f i c a t i o n of free and bound st a t e s . Consider 2 atoms 1 and 2 i s o l a t e d from a l l other atoms, the Hamiltonian assumes the form (2) tf. "Kj + V 1 2 (2.2.1) w h e r e i s the 1-atom Hamiltonian for atom i , and Vj2 i s the pot e n t i a l energy operator between 1 and 2. The pair Hamiltonian 2 ) can be s p l i t into a centre-of-mass part 3*CM and a r e l a t i v e motion part 1<rel ^ ( 2 ) =*CM + K r e l ( 2- 2- 2> and i f 1 and 2 are bound t o g e t h e r , ^ r e l w i l l be the i n t e r n a l energy operator with eigenkets Jb-^  > belonging to negative eigenvalues e D£ K r e l | b ± > = E b i | M >* ( 2 * 2 * 3 ) The projection operator onto the i t h bound state i s defined by P . . = b i > <bi| (2.2.4) DI J and the projection operator onto the f u l l space of the bound states i s - 14 -the sum over a l l Pbi's P b = J P M . (2.2.5) With the d e f i n i t i o n of P^, one may s p l i t up the pair Hamiltonian into a bound part K < 2 ) " • P b i C ( 2 ) - X . < 2 ) P b ( 2 . 2 . 6 ) and a free part * < 2 > f , X < 2 > ( i - P ) D SO 5<1 +J< 2 + V j 2 (2.2.7) X ( 2 ) = K ( 2 ) f + R ( 2 ) b > ( 2 > 2 > g ) i - s t h e modified po t e n t i a l between free molecules. V12 = V12 ~^CM P b ~ I £ b i P b i ( 2 ' 2 ' 9 ) For the N-atom system, the Hamiltonian i s assumed to be a sum of 1-p a r t i c l e Hamiltonians and pair potentials (K^ N N 1fca' = E K + Z V .; (2.2.10) i K j 3 - 15 -n-body p o t e n t i a l s with n ^ 3 are being ignored. One would l i k e to p a r t i t i o n into Harailtonians for i n d i v i d u a l molecules and intermolecular potential terms. However, such a s p l i t t i n g i s channel-dependent while the d e f i n i t i o n of independent channel states i s not unique. The boundedness of any pair of atoms ( i j ) i s c l e a r l y defined by the pro j e c t i o n P b i j • B u t there i s d i f f i c u l t y i n defining when an atom i s fr e e . The obvious choice i s the operator (N) N P f l E 1 - ^ P b l i t 2 ' 2 ' 1 1 ) but t h i s i s not idempotent since the P b l i ' 8 a r e n o t a H orthogonal. A chemically i n t u i t i v e solution to the problem i s to assume that no atom i s simultaneously bound to two other atoms, namely P b i j P b i k = 6 j k P b i j ' < 2' 2- 1 2> t h i s assumption being consistent with the notion that no trimers e x i s t . From the approximation (2.2.12), idempotency immediately follows. It i s (N) also assumed that P ^ and commute: P f i } * i = K i Fff- (2.2.13) This assumption i s chemically reasonable since -Kj i s the Hamiltonian for monomer i and P^^ Is the projection into the space for t h i s (free) - 16 -molecule. The assumption as formulated by equation (2.2.12) t a c i t l y requires that the molecules under consideration are not undergoing (N) c o l l i s i o n s . In other words, defines a molecule free from i n t e r a c t i o n s . The foregoing discussion i s on projection operators acting on kets which only represent pure states. But the non-equilibrium gas i s represented by the density operator p(N). xo i d e n t i f y the p r o b a b i l i t y f o r the gas to be i n any p a r t i c u l a r channel at any instant t requires projection superoperators. The projection superoperator onto the bound i j state i s defined by ^ b i j A -= P b i j A P b i j ( 2 ' 2 ' where A i s an a r b i t r a r y operator. The projection superoperator for the f r e e - i state i n the N-atom operator space i s by analogy f | f A > P < » A P < f . ( 2 . 2 . H ) Atom i i s assumed to be either free or bound: + l F = 5 ( N ) (2.2.16) f i j ; t i b i j - 17 -where i s the i d e n t i t y superoperator i n the N-atom space. The assumptions P f i AP b i. = 0 (2.2.17) and " b i j ^ b i k " 0 ' ^ ( 2 ' 2 - 1 8 ) are imperative for consistency i n presentation. (N) These projectors <P b j j a n ^ ^ f± i d e n t i f y the dimer ( i j ) and monomer i re s p e c t i v e l y , i n p a r t i c u l a r the corresponding density operators are given by P b(12) = ( 2) Tr < f M 2 p ( N ) (2.2.19) 3.. .N p f ( l ) = N Tr f < N ) p ( N ) . (2.2.20) 2...N C. Description of C o l l i s i o n s The system i s considered to be an id e a l gas i n t e r a c t i n g only through c o l l i s i o n s . The i n i t i a l and f i n a l states are assumed to be products of i n d i v i d u a l monomer and dimer density operators. The physical r e a l i s a t i o n - 18 -of such a system i s a d i l u t e gas of molecules i n t e r a c t i n g only through a short-range p o t e n t i a l . The c o l l i s i o n a l types one envisage; of monomers and dimers are non-reactiv c o l l i s i o n s , as well as decomposition a which ei t h e r a monomer or a dimer acts encompassed by (2.3.1): M + M + M + M M + D -*•. M + D D + D + D + D M + M + M + M+ D M + M+ D+ D + D. Here M and D denote monomer and for a reactive system consisting c o l l i s i o n s , rearrangement d recombination c o l l i s i o n s i n as the th i r d body. These are a l l (2.3.1a) (2.3.1b) (2.3.1c) (2.3.Id) (2.3.Ie) C o l l i s i o n s involving more than three molecules are rare occurrences and are thus ignored. 3-fragraent to 3-fragment c o l l i s i o n s are also considered rare events, however as Lowry and Snider point out, part of the 3-fragraent-to-3-fragment non-reactive t r a n s i t i o n operator consists of 3-fragment-to-2-fragment loss terms. It i s thus proposed that the c o l l i s i o n types M + M + M + M + M + M (2.3.If) M + M + D + M + M + D (2.3.lg) - 19 -should be included. The c o l l i s i o n types (2.3.If) and (2.3.lg) are equivalent to the loss t r a n s i t i o n s i n the paper by Lowry and Snider. The termolecular to termolecular t r a n s i t i o n s are rare, nonetheless they are not closed and therefore should be retained. The allowance for (2.3.If) and (2.3.lg) also has the advantage that a termolecular i n i t i a l state does not preclude a termolecular f i n a l state nor vice versa and so should be treated i n the same manner as 3-fragment to or from 2-fragment c o l l i s i o n s . The c o l l i s i o n 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 by the t r i p l e (n, f * , f ) , where n i s the number of atoms involved i n the c o l l i s i o n , f the number of fragments before the c o l l i s i o n , and f the number of fragments In the f i n a l state. The i n i t i a l channel given by ( n , f ) consists of d = n - f* dimers and m = 2f 1 - n monomers, and the f i n a l channel (n,f) has n - f dimers and 2f - n monomers. Including 3-fragment to 3-fragment processes, then the c o l l i s i o n types we study are the ones s a t i s f y i n g 2 ^ n _< 4, n <_ f + f» < 6, and f _< 3, f' _< 3. For each fragmentation channel ( n , f ) , there may be more than one way of p a r t i t i o n i n g the i n d i v i d u a l atoms over the set of molecules so l a b e l l e d channels (a,3 ) are required to catalogue the class i n the same format as i n (2.1.1). For the c o l l i s i o n type (n, f 1 , f) = (3, 2, 2) involving atoms 1, 2 and 3, one p o s s i b i l i t y of the l a b e l l e d channel i s - 20 -(a,3)=l(23) where atom 1 i s free and atoms 2 and 3 are bound. There are 2 possible c o l l i s i o n types 1(23) + 1(23), a non-reactive c o l l i s i o n ; and 1(23) -»• 2(13), an exchange c o l l i s i o n . There are g(n, 2f-n, n - f ) , g as defined by (2.1.3), l a b e l l e d channels for a given fragmentation channel ( n , f ) . The formulation of PCQ-Q» the density operator f o r n c o l l i d i n g atoms i s achieved by using the assumption that the c o l l i s i o n dynamics can be traced back i n time to a time before the c o l l i s i o n began and so to a time when the n-atom density operator could be f a c t o r i s e d into a product of Pf's and p D's. With the assumption that the n-atom c o l l i s i o n i s an i s o l a t e d event, the density operator i s embedded into the ent i r e N-atom space. The c o l l i s i o n types given by (2.3.1a-g) rule out the existence of "ghosts", that i s , where there are atoms helping to determine the outcome of a c o l l i s i o n event, but t h e i r dynamics i s not being e x p l i c i t l y accounted f o r , and "spectators", where one or more atoms are included i n the d e s c r i p t i o n p ( n) but where these atoms play no role i n determining the c o l l i s i o n outcome. Consider an i s o l a t e d n-atom c o l l i s i o n , the time dependence of p^ n,, c o l l representing the c o l l i d i n g atoms i s described by the n-atom von Neumann equation - 21 -,K_9 rt(n) p(n) (n) l f i 3 F P c o l l = I p c o l l v ( n ) (n) (n) (n) . . K P c o l l " P c o l l * ( 2 , 3 * 2 ) where i s the Lio u v i l l e superoperator j.(n) T l n I i=l 1 i<j n n = E I + E i=l 1 K j (2.3.3) Acting on an a r b i t r a r y operator A, i. ± i s the atomic Hamiltonian commutator, that i s , £ ± A = J^A - AK (2.3.4) while i s the interatomic potential commutator, acting to give *±iA = V±.A - A V±y (2.3.5) Equation (2.3.2) i s v a l i d only when the n-atom system i s i s o l a t e d and does not i n t e r a c t with other atoms i n the enti r e N-atom gas. The formal sol u t i o n of (2.3.2) i s P ^ t ) = exp[- ± £ ( n ) ( t - t o ) ] p ^ ( t Q ) . (2.3.6) It i s assumed that long before the c o l l i s i o n ( t •+• -<*»), p^°?, ( t ) o ' c o l l o assumes the form of a product of monomer and dimer density operators, - 22 -There are d i f f e r e n t possible f a c t o r i s a t i o n s and a s t a t i s t i c a l average of a l l the p o s s i b i l i t i e s i s taken. In analogy with equation (2.1.5), t h i s l i m i t i n g value of tp^",,] (t ) i s given by c o l l o lim [p;"},] (t ) = I - 2 — — l i m n P f ( i , t ) H P,(jk,t ). t + — c o 1 1 0 («. B.) n ! f — 16a' f ° (jk)eg' b o o J (2.3.7) The time evolution of the dimer and monomer density operators i s ascribed to that of free monomers and dimers: p f ( i , t Q ) = exp[- • | c f 1 ( t o - t ) ] p f ( i , t ) (2.3.8) P b ( j k , tQ) = exp[-±£(ik) ^ V 0 1 P b ( J k » t ) - (2.3.9) The density operator P ^ ^ l ^ c a n t n u s b e expressed i n terms of p f and P D at time t by taking the mathematical l i m i t t Q •»• - 0 0. This involves the channel M0ller superoperator for the i n i t i a l channel ( a 1 , 8 ' ) , which i s defined by ^ ( a ' , 3 ' ) = l i r a «Plir* ( n ) tQ] e x P [ - ^ ( a ' e - ) S 1 ' (2.3.10) t — °° Here £ ( a ' , 3 ' ) i s the L i o u v i l l e operator for the channel (a',3 ' ) '(a' 3 ' ) 5 E 1 i + 1 * (1k)' (2.3.11) This development p a r a l l e l s that of the channel M i l l e r operator - 23 -V . 0 ' ) " l l m e x P l ^ ( n ) * 0 ] e x p [ - ^ ( a ' $ ' ) t o ] (2.3.12) t •»• -o having channel Hamiltonian * V ^)~= 1 i t ± + 1 * * M k V ( 2 ' 3 - 1 3 ) On the assumption that n ( a ' , 6 ' ) e x i s t s as a strong l i m i t for any a r b i t r a r y i n i t i a l channel (a*3') involving n atoms, i t follows t h a t 1 3 ^cx'S') A " V , 0 ' ) ^ ( a ' . B - / - ( 2 * 3 * 1 4 ) In t h i s way, the Mriller superoperator for n-atom c o l l i s i o n s i s given by (n) s r 2 n - f ( n , ) - ! ^ n e (n) n f ( 2. 3.15) ( a ' , 3 « ) ( a « 3 } i * a ' f i (jk ) € 3 « b j k and the density operator for n c o l l i d i n g atoms i s to be i d e n t i f i e d as P<;01 . <£•> P c > I 2 n" f' ( n ! ) - \ p j 8 M H p ( i ) n P,(jk). (a- ,3*) ^ » p ; i€a» r (jk)eg' 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 (a' , 3 ' ) to the f i n a l channel ( a , ^ ) , the t r a n s i t i o n superoperator 3"( a >6)j(a',3' ) i s defined, - 24 -v i z : *(..»,(.•.»•) 5 »(«.»»(...»•) ^ / f i ( j l £ 6 , " V < 2 - 3 - 1 7 ) where v~( a >g) i s the superoperator representing the intermolecular pot e n t i a l for the channel (a,3): ^ « 5 ^ ( n ) - £ , <n- (2.3.18) (a,3) (a,3) Hence the t o t a l t r a n s i t i o n superoperator into the channel (a,3) i s It has been assumed that c o l l i s i o n s are i s o l a t e d events involving a d e f i n i t e number of atoms, so the N-atom density operator describing a c o l l i s i o n involving n atoms with l a b e l s i , j , k . . . l i s given by r „ W i - / ^ " l r (N) i (N-n) ,„ „ _ x [ p i j k . . . l ] c o l l = (n> [ p i j k . . . l 1 c o l l p • (2.3.20) that i s , the density for the ent i r e gas i s factored into a product of the n-atom density operator for the c o l l i s i o n [p^?? ,] ,., and i t s i j k . . . l c o l l complementary part. C o l l i s i o n s involving d i f f e r e n t sets of atoms are considered mutually exclusive. Therefore an n-atom c o l l i s i o n i s represented by - 25 -[ P ( N ) ] «. n i = S [pi?£ J c o l l . (2.3.21) 1 n-atom c o l l . .. ., n i j k . ,.1 J i<j<k...l J The density operator for c o l l i s i o n i s a sum of density operators in v o l v i n g various numbers of c o l l i d i n g atoms ' " " ' ' . l i • I, " ' " V . W . coll. < 2^"> n>2 Lowry and Snider have studied the structure of the non-reactive t r a n s i t i o n superoperator J ( a, 3) ; ( a, 3) : 3 r ( a , 3 ) ; ( a , 3 ) A = ^ ( a , 3 ) ^ ( a , 3 ) A = t ( a , 3 ) ; ( a , 3 ) A " A [ t ( a , 3); (a,3) 1 ^ + t ( c t , 3 ) ; ( a , 3 ) A [ ( ^ t ) ( a , 3 ) ; ( a , 3 ) l t ~ ^ t ) ( a , 3 ) ; ( a , 3 ) A [ '< a,3) ; (a,3) ] ' ' ( 2 ' 3 ' 2 3 ) Here t ( a ? g ) . ( a^g) ±s the t r a n s i t i o n operator o\ / O N = V, o x fi, O N (2.3.24) (<*,3);(a,3) (a,3) (a,3) and t j ( a ) g ) i s the channel Green's superoperator The f i r s t terms on the r i g h t side of (2.3.23), those which are l i n e a r i n t , are considered to be l o s s terms. This association i s formally v e r i f i e d i n Chapter 4 when a generalised o p t i c a l theorem i s derived. D. Time Evolution of the System The system evolves with respect to time according to the von Neumann equation <Tt = * (2.4.1) Operating on equation (2.4.1) by or we get the time evolution of the dimer density operator P D(12) and the monomer density operator P f ( l ) r e s p e c t i v e l y . In the present work the formulation of Lowry and Snider i s rewritten to emphasize the fact that the molecules are bound according to the channel instead of the opposite l o g i c that has been given i n the aforementioned. These two schemes are mathematically equivalent but the treatment i n which the arrangement channels are regarded as fundamentals i s more convenient to use because i t Is easier to decompose the t o t a l Hamiltonian into the channel Hamiltonian and the corresponding channel p o t e n t i a l than to deal with a l l the one- and two-particle Hamiltonians. We have the commutation r e l a t i o n s ^ ( i j ^ b i j = 6 > b i j £ ( i j ) - 27 -and V f i = f f i < i <2-4-3> which are consistent with (2.2.12) and (2.2.13). Moreover, we also require the approximation l i % " P f j £ i < 2'^> and accordingly the r e l a t i o n * i P f j = P f j ^ i * (2.4.5) For a l a b e l l e d channel (a,3), a projection operator i s defined according to P, Q N = P,. n p ( a ' 6 ) f l (jk) 3 b ^ k = P c (2.4.6) with channel Hamiltonian ^ ( a ^ = t r = 1 (2.4.7) ( a ' 3 ) C i * a 1 (jk)^3 ( j k ) Henceforth the l e t t e r c i s used i n place of (a,3) as the l a b e l for t h i s channel. Within the approximations (2.2.13) and (2.4.5), the commutation r e l a t i o n H P = P c c c c (2.4.8) - 28 -i s v a l i d , and correspondingly, the projection superoperator <P = n c n tp... i«-a (jk)*3 commutes with the channel L l o u v i l l e superoperator defined by C i e a 1 (jk)fr3 U ; where the commutation being written as L 9 " f l -ee c c and which holds within the assumptions (2.4.2) - (2.4.4). The r e l a t i o n (2.4.12) i s then used to obtain k i n e t i c equations for the monomer density operator P f ( i ) and the dimer density operator P D ( j k ) . The use of the la b e l c to replace (a,B) i s i n accordance with the emphasis on l a b e l l e d channels. The import of such a change i n symbolism i s revealed i n Chapter 4 where a sum rule i s derived on the assumption that the projection operator/superoperator commutes with the Hamiltonian/Liouville operator for t h i s channel regardless of how many species are involved. So with the assumptions that have been contemplated so f a r , the Lowry-Snider formulation can be extended to a more general case i n which other molecular species may be present. Then c i s more convenient n o t a t i o n a l l y than (a, 3, ...). (2.4.9) (2.4.10) (2.4.11) (2.4.12) - 29 -Kinetic Equation for the Monomer (N) Acting on (2.A.l) on the l e f t by the projector tf3^ and taking traces over atoms 2 to N gives i n g t " P f ^ 1 ^ N f l P * (2.4.13) 2.. .N The d e f i n i t i o n of the projector into a given channel, (2.4.6) and (2.4.9) t a c i t l y assumes the mutual exclusion of a l l other l a b e l l e d channels for the set of atoms involved. From the approximations l a i d out i n Section B, and the d e f i n i t i o n (2.4.9), i t follows that for a system of n atoms ^ (2.4.14) c l where c l i s the l a b e l for a channel i n which atom 1 i s free. Instead of rigorous formulations, i t i s assumed that the channels and t h e i r corresponding projections can be defined i n the same manner i n the ent i r e N-atom space, f ( N ) _ £ g> (N) _ £ f i ( n ) j ( N - n ) f l c l C l c l C l where J(N-n) ±s the projection into the complementary (N-n) atom space. Corresponding to the channel c l i s the channel L i o u v i l l e and the po t e n t i a l superoperators and t ^ e former given by (2.4.10) and the l a t t e r given by y.(n) = < t ( n ) (n) c l c l - 30 -or i n the en t i r e N-atom space Substituting (2.4.15) into (2.4.13) gives i f i l r P < ° - N Tr E (P P ( N > + N Tr E <P < » V < » > P ( N ) 9 t f 2...NC1 C l C l 2...NC1 C l C l - N Tr I t <?> <P<?> p ( N ) + N Tr E * ( N ) ^ ( N ) P ( N ) 2...NC1 C l C l 2...NC1 C l C l - N Tr £ E p ( N ) + N Tr E <P ( N V W P ( N ) 2...N 1 c l C l 2...N c l C l C l = N Tr L p j « P ( N ) + N Tr E «P ( N V < N > P ( N ) . o M O XT 1 Cl C l 2...N 2...N c l (2.4.18) Since i t has been assumed that an atom cannot be free and bound simultaneously, then < ™ P ( N ) - P f ( D P 0 ^ (2...N), (2.4.19) therefore the equation ± 4 t " p f ( 1 ) = < t l p f ( 1 ) + N T r E eel^S p ( N > (2.4.20) 2...N c l i s obtained. " C^PfO) describes the free evolution of p ^ ( l ) , and so the second term i s attributed to be a c o l l i s i o n a l term. Accordingly p(N) (N) i s replaced by PQQ^ an<* the k i n e t i c equation for P f ( l ) ( f o r small n and large N, M and D) becomes - 31 -i f ^ P ^ l ) = i l P f ( l ) + T r ( f 1 2 T 1 2 P 1 2 + ^ 1 2 3 J123 + ^1(23) J l ( 2 3 ) ) ( p l ( 2 3 ) + 2 p ( 1 3 ) 2 ) + 7 25 ( < f123 J123 + ^1(23) 3 1(23) ) P123 + 2 • ^12(34) ^ 12(34) ( p(12)(34) + 2 p ( 1 3 ) ( 2 4 ) ) + 2 3 4 ^ 1 2 ( 3 4 ) : f l 2 ( 3 4 ) ( P l 2 ( 3 4 ) + p(12)34 + 2 p13(24) + 2 p ( 1 3 ) 2 4 ) + . . . + Tr 2 n _ f ' [(n-1)!] - 1 I <P ( n ) J (?> I p , + . . . o . c l c l . c 2 . . . T 1 c l c' (2.4.21) where c' denotes the i n i t i a l channel. Just l i k e the projector P c/ c c i s defined from P f ^ / ^ . i 6 a, P ^ j ^ / ^ j ^ * ( J k ) «=" &> t l l e arrangement channel density operator i s defined as p = n p ( i ) n P K ( j k ) . (2.4.22) i e a (jk)*B Thus the molecular and arrangement channel descriptions can be translated into one another i n a straightforward manner. K i n e t i c Equation f o r the Dimer The treatment i s the same as that for the monomer. Let c(12) be the l a b e l for a channel i n which atoms 1 and 2 are bound together, then ^ = j f <n> = y f W C2 4 231 1 1 2 _ c ( 1 2 ) ' < » « e ( 12 ) « 1 2 > " ( " - 32 -With d 1 2 acting on the l e f t on (2.4.1) and tracing over atoms 3 to N, we obtain « S t % < 1 2 > " <2> * < M2 * < W > ™ *(12> "b(12) + (2> 3 * H c ( 1 V ° (2.4.24) J . . .N cC12; l ^ P b ( 1 2 ) = X ( 1 2 ) p b(12) + { T r < P ( 1 2 ) 3 T ( 1 2 ) 3 p 1 2 3  + ^ ^(12)3 3 ( 1 2 ) 3 ( p ( 1 2 ) 3 + 2 p ( 1 3 ) 2 ) + Tr 34 (^(12)34 J(12)34 + ( ?(12)(34) 1 (12)(34) ) ( p(12)(34) + 2 p ( 1 3 ) ( 2 4 ) ) + 2 T r 34 ( < ?(12)34 T(12)34 + < P(12)(34) J ( 1 2 ) ( 3 4 ) )  ( P12(34) + p(12)(34) + 2 p13(24) + 2 p(13)24 ) + . + Tr 34..,n 2 n - f ' - 1 [ ( n - 2 ) ! ] - 1 I c(12) c ( 1 2 ) c ( 1 2 ) 1 P c ' c' c + . . . (2.4.25) where n i s small compared with N, M and D. - 33 -The k i n e t i c theory described above does not allow for the existence of intermediates (species i n metastable states) which may undergo subsequent c o l l i s i o n s or decay spontaneously emitting a quantum of energy. I n t u i t i v e l y , one expects that the intermediates can play a s i g n i f i c a n t r o l e in the f l u i d dynamics of a r e a c t i v e system, so the equations of change as obtained in the following chapter may not give a good d e s c r i p t i o n of the behaviour of the system. It i s hoped that r e s u l t s obtained i n the present research w i l l be generalised to a wider scope of s i t u a t i o n s . - 34 -CHAPTER 3 THE FORM OF THE EQUATIONS OF CHANGE The equations of change for monomer and dimer observables are obtained by using the k i n e t i c equations for the respective species as derived by Lowry and Snider. A. Description of Molecular Observables The hydrodynamic d e n s i t i e s of mass, l i n e a r momentum, angular momentum and energy are studied i n t h i s paper. Their corresponding molecular observables can be ascribed to be at t r i b u t e s of a single molecule except for the intermolecular p o t e n t i a l . This chapter thus focuses on the treatment of the single-molecule a t t r i b u t e s , namely mass, l i n e a r and angular momenta and k i n e t i c energy. Let be an observable associated with the monomer i , and <l>(jk) an observable for the dimer ( j k ) . The corresponding observables for the whole N-atom system are given by z b i j ) 4>. (3 .1 .1) for the monomer,and (3 .1 .2) - 35 -for the dimer. The subscripts M and D are used to indicate monomeric and dimeric properties r e s p e c t i v e l y . The expectation values of the physical observables for the species are therefore equal to « 0 = Tr Z ( 5 - Z(P ) * p ( N )  M 1 . . . N 1 j b l j 1 = Tr <j> p (1) (3 .1 .3 ) 1 f o r the monomer, and <<i>n> = Tr E f P ( N )  D 1...N j<k b i j " £ *(12) P b ° 2 ) ( 3 ' 1 - 4 ) f o r the dimer. To describe a reactive system i n which a dimer molecule may be broken up into or formed from two monomer molecules at c o l l i s i o n s , i t i s necessary to be able to treat dimeric and monomeric a t t r i b u t e s on a uni f i e d basis. The physical observables (whether for a monomer or a dimer) can always be treated as atomic a t t r i b u t e s since the basic unit of matter i s the atom for the purpose of t h i s t h e s i s . For the monomer, these atomic observables are unequivocally l o c a l i s e d at the molecular, i . e . , atomic centre of mass. When i t comes to the dimer molecule, one - 36 -may l o c a l i s e these a t t r i b u t e s of the constituent atoms at the centre of mass of the molecule, or at the centres of mass of the respective atoms. The r e l a t i o n s h i p between these two schemes was discussed by Olmsted and S n i d e r 1 1 . In the present work, the l o c a l i s a t i o n Is always to be at the centre of mass of the respective molecule. This choice may be considered a r b i t r a r y , but i t emphasizes the molecular aspect of the system and i s therefore the natural d e s c r i p t i o n used i n chemistry. The l o c a l mass de n s i t i e s for the monomer M^(r_,t) and for the dimer Mrj(r.,t) at macroscopic p o s i t i o n _r and time t are defined by M ^ r . t ) = Tr Z (<f - Z <?,.,) m6 p ( N ) (t) 1 1 " 1...N i j b l J 1 = Tr m 6. p f ( l , t ) , (3.1.5) 1 6± = &(r_ - _£i), _r_i being the p o s i t i o n operator for atom i , and V I ' " s , ^ / b j k < 2 »> SW " W M - Tr 2m 6 ( 1 2 ) p b (12,t), (3.1.6) - i ^k i i -ik x = <5(r ), ^ 2 = L ( b e i n g t h e p o s i t i o n operator f o r — + -  — i + — J ( j k ) = ° ^ " 2 >» 2 = i ( j k ) the centre of mass of the pair ( j k ) . The dependence of macroscopic d e n s i t i e s on _r and t w i l l be understood and w i l l not be e x p l i c i t l y indicated i n the formulae from here onwards. - 37 -For an a r b i t r a r y molecular observable <$>± for the monomer i and <j>(jk) for the dimer ( j k ) , the l o c a l densities per unit mass are res p e c t i v e l y defined by < v ~=\l, T r „ j - . z < s V ( * ± 6 i > s p ( N ) 1...N i j M^1 Tr ( • j f i ^ g P f ( D (3.1.7) and <$n> = M ^ Tr I C... 6 , . . . ) p ( N ) 1...N j<k b ^ k ( ^ k ) ( J k ) 8 = V £ <*<12) 6 ( 1 2 ) ) s Pb ( 1 2 ) ' ( 3 ' 1 - 8 ) When <J> and 6 do not commute, one has to pick the appropriate symmetrisation to ensure h e r m i t i c i t y . For the hydrodynamic densities of mass, l i n e a r momentum and angular momentum, the symmetrised operator i s given by (Ws = i ( V i + W (3a-9> < Wak)^ • r ( W ( j k ) + 6 ( j k ) * ( j k ) > - ( 3- 1- 1°) For mass (<{>£ = m), l i n e a r momentum ( ^ - P±) and angular momentum (<(>£= _r_£ x + where s± i s the atomic angular momentum), the connection between dimer and atomic attributes i s - 3 8 -• < j k ) = * J + V ( 3 - 1 - 1 1 > For k i n e t i c energy, the atomic observable i s the one-particle Hamitonian <fri = ^ i while the dimeric observable i s the sum of the Hamiltonians fo r the two constitutent atoms plus the intramolecular po t e n t i a l V ) =JS +Jfk + V ( 3 - K 1 2 ) The symmetrised k i n e t i c energy operator ( ^ i 6 i ) s for the monomer i i s <*i V s = t IPi 6i + 2£i • 6i-Ei + ( 3 - 1 - 1 3 ) therefore for the dimer (jk), the corresponding symmetrised kinetic energy operator i s ( Kuk) W s = ^  W s + ( ^ W s + V 6 a * ) 6 ( j k ) + 2-P-j ' 6(jk) £j + 6(jk) P j l + - k [ p k 6 ( j k ) + 2 i \ • 6 ( jk)£k + 6 ( j k ) p 2k ] + V 6 ( j k ) <3-1-14> since the pot e n t i a l depends only on positions and so V and 6 commute, V6 = 6V. ( 3 . 1 . 1 5 ) - 39 -B. Formal Equations of Change Consistent with the k i n e t i c equations of Chapter 2 for the states of a t y p i c a l monomer and a t y p i c a l dimer i s the Schrodinger picture i n which the physical observables are e x p l i c i t l y time-independent. For the monomer, the change with respect to time of the density of each of the physical properties i s thus ! t - ( M M < * M » = T ' ( W s i n r p f ( 1 > (3.2.1) The f i r s t term involving the superoperator <£ ± gives r i s e to the change during free f l i g h t (between one c o l l i s i o n and the next), while the l a s t term i s associated with the c o l l i s i o n a l change of <*M>. By analogy, the dimer rate of change i s I F ( M D < V > < W < 1 2 ) > s l t pb< 1 2> " - ^ ( * ( 1 2 ) 6 ( 1 2 ) 8 t ( 1 2 ) P b ( 1 2 > + ^ ( * ( 1 2 ) 6 ( 1 2 ) ) s [4F p f ( 1 2 ) W * ( 3 ' 2 ' 2 ) The free motion terms for dimer molecules have been obtained by Olmsted and Snider and have the same structure as t h e i r counterparts for - 40 -monomers. In t h i s chapter we write down the e x p l i c i t forms of the free motion terms while the c o l l i s l o n a l terms w i l l be discussed i n subsequent chapters. The equations of change for the monomer are to be described f i r s t . Individual properties are discussed separately. The monomer mass density changes because of the mass flux associated with the stream v e l o c i t y while c o l l i s i o n s can lead to a production or loss of monomers and of monomer mass, e x p l i c i t l y Here u i s the macroscopic (stream) v e l o c i t y of the monomer defined by (3.2.3) (3.2.4) The rate of change of monomer momentum density i s given by (3.2.5) where (3.2.6) (_p. <5 J 2 b e i n g the tensor transpose of p 6 p It i s usually convenient to break the k i n e t i c flux of l i n e a r momentum - 4 1 -into convective and conductive contributions to obtain the more f a m i l i a r equation of motion l ^ u + u - V „ - - l £ V i ^ u ] c o n . (3.2.7) (M) Here P i s the monomer k i n e t i c pressure tensor defined by p£ M ) = m"1 1v[(Rl ~ mu) (£j - mu) 6 ^ p f(l). (3.2.8) The angular momentum of any molecule consists of the in t e r n a l angular momentum and the t r a n s l a t i o n a l angular momentum. For the monomer i , the angular momentum observable Jj_ i s accordingly made up of a t r a n s l a t i o n a l part l± = rj_ x and an i n t e r n a l part s , i i = ±i + Ij. = ± ± + 1± x £±- (3.2.9) By d e f i n i t i o n , the monomer angular momentum density per unit mass i s <±H> ^ T r ( i l V s P f ( 1 ) » (3.2.10) whose rate of change i s hence given by Since [ J ^ , ^ ] . . = 0, the free motion term can be rewritten so that - 42 -|F < V 4M» = - V • Tr U 1 6 1 p 1 ) s p f ( l ) + ( V l M > ) ] c o l l (3.2.12) where (ViVs = { [ V A + ^ i 6 i + + W i 1 - (3-2-13) Just as the angular momentum density consists of t r a n s l a t i o n a l and i n t e r n a l parts, that i s , where the k i n e t i c part of the angular momentum flux i s the sum of t r a n s l a t i o n a l and i n t e r n a l angular contributions m"1 T r ( J 1 6 1 p 1 ) s p f ( l ) = r x(g£ M )+ J ^ u u ) * m"1 TrCSj 6j p ^ P f U ) . (3.2.16) On combining the equation of motion (3.2.7) and the equation of change of angular momentum, the equations for the changes of i n t e r n a l angular momentum density and the t r a n s l a t i o n a l angular momentum density are obtained, namely - 43 -T F < V i M f m t » " - 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 ^ c o l l * (3.2.18) The l a t t e r can be rewritten as | ^ ( r x u) + (u • V) r x u = M^1 V • (p£ M )x r) + | ^ (r x u ) l c o l l (3.2.19) which has a form analogous to the equation of motion. Equations (3.2.17) and (3.2.18) are indeed coupled equations as i n t e r n a l and t r a n s l a t i o n a l angular momenta evolve inter-dependently. Likewise, k i n e t i c energy and pote n t i a l energy of a system are not independent q u a n t i t i e s . Since k i n e t i c energy i s a molecular observable, we may write down the equation of change for i t . We define the k i n e t i c energy per unit mass of the monomer as TM = Y ( J V l } s p f ( 1 ) (3.2.20) - 44 -whose rate of change i s given by It ( M M V " - m _ 1 V * ^ i V s P f ( 1 ) + lT* ( W ] c o l l (3.2.21) where m"1 Tr( ^ j f i ^ g P f ( D = m 1 Tr 8 ( W l + K l V l + 6 l ^ l +-P-l V P + 4 (^1 55 + 55 *1> P f ( D (3.2.22) i s the energy f l u x due to free motion of the monomer molecules which can be broken into a convective term, a conductive heat flux and a term due to the energy ca r r i e d by the k i n e t i c momentum fl u x m"1 T r ( K l £ l 6 l ) s p f(l) = ^ » T M + cjf > + £<*>.„ (3.2.23) Here CM) m £1 -P-l -P-l ^ = H T r [ (_L_ u ) . r _ l _ u ) ( - i - u) 6 ] p (1) -TS. z ^ m — m — m — l s f (3.2.24) i s the k i n e t i c contribution to the heat flux from the monomers. The equations of change for the dimer are exactly of the same form as for the monomer. 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 for further reference. - 45 -The dimer mass density changes according to (3.2.25) where _v i s macroscopic (stream) v e l o c i t y of the dimers -1 ^ E M D H ' W c V s p b ( 1 2 ) (3.2.26) The equation of motion for the dimers i s (3.2.27) where p £ D ) = (2m)" 1 T r [ ( £ ( 1 2 ) - 2mv) ( P ( 1 2 ) - 2mv) « ( 1 2 ) ] 8 P b(12) (3.2.28) i s the dimer k i n e t i c pressure tensor, with the symmetrised operator (p,.,v P/.,\ defined by ~ ( j k ) -Hjk) ( j k ) ' s ^jk^jkAjk)^ = 4 • • E C j k A j l O ^ j k ) + £ ( j k ) 6 ( j k ) - E ( j k ) + (^(jk) 6(jk)£(jk) ) , : + 6 ( j k ) £(jk) -E(jk) (3.2.29) As with the monomer, the angular momentum operator for the dimer (jk) •^ Kjk) -3 -k (3.2.30) can be expressed as the sum of a t r a n s l a t i o n a l angular momentum operator for the dimer molecule as a whole - 46 -^ j k ) - ^ j k ) X ^ j k ) ( 3' 2' 3 1> and the i n t e r n a l momentum operator associated with the sum of the i n t e r n a l momenta of 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 constituent atoms j and k, jr-ji^ x _pjk where r . k = r. - r k (3.2.32) i s the po s i t i o n operator of k r e l a t i v e to j , whereas P j k " j ( P j " Pk> (3.2.33) i s the r e l a t i v e momentum operator of k from j . The angular momentum density per unit mass < V = Tr ( J ^ , « ( 1 2 ) ) s P b(12) (3.2.34) can thus be broken into a t r a n s l a t i o n a l part r x v and an int e r n a l part < ^,int> E "D 1 ll + ^2 + Ll2 X *12> 6 ( 1 2 ) ] s P b ( 1 2 ) « ( 3 ' 2 ' 3 5 ) By d i r e c t analogy to the equations of change for the monomer d e n s i t i e s , the rates of change of the t r a n s l a t i o n a l and i n t e r n a l angular momentum de n s i t i e s are res p e c t i v e l y j r i v 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 * % < - i D , i n t > ) - - ( 2 * ) - 1 V • T t [ ( . 1 + s 2 + r 1 2 x p 1 2 ) p ( 1 2 ) 6 ( 1 2 ) ] s % { l 2 ) + It ( MD ^ D . i n t ^ c o l l where t ( £ j + \ + r . k x p. k) ^  j k ) 6 ( j k ) ] s (3.2.37) 4 + <£, + J l k + £ j k x P j k > 5 ( j k ) P ( j k ) + ^ j k ) 6 ( jk ) % + J L k + £ j k x P j k > + 6 ( jk) £ ( j k ) % + - i k + ^ k x ^ k > (3.2.38) As i n the case of the monomer, while the free f l i g h t contribution to t r a n s l a t i o n a l and i n t e r n a l angular momenta are separately conserved, the c o l l i s l o n a l parts are not. F i n a l l y , analogous with the monomer, the dimer k i n e t i c energy density per unit mass i s h=~\l H (*(12) W s P b ( 1 2 ) (3.2.39) whose change i s given by - 48 -^ (M DT D) = - (2m)"1 V • Tr ( K ( 1 2 ) £ ( 1 2 ) « ( 1 2 ) ) 8 ^(12) 12 + l i t ( M D T D ) ] c o l l " V ' [M, v T D + + P<D> • v] + ( M D T D ) ] c o l l . (3.2.40) Here 2^P^ is the kinetic contribution to the heat flux vector from the dimers (3.2.41) and the symmetrised operator (^j^) £(jk) ^(jk)^s * S ^ e t i n e c ^ by ( H ( j k ) ^ j k ) ^ j k ) ^ = ? [ ( J V) 6ak)>s -p-(jk) + £<jk) ^ak) ^ j k ) ^ 1 (3.2.42) The changes in these densities due to free motion a l l have the same structures as in the classic fluid dynamic description. However, since reactions are necessarily quantal in nature, the fluid dynamics of a reactive system, here the dimer/monomer mixture, requires a quantum kinetic theory. - 49 -CHAPTER 4 A SUM RULE The k i n e t i c equations (2.4.21) for the monomer density operator p f ( l ) and (2.4.25) for the dimer density operator 0^(12) are very formal and contain no e x p l i c i t a s s o c i a t i o n with the gain and loss aspects of the c o l l i s i o n processes. For example, for c o l l i s i o n s involving a monomer and a dimer, the loss of a dimer molecule due to decomposition i s not e x p l i c i t l y shown i n the k i n e t i c equation for the dimer density operator: i f J [ l t pb ( 1 2 ) ]M+D = T ^ ( 1 2 ) 3 3 ( 1 2 ) 3 ( P1(23) + 2 p ( 1 2 ) 3 ) * This i s because the channels are c o l l i s i o n a l l y coupled. The time evolution of the system of c o l l i d i n g molecules i s generated by the Hamiltonian of the system, and the f i n a l channel i s represented by the projections onto t h i s channel, therefore i t i s imperative to study the re l a t i o n s h i p between the Hamiltonian of the c o l l i d i n g system and the projections onto the various possible f i n a l channels. A. Channel Coupling and the Generalised O p t i c a l Theorem Consider an i s o l a t e d system of n p a r t i c l e s . Let be the t o t a l Hamiltonian for th i s i s o l a t e d system, and P c be a projection operator onto a given channel c with c and V c being the channel Hamiltonian - 50 -and potential energy for this channel r e s p e c t i v e l y . The sum of projectors over a l l channels i s the i d e n t i t y so that Can be parti t i o n e d according to R ( n ) = E P c H ( n ) = ! P c <K + V ). (4.1.1) c c This i s also equal to & ( n ) ( E P ) = l(i< + V ) P . (4.1.2) c c c c c c For a t y p i c a l channel c, i t s projector P c and Hamiltonian Ji\. commute ** P = P H • (4.1.3) c c c c As a consequence, a sum rule r e l a t i n g the channel projectors and the i r respective potentials i s obtained, namely E P V = E V P . (4.1.4) c c c c c c With the approximations outlined i n Chapter 2, the analogous sum rule f o r superoperators follows: E e KT = E V e . (4.1.5) c c c c c c This sum rule can be expressed i n another form. The i n t e r n a l energy Vint,c °f t n e molecules associated with the channel c i s given by - 51 -n V, . (4.1.6) i n t . c c i ' 7 so the t o t a l Hamiltonian can be broken i n t o E P C ( z V + V C + V ) - E( E K ± + V C + V l n ) P C . (4.1.7) c 1=1 ' c i=l Since E P =1, one has the r e l a t i o n c c E H ± E P C = E P c E}* (4.1.8) i c c i which immediately leads to E P V. . = E V . . P (4.1.9) c i n t . c i n t . c c 7 c c * whose corresponding form for superoperators i s E<P vT = E \ r . . ^ (4.1.10) c i n t . c Int.c c v ' c c ' with V r i n t , c A = t V i n t , c ' A ] « ( 4 ' i a i ) Although whether reaction takes place or not depends on the energies of the c o l l i d i n g molecules, chemists are used to thinking i n terms of the molecular energies of the species involved and the l a t t e r form of the sum r u l e , (4.1.9) and (4.1.10), i s i n accordance with t h i s view. - 52 -T o s e e t h e p h y s i c a l s i g n i f i c a n c e o f t h e sum r u l e , t h e i d e n t i t y ( 4 . 1 . 5 ) i s a l l o w e d t o a c t o n < f V c i , t h e M a i l e r s u p e r 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 w h e n t h e i n i t i a l l a b e l l e d c h a n n e l i s c ' : E P V <W = T, V~ <P ^ , , e c c ' e c c ' c c ( 4 . 1 . 1 2 ) o r e q u i v a l e n t l y f ,2 , , = EIT f , c ' e ' e ' e c c ' c E <P JT , c c c ' c c * c ' ( 4 . 1 . 1 3 ) a n d t h i s m a y b e s e e n a s t h e s u p e r o p e r a t o r f o r m o f t h e o p t i c a l t h e o r e m . A c t i n g o n p c i i t g i v e s < c ' ( t c ' c ' P c« - p c ' = E {v- f p - f [ t t p ,t , ) r - (A , t ,) P , t / ] } . 1 c c c ' c ' c c c ' c ' dec' c c ' tfee' c c ' c ' c c ' 1 c ( 4 . 1 . 1 4 ) T o o b t a i n ( 4 . 1 . 1 4 ) , t h e s t r u c t u r e o f t h e M i l l e r o p e r a t o r h a s b e e n c o n s i d e r e d : c ' c ' cc c ' c ' c ' = P , + ( - / , + i E ) - 1 V , SI , c ' c * c ' c * c ' ( _ < f c ' + : K c ' ~ H c + i e ) _ 1 ( " ^ c ' + ^ c ' " J ^ c + i E ) (~/c, + iO" 1 V c , f t c , - 53 -c' V c' 0 c c ' c' c' vcc' c' c ^ c c ' ( + e l , t , + ft , (tf - * ,) p , <)cc' cc' |jcc' c c' c' (4.1.15a) or U = Z P « , = P , + Z P (A ,t , (4.1.15b) c' c c c c l cc cc c c where P Tl , = 6 ,P , + P (. ,t ,. (4.1.16) c c' cc' c' c ^ c c ' cc' v ' Taking the hermitian adjoint, we have • P C + E ( l c ' t c c ' ) t P c ( 4- 1- 1 7> c and fic' P c = 6 c c ' P C + ^ c c ^ c c ^ V ( 4 - 1 ' 1 8 > If the Hamiltonians for a l l the channels are equal and are denoted by tfo, and i f p c« and equation (4.1.14) are r e s t r i c t e d to be diagonal in energy, then t h i s equation i s reduced to f , ( t , , p , - p , t , , t ) = Z { v - < 0 ^ , p , - 2 T r i ( P t , P , [ 6 U ) t , f] } c' c'c' c' c' c'c' 1 c c c' c' c cc' c' o cc' ' c = Z{VCP e x p [ i j t t ] p c , e x p [ - | i t t ] c - 2irif t p [6(/ ) t / ] } (4.1.19) c cc c o cc - 54 -since ^fo'cc^ pc^cc^ - 'cc'Pc'^fo'cc'^ - ~ 2 i r i t c c ' P c ' [ 6 ( / ' o ) 'cc'* 1 (4.1.20) for the restricted case when CK0,Pc,]_ = 0 and [ f t . t ^ ^ ^ t = 0 \ For pure states, p c i s replaced by P c and (4.1.14) becomes Wc' " 'c'c^  Pc' = 1 c c c' c' c This is to be compared with the optical theorem i s familiar 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 ,t , ) * P V ,P , + P ,V ,P ,} L Occ cc' c c c' c' c' r ' J (4.1.21) expressed in a form which Z c (P c,V cP c + P c C K c - * c , ) P c f ) ( J c c.t c c f) " ^ c c - ' c c ' ^ ( P c V c P c ' + P c ( ^ c -«-c'> P C > - 55 -Z cc" jc c' c c' c c c jcc1 cc' ^cc'cc'* V c W c - V c ^ * [ t t cc' P c ^cc-'cc'* - ^ c c - ' c c ' ^ V c c ' l Equations (4.1.21) and (4.1.22) are equivalent since (4.1.22) Z c t t ,P (U ,t ,) - p (k ,t ,) t t ,P cc' c Jcc' cc c «cc' cc' cc' c tc ,t p t , - p t ,(6 ,t , ) r p ] ' cc' c cc' c cc' 7cc' cc' c z c t* ,p a , - p Q , t* ,p cc' c c c c' cc' c t , - p t ,n\-p ) c' c cc' c cc' c c ( t c ' c ' P 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' z c J ^ . V p n , - p n . ^ . v p c' c c c' c c' c c c - n^.p v n , + p v . S ^ . P c' c c c c c c' c' c = z<P ir-fi .n 1 - , = z ir (P n c c c c c c c c c c (4.1.23) - 56 -Equation (4.1.22) i s concerned with the t r a n s i t i o n of a state i n the i n i t i a l channel c' to a l l accessible f i n a l channels which on comparison with equation (4.1.23) i s found to be effected by the potential superoperator l / " c which couples a l l the channels together. Under the s p e c i f i e d condition to obtain (4.1.19), the on-the-energy-shell o p t i c a l theorem, equations (4.1.21) and (4.1.22) are reduced to the operator form of t h i s theorem. B. Gain and Loss of Monomer Observables It i s found that the a p p l i c a t i o n of (4.1.5) regardless of the i n i t i a l l a b e l l e d 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 ) ] } c o n and {9/<*t[pb(12)] } c o n which immediately leads to the conservation of mass and by ca r e f u l examination, conservation of l i n e a r and angular momenta. However, to see how the species are created or destroyed by the i n d i v i d u a l reactions, i t i s more appropriate to use the o p t i c a l theorem (4.1.13). For the sake of c l a r i t y , the implementation of these two equations involving d i f f e r e n t i n i t i a l fragmentation channels are considered i n separate sections. ( i ) 2-Monomer C o l l i s i o n s The r e s u l t 2M (4.2.1) - 57 -i s immediately obtained from the r e l a t i o n f 1 2 ^ 2 " V12 t f12- ( 4 * 2 ' 2 ) The subscript 2M indicates that two monomers are involved i n the c o l l i s i o n . Since V\2 and 6j commute, the mass density f o r th i s type of c o l l i s i o n i s conserved [kVm = T\ m s i h [ p f ( 1 ) ] 2 M - £ m 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 density of a monomer observable 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 p a i r , [ l F *M <V ]2M = " 2rT H^Ws + (*2 V s ] V n ^ W ( 4 ' 2 ' 4 ) This p o t e n t i a l r e s u l t s i n an intermolecular transfer of l i n e a r and angular momenta when c o l l i s i o n s are non-local. ( i i ) 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 of 1(23) 1(23) 123 123 c c v (12)3 (12)3 (13)2 (13)2 ; (4.2.5) - 58 -int o the appropriate term i n the k i n e t i c equation (2.4.21) for the monomer density operator gives % P f ( 1 ) ] M + D = - sr Tr * 23 ^ cP c - ( ^ ( 1 2 ) 3 T ( 1 2 ) 3 + ( P(13)2 J(13)2 )^ ( A L ( 2 3 ) P 1 ( 2 3 ) + > A'(12)3 P(12)3 + v f V(13)2 p(13)2 ) Tr 23 <Z i r c * c - 2 f ( 1 2 ) 3 7 ( 1 2 ) 3 } c ( JH(23) P1(23) + J V /(13)2 P(13)2 + ^(12)3 P ( 1 2 ) 3 ) 2 Tr P b d 2 ) ] M + D £ Tr IV €c ( ^ ( 2 3 ) P 1 ( 2 3 ) + 2 ^ 1 2 ) 3 p ( 1 2 ) 3 ) li c (4.2.6) where the dimer-monomer term i n the k i n e t i c equation (2.4.25) for the dimers has been used to i d e n t i f y the (12)3 terms as equal to the dimer c o l l i s l o n a l rate of change. So for monomer-dimer c o l l i s i o n s , the mass density changes are related by - 59 -9 9 ["9T VM+D = - [ 9 r VM+D 2 i m Tr 12 2n 1 £ i 2 ±\2 " -1 £ 1 9 2n 2n ^ Z n=2 Z 6 ( 1 2 ) f ( 1 2 ) 3 J ( 1 2 ) 3 [ p l ( 2 3 ) + P(13)2 + P ( 1 2 ) 3 ] 2 i M+D m VV : Tr 123 1 -12 -12 fig, T 2 2 2 ( 1 2 f (12)3 J(12)3 [ P1(23) + P(12)3 + P ( 1 3 ) 2 ] (4.2.7) An expression i d e n t i f y i n g the change associated with i n d i v i d u a l reaction types may be derived by recognising that the non-reactive t r a n s i t i o n superoperator contains the reactive l o s s terms ^ >l(23) : ri(23),l(23) ^ 0 ^ 1 ( 2 3 ) l wU2)3 J(12)3,l(23) (13)2°(13)2,1(23) 123 123,1(23) J (4.2.8) which gives [U Pf ( 1 )WD - 60 -* 23 1 ^ ^ ( 2 3 ) c c ^(12)3^(12)3 + e{ 13)2\ 13)2] Pl(23) + ^ m 1 ^ + t f 5l(23) Tl(23) ) ( P(12)3 + P(13)2 } Tr 23 ( EcV r c < ? c ^ l ( 2 3 ) " 2 < P ( 1 2 ) 3 7 ( 1 2 ) 3 ) p l ( 2 3 ) + 2 ( < ?123 7123 + f > l ( 2 3 ) T l ( 2 3 ) ) P(12)3 (A.2.9) Thus, e x p l i c i t l y monomer 1 i s l o s t due to the reaction 1(23) (12)3 while monomer 1 i s gained i n (12)3 •»• 123 and (12)3 •»• 1(23) c o l l i s i o n s . The rate of change of the density of an a r b i t r a r y monomer observable i s obtained by multiplying t h i s equation by (<Pi ^ 1)s a n c* taking the trace over 1. [ Z V r - ^ ( 2 3 ) " i e< c c (12)3 3"(12)3 ] pl(23) + 2 ( t f > 1 2 3 3 1 2 3 + f > 1 ( 2 3 ) : r i ( 2 3 ) ) P(12)3 Tr 123 (Ws 1 W l ( 2 3 ) c + 2 [ ( * 3 6 3 ) s - ( W s ) ^ ( 1 2 ) 3 2 ( 1 2 ) 3  + « W s + ^ V s ^ m ^ 1(23) (4.2.10) - 61 -The f i r s t term i n (4.2.10) i s analogous to the contribution to the equation of change for 2-monomer c o l l i s i o n s , and a term of t h i s form appears i n each c o l l i s i o n process therefore the transfer of the physical observables other than the transfer from the reactants to the products can be pinpointed. The second term gives the gain and loss of monomer observables for exchange reactions: the loss of a monomer molecule as the reactant monomer (atom 1) becomes part of the dimer (12) and the gain of a product monomer molecule as atom 3 emerges as a monomer. The net gain of two monomers on decomposition i s given i n the l a s t term. Moreover, when atom 1 i s part of the dimer (12) ( f i n a l channel (12)3), any physical a t t r i b u t e associated with i t i s l o c a l i s e d at _ r j ; t h i s i s consistent with the picture that a reactant molecule i s annihilated at the p o s i t i o n of the reactant i t s e l f , not at the product's, so that + k lT13 ( < * ' l 6 l ) s ( f ( 1 2 ) 3 : r ( 1 2 ) 3 P l ( 2 3 ) describes the loss of the reactant monomer. Correspondingly, the product monomer should emerge at the very s i t e i t i s produced; t h i s i s also formally established i n equation (4.2.10). With the loss or production of the monomer and the physical a t t r i b u t e s associated with i t i d e n t i f i e d for each c o l l i s i o n type, the second way of applying the sum rule ( i . e . , acting on the non-reactive c o l l i s i o n superoperator) provides a more complete d e s c r i p t i o n of the c o l l i s i o n s . However, the equations of change for M^ obtained from equation (4.2.6) i s more convenient to use for studying the o v e r a l l e f f e c t of reactions on the f l u i d dynamics of the system. - 62 -( i i i ) 3-Monomer C o l l i s i o n s The same kind of manipulations as in the previous section leads to the r e s u l t s 2 T r [ | - p, ( 1 2 ) ] , M - ^ — Tr I V f «^ o 1 9 t b 3M 2Ti 1 2 3 c I 2 3 P 1 2 3 , (4.2.11) [7t V 3M 2i "[at MD ]3M + ^ t m V V : 123 v2! 2 2 T • • • -J (12)3 < f >(12)3 T(12)3 P123 (4.2.12) and the equation of change [7t M M< V l 3 M - - - K T r 123 -!<•,«,). - (•JV.I F ( , 2 > 3 ^ 1 2 ) 3 123 IK Tr 123 j [ ( * 1 6 1 ) s + ( * 2 6 2 ) s + ( * 3 « 3 ) 8 ] i : u - c P c ^ c 123 (4.2.13) - 63 -Therefore, the a p p l i c a t i o n of formal k i n e t i c theory again s u c c e s s f u l l y shows that the recombination reaction, the monomer a t t r i b u t e associated with the reactant monomers, here, the atoms with l a b e l s 1 and 2, would be destroyed as a r e s u l t . ( i v ) Dimer-Dimer C o l l i s i o n s There cannot be any loss of monomers when two dimer molecules p a r t i c i p a t e i n a c o l l i s i o n : [lt P f ( 1 ) ] 2 D 21 ^ 23 1 2(34)' 712(34) ( p(12)(34) + 2 p ( 1 3 ) ( 2 4 ) ) (4.2.14) giving the corresponding equation of change hTF *M < V J 2 D = " If " Ws + (*2 W ^12(34) ^12(34) ( P(12)(34) + 2 P ( 1 3 ) ( 2 4 ) ) or to express i n a form consistent with the equations i n the previous sections, [ 4 t ^<VJ2D = - * T ' 1234 [ ( W s + ( < t >2 62 ) s ] f12(34) : ri2(34) + 2t(<P 1<5 1) s + < W s ^ l 4 ( 2 3 ) 7 1 4 ( 2 3 ) P(12)(34). (4.2.15) It i s also possible to express the time evolution of P f ( l ) due to 2-dimer c o l l i s i o n s i n terms of that of Pb(12) - 64 -[lt P f ( 1 ) ] 2 D 21 3h. T!, t < P12(34) ^12(34) + ^13(24)^13(24) + ^14(23) J14(23) 1 1234 (P (12X34) + 2 P ( 1 3 ) ( 2 4 ) ) 2 i 3* ^ 4 [ ^ / c " 3<f(12)34<>(12)34 + f ( 12)(34) ^ 12)(34) > J ( J \ l 2 ) ( 3 4 ) P(12)(34) + 2 > A(13)(24) P ( 1 3 ) ( 2 4 ) ) -2 T r [ ^ P b d 2 ) ] 2 D " ^ ^ 3 4 r ^ 1 2 ^ 3 ^ P ( 1 2 ) ( 3 4 ) + 2^13)(24) p(13)(24) )' (4.2.16) (v) Monomer-Monomer-Dimer C o l l i s i o n s The contribution from c o l l i s i o n s involving a dimer and 2 monomers to the k i n e t i c equation for the monomers can be written i n the al t e r n a t i v e forms [7F Pf ( 1 ) ]2M+D ^ Tr Z \T <P 3n 2 3 4 c c c SV 12(34) P12(34) + ^(12)34 p(12)34 + 2 , n(13)24 P(13)24 + 2 v /13(24) P13(24) " 2 2 r [ l t " Pb ( 1 2 ) 12M+D - 65 -= * Tr 234 c c 12(34) c " [ P ( 1 2 ) 3 4 3 ( 1 2)34 + ^(12)(34) 7 ( 1 2 ) ( 3 4 ) P12(34) + 2 ( J13(24) J13(24) + 2 < P(13)24 7(13)24 + 2 I P(13)(24) J ( 1 3 ) ( 2 4 ) ft 234 ^ 1 2 ( 3 4 ) 3 12(34) [ P (12)34 + 2 p13(24) + 2 p ( 1 3 ) 2 4 ] * (4.2.17) From the second form, the gain and loss of a monomer observable are e x p l i c i t l y given by [lt " M <*M >1 2M +D i Tr W 1234 ^ • i V . ^ W . ' S V W * 2 < 3 4 ) + 2 [ (* 4VB " ( W s ] ^(13)24^(13)24 [ ( W s + < * 2 W > ^(12)34^(12)34 i ^ V s + ( * 2 6 2 ) S ] ^(12X34) 3(12)(34) + 2^(13)(24) : r(13)(24) '12(34)' (4.2.18) - 66 -The exchange reaction 12(34) + (13)24 brings about the gain of the monomer 4 which i s compensated by the loss of monomer 1 so there Is no net change i n the number of monomers. Likewise, the simultaneous break-up and recombination 12(34) + (12)34 re s u l t s i n no change i n the number of monomers. The l a s t block of terms r e f l e c t s the loss of the monomer molecules when recombination (with or without exchange) takes place. C. Gain and Loss of Dimer Observables After 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 gain and loss terms for d i f f e r e n t kinds of c o l l i s i o n s are obtained. Since the manipulations and physical meanings are the same as for the monomers, the equations of change are l i s t e d with no der i v a t i o n shown, and except for section ( i ) , no account on the meanings of the terms derived w i l l be given. ( i ) Monomer-Dimer C o l l i s i o n s ( 1 2 ) 6 ( 1 2 ) ) s Z U c c (12)3 c ( 1 2 ) 6 ( 1 2 ) ) s 1(23) Jl(23) p(12)3 " ( * ( 1 2 ) 6 ( 1 2 ) ) s 123^123 (4.3.1) - 67 -A f t e r a chemical reaction has taken place, the i n i t i a l dimer (12) no longer e x i s t s , thus there i s the destruction of the reactant dimer (12) + h T r 3 ( ^ 1 2 ) 5 ( 1 2 ) ) s ( P c V ( 1 2 ) 3 ' C * ( 1 2 ) 3 * For the exchange reaction (12)3 + 1(23), the gain of the dimer (23) and the molecular a t t r i b u t e associated with t h i s dimer i s expected and t h i s i s observed i n (4.3.1). ( i i ) 3-Monomer C o l l i s i o n s [h W]3M = ~ 2S T 5 3 ( * ( 1 2 ) 6 ( 1 2 ) ) s ^(12)3^(12)3^23 ( 4 ' 3 ' 2 ) ( i i i ) 2-Dimer C o l l i s i o n s [ 3 t V V ] 2 D r Tr *1234 TI ( W < 1 2 > > s + ( * ( 3 4 ) W s ] 1 *c*A (12)(34) + ( [ ( * ( 1 3 ) 6 ( 1 3 ) ) s + ( * ( 2 4 ) 6 ( 2 4 ) ) s ] " [ ( , f ) ( 1 2 ) 6 ( 1 2 ) ) s + ( < t ' ( 3 4 ) 6 ( 3 4 ) ) s ] ) f >(13)(24) ! 1(13)(24) 2 ( ( * ( 1 3 ) 6 ( 1 3 ) ) s ' [ ( <' >(12) 6(12) )s + ( * ( 3 4 ) 6 ( 3 4 ) ) s 1 ) P 7 (13)24 J(13)24 " ( * ( 1 2 ) 6 ( 1 2 ) ) s 6 ?12(34) 712(34) P(12)(34) (4.3.3) - 68 -( i v ) Monomer-Monomer-Dimer C o l l i s i o n s [9F WW TfT Tr 1234 ( t ( 1 2 ) 6 ( 1 2 ) ) 8 ^ 0 ^ ( 1 2 ) 3 4 c + ( < | ) ( 3 4 ) 6 ( 3 4 ) ) s f ( 1 2 ) ( 3 4 ) J ( 1 2 ) ( 3 4 ) 2 ( l ( * ( 1 3 ) 6 ( 1 3 ) ) s + ( * ( 2 4 ) 5 ( 2 4 ) ) s ] " ( < f > ( 1 2 ) 5 ( 1 2 ) ) s ) *(13)(24) J(13)(24) + [ ( <f )(34) 6(34) )s ~ ( * ( 1 2 ) 6 ( 1 2 ) ) s ] P12(34) 712(34) + 4 [ ( * ( 1 3 ) 6 ( 1 3 ) ) s _ (' t >(12) 6(12) )s 3 P(13)24J(13)24 p( 12)34 (4.3.4) On comparing the c o l l i s l o n a l contributions to the equations of change for the physical a t t r i b u t e s associated with the monomers and dimers, i t i s found that for a gain of a dimer molecule, there i s the corresponding loss term for the monomers, and a loss for the dimers i s coupled with a gain for the monomers. Thus mass i s conserved for i n d i v i d u a l c o l l i s i o n processes. - 69 -CHAPTER 5 HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES I: NON-REACTIVE COLLISIONS In the preceding chapter, explicit expressions for the coll 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 particular attributes of mass, linear momentum, and angular momentum, the colli s i o n a l contributions to the appropriate hydrodynamic equations are obtained. The energy balance equation w i l l be discussed in Chapter 7 since i t requires a more elaborate formulation. This is 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, linear and angular momentum densities, the treatments of non-reactive collisions is 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 collisions, the mass densities of the dimer and the monomer do not change. As in the previous chapter, different c o l l i s i o n types are treated in separate sections. A. Collisions Involving 2 Monomers The rate of change of the density of a single-molecule property for this c o l l i s i o n type is given by - 70 -(5.1.1) Since d e l t a functions commute with the pot e n t i a l energy operator, the mass density change i s zero. For the dens i t i e s of 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 transfer of the respective a t t r i b u t e from one monomer to the other as (<J>i + <j>2) i n (5.1.1) commutes with the potential f ° r $i = _Pi a n ^ $1 = Jj.• For momentum density change, t h i s i s given i n terms of the momentum f l u x £ 2 M 6 - 6 r i L M. ul = - Tr — — - ( V V ) <f vA/ p  l9t ^1- J2M 2 1 2 1 2 1 2 1 2 1 2 • " v * £ 2 M ( 5 * 1 * 2 ) where P_„, the pressure tensor due to 2-monomer c o l l i s i o n s , i s defined by =2M 2n+l S 2 M = - T r < 6 ( 1 2 ) ¥ + V ( M ) ! r l T 2 ' v 2 n 6 ( 1 2 ) ) 12 n=l ( V V ) tf P . v 12 12' 12 12^12 (5.1.3) - 71 -This pressure tensor gives r i s e to an angular momentum density f l u x , and when the pote n t i a l has spherical symmetry or depends only on po s i t i o n s , i t i s the sole factor f o r the change i n angular momentum density. A s p h e r i c a l l y symmetric potential commutes with the atomic angular momentum [ - i ' V ( r i j ) ] - = °» likewise for a pote n t i a l V i j that depends only on the r e l a t i v e position between I and j , so that = ? • ( P 2 M x r ) . (5.1.4) In general the potential i s not s p h e r i c a l l y symmetric and depends on fac t o r s such as spin or the coupling between js^'s, so i n t e r n a l and t r a n s l a t i o n a l angular momenta are not independently conserved. The rate of change of t r a n s l a t i o n a l angular momentum density consists of a term which corresponds to a net 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 pair [|t (£ x \ i i ) ] 2 M = V * <£ 2M X i> - H ^ T ^ 1 2 X <V12V12 ) f12^12 P12 (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 angular momentum = 2S V ' H ^  \l2) + "'> ( ^ r l 2 ) ^12 P12^12 p12 i 6 l + 6 2 - ± T r ( — - — - — — - ) IT f Jl/ p * 12 2 2 1 2 1 2 1 2 1 2 (5.1.6) to give a t o t a l angular momentum conservation law = V • (P=2M X + Ifi V * T 2 r ( ¥ 6(12) + - > i 2 ) ^12 ^12^12^2-(5.1.7) Equations (5.1.5-7) show that i t i s the s p a t i a l inhomogeneity of the c o l l i d i n g molecules that causes the interchange of in t e r n a l angular momentum and t r a n s l a t i o n a l angular momentum. When the molecules are i n th e i r free f l i g h t , however, the changes of i n t e r n a l and t r a n s l a t i o n a l angular momentum dens i t i e s with time are independent of each other (see, for example, Chapter 3 of 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 that the angular momenta transform into one another. Another point to note i s that the i n t e r n a l angular momentum flux i n (5.1.6) has the form akin to the d i s t r i b u t i o n of the atomic angular momentum over the two c o l l i d i n g molecules. - 73 -For non-reactive c o l l i s i o n s involving a higher number of atoms, the contribution to the equation of change for l i n e a r or angular momentum, l i k e the monomer-monomer c o l l i s i o n s , i s due to the transfer of the at t r i b u t e from one partner to the others. Consider an n-atom c o l l i s i o n , the i n i t i a l channel given by the l a b e l c', the change i n the monomer hydrodynamic density of l i n e a r momentum (<pj = jy±) or angular momentum (<Pi = Jj.) i s , on suppressing the numerical c o e f f i c i e n t due to counting, as follows: n-atom,c',no rxt •h Tr 1.. .n +...+<Pn) 6j] g - [(<p2 +...+<pn) fij] (5.1.8) (here c' i s a channel i n which 1 i s free) and for the dimer, the contri b u t i o n to the hydrodynamic equation for l i n e a r momentum (<|>, > = p, + p.) or angular momentum = J_i + J_.) i s - 74 -l 3 t D F J n atom, c", no rxt = - :=• Tr [(<().+(()-) 6 , 1 0 J V- ..P..«^..p . n . 1 2 (12) s c c c c i • • • n Tr [(*3 + ... + * n) « ( 1 2 )] svr c..P c..J . . .P c „ (5.1.9) i.. .n (here c" contains the dimer (12)). The commutativity of t o t a l l i n e a r or angular momentum of the system with the potential has been used to obtain (5.1.8) and (5.1.9). Let V t o t a i be the sum of a l l inter-atomic p o t e n t i a l s , and l e t c' be a channel i n which ( i j ) , ( i ' j ' ) . . . are bound as dimers, then I*! + + V V C J _ = [<fr + ... + 4, v ] 1 n t o t a l — U j + . . . + 4>N, v ]_ + [<j>1 + . . . + <|>n> v l t , ]_ + = 0 (5.1.10) where = j ) ^ , or = J_^. From equations (5.1.8-9), we see that the l o c a l i s a t i o n scheme that has been adopted here corroborates with the picture that c o l l i s l o n a l transfer i s the transfer of the observable one i s concerned with from the centre of mass of one molecule to the centres of mass of i t s c o l l i s i o n partners. - 75 -B. Collisions Involving a Monomer and a Dimer The rate of change i n the density of a physical observable of the molecules from a non-reactive monomer-dimer c o l l i s i o n i s + V V ^ M + D . n o rxt 123 [ [ 3 (*1 + *(23))]s + [ ( 6 1 ~ 6 ( 2 3 ) ) 3 ] s ' L K1(23) < P1( 23)^(23) P l ( 2 3 ) (5.2.1) There are d i f f e r e n t ways of d i v i d i n g the terms up, but the one used above has the advantage that the term on the l e f t may be interpreted as the contribution to the equation due to the centre of mass motion, and as the 6-functions on the ri g h t gives the r e l a t i v e p o s i t i o n between the molecules, t h i s i s the change due to the r e l a t i v e movement between the c o l l i d i n g molecules. For the monomer, the change of hTt **M — JM+D,no rxt ~ 123 6 l ( V l < 2 3 ) V l ( 2 3 ) ) < P l ( 2 3 ) t / l ( 2 3 ) P l ( 2 3 ) (5.2.2) (with J ^ j k ) = - ^ i ~ "2 ^ — j + —k^ being the pos i t i o n operator for the pair - 76 -(jk) r e l a t i v e to i ) i s compensated by the change i n the dimer momentum density a [lt **D - ]M+D,no rxt = ^3 6 (23) ( V1(23) V1(23) ) f 1(23)^1(23) Pl(23) * (5.2.3) The rate of change of the density of t o t a l momentum for this type of c o l l i s i o n can be expressed as the divergence of the pressure tensor £M+D no rx t ' 3 t e r m a r i s i n& from the intermolecular force between the c o l l i s i o n partners [-5F ( MM H. + MD ^ ) ]M +D,no rxt " " V * SM+D,no r x t ' ( 5 ' 2 ' 4 ) f-2)n -1 —1 f23">n n _ 1 1 1 - 1 I z j n=Z ( V 1(23) V 1(23) ) ^1(23) ^1(23) pl(23) (5.2.5) where R i s the centre of mass of the c o l l i d i n g molecules, and for c o l l i s i o n s involving atoms i , j and k, R - (r_^ + r_j + r^.)/3. From our knowledge i n i r r e v e r s i b l e thermodynamics, we expect a t r a n s l a t i o n a l angular momentum flux (-P„._. ^ x r) and this term i s a c t u a l l y =M+D,no rxt — J i d e n t i f i e d from the equation showing the conservation of t o t a l angular momentum for this c o l l i s i o n type. From the equations f o r the changes In the angular momentum dens i t i e s f o r the i n d i v i d u a l species 3 [~o~t V^I^M+D.no rxt 1 2 3 v - r i ' s u 1(23)^1(23)1(23)^1(23) TT J 2 r 3 [ V l + <£i x J > l V s ] ^l ( 2 3) Pl ( 2 3 ) U l ( 2 3) pl ( 2 3 ) < 5 ' 2 - 6 > ^W'wD.no rxt - 4 ^ 3 ( V 2 3) 5(23 ) ^ ^ l(23)n(23rt(23 ) P l(23) m { " 2 + - 3 + " 2 3 X ^ 2 3 > 6 ( 2 3 > + K £ ( 2 3 ) * W 6(23)U V r K 2 3 ) < P l ( 2 3 ) v / l ( 2 3 ) p l ( 2 3 ) (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 angular momentum densities being e x p l i c i t l y given, the conservation r e l a t i o n for t o t a l angular momentum fit % ^ + V^VD.H O rxt i 2 l i ~ ( £ 2+13) = ~*TU3L 3 ( V 6 ( 2 3 ) ^ s ^ l ( 2 3 / l ( 2 3 ) ^ ( 2 3 ) p l ( 2 3 ) - 78 -= -V (—p x r) + L. =M+D,no rxt — ^M+D,no rxt " ^ 3 (il(23)"° 6 R l f T ( ^ r 2 « 2 - 2 ^ 3 ) V1(23)^1(23)^(23)P1(23) (5.2.8) follows immediately. As (-P„ _ x r) i s the t r a n s l a t i o n a l angular =M+D,no rxt — momentum f l u x , the rest must be the i n t e r n a l angular momentum f l u x . =M+D, no rxt , the molecular angular momentum flux =M+D, no rxt 3 7 1 (£i(23) + 123 K } , } 6R-23 X ( V 2 3 V 1 ( 2 3 ) ) ^ l ( 2 3 ) v A l ( 2 3 ) P l ( 2 3 ) (5.2.9) w i l l be the only contribution to the i n t e r n a l angular momentum flux associated with monomer-dimer non-reactive c o l l i s i o n s when the pote n t i a l depends e n t i r e l y on positions. To see how t r a n s l a t i o n a l and in t e r n a l angular momenta transform into one other, the changes i n the den s i t i e s are considered separately. i T F * * ( " H » + N D ^ W H O rxt = -V • (-p w x r) =M+D,no rxt — 2 6 1 + 6(23) ~ 123 3 -K23) X ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) f l(23) v /V(23) Pl(23) (5.2.10) - 79 -[ 8 T ' V ^ i n t * + V ^ D , i n t > ) ] M f D . n o rxt * 123 (26 +6 ) t 3 ( l 1 + 2 2 + l 3 + i 2 3 X - £ 2 3 ) ] ( 6 r 6 ( 2 3 ) ) ^(23)^1(23)^1(23) Pl(23) - V • L =M+D,no rxt Tr 123 ( 2 5 1 + 6 ( 2 3 ) ) ( } -1(23) X ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) i ( 6 1 _ 6 ( 2 3 ) ) + £ t 3 ( 2 3 ) ( B r 2 s 2 - 2 s 3 ) ] U - i ( 2 3 ) - V ' =M+D, no rxt 6 l ( 2 3 ) < a i ( 2 3 ) P l ( 2 3 ) (5.2.11) Therefore the interchange of the i n t e r n a l and t r a n s l a t i o n a l angular momenta of the c o l l i d i n g pair i s effected by the torque between the molecules. C. Collisions Involving 3 Monomers The form of the equation of change for 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 + [ ( « 2 + [ ( 6 1 ^ ( 6 2 + 6 3 ) ) ( 2 * 1 - # 2 - * 3 ) / 3 ] 8 123 123 123^123 (5.3.1) Since t o t a l l i n e a r and angular momenta commute with the p o t e n t i a l of the system, conservation of these q u a n t i t i e s f o l l o w s immediately. Same as before, the l i n e a r momentum d e n s i t y change can be expressed i n terms of a pressure tensor due to the forces between the c o l l i s i o n p artners 3t —^ 3M+3M V * S3M+3M (5.3.2) where P =3M>3M i s the c o n t r i b u t i o n to the pressure tensor from t h i s type of c o l l i s i o n s E3M+3M - 81 -\- Tr 6 123 6 ( 2 3 ^ 2 3 ( V 2 3 V 1 2 3 ) + 6R^1(23) ( V1(23) V123 ) 2n-l . _£__ 2n-2 2n-2 I ( 2 n - 1 ) n - l - | 3 . v 6 ( 2 3 ) ( V 2 3 V 1 2 3 ) + 1 n=2 +(|) n (n!)"1 [I ( £ ( 1 2 ) 3 n + r ( n ) 2 n ) - (- r 1 ( 2 3 ) ) n ] n-1 n-1 V V V 1 ( 2 3 ) V 1 2 3 ) 9 v/y p 123 123M123 6 123 ( r 1 2 + ...) 6 ( 1 2 ) ( V 1 2 V 1 2 ) + ( r 1 3 + ...) 6( J 3 ) ( V ^ ) + (^23 + — > 6(23) ( V 2 3 V 2 3 ) ^>123%/H23P123* (5.3.3) The second expression for P_.. ... indicates that i f the molecules i n t e r a c t =3M->"3M only v i a pair potentials and i f there i s no reaction taking place, the system of three molecules c o l l i d i n g together behaves as three sets of doublets whereby each pair i s independent from the others. Therefore the equations of change for angular momentum dens i t i e s for t h i s c o l l i s i o n type have structures which are analogous to those for monomer-monomer c o l l i s i o n s : - 82 -lat ^ X " M - ^ S I M M - V * (23M+3M x I) 6 1 2 3 5 + 6 6 + 6 — £ 1 2 x ( V 1 2 V 1 2 ) + - T - r n x ( V 1 3 V 1 3 ) 6 2 +6 + - T - ^ 2 3 X ( V 2 3 V 2 3 ) 123 1 2 3 ^ 1 2 3 ( 5 . 3 . 4 ) C 9 t (V4l,int>:)13M*3M -1 6 - 6 2r7 T ^ 3 " V 2 , ^ " V ^12^123^123^23 6.+6, + * m ^ " £ 1 2 X ( V 1 2 V 1 2 ) e 1 2 3 ^ 2 3 P 1 2 3 ( 5 . 3 . 5 ) and the expressions can be i d e n t i f i e d i n the same manner as previously. D. Collisions Involving 2 Dimers For an a r b i t r a r y dimer physical observable <<pn> t h i s type of c o l l i s i o n s r e s u l t s i n a change i n <$n> according to l 9 t MD < $D > ]2D,no rxt - 83 -2*1234 r6 ( 1 2 ) + 6 ( 3 4 ) , (^(12)~ < ) )(34) ) 1  [ 2 ( * ( 1 2 ) + < ( , ( 3 4 ) ) ] s + [ ( 6 ( 1 2 ) - 6 ( 3 4 ) ) 2 ] s L^12)(34) f(12)(34)' /Vl2)(34)P(12)(34) (5.4.1) This c o l l i s i o n type can be likened to monomer-raonomer c o l l i s o n s : both are non-reactive processes involving two equivalent molecules. Their contributions to the equations of change f or l i n e a r and t r a n s l a t i o n a l angular momentum de n s i t i e s are analogous to each other. For the dimers, they are [ 3 t MD - ]2D,no rxt V • Tr 1234 -(12X34) ( 2 VV(12)(34)V(12)(34) ; 2n+l + Z [ ( 2 n + l ) ! ] " 1 £ < 1 2 > ( 3 4 > 2 " 6 D n-1 2 R < V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) *( 12) ( 3 4 ) ^ 12) (34) P ( 12) (34) V • P =2D,no rxt (5.4.2) (where r. = r, - r,, i s the operator for the r e l a t i v e — ( i j ) ( k l ) -nij) — ( k l ) - o p -p o s i t i o n of (kl) to ( i j ) , and P i s the contribution to the —z IJ j no rx t pressure tensor from t h i s kind of c o l l i s i o n s ) and [|t < £ x %1^2D,no rxt " V ' (P=2D,no rxt X I> _ 6 ( 1 2 ) + 6 ( 3 A ) ^ 1 2 ) ( 3 4 ) X234 2 2 X Q V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ; P. vrt> . . . P (12)(34) (12)(34) H(12)(34)* (5.4.3) The change i n the in t e r n a l angular momentum density i s si m i l a r to i t s counterpart for monomer-monomer c o l l i s i o n s although now there i s an add i t i o n a l flux term given r i s e by the molecular angular momentum of the molecules =2D,no rxt = - I ^ ( £ ( 1 2 2 ° 4 ) + - " ) 6 R ^ 1 2 X V12 V(12)(34)-^34 X V 3 4 V ( 12) (34) > ^(12)(34)^12)(34) P(12)(34)-(5.4.4) a hit MD ^ D . i n t ^ D . n o rxt _ i T r 6 ( i 2 ) - 6 ( 3 4 ) y y y - g * ^ _ v . L - " TT t T r , 2 2 V(12)(34) ^2D,no rxt 1234 < ? ( 1 2 ) ( 3 4 r t l 2 ) ( 3 4 ) P ( 1 2 ) ( 3 4 ) 6 +6 + Tr 123A (12) (34) r(12)(3A) 2 X ( V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) (12)(34) (12)(34) M(12)(34) K. Collisions Involving 2 Monomers and 1 Dimer For t h i s c o l l i s i o n process, we have (5.4.5) [ 3 T ( MM<V + V V ) ] 2 M + D , no rxt - i 2-fi Tr 1234 K6 l + 6 2 + 5 4>i+4>2 + •(34), 2 + °(34)' 2 J! + [ ( — 2 — - « ( 3 4 ) ) 2 ] s « r « 2 12(34) 12(34) 12(34) p12(34) (5.5.1) The conservation laws follow d i r e c t l y from (5.5.1) but we might wish to treat the system as three doublets as before. The equations of motion, for instance, can be expressed i n a manner which p a r a l l e l s - 86 -(5.5.1), or In forms which show how the molecules i n t e r a c t pairwise, i . e . , no rxt y Tr 1234 6 r 6 2 V « 2 [ 2 ( V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) + " V ^ ( V 1 2 V ( 1 2 ) ( 3 4 ) ) ] 12(34) 12(34)^12(34) — Tr 2 1234 61 V 1 ( 3 4 ) ( V 1 3 + V 1 4 ) + 62 V 2 ( 3 4 ) ( V 2 3 + V 2 3 ) 6 -6 1 2 + — — - (V V ) 2 v v 1 2 v 1 2 ; (P 12(34)^2(34) P12(34) and f9t MD —32M+D,no rxt (5.5.2) = T Tr 6,„.,(V 2 1 2 - 3 4 U ( 3 4 ) V V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) ^12(34)^2(34) p12(34) = 2 ^ 6 ( 3 4 ) [ V 1 ( 3 4 ) ( V 1 3 + V 1 4 ) + V 2 ( 3 4 ) ( V 2 3 + V 2 4 ) ] ^12(34) J 112(34) P12(34) (5.5.3) gi v i n g - 87 -[9tT »D2)] 2M+D,no rxt V £2M+D,no rxt* (5.5.4) Accordingly, the presure tensor P 2 M + D > n o ^ can be written i n two forms - 4 T r 1234 6(12)^12(V12V12(34)) + 6R^(12)(34) ( V(12)(34) V12(34) ) 2n+l oo ^. 2n 2n + 2 Z [(2n+l)!]~ 1(Zl 2-) . V 6, *(V V ) n - l 2 (12rV12V12(34); 2n Z [(2n-l)!] 1 = i i n=l 1 2n-l 2n-l • V . 6 2n+l (12) + Z [ ( 2 n + l ) ! ] - 1 -d 2 K34) . v fi) n=l z R ( V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) 'l2(34) V / 112(34) P12(34) 1234 '^134) 6(134) V 1 ( 3 4 ) ( V 1 3 + V 1 4 ) + 6(234)^2(34) V 2 ( 3 4 ) ( V 2 3 + V 2 4 ) + 6(12) 1^2 (V12V12) 2n+l 0 0 r 2n 2n + E [ ( 2 n + l ) ! ] - 1 (=|1) • V 5 ( x 2 ) ( V l 2 V 1 2 ) n=l 12(34) V /12(34) P12(34) - 88 -1234 n=2 l-(-2) n! n -1(34) n _ 1 n - 1 - V * V 6(134) V1(34)< V13 + V14> n r _ „ ^ . _ , v n-l n-l + " V L * V 6(234) V 2 ( 3 4 ) ( V 2 3 + V 2 4 ) 12(34)^2(34) P12(34) ( 5 . 5 . 5 ) where r . . . , . = ( r . + r, + r, )/3 i s the p o s i t i o n operator for the centre — ( i j k ) — i - j —k of mass of atoms i , j and k. To break up the c o l l i d i n g system into a set of i n d i v i d u a l bimolecular i n t e r a c t i o n terms has the advantage that as far as non-reactive c o l l i s i o n s are concerned, when the structures of the fluxes due to bimolecular c o l l i s i o n s are known, t h e i r counterparts for more complex c o l l i s i o n s can be obtained immediately because the l a t t e r can be divided into terms, each for a pair of molecules, which are comparable to the fluxes for analogous bimolecular c o l l i s i o n s . So i t follows from (5.1.5) and (5.2.10) that the t r a n s l a t i o n a l angular momentum change for t h i s c o l l i s i o n type i s [ f ^ r x (Mj, u + M,, v ) ] 2 M + D > n o t x t = V • (P x r) =2M+D,no rxt — - 89 -j Tr 1234 ^6 l + 6 ( 3 4 ) 3 ^1(34) X V 1 ( 3 4 ) ( V 1 3 + V 1 4 ) 2 6 2 + 6 ( 3 4 + — ^2 (34) X V 2 ( 3 4 ) ( V 2 3 + V 2 4 ) 6 l + 6 2 + - T - ^ 1 2 X ( V 1 2 V 1 2 ) 12(34) 12(34) P12(34) (5.5.6) whereas for i n t e r n a l angular momentum density, other than the term that negates the second term i n (5.5.6) i s - V • L„ =2M+D,no rxt + V • — Tr ^ 1234 All (12) ^-1 i 2 ' v 1 2 , £1(34) + ( - ^ . . . ) 6 ( 1 3 4 ) [ ^ - 2 ( 8 3 ^ ) ] ( r 1 3 + i r 4 ) , £2(34) + ...) 6 ( 2 3 4 ) [ s 2 - 2 ( s 3 + s 4 ) ] cr23+u24) k12(34)"l2(34) P12(34) where L , i s the molecular momentum given by =2M+D,no rxt & J - 9 0 -=2M+D,no rxt = Tr 1234 v g • • • • ) 6 (134) -34 X V34 ( V13 + V14> • • * ) <5 (234) r34 x V34 ( V 2 3 + V 2 4 ) 12(34)^2(34)^2(34) (5.5.7) Another molecular observable under study i n th i s research, the k i n e t i c energy of the molecules, unlike l i n e a r momentum and angular momentum, i s not conserved for non-reactive c o l l i s i o n s . Consider the monomer-monomer c o l l i s i o n , the change i n k i n e t i c energy due to this c o l l i s i o n type i s given by To obtain the very l a s t expression above requires the intertwining r e l a t i o n Tr 0 f 1 + ^ 2 > « > 1 2 ^ 2 P 1 2 = - 2 ^ ° V V W l 2 c' c' c' c' ^ c ' c » - 91 -If p\2 i s diagonal in *<i 2, <&12Pl2 vanishes so that k i n e t i c energy i t s e l f i s conserved. However, i f the c o l l i d i n g system i s s p a t i a l l y inhomogenous, i . e . , when P j 2 i s position-dependent (besides being moraentum-depenent) , [**i2» P12J — * 0 a n c* k i n e t i c energy does change at c o l l i s i o n s . Therefore, even for a simple case l i k e 2-monomer c o l l i s i o n s , k i n e t i c energy i s i n general not a conserved quantity owing to s p a t i a l inhomogeneities. For c o l l i s i o n s involving a higher number of atoms In which reactions are possible, k i n e t i c energy i s not conserved for non-reactive c o l l i s i o n s even i f p c» (or p c") i s s p a t i a l l y homogenous because now the projector^ c» (or & c") i s no longer the i d e n t i t y . One would anticipate an interchange of k i n e t i c and po t e n t i a l energies at c o l l i s i o n s so that t o t a l energy i s conserved. Indeed t h i s i s formally v e r i f i e d i n Chapter 7 where the energy balance r e l a t i o n i s studied. CHAPTER 6 HYDRODYNAMIC EQUATIONS FOR ONE-MOLECULE OBSERVABLES II: 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 effecting the change, therefore each of the equations can be divided into two parts. One is analogous to the collisional transfer between the molecules for non-reactive collisions, and on neglecting the numerical factor due to counting, this i s - 5 - Tr (<f>,<5.) V/ tf JV.p , (where c * c' and c' involves T l , 1 1 S C C C C , , x l...n the monomer 1) for the monomers, and ~k , T r ( * ( i 2 ) 5 ( i 2 ) ) s u ' c f c ^ c " p c " {i *?> e: i n v o l v e s l...n the dimer (12)) for the dimers. Same as with non-reactive events, for the observable mass, these terms vanish, and for linear and angular momenta, these correspond to the transfer of the physical attributes between the colliding molecules. This kind of transfer is designated as the "collisional transfer" to be distinguished from the redistribution of the physical attributes from the reactant molecules to the product molecules as the atoms regroup into a different set of molecules. The contribution to the hydrodynamic equations for the monomers from this second kind of transfer is i 1 when there i s no monomer i n the i n i t i a l channel, otherwise i Tr t V W s " < V l W c c ' p c n 1.. .n t Here i i s a product monomer, and for the dimers, t h i s i s i Tr E (<p| l . . . n (jk) (jk) ^ j k ) ^ c3 c c l P c i > or i Tr [ E l . . . n (jk) ( j k A j k ^ s " ( < t , ( 1 2 ) 6 ( 1 2 ) ) s ] < f c : r c c " p c " ' (jk) being a product dimer. The l a t t e r contributing term i s the only term that gives the change i n the number or mass density of either species. For the observables of l i n e a r and angular momenta, t h e i r time evolution i s e x p l i c i t l y shown not only i n the density operator describing the reaction, but also i n the gain and loss of the observables associated r e s p e c t i v e l y with the reactants and products, thus i t can be referred to as the "reactive" part of the hydrodynamic equations. The nomenclature may not be appropriate i n the sense that both " c o l l i s i o n a l transfer" and "reactive" parts are transfer terms at reactive c o l l i s i o n s nonetheless i t d i f f e r e n t i a t e s the kind of transfer common to a l l c o l l i s i o n types from the kind a t t r i b u t a b l e only to reaction besides being convenient to use. - 94 -A. C o l l i s i o n s Involving a Monomer and a Dimer The reactions that may take place are either decomposition whose contributions to the equations of change are a hit MM<*M>^M+D-»-3M n ^ 123 ( W s ^123^123^1(23) + t ( * 2 V s + <*3 63>s l f 123^123 1(23) and l 8 t D D JM+D^3M 1 m =r Tr * 123 ( • ( 2 3 ) 6 ( 2 3 ) ) s ^12 3 **12 3°H( 2 3 ) [ ( < { , ( 2 3 ) 6 ( 2 3 ) ) s < ? 123J123 1(23) ' (6.1.1) (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 given by - 9 5 -[ 9 t "M^M^M+D.exch 21 Tr 123 ^ l V s ^ ( 1 2 ) 3 * 0 2 ) 3 ^ ( 2 3 ) + " W s " ( * l 6 l ) s ^ ( 1 2 ) 3 ^ ( 1 2 ) 3 1(23) and ^ 9t W^MfD.exch (6.1.3) 1 1 123 U(23) 6(23) )s V'(12)3^(12)3 J V K 2 3 ) + [ ( < ,'(12) 6(12) )s ~ ( * ( 2 3 ) 6 ( 2 3 ) ) s ] f(12)3 T(12)3 1(23), (6.1.4) F i r s t , consider the decomposition reaction which involves a net gain of 2 monomers and the net loss of a dimer. These are indeed found i n the equations f o r both species. As for the t h i r d body, i t s role i s unambiguously shown i n Chapter 7. On putting <$>± = ^ i , ^(jk) = j^j +K-k + Vjk> i t provides the energy for or c a r r i e s away the energy generated i n the reaction when the reactant dimer breaks up. In f a c t , there i s an exchange of the k i n e t i c energy associated with the c o l l i s i o n partners and the potential energy associated with the products and the t h i r d body. - 96 -For the equation of change for mass density, the contribution from the break-up reaction i s [TE ( M M + V W 3 M 1 1 2Lo* lo-i 0 0 _ i l o o 2 n 2 n _ 2 2 n _ 2 --4VV: Tr (2m) [ 1 - - £ i + £ [(2n)!] 1 ( - f i ) • V ] 123 Z < Z Z n=2 l 6(23) 123 123 P1(23) = _ i V V : j r m ^ 1 = § 1 6 ^ ^ 2 3 J 1 2 3 p i ( 2 3 ) . (6.1.5) Neglecting terms of higher orders i n VV, a rate of change due to the dispe r s i o n of mass over the product monomers i s obtained. The mass dispersion tensor % i s defined by X = M ^ T r i m r 1 2 r 1 2 « ( 1 2 ) p b(12) = ff}2) + j <r> U . (6.1.6) ( 2 ) 1 U i s the unit second rank tensor and Tft and -j <T> U are r e s p e c t i v e l y the symmetric traceless and the zeroth-weight tensors of'ft. M ^O, the l o c a l density of the scalar moment of i n e r t i a of the dimer molecules has ( 2 ) been associated with 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- to 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 t r a n s i t i o n s 1 1 . - 97 -The rate of change of l i n e a r momentum density i s given by lit % i i + M D Z>W3M = -V • P C t 5M+D+3M - £ W : 1 2 T 3 r ( - 2 r ¥ ¥ + - ) [ % + P 3 > 6(23) 1s« P123^123 Pl(23) i *2 " ^ 3 £ [ 2 ( 62 " V ] s f W l 2 3 P l ( 2 3 ) (6.1.7) ct where ?j^ +rj-»-3M * S t b e c°m si° n al transfer part of the pressure tensor associated with the reaction M+D+3M and i s defined by P C t =M+D+3M ,n r,, „_ N ° n-1 n-1 = T V (r- + 7 l~(-2) -K23) . v . = -Tr ( r - . - ^ v + L — : » — • V ) 123 _ 1 ( 2 3 ) n=2 n ! 3 6 R ( V 1(23) V 123 ) ^123 l Ai(23) Pl(23)* ( 6 ' 1 ' 8 ) The second term i n (6.1.7) corresponds to the change due to the transfer of momentum from the reactant (23) to the product monomers whereas the th i r d term i s the change due to the r e l a t i v e motion of the product monomers. This r e l a t i v e movement i n fact represents a transformation of v i b r a t i o n a l and r o t a t i o n a l degrees of freedom into t r a n s l a t i o n a l degrees of freedom. Neglecting th i r d and higher order terms i n the gradients, the conservation r e l a t i o n i s - 98 -3M = -V • (p ) =M+D+3M' " -V ' £SD*3M " I V V : J 3 R IT [ < P 2 + P 3 ) 6 ( 2 3 ) ] S 6 ?123^23 P1(23) + 4 V ' 6 ( 2 3 ) ^ 2 3 E 2 3 U f 1 2 3 ; i 2 3 p l ( 2 3 ) 2 V X 123 6 ( 2 3 ) 1 2 3 X 1 2 3 ^123^123 P1(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 of (_£.Py) s : (6.1.10) This i s connected with the r o t a t i o n a l and v i b r a t i o n a l t r a n s i t i o n s of the dimers. The zeroth-weight tensor gives r i s e to pure v i b r a t i o n a l t r a n s i t i o n s £ « ( 1 2 ) ( 2 L 1 2 ' £ 1 2 ) s P b(12) - | u : ^TVv i n jXv (v + l ) l / 2 (v + 2 ) 1 / 2 <v + 2, j X r | P b|v j X r> - v 1 / 2 (v - 1 ) 1 / 2 <v - 2, j X r I pfe Iv j X r> (6.1.11) - 99 -^ c o r r e s p o n d s to the production of mass dispersion due to i n t e r n a l motions of the dimer molecules M D ^ ~=2 H 6(12) ( £ l 2 t l 2 Js p b ( 1 2 ) = i T 2 r 6(12) 7 ^ ( 1 2 ) ' i l 2 ^ - P b ( 1 2 ) ' (6.1.12) According to the Heisenberg p i c t u r e , V ^ i s the change of 9^ 1 with respect to time, and hence the notation. For the second weight tensor ^ Ojjj" , as ( ^ j J l l j ) ^ d o e s n o t commute with z • x 2.±y n o r w i t n aa* + k a^a (a and a^ are the destruction and creation operators associated with a harmonic o s c i l l a t o r and k i s constant), t h i s i s related to r o t a t i o n -v i b r a t i o n a l t r a n s i t i o n s of the dimers. Let q w and p w be the components of r-t j and 2±j along the w-axis (w = x, y, z ) , on the a p p l i c a t i o n of the commutation r e l a t i o n [qw. P W L = i*. i t i s found that such v i b r a t i o n a l motions are not the ± 2 quadrupole-induced t r a n s i t i o n s as for pure v i b r a t i o n a l t r a n s i t i o n s . In p a r t i c u l a r , the x y component of *L 2 > i s ~(q p + q p + p q + p q ) 4 ^x ry ^y rx *x^y K y n x i T6n" U , q p + p q ~ ( q p + p q ) ] z yry y y x x x Mx • + [p p , q 2 + q 2] + [p 2 + p 2 , q q 1 Kx*y' H x M y - l l x F y ' 4 x 4 y • - 100 -1 J r 2 2. . 2 2 . i = ~7T* UP P . q + q ] + IP + p , q q 16Ti 1 'x'y' Mx y - x *y' M x M y - J (6.1.13) and the other components can be obtained in a simi l a r manner. Comparing the rate of change of Sv^due to the MrD->-3M reaction [Ji M D ^ M + D * 3 M = "fi J23 6 ( 2 3 > ^-23^23^ ^  123 7123 P1 (23) - Tr [ r 0 0 ( V 0 0 V 1 0 0 ) + ( V „ v J r„J (P 1 0 0^V 123 -23 v 23 123y v 23 123y - 2 3 J 123 1(23)K1(23) (6.1.14) with equation ( 6 . 1 . 9 ) , the contribution to the pressure tensor for t h i s reaction from the loss 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 apparently exhibit o s c i l l a t o r y behaviour when they are formed. S i m i l a r l y the part of the pressure tensor 2h T * 6 ( 2 3 ) -23 X -^23 f 123^123*1(23) x U ["ft V ^ W s M + 6 (23) ^23 X ( V 23 V 123 ) < ? 123 v A ' l ( 23 ) p l ( 23 ) x U (6.1.15) - 101 -represents the transformation of pure r o t a t i o n a l motions of the dimer molecule to the rotation between the product monomers. The molecular angular v e l o c i t y of the dimer i s given by (Mp <r» 1 Tr 6 ( 1 2 ) r 1 2 x p n P b<12). (6.1.16) As for non-reactive c o l l i s i o n s , the pressure tensor gives r i s e to a t r a n s l a t i o n a l angular momentum f l u x . However, there are also flux terms other than ( ~£ M + D+3 M x S) which l s d u e t o the coupling of the distance between the product monomers and t h e i r t o t a l l i n e a r momentum the repercussion of which i s revealed when we study the energy conservation r e l a t i o n i n Chapter 7. The rate of change i n t r a n s l a t i o n a l angular momentum density due to t h i s reaction i s : 123 -23 -23 T— [ ( - £ 2 + ^ 3 ) 6 ( 2 3 ) ] : ri23-123 pl(23) x r) i V 6(23)^23£2 3Js < P123 I123 P1(23) X ^ 1 V X \H 6(23) ^23 X^23 < P123 : I123 P1(23) X ^ - 102 -+ i V • 1 2 ^ - T 2 ^ 23 X [ % + V 6 ( 2 3 ) ] s ^123^23*1(23) 6 o + i 2 3 * 123 ~ 2 3 X ^ 2 3 2 3 s f l 2 3 7 1 2 3 P l ( 2 3 ) 2 6 i + 6 C 9 ^ - Tr — - v ' r x (V V "> P D J23 3 -1(23) X 1 1(23) V123 ; °123 l ( 2 3 ) p l ( 2 3 ) = - V ~ SM+D*3M X -k ^ [ ( T f - TT 0 ~¥+ 2^23 X [ ( - E 2 + - B 3 ) 6 ( 2 3 ) ] s P 1 0 123 123^1(23) Tr 123 i V S 3 •F [£23 x -^ 23 2 ]s f 123^123 + l—k (23) v p 3 -1(23) X V 1(23) V123 ; 123 1(23) 1(23) (6.1.17) Compensating the net change i n t r a n s l a t i o n a l angular momentum are the changes i n i n t e r n a l angular momentum. The gain i n the i n t e r n a l angular momentum of the monomers - 103 -[ 3 t \<lM,int>JM+D->-3M Tr 123 ^ i V ^123^123^(23) + [ ( s 2 « 2 ) f ( s 3 « 3 ) ] ^ 1 2 3 3 1 2 3 p l ( 2 3 ) (6.1.18) i s due to the break-up of the dimers which transforms the i n t e r n a l momentum of the constituent atoms of the decayed dimer molecule into the t r a n s l a t i o n a l angular momentum between the product monomers and the i n t e r n a l (atomic) angular momenta of the products Ut ViD.int^M+D-^M Tr 123 <»2 + ±3 + ^23 X -£23) 6(23) ^123^123^(23) " <I23 X ^23 + -2 + - 3 ) 6(23) C 123*123 1(23), (6.1.19) The net change in atomic angular momentum i s s o l e l y c o l l i s i o n a l transfer i n o r i g i n . The reactive part of the rate of change i n i t s density i s due to the transfer of the physical a t t r i b u t e from the reactant dimer to the product monomers as well as the d i s t r i b u t i o n over the product monomers. [4t" ( MM < JM,int> + MD<4,int > ) ]M +D>3M = " V * SI£D*3M - 104 -Tr 123 6,-6 t - S ^ 2 (^1 - 2 l 2 " 2 l 2>l ^123 f123°i(23) 6 2+6 3 6 2-6 3 + [ ( 8 2 + s 3 ) ( — - « ( 2 3 ) + ( s 2 - s 3 ) )] ^ 1 2 3 J 1 2 3 '1(23) + Tr 123 2 6 l + 6 ( 2 3 ) -1(23) x ( V 1 ( 2 3 ) V 1 2 3 ) ^123^(23) + Ti 6(23)^23 X -223 ^ 123^123 }1(23)' (6.1.20) The r e l a t i o n [s^ + ±2 + —3 + —23 x £.23' = ~^-l(23) X -^1(23)' h a s been used to obtain (6.1.20). The term =M+D+3M 3 ^ ( - r - l ( 2 3 ) - " ) 6R^23 X ( V 2 3 V 1 2 3 ) C ? 1 2 3 ^ ( 2 3 ) p l ( 2 3 ) (6.1.21) which arises from the interchange of i n t e r n a l and angular momentum must be a molecular angular momentum f l u x , having the t r a n s l a t i o n a l and atomic angular momentum fluxes associated with t h i s reaction i d e n t i f i e d . Comparing (6.1.17) with (6.1.20), the net changes i n t r a n s l a t i o n a l and molecular angular momenta are found to cancel each other. Again a term second order i n the gradients i s obtained 2fi Tr 123 L_r23 ^ 3 . v v 2! 2 2 ' (23) £ 2 3 -*-23 1 .E.23 —23 + r x D - — • VV 6 -23 X -P-23 2! 2 2 * V V °(23) 9 1 D 123 J123 P1(23) - 105 -TO V* V H3 r23 ^ 23 X *23 \23)S'l231l23pim) 16ti VV : Tr 123 -23 X ^-23/ : (£23 -23 ^  (2) (2) + (£ 23 W ^23 x *23 (23) 123 123 P1(23) = _ v • C T - T C T ~i V=M+D*3M SM+D+3M'* (6.1.22) We denote t h i s f l u x , which arises from the transformation of molecular angular momentum into the monomer t r a n s l a t i o n a l angular momentum by ct (L - L )„, r i J. 0„ because i t i s the most convenient choice of notation to = = M+D+3M use for f l u x terms of such o r i g i n , i s a fourth-rank tensor obtained by taking the tensor product of two second rank unit tensors i n the manner as below ^D= x\x) + y\%) + z\zj . As for the exchange reaction, there i s no net change i n the mass of the monomer or dimer: *3t *VM+D,exch = " ^  123 ^ " " V f(12)3 : r(12)3 pl(23) 2 i V • Tr m 123 0 0 1 jr. " n-l n-l r 6 + 2 I ( n ! ) ' 1 f • V « n=2 Z J (12)3 J(12)3 P1(23) - 106 -2 i rT V • Tr m 123 2 T (^(12)3 " - 1 ( 2 3 ) ) n + Z (2)» (n..)" 1 ( r ( 1 2 ) 3 n=2 ^ 1 ( 2 3 )) n-1 n-1 • V R (12)3 (12)3*1(23) (6.1.23) U t ^M+D.exch = " n 1 T 2 3 2 m ( 6 ( 1 2 ) ~ 6 ( 2 3 ) ) ^(12)3 J(12)3 P1(23) = ~ IT £ 2 m [ | ^(12)3 " ^ 1 ( 2 3 ) ) + '•• ] 6R?12)3 3U2)3 P1(23> U t ^M+D.exch 21 n~ Tr 2m 123 ,1 -12 -12 v . C"2T"2~"T" °(12) I ^23 -23 *2! 2 2 .) 6 (23) ^12)3^(12)3*1(23) U t ^ M f D . e x c h - 2 V V : U t MD^M+D,exch* (6.1.24) While there i s no net change In the masses of the monomers and dimers, the exchange reaction r e s u l t s i n a change i n the momenta of the molecules - 107 -[ 9 t % -]M+D,exch * 123 ( £ l V s ^(12)3^(12)3^1(23) + [<J> 3Vs - (£l 6l )s ] t f ,(12)3^(12)3 1(23) (6.1.25) ^3t \ -^M+D,exch 2i Tr 123 (-£(23) 6(23) )s V r(12)3 P(12)3 v /l(23) + I ( - E(12) 6(12 ) ) s " (- P-(23) 6(23) )s 1 e ( 1 2 ) 3 T ( 1 2 ) 3 1(23) (6.1.26) Combining the two equations, conservation of t o t a l momentum i s again established for the exchange reaction, t i t ( M M i i + M D ^ ] M + D , e x c h = -V • P =M+D+3M -V • P c t - l i Tr =M+D,exch "h ^< 3 " s + ^ ( 1 2 ) 3 ( 6 ( 1 2 ) - 6 3 > ] s 6 1 + 2 < S ( 2 3 ) _ [ P ( _ L _ i l 3 ) ) ] s _ t P 1 ( 2 3 ) ( 6 1 - 6 ( 2 3 ) ) ] s (12)3 J(12)3 P1(23) - 108 --V • p. ct 2i =M+D,exch ti VV Tr 123 A 12)3 -(12)3 _ -1(23) - 1 ( 2 3 ) , v 1 -i T 5 ) ( - V s f (12)3 3 (12)3 P 1(23) + 2 1 V ' £ ¥ ^ ( 1 2 ) 3 ^ ( 1 2 ) 3 ^ - i l l ( 2 3 > E l ( 2 3 ) J S ] ^(12)3 7 (12)3 P 1(23) " 2 V X V £ ( 1 2 ) 3 X ^(12)3 " -Il(23) X ^1(23) ) ( P (12)3 T (12)3 P 1(23) where L E ^ J ^  — ( i j )k^ i S t* i e t e n s o r i a - ' - i y symmetric part of ( i ( i j ) k ^ ( i j ) k ) s ' t h a t 1 8 (6 .1 .27) ^ij)k-£(ij)k^s = 2 ^  i j ) f c E ( i j ) k ) 8 + [ (-^(ij)k-£(ij)k ) S 1^ t £ ( i j ) k 3 >s * (6.1.28) .ct =M+D exch' t* i e c ° m s i o n a l transfer contribution to the pressure tensor due to the exchange reaction as indicated i n the subscripts i s defined by SM+D.exch S " 2 T ^ ( i l ( 2 3 ) + —> V V 1(23) V (12)3 } *(12)3*1 (23) P 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 of momentum of the molecules from being at those of the reactants to being at those of the products and the changes associated with the change in the r e l a t i v e l i n e a r and t r a n s l a t i o n a l angular momenta. The problem with t h i s sort of expression i s that the i n t e r n a l state variables do not appear here. The at t r i b u t e s can be divided i n another way P 3 6 3 " V l + A l 2 ) \ l 2 ) - £ ( 2 3 ) 6(23) - P 3 6 3 + *2*2 " £(23) 6(23) + -E(12)6(12) " V l ~ V 2 V 6 3 V 6 2 = ^(23) {—2— ~ 6 ( 2 3 ) } + - P - 2 3 ( W + *( 1 2 ) ( \ 12) 2 ' - j»12cvv (6.1.30) so that the conservation r e l a t i o n can be written i n a form which i s analogous to the previous reaction [<E ( MM -52- + MD ^ ) ]M +D,exch - V • P c t =M+D,exch Tr 123 1 -12 -12 2! 2 2 < £ ( 1 2 ) 6 ( 1 2 ) ) 8 1 —23 —23 2 \ — — [^2 + ^ 6 ( 2 3 ) ] s < ?(12)3 : r(12)3 pl(23) - 110 -•12^s 6(12) ^23^23^8 6 ( 2 3 ) ] ^ (12)3^12)3*1(23) (6(12) £l2 X-El2 " 6(23)-23 x ^23 ) ?12)3 3l2)3 pl(23) — — • (6.1.31) The coupling between the i n t e r n a l motions of the dimer and the r e l a t i v e motion between the dimer and the monomer i s evident comparing (6.1.27) and (6.1.31). Indeed, j u s t looking at the products alone (or reactants for that matter), one sees an interchange of i n t e r n a l and t r a n s l a t i o n a l degrees of freedom which arises from the difference i n positions between the molecules. As i n the previous case, t o t a l angular momentum i s conserved for t h i s r e a c t i o n , but t r a n s l a t i o n a l and i n t e r n a l angular momenta are not separately conserved. For t r a n s l a t i o n a l angular momentum, we have = V * (=M+D,exch X £> x ( l V s P(12)37(12)3P1<23) 2i v * T r IIE12I 123 V x Tr 123 2 i + ± ± v • Tr 123 ( ^ ( 1 2 ) 3 + , , , ) -(12)3 ( - l ( 2 3 ) + , , , ) -1(23) - I l l -2i _ * 123 6(12) + 2 6 3 3 -(12)3 X -2(12)3 2 6 1 + 6(23) -1(23) £1(23) ^(12)3J(12)3P1(23) 2 6 1 + 6(23) -2 Tr — — — r , x (V, V , , o N o ) <P 123 3 -1(23) X ( 1(23)V(12)3 ) , P(12)3'Y(23) P1(23) ( P C t x r) + — VV V£M+D,exch x -> + H V -iv A l D l -(12)3 _ -1(23) -1(23), 3 3 3 3" ; 123 J ( ^ R ) s < F ( 1 2 ) 3 J ( 1 2 ) 3 P l ( 2 3 ) X £ 21 "fi V ' ^  V^(12)3£(12)3^s " ^ l ( 2 3 ) £ l ( 2 3 ) ^ 1 ^12)3 7(12)3 P1(23) X -\ V x( Tr ^ 123 6R [-(12)3 X £ ( 1 2 ) 3 " - K 2 3 ) X £ l ( 2 3 ) ] ^(12)3 ^12)3 P1(23) x r) 41 9H V • Tr 123 ^(12)3 £(12)3 £l(23) —1(23) x (Z6R>s t f (12)3 7(12)3 P1(23) - 112 -- 2 Tr 123 2 6 1 + 6 ( 2 3 ) 3 V - 1 ( 2 3 ) X ( V 1 ( 2 3 ) V ( 1 2 ) 3 ) e ( 1 2 ) 3 ^ ( 2 3 ) P l ( 2 3 ) 21 Tr 123 6 ( 1 2 ) + 2 6 3 , (£(12)3 x -^(12)3 3 } 2 6 1 + 6 ( 2 3 ) , " ( - l ( 2 3 ) X -^1(23) 3 } < P ( 1 2 ) 3 J ( 1 2 ) 3 P 1 ( 2 3 ) V * (£ CM +D,exch X ±> 21 VV (Tr 123 2! 2 2 ^-P-(12)°(12) ;s 1 -23 -23 , <• \ TT 2 2 ^ ( 2 3 ) ° ( 2 3 ) ; s < r ( 1 2 ) 3 : r ( 1 2 ) 3 p l ( 2 3 ) X 2 i Tr 123 -12 -12 , f. , ~T~~2~* ^ ( 1 2 ) ( 1 2 ) ; s -23 -23 , t- \ — — x ^P ( 23) 0(23) ;s *U2)3*(12)3 P1(23) X - } 2± V -(Tr [ { £ 1 2 P 1 2 } S « ( 1 2 ) " {£ 23E23^s 6(23) 1 ( S'(12)3 : r(12)3 pl(23) X 123 1 V x ( T ^ [ £ 1 2 X - £ l 2 6 ( 1 2 ) - £ 2 3 X - £ 2 3 6 ( 2 3 ) l f f ( 1 2 ) 3 : r ( 1 2 ) 3 p l ( 2 3 ) X - 113 -6+6 6+6 + ^ f 2 3 ( £12 X ^ 1 2 - V i - ^ 2 3 X £ 2 3 - \ 1 ) s ^(12)3^(12)3*1(23) 2\ + 6 ( 2 3 ) " 2 [23 3 —K23) x ( V1(23) V(12)3 ) < ?(12)3 V^1(23)*1(23) (6.1.32) Al l the terras can be identified in the same way as with the decomposition reaction. Also, just as before, we can identify 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) is combined with the rate of change in the internal angular momentum density due to this reaction [Tt" (ViM , i n t > + V ^ . i n t^M+D.exch = - V • L =M+D,exch + 2i 17. T r * 123 (£l(23)--*) 6R 3 ( £ r 2 ± 2 - 2 ^ 3 ) "(12)3^(12)3*1(23) -l"-2 + [-(r 1 2 ...) 6 ( 1 2 ) 5 + ( £ 2 3 - ° 6 (23) ^(12)3^(12)3. 1(23) + % VV n Tr 123 ,1-12-12 x . , \ {2\—— 6(12) (^1 + ^ 2 ) r r " - * 0 6(23) (^2 + V 6°(12)3:r(12)3Pl(23) - 114 -2 i 1 2 2 3 " 1 2 3 ~ 1 2 X £ l 2 ~ £ 2 3 X ~ 2 3 ~ 1 ~ ) g < f ( 1 2 ) 3 : r ( 1 2 ) 3 P l ( 2 3 ) 2 6 l + 6 ( 2 3 ) 1 2 3 3 - H 2 3 ) X V V 1 ( 2 3 ) V ( 1 2 ) 3 ; ( 1 2 ) 3 J 1 ( 2 3 ) P l ( 2 3 ) * ( 6 . 1 . 3 3 ) T h u s a g a i n t h e r e a c t i v e p a r t d o e s n o t g i v e r i s e t o a n y n e t c h a n g e i n a t o m i c a n g u l a r m o m e n t u m . A s w e l l , t h e c o l l i s i o n a l t r a n s f e r p a r t o f t h e m o l e c u l a r a n g u l a r momentum f l u x o f t h i s r e a c t i o n i s g i v e n b y L C t =M+D,exch = 1 ^ ( £ l ( 2 3 ) + " * ) 6 R ^ 2 3 X ( V 2 3 V ( 1 2 ) 3 ) < r ( 1 2 ) 3 : i l ( 2 3 ) P l ( 2 3 ) -( 6 . 1 . 3 4 ) B. Collisions Involving 3 Monomers R e c o m b i n a t i o n may b e t h o u g h t o f a s t h e r e v e r s e o f d e c o m p o s i t i o n . H o w e v e r , w h e r e a s t h e p a r t o f t h e e q u a t i o n o f c h a n g e d u e t o t h e p a r t i t i o n i n g o f p h y s i c a l a t t r i b u t e s f r o m t h e r e a c t a n t s t o t h e p r o d u c t s h a s a s t r u c t u r e a n a l o g o u s t o t h a t f o r d e c o m p o s i t i o n b u t o f o p p o s i t e s i g n , t h e c o l l i s i o n a l t r a n s f e r p a r t h a s a d i f f e r e n t s t r u c t u r e b e c a u s e t h e s e t o f m o l e c u l e s p a r t i c i p a t i n g i n t h e c o l l i s i o n e v e n t i s d i f f e r e n t . T h e c o n t r i b u t i o n s t o t h e e q u a t i o n s o f c h a n g e f o r t h e m o n o m e r s a n d d i m e r s a r e r e s p e c t i v e l y - 115 -2fT Tr 123 t ( * l V s + ( W s + (*3Vs } U I ( 2 3 ) ^ l(23n23 - K W B + (*3Vs ] fl(23)Tl(23) 123 (6.2.1) and lh W]3M+MH-D - " 2ff ^ 3 t ( 4 , ( 2 3 ) 6 ( 2 3 ) ] s ^1(23) : T1 (23) P123' ( 6 * 2 * 2 ) Except for the c o l l i s i o n a l transfer contributions which are analogous to the non-reactive 3-monomer c o l l i s i o n s , manipulations are the same as with monomer-dimer c o l l i s i o n s , likewise for the i d e n t i f i c a t i o n of terms. The hydrodynamic equations are l i s t e d below. Mass [ 3 T ( MM + V]3M+M+D 2-h VV : Tr 123 1 L 2 3 £ 2 3 °° 1 - 2 3 2 n 2 n ~ 2 2 n " 2 n=z 6 (23) f l(23) T l(23) P 123 1,3 (6.2.3) - 116 -Linear Momentum With monomers being the only species i n the i n i t i a l channel, the c o l l i s i o n a l transfer contribution to the equation of motion comes e n t i r e l y from the equation for the monomers: r — I 9 t "M-^M+M+D n „Ct = - V * P S3M+M+D 2 3 2 T T 123 ( 2 3 ) 2 S K 2 3 ) J 1(23) ^123 6 -<5 + ^ 123 [ " 2 " £ 3 ) ~ ^ J " l s < ? K23) : ri(23) P123 ct i 6 2 + < 53 " V * FoM+M+D + 2tT T 2 3 ^ ( 2 3 ) ~T~~] fl(23)' : ri(23) P123 i 2fi 123 '(23)^23^23^ u 1(23) : T1(23) P123 2 V X ^ 3 6 ( 2 3 ) ^ 2 3 X- E23 < P1(23) : T1( 23) P1 23 (6.2.4) ct Here P „ w _ i s the c o l l i s i o n a l transfer contribution to the pressure tensor associated with the reaction 3M+M+D and i s given by P C t =3M-»-MfD Tr 123 ( 6 ( 2 3 ) £ 2 3 + ( V 2 3 V 1 ( 2 3 ) ) + ( 6 l £ l ( 2 3 ) + • , ° ( V 1 ( 2 3 ) V 1 ( 2 3 ) } 1(23) 123 - 117 -Tr 123 ( r 1 2 + ...) 6 ( 1 2 ) ( V 1 2 V 1 2 ) + ( r 1 3 + ...) 6 ( 1 3 ) ( V 1 3 V 1 3 ) 1(23) 123p123' (6.2.5) Again for t h i s r e action, the change i n momentum density can be expressed i n terras of a pressure tensor 3M+M+D o „ C t =3M+M+D i 1 ^23 *23 + 2f\ W ! T 2 3 2 T " 2 ~ " 1 ~ ( ^ ( 2 3 ) 6 ( 2 3 ) ) s < P1(23) 71(23) P123 i 2r7 123 6 ( 2 3 ) ^ - 2 3 £ 2 3 ^ s P1(23) T 71(23) P123 " ^ V X 123 6 ( 2 3 ) ~ 2 3 X ~ 2 3 | PH23) : I1(23) P123 (6.2.6) Whereas for the break-up reaction the i n t e r n a l motions of the reactant dimer i s transformed into 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 reverse i s observed. And the dimer undergoes v i b r a t i o n s and rotations when i t i s formed. 118 -Angular Momentum ,lnt>^3M->-M+D 2ti Tr 123 [ ^ S l > s + ( l 2 S 2 > S + <l3Vs ' ^ ( 2 3) Pl ( 2 3 ) ^ 2 3 - t < £ 2 V s + ( s 3 6 3 ) s ] P 1 ( 2 „ 3 , 123 (6.2.7) r l"8t ( MM <^D,int > ) ]3M^M+D = " ** 123 [ - 2 + 1 3 + X £23>' 6 ( 2 3 ) ] s ^ 2 3 ^ , r „ , (6.2.8) (23) Js y l ( 2 3 ) J l ( 2 3 ) P 1 2 3 i [lt ( MM<^,int> + V ^ D . l n t » l 3M+M+D • + 2 W ' • ^ , < £ 1 2 - • » ) S U 2 ) < i r V V 1 2 P l ( 2 3 / W 123 So - s„ 123 2 (23) 1(23) J1(23) P123 *?23 + 6 ( 2 3 ) l 2 3 X ^23 ^ 1(23)^1(23) 123' (6.2.9) - 119 -Just as for the decomposition reaction, the only term that gives r i s e to the net change in atomic angular momentum i s the c o l l i s i o n a l transfer term. But now since only monomers are involved i n the i n i t i a l channel, no molecular angular momentum flux due to c o l l i s i o n a l transfer i s id e n t i f i e d . Combining with the equation of change for t r a n s l a t i o n a l angular momentum [ - ^ r x ( M M U + M D v ) ] 3 M > M+D = V =3M>MfD X -IK 1 2 3 t ( = ^ 1 + ' • ° ^ X t ( V - B 3 ) 6 ( 2 3 ) ] s d 3 l ( 2 3 ) T l ( 2 3 ) p 1 2 3 ] ^23 7 T r  L 123 61 + 62 2 £l2 X ( V 1 2 V 1 2 ) + - ^ T - U l 3 x ( V 1 3 V 1 3 ) ^1(23)^23 i 6 2 + 6 3 " ti l-23 X£23 ~2~ 1 s ^ l ( 2 3 ) 3 l ( 2 3 ) 123 ct V • P =3M*M+D " 2 ^ W : lH3ll T-T-T-f- (-H(23) W s f l ( 2 3 ) % 2 3 ) P 1 2 3 X £ ] - 120 -+ 4 V * 6(23)^23£23^s P1(23) J1(23) P123 X *> T V X ( T ^ 3 6(23) ^ 23 X-£23 P1(23) J1(23) P123 X ±> r r i —23 —23 " V ' 123 ~ ~ X ~ ( 2 3 ) 6 ( 2 3 ) ) s ^K23)^1(23) P123 y Tr 123 « + 6 6 + 6 2 -12 X ( V 1 2 V 1 2 ) + — — ^13 X <713V13> P , ^ p 1(23) 123^123 6„ + 6 i 2 3 + 2rT T ! L [r-23 X ^23 2 1 s ^ 1 ( 2 3 ) J l ( 2 3 ) P 1 2 3 ' 123 (6.2.10) one finds that the molecular angular momentum flux a r i s e s from the formation of a molecular angular momentum and the destruction of a t r a n s l a t i o n a l angular momentum + 2rT12T3* ^ 2 3 X - E 2 3 ( 6 ( 2 3 ) 5 „ + 6 o 2 3 2 ' J s u 1(23T1(23)^123 T6n" VV Tr 123 -23 X -^ 23 r 2 3 / 3 + 2 ^ 3 ^ 2 3 / / : ^-23^23 + 2 -23 -23 -23 X ^23 6(23) ^ 1(23)^1(23) P123 - V • T =3M+M+D (6.2.11) -"121 -C. Dimer-Dimer Collisions With two dimers, the reactions that may take place are decomposition of a dimer without exchange, decomposition with exchange, and exchange without decomposition. Their respective contributions to the equations of change for the molecules are as follows: Decomposition Without Exchange hTt W^D-^M+D.no exch " * [ ( * l V s + ( < W s ] f ' l 2 ( 3 4 ) 3 i 2 ( 3 4 ) p ( l 2 ) ( 3 4 ) ' ( 6 - 3 - 1 } 1234 r— M <$ >i l 9 t D D J2D->2M+D,no exch * 1234 ^hll^in^s + U ( 3 4 ) 6 ( 3 4 ) ) s 1 U12(34)<:>12(34)v/Vl2)(34) " (*(12)6(12))s ^12(34)^2(34) P ( 1 2 ) ( 3 4 ) 5 (6.3.2) Exchange Decomposition U t W ^ D ^ M + D . e x c h I (*1 61 >B + ( W s ^ l 3 ( 2 4 ) ^ 1 3 ( 2 4 ) p ( 1 2 ) ( 3 4 ) ' ( 6 ' 3 - 3 ) 1234 - 122 -Ut MD<$D>-I2D+2M+D,exch 2 1 „ zr- Tr 1234 [ i \ \ 2 ) \ \ 2 , \ + ( < ( >(34) < S(34) )s 1 ^3(24)^13(24) ^ 12)(34) + [<*(24) Ws - «*<12> Ws + <*(34) 6(34) > s>3 ^13(24)^13(24) (12)(34)' (6.3.4) Simple Exchange ^1234 {{\\2)\\2)\ + ( < | ) ( 3 4 ) 6 ( 3 4 ) ) s 1 U r i 3 ) ( 2 4 ) X l 3 ) ( 2 4 ) ^ 1 2 ) ( 3 4 ) ^(W^n^B + ( < t > ( 2 4 ) 6 ( 2 4 ) ) s ( < t > ( 3 4 ) 6 ( 3 4 ) ) s f(13)(24)' J(13)(24) (12)(34). (6.3.5) For the simple decomposition reaction, the changes i n the dens i t i e s of mass, l i n e a r momentum and angular momentum other than that due to c o l l i s i o n a l transfer are found to be e s s e n t i a l l y the same as that for the - 123 -M+D+3M reac t i o n except that now i t i s a dimer molecule which acts as the t h i r d body. Mass Density L 8 t 2D->-2M+D,no exch r r i 1 —12 —12 li V V : ,15, 2 m ( 2 T " 2 ~ ~ + ~ ' ) 6 ( 1 2 ) r i 2 ( 3 A ) ^ 2 ( 3 4 ) p ( 1 2 ) ( 3 4 ) 1234 2 ' l 8 t D = 2D*2M+D,no exch Linear Momentum ["at ( MM ± + % £ ) ]2D>2M +D,no exch = -V • P =2D+2M+D,no exch = - V • P c t =2D->-2M+D,no exch 1234 < i12(34) : ri 2(34) P12(34) (6.3.6) ' " 6<12)^122l2 }s < ri2(34) J12(34) p(12)(34) 1234 " I V X ?5, 6(12) £l2 X £ l 2 < P12(34 ) l 2(34) p(12)(34) 1234 + ( t h i r d and higher order terms i n V) (6.3.7) - 124 -£2D>2M+D,no exch' t h e c o l l i s i ° n a l transfer part of the pressure tensor for the reaction i s given by P C t =2D+2M+D,no exch - Tr 1234 ^ 1 2 X 3 4 ) 6 R 2n+l n=l (2n+l)! 2 ( V(12)(34)V12(34) ) f12(34)X /Vl2)(34)P(12)(34) Angular Momentum r x u + ^  v ) ] 2 D + 2 M + D j n o e x c h V • ( P C t x =2D+2M+D,no exch ±-} (6.3.8) r r 1 -12 -12 + if " ' < ! ? , 2 ! — — • < , 2 ) > 1 . ' I ( 2 3 ) ^ l ( 2 3 ) ' l 2 3 * I> I f l V ' ( 1 2 M 6 ( 1 2 ) ^ 1 2 ^ 1 2 J s f12(34) J12(34) p(12)(34) X ±> -1 2 V X ( T ^ 3 4 6(12) £l2 X ^ 1 2 f12(34)^12(34) p(12)(34) x I> - 125 -< ri2(34) T12(34) P(12)(34) . T r S ( 1 2 ) + 5 ( 3 4 ) r x ( v V ) 1 2 3 4 2 -(12)(34) X ^ V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ; ^12(34)^ 12)(34) P(12)(34) 6 +6 i 1 2 *T J234 " 1 2 X' £12 ~~2~]s f12(34) ^ 2(34) P ( 12) (34) U t ( MM ^ M . i n ^ + % ^D.int^UD-^M+D.no exch (6.3.9) k v v : 1 2 T 3 r 4 TT ~T~ ~1T < W V ^ s ^12(34)^: 12(34) J12(34)^(12)(34) 'h Tr 1234 (£j + ±2) - (£3 + J4) [ ( 6 ( 1 2 ) ~ 6 ( 3 4 ) ) 2 ] s ^12(34)^2(34)^12X34) - l~-2 + [C61-«2) — 2 — ] s ^12(34) ^ 12(34) P(12)(34) c t - V • L =2D+2M+D,no exch - 126 -+ Tr 1234 6 ( 1 2 ) + 6 ( 3 4 ) 2 £ ( 1 2 ) ( 3 4 ) X ^ V (12) (34) V 12(34) ; r l 2(34rU2 ) ( 3 4 ) + -fi 6 (12) ^12 X £ l 2 ^12(34)^12(34) (12)(34) (6.3.10) The c o l l i s i o n a l transfer contribution to the molecular angular momentum flu x Is defined by =2D+2M+D,exch - 1 ^ 4 ( £ ( 1 2 ) ( 3 4 ) + 6 ^12 X ( 7 1 2 V 1 2 ( 3 4 ) ) ~ £ 3 4 X ( 7 3 4 V 1 2 ( 3 4 ) ) < ? 12(34) v / 2 d2)(34) P (12)(34)* (6.3.11) The term due to the transfer of the molecular angular momentum of the reactant to r e l a t i v e t r a n s l a t i o n a l angular momentum between the products 2rT vv Tr 1234 ( J _ r l 2 z l 2 . 2! 2 2 ' r r 1 —12—12 + ( 2 f T " ~ -12 X-£l2 ^12(34) ^12(34) P(12)(34) = - V • (L - L C t ) 2D->-2M+D,no exch (6.3.12) c l e a r l y p a r a l l e l s the flux term a r i s i n g from such transfer of angular - 127 -momentum from the reactant dimer to the product monomers for the M+D+3M reaction. For the exchange-decomposition reaction, the appropriate equations of change are: Mass d l3t ( MM + V]2D->2M+D,exch 2i tr VV : Tr 2m 1234 r r 1 -12 -12 ( — — ^ + ^5 + (l -34-34 K2l 2 2 + ~'>\l2) + ( 2 T ~ ~ r r _ r l 24 24 K2\ 2 2 6(24) ^13(24) J13(24) P(12)(34) .) 6 (34) 1 VV : [|- M Oil] 2 9t D =J2D*2M+D,exch' (6.3.13) Linear Momentum Mr U + M,, v)j 2 D + 2 M f D ( e x c h = -V • P =2D+2M+D,exch a - V • P C t =2D-»-2M+D,exch - 128 -21 VV : Tr 1234 r r 1 .£12 £l2 2 T ( — — + ••' ) " P i + £ 2 ) 6 ( 1 2 ) ] s 1 ^3 A "~3 A + ( 2 l — — [ ( P 3 + - E 4 ) 6 ( 3 4 ) ] s . 1 - 2 4 - 2 4 w , ~ ( T T ~ 2 " 0 [ (?2 +V 6(24 ) ] s S°13(24) : J1 3(24) P(12)(34) 21 V ' m 4 t 6 ( 1 2 ) ^ 1 2 ^ + 6 ( 3 4 ) ^ 3 4 H 3 4 ^ s " 6(24) ^ 24*24 >s] ^ 1 3 ( 2 4 ) : r i 3 ( 2 4 ) P ( 1 2 ) ( 3 4 ) 4- V x Tr 1234 6 ( 12)^12 X ^ 1 2 + 6(34) ^ 34 X-P-34 - 6 (24) -24 X -^ 24 ^13(24) 313(24) P(12)(34) (6.3. ct where P„„ _„ „ - i s the c o l l i s i o n a l transfer part of the pressure =2D-»-2M+D,exch tensor a r i s i n g from the exchange decomposition involving two dimer molecules, namely - 129 -=2D+2M+D,exch 2 Tr 1234 (r -(12)(34) °R 6 D + •-.) (V (12)(34) V13(24) ) ' P13(24) v /]fl2)(34) P(12)(34) Angular Momentum (6.3.15) [1>F ( MM <iM,mt> + "D <4 , int>^ 2D-»-2M+D,exch -2 Tr 1234 6 ( 1 2 ) + 6 ( 3 4 ) 2 £(12)(34) X ( V ( 1 2 ) ( 3 4 ) V 1 3 ( 2 4 ) } 13(24) (12)(34) P(12)(34) + k ^24 X ^ 2 4 6 ( 2 4 ) ~ £ l 2 X £ l 2 6 ( 1 2 ) " -^ 34 x P 3 4 6 ( 3 4 ) ] ^13(24) ^ 3 ( 2 4 ) ^ 1 2 X 3 4 ) V • L C t =2D>2M+D,exch 21 •h Tr 1234 r ( 8 + s - s - 8 ^ ( 1 2 ) ( 3 4 > i ^±l+±2 -3 V 2 ] s ^ 3 ( 2 4 ) ^13(24)^12X34)^12X34) - 130 -2i Tr 1234 - l " - 2 V 6 2 V ^ V V + < - V V ("V2-- 6<12>> + - i z ± ( V 6 3 ) + ( S 3 + S a ) ( _ i _ J L _ 6 ( 3 A ) ) ^ ( v v - ( y y (-V-- w ^13(24) 713(24) P(12)(34) ; ( 6 . 3 . ,ct the structure of L«" . o v l, . i s referred to the molecular angular =2D->-2M+D, exch momentum flux due to non-reactive dimer-dimer c o l l i s i o n s , and [-37 * X ( MM 2 + ^)]2D+2M+D,exch = V • (P x r) =2D+2M+D,exch Tr 1234 (~T (Tr~l! ) +"-) ~ r x (£(12) 6(12) }s + ( ¥ ( T T - T T > + ' " ) : 3 r x ( £ ( 3 4 ) 6 ( 3 4 ) ) ! - <^T ( TT " IT* + - > ^ ( ^ ( 2 4 ) 6 ( 2 4 ) ) s f13(24) 713(24) P(12)(34) - 131 -2 i r 6 d 2 ) + 6 ( 3 4 ) f 1234 2 £(12)(34) X ( V ( 1 2 ) ( 3 4 ) V 1 3 ( 2 4 ) ) ^13(24)^12)(34) P(12)(34) 2 i * 1234 ~ 1 2 X J £ 1 2 6 ( 1 2 ) + £ 3 4 X £ 3 4 6 ( 3 4 ) " r24 X - E 24 6(24) 1 < ?13(24)Jl3(24) P(12)(34) V • ( L - L C t ) 2D-*-2M+D,exch' (6.3.17) For the simple exchange, conservation laws are again established. Mass [ 3 t ^ 2 D , e x c h ^ VV : Tr 2m 1 4 1234 ( i - r i ! £ 1 2 v 21 2 2 • • • z 0 (12) ^2! 2 2 6(34) 1 -13 -13 - (— Zi£ } * /I -24 ^24 N . 4 , 2 2 ...) 6 ( 1 3 ) - ( _ _ _ . . . ) ,5 (24) 6R f(13)(24) 1(13)(24) P(12)(34) I " [ 3 V " ^ D . e x c h (6.3.18) - 132 -Linear Momentum [Jt % - ] 2D,exch = ~ V * S 2D,exch V • P c t =2D,exch - i VV • Tr r i - £ ( 1 3 > ( 2 * > £(13)(24) 1 £ ( 1 2)(34) £ ( 1 2)(34) A * • , : o , l 2 ! 2 2 2T 2 2 + * 1 2 3 4 ..] ( P 6 R ) s f ( 1 3 ) ( 2 4 ) J(13)(24) p(12)(34) Tr 5 1 2 3 4 1 ^<13)(24) -E(13)(24)U " ^{12)(34)- P-(12)(34) *(13)(24) 3(13)(24) p(12)(34) 1 - 4 V x Tr 6 2 1234 R £<13)(24) X-E(13)(24) - r £(12)(34) X £(12)(34) U 3 ) ( 2 4 ) J ( 1 3 X 2 4 ) P ( 1 2 ) ( 3 4 ) - V • P c t =2D,exch + * VV Tr 1234 1 £l3 -13 , K \ 2\ ~T 2 ^(13)°(13) ;s 1 £24 £24 . . . I f 2 2 l-P-(24)°(24);s r r 1 £-12 £12 (P .) -1 £34 £34 .) 2! 2 2 V J i(12) (12) s 2! 2 2 x ( 3 4 ) (34) s X13) (24)^(13) (24) P ( 12X34) - 133 -i V • Tr 1234 U l s P j A 6 ( 1 3 ) + ll24£24^s 6 ( 2 4 ) " { r 1 2 P 1 2 } s « ( 1 2 ) - [ r 3 4 P 3 A } s 6 ( 3 4 ) ^ ( 1 3 ) ( 2 4 ) T ( 1 3 ) ( 2 4 ) P ( 1 2 ) ( 3 4 ) -~ V x Tr 1234 ^13 X ^ 1 3 6 ( 1 3 ) + £ 2 4 X ^ 2 4 6 ( 2 4 ) -111 x - 2 l 2 6 ( 1 2 ) " ^ 3 4 X ^ 3 4 6 ( 3 4 ) Q l 3 ) ( 2 4 ) : r ( 1 3 ) ( 2 4 ) p ( 1 2 ) ( 3 4 ) (6.3.19) Here c t £2D,exch 5 " 1 2 ^ 4 ( 6R^ ( 1 2)(34) + " ^ ( V ( 12) ( 3 4 ) V ( 12) (34) } ^(13)(24) v / Vl2)(34) P (12)(34) (6.3.20) i s the p a r t o f the p r e s s u r e t e n s o r 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 t r a n s f e r f o r the exchange r e a c t i o n i n v o l v i n g two di m e r s . The t e n s o r i a l l y symmetric p a r t o f ( r ( i j ) ( k l ) P ( i j ) ( k l ) > s 1« g i v e n by - 134 -te<ij)<ki> -E(ij)(ki>U - i ( - ( i j ) ( k l ) -£(ij)(kl))s + ( £ ( i j ) ( k i ) -BcijXkl)^ - i + - j ~ -k ~ -1 = ^ i j ) ( k l ) 2 (6-3.21) Angular Momentum The rate of change i n t r a n s l a t i o n a l angular momentum density associated with the 2-dimer exchange reaction i s ljt±* K j ^ D . e x c h = V • (p x r) =2D,exch — ti Tr 1234 1 \ -13 £13 [ ( 1 " f i 0 ~T-"] ~T x (-H(13) \ \ 3 ) h + K i -jr) -IA...] -fAx ( £ ( 2 4 ) 6 ( 2 4 ) ) ( r r [ ( 1 = l i — 1 Z = T X ^ ( 1 2 ) 6 ( 1 2 ) ) s 1 ^ 3 A "~"3 A [ ( l - 2f> V*-*1 T - x V-u^. -(34) u(34)'s < F U 3)(24) : r(13)(24) P(12)(34) 6 +<5 Tr (12) (34) ( 7 v ) 1234 2 £ ( 1 2 ) ( 3 4 ) X (' V(12)(34) V(13)(24) ; < ?(13)(24)^12)(34) P(12)(34) - 135 -i „ + zr Tr * 123A £l3 X ^ 1 3 61 + 63 2 • + ^24 X ^24 62 + 54 " £ l 2 X ^ 1 2 61 + 62 " £34 X £34 63 + 64 < P(13)(24) J(13)(24) P(12)(34)' (6.3.22) Comparing the terms i n (6.3.22) that corresponds to a net change i n i n t e r n a l angular momentum with the rate of change i n i n t e r n a l angular momentum density, one finds that the molecular angular momentum flux Lor, • i s given by =2D,exch 6 3 - V • L =2D,exch = V ' * 1234 - ( 1 2 ) ( 3 4 ) + - ) [ £ l 2 * ( 7 1 2 V < 1 3 ) < 2 4 > > - ^ 3 4 x ( V 3 4 V ( 1 3 ) ( 2 4 ) ) ] ^ 1 3 X 2 4 ) ^ 1 2 X 3 4 ) P ( 12) (34) "4T T r 1234 £-13 X ^ 1 3 ( 6 ( 1 3 ) " £ 1 2 x P i 2 ( 6 ( 1 2 ) 6 1 + 6 3 2 •> + -E 24 X - E 2 4 ( 6 ( 2 4 ) 6 l + 6 2 6 2 + 6 4 . 2 ) - r 3 4 x p 3 4 ( 6 ( 3 4 ) e{ 13 ) ( 2 4 ) ^ 13)(24) p ( 12)(34)* (6.3.23) - 136 -D. Collisions Involving 2 Monomers and 1 Dimer There are four possible kinds of reactions: simultaneous decomposition and recombination, recombination without exchange, exchange-recombination, and exchange without recombination. By analogy to the reaction types dealt with e a r l i e r i n this chapter, the hydrodynamic equations for mass, l i n e a r momentum and angular momentum de n s i t i e s can be obtained immediately. Simultaneous Decomposition-Recombination The general equations of change are: ^9t S^i ^M^M+D.recomb, d ecomp 2 * 1234 [ ( W s + ( W s ] ^(12)34^(12)34^2(34) + [ ( • 3 8 3 ) s + ( W s " <*lVs " ( * 2 6 2 ) s ^ l 2 ) 3 4 : 3 ( 1 2 ) 3 4 12(34) ' and f — M <$ >1 3t D D 2M+D,recomb,decomp (6.4.1) 2fT Tr 1234 ( < | )(34) 6(34 ) ) s V r(12)34 4°(12)34 v /12(34) + t ( < f >(12) 6(12 ) ) s " ( < t >(34) 6(34 ) } s ] ^(12)34^12)34 P12(34)' (6.4.2) - 137 -Substituting into the above equations the appropriate observables, and performing c a l c u l a t i o n s analogous to the M+D+3M and 3M+M+D reactions, conservation laws are established for th i s rea c t i o n . Mass hnr ( M M + V 3 2M+D,recomb,decomp 2f7 vv Tr 2m 1234 1 £34 £34 , U ~2 1~ °(34) + " 1 £l2 £l2 , , ~ 2T ~2 2~ °(12) + J(12)34 : : r(12)34 p12(34) « _ J. VV : rl— M ^ 1 2 9t D =J2M+D,recomb,decomp Linear Momentum (6.4.3) ["aT ( M M i i + ^^^M+D.recomb.d ecomp = - V • P =2M+D,recomb, decomp r? „ct cr _ V • P =2M+D,recomb,decomp 2rT VV : Tr 1234 2! £-34 £34 r , , v r , — — t<P 3 + 2 4 ) 6 ( 3 4 ) ] s £l2 £l2 r, . N , , - — — l ( £ l + ^ 2 ) 6(12) ] ! ^12)34^(12)34 P12(34) - 138 -T5 [ ^ M I I ^ J 1234 l l l ? P l ? } j (34) ^ 34^34 Js u ( 12) ^ 12-^12 ; s P ? (12)34 J(12)34 p12(34) 2 V X 1 2 3 4 [ 6 ( 3 4 ) £34 X P34 " 6(12)£l2 X P l 2 ] e > 1 (12)34 d(12)34 p12(34) =2M+D,recomb,decomp 1234 ( 6 ( 1 2 ) £ l 2 + ( V 1 2 V ( 1 2 ) 3 4 ) + ( 6R£(12)(34) + - * 0 ( V(12)(34) V(12)(34) — Tr 2 z 1234 P n (12)34 12(34) P12(34) ( £ 1 ( 3 4 ) + '••> 6(134) ( V1(34) ( V13 + V 1 4 » + ( £ 2 ( 3 4 ) + '•• ) 6(234) ( V2(34) ( V23 + V2A» <r vfi-' p (12)34 12(34) 12(34) (6.4.4) (6.4.5) i s the contribution to the pressure tensor associated with t h i s reaction from c o l l i s i o n a l transfer. - 139 -Angular Momentum ht - X ( MM - + -^2M+D,recomb,decomp = V • (p x r) =2M+D,recomb,decomp — + i - V 2n Tr 1234 1 1 N -34 -34 (TT - IT 5 — — X f (P 3 +PA> 6(34 ) ] s 1 1 N -12 -12 ~ ( T T ~ TT> — — x t ( - E i + £ 2 ) 6 ( 1 2 ) ] J Tr z 1234 < P(12)34 : i(12)34 P12(34) 26 + 6. . 3 — L - 2 - £ l ( 3 4 ) x V l ( 3 4 ) ( V13 + V14> 2h" Tr 2^ 2 + S ( 3 A )  + 3 -2(34) x V2(34) ( V 2 3 + V 2 4 ) 4(12)34 /*i2(34) P12(34) 6 3+6 4 6+6 (r.34x-E34 — r ~ > s ~ (£l2 x-El2 —2~\ 1234 _ (12)34 J(12)34 P12(34) (6.4.6) As with the c o l l i s i o n types that have been studied e a r l i e r on, the l a s t two terms contribute to the change i n i n t e r n a l angular momentum and the t o t a l angular momentum i s conserved: - 140 -Ut ^M^iM.int/* + MD<^D,int>^2M+D,recomb,decomp 2fi Tr 1234 6 + 6 6 l + 6 2 - K s ^ ) ^ - 6 ( 1 2 ) ) ) ^12)34" I(12)34 P12(34) V • L C t =2M+D,recomb,decomp 2 6 1234 s - 2s. - 2s. < V W 3 ^ 1 3 + ^ 4 * + < V W S' 'I ^  ( ^ 3 + U 2 4 ) P(12)34 J12(34) P12(34) ~ 2* < 6(12>^12 x-£l2 6(34) -34X-P-34) ^(12)34 T(12)34 P12(34) + j Tr Z 1234 2 ( 5 i + $ri/\ 3 -1(34) X V l ( 3 4 ) ^ 1 3 + V 1 4 ; 2 6 o + + 1 ^5*> T x V CV + V ) + 3 £2(34) X 2(34) ^ V23 V24 ; ^(12)34 yi /2(34) P12(34)* (6.4.7) - 141 -The molecular angular momentum flux L , „ , , for t h i s =2M+D,recomb,decomp reaction i s given by the c o l l i s i o n a l transfer part =2M+D,recomb,decomp \ Tr 1234 ( £ 1 ( 3 A ) + — > < S(134) £34 X t V34 ( V13 + V 1 4 ) ] - <r 2 ( 3 4 ) + . . O \ 2 3 A ) £ 3 4 x [ V 3 4 ( V 2 3 + V 2 4 ) ] '(12)34 12(34)^12(34) (6.4.8) and the flux term obtained by summing the production loss terms i n ct, (6.4.6-7) i . e . , (L - L ) 2M+D,recomb,decomp' - V • (L - L C t ) 2M+D,recomb,decomp 2 f l 1234 61 + 62 [ £12 X ^ 1 2 ( 6 ( 1 2 ) 2 ) ] s 63 + 64 ~ t £ 3 4 x p 3 4 ( 6 ( 3 4 ) ^ ) ] g (12)34 (12)34 P12(34) (6.4.9) Recombination Without Exchange The di f f e r e n c e between recombination-decomposition and simple recombination i s that 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 disi n t e g r a t e but acts as a th i r d body. C l e a r l y the conservation for the physcial a t t r i b u t e s of mass, l i n e a r momentum and angular momentum holds. The rates of change of the density of an a r b i t r a r y single-molecule observable due to t h i s reaction are 2M+D+2D,no exch i 2b Tr 1234 l ( * l V s + ( W s ] V r(12)(34)^(12)(34)i /2(34) - K ^ V s + <*(12)(34) 7(12)(34) P12(34)* (6.4.10) and a t D D J2M+D>2D,no exch i Tr 1234 ( 3 4 )6 ( 3 4 ) ) s V /'(12)(34) f(12)(34) 12(34) " " 2rT + (<t> ( 1 2 ) 6 ( 1 2 ) ) s tf'(12)(34):I(12)(34) P12(34)* (6.4.11) The hydrodynamic equations of change for mass, l i n e a r momentum and angular momentum are also l i s t e d . - 143 -Mass Ut ( MM + MD)]2M+D+2D,no exch r r 2* 1 2 3 4 2! 2 2 (12) (12)(34) J(12)(34) M12(34) 1 r 3 ~ 2 Ut V V 1 ( MD^UM+D>2D,no exch Linear Momentum UrT ( MM + *D £ ) ]2M +D+2D,no exch = -V • P =2M+D-»-2D,no exch - V • P C t =2M+D*2D,no exch r r 1 ,-12 -12 + VV : Tr ^ - ( 1234 2! v 2 2 + < £ ( 1 2 ) 6 ( 1 2 ) ) 8 ^12)(34)" 7(12)(34) P12(34) (6.4.12) i 2ti 1234 ' ( 1 2 ) l i - i 2 - i i l 2I s V(12)(34) : I(12)(34) P12(34) " 2 V X 6(12)-12 X -El2 ^(12)(34) J(12)(34) p12(34) 1234 (6.4.13) ^ _ , i s defined by =2M+D-»-2D,no exch - 144 -P C t =2M+D+2D,no exch y Tr 1234 ( 6(12)£l2 + ( V 1 2 V ( 1 2 ) ( 3 4 ) ) + ( < S R£(12)(34) + ' " } ( V ( 1 2 ) ( 3 4 ) V ( 1 2 ) ( 3 4 ) ) 5(12)(34) v 0f2(34) P12(34)* Angular Momentum [•§7 £ x ( MM i + ^ ^ ) ]2M +D-2D,no exch (6.4.14) V • (P 2M+D-»-2D,no exch x r) - • = r V . Tr C ^ - ^ ) I i i I i i x (p / 1 0,6, i 0,). 2n 1234 2 2 ^ 1 2 ) (12)'s f !(12)(34) T(12)(34) P12(34) y Tr * 1234 2 6 1 + 6(34) £-1(34) X V l ( 3 4 ) ( V 1 3 + V14> 2 6 o + + 1 r x V (V + V ) + 3 £2(34) X 2(34) ^ 2 3 24 ; e(12)(34) J(12)(34) p12(34) i V ! i + 2h" J ' / ^ ^ ^ n T ^ s ^(12)(34) J(12)(34) P12(34) 1234 (6.4.15) - 145 -TF ^M^M . i n ^ + MD<J-D,int>)^2M+D-2D, no exch ^ 1234 6 + 6 6 - 6 (P, (12)(34) 7(12)(34) p12(34) 2ti Tr 1234 J , ~ 2s - 2s. < V '»*)> 1 ^ u + ' V £„ - 2s - 2s, + < V «(34>> 1 ^23 + V P(12)(34) V / 212(34) P12(34) 2n 3 £12 X ^ 1 2 ] s P ( 1 2 ) ( 3 4 ) ; T ( 1 2 ) ( 3 4 ) P 1 2 ( 3 4 ) V • L =2M+D-»-2D,no exch + f Tr 1234 2 6 1 + 6(34) 3 £1(34) x V l ( 3 4 ) ( V13 + V14> 2 62 + 6(34) + 3 ^2(34) x V2(34) ( V23 + V 2 4 ) (12)(34) 12(34) P12(34) - 146 -Simple Exchange Reaction The contributions to the equations of change from t h i s reaction are: 2M+D->-2M+D,exch * 1234 ( » l V s + ( W s ] ^(13)24X13)24^12(34) + " W s " ( * l 6 l ) s ^ ( 1 3 ) 2 4 J ( 1 3 ) 2 4 12(34) (6.4.17) and L 8 t D D 2M+D+ 2M+D, ex ch 2 i Tr 1234 ( X 3 4 ) 6 ( 3 4 ) ) 8 ^(13)24^(13)24^2(34) + I ( * ( 1 3 ) 6 ( 1 3 ) ) s " ( < | ,(34) 6(34) )s ]Xl3)24J(13)24 12(34)' (6.4.18) Besides the c o l l i s i o n a l transfer part, t h i s i s e s s e n t i a l l y the same as the exchange reaction involving a dimer and a monomer, only now there i s an extra 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 react i t s e l f . E x p l i c i t l y , the rates of change of the hydrodynamic d e n s i t i e s of mass, l i n e a r momentum and angular momentum associated with t h i s reaction are as below. - 147 -Mass [ 37 ( M M + V W D •»-2M+D,exch 2i 4r- VV : Tr * 1234 2 m [ ? ( £(13)4 £(13)4 " £ 1 ( 3 4 ) £ 1 ( 3 4 ) ) + — 1 6 P T '(13)24 J(13)24 P12(34) (134) 21 VV Tr 2m 1234 (1 £13 -13 V2! 2 2 6(13) - (1_ =34 =34 2! 2 2 •••^o (34) tf 7 (13)24 J(13)24 P12(34) - I VV • f— M % 2 * l 3t llD =M2M+D>2M+D,exch (6.4.19) Linear Momentum The contribution to the equation of motion from t h i s reaction i s [•§£ R + % - ) ] 2M+D+2M+D, = ~ V S2M+D*2M+D,exch ct ~ ~ V * S2M+D->-2M+D,exch exch 2i VV : Tr 1234 (£ ( 1 3 ) 4 £ ( 1 3 ) 4 _ £1(34) £U34) 3 3 3 3 } (- E(134) 6(134) )s r(13)24 T(13)24 P12(34) - 148 -2i V • Tr 1234 6(134) ( 11(13)42(13)4 Js " (ll(34)-El(34) U) *( 13)24 ^(13)24p12(34) V x Tr 1234 6R (£(13)4 x 2(13)4 " £l(34) x -El(34) ) ^(13)24-r(13)24p12(34) - V • P c t =2M+D+2M+D, exch + 2i w Tr 1234 1 —13 —13 { U ~ ~ T ) ( £ ( 1 3 ) 6(l3))s X 3^ A 3^ A ~ (2T ~T~ ~T~) (2(34) 6(34) )s ?13)24J(13)24P12(34) 2i V ' m 4 t 6 ( 1 3 ) ^ ^ 1 3 ^ 8 - 6 (34)t l34£34U (13)24J(13)24P12(34) * V X 1234 f 6 ( 1 3 ) £ l 3 X £ l 3 " 6 ( 3 4 ) L U X " 3 4 1 (13)24J(13)24P12(34) (6.4.20) where _P(ijk) ^ s the total momentum of atoms i , j , and k, and is defined by - 149 -J k i j i o = £ i + £ j + £ v (6.4.21) - C t a n c * S2M+D-»-2M+D exch' t** e c o - ' - l : ' - s i o n a - ' - transfer part of the pressure tensor associated with the monomer-monomer-dimer simple exchange reaction i s defined by P C t =2M+D*2M+D,exch 2 Tr 1234 ( a(12>^12 + — } ( V 1 2 V ( 1 3 ) 2 4 ) + ( 6 R - ( 1 2 ) ( 3 4 ) + ( V(12)(34) V(13)24 f ( 13)24^12(34) P12(34)* (6.4.22) Angular Momentum l i t £ X % + *D ^-)]2M+D+2M+D,exch V *^ 22M+D->-2M+D,exch x 2 i 1 1 Tr (- — ) 1! 2! 1234 -13 -13 , , . T - ~T x ^ ( l s A l S ^ s -34 -34 , . . — — x <P(34) 6(34) )s '(13)24^13)24^2(34) - 150 -+ ^ 1S4 t 6 ( 1 3 ) £ l 3 X £ l 3 " 6 ( 3 4 ) 1 3 4 X £ 3 4 1 *a3)243(l 3)24 p12(34) 2 Tr 1234 6 1 + 6 2 £ 1 0 X ( 7 , n V , J 2 -12 " w 1 2 v 1 2 2 6 1 + 6(34) 3 -1(34) x ( V1(34) V 1 4 } 2 6 2 + 6(34) + 3 £2(34) x V2(34) ( V23 + V 2 4 ) < ?(13)24 v /T.2(34) P12(34) r ° ["at ( M M < 4 l , i n t > + MD <-D,int > ) ]2M+D*2M+D,exch (6.4.23) 21 -h Tr 1234 6.-6„ s. - 2s_ - 2s. _s„ - 2s. - 2 ^ + <V W ~ 1"^ (^23 + V ^(13)24^2(34)^2(34) - 151 -21 Tr 1234 61 + 63 - 1 ~ -1 6, •+ 6, js - s, " ( 6 ( 3 4 ) " - V"^  (^3 + V + ( 6 3 " V "V* P(13)24 : r(13)24 P12(34) - V • L C t =2M+D->-2M+D,exch 21 T~ 1 2 ^ 4 v"(13)-13 " A 3 "(34)-34 A -^34y " ( 13)24 J( 13)24"l2( 34) + 2 Tr 1234 6 1 + 6 2 L , x (V 1 0V,„) 2 -12 x 12 12 2 6 1 + 6(34) 3 -1(34) X V l ( 3 4 ) V14 26 + 6. . + 3 — ^ ^ 2 ( 3 4 ) X V2(34) ( V23 + V24> (13)24 12(34) M12(34) (6.4.24) The physial associations of the terms can be referred to the exchange reac t i o n involving a dimer and a monomer, and the flux can be obtained as before. (L - L C t ) 2M+D-»-2M+D,exch Recombination with Exchange Corresponding to th i s reaction are the equations of change for the monomers - 152 -Ut MM<$M>UM+D->-2D,exch -1 Tr 1234 [ ( W s + ( * 2 W V /(13)(24) e(13)(24A 2(34) [<4>lVs + ( * 2 6 2 ) S ] P(13)(24) J(13)(24) P12(34)' (6.4.25) and for the dimers T — M < $ >1 l 3 t D D 2M+D+2D, exch 1234 ( < t > ( 3 4 ) 6 ( 3 4 ) ) s UU3)(24) < SU3)(24)°12(34) p12(34) + [ ( * ( 1 3 ) 6 ( 1 3 ) ) s + ( < t > ( 2 4 ) 5 ( 3 4 ) ) s " ( < f ,(34) 6 ( 3 4 ) ) s ] *(13)(24) J(13)(24) P12(34) (6.4.26) This may be compared with the exchange decomposition involving two dimers. Since c o l l i s i o n a l transfer of l i n e a r momentum and angular momentum obeys the conservation laws for these a t t r i b u t e s , the conservation of mass, l i n e a r momentum and angular momentum follows. Mass U F ( M M + V ] 2M+D-»-2D,exch " " 2 W : [¥t (MD^')12M+D->-2D,exch (6.4.27) - 153 -Linear Momentum ["37 ( M M ^ + MD£ ) ]2M +D+2D,exch V * S2M+D-»-2D,exch « _ y • P C t =2M+D+2D,exch r- VV : Tr * 1234 2! r r 6 ( l 2 ) ] s + ( £(13)(24) £ ( 1 3 X 2 4 ) £(12)(34) £(12)(34) ( !(13)(24) 3(13)(24) P12(34) +4 V • Tr 1234 " 6(12) ^ lzWs + 6R (^£(13)(24)£(13)(24)^(12)(34)£(12)(34)K )  tf(13)(24):I(13)(24)P12(34) V x Tr 1234 = 6(12) £l2 X £ l 2 + 6R (£(13)(24) X£(13)(24) ~ £( 12)(34) X£( 12)(34) } '(13)(24) (13)(24) p12(34) - 154 --V • P c t =2M+D+2D, exch + -=y- VV : Tr ^ 1234 i l ^ n ^ O s A l S ^ s + £ 2 4 £ 2 4 ( £ ( 2 4 ) 6 ( 2 4 ) ) s -34-34 (- P-(34) 6(34) )s < r(13)(24) : I(13)(24) P12(34) i H r 1 3P 1 3} 6 ( 1 3 ) + { r ^ } 6 ( 2 4 ) - { r ^ } « ( 3 4 ) ] (13)(24) J(13)(24) P12(34) 1234 4- V x Tr 1234 £13 X ^ 1 3 6(13) + £ 2 4 X £ 2 4 6(24) - £ 3 4 x £ 3 4 5 ( 3 4 ) P. (13)(24)' ; I(13)(24) P12(34) (6.4.28) The c o l l i s i o n a l transfer part of the pressure tensor associated with the exchange-decomposition involving two dimer molecules i s defined by p c t =2M+D-»-2D,exch - Tr 1234 ( 6(12)-12 + •"• ) ( V 1 2 V ( 1 3 ) ( 2 4 ) } + ( 6 R - ( 1 2 ) ( 3 4 ) + - * 0 ( V(12)(34) V(13)(24) T. ^ o (13X24) 12(34)^12(34)* (6.4.29) - 155 -Angular Momentum 2M+D>2D,exch = V • (p x r) =2M+D-»-2D,exch V • Tr 1234 —13 1 1 —13 (— (yr - Tf) + ...) — x [(^+£3) fi(13)l8 . ,-24 .1 1 . J . 2^4 r / + ( — (TT " Yf) + •••) — x [(£ 2+E4> 6(24) ]s _ ,-34 1 1 £34 r / A x , , v ( ) + ...) x [ ( p + p ) 6 ] 2 1! 2! 3 4 (34) s (13)(24) J (13)(24) P 12(34) Tr 1234 6 1 + 6 2 I n x ( V 1 0 V 1 0 ) 2 -12 v 12 v12 2 6 1 + 6(34) + - £1(34) x Vl(34) V14 + —2 1341 r x V V 3 -2(34) X V2(34) V23 ' (13)(24) v / l2(34) P 12(34) 1 V 6 - * V 6 A + H " ^ I S " T ^ s + (£24X-B24 "V^s 1234 6 +6 ( £34X-%4 " V ^ s 1 ^(13)(24) 3 (13)(24) P 12(34) (6.4.30) - 156 -[ 3 t ^ l A l . i n t ^ + V - D . i n t ^ W D ^ D . e x c h * 1234 6 - 5 ±1 - 2s - 2s^ JLo ~ 2jjn ~ 2s, + ( V W ~ 1 =^^23 ^(13)(24)' /12(34) p12(34) , 6 - 6 . 6-6. 6-6 X l 3 ) ( 2 4 ) ^13)(24) P12(34) + Tr 1234 V 6 2 - 2 — ^ 1 2 X ( V 1 2 V 1 2 ) 26. + 6 , . . . 1 (34) 3 £-1(34) X 1(34) V14 2 6 2 + 6(34) 3 £2(34) X V2(34) V23 r ( 1 3 ) ( 2 4 M 2 ( 3 4 ) p 1 2 ( 3 4 ) V * =2M+D>2D,exch i f 6 l + 6 3 + V 6 4 V 6 4 Tr t — 0 — r , . x p, . + — — r_ O A x _p_„ f ^ [—2— ^13 X *13 " ~1 ^24 * -^ 24 " ~"~2 ^34 X ^34 J '(13)(24r (13)(24) P12(34) (6.4.31) - 157 -It has been the conjecture of chemical k i n e t i c i s t s that the d i s s o c i a t i o n reaction i s preceded by vigorous rotations and v i b r a t i o n s o the reactants. These i n t e r n a l motions of the dimer molecules have been i d e n t i f i e d for both reactive and non-reactive c o l l i s i o n s . These are i n f a c t a necessary feature for c o l l i s i o n s involving molecules possessing i n t e r n a l states. Whether these i n t e r n a l motions are "vigorous" or not i unimportant: i t i s the k i n e t i c energy of the c o l l i d i n g molecules that determine the a c c e s s i b i l i t y of reactions. - 158 -CHAPTER 7 THE ENERGY BALANCE EQUATION The k i n e t i c equations for P f ( i ) , the density operator representing a t y p i c a l monomer i , and P D ( j k ) , the density operator for the dimer molecule (jk) have been used to obtain equations of change for the physical observables mass, l i n e a r momentum and angular momentum. But i t i s how the en t i r e system p(N) evolves with time instead of how i n d i v i d u a l molecules change that describes the change i n the energies associated with the species because the intermolecular p o t e n t i a l renders the molecules interdependent. A. The Production and Loss of Energy Associated with the Monomers and  with the Dimers In t h i s section, the energy observables for the two species w i l l be defined. The equations showing the gain and loss aspects of the energy associated with the dimers and monomers expressed i n an analogous form to those i n Chapter 4 w i l l be obtained. From these equations, energy conservation i s established for each kind of c o l l i s i o n . 1. The Energy Operators The observables associated with the monomers and the dimers of the system have been defined i n Chapter 3: - 159 -1*± 6 = Z tf* A, 9D . < k b j k 9 ( j k ) These equations have to be modified for the energy operators to include intermolecular p o t e n t i a l s . The operator describing the energy of the monomer i n the system i s given by tf-j'a <*i*T 1 V <7-'-1> i J so the expectation value of the monomer energy i s 1 • • «n i j = N Tr ( K 1 + I Z V u ) <f p ( N ) 1. . .N 1=Z = T r K 1 p f ( l ) + N T r j J v ^ S f t P ^ W ' (7-1.2) 1 1...N i=z Having i d e n t i f i e d the observable of k i n e t i c energy and i t s expectation value (Chapter 3), the pote n t i a l energy operator and i t s expectation value can be defined. For the dimer, the operator describing i t s energy and the expected value are re s p e c t i v e l y - 160 -(7.1.3) l * j * k and (N) bl2 p(N) c o l l * 1... N i=3 (7.1.4) The k i n e t i c energy and pote n t i a l energy can be separately assigned i n the same way as the monomer. The sum of (7.1.1) and (7.1.3) indeed gives with the monomer or dimer i s adequately defined re s p e c t i v e l y by the density operators describing a t y p i c a l monomer P f ( i ) and a t y p i c a l dimer P D ( j k ) . On the other hand, the potential energy has to be defined by [ p ( N > ] c o l l . For the time being, focus i s put on the energy associated with either of the species. The energy balance equation for either species should be reduced to the equations f o r k i n e t i c and potential energy d e n s i t i e s for thi s species. The energy associated with the monomer or with the dimer has been recognised although t h i s i s not a one-molecule property. Having t h i s defined, we may l o c a l i s e i t at where the molecule i s . Such the Hamiltonian for the system. The k i n e t i c energy associated - 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 In t h i s t h e s i s and with the f i n d i n g e a r l i e r on that the p h y s i c a l a t t r i b u t e s are l o c a l i s e d at the p o s i t i o n s of the molecules 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 Is i n turn 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 throughout. 2. Energy Balance for the Monomer The l o c a l energy d e n s i t y per u n i t mass of the monomer i s defined by - t f N Tr [(^ + I I V l i ) 6^9™ p 1. .. N 1=2 ( 7 . 1 . 5 ) The time e v o l u t i o n of the energy d e n s i t y i s ther e f o r e 1. .. N 1=2 N ( 7 . 1 . 6 ) On applying the von Neumann equation i f i and using the r e l a t i o n s <P^; = E ^ ^ , <PjX c l = ^ c l ^ > « " 1 ^ = £ c l + U c l ' the energy balance equation f o r the monomers, namely - 162 -I'M 4rr 3F ( W N Tr 1...N ^ l V s ^ / f l p + ?,", ( V l l V . Z / c l £ c l p ( N > 1=2 cl + t ( V i > s + i A v uV E / c i ^ c i p ( N ) 1=Z Cl = Tr (JI 6,) JC, p_(l) ^ 1 1 s 1 i + N Tr (K.6.) I <P ,\T ,p , „ 1 l's . c l c r 1.. .N c l N (N) + J * ( V l i 6 l } s V ^ l + ^ c l ^ l > p W 1 . . . N 1=2 c l (7.1.7) i s obtained. As Tr l ^ i ^ s d ^ i P f ( 1 ^ describes the free motion change i n energy density of the monomers, the remaining terms must be due to c o l l i s i o n s : the f i r s t of these i s attributed to be the c o l l i s i o n a l contribution to the k i n e t i c energy density change, and the second one to the change i n pot e n t i a l energy. Using the same arguments as i n Chapter 2, the equations of change for k i n e t i c energy and po t e n t i a l energy d e n s i t i e s of the monomers are obtained. The change i n k i n e t i c energy density Is given by l f i l t < W " f < W s V f ( 1 ) + f < * l V s ^ 1 2 ^ 1 2 + - 163 -+ I c' (n-1)! Tr (J*. 6,) I f . i r . p ,+ 1 I s , c l c l c' 1...n c l (7.1.8) t h i s has the same structure as the equations of change for any single-molecule observables for the monomers, and the form of the equation showing the gain and loss aspects of various kinds of c o l l i s i o n a l processes i s referred to Chapter 4, putting <j)^=i^. For the pot e n t i a l energy, the potential energy density per unit mass of the monomers i s defined as Necessarily monomer 1 must be i n the process of c o l l i s i o n , otherwise a l l terms vanish. Now the sum rule derived i n Chapter 4 may be applied to derive the equation of change for monomer potential energy density i n a form that gives e x p l i c i t gain-loss terms, which i s c o l l * (7.1.9) if i IF ( M M V - H 6 V 1 1 2 2 ^12^12^2 + f12 J12> p12 V +V 12 13 2 + <P p123 123J123 1(23) 1(23) - 164 -+ Tr 6 123 V 1 2 + V 1 3 1 2 + ^ l ( 2 3 f l ( 2 3 ^ ( 3 ) + ( P123 7123 + ( P1(23) J1(23) ( P1(23) + 2 P ( 1 2 ) 3 } + Tr 6 1234 v +v +v 12 V13 14 ^ 2 ( 3 4 ) < F 1 2 ( 3 4 ) ^ 4 ) + *T.2(34) J12(34)} ( P12(34) + p(12)34 + 2 P13(24) + 2 p ( 1 3 ) 2 4 } + H <1-T ^ 1 2 ^ 2 ^ 2 + V +V 2 123 1 2 1 123 123 123 ^1(23) 1(23) 123 (12)3 (12)3 ; ^123 + Tr 123 V12 + V13 61 2 [^123^123 + t e i ( 2 3 ) < P l ( 2 3 ) ] v / l ( 2 3 ) V +V V +v + fx 12^ V23 , . V13^ V23, + ( < S2 2 + 63 2 } [ ( ?123 J123 + <*123 < P123" ll(23) 1 V +V V +v 4 . of A, 12 23 x 1 2 v13, f + 2(6 3 ^ 5 l 2 } < P(12)3 7(12)3 ^13 + 723 + 2 63 2 ^ (12)3 < f(12)3 y Li(23) 1(23) - 165 -+ Tr 6 1234 V12 V13 V14 2 ^12(34)^12(34)^(12) (34) + 4J12(34)712(34) ] + A[£13(34) (H13(24)'n'(12)(34) +6313(24)^13(24) 1 (12)(34) + Tr 1234 v +y +y v +v +v (fi 1 2 2 3 3 4 fi 12 V13 1 4 ^ „ V 3 2 1 2 ; (12)34J(12)34 9 rfi 14 24 34 12 13 14, T 1 K 4 2 " °1 2 ; (13)24 (13)24 - 6 V12 + V13 + V14 1 2 [6"(12)(34)J(12)(34) + 2 f (13)(24) T (13)(24) ] V12 + V13 + V14 I f12(34)*12(34) + 2*13(24) <l3(24) ] v.,+vn+v,. . r 12 13 34 p fp 3 2 (12)34 (12)34 + 26 V ^ 4 + V 2 4 + V 3 4 / + ^°4 2 ^(13)24 (13)24 12(34) 12(34)* (7.1.10) - 166 -For non-reactive c o l l i s i o n s , the term i n V l i 1.. .n 1=2 i s the change of the density of the potential energy associated with a t y p i c a l monomer when t h i s molecule undergoes c o l l i s i o n s . This may be compared with the change i n k i n e t i c energy density at non-reactive c o l l i s i o n s n , 1 1 s c' V c' c' 1.. .n What these two terms correspond to can be inferred by the commutation r e l a t i o n [ j 2 7 V l i 6 l ' n : ' ] - = 2 }j^U[8l>V-\+ " V ^ . L V s > ' i=2 (7.1.11) Here the symmetrised operators are defined by _ 1 (V [6 ,*±]Jg = ^  T V i « ) • £ ± V + V £. • (V ±6) " + 3 £ ± • V(V ±6) + 3(V ±6) • yp^ = I [ £ i . ( V i 6 ) V ] s , (7.1.12) and ( t V , ^ ] _ 6 ) s = {{[V,« ±]_ 6 + 6 [ V , ^ ] _ + 2 [ V , £ i • 6 £ i]_} - 167 -+ 1_ 8m 1_ 8m (V ±V) • 2±6 + 6 P i • (7 ±V) + 3 P ± • 6 ( 7 ^ ) + 3(V ±V) • 6^ (7.1.13) Add (7.1.11) to [0* 6 ) , V , ] , one gets 1 1 s c (7.1.14) Therefore, the change In energy density owing to non-reactive c o l l i s i o n s involving a monomer i s due to the interchange of k i n e t i c and potential energies of t h i s molecule as well as the s p a t i a l inhomogeneity of the gas. Due to such inhomogeneity, the potential energy density change, unlike the hydrodynamic density of l i n e a r or angular momentum, i s non-zero even i f c o l l i s i o n s are l o c a l . For reactive c o l l i s i o n s , the monomer po t e n t i a l energy density change i s i Tr 1.. .n w. + * 1 ] c' c c n - E 6 i=2 - 168 -The gain i n monomer pote n t i a l energy when the product monomers j are formed and the loss i n monomer potential energy when the reactant monomer (1) i s destroyed can be assigned pretty straightforwardly. However, the f i r s t term, which p a r a l l e l s the change i n pot e n t i a l energy density for non-reactive c o l l i s i o n s , i s associated with the product monomers whereas for single-molecule observables, the association i s with the reactants: The s i g n i f i c a n c e of t h i s c h a r a c t e r i s t i c of pot e n t i a l energy i s apparent i n Section C when the energy balance equation for reactive processes i s studied i n d e t a i l . 3 . Energy Balance for the Dimer The l o c a l energy density per unit mass of the dimer Is defined by Thus the r i g h t hand side of equation (7.1.14) becomes E = M - 1 D D Tr Z <T 1...N j<k (N) bjk (N) (7.1.15) - 169 -which can be broken into the potential energy density VD 5 M D 1 (2> , ( V l i + V 2 i ) 6 ( 1 2 A l 2 p ) i . . . N 1= j = "D1 (2> , T r N 1 * ( V l i + V2i> 6( 12)^12^011 ( 7 ' 1 ' 1 6 ) 1...N 1=3 and the k i n e t i c energy density T D. The same procedure as for the monomer can be car r i e d out to obtain the energy balance equation for the dimer which can be separated into two parts: that of the k i n e t i c energy, and that of the pote n t i a l energy h ( W - I t ( M D V + I F ( M D V - ( 7 - 1 - 1 7 ) On putting 4*^^ = ^ j + ^ k + V j k ' t* i e e c l u a t i ' - o n o r" change for M D<$ D>, the density of an a r b i t r a r y dimer property i n Chapter 4 gives the equation of change of the k i n e t i c energy density ^ (M DT D) = ft [ ( ^ + V 1 2 ) \ n ) ] s i r P h ( l 2 ) . (7.1.18) Coupled with t h i s equation i s the equation of change for the density of the p o t e n t i a l associated with the dimers h ( W - <2> , TrJ<\ ( V U W 2 i > 6 ( 1 2 ) ^ b l 2 I F ^ c o l l ' 1. . . N 1=3 (7.1.19) The changes i n the pote n t i a l energy density of the dimers for d i f f e r e n t c o l l i s i o n a l types are e x p l i c i t l y given by - 170 -3 l v n + v ? 3 i n 3t V " D V 2 ^ °(12) 2 ( 1(12)3 f r(12)3^123 + < r'(12)3 7(12)3 ) P123 Tr 123 V +V 6 13 23 f. (12) 2 < ri23 J123 + 2 f 6 V12 + V13 . V 1 3 + V 2 3  + 2 ( 6 ( 2 3 ) 2 6(12) — ' 1(23) J1(23) V 1 3 + V 2 3 + 6(12) 2 • C(12)3 < P(12)3 v A /(12)3 7 1 2 + 7 1 3 + 2 6 ( 2 3 ) 2 ^1(23)^1(23)^(12)3 (12)3 Tr 1234 - 6 V 1 3 + V 1 4 + V 2 3 + V 2 4 f (12) 2 12(34) J12(34) v 1 9 +v +V„,+V„. + offi 12 14 23 34 fi + 2 ( 6 ( 1 3 ) 2 6(12) \ l 3 ) ( 2 4 ) J(13)(24) V 1 3 + V 1 4 + V 2 3 + V 2 4 . + 2 ( 6 V 1 2 + V 1 4 + V 2 3 + V 3 4 V 1 3 + V 1 4 + V 2 3 + V 2 4 , + 2 ( 6 , , 0 , - 2 6 ( 1 2 ) - ) (13) (13)24^(13)24 (12)(34) 171 -+ Tr 1234 V 1 3 + V 1 4 + V 2 3 + V 2 4 (12) ^(12)34X12)34 + < d 2 ) ( 3 4 ) X l 2 ) ( 3 4 ) } + 26 (13) V +v +v +v 12 V14 V23 34 (^(13)24 < P(13)24 + 2 ^ 1 3 ) ( 2 4 ) u ( 1 3 ) ( 2 4 ) (12)(34) p(12)(34) + y Tr 1234 6 V13 + V14 + V23+ V24 ff °(34) 2 V (12)(34) J(12)(34) + (6 (34) V 1 3 + V 1 4 + V 2 3 + V 2 4 - 6 (12) V 1 3 + V 1 4 + V 2 3 + V 2 4 . 12(34) J 12(34) + 4(6 (13) V 1 2 + V 1 4 + V 2 3 + V 3 4 - 6 (12) V 1 3 + V 1 4 + V 2 3 + V 2 4 . X 1 3 ) 2 4 J ( 1 3 ) 2 4 + 2 ( ( 6 , _ + 6 , 0 , N ) V l 2 + V l 4 ^ V 2 3 + V 3 4 _ - 13 -'(13)^(24)' (13)(24) J(13)(24) (12) P(12)34 - 172 -+ 2 T r 1234 V 1 3 + V 1 4 + V 2 3 + V 2 4 (12) ( £ ( 1 2 ) ( 3 4 ) 6 > ( 1 2 ) ( 3 4 ) + £ ( 1 2 ) 3 4 < ? ( 1 2 ) 3 4 ) + 4 6 12 14 23 24 f p 4 °"** % 0 £ ( 1 3 ) 2 4 ° (13)24 '(13)+ 6 V 1 3 + V 1 4 + V 2 3 + V 2 4 (34) ( < f 1 2 ( 3 4 ) P 1 2 ( 3 4 ) + £ ( 1 2 ) ( 3 4 ) P ( 1 2 ) ( 3 4 ) ) + 2 ( 6 ( 1 3 ) + W V 1 2 + V 1 4 + V 2 3 + V 2 4 3 ) ( 2 4 ) ^ ( 1 3 ) ( 2 4 ) (12)34 p ( 1 2 ) 3 4 ' B. Non-Reactive Collisions (7.1.20) The change i n energy d e n s i t y 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 s , as shown e a r l i e r due to the t r a n s f e r o f energy between the m o l e c u l e s p l u s the p o t e n t i a l energy f l u x o f the m o l e c u l e s which i s not c o l l i s i o n a l t r a n s f e r i n o r i g i n . The e q u a t i o n s f o r k i n e t i c energy and p o t e n t i a l energy w i l l be - 173 -dealt with separately and one finds that the net gain of one is compensated by the loss of the other. (i) Monomer-Monomer Collisions The contribution from this type of coll i s i o n to the energy balance equation is as follows: For the kinetic energy density, we have [h W 2 M = -k H ^ i V s ^ 1 2 ^ 1 2 2* 1 2 6 *T* 6 u 1 2 ; 2 Js 12*12 12 '12 = - V P • u =2M -r P - Tr ( = i i + ...) [(=-12 2 2 m ( 7 1 2 V 1 2 ) 12 12^12 2ra Tr 12 6+6 ^ 1 2 - V ^ B ' <V12V12> 6+6 P 1 2 ^ 2 P 1 2 ' ( 7 - 2 ' 1 ) - 174 -P^^ • ju i s the contribution to the energy flux due to the mechanical work done to the system. The term - Tr ( - i 2 - + ...) [ ( - - u) 6 ( 1 2 ) ] s • ( V 1 2 V 1 2 ) 1 2 1 2 p 1 2 , P^  being the t o t a l momentum of the c o l l i d i n g molecules, i s the conductive contribution to the heat flux a r i s i n g from the intermolecular force between the two monomers. For p o t e n t i a l energy,we have r — M V 1 L 3 t M M 2M ^ ^ ^ 1 2 ^ 1 2 = - v [ MM VM^2M + T r [ ( ^ - - u ) 5 1 V . 9 ] o ( P 1 9 A 9 P j 2 m — 1 12 s 12 12 12 + y - Tr 2m 1 2 ( £ 1 2 V V 2 ; s ( V 1 2 V 1 2 ) + ( V12 V12> ' (*12 V V 2 ; s 12 12^12 (7.2.2) where [p.6.V..1 i s defined by — i i i j s - 175 -[ £ i 6 i v i j ] s =~ ? ^ i V i j + W l j + V i ^ i 6 i + 3vu\Zi]> ( 7 - 2 - 3 ) and [M^M^^M A N D T r i 2 ^Z^m ~ ^  6 i V12^ s *^  12<f>12P12 a r e r e s P e c t i v e l y the convective and conductive parts of the potential energy flux due to monomer-monomer c o l l i s i o n s . The l a s t term on the ri g h t side of equation (7.2.2) i s equal and opposite to the l a s t term of equation (7.2.1). Thus the k i n e t i c energy flux arises from c o l l i s i o n a l transfer of energy ( i . e . , between the c o l l i s i o n partners) whereas the pot e n t i a l energy flux does not involve any intermolecular energy t r a n s f e r . The interchange of k i n e t i c energy and pote n t i a l energy associated with ei t h e r molecule i s due to the coupling between the r e l a t i v e momentum between the c o l l i s i o n partners and the force between them. It i s nonzero even when the c o l l i s i o n partners occupy the same l o c a l i t y because the c o l l i s i o n density operator i s not diagonal i n £12- Summing (7.2.1) and (7.2.2), the rate of change i n energy density due to monomer-monomer c o l l i s i o n s can be expressed i n terms of a flux [h ¥ M ] 2 M • "V • t =2M * JI + T MM VM^ ]2M + W ( 7 ' 2 ' 4 ) where _q^ M * s t n e part of the conductive heat flux which i s associated with the c o l l i s i o n s involving two monomers, and i s given by e i 2^ 2P 1 2. 2M Tr 12 m — 1 12 s ( V12 V12> (7.2.5) - 176 -It i s s a t i s f y i n g that for monomer-monomer c o l l i s i o n s , and in f a c t , non-reactive c o l l i s i o n s , the convective potential energy flux i s i d e n t i f i e d so that we can i d e n t i f y the complete convective energy flux (when no reactions take place) and e x p l i c i t l y and formally connect the k i n e t i c energy contribution to free motion of the molecules and the pote n t i a l to c o l l i s i o n s . ( i i ) Monomer-Dimer C o l l i s i o n s The equations of change for k i n e t i c and potential energy d e n s i t i e s associated with t h i s c o l l i s i o n are r e s p e c t i v e l y ["3T ( MM TM + MDTD)]M+D,no rxt " - i f23 I C V l > s + (^(23) 6 ( 2 3 ) ) s ] V / l ( 2 3 ) ^ l ( 2 3 f t ( 2 3 ) P l ( 2 3 ) y- Tr 123 ^1(23) 5 l ( 2 3 ) r i i ( 2 3 ) p l ( 2 3 ) V ' 1 2 * ( £ l ( 2 3 ) + — > - (P/3m 6 R ) s • ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) + ^t%3 ' ( V 2 3 V 1 ( 2 3 ) ) + ( V 2 3 V 1 ( 2 3 ) ) '^23 ^ ( 2 3 ) n i ( 2 3 ) P l ( 2 3 ) - 177 -1 32m [ ( ^ 1 ( 2 3 ) " - ) 6 R £ l ( 2 3 ) + 3 ^ l ( 2 3 ) ( ^ l ( 2 3 ) - " ) V t ] ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) + ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) * [ 3 ( ( r 1 ( 2 3 ) . . . ) 6 R 2^2^)* + P i ( 2 3 ) ( ^ l ( 2 3 ) - - - ) 6 R 1 fl(23A(23) Pl(23) i 61 + *OV **[<*1 ^ 2 3 ) > 2 ( \ * 1 ( 2 3 ) « 123 1(23)M(23) , / H(23) P1(23) (P • v M V=M+D, no rxt - ; £ ^ l ( 2 3 ) - ) I ( 3 l T - ^ > V s ' ( V 1 ( 2 3 ) V 1 ( 2 3 ) : Tr 123 ^1(23)^1(23)^(23) ( - l ( 2 3 ) * , , ; ) 6-1(23) X ( V 1 ( 2 3 ) V 1 ( 2 3 ) ) v/l/ 1(23) 1(23)^1(23) + ^ ( ^ l ( 2 3 ) " - ) — ^ 2 3 X ( V 2 3 V l ( 2 3 ) ) * ' l ( 2 3 f i ( 2 3 ) p l ( 2 3 ) - 178 -• Tr 123 ( - l ( 2 3 ) " ° 6 R £23 £-23 , 9_ ra ' r 2 3 U r ^ 1(23) ) + ( IT X p "23 -23 _ y 2 ) . x V 2 3 V 1 ( 2 3 ) ) mr 23 < - l ( 2 3 ) + * * * ) ~ 3 £l(23) . 3 v 4m£l(23) * r 1 ( 2 3 ) l 3 r 1 ( 2 3 ) V l ( 2 3 ) ; + ( 3 £ 1 (23) X -El(23) -VD> 4mr 1(23) ' £l(23) X ( V1(23) V l ( 2 3 ) ) —I s 1(23) 1(23X1(23) I * 1<V*(23)> ^ F 1 1 1 ^ ^ 1 ( 2 3 ) ^ 1 ( 2 3 ) ^ ( 2 3 ) ^ ( 2 3 ) ( 7 ' 2 ' 6 * £ 3 - l - ( 2 3 ) ( MM VM + MD VD ) 3M +D,no rxt i V 1 2 + V 1 3 ^ 1 2 3 ( 6 l + 6< 2 3> ) _ 2 * l ( 2 3 ) f l ( 2 3 ) * l ( 2 3 ) P l ( 2 3 ) - 179 -= _ V • T r y ! ! 6 l + ^ g i 6 ( 2 3 ) ) ^ l ] g P 1 ( 2 3 ) ^ ( 2 3 ) P 1 ( 2 3 ) k f 2 3[ ? V U 2 3 ) ' ( ( V 6 ( 2 3 ) ) ^ l ( 2 3 ) ) s ] - ^ 1 ( 2 3 ) ^ 1 ( 2 3 ) ^ ( 2 3 ) = - V + Tr 123 t ^ " ^ 61 2 J s , ,,£(23) , , + K - S T - " ^ 6(23) V +v 12 13n 1(23) 1(23) P1(23) i V 6 ( 2 3 ) " * 1 2 3 t V l ( 2 3 ) > ( 2 ^ K 2 3 ) ) s 1 - < f 1 (23)^(23) P l (23)' (7.2.7) Here the symmetrised operator (£,( i j )) "5(ij )V) s i s given by 1 ( * ( i j ) 6 ( i j ) V ) s = 8 3*(ij)W + W(ij)V  + V ^ < i j ) 6 ( i j ) + 3 V 6 ( i j A i j ) (7.2.8) We define the symmetrised k i n e t i c energy operators r e s p e c t i v e l y for 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 ^ + . f p J ^ . p r ' + 3 l ' ( ¥ J ? ' l + 3 p r (7.2.9) - 180 -and r x p r- x (4-.. V)l — or s 1 T r x p r" x V) + r" x (|p. V) + 3 r« r" x (|p. V) r_ x _p jc x _p — + 3 — As well to •=M-D ± x _p r" x (-|p. V) r' (7.2.10) S % r i ( 2 3 ) V f ( 1 ) p b ( 2 3 ) ] _ 1 H3 £-1(23) X *1(23> W 1 ) p b < 2 3 > (7.2.11) i s 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 v e l o c i t y u + 2v (7.2.12) i s the average value of the stream v e l o c i t i e s of the monomer and the dimer associated with monomer-dimer c o l l i s i o n s . The term ( g M + D n Q r x t • _v') i s the contribution to the energy flux associated with the pressure tensor for t h i s c o l l i s i o n type. The conductive energy f l u x carried along by L „ . _ i s i d e n t i f i e d by noting that =M+D,no rxt J & - 181 -a x (b x c) • d = a • (b x c) x d = a • (b c) • d - (a • b) c • d, (7.2.13) and by comparing (7.2.6) with (5.2.10-11). The k i n e t i c energy flux contributions from v i b r a t i o n a l motion (the term with the r a d i a l components p ^ • r ^ 3 / 3 ^ V 1 ( 2 3 ) + ( 3 / 3 ^ V 1 ( 2 3 ) ) • p ^ ) , c f . equation (6.1.11), does not contribute to the heat flux due to bulk motion of the gas. As for dimer r o t a t i o n , c l e a r l y i t i s conducive to the energy flux associated with the angular momentum flux L, which i n turn i s associated not only with _u>, but also with jo^ D > the r e l a t i v e angular v e l o c i t y between a monomer and a dimer. As a monomer i s involved i n the c o l l i s i o n , the ro t a t i o n of t h i s monomer i s n a t u r a l l y included i n the L • kind of k i n e t i c energy f l u x term. Combining equations (7.2.6) and (7.2.7), one obtains the energy conservation r e l a t i o n for t h i s c o l l i s i o n [1TF ^ M + W W n o rxt = - V (P • v') + a v=M+D,no rxt — ' -^ M+D,no rxt + (M V u + M V v) MM - D D -'M+D.no rxt 3w + to - -M-D no rxt (7.2.14) Here q„, ^  i s the conductive heat flux associated with t h i s •^ M+D no rxt c o l l i s i o n . - 182 -( i i i ) C o l l i s i o n s Involving 3 Monomers F i r s t look at the change i n the k i n e t i c energy density [ T t MMTM]3M+3M 6fi Tr 123 V 6 3 + 3 [ ( 2 ^ - ^ ) ) ( 6 1 - ^ ) ] s + j t o v v ^ v ( 6 i + 62 + 6 3 ) ] i v123 123 123p123 V • (P • u) =3M+3M -- V Tr 123 - (£i(23)**') [L/3m ~ 6 p l R Js r r - (JT^-:fL ' V...)[(P/3m- u) 6 ( 2 3 ) ] s , ( V 1 ( 2 3 ) V 1 2 3 ) ' " ( £ 2 3 - " ) K ^ r n ^ - ^ Ws ' < 7 2 3 V 1 2 3 ) A 1 r P23 _,_ , P23 N t , . + 4 [ £ 2 3 — + ( — £ 2 3 ) 1 6(23) ( V1(23) V 1 2 3 ) I —23 —23 l ( - r - l ( 2 3 ) * , * ) 6 R + ( ~ ~ — ' v - - - ) R v2! 2 2 (23) J'3 ^23 2\ LL23)_ . / V V N _ / l 4m ^ V1(23) V123 ; m 123 123 123 ( V 2 3 V 1 2 3 ) 1 —is - 183 -" m f23 [ — <«i + 62 + V J s Wl23*f23 p123 - V ' %M*3M • »> + j V • Tr 123 (^12 ••' ) ^ 2 m - - « > ' ( V 1 2 V 1 2 ) + ( r 1 3 ...) 0 V-P-3 + ( r 2 3 ...) (. 2m - £ > ' <V13V13> 2 3 2 m " «> * (v 2 3 v 2 3 ) f123^i23 p123 ^ 1 2 3 6 1 + 6 9 61 + 6 3 + [ ( ^ + ^ 3 ) - J - 2 _ 2 ] s ^ + [ C V 2 + * 3 ) - ^ T - ^ ] s i r 2 3 123 123P123' (7.2.15) The symmetrised operators i n the second equality i s defined i n equation (7.1.13). As for the change i n the pote n t i a l energy density, i t i s given by 3M+3M 123 1 Z n 23 123 123P123 - 184 -V +V V . Tr ( 6 l P l 1 2 1 3 ) s> o , „ 1-1 2 ; s 123 123P123 i V 1 2 + V 1 3 1 5 IU [~~2~~' ( yi23 6l> s J-^123^ i 2 3 p123 = - V ( MM VM ^ 3M+3M + Tr [(-I - U) & - 1 2 1 3] *> ^ p 123 m — 1 2 8 123 123 pl 123 123^123 6fi Tr 123 [ V 1 2 ' ( ^ 1 +^2 ) - V i > s l -+ [ V13' ( (*1 + ^ 3 ) - T ^ > s L s -123 123p123 (7.2.16) As before, by introducing the heat flux q3M>3M» the energy balance r e l a t i o n can be expressed as [ii (MMEM)13M-3M (7.2.17) - 185 -(i v ) C o l l i s i o n s Involving 2 Dimers Except for the change associated with the i n t e r n a l energy f l u x , the equation of change for the k i n e t i c energy density due to t h i s kind of c o l l i s i o n has the same form as that for monomer-monomer c o l l i s i o n s . In both cases, two equivalent molecules are involved, but now, the molecules possess molecular energy which gives r i s e to an i n t e r n a l energy f l u x . Using the same method as i n ( i i ) , the energy flux associated with the molecular angular momentum flux can also be i d e n t i f i e d . ^9t MDTD^2D,no rxt = -7 (P • v ) =2D,no rxt — Tr 1234 A l 2 ) ( 3 4 ) , , V s • • •) [ ( 4 ^ " ^ Vs* ( V(12)(34) V(12)(34) 1 -El 7 + T V i T • ( V 1 2 V ( 1 2 ) ( 3 4 ) ) ) -12, 12 v(12)(34) / m ~l6Jlr' ( V 3 4 V ( 1 2 ) ( 3 4 ) ) £34x + ( V 3 4 V ( 1 2 ) ( 3 4 ) ) * I T > ^(12)(34) s / Ltl2)(34) p(12)(34) - 186 -1 Zh V'(12)(34) < P(12)(34) v / 1(12)(34) P(12)(34) For po t e n t i a l energy, we have (7.2.18) no rxt V 1 3 + V 1 4 + V 2 3 + V 2 4 ] s P(12)(34)S JYl2)(34)P(12)(34) i T r r V 1 3 + V 1 4 + V 2 3 + V 2 4 „ ^ tTjf.t 2 > ( 6 ( 1 2 A l 2)(34) )s ]-1234 (12)(34) (12)(34) M(12)(34) = - V ( M D V D ^ 2 D , no rxt + Tr 1234 r A i 2 ) . . V 1 3 + V 1 4 + V 2 3 + V 2 4 . (12)(34) v / 1'(12)(34) p(12)(34) Tr [V r 6 ( 1 2 ) + 6 ( 3 4 ) ,,0 2n 1 2 3 4 L V ( 1 2 ) ( 3 4 ) > 1 2 3 ( 1 2 ) ( 3 4 ) ] s ] -(P (12)(34)^(12)(34) P(12)(34)' (7.2.19) - 187 -Again, t o t a l energy i s conserved, and the energy flux can be broken into a part due to P-V work, a convective (potential) energy f l u x , a conductive energy f l u x , and a flux carried along by the in t e r n a l angular momentum fl u x L__ =2D, no rxt l 7 t ¥ D ] 2 D , no rxt = - V (P • v v=2D,no rxt — + ( MD VD^2D,no rxt + =2D,no rxt * — + -^2D,no rxt (7.2.20) (v) C o l l i s i o n s Involving 2 Monomers and 1 Dimer The change i n the k i n e t i c energy density for t h i s c o l l i s i o n i s ["§T ( MM TM + MD TD ) ]2M+D,no r x t V • -5- Tr 1 1234 ( - 1 2 , , , ) ( _ 2 m ~ 6 ( 1 2 ) / s • v '12*12' , w £ ( 1 3 4 ) . ^ - l ( 3 4 ) * * * n 3m °(134)'s 1'1(34) v" 13 '' 14' •±-(131*} ~ ^2(34)' " ^ - S i r 2 - 6 ( 2 3 4 ) ) s * t V 2 ( 3 4 ) ( V 2 3 + V 2 4 ) ] 12(34) 12(34) M12(34) - 188 -+V« i Tr J 1234 K£ 1 ( 3 4 ) - - - ) 6 ( i 3 4 ) r i 3 £ K34), n l2 4m l ( 3 4 ) v v 1 3 v14 ; >(v,,+v1A) - ^ 3^34 2 m V 3 4 ( V 1 3 + V 1 4 ) ] — i s ( £ 2 ( 3 4 ) * , , : ) 6(234) 1 3-P-2(34) [2" 4m" V 2 ( 3 4 ) v ' 2 3 , , 2 4 / 2 m '34 v ,23' v24 3 £34 ^12(34) V l12(34) P12(34) —Is 2fi Tr 1234 6+6 6+6 12(34) 12(34)^12(34). (7.2.21) The second block of the flux terms gives the energy f l u x associated with i2M+D no r x t ' a n c* ^ introducing v", the average stream v e l o c i t y associated with t h i s c o l l i s i o n given by v" = I (u + v) = { (u + 3v'), (7.2.22) the energy flux associated with P o w l T k can also be i d e n t i f i e d . =2M+D, no rxt - 189 -The net change In k i n e t i c energy i s compensated by the net change i n po t e n t i a l energy whose density changes according to [ 9 t ( MM VM + MDVD^2M+D,no rxt = - V ( V M u + M DV D v ) 2 M + D j n o r x t •P-l V 1 2 + V 1 3 + V 1 4 + Tr K ^ - u ) \ 2 18*12<34)<12<34>P12<34) 1234 4 . 1 T T - r/-(34) x - 13 14 23 24. + 2 ^ 4 [ ( _ 2 m - " ^  6(34) 2 ] s tf>12(34)"l12(34)P12(34) Zh" Tr 1234 «1 + 5 2 [ V 1 2 , ( 2 ^ i 2 ( 3 4 ) ) s 1 -+ t V13 + V14« ^ ^ 1 2 ( 3 4 ) ^ -+ t V 2 3 + V 2 4 ' ( - M l i l K 1 2 ( 3 4 ) ) s l -f12(34)v/Y2(34)p12(34) = - V (=2M+D,no rxt^ " — + -^M+D.no rxt + =2M+D,no rxt 4 + (M V u + MV v)„ V V M - D D -2Mf-D.no rxt [TE ( M M T 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 rxt * -"M ± + M D £ + P • v' + P • v" = P • — — =M+D,no rxt — =2M+D,no rxt — =no rxt + (P + P + — P + — P ) M =2M =3M,no rxt 3 =M+D,no rxt 2 =2M+D,no r x t 7 D ? 1 - CP + — P + — P ) M v=2D,no rxt 3 =M+D,no rxt 2 =2M+D,no r x t y (u - v) "M* V the k i n e t i c energy flux carried along by the pressure tensor due to non-reactive c o l l i s i o n s can be i d e n t i f i e d . As they are two d i f f e r e n t species i n the system, there i s an add i t i o n a l contribution to the heat flux which i s connected to the difference i n the v e l o c i t i e s of the two species. Likewise, owing to the presence of monomers, the energy flux c a r r i e d along by the molecular angular momentum flux ( i n the non-reacting case) i s not associated with u>, the angular v e l o c i t y of the dimer. Obviously, when dimers are the only species i n the system, and when no reactions take place, the k i n e t i c energy flux associated with molecular angular momentum flux i s L _ . 4 . , w = L , c o . ° =no rxt — = — C. Reactive C o l l i s i o n s For the hydrodynamic densities of mass, l i n e a r momentum and angular momentum, the contributions to the equations of change from c o l l i s i o n a l transfer are comparable to those for non-reactive processes. However, - 191 -the c o l l i s i o n a l transfer parts of the energy balance equations for reactive and nonreactive c o l l i s i o n s are d i f f e r e n t from each other, as the interchange of k i n e t i c and potential energies give r i s e to a flux term only i n the former case. This flux arises by way of the differe n c e i n l o c a l i t i e s of the products and reactants. ( i ) C o l l i s i o n s Involving a Monomer and a Dimer F i r s t , consider the decomposition reacti o n . From equations (6.1.1) and (6.1.2), we have, for k i n e t i c energy density, the rate of change [ i ( MM TM + V D ) ]M +D+3M = - V P C t =M+D +3M * ? + ki-ct 3 ^ + ^ D [+D+3M + -SM+D+3M ,ct, + <E - S W 3 M ' £ ~ * 1 2 3 [ ( - 2 3 + * " ) 6(23) ( f a # - - i i ) • i 2 3 1 a * , 1 2 3 : r i 2 3 P l ( 2 3 ) + T - Tr 7 1 123 1 -^ 23 ^ 23 ("2T ~~1 T~ + •••)t(£2+-p-3^ 5(23)^ 123 J123 P1(23) (V u ) r - VV : " ( T T = 2 1 4 3 - + 6(23) 2 m ^123^123^(23) T u 1 ..2 123 - 192 -VV : Tr 123 1 £ 2 3 £ 2 3 . £ 2 + P 3 - 2 m £ . , . n  { 2 \ — — ) [ 4 m - ( £ 2 + £3 2mu) 6 ( 2 3 ) ] g k^ * 123 . r , l -23 £23 v £23 + K 2 T ~1 2~ ' " } ~ ^ 3 P ; x 123 123 1(23) V 6 ( 2 3 ) [ 2 K 1 2 3 ] s ^123^123^1(23) + V23 6(23)^123 7123 £23 ]s 6(23) 1(23) (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) (7.3.2) has been used. The structure of J^J+D^JJ * s referred to (7.2.6). The symmetrised operators are defined by t ( £ ± j . . . ) ^ + 2,) ' £ 1 : j 5 s = 7 t ( r ± j . . . ) p ±. + (jByUij...))'] • ( 6 ( l j ) (£ ± + P j ) ) s , (7.3.3) and =- i l £ ± j £ ± j ^ * £ ± j + £ ± j ' £ £ j £ ± j + 2 ( 2 i j £ ± j ) t . ( r^ P ^ ) ^ (7.3.4) - 193 -The c o l l i s i o n a l transfer contribution to the energy flux has the same structure as for non-reactive monomer-dimer c o l l i s i o n s except for the channel potential and projection superoperators. As with the non-reacting case, t h i s flux arises from the r a d i a l and r o t a t i o n a l motions caused by the forces and torques between the c o l l i s i o n partners (molecules i n the i n i t i a l channel). Because of the association with the c o l l i s i o n partners, that part of the energy flux which i s related to M^+D^ 3M a n d 2M+D-3M m U S t b e a s s o c i a t e d (3j£ + J V D ) / 4 A N D - ' T H E 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 . The term VV * 123 2 f ~ ~ ( 2 3 ) 123 J123 P1(23) 2 U (7.3.5) i s the rate of change i n k i n e t i c energy density r e s u l t i n g from the change in mass dispersion due to th i s reaction (M+D+3M). The flux 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 of the product monomers are coupled together •£(23) n ^3-23 6(23) 2m * -&23]s P 123^23 P1(23) Tr 123 1*23 ' (£(23) 6 ( 2 3 ) ) s 2 (£23 X-H23 ) X (- P-(23) 6(23) )s ^123^123 P1(23: (7.3.6) - 194 -Is associated with the transformation of i n t e r n a l motions of the reactant dimer into 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. This contributes to the energy flux connected with the pressure tensor for t h i s r e a c t i o n : compare this with equation (6.1.9). Besides such transformation of energy, the heat flux also contains the term associated with the transfer of t o t a l momentum from the reactant to the products i _ r J _ z 2 3 M = l + - 3 " * 123 2 5 2 2 4m <£ 2 + P 3 2 m ^ Ws f123~123 Pl(23)* With regard to the flux term associated with -(V , i t i s connected with the v i s c o s i t y of the monomers and to the monomer angular v e l o c i t y 1/2 V x _u. When only the second order term i n rj>3 i s kept, on comparing with equation (6.1.17), t h i s energy flux i s found to be related to the t r a n s l a t i o n a l angular momentum flux due to the coupling of the r e l a t i v e p o s i t i o n between the product monomers with t h e i r t o t a l l i n e a r momentum r r I H3 TT^f-^-* (^(23) 6 ( 2 3 ) ) s ^ l 2 3 J 1 2 3 P l ( 2 3 ) * ( _ T> V X Tr 123 I JE.23 JI-23 —23 4 ~T~ [~2~ (- P-(23) 6(23) )s + (-£(23) 6(23) )s ~T~ ] ^123~123 P1(23) -(Vu + (Vu) ) The energy flux carried along by the angular momentum flux due to the transfer of the molecular angular momentum of the reactant dimer to the - 195 -r e l a t i v e angular momentum between the product monomers i s also 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 _g_ £L 2 2 " £ • £ " £ 2~ * £ ' (7.3.7) q q q _g_ being the po s i t i o n operator. In Chapter 6, (p-j j • S i j ) s has been connected with ± 2 pure v i b r a t i o n a l t r a n s i t i o n s for the dimer ( i j ) . From 2 equation (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^ /m can be broken into the v i b r a t i o n a l k i n e t i c energy JJij * < m ^ ) _ 1 £ij £ij ' £i i and r o t a t i o n a l energy £ ± j x j ^ j • ^  x £ i./m r 2 . . The decomposition of j> • j> into 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 before . However, the structure of 2 the r a d i a l motion term p does not resemble what i s obtained here and q no physical association of i t has been given. In point of f a c t , the r e l a t i o n (7.3.7) can be substituted into equation (7.2.6) to obtain the contributions to the energy flux due to r o t a t i o n a l and v i b r a t i o n a l e x c i t a t i o n s i n the same manner as i t i s applied to I I 1-22 —23 —23 " h V V : i 2 3 [ < 2 T ~2~ ~ ' " ^ ~ ' £ 2 3 ] s 6(23) 123 123 P1(23) - 196 -kVV: Tr 123 , J _ ^23 ^23 . . 4.' 2 2 (23) £ 2 3 X -P-23 1 . , m 2 . ( — ~ 2 ^ * ( £ 2 3 X £23 _ 2 r23 ^ m r 23 r r £23 £23 + £23 1 " * £23 m r 2 3 123 J123 P1(23) —I s T f i - Z i i l i 3 - > fi 2 m 2, J23 2 1 2 2 * <23> r 2 3 1 2 3 I 2 3 1(23) 2 ' + j Tr 123 £23 X £23/ , J _ £23 £23 V 2! 2 2 ***' .1 £23 £23 + ( 2, 2 2 £23 X £23 6(23) < S'l23 : ,123 pl(23) (7.3.8) so that one can i d e n t i f y the contribution to the energy flux from the transformation of i n t e r n a l , i . e . , r o t a t i o n a l and v i b r a t i o n a l energy of the dimer into the energy due to the r e l a t i v e motion between the product monomers. Whereas the v i b r a t i o n a l term shows up only i n the heat f l u x , the r o t a t i o n a l part, by associating i t with _w, gives r i s e also to a term of which structure i s analogous to the dispersion of <D co /2 as well as a term where the molecular angular momentum of the reactant dimer i s dotted into OJ. As one i s dealing with the transformation of i n t e r n a l and t r a n s l a t i o n a l states, the association with u) i s i n fact a r b i t r a r y . One - 197 -may introduce u^, such that the energy flux due to L i s L • , which may be an appropriate choice i n this case. As for the part that gives a net change i n k i n e t i c energy, the term •fi ^ 3 6< 2 3> V23 ^ 123^123^(23) corresponds to the loss i n i n t e r n a l energy of the dimer (23) when i t i s broken up to give two monomers. This i s transformed into the po t e n t i a l energy between the products. This gain term i n the po t e n t i a l energy density associated with the monomers i s e x p l i c i t l y shown i n the equation *~3t VM'M+D+3M 123 V +V V +V V +v ( K 12 13 . * 12 23 . V 1 3 + V 2 3 N p p j v ( 6 1 2 + 62 2 + 63 2 ^123^123^(23) V +V V +v 23 1(23) (7.3.9) For the dimers, the rate of change i n the po t e n t i a l energy density i s [.£_ M V 1 L 8 t D D M+D+3M £ - 12 13^  A? = ^ 123 ( 2 3 ) 2 123 J123 P1(23)* (7.3.10) These give - 198 -- V ( MM VM-^M+D+3M Tr 123 P i V 1 2 + V 1 3 -P-2 [(-£ - u) 6 1 2 1 3 ] + 2 [ ( - | m — 1 L s m 123 1(23) P1(23) ^ 62 1"]s 123 -23 12 n r 9 . r n - l n - l + nT<-T> ' 7 > 5(23) V13 123 123 pl(23) * 123 6+6 6+6 6 „ + 6 >- + ^23 (-V *123>8> '12 3VH(23) + «2+63 V <P T 23 123 J123 P l ( 2 3 ) * (7.3.11) As expected, the displacement of energy from the reactants to the products r e s u l t s i n a pote n t i a l energy flux (which i s part of the heat f l u x ) . For c o l l i s i o n a l transfer the pote n t i a l i s associated with the products whereas the molecular a t t r i b u t e s l i k e mass, l i n e a r and angular momenta, and k i n e t i c energy are associated with the reactants. This i s - 199 -not s u r p r i s i n g r e a l l y . For non-reactive processes, the interchange of k i n e t i c and pot e n t i a l energies has a contributing part which i s not due to n o n l o c a l i t y of the c o l l i s i o n . When the c o l l i s i o n s are reac t i v e , t h i s kind of interchange provides the energy for as well as dissipates the energy generated during the reaction. So the energy terms cannot be only reactant energies. Evidently, conservation of energy i s established for t h i s reaction [ 1 F ( V M + W W 3 M « _ v 3(0 + 4l-D n c t , . T ct =M+D-»-3M - %+D+3M + '<£ - 2 C t W 3 M * i i + (= " ^  W 3 M * i2 + % V M ii}M+D>3M + -3M+D^3M + * £3U~T--T- [ (^2 + *3> 6 ( 2 3 ) ] s n 2 3 J l 2 3 P l ( 2 3 ) : ( V i i ) 1 + i l l ? ^23 ^23 ^23 X ^23 6(23) <?123 3123 pl(23) : ( V - } t + l l n " ^  V^23 X ^ 2 3 r - 2 3 I 2 3 > / 6(23) ^ 123 ^ 23 P l ( 2 3 ) : ( V - } ~k W : V z T T 1 ^ [f r23 «2/2 + V 2 3 6 ( 2 3 ) K 123^123^(23) 123 (7.3.12) - 200 -The flux that i s associated with V _OJ and (V ut)t can be reduced to a term which i s connected with the c u r l of the dimer angular v e l o c i t y (the coupling between the rotation of the reactant dimer and the rot a t i o n of the gas) and another term connected with the symmetric traceless part of the tensor V u) (energy flux due to resistance to rotation of the dimer) 16h Tr 123 r 2 3 r 2 3 x ( r 2 3 x £ 2 3 ) " \<r 2 3 XJ> 2 3) x r 2 3 /• r 2 3 6 ( 2 3 ) r i 2 3 ^ 2 3 P l ( 2 3 ) * 2~ V X -^Tr [ r 2 3 ( r 2 3 x £ 2 3 ) ( 2 ) + ^ x £ 2 3 r 2 3 ) ( 2 ) 6(23) t f123' 7123 pl(23) V oi + (V w) ( 2 j V • V u U) . In other words, t h i s i s the angular momentum analogue of the energy f l u x a r i s i n g 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 which i s i n turn connected with the l i n e a r momentum of the molecules. As a monomer i s s p h e r i c a l , there i s no r o t a t i o n a l term analogous to the terms above which i s associated with this species. If we use V IUQ and ( V a ) Q ) t instead, the same argument goes although now one i s concerned with the resistance to the rotation between the product monomers. In addition, there i s a term which corresponds to the dispersion of the po t e n t i a l energy between the atoms with labels 2 and 3. This term and the l a s t term of equation (7.3.12) constitute the rate of change i n energy density for this reaction r e s u l t i n g from the bulk flow of the gas. - 201 -For the exchange reaction, to express the energy flux due to mechanical work. In terms of _v' , the v e l o c i t y associated with t h i s event both before and after c o l l i s i o n , we note that 3 ( £ i + JLj +-P-k) + £ d j ) k = 3 <£i + £j + P k - 3m v') + p ^ ^ + 2m v« = ( £ i + £. - 2m _v') + 2m v' , therefore the rate of change i n k i n e t i c energy density i s {1£ ( MM TM + W Wexch (7.3.13) , ^ O t 3 03 + P • v + L. • =M+D,exch — =M+D,exch 4 ct + ( i ~ h ^M+D.exch * - + -^M+D.exch * 123 [ ( r 1 2 . . . ) 5 ( 1 2 ) v — g j j - - ^ 12 Js -H23) " K £ 2 3 * " ) 6(23) (^m— " - ^ * £ 2 3 ] s <P 7 P (12)3 J(12)3 1(23) - 202 -2i Tr 123 , 1 £l2 -12 x r . , ("2T ^ ~ - 2 ~ l- P-(12) 6(12) Js 1 —23 —23 - (T r- • • •) f_P £(23) 6(23) ]s P "7 (12)3 (12)3 P1(23) Tr 123 £l2 £-12 ! l 2 X *12 6(12) ~ £.23 £23 £23 x -^ 23 6(23) ^(12)3 : f(12)3 Pl(23) :(V to)* Tr 123 (V v ' ) C : (V oi) P(12)3 : I(12)3 P1(23) - VV : { [ ^ M D ^ J M + D ) e x c h + ...} + |*VV Tr 123 j _ ^12 -12 2! 2 T~ 6(12) r12 ,1 -12 -12 x , 2 " kU~2 2" (12) r12 ? 7 P (12)3^(12)3 1(23) m 2 / n "2 w /2 - 203 -+ 2 i v v . 123 ,1 -12 -12 s . l2T~2~~2~ °(12) £l2 x -El2 1 m _2 ( - 7 jo) • ( r 1 2 x p 1 2 - T r 1 2 <£) m r + Pl2 12 -12 -12 m r 2 *£l2 12 ,1 -23 -^ 23 . . (~2T ~2 2~ ' " ' °(23) (-23 X £23 _ 1 £ ) . ( £ z 3 X £ 2 3 m r m 2 N 2 r23 •£> 23 £23 £ 2 3 + £ 2 3 * m '1 ' £23 m r23 —I s f(12)3 : i(12)3 Pl(23) 2 i Tr 123 ( V12 6(12) " V 2 3 6 ( 2 3 ) ) ^(12)3^(12)3 6 l + 6 ( 2 3 ) + [^(12)3 2 ] s ^ ( 1 2 ) 3 f ( 1 2 ) 3 V / ) l ( 2 3 ) 1(23) (7.3.14) - 204 -The term Tr 123 [ — ' £ 1 2 x £ l 2 ( 6 ( 1 2 ) - — j - > 1 61 + V m r 12 - i -23 X -P-23 m r 2 * £23 X - P-23 ( 6(23) 6 + 6 2 3, 2 J s 23 (12)3 (12)3 P1(23) represent the gain and loss i n r o t a t i o n a l energy as the product dimer i s formed and when the reactant dimer i s destroyed. But the reactive part of the k i n e t i c energy equation can be broken i n such a way that 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 . equation (6.1.27) and the second l a s t equality of equation (6.1.32)) i s revealed, and hence expressed i n terms of r e l a t i v e angular v e l o c i t y between the molecules. As for the po t e n t i a l energy equation, we have [!£ ( MM VM + WWexch = - V <VM^+ VD ^MfD.exch + 2 Tr 123 rV13 + V23 A , . , [ 2 6 3 ] s + rV l 3 + V " r £ < 1 2 > ^ A i (12)3 1(23) Pl(23) - 205 -_ V • i Tr r ( 1 2 ) 3 - (- lEli2!^ + ) 6 v ? 1 0 * 123 ~ 3 R 13 (12)3 J(12)3 P1(23) 21 H Tr 123 ( 6(12) + 63 v  v 2 v23 61 + 6(23) V 1 2 } P ( 1 2 ) 3 3 ( 1 2 ) 3 _ r 6(12) S i .y a ^ 1 2 ^ ( 1 2 ) 3 ^ ^ ( 1 2 ) 3 X 1 2 ) 3 1(23) P l ( 2 3 ) * (7.3.15) Cl e a r l y , t o t a l energy i s conserved for th i s r e a c t i o n . We may also express the potential energy equation i n an analogous form to (7.3.11) i n which the transformation of potential energy from k i n e t i c energy i s e x p l i c i t . ( i i ) C o l l i s i o n s Involving Three Monomers The only reaction that may take place i s the recombination reaction with a monomer molecule as the th i r d body. The energy balance for t h i s reaction i s given by hnr ( MM TM + MDV ]3M+M+D = - V ct =3M->-M+D — ?X~ • u + ( P - Pc t ) „ • v '3M+M+D + L • a) + a' =3M+M+D - -±3M+M+D + 2% ^ 2 3 — > 6(23) ^23 ' 0 123 1(23)*1(23) P123 (23) 2m v ' ) ] 8 - 206 -Tr 123 1 £-23 -23 >. , KZT~T~ • * , ; ° ( 2 3 ) ( £ 2 3 J 1 ^ 3 _ . x - - r 2 3 m r ^23 23 -23 -23 m r 7— * £23 23 p 7 1(23) 1(23)P123 + Tr ( 123 1 -23 -23 2T 2 2 r £(23)' -2m v' 4m . ( p ( 2 3 ) - 2 m v ' ) 6 ( 2 3 ) J 6(23) ( m v ' 2 +T r23 1(23) 1(23)P123 1 ~*23 ""23 T r ("5T 5 - • • 0 ( £ / , o ^ r ? 3 . ) t P u ? 3 . T , p 123 (23) u (23 ' s u l (23) 0 l (23) K l (23) : ( V -+ —- Tr 1 6 123 -23 -23 -23 X -223 6(23) *1(23)-31(23)P123 ( V u))' 1 6 123 vr„ 3 x p 2 3 _ r 2 3 r (23) *1(23) J1(23) P 123 : ( V a)) - 207 -2rT Tr 123 + * 2 + *3> 6 l + V \ N ( 2 3 ) ^ 1 ( 2 3 / I 2 3 + V23 6 ( 2 3 ) 6 ' l ( 2 3 ) 7 l ( 2 3 ) 123 (7.3.16) • 3 T ( MM VM + MD VD ) ]3M^D - V ( V M - ^ V D ^ W D 2 h 123 (23) 2 2m / J s + F6 V12 + V13 r P l „ + [ 61 2 ( i - " £ ) ] s fl(23) U Ol23 P123| 4fT Tr 123 -£ 23 (~~2~~ + 6(23) V 12 + <-|l + ...) 6 ( 2 3 ) V 1 3 D -T 1(23) J1(23) P123 L 2 ^1 2 3 J s ^1(23)^1(23) 123 + 2 V23 1(23) Jl(23) 123 (7.3.17) - 208 -A l l the terms can be Id e n t i f i e d as for monomer-dimer c o l l i s i o n s , likewise for c o l l i s i o n s that involve four atoms whose contribution to the equations of change for k i n e t i c and potential energies are given below. ( i i i ) Dimer-Dimer C o l l i s i o n s Decomposition with no Exchange [ l t ( M M T M + M D T D ) ] 2D+2M+D,no exch « _ V ,ct ,ct, P:: • v + (p - p )_ 2 • v =2D->-2M+D,no exch 2D>2M+D,no exch — =2D*2M+D,no exch — 2D->-2M+D,no exch + 4- Tr n 1234 1 £-12 £-12 2~ (- P-(12) 6(12) )s 2T 2 f12(34) 312(34) p(12)(34) : (V u)' i ^ l 2 <%ST-'2> ^ 1 2 Ws f12(34) J12(34) p(12)(34) + ^ 1234 L n L 1 2 L 1 2 x 2n 6(12) : ( v •u>t ^12(34)^12(34) p(12)(34) m 1234 x l l 2 X £-12 -12 -12/' 6(12) f12(34) : 712(34) p(12)(34) : ( V cu) - 209 -- vv 1 TV 1 ^12 -El2 A l l ) ~ 2m^" * 1 2 3 4 2! 2 2 Am f12(3A) 312(3A) P(12)(3A) ( P ( 1 2 ) - 2m v") 6 ( 1 2 ) ] g " 1 & " D ^ J 2D->-2M+D,no exch v"2/2 i • A -23 -23 . . Tr (.-^—, ^— ...; 6 1 1 2 3 4 2! 2 2 2 m 2 ,„ (23) r23 2 U 7 2 (P T 12(3A) J12(3A) P(12)(3A) - ^  VV: Tr 123A ,1 -12 £12 v , ( 2 T " T ~ ~ °(12) r 2« ^  _P| 2 1 m 2 = — - 7 JH> • (£12 x -P-12 ~ T r12 ^ m r 12 + P 1 2 * -12 -12 m r 12 ^12(3A)^2(3A) P(12)(3A) — i s Tr 123A r,« +« . 6(12) + 6(3A)  [ C K(12) + 1 t 3 A ) } 2 J s V12(3A) f'l2(3A)^12)(3A) p(12)(34) _ V12 6(12) ^12(34) J12(34) P(12)(3A) (7.3.18) - 210 -As with the monomer-dimer exchange reaction, we can associate the energy ct f l u x carried along by the pressure tensor (P - P , with b J r = = 2D->-2M+D,no exch v" by using the i d e n t i t y £i + 2 j =J (£i + Pj + -Ek + -El> + -E(ij)(kl) = (j>^  + -Ej ~ 2 m .Y.") + 2 m .Y." P k + 4 < P i + Pj + P k + Pl> + £ i ( j k l ) (7.3.19) With t h i s relationship,.the i d e n t i f i c a t i o n of P_ w „ • v" from p ' =2M+D,no rxt — the term 1 T< 1234 £12 W. • ( V12 V 1 2 ) - r (-He2- fi„„.o„ • [ v 1 / 0 / . x (v 1 0 + v w.)] -1(34) v 3m (134)'s 1 1(34) v v13 14y - £ 2 ( 3 4 ) < T T I * ( 2 3 4 ) ) 8 * [ V2(34) ( V23 + V 2 4 ) ] 12(34) 12(34)^12(34) i n (7.2.19) can be made in a more d i r e c t way. The rate of change i n the pote n t i a l energy density due to this reaction i s : - 211 -*3t ( MM VM + MDVD^2D+2M+D,no exch V ' t ™ M V M ^ + MD VD^2D>2M +D,no exch + Tr 1234 -El V12 + V13 + V14, 2 "iT"^ 6 i — 4 ~ ] s V13 + V14 + V23 + V24. 2m "(34) 12(34) J(12)(34) P(12)(34) + -i- Tr 1234 ( : r i + T r ¥ ¥ - - ) 6 ( i 2 ) ( v i 3 + V r r r 2 2! 2 2 ~P T n 12(34) J12(34) P(12)(34) (12) v 23 2 1234 12(34) 61 + 6(34) >. ^ 1 3 + V + (^ 12(34) 62 + 6(34) ; s ^ 23 24 ; 4 61 + 62 + ^12(34) f ^ s ^ 12 ri2(34Al2)(34) p(12)(34) 6 t + 6 2 2 V12 ^12(34)^12(34) P(12)(34) - 212 -Compared with the equation of change for k i n e t i c energy density conservation of energy i s again v e r i f i e d . Exchange Decomposition [!Tt ( MM TM + MDTD)]2D>2M+D,exch - V P C^ • v + L • 0) =2D+2M+D,exch — =2D+2M+D,exch — c t » t + ( I " £ )2D>2M+D,exch ' - + £2D->-2M+D,exch + 1^24 ' P 2 4 Ws 1234 r A 1 2 ) \ , , " [ £ l 2 ( — S S T " ^ ' £ l 2 6(12) ]s [£34 < ^ ~ 1 > *£34 W * 13(24) J13(24) P(12)(34) , 2 i T 1 £ 2 4 £24 , . v ,„ „,t + ~fi 1 2 * 4 2T — — (£(24) 6 ( 2 4 ) ) s : ( V £ > 13(24) J13(24) P(12)(34) - 213 -+ 2J T r * 1234 1 £l2 £l2 , R v 21 2 2 A 1 2 ) (12 ) ; s , 1 £34 £34 , . , 2 l 2 2 A 3 4 ) ° ( 3 4 ) ; s < P13(24) : f13(24) p(12)(34) (V v ) * 21 Tr 1234 £12 £12 + ( £ i 2 £ i 2 } t 6(12) + £34 £34 + (£34 £ 3 4 } t < f13(24) : I13(24) p(12)34) 6(34) 8n Tr 1234 £ 2 4 £ 2 4 £ 2 4 X £ 2 4 6 (24) ~ £ l 2 £ l 2 £ l 2 X £ l 2 6(12) ~ £34 £34 £34 X £34 6 (34) u - v (V a))' 13(24) 313(24) P(12)34) 8h Tr 1234 ^24 X £ 2 4 £ 2 4 £ 2 4 7 * 6(24) : (V co) \ r34 x £34 £34 £347* 6(34) 13(24) 13(24) p(12)34) - 214 -2 i Tr 1234 1 ^24 ^24 -£(24) " 2 m £ " TC 2 2 4m • ( F v 2 4 ) - 2m v") 6 ( 2 4 ) 1 -12 -12 £(12) 2! 2 2 - 2m v 4m 1 £ 3 4 £34 £(34) ~ 2 m ^ 2T 2 2 4m • ( P ( 1 2 ) - 2m v) 5 ( 1 2 ) (P(34) - 2 * v) 6(34) o -1 13(24) J 13(24) p(12)(34) T [ 4 t V ^ W ^ D . e x c h + 21 T r i _ (^12 £l2 £34 £ 3 4 } ( v " 2 - v 2 ) ft 1 2 3 4 2! 2 2 2 2 ; ^ 2 ; P13(24) J13(24) P(12)(34) + ^ T r 1 1 1234 ,1 £24 £24 . . 2 ^ T T T " T ~ (13) r i 3 (1_-12 £l2 w 2 K2\ 2 2 "'} °(12) r12 ,1 £34 £34 V . "21 2 2— * * * (34) r34 S*2/2 13(24) 13(24) p(12)(34) - 215 -+ 2 i VV: Tr 1234 1 £24 £24 V . ( T F T T m " } 6(24) £24 x £24 1 Mv . , v „ m 2 Mv ( 2 7 JS> • (£24 x £24 " 7 r24 m r; 24 £24 £24 + £24 ~2 * £24 m r 2 4 1 £12 £l2 v . l 2 T T T 6(12) (-12 X f 1 2 - | . ) • ( r 1 2 x p 1 2 - f r 2 , «) m r 1 2 £l2 £l2 + £12 ' ' £12 m r 1 2 —Is f 1 -34 -34 N . {2T~ ~ °(34) i ^ l ^ L - ^ ^ . ( £ 3 4 x p 3 4 - f r 2 4 ») m r 3 4 -34 -34 „ + £34 — ' -£34 m r34 —Is 13(24) J13(24) P(12)(34) - 216 -1234 ^ (12 )"^"^(34) [ (^13(24) ~ 2 ) s 1 ^ 12(24) ^ 3 ( 2 4 ) ^ 1 2 ) ( 3 4 ) P(12) (34) + [ V24 6(24) " V12 6(12) " V34 6 ( 3 4 ) ] ( ?13(24) : J13(24) P(12)(34) Ut ( MM VM + ^V^D-^M+D.exch (7.3.21) = - V ( M M V M £ + MD VD ^D^M+D.exch + 2 Tr 1234 2[(=i - U ) 6 m — l V12 + V13 + V14, + f&ZSl - v) 6 V12 + V14 + V23 + V 3 4 ] + l ( — 5 — 1> °(24) 2 J s ^13(24) >(12)(34) P(12)(34) 2 i Tr 1234 A + 63 6(12) + 6 ( 3 4 ) > _ K 2 ~ 2 ' 13 . *1 + 6(24) 6(12) + 6 ( 3 4 ) , _ ^ 2 2 ; V14 + / 3 * 6(24) 6(12) + 6 ( 3 4 ) . v 2 ~ 2 ; v23 ( i13(24) : : I13(24) P(12)(34) - 217 -2 i 1234 61 + 63 ( i <13(24) 2 } s ^13 +<*13(24) 6 l V ( 2 4 \ ^12 + ^14> + ^13(24) S + 2 ( 2 4 ) ) s ^23 + V ri3(24) v / l /(12)(34) P(12)(34) + [6 ( 1 2 ) + 5 ( 3 4 ) v _ V 5 ( 2 4 ) v V 6 ( 2 4 ) y . 1 2 V24 2 V12 2 V 3 4 ] 13(24) J13(24) P(12)(34) (7.3 Exchange with no Decomposition [ 9 t W2D,exch * " V •[=2D,exch * i + i 2 D , e x c h ' » + q 2 D > e x c h ] 218 -4 Tr n 1234 1 -13 -13 , * x 2T ~1 2~ Al3) °(13) ;s 1 1.24 -2h i 1 — ^ —£H t p \ "2T ~~2 2~~ C£(24) 6 ( 2 4 ) ; s 1 £-12 £ l 2 , , x "2T ~T ~T A l 2 ) °(12) ;s 1 £34 £34 , , N "2T"T~~2~ A34) °(34) ;s f >(13)(24) I I(13)(24) P(12)(34) (V v)' Tr 1234 £13 '£l3 6 ( 1 3 ) ] s + £ 2 4 <HST--2> '-£24 Ws £12 <=TSp--2> * £ l 2 6 ( 1 2) ]s £34 *£34 W-f(13)(24) 3(13)(24) P(12)(34) - 219 -TF Tr 1234 i l 3 -13 -13 x £l36(13) + £24 ^24 -24 x £24 6(24) ~ £12 £l2 -12 X £l26(12) " £34 £34 -34 X £346(34) (V to) 1 6 1234 (13)(24)J(13)(24) P(12)(34) r 2 3 x p 2 3 £ 2 3 r 5 ( 2 3 ) + \£ 24 x £ 2 4 ^ 2 4 £2 v£l2 x £12 £12 £1 (24) '(12) " \ ^ 4 * . £ 3 4 i 3 4 £ 3 4 / 6(34) (13)(24) J (13)(24) p(12)(34) ( V <o) j_£l3_£l3 ^ 1 3 )  v2! 2 2 ' 1 4m * ( £ ( 1 3 ) ~ 2m v) 6 n , J (13) Js . ,1 ^24 £24, r£(24) " 2 m £ , ^2T ~2~ ~2~) 1 Am * (P r,„ r - f 1 ~ 1 2 ~ 1 2 ^ r ± Q 2 ) v 9 ) 4m - 2m v 2! 2 2 ) [- 4m (24) " 2 r a £ > 6 ( 2 4 ) ] s ( P ( 1 2 ) - 2m v) o , 1 0 J "(12) Js 1 -^ 34 ^34, r-£(34) ~ 2 m £ , „ x x , } [ = (^(34) - 2 m £ > 6 ( 3 4 ) ] s (i v2! 2 2 4m (13)(24) J(13)(24) P(12)(34) - 220 -+ £ VV: Tr 1234 fl -13 -13 . . v^r — — o ( 1 3 ) ( £ l 3 " f 1 3 co) • ( r 1 3 x p 1 3 - | r 2 3 i £ ) m r + -El3 12 £-13 £-13 m r 2 — * £13 13 —I s , 1 £24 £24 . . (24) (~ 2 4 X f 2 4 - j jo) • ( £ 2 4 x p 2 4 - ? r 2 , co! m r 24 £ 24 £ 24 + £24 • -Jhr-' £24 m r 2 A —I s , 1 -12 -12 , -{1T ~T~ ~T~ "m) °(12) ("12 **12 _ 1 £ ) . ( £ i 2 x _m ^ ^ m r + P 12 £l2 £l2 -£12 12 ' £l2 m r 1 2 —I s (13)(24)- i(13)(24)Xl2)(34) - 221 -VV: Tr 1234 1 £34 £34 . . ( 7 T T T ••• ) 6(34) .£34 x £34 1 m _2 (— 2 — " 7 ^ * (£34 x £34 ~ T r34 2$ m r + £34 * 34 £34 £34 m r 2 — * £34 34 ( F U 3 ) ( 2 4 ) 3 ( 1 3 ) ( 2 4 ) p ( 1 2 ) ( 3 4 ) | V V : { [ f e ^ ] , „ c h v 2 / 2 } - £ vv Tr 1234 1 -13 -13 * 2 "2T ~~2 2~~ °(13) r13 + 1 £24 £24 . 2 TT ~2 2~ (24) r24 1 £l2 £l2 . 2 2! 2 2 (12) r12 1 £34 -34 . 2 2! 2 2 (13) r13 (13)(24)°(13)(24) P(12)(34) f a,2/2 - 222 -Tr 1234 [ V13 6(13) + V 2 4 6(24) " V12 6(12) " V34 6(34)^ f(13)(24)3(13)(24)P(12)(34) + +y2 + K 3 + # 4 ) 6 ( 1 2 ) + 2 5 ( 3 4 ) ] S \ ^ ( s w (13)(24) (13)(24)v/U(12)(34)p(12)(34) Ut 2D, exch (7.3.23) « _ V ( MD VD 2D,exch + 2 Tr [(-^l3-2- - v) 6 1234 2 m ~ ( 1 3 ) V12 + V14 + V23 + V24, ^(13)(243J(13)(24)P(12)(34) 1234 z < , £ (13X24) £ ( 1 3 ) ( 2 4 )  v 2 2 £ (12 ) (34 ) £ ( 1 2 ) ( 3 4 ) 2 2 ( V14 + V23> P(13)(24)1(13)(24)P(12)(34) + — Tr * 1234 *(13) + 6(24) (13)(24) 2 Js V(13)(24)X13)(24)%/2tl2)(34)p(12)(34) - 223 -5(13) + 6(24) f . 6(12) + §(34) , , v "K 1 2 ( V12 + V 3 4 ) 2 ( V 1 3 + V 2 4 ) ] 1234 VC13)(24) J(13)(24) P(12)(34) ( i v ) Monomer-Monomer-Dimer C o l l i s i o n s  Recombination-Decomposition *3t ( MM TM + MDTD^2M+D,recomb,decomp (7.3.24) m - V P M ^ C L L ^  ^) * w =2M+D,recomb,decomp — = 2M+D,recomb,decomp — + L ct 3 ^ + M^-D 2M+D,recomb,decomp + q ' 4 —2M+D,recomb,decomp TfT Tr 1234 t£l2 < ^ - r ) * £12 W : " ^ 3 4 ^ - ^ ) * P 34 6 ( 3 4 ) ] s (12)34 (12)34 p12(34) "33S Tr 1234 \^2J^El2 ^ 12 £12/ 6(12) (34) (V w) f(12)34^(12)34 p12(34) - 224 -- V "3"2TT Tr 1234 £l2 £l2 £l2 x £l2 6(12) .34 £34 £34 x £34 6(34) - r. : (V co)' *(12)34 J(12)34 P12(34) + V 2h Tr 1234 1_ 2! £l2 £l2 2 2 (£(12) 6 ( 1 2 ) ) s 1 £34 £34 , , . TT ~T ~2~ A34) °(34) ;s (12)34 (12)34 P12(34) (V v") + VV : ( r — M ^ l v" 2/ 2} ' 1 1 3 t D - 2M+D,recomb,decomp ' ' + -=£ VV : Tr z 1234 r,„ r,„ p,,„ x - 2m v" 1 -12 -12 Ml2) 2! 2 2 1 4m (P(12) " 2 m £ " > 6 ( i 2 ) ] s i _ £34 £34, r £ w " 2 m v " 2! 2 2 -) [- 4m ( P ( 3 4 ) " 2 m £ " ) 6 ( 3 4 ) ] ( 1 2 ) 3 ^ — ^ (12)34 K12(34) - 225 -+ ±=- VV : Tr Z n 1234 ,1 -12 -12 . . ( 2 f l " T ••• } 6(12) ( £ l 2 X f 1 2 - 2 ^ * (^12 X ^ 1 2 m r m 2 v 2 r12 ^ + P i 2 12 -12 -12 m r 2 ' £l2 12 , 1 -34 -34 v . ("2T ~2~ ~2~ *••-' °(34) m r34 -34 -34 + £34 — * £34 m r34 -4s ^(12)34 J(12)34 P12(34) - VV 2n Tr 1234 1 ,-12 ^12_ 2 . _ fv34 ^34 I T 1 2 2 12 (12) 2 2 -34 -34 2 . v m ,2, — — r34 6 ( 3 4 ) ) 2 w ' l $(12)34 : j(12)34 P12(34) - 226 -IT Tr 1234 F < * l + W 6 l + 2 ( 3 4 ) ] s ^ 1 3 + ^ 1 4 > + ^ 2 + % 4 ) > " 2 % 6 ( 3 4 ) ] s ( ^ 3 + ^ ) + ( j f(34) 6 ( 3 4 ) } s ^ 3 4 6 ?(12)34 , r t12(34) p12(34) + ( 6 ( 1 2 ) V12 ~ 6(34) V34 ) P(12)34 : r(12)34 p12(34) ["9t + ^VVUM+D.recomb,decomp (7.3.25) - V % V M - + MD VD —^2M+D,recomb, decomp + -y Tr z 1234 , r ^ 3 , x V13 + V23 + V34, 2 [ ( ^ - u) 63 j ], + ttyp-' v) « ( 1 2 ) « ( 1 2 ) ' 1 3 ^ 1 4 ^ 2 4 , (12)34 V /i'2(34) pl 2(34) 2fi Tr 1234 6(12) " 6(34) ( V 1 3 + V14 + V23 + V 2 4 ) 6_-6, 6.-6. 6_-60 6,-6_ v + ( 3 1 „ . _4 L V + 3 2 v + — 2 V } + V13 V14 + 2 V23 + 2 V24 (12)34 (12)34 P12(34) - 227 -2 * 1234 ^(12)34 ^ P1 ^ ^ 3 + ^ 2 3 > 6 ( 1 2 ) + 6 4 (12)34 2 's v 14 24 6 + 6 + — — V 2 V34 ^(12)34^2(34) P12(34) + ( 2 V12 2 V 3 4 ) ^ (12)34 J(12)34 P12(34) (7.3.26) Recombination Without Exchange [~9t ( MM TM + MDTD)J2M+D^2D,no exch » _ V P C t • v" + (P - P C t ) • u 2M+D+2D,no exch — v= = '2M+D->-2D,no exch -ct 3uo + 4l-D + =2M+D+2D,no exch 4 + CT - L C*'') • a) V= = ;2M+D>2D,no exch -+ -3-2M+D>2D,no exch r t (12) x , , + 2i? "12 ( ^ T - - ^ * £ 1 2 6 ( 1 2 ) ] s 1234 f(12)(34) T(12)(34) P12(34) - 228 -- V 2TT Tr 1234 1 £j_2 £l£ 2T 2 2 (£(12) (V v ) 1 e. (12)(34) 3(12)(34) p12(34) 1 37R" Tr 1234 S£l2 x -El2 £l2 £l2 6(12J (12)(34r(12)(34) p12(34) : V to + Tr 1234 -\2 £l2 6(12) £l2 x -El 2 Xl2)(34) J(12)(34) p12(34) (V co)* VV Tr •ar ^34 2 1 1 £12^12 £(12) " 2 m £ 2 2 4m (£(12) 2m v) 6 ( 1 2 ) ] s < P(12)(34) : I(12)(34) P12(34) " 1 lit 2M+D->-2D,exch v2/2 IfT T r ^12 —12 6(12) 2 r12 < P(12)(34)^(12)(34) P12(34) 0 ) 2 / : 1234 - 229 -- VV 1234 1 -12 -12 2! 2 2 (12) Al x - ^ i 2 m r " 2 $ ' ( r-12 X-P-12 " m 2 12 -12 -12 + £l2 ' 2 ' *12 m r 12 ^(12)(34) 7J2)(34) P12(34) 2 T Tr 1234 ^+ % 4 ) 6 l + 2 ( 3 4 \ ^13 + ^4> U 2 ^(34) 2 Js W 2 3 V 2 4 ; ^(12)(34)^12(34) P12(34) + 6(12) V12 ^(12)(34) ^12)(34) P12(34) [ 3 t ( MM VM + W*2M+D-»-2D,no exch ( MD VD-^M+D^D.no exch + Tr [ ( - ^ ~ v) 5 1234 2 m " ( 1 2 ) V + V + V + V 13 14 23 24. ^(12)(34 )°T2(34 ) P 12(34) 2 3 0 -2¥ J 3 r 4 [ ^ ( 1 2 ) - V < V 1 3 + V 1 4 > + < 6 ( 1 2 ) - V < V 2 3 + V 2 4 > 3 ( 1 2 ) ( 3 4 ) J 1 2 ( 3 4 ) P 1 2 ( 3 4 ) + i- T r * 1 2 3 4 ( 1 2 ) ( 3 4 ) 6(12) + 6(34) V ^ 1 2 ) ( 3 4 ) ^ ( 1 2 ) ( 3 4 ) * ° 1 2 ( 3 4 ) p 1 2 ( 3 4 ) \ + 6 2 + 2 V 1 2 f ( 1 2 ) ( 3 4 ) 7 ( 1 2 ) ( 3 4 ) p 1 2 ( 3 4 ) E x c h a n g e - R e c o m b i n a t i o n [ 8 t ( M M T M + M D T D ) ] 2M+D-»-2D,exch ( 7 . 3 . 2 8 ) - V =2M+D>2D,exch c t , , v " + (£ - P " u ) 2M+D+2D,no e x c h — + L C t . 3 ~ + i ^ D =2M+D-»-2D,exch 4 + ( L - L C t ) 2M+D->-2D,exch * - + £2M+D-»-2D,exch 1 2 3 4 [£l3 £l3 * '"Tm1 -Z> 5 M 3 ^ ( 1 3 ) J s + t £ 9 A p.. • (^2£> - v ) i-L24 -^ 24 " V " T T " " ^ u(24)]s ~ t £ 3 4 p 3 4 * ^ " v") « ( 3 4 ) ] 8 ( 1 3 ) ( 2 4 ) ( 1 3 ) ( 2 4 ) p 1 2 ( 3 4 ) - 231 -- V - Tr 1234 £34 £34 + (-P-34 ^ 34 ) 2 (34) 6 >(13)(24) : I(13)(24) P12(34) 1 u - v Tr 1234 1 -13 -13 ( 2 / 2! 2 2 V-H13) (13)'s , 1 -24 -24 , . . + " 2 T ~ 2 ~ ~ ^(24)°(24) ;s (V v) f c (13)(24) (13)(24) p12(34) + 7 Tr 1 £34 -34 , - . „,t 1234 U~2 2~ ^ ( 1 3 ) (13 ) ^ 8 : ( V ^ > ^13)(24) J(13)(24) P12(34) T6n" Tr 1234 -13 -13 -13 X -El3 6(13) + £24 £24 -24 X -2246(24) -34 -34 -34 x -p-346(34) ^(13)(24) J(13)(24) P12(34) (V w)' - 232 -- V Tr 1234 S2Li3 x P13 £13 Ln/' 5(i3) + \ I 2 4 X £ 2 4 £ 2 4 j 2 > / ' 6(24) -VI.34 x £34 £34 JL-. (34) V to (13)(24) J(13)(24) P12(34) + VV: i Tr 1234 ,1 ^13 -^ 13 X , ( T T I * 75— . . . ) 0 (13)2 • £ l 3 - ~ 1 3 " 7 ^ " ^13 x-£l3 m r m 2 x 2 r13 »> + P i 3 13 ^13 ^13 m r T ~ ' £i3 13 —Js . 1 -^ 24 -2k v . C 7 T ~ 2 ~ T " (24) r.24 x £ 2 4 1 m _2 j - L - T to) • ( r 2 4 x p 2 4 - 7 r 2 4 to) m r 24 •^ 24 ^24 -£24 2 ^ * £24 m r24 —J s ^(13)(24)^(13)(24) P12(34) - 233 -+ VV : Tr 1234 1 1-34 -34 . . " {TT — ~T 6(34) —34 X -H34 1 m _2 2 • ( £ 3 4 x P 3 4 - 7 r 3 4 ^ m r 34 £ 3 4 £34 + £ 3 4 ' n 1 ' £34 M R34 P(13)(24) J(13)(24)P12(34) + VV : 2 L 8 t D = 2M+D+2D,exch V 1234 r r 1 £l3 £l3 r,,„.2 2! 2 2 (13) (13) r ^ / N 2 2! 2 2 (24) "(24) 1 -34 -34 r , , „ x 2 £ « o 2 / 2 2! 2 2 (13) "(13) P(13)(24) 7(13)(24)p12(34) 1 T V -34 -34 m(v 2 - v " 2 ) , p ^ 2 2 2 J r(13)(24) J(13)(24) P12(34) 1234 - 234 -Tr 1234 l ^ V ^ T - V l Z * ^ l + * ( 3 4 ) > 2 > s ^ 4 + ^ 2 + ^ ( 3 4 ) > ^ + 2 ( 3 4 ) ] s ^ 2 3 + (*(34) W 3 ^ 3 4 (13)(24)' /12(34) P12(34) + ( 6 ( 1 3 ) V13 + 5(24) V24 - 6(34) V 3 4 > (13)(24r(13)(24) p12(34) 2M+D->-2D,exch (7.3.29) ^ VD —^2M+D+2D,exch + 2 1234 6 < » > V 1 2 + V 1 4 + V 2 3 + V 3 4 -^ >(13)(24T ! 12(34) P12(34) - Tr l1234 Al3) + 6(24) V _ V _ ^ 2 2 ' v12 . f 6 ( 1 3 ) + 6(24) 61 + 6 ( 3 4 ) , v 2 " 2 ' v14 r6 ( 1 3 ) + 6 ( 2 4 ) 62 + 6 ( 3 4 ) y , ^ 2 2 ' 23 P(13)(24) 3(13)(24) p12(34) - 235 -+ i Tr 1234 [ t t 6(12) + 5 ( 3 4 ) 1 2 J s V(13)(24) (12)(34) f(13)(24r V12(34) P12(34) . r 6 l * 6(34) _ . 62 + 6(34) 6(13) + 6(24) . + l 2 V13 2 V24 .2 V34 J f'(13)(24) : I(13)(24) p12(34) (7.3.30) Exchange Without Recombination Ut ^ T M + ^TD^2M+D-*-2M+D,exch m - V =2M+D+2M+D,exch — =2M+D+2M+D,exch 4 ct + (L - L )2M+D>2M+D,exch * — + -^ 2M+D->-2M+D,exch 1234 [£l3 ^ " Z " ) *£l3 6 ( 1 3 ) ] s r /- E(34) ... . . f T 0 (13)24 J(13)24 P12(34) 34 (34) Js - 236 -- V 2i Tr 1234 1 -13 -13 , (13) hn^s 1 -3h £34 " TT ~2 T~ A34) ° (34 ) ; s Xl3)24:r(13)24P12(34) : (V v")' Tr 1234 -13 -13 -13 x -Bl36(13) - r : (V co) Tr 1234 £34 £34 £34 x £34°(34) 6>(13)24:J(13)24P12(34) £l3 x £13 £13 £13 * 6(13) - £34 x £34 £34 £34 " "(34) : (V co) e 1 D (13)24 (13)24P12(34) 2i + VV : 4F- Tr * 1234 1 £i 3£i3, r £ ( i 3 ) " 2 m -<7T 2! 2 2 -) [ 4m ' ( £ ( 1 3 ) - 2 m £ " ) 6 ( 1 3 ) ] s . r_. r_. p, 0 / N-2m v f ^ f l 1 ' ^ " -<l(34)- 2 "^ ) 5 (3 4 )' S (13)24J(13)24P12(34) - 237 -- VV 2 i 1T 1 ,1-13 -13 . 2 -34 -34 . 2 . J , . 7T l~2~" ~2~ °(13) r13 ~ ~T~T ° ( 3 4 ) r 3 4 ; 1234 ? J (13)23 (13)24 p12(34) + 1 fit V^2D*2MfD.exch V " 2 / 2 f . 2/2 + VV : 2 i Tr 1234 ,1 -13 -13 v . ^7T ~2 2~~ 6 ( 1 3 ) ,-13 x ^13 1 m -2 < - " 7 »> * (^13 x Pl3 " 7 r 1 3 »> m r + Pl3 13 -13 -13 . „ J— * -P-13 m r 13 1 £34 £34 ( 2 T - 2 " T " 5 ( 3 4 ) X f 3 4 - \ »> ' (£34 x P34 " 7 r34 »> m r 3 4 £34 £34 . „ + £34 2 — * -234 m r34 —is (13)24 (13)24 P12(34) - 238 -Tr 1234 6 , + 6 / + X 2 ) - V i l 8 V 1 2 + [ ( ^ ^ ( 3 4 ) ) -M ^ ' B " l 4 + ^ 2 + ^ 3 4 ) ) ^ T 1 1 ^ ( t f23 + U24> + (^(34) 6(34) > s ^ 34 (13)24 , n ;12(34) pl 2(34) + ( 6(13) V13 " 6(34) V ( 3 4 ) } ^(13)24^(13)24^2(34) (MMVM + MDVD)12M+D-2M+D,exch (7.3.31) (¥M + MD VD ^ )2M+D^2M+D,exch + 2 Tr 1234 .A , , V14 + V24 + V34. [ ( _ _ u) 64 -2 ] g 2rm V12 + V14 + V23 + V34-P(13)24 J 1l2(34) p12(34) 2i Tr 1234 2 -( 6 ( 1 3 ) - 6 l ) V 1 2 + l ( 6 ( 1 3 ) - 6 ( 3 4 ) ) V 2 3 + 2 ( 64 ~ 6(34) } V24 + ( 2 2 } V (13)24J(13)24P12(34) 14 - 239 -1234 62 + 6('n,) + [ O f 2 + ^ ( 1 3 ) ) 2 ( 1 3 ) i s t r 2 23 6 , + 6, ^(13)24^12(34) p12(34) + 81 * S(34)  1 2 13 64 + 6(13) V 3 4 ] < P(13)24^(13)24 P12(34) (7.3.32) Thus, when chemical reactions take place, owing to atom rearrangement, in addition to the heat f l u x and the and L • cu = —o types of terms, the energy flux also contains 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 physical a t t r i b u t e from the reactants to the products. - 240 -CHAPTER 8 SUMMARY In this work, conservation laws for the physical observables of mass, linear and angular momenta, and energy are attained for a dimer-monomer reacting gas as described by the Lowry-Snider kinetic theory. Following Olmsted and Snider, the physical observables are visualised as molecular attributes and are localised 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 which is related to the strong orthogonality assumption, which is in turn a crucial feature of the Lowry-Snider kinetic theory. From the sum rule, the contribution to the equations of change from each collision type is identified. For a more general situation in which coherences between the (different channels play a significant role, the completeness relation is no longer valid and the sum rule w i l l thus also need modification to allow for coherence effects. C v C C * C c c is derived, assuming the validity of the completeness relation - 241 -The equations of change obtained i n th i s work can be compared with those obtained using a macroscopic point of v i e w 1 5 . In the l a t t e r case, the mass flux i s due e n t i r e l y to molecular v e l o c i t i e s . In the present treatment, when chemical reactions occur, there i s also a term associated with the change i n mass l o c a l i s a t i o n . This a d d i t i o n a l term vanishes when the system i s homogeneous. Another feature of the Lowry theory i s that c o l l i s i o n s are considered to be isolated and independent events. When c o l l i s i o n s are no longer i s o l a t e d or when intermediates are involved, the k i n e t i c theory needs to be generalised and the Evans theory i s a reasonable st a r t i n g point for th i s g e n e r a l i s a t i o n . It i s surmised that the sum r u l e , with modification, and the l o c a l i s a t i o n scheme used i n t h i s thesis w i l l s t i l l form a basis on which the general theory can be b u i l t . - 242 -BIBLIOGRAPHY 1. J.O. Hirschfelder, C F . Curtiss and R. B. Bi r d , Molecular Theory of Gases and Liquids (Wiley, New York, 1954). 2. J.H. Irving and J.G. Kirkwood, J . Chem. Phys. 18_, 817 (1950). 3. J.T. Lowry and R.F. Snider, J . Chem. Phys. 6J_, 2320 (1974). 4. R.F. Snider and K.S. Lewchuk, J. Chem. Phys., 46, 3163 (1967). 5. N.N. Bogoliubov, Problems of a Dynamic Theory i n S t a t i s t i c a l Physics, i n Studies i n S t a t i s t i c a l Mechanics, J . de Boer and G.E. Uhlenbeck (eds.), Vol. I (North-Holland Publishing Co., Amsterdam, 1962); M. Born and H.S. Green, Proc. R. soc. 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 Theorie des f l u i d e s et de l'equationd'etat (Paris: Hermann et Cie, 1935). 6. R.F. Snider and B.C. Sanctuary, J . Chem. Phys. J55, 1555 (1971). 7. L. Waldmann, Z. Naturforsch. A 12, 660 (1957); R.F. Snider, J . Chem. Phys. 32, 1051 (1960). 8. M.W. Thomas and R.F. Snider, J . Stats. Phys. 2_> 6 1 (1970). 9. R.D. Olmsted and C F . Cu r t i s s , J . Chem. Phys. 62, 903 (1975); j>2_, 3979 (1975); 63, 1966 (1975). 10. J.A. McLennan, J . Stats. Phys. 28, 257; 28, 521 (1982). 11. R.D. Olmsted and R.F. Snider, J . Chem. Phys. (1976). 12. D.K. Hoffmann, D.J. Kouri and Z.H. Top, J . Chem. Phys. 70_, 4640 (1979); J.W. Evans, D.K. Hoffman and D.J. Kouri, J. Math. Phys. 24, 576 (1983); J . Chem. Phys. 78, 2665 (1983). 13. J.M. Jauch, B. Misra and A.G. Gibson, Helv. Phys. Acta 41_, 513 (1968). 14. A. Messiah, Quantum Mechanics, Vol. I, English t r a n s l a t i o n by G.M. Temmer, (Wiley, New York). 15. S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Publishing Co., Amsterdam, 1962). 

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