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12 Chemistry of High Energy Atomic Fluorine: Steady State Kinetic Theory Model Calculations for the 18

F + H Reaction III

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EDWARD R. GRANT, DA-FEI FENG, and JOEL KEIZER Department of Chemistry, University of California, Davis, CA 95616 KATHLEEN D. KNIERIM, and JOHN W. ROOT Department of Chemistry and Crocker Nuclear Laboratory, University of California, Davis, CA 95616 As reviewed elsewhere in this volume (1,2) considerable recent progress has been achieved with respect to our understanding of the dynamics of atomic fluorine reactions. A central objective of chemical dynamics research involves the elucidation of coupling mechanisms by which various forms of energy affect reaction rates (3).Vibrational excitation has been shown to promote certain bimolecular processes (4). In cases that involve mode selective reagent excitation, the rate enhancement may be sufficient to provide a basis for isotope separation (5). Reactivity and energy transfer characteristics of novel electronically excited species have also been investigated (6-8). Transient hot atom or hot radical momentum distributions arise in many nonequilibrium kinetic situations. Some examples include flash photolytic, combustion (9), explosion (10), chemical laser (11,12), radiolytic, photochemical recoil, nuclear recoil(13-15),and thick-target accelerated ion or atomic beam (l6) experiments. Principally because of limitations in the available methodology, the separate consideration of translational excitation in non-ionic systems has received comparatively little attention. Molecular beam experiments have provided much of the available information pertaining to collision energy effects in reactive scattering (17). However, the beam technique can be successfully utilized only for systems that involve large reaction cross sections (1), and the energy range characteristics of available beam sources are limited (18) Atomic recoil experiments offer an alternative procedure for investigating the chemical effects of trans lational excitation(19,20).Thermal (21,22) and energetic gas (23-25) ‡

†Present Address: Department of Chemistry, Cornell University, Ithaca, NY 14850. ‡University of California Regents Fellow 1977-78. © 0-8412-0399-7/78/47-066-314$10.00/0

Root; Fluorine-Containing Free Radicals ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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and l i q u i d phase (26,27) nuclear r e c o i l F studies have recentlybeen reported. Photochemical r e c o i l techniques (19«2*0 have not yet been u t i l i z e d for the study of hot fluorine atom reactions, and translationally excited non-ionic polyatomic reagents have been investigated only to a limited degree (28-50). The potential content of atomic r e c o i l experiments has recently been shown to include useful dynamical information (13-16,19,23-27,51»32). A central interpretive d i f f i c u l t y , however, follows from the need for unique and accurate s p e c i f i cations of the reactant non-Boltzmann momentum distributions. In the present work b r i e f reviews are given of the theoretical progress achieved thus far toward the resolution of this problem and of those aspects of the steady state hot atom kinetic theory (13) that are pertinent to modeling calculations. De t a i l e d modeling results are presented for nuclear r e c o i l F atoms reacting with H (15,31.53-35) i n order to i l l u s t r a t e the l e v e l of dynamical undersianding that can be obtained. Specific topics treated include the nature and significance of microscopic time dependent phenomena, the nature of the coupling between reactive cross section structure and energy ranges for hot atom reactions, and perturbations of hot reactive energy ranges by inert additives i n mixture experiments. 1 8

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Background Theory. The remainder of this a r t i c l e i s mainly concerned with the theoretical analysis of nuclear r e c o i l hot atom chemistry experi ments . Under t y p i c a l laboratory conditions the r e c o i l species are generated consecutively through irradiations having much longer duration than the characteristic hot atom mean free lifetime (25-27). It i s not unusual for the individual r e c o i l events to be isolated i n r e a l time. On this basis the early hot atom kinetic theories u t i l i z e d stochastic formulations for independent r e c o i l particle c o l l i s i o n cascades occurring i n thermally equilibrated molecular reaction systems. In a pioneering effort ( 5 6 - 5 8 ) Estrup and Wolfgang ( E W ) adapted c l a s s i c a l hard sphere neutron cooling theory i n order to obtain a time independent stochastic kinetic treatment of hot atom reactions. The primary hot reaction y i e l d followed from the Miller-Dodson equation (2^,2?) as the integral of the center-of-mass reaction probability [ P ( E ) ] over the c o l l i s i o n density distribution function [ N ( E ) ] . Because of the explicit neglect of time dependent relaxation processes, stochastic approaches of this general type cannot y i e l d dynamical information (ik). In most EW-theory applications c l a s s i c a l central force descriptions for elastic scattering have been adopted i n order to f a c i l i t a t e the evaluation of N ( E ) . In the absence of reactive perturbations, such treatments lead to N ( E ) distributions having the general form (l/αΕ). For c l a s s i c a l hard spheres the logarithmic energy loss constant ct follows from the hot


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atom and r e s e r v o i r p a r t i c l e masses ( 3 6 , 5 7 ) . For c l a s s i c a l s o f t spheres the b a s i c (l/oiE) form f o r N ( E ) i s p r e s e r v e d , but a must be determined from the r e p u l s i v e p a r t o f the i n t e r m o l e c u l a r p o t e n t i a l ( 3 8 ) . For more r e a l i s t i c s c a t t e r i n g d e s c r i p t i o n s and p o t e n t i a l s and f o r c o l l i s i o n energy regimes i n which r e a c t i o n s o c c u r , t h i s simple c l a s s i c a l d e s c r i p t i o n f o r N ( E ) r e quires modification. Although Wolfgang o f t e n l i k e n e d i t t o a n o n e q u i l i b r i u m counterpart o f the A r r h e n i u s e q u a t i o n , the E W - t h e o r y does not t r e a t temperature c o u p l i n g phenomena. In f a c t , u n t i l r a t h e r r e c e n t l y ( 1 3 , 1 5 , 2 5 , 5 1 ) temperature i n s e n s i t i v i t y has been accepted as a g e n e r a l c h a r a c t e r i s t i c o f w e l l behaved hot atom systems ( 3 9 ) . The p r i n c i p a l a p p l i c a t i o n of - t h e o r y has i n v o l v e d the e m p i r i c a l a n a l y s i s o f absolute hot y i e l d data measured f o r two component systems. I n d i v i d u a l r e a c t a n t s are t y p i c a l l y studied at progressively increasing d i l u t i o n using i n e r t gas a d d i t i v e s under c o n d i t i o n s of uniform sample s i z e , t o t a l p r e s s u r e , temperature and nuclear r e c o i l p r o d u c t i o n method ( l 9 , 2 0 ) . t The r e p o r t e d i n v e s t i g a t i o n s o f t h i s type are t o o numerous t o be catalogued h e r e . Reactive mixtures have a l s o o c c a s i o n a l l y been s t u d i e d ( 3 7 , ^ 1 , ^ 2 ) . The E W - t h e o r y data a n a l y s i s i n v o l v e s reduced p l o t t i n g procedures, which are s u p posed t o y i e l d i n f o r m a t i o n p e r t a i n i n g t o i n t e g r a t e d hot r e a c t i o n cross s e c t i o n s , r e a c t i o n energy range e f f e c t s and r e a c t i v e shadowing.* Refined cascade models have been i n c o r p o r a t e d i n s e v e r a l m o d i f i c a t i o n s of the o r i g i n a l EW-theory. K o s t i n and coworkers examined some o f the b a s i c r e l a x a t i o n a l assumptions u s i n g a mathematical model f o r a s t a t i c c o l l i s i o n d e n s i t v maintained a t steady s t a t e by a constant source term (k~5-h5) ^ P o t e n t i a l l y s e r i o u s experimental complications t h a t can a r i s e a t l a r g e moderator c o n c e n t r a t i o n i n c l u d e excessive r e c o i l l o s s (ho) and incomplete r e c o i l i o n charge exchange. Recoil l o s s enhances r a d i o l y t i c sample degradation and reduces the a v a i l a b l e r a d i o a c t i v i t y , thereby l e a d i n g t o diminished e x perimental accuracy. The s e v e r i t y o f these e f f e c t s c o u l d be c o n t r o l l e d through the technique o f i n c r e a s i n g the sample s i z e i n order t o accommodate the a d d i t i o n o f moderator. ^Shadowing i n t e r p r e t a t i o n s have o f t e n been employed i n m u l t i channel r e a c t i o n systems i n order t o account f o r r e l a t i v e product y i e l d v a r i a t i o n s w i t h sample composition. Reactive a d d i t i v e s "shadow" (deplete) y i e l d s from lower energy processes through e n e r g e t i c cross s e c t i o n components t h a t s e l e c t i v e l y i n t e r c e p t the cascading hot atoms. ^ T h i s more c o n v e n t i o n a l use o f the term steady s t a t e , which simply connotes time-independence f o r N ( E ) , i s fundamentally d i f f e r e n t from the dynamical d e f i n i t i o n t h a t a r i s e s i n the steady s t a t e hot atom k i n e t i c theory ( l 5 , l U ) .


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C o l l i s i o n d e n s i t i e s computed f o r more r e a l i s t i c molecular p o t e n t i a l s have supported the g e n e r a l a p p l i c a b i l i t y o f EW-theory with the p r o v i s o t h a t the energy l o s s parameter a must be allowed t o vary w i t h c o l l i s i o n energy. Porter's integral r e a c t i o n p r o b a b i l i t y (iRP) s t o c h a s t i c f o r m u l a t i o n u t i l i z e s genera l i z e d momentum t r a n s f e r cross sections (k6-k9). In photodisso c i a t i o n r e c o i l experiments the nonequilibrium reagent i s i n j e c t e d d i r e c t l y w i t h i n the hot r e a c t i v e c o l l i s i o n energy zone. A d e t a i l e d IRP-theory a n a l y s i s of t h i s s i t u a t i o n suggests t h a t d e convoluted r e a c t i o n cross sections can be d e r i v e d from a c c u r a t e hot y i e l d measurements c a r r i e d out over a range o f i n i t i a l c o l l i s i o n energies. However, t h i s a t t r a c t i v e procedure f o r c h a r a c t e r i z i n g the r e a c t i v e cross s e c t i o n energy dependence has remained c o n t r o v e r s i a l (50-5**-) » Other s p e c i a l i z e d extensions of s t o c h a s t i c hot atom k i n e t i c theory have a l s o appeared (55*56). S t o c h a s t i c hot atom k i n e t i c treatments have demonstrated v a r y i n g degrees of u t i l i t y with respect t o the s y s t e m i z a t i o n o f experimental r e s u l t s , c o n t r i b u t i n g t o the p a r t i a l c h a r a c t e r i z a t i o n o f nonreactive energy l o s s . Although c l a s s i c a l e l a s t i c s c a t t e r i n g models have been g e n e r a l l y recognized as s e r i o u s l y o v e r s i m p l i f i e d , no more complete treatment f o r nonreactive c o l l i s i o n s has y e t appeared ( 5 6 ) . A common feature o f these t h e o r i e s i s the n o t i o n t h a t an approximate d e s c r i p t i o n f o r hot atom moderation can be i n c o r p o r a t e d i n t o an i n t e r p r e t i v e model which then allows the c h a r a c t e r i z a t i o n o f r e a c t i v e energy range and shadowing e f f e c t s from measured product y i e l d s . However, Feng et a l . have questioned the v a l i d i t y o f t h i s approach (51), and EW-type t h e o r i e s have been shown t o be i n s e n s i t i v e t o the f a i l u r e o f u n d e r l y i n g assumptions concerning nonreactive c o l l i s i o n s ( 5 5 - 6 l ) . These r e s e r v a t i o n s are compounded by other conceptual d i f f i c u l t i e s i n c l u d i n g the neglect o f temperature c o u p l i n g and microscopic time dependent phenomena. Post hot r e a c t i o n unimolecular e f f e c t s ( l 9 . 2 5 . 6 2 . 6 5 ) c o n s t i t u t e a p o t e n t i a l l y s e r i o u s p r a c t i c a l problem (vide i n f r a ) t h a t has o f t e n been ignored i n moderator experiments w i t h polyatomic sub stances . Because the primary hot r e a c t i o n y i e l d ( 2 ί , 2 7 ) follows from the M i l l e r - D o d s o n e q u a t i o n , a l l open secondary decomposition channels must be d i r e c t l y monitored. Other e f f e c t s t h a t can mask the s i g n i f i c a n c e o f measured r e s u l t s i n c l u d e unimolecular c o l l i s i o n a l energy t r a n s f e r (6^-66) and c o l l i s i o n induced d i s s o c i a t i o n o f i n t e r n a l l y e x c i t e d primary hot r e a c t i o n products (67.68).t I n summary, recent t h e o r e t i c a l c a l c u l a t i o n s have supported the e m p i r i c a l u t i l i t y o f the reduced data p l o t t i n g procedures p r o s c r i b e d by EW-theory. However, the q u a n t i t a t i v e s i g n i f i c a n c e tRecent unpublished q u a s i c l a s s i c a l t r a j e c t o r y c a l c u l a t i o n s suggest t h a t c o l l i s i o n induced product d i s s o c i a t i o n may be o f minor importance i n the nuclear r e c o i l H v s . H r e a c t i o n system ( 6 9 ) » 3

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o f d e r i v e d k i n e t i c parameters i s u n c e r t a i n . I n a d d i t i o n t o these fundamental r e s e r v a t i o n s , i t a l s o seems apparent t h a t an adequate l e v e l of d e t a i l has g e n e r a l l y not been provided i n hot atom moderator experiments (vide i n f r a ) . Reasonable agreement has r e c e n t l y been achieved between t h e o r e t i c a l l y and experimentally d e r i v e d r e l a t i v e i n t e g r a t e d r e a c t i o n p r o b a b i l i t i e s f o r the nuclear r e c o i l H v s . H ( D ) system ( J 6 ) . Even though a r a t h e r l a r g e number o f such claims have appeared i n the l i t e r a t u r e , we s e r i o u s l y q u e s t i o n whether d e f i n i t i v e a p r i o r i i n f o r m a t i o n p e r t a i n i n g t o hot r e a c t i v e c o l l i s i o n energy d i s t r i b u t i o n s can be obtained from t h i s type of k i n e t i c a n a l y s i s . 3

2

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The Steady State Hot Atom K i n e t i c Theory. In order t o e l u c i d a t e the dynamical features of a hot atom r e a c t i o n , the microscopic time dependent c o l l i s i o n cascade must be modeled. Provided that the r e q u i r e d r e a c t i v e and nonreactive cross s e c t i o n data are a v a i l a b l e , t h i s can be accomplished u s i n g the steady s t a t e hot atom k i n e t i c theory ( 1 5 ). This section contains a b r i e f i n t r o d u c t i o n t o . t h e mathematical apparatus. L a t e r s e c t i o n s describe the modeling procedure and r e s u l t s o b t a i n e d f o r nuclear r e c o i l F atoms r e a c t i n g w i t h pure (jo) and i n e r t gas moderated ( j l ) H . These r e a c t i o n systems have been chosen f o r study because of the a v a i l a b i l i t y of q u a s i c l a s s i c a l t r a j e c t o r y r e a c t i o n cross s e c t i o n s C51) together w i t h r e s u l t s from thermal (21,72) and nonthermal (2§1 nuclear r e c o i l experiments. The steady s t a t e theory begins w i t h a f i c t i o n a l i z e d r e p r e s e n t a t i o n o f the hot atom cascade i n which the e n t i r e c o l l e c t i o n of atoms i s assumed t o be present i n i t i a l l y . This mathematically convenient model i s a l s o r i g o r o u s l y a p p l i c a b l e , provided t h a t the r e c o i l atoms are mutually n o n - i n t e r a c t i n g and t h a t they are produced w i t h very s m a l l t o t a l c o n c e n t r a t i o n and w i t h s p a t i a l u n i f o r m i t y throughout the host r e s e r v o i r . The time dependent r e l a x a t i o n of the hot atom momentum d i s t r i b u t i o n i s followed w i t h the a i d o f the Boltzmann e q u a t i o n . We b e g i n w i t h the f o r m u l a t i o n of the l a b o r a t o r y momentum d i s t r i b u t i o n f o r hot atoms (A) undergoing r e l a x a t i o n i n a bath o f pure r e s e r v o i r molecules (R) that are i n thermal and mechanical e q u i l i b r i u m w i t h each other and w i t h t h e i r surroundings. For s i m p l i c i t y the r e s e r v o i r species are assumed t o be i n i t i a l l y present i n a s i n g l e quantum s t a t e designated by s u b s c r i p t j . The g e n e r a l i z a t i o n t o systems o f g r e a t e r complexity i s g i v e n later. The e q u i l i b r i u m and time I t ) dependent n o n - e q u i l i b r i u m momentum d i s t r i b u t i o n s f o r species R and A are denoted f . ( P ) and g. ( P . , t ) . The time d e r i v a t i v e o f the hot atom momentum^ d e r i s i t y ~ i s given by the Boltzmann equation m o d i f i e d f o r the i n c l u s i o n of i n t e r n a l states (73). 1

8

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R

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I n E q . 1 n ( t ) denotes the number d e n s i t y o f the hot atoms; Ρ and μ the s c a l a r center-of-mas s momentum and reduced mass; σ ( j / k ) , the t o t a l cross s e c t i o n f o r e l a s t i c s c a t t e r i n g and i n e l a s t i c s c a t t e r i n g t o f i n a l quantum s t a t e k ; the d e t a i l e d s t a t e s p e c i f i c t o t a l r e a c t i v e cross s e c t i o n ; and the primed terms designate energy r e s t o r i n g c o l l i s i o n s . The second term, which accounts f o r hot atom d e p l e t i o n due t o r e a c t i o n s , can be i n t e g r a t e d over P^ t o y i e l d a time dependent chemical r a t e equation. ~ A

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The r a t e c o e f f i c i e n t [K ( t ) ] i n t e g r a l i n E q . 2 i s time dependent through the momentum d i s t r i b u t i o n f u n c t i o n g Equation 1 r e l a t e s the changing hot atom momentum density t o the d i s t r i b u t i o n functions and the cross s e c t i o n s f o r e l a s t i c , i n e l a s t i c and r e a c t i v e s c a t t e r i n g . We next r e q u i r e an e x p l i c i t expression f o r g » The r a t e o f c o l l i s i o n a l r e l a x a t i o n away from the i n i t i a l a n i s o t r o p i c l a b o r a t o r y v e c t o r momentum d i s t r i b u t i o n i s extremely r a p i d . The s p a t i a l part o f t h i s d i s t r i b u t i o n i s q u i c k l y randomized, so t h a t the average vector l a b o r a t o r y momentum becomes v a n i s h i n g l y s m a l l a f t e r only a few c o l l i s i o n s . ΐ In the s p i r i t o f the Chapman-Enskog treatment o f nonequilibrium trans port processes (jk) a zeroth order approximation f o r g a f t e r a very s m a l l number o f nonreactive encounters i s g i v e n by a Gaussian d i s t r i b u t i o n centered about the value z e r o . A

A

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Here the q u a n t i t i e s k , m and T . denote Boltzmann s constant, the hot atom mass and the time dependent hot atom temperature. A

!

ÎDepending upon the experimental method, the i n i t i a l hot atom l a b o r a t o r y v e c t o r momentum d i s t r i b u t i o n may or may not be a n i s o t r o p i c . The H e ( n , p ) H and o r d i n a r y u l t r a v i o l e t p h o t o d i s s o c i a t i o n techniques produce i s o t r o p i c d i s t r i b u t i o n s , whereas P(n,2n) F and p o l a r i z e d l i g h t induced phot odis s oc i a t ions do n o t . 3

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FREE

RADICALS

T h i s r e s u l t i s e s s e n t i a l l y equivalent t o the Chapman-Enskog l o c a l e q u i l i b r i u m approximation, which has proven q u i t e s u c c e s s f u l f o r the t h e o r e t i c a l r e p r e s e n t a t i o n of i r r e v e r s i b l e t r a n s p o r t processes f o r r e a l gases. Reasoning by analogy, the p h y s i c a l b a s i s f o r E q , 3 i n v o l v e s the simple n o t i o n that t r a n s l a t i o n a l r e l a x a t i o n occurs i s o t r o p i c a l l y and much more r a p i d l y than other r e l a x a t i o n modes, notably i n c l u d i n g nonthermal chemical r e a c t i o n s . The v a l i d i t y and l i m i t a t i o n s o f E q . 3 are of c e n t r a l i m portance t o the remainder o f t h i s d i s c u s s i o n . The steady s t a t e theory represents a fundamental departure from the s t o c h a s t i c methodology t h a t has been widely accepted by workers i n none q u i l i b r i u m chemical k i n e t i c s ( ΐ ί ) . We t h e r e f o r e d i g r e s s b r i e f l y t o consider the nature of l i m i t a t i o n s and p o s s i b l e improvements f o r E q . 3 along w i t h the r a t i o n a l e f o r steady s t a t e theory modeling c a l c u l a t i o n s . The appropriateness and u t i l i t y of the l o c a l e q u i l i b r i u m approach f o l l o w from the r e c o g n i t i o n t h a t non thermal chemical r e a c t i o n s can be meaningfully described as time dependent r e l a x a t i o n processes t h a t p r i n c i p a l l y compete w i t h momen tum t r a n s f e r . I t i s the d e t a i l e d b a l a n c i n g between these e f f e c t s t h a t comprises the core of a dynamical t h e o r y . Equation 3 represents a good approximation f o r s i t u a t i o n s i n which momentum r e l a x a t i o n takes p l a c e c o n s i d e r a b l y f a s t e r than nonthermal r e a c t i o n . The l o c a l e q u i l i b r i u m model becomes i n c r e a s i n g l y inadequate as these r a t e s approach one another, so t h a t the present form of the steady s t a t e theory w i l l be l e a s t accurate f o r systems t h a t i n v o l v e very r a p i d r e a c t i o n s . Higher order Chapman-Enskog s o l u t i o n s o f the Boltzmann e q u a t i o n , which provide successive degrees of refinement, c o u l d be i n c o r p o r a t e d i n t o the t h e o r y . Such m o d i f i c a t i o n s would introduce a d d i t i o n a l mathematical s t r u c t u r e i n E q . 3 , which i s probably not needed except f o r the d e s c r i p t i o n of systems t h a t c l o s e l y approach t r u e steady s t a t e b e h a v i o r . T h i s does not occur f o r any of the cases of present i n t e r e s t (vide i n f r a ) o r , indeed, f o r any known nuclear r e c o i l r e a c t i o n system. For t h i s fundamental reason and a l s o because of the crude l e v e l of approximation i n v o l v e d i n our t r e a t ment of nonreactive c o l l i s i o n s , the f u r t h e r refinement of E q . 3 has not y e t been considered t o be worthwhile. A paramount advantage of the above f o r m u l a t i o n f o r g^ i s conceptual. As i l l u s t r a t e d below, a s i n g l e i n t u i t i v e l y meaningful parameter, the hot atom temperature, provides a s u i t a b l e b a s i s f o r c h a r a c t e r i z i n g a l l of the s a l i e n t dynamical a t t r i b u t e s o f hot atom r e a c t i o n s through standard nonthermal r a t e c o e f f i c i e n t formalism. We f e e l t h a t the p r i n c i p a l and p o s s i b l y unique value of steady s t a t e theory modeling i s t o provide d e f i n i t i v e q u a l i t a t i v e d e s c r i p t i o n s f o r hot atom dynamical phenomena such as r e a c t i o n energy t i n the present context steady s t a t e s i g n i f i e s t h a t the r e a c t i v e and nonreactive r e l a x a t i o n r a t e s e x a c t l y b a l a n c e , so that both g and Τ cease t o v a r y w i t h t i m e . A

Δ


Chemistry

GRANT ET A L .

12.

of High

Energy

Atomic

321

Fluorine

range e f f e c t s , r e a c t i v e shadowing, nonthermal k i n e t i c isotope e f f e c t s , i n t e r n a l s t a t e c o u p l i n g ( l 5 , 2 5 ) , and other ambient temperature c o u p l i n g mechanisms (15). At the present stage o f t h e o r e t i c a l development, we b e l i e v e t h a t the c a l c u l a t e d r e s u l t s have semiquantitative numerical s i g n i f i c a n c e and t h a t the i n formation content o f the modeling c a l c u l a t i o n s has surpassed those of a l l but the most s o p h i s t i c a t e d hot atom experiments, A l i m i t a t i o n of the theory i s t h a t i t i s not u s e f u l f o r d i r e c t data m a n i p u l a t i o n . Even s o , the comparison of modeled v s . measured r e s u l t s can s t r o n g l y i n f l u e n c e the i n t e r p r e t a t i o n of nonthermal experiments. Returning now t o the development of mathematical apparatus, we next consider the important c h a r a c t e r i s t i c s of the hot atom temperature. E q u a t i o n 3 i s Maxwellian, so t h a t the average instantaneous hot atom l a b o r a t o r y k i n e t i c energy (E^) i s given by E q . k.

M u l t i p l i c a t i o n of t h i s r e s u l t by the t o t a l number of hot atoms [N^(t j] y i e l d s an expression f o r the t o t a l k i n e t i c energy ( E ^ ) , which can be solved f o r T and d i f f e r e n t i a t e d i n order t o o b t a i n a d e f i n i t i o n f o r the r a t e of T r e l a x a t i o n . A

A

dn^

2_

5

(5)

dt

3k

dt

\ Λ

2

Η

The q u a n t i t y (dE / d t ) follows from averaging [ P / 2 m l over g and then t a k i n g the time d e r i v a t i v e v i a the Boltzmann e q u a t i o n . A

2

A

A

dK

^

=

y t )

u (t) n

(6)

c (t)]

+

r

S u b s c r i p t s r and η i n E q . 6 denote the r e a c t i v e and n o n - r e a c t i v e terms, which f o l l o w d i r e c t l y from the equivalent energy expression t o E q . 1.

Ç/*J(P 7*n )

ξ

η

=

ξ

Γ

= -f*f

JdP

A

A

R

σ(όΑ)

(Ρ/,)

[ ;ΐ£ β

-

g^]

% (P 72m )f dP a*( ) (Ρ/μ) g f ^ A

A

R

d

A

(8)

The hot atom d e n s i t y d e r i v a t i v e i n E q . 5 i s obtained from E q . 2. dT


(7)

322

FLUORINE-CONTAINING

FREE

RADICALS

The s i g n i f i c a n c e o f the terms i n E q . 9 has been considered elsewhere (13). The q u a n t i t y ξ ( t ) describes t h e r a t e o f c o o l i n g o f the E d i s t r i b u t i o n induced By e l a s t i c and i n e l a s t i c c o l l i s i o n s . The other terms describe the r e l a t i o n s h i p between T^ and t h e average energy o f r e a c t i o n [ ] . F o r values smaller than 1 . 5 k T , the combined r e a c t i v e ïerms have a n e t h e a t i n g e f f e c t . For systems c h a r a c t e r i z e d by the abrupt onset o f h i g h l y e f f i c i e n t r e a c t i o n over a narrow T . i n t e r v a l , i t i s p o s s i b l e i n p r i n c i p l e f o r the c o o l i n g and h e a t i n g r a t e s t o become e x a c t l y b a l a n c e d . The d e r i v a t i v e (dT^/dt) then v a n i s h e s , corresponding t o the establishment o f a t r u e steady s t a t e hot atom momentum d i s t r i b u tion. As noted above, no r e a l p h y s i c a l system has y e t been i d e n t i f i e d t h a t i s b e l i e v e d t o f u l f i l l t h i s c o n d i t i o n . However, many nuclear r e c o i l systems probably e x h i b i t s u f f i c i e n t l y e n hanced r e a c t i o n r a t e s t h a t t h e i n h i b i t i o n o f c o o l i n g leads t o q u a s i steady s t a t e behavior (13,15,23»31»32-5*0* The nonthermal F v s . H ( D ) cases are o f t h i s l a t t e r t y p e . In general, quasi steady s t a t e (time independent) nonthermal r a t e c o e f f i c i e n t s can be a n t i c i p a t e d t o be u s e f u l f o r the a n a l y s i s o f data obtained f o r such systems. The time dependence i n E q . 9 o r i g i n a t e s e x c l u s i v e l y from the momentum d i s t r i b u t i o n g^, which i n t u r n depends upon T ^ . A

r

A

1 8

2

2

dT.

ΈΓ

F(

=

(10)

V

T h i s s i n g l e v a l u e d , unique r e l a t i o n s h i p can be i n v e r t e d i n order t o d e s c r i b e the c o u p l i n g between d i f f e r e n t i a l time and T^ i n t e r vals . dt

=

F "

1

^ ) dT

(ll)

A

The i n t e g r a t e d form o f E q . 11 s p e c i f i e s t h e elapsed time f o r a p a r t i c u l a r T^ change (75).

at

f

=

/

A F "

τ

1

X

A

1

(12)

( T ) dT A a

Here the i and f s u p e r s c r i p t s denote i n i t i a l and f i n a l c o n d i t i o n s . For multicomponent systems the only r e q u i r e d formalism change i n v o l v e s the i n c l u s i o n o f a d d i t i o n a l c o o l i n g and h e a t i n g terms.

dt

3k

i ^ni L

S i

2

i

J


12.


GRANT ET A L .

323

Here the index i s p e c i f i e s i d e n t i t y and X . denotes component mole f r a c t i o n . Further g e n e r a l i z a t i o n t o a l l o w a more complex i n i t i a l quantum s t a t e dependence r e q u i r e s the i n c l u s i o n o f p o p u l a t i o n d i s t r i b u t i o n c o r r e c t i o n s and an a d d i t i o n a l summation over index j . For r e a c t i v e a d d i t i v e s a n 3 terms i n I q . 13 must be e v a l u a t e d , but only the c o o l i n g term c o n t r i b u t e s t o (dT^/dt) f o r i n e r t moderators. Steady State Theory Model C a l c u l a t i o n s . The s i m u l a t i o n procedure f i r s t r e q u i r e s the d e t a i l e d s p e c i f i c a t i o n of the system t o be modeled i n c l u d i n g sample com p o s i t i o n , t o t a l p r e s s u r e , ambient temperature, molecular c o n s t a n t s , r e a c t i v e cross s e c t i o n s and a set of T^ v a l u e s . ΐ Equation 9 i s then evaluated at each T. based upon s u i t a b l e approximate r e p r e s e n t a t i o n s f o r E q s . 2 , 7 and 8 . The energy dependent e l a s t i c and i n e l a s t i c cross s e c t i o n data r e q u i r e d f o r use i n E q . 7 are not a v a i l a b l e . A c c o r d i n g l y , i n our i n i t i a l c a l c u l a t i o n s E q . J has been approximated by combining d e r i v e d expressions f o r hard sphere e l a s t i c energy l o s s and hot atom v s . e q u i l i b r i u m Maxwellian r e s e r v o i r p a r t i c l e c o l l i s i o n f r e quency. An ad hoc zeroth order c o r r e c t i o n f o r r e s t o r i n g c o l l i s i o n s , which prevents T. from r e l a x i n g below ambient tempera t u r e ( T j , has been i n c o r p o r a t e d i n E q . 7 through replacement o f T by the q u a n t i t y ( T - T ) . Although r e s t o r i n g c o l l i s i o n s should be r e l a t i v e l y u n important a t energies s u b s t a n t i a l l y l a r g e r than k T , the above treatment of nonreactive energy l o s s represents our most d r a s t i c assumption. The present d e s c r i p t i o n of hot atom moderation i s s i m i l a r t o those p r e v i o u s l y employed i n the EW-theory ( 5 6 , 5 7 ) and i n Monte C a r l o model c a l c u l a t i o n s by Koura ( l £ ) . Classical t r a j e c t o r y c a l c u l a t i o n s c o n s t i t u t e the most probable future source f o r t o t a l i n e l a s t i c cross s e c t i o n s . Available approxi mate treatments account f o r the c o l l i s i o n energy dependence of e l a s t i c cross s e c t i o n s over narrow energy ranges ( 7 6 ) . To some e x t e n t , f o r polyatomic substances the e r r o r s a s s o c i a t e d w i t h our incomplete d e s c r i p t i o n s of e l a s t i c and i n e l a s t i c energy l o s s are i n t e r n a l l y compensating. At moderate t o l a r g e c o l l i s i o n energies the e l a s t i c cross s e c t i o n s have been overestimated, while the i n e l a s t i c values have been underestimated. The average energy t r a n s f e r [] between a hot atom w i t h l a r g e average energy ΤΪΓ ) and a s t a t i o n a r y r e s e r v o i r p a r t i c l e i s given by E q s . Ik and 1 5 . A

A

=

ο

(HO

îThe use o f r e s e r v o i r d e n s i t y reduced time u n i t s permits a s i m p l i f i e d pressure independent r e p r e s e n t a t i o n o f the primary hot atom r e a c t i o n s and momentum r e l a x a t i o n processes.


324

FLUORINE-CONTAINING

FREE

RADICALS

(15) The hot atom c o l l i s i o n frequency ( z ^ ) follows from i n t e g r a t i o n of the hard sphere e l a s t i c cross s e c t i o n over the d i s t r i b u t i o n functions f o r n^ and n^

•3/2 Z

AR

=8π η σ μ

Ε

Α Ε

[2ΤΤ

μ Κ

Γ ]- / Α

5

(16)

2

MT AJ

M

In E q s . 1^-16 M denotes the t o t a l mass of c o l l i d i n g p a r t i c l e s and θ i s given by E q , 17.

Γm kT 1

μ h

R

Ί

1 m

A

k

1

1 kT

(IT)

\ kT

I n u n i t s o f temperature the s i n g l e component r e s u l t f o r ξ follows from E q s . Ik-17: « (t)

=

n

2Si M

Z

(t)

(18)

AR A T

Hard sphere e l a s t i c cross sections [ σ ] have been obtained from averaged molecular force constants as determined from e x perimental equation of s t a t e and t r a n s p o r t property data ( 7 6 , 7 7 ) . The 2k.2 A value f o r F represents the s e l f - c o l l i s i o n e l a s t i c cross s e c t i o n f o r Ne. The mixed values f o r ^ F v s . H , A r and Xe then f o l l o w as 2 5 . 9 A , 3 0 . I A and 3 6 . 6 A , r e s p e c t i v e l y . Reactive cross sections s u i t a b l e f o r the s i m u l a t i o n o f energetic r e a c t i o n s through E q s . 2 and 8 have been c a l c u l a t e d v i a q u a s i c l a s s i c a l t r a j e c t o r y methods (78). I n a preceding paper Feng et a l . (31) demonstrated t h a t knowledge of the p o t e n t i a l energy surface topology (79) f o r the FHH c o l l i n e a r c o n f i g u r a t i o n i s s u f f i c i e n t t o determine r e a c t i v e cross sections t o the p r e s e n t l y d e s i r e d l e v e l o f accuracy. The q u a n t i t a t i v e v a l i d i t y o f t h i s model f o r d e s c r i b i n g epithermal c o l l i s i o n processes has been severely c u r t a i l e d by the approximate formu l a t i o n f o r ξ ( t ) . For t h i s reason i t has not been necessary t o o b t a i n r e f i n e d r e a c t i v e cross sections corresponding t o thermally a c c e s s i b l e c o l l i s i o n s . These simulations have u t i l i z e d r e a c t i v e cross sections computed f o r Muckerman s optimized LEPS surface V ( 8 0 ) . A n a l y t i c a l equations have been f i t t e d t o the t r a j e c t o r y cross s e c t i o n data through nonlinear r e g r e s s i o n analysis. The f i n a l r e s u l t f o r ξ ( t ) i s given by E q s . 19 and 2 0 . Α Τ ?

2

1

8

8

o

2

2

2

2

1


12.

Chemistry

GRANT ET A L .

ξ

(t)

of High

Energy

Atomic

325

Fluorine

=

E

2

a*(j)

βχρ(-θΕ) dE

-5/2

K]χ

/

Ε

5 / 2

[?

^

+

exp(-OE) dE

(19)

Ό

θ'

=

ά θ

d(l/kT J A

μ = — Similarly,

K(t)

2μ(Τ (m^

+

r(t)

=

Α

- Τ) + m T) A

follows

^

m

A

^ A T

( ^

+

"

Τ

^

A

from E q . 2 .

Ι^μη^ίΓμΗ^]" / 3

2

-3/2

M 4J / * +

(20)

+ m T;-=

Εσ

ω exp(-9E)

π

(21)

Equations 16-21 permit the e v a l u a t i o n o f E q . 9 at each d e s i r e d T. v a l u e . The r e s u l t i s then i n v e r t e d and i n s e r t e d i n t o E q . 12 followed by numerical i n t e g r a t i o n over the d e s i r e d T^ range i n order t o c h a r a c t e r i z e the time dependent r e l a x a t i o n of the hot atom momentum d i s t r i b u t i o n . The Pure H

g

R e a c t i o n System.

C a l c u l a t e d T. r e l a x a t i o n r e s u l t s obtained at 300°Κ ambient temperature have been shown i n Table I and F i g . 1. In order t o remove an a r t i f i c i a l dependence upon the r e s e r v o i r p r e s s u r e , d e n s i t y reduced time u n i t s [ ( t - n ^ ) , molecule sec cm" ] have been employed throughout the present d i s c u s s i o n . As shown i n


326

FLUORINE-CONTAINING F R E E

RADICALS

Table I . R e a c t i o n Rate C o e f f i c i e n t s and T. R e l a x a t i o n R e s u l t s f o r Pure H a t 300°Κ Ambient Temperature. 2

(°Κ) χ 10

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 500 400 300 200 100 50 4o

30 20 10 5 4

3

2 1 0.5

(cm

3

molecule

sec

0.077 O.O9O 0.107 0.131 0.164 0.216 0.299 ΟΛ52 0.763 Ο.97Ο 0.985 Ο.876 Ο.658 0.448 Ο.388 Ο.318 Ο.236 O.I34 0.076 0.064 0.052 0.040 0.028 0.022

) χ 10

9

(molecule

sec c m

3

) χ 10

0.0215 0.0470 Ο.Ο78Ο 0.116 0.167 0.235 Ο.338 0.517 0.9^9 1.55 1.76 2.05 2.48 3-33 4.42 4.84 5.46 6.47 8.79 12.1 13.5 15.5 18.9 26.5 38.6

a. The i n i t i a l value corresponding t o each t a b u l a t e d f i n a l Τ i s g i v e n by the p r e c e d i n g e n t r y . b. C f . F i g . 2.


12.

Chemistry

GRANT ET A L .

of High

Energy

Atomic

327

Fluorine

F i g . 1 T e x h i b i t s an i n i t i a l l y r a p i d decrease f o r pure H , l e v e l l i n g o f f a t values below 1 0 K j f o r r e l a x a t i o n times longer than 5 χ 1 0 molecule sec c m " . The T. r e l a x a t i o n r e s u l t s f o r mixed A r / H systems are considered; i n the f o l l o w i n g section. The reduced time f o r m u l a t i o n f o r the nonthermal r e a c t i o n r a t e i s given by E q . 22, A

2

5 o

3

9

2

-*(t)n (t)

(22)

A

which can be i n t e g r a t e d i n order t o o b t a i n the time of the hot atom d e n s i t y .

n (t) A

=

n° exp

*(t)

dt

dependence

J

(23)

Time dependent r a t e c o e f f i c i e n t s c a l c u l a t e d v i a E q . 21 ( c f . F i g . 2) are numerically i n t e g r a t e d u s i n g E q . 23 i n order t o o b t a i n n « ( t ) . Reaction r a t e s , which then f o l l o w from E q . 22, are i n t e g r a t e d i n a s i m i l a r f a s h i o n i n order t o produce time dependent hot yields [ Y ( t ) ] . f

J

Y(t ) f

K(t) n ( t ) A

(2k)

dt

The time dependent hot atom d e n s i t i e s , r e a c t i o n r a t e s and y i e l d s d e p i c t e d i n F i g s . 3 and k r e v e a l that nonthermal r e a c t i o n com mences almost immediately i n pure H because of the l a r g e mag nitude of the h i g h energy cross s e c t i o n . The maximum r e a c t i o n r a t e occurs at an elapsed time of ç a . 1.0 χ 1 0 molecule sec cm , corresponding t o a r e a l r e l a x a t i o n time of c a . 30 picoseconds at 300°K and 1000 T o r r p r e s s u r e . Ninety percent o f the h o t y i e l d i s produced a t times s h o r t e r than 3 χ 1 0 molecule sec cm , and the r e a c t i o n has n e a r l y cegsed f o l l o w i n g an elapsed time of 5 χ 1 0 molecule sec cm * A 99.8$ c a l c u l a t e d t o t a l hot y i e l d follows somewhaj a r b i t r a r i l y by t r u n c a t i n g E q . 2k at a t o t a l elapsed time (t ) value of 2.65 χ 1 0 molecule sec cm" . From Table I t h i s Y ( t ) c u t o f f corresponds t o an equivalent f i n a l hot atom temperature ( T . ) o f 1000°K. I n agreement with the range of p l a u s i b l e values deduced from experiment (23.), the thermalized F atom f r a c t i o n i n pure H i s n e g l i g i b l y s m a l l . Because the present £ (t) model exaggerates the e f f i c i e n c y of c o o l i n g processes at large T , t h i s c a l c u l a t i o n tends t o overestimate the f r a c t i o n of the i n i t i a l F atom d i s t r i b u t i o n that f a i l s t o undergo nonthermal r e a c t i o n . 2

9

9

3

9

f

1 0

1

8

2

A

1

8


328

FLUORINE-CONTAINING

Figure 1. Rehxation time behavior of the hot atom temperature. Key to curves: A, Pure H ; B, 50 mol %; C, 99 mol % Ar-moderated H at 300°Κ ambient tem perature.

F R E E RADICALS

2

2

t-n

R

0.0 1.0 2.0 3.0 (molecule sec crrr )x10~

H o t A t o m T e m p e r a t u r e (°K)

0.0 t-n

R

0.2 0.4 0 . 6 0 . 8 1.0 ( m o l e c u l e s e c c m ) χ 10" 3

Figure 2. Time (curve A) and T (curve B) dependent nonthermal rate coefficients for the F + Η reaction at 300° Κ ambient temperature A

18

2


3

1 0

12.

GRANT ET A L .

Chemistry

of High

Energy

Atomic

Fluorine

Figure 3. Time-dependent hot yields (curve A) and surviving F atom fractions (curve B) for pure H at 300°Κ ambient temperature 18

2

Figure 4. Time-dependent hot yields (curve A) and reaction rates (curve B) for pure H at 300°Κ ambient temperature 2


329

330

FREE

FLUORINE-CONTAINING

RADICALS

From F i g s . 2 and k the r a p i d d e c l i n e i n n^(t) a t l a r g e time e f f e c t i v e l y determines the width of the r e a c t i o n r a t e d i s t r i b u t i o n ( c f . E q . 2 2 ) . A problem of c e n t r a l importance i n hot atom chemistry concerns the c h a r a c t e r i z a t i o n of nonthermal r e a c t i v e c o l l i s i o n energy ranges. F i g u r e 1 and E q s . 1 0 - 1 2 show t h a t the r e l a t i o n s h i p between T ^ and t i m e , w h i l e s t r o n g l y n o n l i n e a r , i s s i n g l e - v a l u e d and unique. Thus, as shown i n F i g . 2 and Table I , the time dependent r a t e c o e f f i c i e n t s can a l s o be e x pressed i n terms of e q u i v a l e n t T . dependences. In a s i m i l a r f a s h i o n the temperature dependent y i e l d s and the f a U o f f i n hot atom d e n s i t y f o r the r e l a x i n g d i s t r i b u t i o n have been shown i n F i g . 5? and the corresponding r e a c t i o n r a t e data i n F i g . 6. Since the average F l a b o r a t o r y k i n e t i c energy i s simply 1 . 5 k T ^ , these r e s u l t s c l e a r l y c o n t a i n r e a c t i v e c o l l i s i o n energy dependence i n f o r m a t i o n . However, t h e i r p r e c i s e quan t i t a t i v e s i g n i f i c a n c e i s somewhat u n c e r t a i n because the c a l c u l a t e d r e a c t i o n r a t e s and y i e l d s represent the behavior o f r e l a x i n g Maxwellian d i s t r i b u t i o n s o f F atom l a b o r a t o r y k i n e t i c energies. I n order t o s p e c i f y the corresponding center-of-mass r e a c t i v e c o l l i s i o n energy d i s t r i b u t i o n s , the coordinate t r a n s formation and d i s t r i b u t i o n u n f o l d i n g problems must be s o l v e d . T h i s has been accomplished a n a l y t i c a l l y f o r the case of s m a l l mass hot atoms r e a c t i n g with l a r g e mass r e s e r v o i r species ( 1 3 ) . Neither a n a l y t i c a l nor numerical s o l u t i o n s are yet a v a i l a b l e f o r the F v s . H system. A s l i g h t a d d i t i o n a l ambiguity a r i s e s from the i n t r i n s i c n o n l i n e a r i t y o f the T . v s . r e l a x a t i o n time r e l a t i o n s h i p , which gives r i s e t o exaggerated r e l a t i v e weighting o f low temperature r e a c t i o n processes i n Τ dependent r e a c t i o n r a t e or y i e l d p l o t s . 1

8

1

1

8

8

2

A simple procedure f o r i l l u s t r a t i n g the s i g n i f i c a n c e of the T^ dependent r e s u l t s has been shown i n F i g . 7 · The q u a n t i t y -(dY/dLogT ) follows from d i r e c t numerical d i f f e r e n t i a t i o n o f the c a l c u l a t e d Y ( T ~ ) r e s u l t s . Integration of t h i s d i s t r i b u t i o n over any i n t e r v a l on the Log T^ s c a l e gives the f r a c t i o n of the hot y i e l d corresponding t o the s p e c i f i e d T. range.

(25)

P l o t s of - ( d Y / d L o g T . ) v s . Log T. thus c l e a r l y r e v e a l the nature o f the average F l a b o r a t o r y k i n e t i c energy dependence o f the hot y i e l d . T h i s data p r e s e n t a t i o n technique i s e s p e c i a l l y u s e f u l f o r the a n a l y s i s of r e s u l t s obtained f o r multicomponent and m u l t i c h a n n e l r e a c t i o n systems. We next consider how c l o s e l y the nuclear r e c o i l F vs. H system may have approached t r u e h i g h temperature steady s t a t e 1

8

1

8

2


12.

Chemistry

GRANT E T A L .

of High

Energy

Atomic

Fluorine

in

1.0

\

~ ο 0.4 I

0.2 -/ 0.0

/ 0.0

t-n

-

JBWA

R

\ ι 0.2

ι""~*Ί 0.4 0.6

( m o l e c u l e sec

0.8

0.0 1.0

cm )x10 _ 3

- 1 0

Figure 8. Time-dependent scaled hot yields (curve A) and reaction rates (curve B) for 50 mol % Ar-moderated H at 300°Κ ambient temperature 2


336

FLUORINE-CONTAINING

1.0

ι

-A—r |\

0.8

ι

FREE RADICALS

1.0

1

ω

π o.8 a

300°

Φ

CL

"D

0.6

-

\

B

/

-

k

0.4

Ο 0.4 I

5

D

0.2 ο

0.2 0.0

0 6 J

/

I

OO t-n

0.2 D

I

I

0.4

0.6

0.8

0.0 1.0

(molecule sec cm" )xlO 3

-10

Figure 9. Time-dependent scaled hot yields (curve A) and reaction rates (curve B) for 99 mol % Ar-moderated H at 300°Κ ambient temperature g

Figure 10. Time-dependent scaled hot yields (curve A) and reaction rates (curve B) for 99 mol % Xe-moderated H at 300°Κ ambient temperature s


12.

GRANT E T A L .

337


the importance o f H r e a c t i v e self-shadowing, which r e s u l t s from the r a p i d and e f f i c i e n t d e p l e t i o n o f the hot atom den s i t y a t l a r g e c o l l i s i o n energy i n the absence o f moderator. The nonthermal r e a c t i o n r a t e s e x h i b i t s i g n i f i c a n t time dependence v a r i a t i o n s a s s o c i a t e d w i t h t h e presence o f i n e r t moderators. From F i g s . k 8 and 9 the a d d i t i o n o f 99 mole $ A r t o H causes the maximum r e a c t i o n r a t e t o be e s t a b l i s h e d roughly twofold more q u i c k l y . Presumably as a r e s u l t o f t h e rough s i m i l a r i t y i n moderating e f f i c i e n c i e s between H and Xe Tcf. Table I I I ) , a s i m i l a r e f f e c t does not occur i n 99 mole $ Xe moderated H . T h i s shortening o f t h e nonthermal r e a c t i o n i n d u c t i o n p e r i o d seems p r i m a r i l y t o r e f l e c t l a r g e t o t a l c o o l i n g rate increases. Table IV and F i g s , k and 8 - 1 0 show t h a t the progressive a d d i t i o n o f moderator a l s o causes the r e a c t i o n r a t e curves t o become i n c r e a s i n g l y skewed t o longer r e l a x a t i o n t i m e s . The t a b u l a t e d τ values were obtained from E q . 26 w i t h a t c u t o f f corresponding t o 1000°Κ T ^ . A convenient measure o f the r a t e d i s t r i b u t i o n width i s provided by - C t ^ 2

9

2

2

2

f

2

J 2

t

2

K ( t ) n ( t ) dt A

= ~~£?

J

(27) K(t) n ( t ) dt A

As r e v e a l e d by these τ and r e s u l t s , the skewing o f the r a t e d i s t r i b u t i o n s i s s t r o n g l y enhanced f o r Xe r e l a t i v e t o A r . As t h e t o t a l y i e l d f a l l s o f f w i t h i n c r e a s i n g moderator c o n c e n t r a t i o n , the hot atom d e n s i t y s u r v i v e s t o sample t h e r a t e c o e f f i c i e n t a t longer r e l a x a t i o n times than i s p o s s i b l e i n pure H . Since r a p i d c o o l i n g through the epithermal r e g i o n l i m i t s the e f f e c t i v e n e s s o f these "slow" r e a c t i o n s , the skewing e f f e c t i s l e s s pronounced f o r A r i n comparison w i t h Xe. We conclude t h a t i n e f f i c i e n t l y moderated systems the l e n g t h o f the i n d u c t i o n p e r i o d r e q u i r e d t o e s t a b l i s h the maximum r e a c t i o n r a t e decreases, whereas the average r e a c t i v e lifetime τ increases. The a n a l y s i s of T . dependence r e s u l t s r e v e a l s the p h y s i c a l b a s i s f o r the above noted changes i n the hot atom r e a c t i v e l i f e t i m e d i s t r i b u t i o n s , i n c l u d i n g the ambient temperature e f f e c t s shown i n Tables I I I and I V . The T^ dependent s c a l e d r e a c t i o n r a t e s , y i e l d s and -(dY/dLogT^) d i s t r i b u t i o n s have been p l o t t e d i n F i g s . 11-13. The r a t e data have been s c a l e d as noted p r e v i o u s l y . Y i e l d s have been p l o t t e d as f r a c t i o n s of the t o t a l values c a l c u l a t e d f o r t h e r e s p e c t i v e 300°Κ r e a c t i o n systems. The - ( d Y / d L o g T . ) d i s t r i b u t i o n s then f o l l o w from d i r e c t numerical d i f f e r e n t i a t i o n o f the s c a l e d y i e l d s . The most probable - ( d Y / d L o g T . ) values g e n e r a l l y e x h i b i t d i s 2

2


338

FLUORINE-CONTAINING

F R E E RADICALS

Figure 11. T dependent scaled hot yields (curves A), reaction rates (curve B), and —(dy/ d log T ) distributions (curves C) for 50 mol % Ar-moderated H A

A

2

1.0

ι

y

1.0

3 0 0 ° /

-

Ο

T3

^

0.6

-

0.4

-

0.6

/

0.4

/c /

VNN^OO^

0.2

ο 0.0

I

0.8

/ Β\

\\IO°

0.2

1 · 0.0 10 10 10 10 10 Hot A t o m T e m p e r a t u r e (°K) 7

6

5

4

3

Figure 12. T dependent scaled hot yields (curves A), reaction rates (curve B), and —(dY/ d log T ) distributions (curves C) for 99 mol % Ar-moderated T A

A

2


12. GRANT ET AL.

Chemistry

of High

Energy

Atomic

Fluorine

Figure 13. T dependent scaled hot yields (curves A), reaction rates (curve B), and —(dY/ d log Ί ) distributions (curves C) for 99 mol % Xe-moderated H A

Α

2


339

340

FLUORINE-CONTAINING

FREE

RADICALS

Table IV*. P r o p e r t i e s of Time Dependent Nonthermal Reaction Rate D i s t r i b u t i o n s a t 300°Κ and 10°Κ Ambient Temperatures. Ambient Temperature

Argon Concentration (Mole f ) -

300°Κ

10°Κ

0

τ

τ

(molecule sec cm ) χ 10 3

0.0 10.0 30.0 50.0 70.0 90.0 99.0 99.0a a.

1.68 1.80 2.15 2.53 2.85 3.10 3.19 5.32

0.59 Ο.78 1.23 1.65 1.97 2.17 2.22 6.2k

1.6k 1.70 1.85 1.99 2.07 2.10 2.10 3^7

9

0.51 0.60 Ο.76 0.86 Ο.89 0.88 0.87 2.36

Xe moderator.

placement t o reduced T . , showing t h a t the average F laboratory k i n e t i c energy sampled by nonthermal r e a c t i o n decreases w i t h i n c r e a s i n g moderator c o n c e n t r a t i o n . A t A r concentrations below 50 mole t h i s e f f e c t i s accompanied by simultaneous i n c r e a s e s i n the - ( d Y / d L o g T . ) d i s t r i b u t i o n widths r e l a t i v e t o the r e s p e c t i v e most probable T^ v a l u e s . As noted above we have not y e t attempted t o c a l c u l a t e average r e a c t i v e c o l l i s i o n energies i n the center-of-mass coordinate system. The c l o s e s t present approximation t o t h i s q u a n t i t y follows from the most probable hot atom temperature [] obtained, f o r each -(dY/dLogT^) d i s t r i b u t i o n . These r e s u l t s have been expressed as equivalent average F l a b o r a t o r y k i n e t i c energies [1.5k] and center-of-mass c o l l i s i o n energies i n Table V . As estimated i n t h i s f a s h i o n , the most probable r e a c t i v e c o l l i s i o n energy v a r i e s from 9*7 eV molecule i n pure H t o 1.2 eV m o l e c u l e " i n 99 mole °jo He, A r or Xe moderated H . Reactive c o l l i s i o n energy d i s t r i b u t i o n widths a t h a l f maximum, which have been estimated i n an analogous manner, increase by more than t h r e e f o l d accompanying the a d d i t i o n o f 99 mole # He, A r or Xe t o pure H . The above r e s u l t s c l e a r l y r e v e a l the nature o f the r e a c t i o n energy range changes t h a t take place i n moderated nuclear r e c o i l hot atom chemistry experiments. In s i t u a t i o n s c h a r a c t e r i z e d by broad Κ (τ^) functions and l a r g e i n t r i n s i c r e a c t i v i t y , the a d d i t i o n o f moderator increases the r e l a t i v e width o f the r e a c t i v e c o l l i s i o n energy d i s t r i b u t i o n and diminishes i t s average magnitude. 1

8

1

1

2

8

1

2

2


GRANT ET AL.

12.

Table V .


341

P r o p e r t i e s of -(dY/dLogT^) D i s t r i b u t i o n s a t 300°Κ Ambient Temperature.

Moderator (Concentration, Mole $)

A

(°K)

xlO"

(eV

5

Molecule" ) 1

e Distri bution Width

3.27 Ο.89

96.7 U2.3 11.5

9.7 ^3 1.2

22.2 15.2

Xe (99.0)

Ο.89

11.5

1.2

9Λ

He (99.0)

Ο.89

11.5

1.2

9Λ

Ar(O.O)

7Λ8

(50.0) (99.0)

9Λ

Another c h a r a c t e r i s t i c of the - ( d Y / d L o g T . ) d i s t r i b u t i o n s t h a t we b e l i e v e t o be g e n e r a l f o r systems of t h i s type follows from F i g s . 12 and 13. These d i s t r i b u t i o n s approach a pure moderator l i m i t i n g form that i s independent of moderator i d e n tity. The average r e a c t i v e c o l l i s i o n energy thus does not d e crease i n d e f i n i t e l y with i n c r e a s i n g moderation. For the non thermal F v s . A r / H system i t achieves an e s s e n t i a l l y c o n stant value at Ar concentrations i n excess of c a . 90 mole $ ( c f . Table I V ) . The form of the l i m i t i n g -(dY/dLogT^) d i s t r i b u t i o n i s c o n t r o l l e d by Κ (τ ), which thus governs the e f f e c t i v e c o l l i s i o n energy r e s o l u t i o n that can be achieved i n moderated hot atom chemistry experiments. W i t h i n the framework of the present nonreactive c o l l i s i o n model, i n the complete absence o f r e a c t i v e shadowing the l i m i t i n g d i s t r i b u t i o n should be the same f o r a U i n e r t moderators. We now consider the upturn i n - ( d Y / d L o g T . ) observed a t s m a l l T . values f o r h i g h l y moderated 300°Κ ambient temperature systems. The increases that occur f o r T values below 5000°Κ i n d i c a t e the onset o f contamination by epithermal r e actions. Both the ξ (t) and hot atom c o l l i s i o n frequency (z ) models i n c l u d e a d hoc zeroth order c o r r e c t i o n s f o r t r a n s lational restoring collisions. The c a l c u l a t e d low energy r e a c t i o n y i e l d s can be reduced—though not eliminated—through removal of the r e s t o r i n g c o r r e c t i o n term from ξ ( t ) . How e v e r , a more e f f e c t i v e suppression technique i n v o l v e s r e d u c t i o n of the ambient temperature t o 100°Κ or below. From Table I I I and F i g . 12 the epithermal contamination approaches 20$ of the t o t a l hot y i e l d at 300°Κ and 99 mole $ Ar c o n c e n t r a t i o n . In order t o i l l u s t r a t e the q u a n t i t a t i v e importance of t h i s e f f e c t , the 10°Κ γ(τ^) and -(dY/dLogT^) r e s u l t s shown i n F i g s . 11-|3 have been s c a l e d r e l a t i v e t o the corresponding 3 00° Κ Y ( T ) values. Because of the l i m i t a t i o n s inherent i n the p r e sent t h e o r e t i c a l model, the numerical s i g n i f i c a n c e o f 1

8

2

A

n

AR

a


FLUORINE-CONTAINING

342

F R E E RADICALS

these c a l c u l a t e d e p i t h e r m a l y i e l d s i s u n c e r t a i n . However, the q u a l i t a t i v e i n d i c a t i o n s are v a l i d ( i ) t h a t e p i t h e r m a l r e a c t i o n s may cause i n t e r f e r e n c e i n moderated hot atom chemistry e x p e r i ments, ( i i ) t h a t the e p i t h e r m a l y i e l d s w i l l be s t r o n g l y d e pendent upon the ambient temperature, and ( i i i ) t h a t these c o n c l u s i o n s have g e n e r a l a p p l i c a b i l i t y except f o r s i t u a t i o n s t h a t i n v o l v e t h e r m a l l y i n a c c e s s i b l e hot r e a c t i o n s . Energetic H and F atomic s u b s t i t u t i o n p r o c e s s e s , f o r example, have center-of-mass t h r e s h o l d energy requirements o f 1.5 eV or more J2.k). The e f f i c a c y o f competitive r e a c t i o n techniques f o r c o n t r o l l i n g e p i t h e r m a l contamination w i l l be determined i n f u t u r e model c a l c u l a t i o n s . An important consequence of the r e s u l t s shown i n Table V i n v o l v e s the p o s s i b l e s i g n i f i c a n c e o f hot y i e l d s measured i n moderator experiments w i t h polyatomic r e a c t a n t s . For nonthermal r e a c t i o n s t h a t sample broad Κ ( T . ) under c o n d i t i o n s of l a r g e t o t a l hot r e a c t i v i t y , the r e l a t i v e decrease i n the average r e a c t i v e c o l l i s i o n energy a t the moderated l i m i t may approach one order o f magnitude. The secondary decomposition behavior of a polyatomic primary r e a c t i o n product would t h e n l i k e l y e x h i b i t a marked dependence upon the moderator c o n c e n t r a t i o n . The present r e s u l t s t h e r e f o r e demonstrate t h a t such experiments must i n c l u d e the c h a r a c t e r i z a t i o n and monitoring o f a l l open secondary decomposition channels under the f u l l range of i n vestigated conditions. F i n a l l y , we b r i e f l y consider the p h y s i c a l s i g n i f i c a n c e o f the T dependent s c a l e d r e a c t i o n r a t e s shown i n F i g s . 6 and 11-13. Accompanying the a d d i t i o n of i n e r t moderator, the s c a l e d r a t e s c o n s i s t e n t l y i n c r e a s e a t a l l T. values below 10 K. T h i s e f f e c t simply r e f l e c t s the f a c t t h a t the hot atom d e n s i t y i s not s i g n i f i c a n t l y d e p l e t e d d u r i n g the r e l a x a t i o n process a t l a r g e moderator c o n c e n t r a t i o n . It i s p a r t i c u l a r l y dramatic a t s m a l l values t h a t do not c o n t r i b u t e a p p r e c i a b l y t o the hot y i e l d i n pure H . However, r e l a t i v e t o pure H even the most probable s c a l e d r a t e e x h i b i t s a twofold i n c r e a s e i n the presence of 9 9 mole °j A r . These r e s u l t s provide a u s e f u l b a s i s f o r e l u c i d a t i n g the q u a l i t a t i v e nature and r o l e of r e a c t i v e shadowing e f f e c t s i n s i n g l e - c h a n n e l nonthermal r e a c t i o n systems. More complex s i t u a t i o n s can best be analyzed i n terms of the dynamic i n t e r p l a y between the hot atom d e n s i t y and the a v a i l a b l e nonthermal r a t e c o e f f i c i e n t s . I n f u t u r e r e s e a r c h we p l a n t o seek the development o f improved models f o r e l a s t i c and i n e l a s t i c s c a t t e r i n g and t o apply the l o c a l e q u i l i b r i u m steady s t a t e theory f o r a d d i t i o n a l s i m u l a t i o n s o f nuclear and phot odi s s oc i a t i on r e c o i l H , F , and C 1 r e a c t i o n systems. 3

1

8

A

7 o

2

2

0

3

1

8

3 8


12.

GRANT ET AL.

Chemistry of High Energy Atomic Fluorine 343

Acknowledgement. The authors express appreciation to Dr. James Harrison of the Crocker Nuclear Laboratory for invaluable assistance with computer programming; to the U.C.D. Computer Center for assistance with computing costs; to Dr. James Muckerman for sponsoring the quasiclassical trajectory reaction cross section calcula tions; and to the U.S. Energy Research and Development Admini stration for financial support under contract E-(04-3 )-34, agreement no. 158. H. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Literature Cited. Farrar, J. Μ., and Lee, Y. T., this volume. Bogen, D., and Setser, D. this volume. See for example State to State Chemistry (American Chemical Society Symposium Series Monograph, Washington, D.C. 1977), Brooks, P., Editor. Ding, A.M.G., Kirsch, L. J., Perry, D. S., Polanyi, J. C., and Schreiber, J. L . , Disc. Faraday Soc. (1973), 55, 252, and references therein. Letokhov, V. S., and Moore, C. B., Sov. J. Quant. Electron. (1976), 6, 129, 259. Bergmann, Κ., Leone, S. R., and Moore, C. B., J. Chem. Phys. (1975), 63, 4161. Houston, P. L . , Chem. Phys. Letters (1977), 47, 137. Stedman, D. Η., and Setser, D. W., Prog. Reaction Kinetics (1971), 6, 193. Riley, Μ. Ε., and Matzen, M. K., J. Chem. Phys. (1975), 63, 4787. Sullivan, J. Η., Invited Lecture Presented at 169th National American Chemical Society Meeting, Philadelphia, 1975. Greiner, N. R., IEEE J. Quantum Electron. (1973), QE-9, 1123. Gerber, R. Α., Patterson, E. L . , Blair, L. S., and rtreiner. N. R.. Appl. Phys. Letters (l974), 25, 281. Keizer, J., J. Chem. Phys. (1973), 58, 4524. Keizer, J., J. Chem. Phys. (1972), 56, 5958. Koura, K., J. Chem. Phys. (1977), 66, 4078; ibid. (1976), 65, 3883. Menzinger, Μ., and Wolfgang, R., J. Chem. Phys. (1969), 50, 2991. See for example Proceedings of the VI International Symposium on Molecular Beams (Noordwijkerhout, Netherlands, 1977). Fluendy, M.A.D., and Lawley, K. P., Chemical Applications of Molecular Beam Scattering (Chapman and Hall, London, 1973)·


FLUORINE-CONTAINING F R E E

344 19. 20. 21. 22. 23.

RADICALS

Spicer, L., and Rabinovitch, B. S., Ann. Rev. Phys. Chem. (1970), 2 1 , 349. Wolfgang, R., Prog. React. Kinetics (1965), 3, 97. Mo, S. Η., Grant, E. R., L i t t l e , F. Ε., Manning, R. G., Mathis, C. Α., Werre, G. S., and Root, J. W., this volume. Rowland, F. S., Rust, F., and Frank, J. P., this volume. Grant, E. R., and Root, J. W., J . Chem. Phys. (1976), 6 4 ,

417. 24. 25· 26. 27· 28. 29. 30. 31. 32. 33.

Manning, R. G., Krohn, Κ. Α., and Root, J. W., Chem. Phys. Letters (1975), 35, 544, and references therein. Manning, R. G., Mo, S. Η., and Root, J. W., J . Chem. Phys. (1977). 67. 636. Manning, R. G., and Root, J. W., J . Chem. Phys. (1976), 64, 4926. Manning, R. G., and Root, J. W., J . Phys. Chem. (1975), 79, 1478; ibid. (1977), 81, in press. Riley. S. J . , and Wilson, K. R., Chem. Soc. Faraday Disc. (1972), 53, 132. Ting, C. T., and Weston, R. E., J . Phys. Chem. (1973), 77, 2257. Chapman, S., and Bunker, D. L., J . Chem. Phys. (1975), 62, 2890. Feng, D. F., Grant, E. R., and Root, J. W., J . Chem. Phys. (1976), 64, 3450. Stevens, D. J . , and Spicer, L. D., J . Chem. Phys. (1976),

64, 4798.

Grant, E. R., Ph.D. Dissertation (University of California, Davis, 1974, University Microfilms No. 75-15434). 34. Root, J. W., Nuclear Recoil Studies of Hydrogen Abstraction Reactions by Atomic Fluorine (U.S. Atomic Energy Commission Technical Report No. UCD-34P158-74-2, University of California, Davis, 1975). 35. Knierim, K. D., Grant, E. R., and Root, J. W., manuscripts i n preparation. 36. Estrup, P. J., and Wolfgang, R., J. Amer. Chem. Soc. (1960), 82, 2661, 2665. 37. Wolfgang, R., J . Chem. Phys. (1963), 39, 2983. 38. Estrup, P. J . , J . Chem. Fhys. (1964), 41, 164. 39. Chang, Η. Μ., and Wolfgang, R., J . Phys. Chem. (1971), 75, 3042. 40. Root, J . W., and Rowland, F. S., Radiochimica Acta (1968), 10, 104. 41. Root, J. W., and Rowland, F. S., J . Chem. Phys. (1967), 46, 4299. 42. Root, J. W., and Rowland, F. S., J . Phys. Chem. (1970),

74, 451. 43.

Chapin, D. M., and Kostin, M. D., J . Chem. Phys. (1968), 48, 3067; i b i d . (1967), 46, 2506. 44. Felder, R. M., and Kostin, M. D., J . Chem. Phys. (1967), 46, 3185; i b i d . (1965), 43, 3082.


12.

GRANT ET AL.


345

45. Kostin, Μ. D., J. Appl. Phys. (1966), 37, 791, 3801; ibid. (1965), 36, 850. 46. Porter. R. N., J. Chem. Phys. (l966), 45, 2284. 47. Porter, R. N., Disc. Faraday Soc. (1968), 44, 84. 48. Porter, R. Ν., and Kunt, S., J. Chem. Phys. (1970), 52, 3240. 49. Adams, J. T . , and Porter, R. Ν., J. Chem. Phys. (1973), 60, 3354; ibid. (1973), 59, 4105. 50. Kuppermann, Α . , and White, J. Μ., J. Chem. Phys. (1966), 44, 4352. 51. Kuppermann, Α . , in Fast Reactions and Primary Processes in Chemical Kinetics (Interscience. New York. 1967), Claesson. S., Editor. 52. Greene, E. F., and Kuppermann, A . , J. Chem. Educ. (1968), 45, 361. 53. Gann, R. G . , Ollison, W. Μ., and Dubrin, J., J. Chem. Phys. (1971), 54, 2304. 54. Melton, L. Α . , and Gordon, R. G., J. Chem. Phys. (1969), 51, 5449. 55. Malerich, C. J., and Spicer, L. D., J. Chem. Phys. (1973), 59, 1577. 56. Truhlar, D. G., and Wyatt, R. E., Ann. Rev. Phys. Chem. (1976), 27, 1. 57. Urch, D. S., Radiochimica Acta (l970), 14, 10. 58. Urch, D. S., M.T.P. Internat. Rev. Sci. Ser. 1 (1972), 8, 149. 59. Carlson, R., Freedman, A . , Press, G. Α . , and Malcolme-Lawes, D. J., Radiochimica Acta (1972), 18, 167. 60. Malcolme-Lawes, D. J., J. Chem. Phys. (1972), 57, 2476, 2481; ibid. (l972), 56, 3442. 61. Johnston, A. J., and Urch, D. S., J. Chem. Soc. Faraday I (1974), 70, 369. 62. Krohn, Κ. A . , Parks, N. J., and Root, J. W., J. Chem. Phys. (1971), 55, 5771, 5785. 63. Urch, D. S., Radiochemistry (London) (1975), 2, 1. 64. Nogar, N. S., Dewey, J. Κ., and Spicer, L. D., Chem. Phys. Letters (1975), 34, 98. 65. Nogar, N. S., and Spicer, L. D., J. Phys. Chem. (1976), 80, 1736. 66. Nogar, N. S., and Spicer, L. D., J. Chem. Phys. (1977), 66, 3624. 67. Malcolme-Lawes, D. J., J. Chem. Phys. (1972), 57, 5522. 68. Malcolme-Lawes, D. J., J. Chem. Soc. Faraday II (1975), 71, 1183. 69. Valencich, T . , private communication. 70. Grant, E . R., Knierim, K. D., and Root, J. W., submitted for publication. 71. Knierim, K. D., and Root, J. W., submitted for publication. 72. Grant, E . R., and Root, J. W., J. Chem. Phys. (1975), 63, 2970.


346

FLUORINE-CONTAINING FREE RADICALS

73. Snider, N. S., and Ross, J . , J. Chem. Phys. (1966), 44, 1087. 74. Huang, Κ., Statistical Mechanics (Wiley, New York, 1963), Ch. 6. 75· Wemmer, D., and Keizer, J., unpublished results. 76. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. Β., Molecular Theory of Gases and Liquids (Wiley, New York, 1964). 77. Root, J. W., Ph.D. Dissertation (University of Kansas, Lawrence,1964,University Microfilms No. 65-7004). 78. Muckerman, J. T., J. Chem. Phys. (1972), 57, 3388. 79· Bender, C. F., and Schaefer, H. F., this volume. 80. Root, J. W., and Muckerman, J. T., unpublished results.


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