O N T H E T H E O R Y O F M O N O M O L E C U L A R D E C O M P O S I T I O N

F O L L O W I N G A C T I V A T I O N O N T H E W A L L S

N. M. K u z n e t s o v UDC 511.10

The m o n o m o l e e u l a r decompos i t i on in the vo lume of a gas fol lowing the ac t iva t ion of the m o l e c u l e s only on the wa l l s of the v e s s e l i s r e a l i z e d when the fol lowing two cond i t ions a r e met: 1) The gas is r a r e f i e d so much that the length of the f r e e path is not dependent on the d e n s i t y and is d e t e r m i n e d only by the d i m e n s i o n s of the v e s s e l , and 2) when the e n e r g i e s fo r the d e c o m p o s i t i o n on the wa l l s and in the bulk a r e c o m p a r a b l e , the r a t io of the number of u n a d s o r b e d m o l e c u l e s N to the n u m b e r of a d s o r b e d m o l e c u l e s N a is much g r e a t e r than un i ty . In a c o n s i d e r a t i o n of the k i n e t i c s of t h e r m a l m o n o m o l e c u l a r de c ompos i t i on in the t e m p e r a t u r e range of p r a c - t i c a l i m p o r t a n c e (T ~ 1000~K), the second condi t ion is u s u a l l y fu l f i l led fo r phys i ca l adso rp t ion . In fact , at e q u i l i b r i u m with r e s p e c t to the o r d e r of magni tude we have

N - V -e/kr ,

where V and S a r e the vo lume and s u r f a c e of the v e s s e l , r e s p e c t i v e l y ; d is the t h i c k n e s s of a m o n o m o l e e u l a r a d s o r b e d l ayer ; E is the e n e r g y of adso rp t i on . Usua l ly the v a l u e s of V /Sd a r e v e r y l a r g e n u m b e r s on the o r - d e r of 109-10 l~ T h e r e f o r e , at T ~ 1000~K and any r e a l i s t i c e n e r g i e s fo r p h y s i c a l adso rp t ion due to a van d e r Waa l s i n t e r ac t i on , N >> N a.

In o r d e r to c a l c u l a t e the r a t e cons t an t for m o n o m o l e c u l a r decompos i t i on , we m u s t know t h e r a t e con - s t an t s fo r the spon taneous decompos i t i on of the s t a t e s of the ac t ive m o l e c u l e k i and the r a t e c o n s t a n t s k i / f o r the t r a n s i t i o n s f r o m s t a t e i to s t a t e l fo r a l l i and l~ When the va lue s of k i and ki/ a r e a s s i g n e d , the p r o b l e m , a s we know, r e d u c e s to the d e t e r m i n a t i o n of a s t e a d y - s t a t e d i s t r i bu t ion function with r e s p e c t to the s t a t e s of the ac t ive mo lecu l e fi and to the c a l c u l a t i o n of the sum

K = ~ fiki. (1) i

The d i f f e rence be tween the m o n o m o l e c u l a r decompos i t i on r e a c t i o n under c o n s i d e r a t i o n f r o m the s a m e r e a c t i o n o c c u r r i n g c o m p l e t e l y in the ga seous phase l i e s in the m e c h a n i s m of ac t iva t ion and, t h e r e f o r e , in the e x p r e s s i o n of fi" In addi t ion , in the c a s e of ac t iva t ion on the wal l s the d i s t r i bu t i on function is not cons tan t in space . F o r example , t he re a r e m o r e ac t ive m o l e c u l e s nea r the wa l l s than in the c e n t e r of the v e s s e l . T h e r e - fo re , the r a t e cons t an t fo r decompos i t i on (1) is a l so dependent on the coord ina te . However , the o v e r a l l r a t e of the c h e m i c a l c o n v e r s i o n in the e n t i r e v e s s e l is our ma in p r a c t i c a l conce rn . To ca l cu l a t e th i s r a t e , it is su f - f i c i en t to know the d i s t r i b u t i o n funct ion a v e r a g e d ove r the vo lume of the v e s s e l . The a v e r a g i n g of fi should be c a r r i e d out so tha t the subs t i tu t ion of the r e s u l t in Eq. {1) would y ie ld the c o r r e c t a v e r a g e r a t e cons tan t , which c h a r a c t e r i z e s the r a t e of the c h e m i c a l c o n v e r s i o n in the e n t i r e vo lume with r e s p e c t to one mo lecu le . Since the r e l a t i v e concen t r a t i on of the ac t ive m o l e c u l e s is s m a l l , the dens i ty of the number of m o l e c u l e s in the v e s - se l m a y be c o n s i d e r e d iden t ica l . The r a t e cons tan t s fo r the spontaneous decompos i t ion k i a r e t o t a l l y indepen- dent of the method of ac t iva t ion .

In the fo l lowing s e c t i o n s we sha l l c a l c u l a t e the a v e r a g e d i s t r ibu t ion function and the r a t e cons t an t of mono- m o l e c u l a r decompos i t i on on the b a s i s of a mode l of comple t e accomoda t ion and the s t a t i s t i c a l t h e o r y fo r k i.

A c t i v a t i o n M o d e l a n d A v e r a g e D i s t r i b u t i o n F u n c t i o n

The "bo t t l eneck" in the ac t iva t ion k i n e t i c s of a m o l e c u l e at not e x t r e m e l y high t e m p e r a t u r e s (kT ~ 0.04 - e a [1-3], where e a is the ac t iva t ion energy) is l oca ted in the range of e n e r g i e s on the o r d e r of kT nea r the

Ins t i tu te of C h e m i c a l P h y s i c s , A c a d e m y of Sc iences of the USSR, Moscow. T~:anslated f r o m T e o r e t i c h e s - kaya i F, k s p e r i m e n t a l ' n a y a Kh imiya , Vol. 12, No. 1, pp. 3-12, J a n u a r y - F e b r u a r y , 1976. Or ig ina l a r t i c l e sub- mi t t ed F e b r u a r y 17, 1975.

This protected by copyrigllt registered in the name o f Plenum Publishing Corporation. 227 West 17th Street. New York, 1~: Y. 10011. No part material is o f this publication may be reproduced stored in a retriel,al system or transmitted in any form or by any means electronic mechanical, photocopying, I microfilming, recording or otherwise, ,~i'thout written pernT,ssion o f the publisher. A ~:opy o f thi~ article is available-from the p,tbli~her .for $ Z50 ]

b o u n d a r y of the a c t i v a t i o n b a r r i e r [4]. B e c a u s e of the high d e n s i t y of v i b r a t i o n a l l e v e l s in a s t r o n g l y e x - c i t e d m o l e c u l e , the t r a n s i t i o n s in the v i c i n i t y of the " bo t t l e ne c k" with v i b r a t i o n a l e n e r g y c h a n g e s on the o r - d e r of kT a r e c h a r a c t e r i z e d b y e f f ec t i ve c r o s s s e c t i o n s , which a r e c o m p a r a b l e to the g a s - k i n e t i c c r o s s s e c - t i ons . T h e r e f o r e , in the c a s e of kT << e a the known v a r i a n t s of the t h e o r y of m o n o m o l e c u l a r d e c o m p o s i t i o n , in which the e f f ec t i ve d e a c t i v a t i o n c r o s s s e c t i o n s a r e a s s u m e d to be l a r g e [5, 6], a r e in s a t i s f a c t o r y a g r e e - m e n t with the e x p e r i m e n t (see [6, 7]). The e f f ec t i ve d e a c t i v a t i o n c r o s s s e c t i o n s in th i s c a s e a r e e i t h e r a s - s u m e d to be independen t of the i n i t i a l and f ina l s t a t e s (the m o d e l of s o - c a l l e d s t r o n g c o l l i s i o n s ) o r a r e c a l - c u l a t e d on the b a s i s of the m o d e l of the s t a t i s t i c a l m o l e c u l e + m e d i u m p a r t i c l e c o m p l e x f o r m e d in each c o l - l i s i o n .

T h e s e two a c t i v a t i o n m o d e l s a r e i d e n t i c a l when t h e y a r e a p p l i e d to c o l l i s i o n s with the wa l l s , s i n c e in th i s c a s e e a c h of t h e m l e a d s to e q u i l i b r i u m popu la t i ng of the e n e r g y l e v e l s of the m o l e c u l e in one c o l l i s i o n . A m o l e c u l e d e p a r t i n g f r o m the wai l a s a r e s u l t of the d e c o m p o s i t i o n of the m o l e c u l e + wa l l c o m p l e x has an e q u i l i b r i u m d i s t r i b u t i o n of the v i b r a t i o n a l , r o t a t i o n a l , and t r a n s l a t i o n a l e n e r g y with the t e m p e r a t u r e of the wai l , i . e . , c o m p l e t e a c c o m m o d a t i o n of the m o l e c u l e t a k e s p l a c e d u r i n g the c o l l i s i o n . Th i s o v e r a l l r e s u l t of the a p p r o x i m a t i o n s po in t ed out then s e r v e s a s a s t a r t i n g po in t in c a l c u l a t i n g the d i s t r i b u t i o n funct ion and the r a t e c o n s t a n t s of the r e a c t i o n . The exchange of e n e r g y b e t w e e n a p o l y a t o m i c m o l e c u l e and a wal l is m o r e e f f ec t ive in m a n y c a s e s * than with the a t o m s and s i m p l e r m o l e c u l e s of the c h e m i c a l l y i n e r t m e d i u m . Th i s f ac t and the known s u c c e s s of the a p p l i c a t i o n of a s t a t i s t i c a l c o m p l e x even in the gas t h e o r y of m o n o - m o l e c u l a r d e c o m p o s i t i o n [6] s e r v e s a s a q u a l i t a t i v e b a s i s of the a p p r o a c h u n d e r c o n s i d e r a t i o n .

In the gas phase the d i s t r i b u t i o n of i nac t i ve m o l e c u l e s a f t e r t h e y c o l l i d e wi th the wa l l r e m a i n s the e q u i l i b r i u m d i s t r i b u t i o n , but the a c t i v e m o l e c u l e s have a n o n e q u i l i b r i u m d i s t r i b u t i o n owing to the s p o n - t aneous d e c o m p o s i t i o n . We s h a l l f ind the a v e r a g e d i s t r i b u t i o n of the m o l e c u l e s &i, a s s u m i n g tha t the r a t e c o n s t a n t s f o r s p o n t a n e o u s d e c o m p o s i t i o n k i a r e known. The r e l a t i v e popu la t ion x i of e a c h i - t h s t a t e a t the t i m e t, which is m e a s u r e d f r o m the t i m e of the c o l l i s i o n , is de f ined b y the e x p r e s s i o n

xi = f~e -k : , f~ ~-e-~ilZv (2)

w h e r e e i is the e n e r g y of s t a t e i d i v ided b y kT, and Z i i s the v i b r a t i o n a l and r o t a t i o n a l s t a t i s t i c a l sum fo r the m o l e c u l e . F o r the n u m b e r of m o l e c u l e s AN{ d e c o m p o s i n g f r o m s t a t e i du r ing the t i m e of one f l igh t r l a long the t r a j e c t o r y l , we have

AN'i = Nf; (1 - - e-a:~). . (3)

If we sum Eq. (3) o v e r m a n y f l i gh t s of one p a r t i c l e (with a c o n s t a n t N), d iv ide the r e s u l t b y the to ta l t i m e the p a r t i c l e i s in m o t i o n t = Z~ l , and then m a k e the t r a n s i t i o n to the l i m i t t ~ ~o and to the p r o b a b i l i - t i e s dW f o r the f l igh t of the p a r t i c l e f r o m an a s s i g n e d s u r f a c e e l e m e n t in an a s s i g n e d s o l i d - a n g l e e l e m e n t , we f ind the a v e r a g e r a t e of d e c o m p o s i t i o n f r o m s t a t e i

dNi f~ lira (1 - - e-kin)! xl (1 - - e -ki~) dig, (4) Ndt ~ :=l t-I

w h e r e

v~-~- I/IxdW

i s the a v e r a g e n u m b e r of c o l l i s i o n s be tween one m o l e c u l e and the wal l p e r uni t t i m e . ? be tween u and the g e o m e t r i c d i m e n s i o n s of the v e s s e l s ee Appendix 1.

By de f in i t ion the func t ion sought , ~' i , s a t i s f i e s the r e l a t i o n s h i p

dt

C o m p a r i n g (4) with (5), we f ind

(4a)

F o r the r e l a t i o n s h i p

(5)

* A c c o r d i n g to the e v a l u a t i o n s in [8], which w e r e b a s e d on e x p e r i m e n t a l da ta on the m o n o m o l e c u l a r d e c o m - p o s i t i o n of iso-C3HTI fo l lowing a c t i v a t i o n on the wal l of a q u a r t z v e s s e l , the a v e r a g e v a l u e of the e n e r g y r e - c e i v e d o r g iven off du r ing one c o l l i s i o n with the wal l a t T -~ 1000~ is about 4 .5 kT . ~The t r a n s i t i o n to the p r o b a b i l i t i e s W in Eq. (4) was a c c o m p l i s h e d b y a v e r a g i n g the mo t ion with r e s p e c t to t i m e f o r one p a r t i c l e . The s a m e r e s u l t can , of c o u r s e , a l so be ob t a ined b y the c o r r e s p o n d i n g a v e r a g i n g o v e r an e n s e m b l e of p a r t i c l e s a t a g iven m o m e n t in t i m e .

r = T f, ~ (1 - e -k'') dW ~6)

The calculation of (6) for a gas enclosed between two infinite paral lel planes in a spherical vessel (see Ap- pendix 1) leads, respect ively , to the following resul ts :

[ ? - [~ v 1 ki 1 e_X dx , (7) O i -~- -~- (1 - - e -ki/~v) 2F --~ e - k i m - - 4"-v x

k i /2v

3 ki 2 3 ) , v 8 v 'z 4 ~ e - Y ; - + f f _ x . ~ _ e 8 v __~ ~A ( 8 )

It is interest ing to compare (7) and (8) with the distribution function

fi = f7 ,z(k, + v), (9)

which is obtained in some var iants of the theory of gas-phase monomolecular decomposition. In the case of strong coll is ions in the gas phase, resul t (9) is exact. In the f ramework of the theory of activation with the formation of an intermediate complex [6], function (9) is a f i r s t approximation of the solution of the integral equation and differs slightly f rom the exact solution (to g rea te r degrees, the g rea te r is the energy of state i). As a l ready noted, the models of s t rong collisions in a gas and of coll isions involving the format ion of a stat ist ical molecule + wall complex yield identical distribution functions for the energy of the molecule at t = 0, i . e . , immediately after a collision. Therefore , the differences between functions (7), (8), and (9) a re related only to the different distributions of the lengths of the paths. In a homogeneous gas the probabil i ty of a collision of a part icle does not depend on the time or the direction of its motion, but when there are coll isions only between the molecules and the walls, the f ree-pa th length is determined uniquely by the site and direction of the flight of the molecule f rom the wall.

Despite the differences in the analytical expressions, functions (7) and (8) differ only slightly f rom (9) (s~ee Fig. 1), all three formulas yielding the identical limiting resul ts

fi = Oi : f i ' V/ki --~ 00; (10)

fi = ep~ = f ~ v/k~, v /k i -~ O.

The relative value of the difference between functions (8) and (9) is maximal when the values of k i /v are in the 1-2.5 range and amounts to about 25%. The difference between (7) and (9) is about as small.

In the case of a gas enclosed in a cyl indrical tube, the determination of ~i involves more complicated calculations. The calculation of v and the original formulas for r in the case of a cylindrical vesse l are given in Appendix 1. However, a l ready f rom a compar ison of the numerical values of (7), (8), and (9), we can conclude that the function &i for a gas in a cyl indrical vesse l is also quantitatively close to (9).

R a t e C o n s t a n t f o r M o n o m o l e c u l a r D e c o m p o s i t i o n

The compar ison of distribution functions (7), (8), and (9) given in the preceding section suggests that the difference between the rate constants for monomolecular decomposition (1) calculated f rom each of these functions is g rea tes t in the region of the transit ion f rom a f i r s t - to a second-orde r react ion, but it is always small. In relative units this difference is smal le r than the maximum difference between the dis- tribution functions, which amounts to about 25%. Such a difference could hardly be considered at all s ig- nificant on the background of the assumptions which are made by necess i ty in any theory of mon0molecular decomposition. Thus, the rate constant for monomolecular decomposition following activation on the walls with complete accommodation sca rce ly differs f rom the rate constant for the gas-phase monomoleeular de- composit ion calculated in the approximation assuming strong collisions.

A simple variant of the theory, which employs the model of s t rong collisions, is the c lass ica l s ta t is - tical theory of Kassel . However, the rate constants for spontaneous decomposition k i and the density of the energy levels of the molecule ~2 in this theory are defined too approximately without considerat ion of the concrete spect rum and the anharmonic nature of the internal motion of the activated molecule. A more s tep-by-s tep calculation of k i and ~2 has been ca r r i ed out in the f ramework of a stat ist ical theory of the R i c e - - R a m s p e r g e r - - K a s s e l - - M a r c u s type for "modera te ly complex" molecules [6]. Because of the dif- ferences of any of the functions (7), (8), and (9) f rom the distribution function in [6], the rate con- stant found there for monomolecular decomposition does not apply direct ly to the problem under consideration.

8,8 " ~

7

~ J 2

r

Fig. 1. Dependence of the distr ibution function for any s tate of an act ive m o l e - cule on the ra te of the ra te constant for the spontaneous decomposi t ion to the ave r age number of col l is ions the m o l e - cule has with the walls pe r unit t ime: 1) gas between infinite p a r a l l e l planes [Eq. (7)]; 2) gas in a spher ica l ve s se l [Eq. {8)]; 3) model of s t rong col l is ions [Eq. (9)].

However, many of the computat ional methods in [6] r ema in in fo rce . Calculat ions of the ra te constant of monomolecu la r decomposi t ion with dis tr ibut ion function (9) a r e p resen ted in Appendix 2.

The r e su l t s of these calcula t ions a re r ep re sen t ed (after [6]) as the sum

K = KI + K~. (ii)

The terms K I and K 2 are expressed in terms of the molecular parameters, the temperature, and the col- lision frequency in the following manner

Ki=Ate-6 X e-~"(s+ 1,6"- -6- -8u) ! , (12) Zu-.<6*--6

.4e_8. a - - I

K,= o0v ECa"+e~ ~ -6--6;- rnl (13) m~0

The following notation (which was taken f r o m [6]) was employed in Eqs. (12) and (13):

g% c23 % Qs 8~eJkT; At~ (s+ 1)lzi

2~

% = ]/'8 kT/aM; Q~=(~hT/2~th2)a/2; Q~ =;~-~ [1 ( ~ ) r

Here g is the number of different pathways for the reac t ion Mt -~ M2 + M3; e . g . , for the react ion CHa CH 3 + H, g = 4. The quanti ty a23 takes on the following values: I f M 2 or M 3 is an atom, a23 -~ 10 -16 cm 2. If at l eas t one of the pa r t i c l e s M 2 and M3 has cyl indr ica l s y m m e t r y , and the react ion involves the c leavage of only one chemica l bond, cr23 ~ 10 -17 c m 2. In all o ther c a s e s cr23 - 2- 10 -18 cmz. In addition, epkT, 2s, and B i a re , r espec t ive ly , the v ibra t iona l quanta, the number of rotat ional degrees of f reedom, and the ro - tat ional quanta of the s y s t e m of the two unbound pa r t i c l e s M 2 and M3; l = 1, if at l eas t one of the two p a r - t ic les M 2 and M~ is an atom, and in all o ther c a se s l = 2; # is the reduced m a s s of pa r t i c l e s M 2 and M3; z 1 is the s ta t i s t ica l sum of the ro ta t ions and v ibra t ions of the or iginal molecule Mr; n = v + r , where v and 2r a re the n u m b e r s of v ibra t ional and rotat ional degrees of f r e edom in Ml, respect ively; eokT and 0 v a re the ene rgy of the zero v ibra t ions and the v ibra t iona l s ta t i s t ica l sum of the molecule of MI; 1/0 v is the c l a s -

b

sical l imi t of the v ibra t iona l s ta t i s t i ca l sum 0v; (a, b)! -~S xae-Xdx is an incomplete g a m m a function. 0

The value of 5" is de te rmined f r o m the equation

(n - - 1)l Al (6" + e0) 1-~ V . - - 0o 0~ z...J (6* - - 8- - g.).)*+' (14)

eu<6*--~

with

and together with Eq.

The t e r m s K 1 and K 2 de te rmined f r o m Eqs. in [6], r espec t ive ly , by the f ac to r s

~ ' ~ (n - - 1)1(6" + %) ~< 112 (15)

(13) we obtain with good accuracy

v (6* + ~o~"-' K , = - 0o - - 1)l e--~'" (16) 0o o(n (1-- ~')

(12) and (13) a re g r e a t e r than the analogous expres s ions

4

(1 + n)s/(1 + 3T])and(l + ~l)S/(l -- TI' ) (1 + 3~l), (17)

w h e r e ~1 - (n + 1) / (6 + e0). The l a t t e r of t h e s e f a c t o r s v a r i e s f r o m 1 to 2.7 a s 7/' i s v a r i e d f r o m 0 to 1 /2 . ~qlen ~/' i s i n c r e a s e d f u r t h e r , the d i f f e r e n c e be tw e e n the act i ;eat ion m e c h a n i s m s in the p r o b l e m u n d e r c o n - s i d e r a t i o n h e r e and in [6] p r o d u e e s an even g r e a t e r d i f f e r e n c e b e t w e e n the r a t e c o n s t a n t s , but i t i s no l o n g e r d e t e r m i n e d b y the f a c t o r s in (17).

B e s i d e s the n u m b e r of c o l l i s i o n s wi th the wall, in o r d e r to c a l c u l a t e the r a t e c o n s t a n t f o r m o n o m o l e e - u l a r d e c o m p o s i t i o n i n t h e c a s e of a c t i v a t i o n on the w a i l s (11), we m u s t know the s a m e da ta a s in the c a s e of a c t i v a t i o n in the g a s p h a s e I6]. T h e s e da ta inc lude the v a l u e s of the m a s s e s , the r o t a t i o n a l and v i b r a t i o n a l quanta of the o r i g i n a l m o l e c u l e and the r e a c t i o n p r o d u c t s , the hea t of r e a c t i o n , the f a c t o r g, and the e f f e c - t i ve c r o s s s e c t i o n cr23 which c h a r a c t e r i z e s the d i m e n s i o n s of the a c t i v a t e d m o l e c u l e and the s y m m e t r y of the r e a c t i o n p r o d u c t s .

In c o n c l u s i o n , we s h a l l p r e s e n t the e x p r e s s i o n s f o r the t ion o b t a i n e d f r o m Eqs . (12) and (13) at the l i m i t s of f r e q u e n t wal l , r e s p e c t i v e l y ,

r a t e c o n s t a n t s of m o n o m o l e c u l a r d e c o m p o s i - ( v - * oo) and r a r e (v ~ O) c o l l i s i o n s wi th the

K = K r (reac~3Q~az~ e-5; K2 = 0 , Zi

K = Ko ~- K~ for 6*=6; K l = 0 ,

w h e r e z23 is the s t a t i s t i c a l sum of the r o t a t i o n s and v i b r a t i o n s of the s y s t e m of two unbound p a r t i c l e s M 2 and M 3.

A P P E N D I X 1

D i s t r i b u t i o n F u n c t i o n

~The d i r e c t i o n of the v e l o c i t y v e c t o r of a m o l e c u l e f ly ing out f r o m the s u r f a c e i s u n i q u e l y de f ined by the angle 0 b e t w e e n the v e l o c i t y v e c t o r and a n o r m a l to the s u r f a c e (the i n t e r n a l s u r f a c e with r e s p e c t to the v e s s e l ) and the a z i m u t h a l ang le go m e a s u r e d in the p lane t angen t to the s u r f a c e a t the po in t of d e p a r t u r e . A s s u m i n g tha t the p r o b a b i l i t y of f l igh t is p r o p o r t i o n a l to c o s 0 o v e r an e n t i r e s o l i d ang le equa l to 2v (a h e m i s p h e r e ) , we r e p r e s e n t the p r o b a b i l i t y of f l igh t in an e l e m e n t of the s o l i d ang le f r o m the s u r f a c e e l e - m e n t dS in the f o r m

1 dW ~ -~-~ cos O sin O dO dcp dS. (AI.1)

The r e l a t i o n s h i p be tween the t i m e of f l igh t 7, the a n g l e s 0 and go, and the poin t of d e p a r t u r e f r o m the s u r f a c e depends on the g e o m e t r i c p a r a m e t e r s of the v e s s e l . D i s t r i b u t i o n funct ion (6) is c a l c u l a t e d f o r a g a s l o c a t e d (1) be tween two inf in i te p a r a l l e l p l a n e s , (2) in a s p h e r i c a l v e s s e l , and (3) in an i n f i n i t e l y long c y l i n d r i c a l tube (in th i s e a s e the d i s t r i b u t i o n func t ion is r e p r e s e n t e d in the f o r m of an i n t e g r a l , which m u s t be c a l c u l a t e d b y n u m e r i c a l m e t h o d s ) . F o r a l l t h r e e t y p e s of " v e s s e l s " T is not dependen t on the pos i t i on of the poin t of d e p a r t u r e . T h e r e f o r e , the i n t e g r a t i o n of (6) with d i s t r i b u t i o n (A1.1) o v e r the v a r i a b l e dS is t r i v i a l .

In the e a s e of a g a s e n c l o s e d be tw e e n two inf in i te p a r a l l e l p l a n e s s e p a r a t e d b y the d i s t a n c e h, we 1. have

= h[u cos 0, (A1.2)

w h e r e u is the a v e r a g e t h e r m a l v e l o c i t y of the m o l e c u l e s . * Us ing (A1.1) and (A1.2), we f ind the d i s t r i b u - t ion with r e s p e c t to T

dW = 2t~ dx/x 3, to ~- Mu. (Ai .3)

Subs t i tu t ing (A1.3) into (6) and m a k i n g the t r a n s i t i o n f r o m 0 to the new v a r i a b l e x -= k iT , we have

l - - e - " (AI.4) . x------- ~ - dx.

. I kit*

�9 The s m a l l c o r r e c t i o n to the d i s t r i b u t i o n funct ion due to the d i f f e r e n c e be tween the t r ue v a l u e s of the v e - l o c i t i e s and the a v e r a g e t h e r m a l v e l o c i t y has not been c o n s i d e r e d h e r e .

The time t o may be expressed in t e r m s of the frequency of coll isions between a molecule and the wall v. After the integration of (4a) with considerat ion of (A1.2) and (A1.3), we obtain

2t 0 ~ 1/~. (A1.5)

The integration of (A1.4) and the replacement of t o by 1/2 u in the resul t obtained yield the express ion sought (7), which contains a tabulated integral.

In the case of an isotropic distr ibution of the direct ions of the departure of the molecules f rom the surface (AI.1), the number v can be found without resor t ing to the integration of (4a). In fact, in this case all the directions of the inactive molecules (i. e . , p rac t ica l ly all the molecules) at any point within the ves - sel are equally probable. The density of the number of par t ic les is also independent of the coordinate. As we know, under such conditions the number of coll isions v 1 of all the molecules f rom a unit of surface is determined by the relationship

v t = N u / 4V .

Multiplying vl by the a rea of the entire surface S of the vessel and relat ing the resul t to one part icle, we find

�9 ~ - S u / 4 V . (A1.6)

This resul t is valid for a vesse l of any shape. F rom (A1.6) it is seen that the mean free path length of a molecule in an empty vesse l with isotropic distribution (AI.1) is equal to the ratio of quadrupled volume to the surface.

2. In the case of a gas located in a spherical vessel of radius R, we have

"~ = ti cos 0, ti ~ 2 R / u , (A1.7)

Equations (A1.7) and (AI.1) yield the distribution with respect to T

d W = - - 2zd 'Ut~. (A1.8)

The substitution of (A1.8) into Eq. (6) and integration with respec t to 7 in the l imits t i -- 0 yields

�9 + fl+ ap~ = 1 - - ~ R ~ \ ki t i / J"

(A1.9)

According to (A1.6) or (4a), the f requency v is related to t 1 by the equation

-.~ 3 / 2 t t . (AI.10)

Distribution function (8) is obtained as a resul t of the substitution of t 1 f rom (AI.10) into (A1.9).

3. In the case of a cyl indrical vesse l (an infinitely long cyl inder of radius r) the axis 0 = 0 of the spherical coordinates is conveniently directed along that forming the cylinder. In addition, the azimuthal angle q~ is measured f rom the tangent to the surface of the cylinder drawn in the plane of the c ross section of the cyl inder passing through the point of departure. In such coordinates the angle between the velocity vector and the normal to the surface equals sin 0 sin ~. Therefore , instead of (AI.1), after the integration with respec t to dS we have

d W --~ 2 sins 0 sin q~ dO dq~ (A 1.11 )

The time of free flight is expressed in the fo rm

T ~-~ t~ sincp/sin0, t S ~ 2 r / u . (A1.12)

The calculation of v with the aid of ei ther of the formulas (A1.6) and (4a) yields

- ~ 1~Iv (Al.13)

Subst i tut ing (A1.11) and (A1.12) into (6) and taking (A1.13) into account, we obtain

O~= 2~f~ fsinepd(p f 1--exp sin~0d0. (Al.14) ~/~ d

o o

The in tegra t ion of (A1.14) involves n u m e r i c a l ca lcu la t ions , which we intend to c a r r y out in the following. It is e a s y to see that d i s t r ibu t ion funct ion (A1.14) s a t i s f i e s the a sympto t i c r e l a t ions in (10).

A P P E N D I X 2

C a l c u l a t i o n o f t h e R a t e C o n s t a n t o f M o n o m o l e c u l a r D e c o m p o s i t i o n

In the s t a t i s t i ca l app rox ima t ion the va lues of k i and r f o r all the s t a t es with the s a m e e n e r g y e a re ident ical , and a f te r the r e p l a c e m e n t of fi by the a v e r a g e d i s t r ibu t ion funct ion ~i and of the sum o v e r the s t a t e s by the in tegra l with r e s p e c t to the e n e r g i e s , Eq. (1) b e c o m e s

g = ~ (I) (e) fl (~) k (e) de, (A 2. 1 )

5

w e r e f~(e) -- d i / de is the n u m b e r of s t a t e s in a unit in te rva l of ene rgy . def ined by the e x p r e s s i o n s

-q (0 -~ (~ + %)~-1 Q, ( n - - 1)I Og "

k(8) -- ( N - - 1)]Q23a23c2aQsO ~ 2 (s + 1 ) IQ, (e+%) ~-1 ,,<~--8

Q, ~ a'~l l-] (kTIB/)Y 2" i=z

Accord ing to [61, a(e) and k(e) a r e

(A2.2)

(e - - 8 - - e,) '+1, (A2.3)

In these equat ions Bj and Q r a r e the ro ta t ionM quanta and the ro ta t iona l sum of the o r ig ina l molecu le .

Fol lowing [6], we divide the in tegra t ion l imi t s in (A2.1) into two (from 5 to 5" and f r o m 6*to ~o) and deter~nine the value of 5 " f r o m condi t ion (14). The d i s t r ibu t ion funct ion O(e) appea r ing in (A2.1) [any of the funct ions (7), (8), o r (9)] v a r i e s mono ton i ca l l y and f a i r I y r ap id Iy with smal l changes in e in the v ic in i ty of the point e = 6 " . In ca lcu la t ing (A2.1), this p e r m i t s the r e p l a c e m e n t of the funct ion ~(e) by its a s y m p - totie e x p r e s s i o n s

r (0 -- ~* (0, s < 6", (A2.4)

q:, (8) = f* (8) s 8 >

It can be shown that the r e l a t ive e r r o r in the ca lcu la t ion of (A2.1) due to this subst i tu t ion is m a x i m u m in the c a s e of 6 " - 6 + 1, i . e . , at the value of v c o r r e s p o n d i n g a p p r o x i m a t e l y to the middle of the t r ans i t i on range f r o m the f i r s t - to s e c o n d - o r d e r r eac t ion , and amounts to about [e(n + 1)] -1 at the m a x i m u m . At the l imi t of l a r g e and sma l l ~ the e r r o r in app rox ima t ion (A2.4) equals ze ro .

With cons ide ra t i on of (A2.4) r a te cons tan t (A2.1) is r e p r e s e n t e d as (11), and we have

6"

K~_~._Z( ~2(e)k(e)e-e de, (A2.5)

6

with all the va lues e u -< 6" - - 5. [ s e e (12a)]

~7 S (A2.6) g~ = ~ s2 (8) e -~ cts.

5"

Afte r subs t i tu t ing (A2.2) and (A2.3) into (A2.5), the ca lcu la t ion of KI r educes to the ca lcu la t ion of the inte- g r a l s

5"

1 ( 8 ~ ) ~ 5 ( 8 - - 6 - - 8u) ~+'e - ' d 8 6 + ~ u

These in t eg ra l s a r e e x p r e s s e d in t e r m s of the incomple te g a m m a function

I (e~) -~- e -(~"+5) (s 4- I, 5" - - 8 - - eu)l. (A2.7)

The summation of (A2.7) over all e u -< 5" -- 6 and the considerat ion of the energy-independent fac tors in (A2.5) yield the express ion sought (12). The substitution of (A2.2) into (A2.6) yields an integral of the same type as in the c lass ica l Kassel theory at low p res su res . The exact calculation of the integral yields se r ies (13). Result (16) is obtained by calculating the same integral with considerat ion of the approximate relation

(e + 8o)"-' ___ (6" + 8o) '~-l e n ' (~ '~ ,

in which ~' is defined by identity (15).

L I T E R A T U R E C I T E D

1. N . M . Kuznetsov, Dokl. Akad. Nauk SSSR, 164, 1097 (1965). 2. N . M . Kuznetsov, Teor Eksp. Khim. , 7, 22 (1971). 3. N . M . Kuznetsov, Dokl. Akad. Nauk SSSR, 208, 145 (1973). 4. E . E . Nikitin, Theory of Elementary Atomic-Molecular P r o c e s s e s in Gases [in Russian[, Khimiya,

Moscow (1970)o 5. V . N . Kondrat 'ev and Eo E. Nikitin, Kinetics and Mechanism of Gas-Phase Reactions [in Russian],

Nauka, Moscow (1974)~ 6. J. Keck and A. Kalelkar, J. Chem. Phys . , 4__99, 3211 (1968). 7. T . F . Thomas, P. J. Conn, and D. F. Swinehart, J. Amer. Chem. Soc., 9_.1.1, 7611 {1969). 8. S . W . Benson and G. N. Spokes, Jo Amer. Chem. Soc., 8_.99, 2525 (1967).

I S O T O P I C E F F E C T S IN T H E K I N E T I C S OF H E T E R O G E N E O U S

C A T A L Y T I C R E A C T I O N S

S. L . K i p e r m a n UDC 541.128

The measurement of kinetic isotopic effects provides valuable information about the mechanism of heterogeneous catalytic react ions and the rat ios between the react ions ra tes of the individual stages. The available data show that in most cases the values of the isotopic effects are g rea te r than unity, i . e . , that replacement of the light isotope with a heavier (for instance, of prot ium by deuterium or tri t ium) leads to a reduction in the reaction r a t e . In some cases , however, isotopic effects smal ler than unity have been found which in the fur ther text will be called r eve r se effects. We have found such effects [1-3] in the ac- curate measurement of reaction ra tes of the hydrogenation of benzene, toluene, and acetone on nickel cata lys ts in g rad ien t ' f r ee systems; others [4] have observed them when studying the synthesis of hydro- carbons f rom carbon monoxide and hydrogen on a coba l t - - t ho r ium oxide cata lyst and in the hydrogenation of ethylene near the maximum reaction rate [5, 6].

On the f i rs t glance such r eve r se values are in contradiction with the e lementary theory of isotopic effects which is based on the in termedia te-s ta te method [7-9]. No explanation is offered for the r eve r se effects in [4-6], while some authors [10, 11] assume qualitatively that such effects are possible in the case of homogeneous reactions, without offering a quantitative t reatment of the problem. We shall therefore discuss the possible causes for the appearance of r eve r se kinetic isotopic effects in heterogeneous catalytic react ions.

The kinetic isotopic effect 13 is general ly defined as the ratio of reaction ra tes involving a light and heavy isotope (protium and deuterium only) at a given composit ion of the reaction mixture, i . e . , at the same initial conditions and degrees of convers ion x:

N. D. Zelinskii Institute of Organic Chemist ry , Academy of Sciences of the USSR, Moscow. T rans - lated from Teoret icheskaya i ]~ksperimental 'naya Khimiya, Vol. 12, No. 1, pp. 13-17, J anua ry -Februa ry , 1976. Original ar t ic le submitted Feb rua ry 13, 1975.

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