R E L A T I O N B E T W E E N R A T E C O N S T A N T S

O F O P P O S I T E R E A C T I O N D I R E C T I O N S

N. M. K u z n e t s o v UDC 541.124

Measurements o r computations of the rate constants often r e f e r just to 1 of the 2 opposite reaction direct ions (for general i ty , we agree to call a reaction a change in any internal s tates of a par t ic le o r group of par t ic les averaged over the remaining var iables of the system; here , for example, these are chemical o r e l e c t r o n - i o n react ions , vibrational t ransi t ions , and the excitation of electronic levels of par t ic les) . In this connection, an interest ing question is whether the rate constant of the second react ion direct ion can be found on the basis of such data. Pa r t i cu l a r cases of this general question have been examined repeatedly in application to s ingle-s tage and some mult is tage react ions [1-6]. However, react ions consist ing of an a rb i t r a ry set of se r ies and paral le l s tages with an a rb i t r a ry t empera tu re dependence fo r the s tat is t ical sums of the in termedia te products , as well as mult is tage react ions in gas mixtures consist ing of subsys tems withtwo d i s t inc t t empera tu res , or spoilage of thermodynamic equilibrium because of the react ion i tse l f in the more general case , have been investigated slightly in this respect .

An attempt is made in this survey at a sys temat ic examination of the question as a whole, including even the mentioned, little studied, aspects .

R a t i o b e t w e e n T h e R a t e C o n s t a n t s o f t h e F o r w a r d a n d R e v e r s e R e a c t i o n D i r e c t i o n s in a M e d i u m in T h e r m o d y n a m i c E q u i l i b r i u m

Thermodynamical ly Equil ibrium Medium. Let us agree to denote any react ion convert ing a set of initial par t ic les A 1 into reaction products A m in a medium M as

A , + M .~- A m + M (I)

The role of the medium is that energy l iberated during the react ion is t ransmit ted therein at once o r as a resul t of success ive coll isions between the par t ic les A and M. In addition to the others , the par t ic les f rom A t and A m can also be par t of the composition of the medium. Let us assume that the medium and the react ing substance a re not subjected to any external effect, and that the sole reason for the change in their macroscop ic state is the react ion itself . Let us also cons ider the react ing par t ic les to form an ideal gas (or an ideal solution), i .e . , they do not in terac t at a dis tance f rom each other.

The passage of the energy l iberated during the react ion into kinetic energy of the medium par t ic les general ly resul ts in spoilage of the Maxwell distribution in the medium. In o r d e r for such spoilage to be slight, rapid distribution of the l iberated energy over the translational degrees of f reedom of all the par t i - c les is neces sa ry . F o r commensura te masse s of medium and react ing-component par t ic les the condition for slight spoilage of the Maxwell distribution during react ion can be formulated as the inequality

iQ]~l*<< qT, (1)

in which Q, ~*, and ~ denote the heat of reaction, the number of reaction acts per unit time divided by the number of reacting particles, and the number of collisions of one particle per unit time, respectively, and T is the temperature in energy units. The inequality (1) will be satisfied better, the smaller the relative concentration of the initial reacting particles and the lower the probability of reaction during their collisions. The reactions of atom ionization and molecule dissociation during collisions with atoms are an example of

Moscow. Translated f rom Fizika Goreniya i Vzryva, Vol. 9, No. 5, pp. 683-699, September-October , 1973. Original ar t ic le submitted May 28, 1973.

�9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

595

reac t ions sat isfying (1). The change in the e lec t ronic energy of the a tom o r the v ibra t ional energy of the molecule , which resu l t s in the ionizat ion o r d issocia t ion, r espec t ive ly , occu r s in the reac t ions ment ioned as a r e su l t of col l i s ions between the reac t ing p a r t i c l e s and the a toms of the med ium whose veloci t ies a r e in Maxwell d is t r ibut ions . This i s a p a r t i c u l a r case of a react ion in an equi l ibr ium medium. Here the medium is a subsys t em of the t rans la t iona l deg ree s of f r eedom of the pa r t i c l e s . If not only the t r ans l a t ion - al, but also the o the r deg ree s of f r eedom of the pa r t i c le , the rotat ional , say, play a pa r t during the react ion, then the s u b s y s t e m of all such d e g r e e s of f r eedom r e f e r s to the medium in which the react ion occu r s . With- out going into the de ta i l s on the equi l ibr ium conditions fo r any specif ic med ium during a react ion, le t us jus t note that in addition to (1), o the r inequal i t ies denoting the rapid re laxat ion of the medium as compared with the reac t ion ra t e should also be sa t is f ied fo r such equ i l ib r i a .

A r e m a r k a b l e p r o p e r t y of equi l ibr ium media is that the ra t io between the r a t e constants of the f o r - ward and r e v e r s e reac t ion d i rec t ions in such media equals the react ion equi l ibr ium constant calcula ted fo r va lues of the the rmodynamic p a r a m e t e r s governing the s ta te of the medium. In those ca se s where i t i s imposs ib l e to c h a r a c t e r i z e the react ion by the r a t e constant independently of the concent ra t ions of the medium components , the in t e r re l a t ion between the r a t e s of the two reac t ion d i rec t ions is e x p r e s s e d to a m o r e genera l f o r m : the equi l ibr ium constant equals the ra t io between the " reac t ion ra te coeff ic ients" which a re re la ted in a definite m a n n e r to the concent ra t ions of the medium components and to the r a t e constants of the s ing l e - s t age reac t ions . The proof of this p rope r ty i s g iven below.

A s ing le - s t age react ion denotes the d i rec t convers ion of a substance A 1 into react ion products A 2 without going through any in t e rmed ia t e s tage. Such reac t ions a r e cal led s imple in c h e m i s t r y . It i s s imp l - e s t to obtain the re la t ion between the r a t e cons tants of opposi te s ing l e - s t age rea0t ion d i rec t ions by s ta r t ing f rom the phenomenological k ine t ics equations, in conformi ty with which the ra te J+ of the reac t ion in the d i rec t ion f r o m A 1 to A 2 and the r a t e J_ of the r e v e r s e reac t ion a r e e x p r e s s e d as

. f+=k, : f , [M], .f_----k.~fg_ [M], (2)

ft = II N ~'i H E ~2~ ,i , f-_ - - 2i (2a) i i

H e r e Nli and N2i a r e the n u m b e r of pa r t i c l e s of the reac t ing component i p e r unit volume, the num- b e r s uli and ~ i equal the absolute va lues of the cor responding s to ich iomet r i e react ion coeff ic ients , the subsc r ip t s I and 2 denote quant i t ies r e f e r r i n g , r e spec t ive ly , to the ini t ial subs tance and the react ion p r o - ducts A 1 --*A2; [M] i s the n u m b e r of p a r t i c l e s of the medium p e r unit volume (or the product of such num- bers) taking pa r t in the reac t ion as a ca ta lys t , and kl2 and k21 a re the reac t ion ra te constants which a r e independent of Nli, N2i and [M].

Taking into account the independence of kl2 and k21 f r o m the concent ra t ions of the reac t ing subs tance components and the med ium and also that in comple te the rmodynamic equi l ibr ium

]+=1_,

we obtain the des i r ed re la t ionship between the reac t ion ra te constants in the two d i rec t ions f r o m (2)

k,~ g --K,~. (3) k,1 h

Here- f 1 and ~ a r e equi l ibr ium va lues of fl and f2, and K12 is the equi l ibr ium constant of the reac t ion dependent only on the t e m p e r a t u r e of the medium.

Bes ides the der iva t ion of the re la t ionship (3) p resen ted above which i s based on using the phenomeno- logical equations of m a c r o s c o p i c k ine t ics and the equi l ibr ium equation of the reac t ion , let us give sti l l another der iva t ion of (3) by s ta r t ing f r o m a m i c r o s c o p i c descr ip t ion of the s y s t e m dynamics .

The ra t e of the s ing le - s t age reac t ion (I) in the d i rec t ion A 1 -*A 2 i s the ra te of the mechanica l p a s s a g e ofwik, equal to one d iv idedby the t ime of pa s sage , f r o m a mechan ica l ly unique specif ic ra te i of the s y s t e m phase space to a react ion (the s y s t e m 1) in a uniquely defined s ta te of the s y s t e m phase space a f t e r one e l e m e n t a r y reac t ion act (the s y s t e m 2) ave raged o v e r all poss ib le segments of a qua~ic lass ica l phase t r a - j e c to ry s ta r t ing in ~he s y s t e m 1 and te rmina t ing in the s y s t e m 2, and then averaged o v e r all phase t r a j e c - t o r i e s . If one takes into account that all s t a tes a re e q u a l l y p r o b a b l e o n t h e s y s t e m phase t r a j e c t o r y (Liouvil le theorem) , :the f i r s t averag ing yie lds

596

{p,s~_ \ " , ; ; = __ ~ ' , i< l t l . t sT~ )>

i , /e

(4)

where j is the number of the phase t ra jec to ry , n~ is the number of states on the phase t r a j ec to ry j in the sys tem 1, n~ is the number of states on the same t ra jec tory* in the sys tem 2o Per fo rming the second averaging, we a r r ive at the following express ion fo r the reaction rate in the direct ion A t - -A2:

7

O{ := exp ( - - s } i T ) / Z ~ . (5)

Here A are the distribution functions over the states of the sys tem 1 taken on the t r a j ec to ry j, sj is the state energy on the t ra jec tory j (all the states on the phase t r a j ec to ry have identical energies) , Z i is the s tat is t ical sum of the sys tem I per unit volume. The r eve r s e reaction rate is determined by the same expression with the subscripts commutated:

, ,;)) J - - z . , - - i ~ r ' _ e , (l!'r!_ = _* r ~ g , ~ , l ( n { n , (6) - I l e ' , ~"

where the s tates i ' and k' differ f rom the s t a t e s i and k just by a change in the sign of the time, and Z 2 is the stat is t ical sum of the sys tem 2 per unit volume which is obtained f rom the sys tem 1 as a resul t of an e le- mentary reaction act.

The equality

~'~s: = ~'s~',~', (7)

is the basis fo r the relationship between J+ and J_, which is a resul t of the invar iance of the laws of me - chanics relat ive to change in the sign of the time. F rom (7), (5) and the second formula in (6) there also resul ts

It :'s . . . . IVY.

Substituting (8) and (6) for J_ and then comparing (6) to (5), we find

(s)

! ] _ / ] - ~ Zz t Z ~.

In an ideal gas the stat is t ical sum of a sys tem with unit volume is expressed as

(9)

0 V Z : : fl ( z i ) i!A",,-!. (10)

i

0 Here N i is the number of par t ic les of the species i per unit volume, and z i is the stat is t ical sum of one par t ic le of the species i occupying unit volume. Substituting the expression fo r the s tat is t ical sums (10) for the sys tems 1 and 2 into (9) and cancelling identical fac tors in the numera to r and denominator , we obtain

J./s_ = (zUz,) (i2/},) ~Kl~[dfL, (11) Y I ' 0 V l" 0 ~" �9 z~ ::: Jl (zii) z . . . . H (z20 e~. (12) i i

The quantities in (11) are defined by the identit ies (2a). Comparing (11) with (2), we again a r r ive at the relation between the react ion rate constant and the equilibrium constant (3).

Taking account of (2) and (3) the J12 ~ J+ - J - is represented as

]12=Ie21 (f l Kig--f2) . (13)

* In c lass ica l mechanics , integrat ion over the phase volumes dFi and dF k cor responds to summation over i and k, and an infinitely tkin cu r ren t tube should be considered in place (if the quasic lass ical t ra jec tory .

597

F o r the sequel it is convenient to give (13) a more symmet r i c form by substituting the ratio between the stat ist ical sums defined by (I2) into (11):

]~=k~,z~(X,--X~) [M], (14)

XI ==-fdzl, X - - f 2 / z 2 (14a)

[To avoid misunderstanding, let us note that the stat ist ical sums (12) in (14) are denoted by z in cont ras t to the total stat ist ical sums.]

The stat ist ical sums (12) are

zl Zi (T)e-- E~ r (i:1, 2L (i5)

where xi(T) is some function of the tempera ture independent of El, and E i is the total internal energy of the par t ic les A i talcing par t in one e lementa ry react ion act without taking account of those degrees of f r ee - dom over which the thermodynamic average is taken; in o ther words , E i is the energy of the mentioned par t ic les at absolute zero . The form of the function )r is determined by the s t ruc ture of the phase space occupied by the initial substance (i = 1) o r the products (i = 2) of the e lementary reaction act. As an i l lus- trat ion, let us p resen t values of xi(T) and z i for s ingle-s tage diatomic molecule dissociat ion and atom ioni- zation react ions .

1. Single-stage dissociat ion react ion of molecules AB in the vibrational state with vibration energy E 1 (with thermodynamic averaging over all the remaining degrees of f reedom):

.IB~+M ~ , t+B+M,

.. {mABT ~3 2 zA B'AB , Z1 (T) " k ~ ] r~

zt .: Zt (T) e -F-' r,

: (mAT ~3/2( mt,T 't32 zl J

z,. : Z~. (T) e - e d r .

Here mA, mB,and mAB are the atom and m o l e c u l e m a s s e s , E 2 i s the d i s soc ia t ion energy, Zro t i s the rotational s ta t i s t i ca l sum of the m o l e c u l e , and the subscr ipt e r e f e r s to the e l ec tron ic s tat i s t ica l sum of the m o l e c u l e or atom.

2. S ing le - s tage ionizat ion react ion of an atom in an exci ted stage with exci tat ion energy E 1 and ion formation in a given electronic state 2

AI-~- M -> A~L + e + M,

~ "r" [ mAT \3,2

zx :: Xx (T) e -el:r ,

( mA T ~32{ met ~3:~

z~ :-: Z~ (T) e -~'-Ir.

Here gl and g2 a re the s tat is t ical weights (degree of degeneration) of the initial electronic level of the atom and the final e lectronic level of the ion, E 2 is the ionization energy of the atom A 1 with ion formation in a given electronic state 2, and m e i s the e lectron mass .

In i ts s t ruc ture (14) i s analogous to Ohm's law for an e lec t r ica l cur rent . The role of the cur ren t intensity, the potential difference, and the res i s tance is played by J12, X1 - X 2 , and (k21z2 [M]) -1, respect ively . Let us la te r use this analogy to analyze mult is tage react ions .

Multistage Reaction. Let us turn f i r s t to a pa r t i cu la r ease of a mult is tage react ion, which is a se - quence of s ingle-s tage react ions

A I + M ' ~ A 2 + M ~ - . . . A ,~+M. (II)

598

The reac t ions (I) and (II) d i f f e r not only by a s imple magnif icat ion of the n u m b e r of s tages . The fact i s that the va r ious reac t ion s tages can also have d i f ferent va lues of [M]. Thus, fo r example , the quantity [M] has the sense of pa r t i c l e concent ra t ion in the medium fo r the f i r s t s tage of a monomolecu la r d i ssoc ia t ion react ion (the p a s s a g e of the inac t ive molecu le A 1 into the ac t ive s ta te A 2 during col l i s ions with pa r t i c l e s of the medium)

A~ + M ~ - A 2 + M

The second stage of the reac t ion , the spontaneous d issoc ia t ion of a mul t ia tomic molecule f r o m a specif ic act ive s ta te A 2 into compos i t e pa r t s A 3

A2 ~ A3 (111)

o c c u r s without col l i s ions . Hence, the value of [M] is one in the k inet ics equation (14) in the case of the react ion (KI).

Still ano ther essen t ia l d i f fe rence between reac t ions (II) and (I) i s that f o r a fixed n u m b e r of s tages i there may be not one but s eve ra l s ing le - s t age reac t ions

A ~+M j ~ A ,~+ ~ -}-M j, ] = 1, 2 . . . . .

in the sequence (II), each of which is c h a r a c t e r i z e d by i ts values [M] = [MJ] and the ra te constants k 3 . - i + i , i

An i l lus t ra t ion of such a reac t ion i s the d ia tomic molecule d issoc ia t ion reac t ion in a gas mix tu re which is rea l ized by sequential v ib ra t iona l exci ta t ions of the molecule .

A mul t i s t age reac t ion with s eve ra l M j and a r b i t r a r y values of the r a t e cons tants ~ + 1, i , evidently cannot be r ep re sen t ed as the sum of mul t i s tage reac t ions , each of which r e f e r s to one specif ic M] in each s tage.

Taking into account that fo r each s ing le - s t age reac t ion

.4, + .~P~ A~ ~ w 3: ' (IV)

(14) i s val id , and assuming the sum of the r a t e s of all r eac t ions (IV) with a fixed n u m b e r of s tages i ("the c u r r e n t in tens i ty" in any sect ion of a "conduc tor ~) to be ident ical

\ ' , / i w V \--J~,, J_ _, / : ! :~ . . . . . . , ::~ J ' ( 1 6 )

we obtain

where

- \ t - - -Y,:; J

m - I

R : : ,.:,\"l~" z, i i { ~-~l kf. ,.~IM/] ) - ' . (17)

F o r m u l a (19) r e m a i n s app rox ima te ly t rue even i f the l e s s r igorous conditions*

]-- (Jr ~1, i - ' 2 -- F / I - - 2. ( 18 ) 1

a r e sa t is f ied ins tead Of (16). If the inequal i t ies (18) a r e not sa t i s f ied , then the ra te of the mul t i s tage react ion

* Inequal i t ies of the type (18) under l ie the method of quas i s t a t i ona ry concent ra t ions of M. Bodenshtein and N. N. Semenov [7], which is used extens ive ly in chemica l k inet ics .

599

i s subs tant ia l ly a s soc ia ted with the dynamics of i n t e rmed ia t e p roduc t fo rmat ion and cannot be e x p r e s s e d in t e r m s of any p a r a m e t e r s independent of the concent ra t ions of these p roduc ts .

The mul t i s t age reac t ion (II) p r o c e e d s jus t in one d i rec t ion 1 - - m o r m --* 1 if X m = 0 o r X 1 = 0. In conformi ty with (17), the reac t ion r a t e s (J+ and J ) hence equal, r e spec t ive ly ,

:.=X~/R, ]_=X, , , /R, (19)

The phenomenologica l k ine t i c s equations of each d i rec t ion of the mul t i s tage reac t ion a r e c h a r a c t e r - ized by the r a t e constant k tm (or kml) independent of the concent ra t ions of the medium components except when MJ actual ly has one value, i .eo, when the ident ical med ium pa r t i c l e s take p a r t in all the s ing le - s t age reac t ions introducing a s ignif icant contr ibut ion to the total " r e s i s t a n c e " of the chain R [Eq. (17)]. In such a case the phenomenological equations of the reac t ion kinet ics a r e

J~:.=k~,.L [M]. :_=k,,,:f~ [M]. (20)

Compar ing (19) to (20), taking account of (17) fo r R and the definition of the "potent ia ls" (14a), we find

I i k, ,n: ,,, I , k,,: - ,,~ ~ (21)

Zl \ " ( k i i , : z i : . t , ~ i - ~ - -~ l ( ] ' s I) .l i:.=~l ' . , ~11~ f _

/e1,./k.,l = f(.,~, (22)

w h e r e Krm is the equi l ibr ium constant of the reac t ion conver t ing the ini t ial subs tance A t into the final

p roduc t s A m .

If M j t akes s eve ra l va l ue s , then (20) does not hold, but J+ and J can be r ep re sen t ed as

] . . . . ,~-~ L, : =~:., ,# .... (23)

where a i m and O~ml a r e coeff ic ients* dependent on the t e m p e r a t u r e and on [MJ]. By analogy with the known definit ion of the i m p a c t recombina t ion coeff ic ient of charged p a r t i c l e s , which depends on the concentra t ion of the med ium a toms , le t us a g r e e to cal l C~m and ~ml the reac t ion ra t e coeff ic ients . A compar i son between (23) and (17) fol lows:

a l . , = l l z IR, a,.1 = 1/z.,R, (24)

a~,./ami=K,,~. (25)

The re fo re , the ra t io between the r a t e coeff ic ients of the forward and r e v e r s e d i rec t ions of the mul t i - s tage reac t ion (II) equals i t s equi l ib r ium constant . If ident ical p a r t i c l e s of the medium par t i c ipa te in all the s ing l e - s t age reac t ions introducing a not iceable contr ibut ion to the total " r e s i s t a n c e " of the chain, then the ra t io between the ra te cons tants of the fo rward and r e v e r s e d i rec t ions of the mul t i s tage react ion is also equal to the equi l ibr ium constant .

Exac t ly as (12), the re la t ionsh ips (32), (25) a r e a consequence of the equi l ibr ium of the medium. They a r e sa t i s f ied independent ly of whe the r the populat ions of the i n t e rmed ia t e reac t ion p roduc t s are o r a r e not

in equi l ibr ium.

Let us p r e s en t (21) fo r the r a t e constant of the mul t i s tage reac t ion (II) when the p re -exponent la l f ac to r xi(T) in the s ta t i s t i ca l sums (15) d i f f e r only by the d i f ferent constant s ta t i s t ica l weights gi:

rn- - I Eiq_l/T J[? ] - - 1 .

i (26)

* The fo rma l introduct ion of the concept of a mul t i s t age reac t ion r a t e constant in the case of s eve ra l va lues of MJ can re su l t in a f rac t iona l o r d e r of the reac t ion [fractional va lues of v i in (2a)]. This well known fact in c h e m i s t r y has r ecen t ly been named the "dilution effect" in applicat ion to the v ibra t iona l r e - l axa t ion and d issoc ia t ion of mo lecu l e s [8-10].

600

F o r example , (26) def ines the ra te constant of the impac t ionizat ion of a toms in s ing le -component a tomic g a s e s with the low f ree e lec t ron concent ra t ions neglected in the approximat ion of the s ing le -quantum excitat ion sequence of s eve ra l of the lowest e lec t ron ic l eve ls , which has meaning at high t e m p e r a t u r e s (T ~ 1 / 2 0 , where I i s the a tom ionizat ion potential ,* and ~he d issoc ia t ion ra te constant of d i a t o n i c molecu les in a s ing le -component a tomic medium with one-quantum vibra t ional t rans i t ions as i n t e rmed ia t e s tages of the react ion. Moreove r , in the ca se of d issocia t ion , all the preexponent ia l f ac to r s of the s ta t i s t i ca l sums of the i n t e rmed ia t e s tages of the react ion (vibrat ional ly excited molecule states) a re ident ical to the accu r acy of a weak dependence of the m om en t of i ne r t i a on the n u m b e r of the v ibra t ional level . The known express ion fo r the d issoc ia t ion ra t e constant? of d i a t o n i c molecu les in a monatomic gas [12-15]

(27)

follows f rom (21) to this accuracy .

In the genera l c a se of the mul t i s tage react ion A 1 + M ~- A m + M s ing le - s t age r eac t i onsAi+ M ~ A k + M a r e poss ib le fo r a r b i t r a r y va lues of i and k f r o m the sequence 1, 2 . . . . . m.

It i s convenient to use the analogy between (14) and Ohm' s law to analyze the ra t ios between the ra te constants (or coefficients) of a mul t i s tage react ion, and to c o m p a r e the kinet ic s cheme of the react ion to the equivalent e l ec t r i ca l c i rcui t , where a point c o r r e s p o n d s to each value of i o r k of the above-ment ioned sequence. Any two points i and k a re connected a conductor whose r e s i s t a n c e Rik i s defined by the ex- p r e s s i o n

where ~ ~ k is the r a t e constant of the s ing l e - s t age reac t ion

A~ + M i ~ Ah + Mi. (V)

The equivalent c i rcu i t of the react ion with two in t e rmed ia t e s tages A 2 and A 3 is shown in the f igure .

Any set of s e r i e s and pa ra l l e l r e s i s t o r s connecting 2 points of the e lec t r i ca l loop (1 and m in this case) is equivalent , as is known, to s o m e one r e s i s t a n c e R* connecting these points . The re fo re , the de - pendence of the total mul t i s t age reac t ion ra t e J i m on the "potential" d i f fe rence i s exp re s sed by the fo rmu la

y , = ( X , - - X , , , ) / R * (28)

which ag ree s with (14) and (17) to the accu racy of the magnitude of the " r e s i s t a n c e . " It hence follows d i - r ec t ly that the ra t io between the ra te coeff ic ients O~lm and V~ml of any mul t i s t age react ion in an equi l ibr ium medium equals the equi l ibr ium constant of the react ion and is defined by (25)~ The expres s ions f o r the reac t ion ra t e cons tants a r e Obtained f r o m (24), r e spec t ive ly , by replacing R by R*o If ident ical p a r t i c l e s of the med ium (identical va lues of [M]) pa r t i c ipa te in all the s ing le - s t age reac t ions , then the mul t i s tage reac t ion is cha r ac t e r i z ed by ra te constants independent of [M], which a r e re la ted to the " r e s i s t a n c e " R* by the fo rmu la s

k .~ [ M ] / R * z . k ~ = = ,,,1 [M]/R*z,~

*An e s t ima t e of the domain of the h i g h - t e m p e r a t u r e approximat ion fo r the ra te constant of ionizat ion by a tomic i m p a c t follows f r o m a compar i son between the effect ive excitat ion cross , s ec t i ons of the ground and f i r s t exci ted s ta tes of an a tom jus t as this i s done i n [1] in de te rmin ing the l imi t s of the approximat ion of ins tantaneous ionization of a toms by electrons. ? The d issoc ia t ion ra t e is o rd ina r i l y r ep re sen t ed as the product of the r a t e constant by the pa r t i c l e con- cent ra t ion in the med ium and in all, not only the unexcited, molecu les . In such a normal iza t ion the r a t e constant (26) should be divided by the ra t io between the n u m b e r of all molecu les and the n u m b e r of m o l e - cules in the ground s ta te , which approx ima te ly equals the v ibra t iona l s ta t i s t i ca l stun~

601

Fig. 1. Equivalent c i rcui t of the reaction A 2 -'~ A t with the in termedia te s tages A 2 and A 3.

and sat isfy a relationship analogous to (22)

lel~/k~,= Ka,~. (29)

Only the magnitude of the " res i s tance" R* and i ts relation to the ra te constants of the s ingle-s tage react ions , which can be expressed by quite awkward formulas , depend on the specific scheme of the r e a c - tion.

Diffusion Approximation. Fokker- Planck Equation

If the energies E i + I - Ei of the adjacent reaction stages* are much less than T and if only those s ingle-s tage react ions for which E i + 1 - Ei << T are equally probable, then it is possible to go f rom the difference relat ionship (14) to a differential relationship by replacing Xi + 1 - Xi by (E i + 1 - Ei)dX/dE. Hence, (12), (22), (25), and (29) for the ratio between the reaction rate constants o r coefficients , which a re genera l ly independent of the magnitude of the interval E i + i - E l , evidently remain valid. Only express ions fo r the reaction rate constants (or coefficients) vary; i n p a r t i c u l a r , a f ter taking account of all paral le l and se r ies paths of the reaction, the sums over i in formulas of the type (17) and (21) go ove r into in tegra l s in the energy E.

The "cur ren t intensity" associa ted with the s ingle-s tage react ion A i + M ~ A k + M is expressed, according to (14), as

Jlk : X,. -- X k ! . /~.,. , R~ i ..... &, .~.~ kiz: [M]I. (30)

1

The react ion Ap + M ~ A p + k- i + M yields the same "cur ren t" at the point k for any-~ value of p be- tween i and k. The number of such points p equals ~'(E k) I E k - E i l, where ~p(E k) is the number of all in- te rmedia te react ion s tates per unit energy interval . Multiplying (30) by the number of points mentioned and replacing X i - X k by (E i - E k) (dX/dE), we find the total " cu r ren t intensi ty n at the point k for a fixed value

of the interval k - i

(Ek) dX (Ek - - E i ) ~. Jik= Ri k dE

(31)

(The direct ion of the "cur ren t n Jik is considered posit ive if E k > El.) To calculate the total " cu r - rent" at the point k it is still n e c e s s a r y to in tegra te k over all i < k. We consequently obtain

dX (32) J = - ~ ( (hE)'- > z ~ ,

((AE)2) ~ X'/~AE ~. \ i

((hE) ~) _= [Mi] ~ k{k (E~ - - E~) ~" dE~. (33) - - co

The argument E -= Ek, on which all the fac tors in the right side of (32) depend, is omitted in (32). The coefficient l/2 is introduced in (32) because the integrat ion of (31) is not per formed ove r E i < E k, but ove r all E i. The quantity < (AE) 2 > has the meaning of the roo t -mean - squa re change in E pe r unit t ime.

Integrating (32) under the s ta t ionar i ty condition J = const, we find

X 0 - - X m J - -

Em dE 2 ,( z~ <(A~)%

Eo

* The react ion s tages a re numbered in the o r d e r of increas ing energy E.

f i t is assumed that the values of kp, p + k - i va ry slightly in the interval i -< p < k.

602

This express ion d i f fe rs f r o m (28) only by the in tegra l

E dE (34)

R*=2 ,! zr~, ~(hE)% ' Ez

now playing, the p a r t of the " r e s i s t a n c e n R* . Correspondingly , the r a t e cons tants of the two reac t ion d i r e c - tions (if [MJ] in (33) has jus t 1 value) o r the ra te coeff ic ients (if the re a r e s eve ra l va lues of [MJ]) a r e de - t e rmined by (21), (17), o r (24) in which R mus t be replaced by (43). The rat io between the reac t ion ra te constants o r coeff ic ients i s independent of the f o r m of R and i s de te rmined by (22) o r (25), as has a l r eady been noted.

Going f r o m the ~potentials" X o v e r to the product of the concent ra t ions f in (32) according to (14a), and taking account of the dependence (15) of the s ta t i s t i ca l sums on the n u m b e r of react ion s tages e x p r e s s e d in t e r m s of the energy E, we obtain

d~

The reac t ion ra te and the ~potential ~ g rad ien t in the reac t ion energy space E a re l inea r ly dependent (32), jus t as a r e the diffusion flux and the concentra t ion gradient of diffusing p a r t i c l e s . In subs tance , the s y s t e m motion o v e r the reac t ion s tages is a diffusion p r o c e s s , but in the reac t ion energy space r a t h e r than in the c u s t o m a r y space . To de s c r i be the development of such a p r o c e s s in t ime , (32) o r (35) should be supplemented by the continuity equation of the diffusing substance

0qJ 0

~x ~ - = - - 3T] .

This equation with the express ion (35) fo r the c u r r e n t i s of the Fo ld~e r -P l anck type [16, 17], by which the diffusion motion of a s y s t e m in the space of any genera l i zed coordinate o r group of coord ina tes ave raged o v e r the remain ing phase space coord ina tes , i s desc r ibed . The condition of constant flux used in the theory of reac t ion ra t e constants c o r r e s p o n d s to s t a t ionary diffusion.$

The reac t ion ra te constants in an equi l ibr ium medium w e r e calculated in the diffusion approximat ion by P i taevsk i i and Gurevich [20, 21] ( e l e c t r o n - i o n recombina t ion in an a ~ m o r e l ec t ron medium), Stupo- chenko and Safaryan [4] (diatomic molecule d issocia t ion in an i ne r t l ight gas , see. a lso [22]-[25]), B ibe rman , Varob ' ev, and Yakubov (atom ionization by e lec t ron impac t , see the su rvey [26]), Dalidchik and Sayasov [27], Denisov and Kuznetsov [28] (e lec t ron- ion recombina t ion in a d ia tomic molecu le medium), London and Keck [29] (ion recombinat ion during t r ip le col l is ions) .

I n t e r r e l a t i o n b e t w e e n t h e R e a c t i o n R a t e s f o r t h e F o r w a r d a n d R e v e r s e D i r e c t i o n s i n a N o n e q u i l i b r i u m M e d i u m

T w o - T e m p e r a t u r e Gas. One of the local ly nonequi l ibr ium media of p rac t i ca l in t e res t , whose non- equi l ibr ium is mainta ined by the c o u r s e of the reac t ion i tse l f , i s a med ium consis t ing of two in te rna l equi l i - b r i um s u b s y s t e m s . The nonequi l ibr ium of the whole medium is e x p r e s s e d only by the fact that the t e m p e r a - tu res of these s u b s y s t e m s a re dis t inct . Subsys tems of e lec t rons and heavy pa r t i c l e s (atoms and ions) in a mona tomic gas o r s u b s y s t e m s of v ibra t ional and t rans la t iona l d e g r e e s of f r eedom in a m o l e c u l a r gas can have dif ferent t e m p e r a t u r e s under specif ic condit ions, fo r example~

If all the reac t ion s tages and the the rmodynamic averaging of the s ta tes of the ini t ial subs tance and the products A i a r e due to in te rac t ion of the reac t ing components with jus t one of the s u b s y s t e m s , then (24), (29) r e m a i n valid, but it i s ce r t a in ly n e c e s s a r y to subst i tute the t e m p e r a t u r e of this subsys t em as the t e m p e r a t u r e of the medium there in . F o r example , the ra t io between the a tom ionizat ion ra te constant of the a toms k12 and the ion recombina t ion k21 in an e lec t ronic medium

~ K r a m e r s [18] f i r s t fo rmula ted the diffusion theory in appl icat ion to chemica l reac t ions . In a n u m b e r of subsequent p a p e r s a m o r e r igorous bas i s was given f o r the theory as was an invest igat ion of the l imi t s of i t s appl icabi l i ty . A bib l iography and exposit ion of the diffusion theory of chemica l reac t ion k ine t ics i s con- tained in the Nikitin book [19].

603

A + e ~- A § + 2e (VI)

equal s

i

I meT \3'~ z~ e) k21

Here Te is the t empera tu re of the f ree and bound e lec t rons , Z(Te) and Z(Te) are the electronic s ta-

t is t ical sums of the atom and ion. The quantity K(T_~ is none o ther than the equilibrium constant of the react ion (VI) under the nonequil ibrium conditions being considered, T e ~ T. Since the atoms and ions have the identical t empera ture T, this equil ibrium constant is genera l ly independent of T. (The pre-exponential fac tors of the s tat is t ical sums of the initial substance and the products of the react ion (VI) depend on two t empera tu res , but the ratio between these fac tors contains just the e lectron temperature' .)

The kinetics of e l e c t r o n - i o n recombinat ion with the part icipation of an atom as the par t ic le of the medium M

A+ + e + M - + A + M (vii)

is considered in [30] for unequal e lectron T e and atom T tempera tu res . Sufficiently low tempera tures , where the react ion is like continuous diffusion of a weakly bound electron in energy space, are kept in mind. However, the v e r y formulat ion of such a problem in application to the react ion (VII) can apparently be of just methodological in te res t , because the condition for the existence of a Maxwell e lect ron distribution with a t empera tu re different f rom the atom tempera ture , and the condition under which diffusion recombination proceeds according to the scheme {VII) r a the r than (VI) a re contradic tory . The f i rs t condition is expressed by

Tc~ << Te~, (36)

where Tee and Tea are the cha rac te r i s t i c buildup t imes for the Maxwell distribution in the electron sub- sys tem and in the total e l e e t r o n - a t o m s y s t e m , respect ively. The second condition [21] is d i rect ly opposite to (36). Removing the mentioned contradict ion and keeping in mind the purely methodological aspect of the question, let us note that the dependence a ~ T/(TeT/2) i s found in [30] for the recombination coefficient of the reac t ion (VII) in a two- tempera tu re gas. Such a resul t is obtained if the difference between the t empera tu res T e and T is taken into account only in calculating the diffusion coefficient [20]. However, the medium tempera ture , and not only the diffusion coefficient, should enter throughout in the F o k k e r - P l a n c k equation (35) for the react ion (VII). The electron tempera tu re hence f igures only in the l imit condition at the point E = 0. It can be shown that the c o r r e c t resul t has the fo rm (~ = a T / T e 3/2) with the coefficient a found in the Pi taevski i paper [20] for the case T = T e.

The in ter re la t ion between the kinetics of the two react ion di rect ions is somewhat more complex if the sequence of the react ion s tages is separated by the boundary i = k o r by a narrow band Ak on one side of which the subsys tem 1 plays the par t of the medium and the subsys tem 2 on the other side. For example, because of the different dependence of the effective excitation c r o s s section of an atom on the energy of the initial state of this atom,* for some relat ive f ree e lectron concentrat ions the t ransi t ions over the lower atom levels at sufficiently high tempera tures can be due mainly to electron, and the ionization of the excited pa r t i c les , to atom impacts . The s tat is t ical sums z i of the initial substance i = 1, of the in termediate sub- stance at the boundary mentioned i = k, and of the final react ion products i = m, defined by (12) and (15) in a monotempera ture gas, a re under the considered conditions:

= , z~ = Z,n exp I. T~ '

where T 1 and T 2 a re the t empera tu res of subsys tems 1 and 2.

(37)

*A quite s t rong dependence of the effective c r o s s section of the fu r the r excitation on the electron energy of the initial atom state hold if no in tersect ion of the te rms of the initial and final s tates occurs during the coll is ion of slow atoms.

604

The t e m p e r a t u r e of that subs tance , in te rac t ion with which r e su l t s in the rmodynamic averag ing of the populat ions of the s ta tes A i, mus t be subst i tuted as the a rgumen t in the p re -exponen t i a l f ac to r s X e, X m. However , s ince the t e m p e r a t u r e dependence of the f ac to r s X i i s r e la t ive ly slight, and the s u b s y s t e m t e m p e r a t u r e s do not o rd ina r i ly d i f fe r in o r d e r of magnitude e i ther of the two t e m p e r a t u r e s can be the a rgu- ment in a s a t i s f ac to ry approximat ion . Represen t ing the reac t ion )/i as two success ive mul t i s tage reac t ions

A~+M'~-A, , -}-M ~, A~+M(2)~A,~+M(:) , (VIII)

we eas i ly obtain the following exp re s s ions fo r the reac t ion ra te coeff ic ients a i m , ~mi and for the i r ra t io

~lk ~ rJ'mk ~kl =='- ' am1 (38) ~ l m O:,lt I 1- ~km if'k: "~- ~'lr

Taking into account that f o rmu la s of the type (24) a re valid fo r each of the reac t ions (VIII), we find

otl.,]a.,l ~ z,,,/z~, (39)

where z I and z m are defined by (37). Let us note that since the quantity [MJ] in the reactions (VIII) is char- acterized by at least two values of j, the ratio between the.coefficients in (39) cannot be reduced to the cor- responding ra t io between the react ion ra te cons tan ts , except fo r the c a s e s C~kl >>akin and akl << q~mo

F o r m u l a (39) can be r ep re sen t ed as

E { I ~ ' %. ~ ( ~ - ~-,) (40) (To) 8 (Zml -

where KIm (T2)is [he equi l ibr ium constant of the reac t ion at the t e m p e r a t u r e T 2. If 'X~, )/k, and : /mdepend on the t e m p e r a t u r e T 1 o r on d i f ferent t e m p e r a t u r e s , then (40) i s t rue to the accu racy of a weak dependence of Xm,/Xi, X k on the t e m p e r a t u r e d i f fe rences which, as has a l r eady been r e m a r k e d , a r e usual ly slight. The rat io between the react ion coeff ic ients can also be r ep re sen ted as the product of the equi l ibr ium constants of the reac t ions (VIII) to app rox ima te ly the s a m e accu racy as in (40)

ulm _= Kik (T~) I(~,~ (T.). (41) ~ I

In con t r a s t to (25), the ra t io between the reac t ion ra te .coef f ic ien ts (40), (41) depends on t~v t e m p e r a - tu res . As is easy to see , with the poss ib le sl ight i naccu racy in the p re -exponen t ia l f ac to r noted above, the right side of (40), (41) i s the equi l ibr ium constant A I ~ A m in a t w o - t e m p e r a t u r e medium. There fo re , in the t w o - t e m p e r a t u r e medium under cons idera t ion , as in the rmodynamica l ly equi l ibr ium media , the ra t io between the reac t ion ra te constants equals i t s equi l ibr ium constant which is dependent on p a r a m e t e r s cha rac t e r i z ing the s ta te of the medium.

In the case Ski >7 a~m Eqs. (38) go o v e r into

(42)

M o r e o v e r , the reac t ion ra t e in this ea se i s independent of [M i] and, t he re fo re , i t i s poss ib le to go f r o m the coeff ic ients a i m , C~mlover to the react ion ra t e cons tants according to the re la t ions

k,m = K,k (Tx) akm/[M(2)], (43) kin, = am'-/[M(2)].

where

km 1 (44)

605

Equations (42), (43) co r r e spond to the equi l ibr ium of a fas t reac t ion A t ~-A k at the t empe ra tu r e T 1 which ex i s t s during the slow total reac t ion (VIII).

F o r example , (43) and (44) a r e appl icable to the d issocia t ion and recombina t ion react ion ra te con- s tants of d ia tomic molecu les

AB --~ AB ~-- A + B -[- AB, (IX)

i f the t e m p e r a t u r e of the t rans la t ional motion of the a toms and molecu les equals T 2 and the v ibra t ional energy E of the molecu les i s a Bol tzmann dis t r ibut ion fo r E -< E k [31, 32] with the t e m p e r a t u r e Tt.* The posi t ion of the boundary E k fo r the reac t ion (IX), e x p r e s s e d in t e r m s of the v ibra t ion f requency w k of the molecu le at the level k, depends sl ightly on the specif ic p r o p e r t i e s of the molecu les and is de te rmined by the condition Wk/W 1 .~ 0.7-0o8.[32]. However , the absolute value of E k depends on the spec ies of molecule .

An exper imen ta l p roof of the fact that the v ibra t iona l t e m p e r a t u r e during the dissocia t ion of d ia tomic molecu les in s t rong shocks does not equal the t e m p e r a t u r e of the pa r t i c l e t rans la t iona l and rotat ional mo- t ions, and the inves t igat ion of the d issoc ia t ion k ine t ics under such conditions has been p e r f o r m e d by Losev , Gene~alov, Ovezhkin, Yalovik, and Maks imenko in [34-38]. The r e su l t s of the inves t iga t ions [34-38] ag ree both qual i ta t ive ly and quant i ta t ively with the theore t ica l ana lys i s of these phenomena [31, 32]~

One Of the in te res t ing poss ib i l i t i e s of a f u r t he r exper imenta l study of the k inet ics of the reac t ion (IX) i s a s soc ia ted with the independence of va r i a t ions in the v ibra t iona l t e m p e r a t u r e f rom the t rans la t ional m o - tion t e m p e r a t u r e by int roducing some additional v ibra t iona l ene rgy sou rees into the gas c o m p r e s s e d by the shock. In p a r t i c u l a r , t e s t s of t h i s k i nd would p e r m i t r e f inement of the p a r a m e t e r E k in the re la t ionship (44) which gove rns the recombina t ion r a t e cons tant at any t e m p e r a t u r e s T 1 and T 2 if the d issocia t ion ra t e con- stant i s known at any point on the T1, T 2 plane.

I n f l u e n c e o f S p o i l i n g t h e M a x w e l l D i s t r i b u t i o n on t h e R a t i o b e t w e e n t h e F o r w a r d a n d R e v e r s e R e a c t i o n R a t e C o n s t a n t s

The connection between the r a t e cons tants of the s ing l e - s t age react ion (3) c h a r a c t e r i z e s the react ion k inet ics in a t he rmodynamica l l y equi l ibr ium medium. If the medium in which the reac t ion occu r s is non- equi l ibr ium, then (3) becom es empty . This is a l r eady c l e a r f r o m the fact that the t e m p e r a t u r e has a deft- uite meaning only f o r equi l ibr ium s ta t e s of the medium. It a lso follows f r o m the above that the re la t ion- ships (22), (25), (29) for mul t i s t age reac t ions a r e indeed inappl icable to the case of the rmodynamica l ly non- equi l ibr ium media . As r e g a r d s the reac t ion r a t e cons tants t h e m s e l v e s , they can even by meaningful in appl icat ion to r eac t ions in nonequi l ibr ium media . F o r this , i t i s n e c e s s a r y that the nonequi l ibr ium d i s t r i - bution function of the med ium be s t a t ionary o r that i t s t ime change be sufficiently slow in compar i son the reae t ion r a t e . In such e a s e s , the ra t io between the reac t ion ra te cons tants does not depend on the t e m p e r a t u r e (or not only on the t empera tu re ) but on the specif ic va lues of the externa l p a r a m e t e r s w h e r e - upon s ta t ionar i ty of the nonequi l ibr ium dis t r ibut ion function is a s su red . The external e l ec t r i ca l o r e l e c t ro - magnet ic f ields can be such p a r a m e t e r s , for example . Among the ex te rna l conditions on whichthe reac t ion ra te constants can depend i s a lso the s t rong spat ia l inhomogenei ty of the s y s t e m resul t ing in t r a n s p o r t pheno- mena. The question of calculat ing reae t lon ra t e cons tants in externa l f ie lds o r for some other s y s t e m p e r - turbat ion except the effect of the reac t ion i t se l f is beyond the scope of this su rvey . Let us just note the evident c i r c u m s t a n c e that the radia t ion yield of the exci ted a toms (molecules) outside the s y s t e m l imi t s s lows down the ionizat ion (dissociat ion) , a c c e l e r a t e s recombinat ion , and r e su l t s in the s ta t ionary ionizat ion (dissociation) level beirlg l e s s than the equi l ibr ium value. The r e v e r s e p ic tu re holds fo r the s y s t e m sub- jected to a s t rong e l ec t r i ca l f ield o r ex terna l l ight flux in the appropr i a t e wavelength range.

The solution of a n u m b e r of functions about the e lec t ron dis t r ibut ion function and about the k inet ics of a tom excitat ion and ionizat ion by e lec t ron i m p a c t in an externa l e l ec t r i ca l o r e lee t romagne t ic field can be found in the Granovski i book [39], in the Kagan su rvey [40], and in [41-43]. The influence of radiaht l o s s e s of an opt ica l ly thin gas l a y e r on the r a t e of e lec t ron ic recombina t ion and ionizat ion of a toms has been inves t iga ted by Bates et al . , [44-46] and Collins [47], and by Biberman , V0rob ' ev , and Yakubov (modi- fied diffusion approximat ion in which the continuous e lec t ron diffusion in the ene rgy space i s taken into account fo r low, and the d i s c r e t e ~tom level configurat ion fo r high e lec t ron binding ene rg ies , see the s u r - vey [26]).

* Osipov [33] obtained (44), wr i t t en somewhat d i f ferent ly , in appl ieat ion to the reac t ion (IX) fo r the model of a t runcated ha rmonic o s c i l l a t o r (E k is the d issocia t ion energy) .

606

In connection with the deve lopment of l a s e r i n f r a r ed spec t roscopy quest ions of the optical pumping of v ibra t iona l energy have recen t ly become vital in addition to the c l a s s i ca l method of obtaining a toms and rad ica l s by exposing a ga s ' t o e l ec t romagne t i c quanta of the v is ib le and u l t rav io le t wavelength bands (photo- lys i s ) . The influence of opt ical sou rces of v ibra t iona l excitat ion and photodissociat ion of v ib ra t iona l ly excited molecu les on the ra te of gas d issoc ia t ion has been studied by Gordietn, Osipov, and Shelepin (see the s u r v e y [48]) and by Safa ryan [49]~ Some expe r imen ta l inves t iga t ions on this topic have been mentioned in these p a p e r s .

Relat ionships of the type (3), (22), (25), and (29) with a t empe ra tu r e -dependen t equi l ibr ium constant a r e meaningful as the s ta r t ing point of the ana lys i s and for reac t ions in nonequi l ibr ium media if the equil i - b r i um as a whole is s l ightly d is turbed. The s m a l l n e s s of the deviation f r o m equi l ibr ium can be e x p r e s s e d e i the r by the fact that the d is t r ibut ion function of the medium~ not much di f ferent f r o m the equi l ibr ium function fo r all ene rg i e s , o r by the fact that i t i s s t rongly nonequi l ibr ium jus t in some bounded energy range to which a re la t ive ly smal l p a r t of the phase space occupied by the medium co r r e sponds . F o r example , during the t he rm a l ionizhtion of a toms by e lec t rons (VI) at high t e m p e r a t u r e s (T is g r e a t e r than 1/20 of the a tom ionizat ion potent ial [11]), the e lec t ron dis t r ibut ion function i s essen t ia l ly l e s s than the equi l ibr ium function only at ene rg ie s sufficient fo r a tom excitat ion. The nonequi l ibr ium of the d is t r ibut ion is a s soc ia ted with the fac t that the e lec t rons with energ ies h igher than the a tom excitat ion ene rgy E 2 rapidly lose i t in an e las t ic col l is ions with a toms . These l o s s e s a r e only pa r t i a l ly filled in because of e las t ic e l e c t r o n - a t o m and e l e c t r o n - e l e c t r o n col l i s ions . Incomple te compensa t ion of the l o s s e s occu r s because the kinetic energy of the a toms is t r ansmi t t ed c o m p a r a t i v e l y slowly to the e lec t rons because of the g r ea t d i f ference between the pa r t i c l e m a s s e s , and the in tens i ty of mutual e l ec t ron ' in t e rac t ion is also suff icient ly smal l i f the re la t ive e lec t ron concentra t ion c i s low. The ra t io between the n u m b e r of e lec t rons with energ ies E > E 2 and the i r total quanti ty at the t e m p e r a t u r e T << E 2 i s exponential ly smal l , but the excitat ion and ionizat ion r a t e con- s tant i s de te rmined p r i m a r i l y by name ly this ra t io . E lec t rons with energ ies l e s s than E 2 a r e cha ra c t e r i z ed during this reac t ion by dis t r ibut ion functions c lose to the equi l ibr ium function with the e lec t ron t e m p e r a t u r e [11] ( large c case) o r with the a tom t e m p e r a t u r e [50] (small c case) .

It i s conven ien t to r e p r e s e n t the r a t e constant of a single o r mul t i s tage reac t ion in such c a s e s of slight spoi lage of the med ium equi l ibr ium as the equi l ibr ium value of this constant mult ipl ied by some coeff ic ient v(T, Ni) dependent on the t e m p e r a t u r e and concent ra t ions of the react ing components , whose d i f fe rence f r o m one is a s soc ia ted with the spoi lage of medium equi l ibr ium during the react ion. It i s e s - sent ial that the ra te cons tants of the forward and r e v e r s e reac t ions a r e genera l ly desc r ibed by dist inct p a r t s of the dis t r ibut ion function. Thus, only that p a r t of the dis t r ibut ion function which c o r r e s p o n d s to energy va lues above the threshold i s ' i m p o r t a n t fo r the endothermal reac t ion di rec t ion. Hence, the c o r - rec t ion f ac to r s v(T, Ni) to the r a t e constants of the two reac t ion d i rec t ions , calculated fo r even the s ame dis t r ibut ion functions co r respond ing to a given nonequi l ibr ium gas composi t ion , a re dis t inct . Moreove r , the f o r m of the noneqni l ibr ium dis t r ibut ion function of the med ium depends on the d i rec t ion of the reac t ion . F o r example , i f the reac t ion (VI) p roceeds p r i m a r i l y on the ionizat ion side, then the n u m b e r of e l ec t rons with ene rg ie s E > E2 is l e s s than the equi l ibr ium value, as has a l ready been r e m a r k e d , and this r e su l t s in a co r respond ing diminution in the ionizat ion ra t e constant (~r < 1). F o r the r e v e r s e d i rec t ion of the reac t ion , the n u m b e r of such e l ec t rons i s g r e a t e r than the equi l ibr ium value (Maxwell equi l ibr ium is kept in mind fo r a given total quanti ty of f r ee e lec t rons) , but the role of these h igh -ene rgy reac t ions is not essen t ia l in recombinat ion . Hence, ' : the ' ionizat ion k and recombina t ion k ' r a t e constants of reac t ion (VI) sa t i s fy the re la t ionships

k =-~(y)n(y,c,~), k '= ~'(y),

k/k ' = I~ (y, c, ~) K (y).

H e r e y = T e f o r l a r g e and y = T f o r low re la t ive e lec t ron concent ra t ions c, k and ~t a r e the equi l ibr ium va lues of k and k ' , r e spec t ive ly , c i s the equi l ibr ium value of c, and K(y) i s the equi l ibr ium constant of the reac t ion {VI). If the reac t ion (VI) i s s t rongly r e v e r s i b l e (i .e. , the gas composi t ion is'~far f r o m the equi l i - br i tun ~alue c <<c),then ~r depends only on y and c~ The;ionizat ion ra t e constant k fo r the reac t ion (VD at y = T e and va r ious dependences of the effect ive a tom ionizat ion c r o s s ' sect ion on the e lec t ron energy with spoilage of the e lec t ron Maxwell d is t r ibut ion taken into account has been calcula ted in [51-53]. The co- efficient ~ (y, c) in the ca se of g r e a t e s t p r ac t i ca l i n t e r e s t of a l i n e a r threshold dependence of the effect ive a tom exci tat ion c r o s s sect ion on the e lec t ron energy fo r the reac t ion (VI) has been calculated in [11] and [50], r e spec t ive ly , fo r high (y = Te) and low (y = T) re la t ive e lec t ron concentrat ions~

607

Because of the strong dependence of the coefficient 1-[ onthe degree of spoilage of the ionization equili- brium (i~ bn the difference XjX2), we determine the rate of a single-stage atom excitation reaction at high temperatures (20T >- I [11]) and it is impossible to express the rate of the whole ionization reaction WI) as a l inear relation of the Lype (14) with the quantity kl2z 2 which would be independent of X 1 - X 2. Limit cases of a strongly i r revers ib le reaction X 1 ~" X 2 and a small deviation from'ionization equilibrium [ X 1 - X 2 [ << X1, as well as the case of very large electron concentrations (almost complete single ionization) are the exceptions. Inthe f i rs t of these cases Jl2 is determined by the expression

/12=H(y, c)k~lz2X,

(with the equilibrium value k21z2) instead of (14) while (14) remains valid in the second and third cases.

C O N C L U S I O N S

1. The concept of rate constants (coefficients) of mutually reversible reaction directions in an ideal gas (or in an ideal solution) and the question of thbir connection with the equilibrium constant are quite meaningful upon compliance with the following two respective conditions: a) the progress of the reaction in each of the two directions is such that the rate of change in the concentration of intermediate products is low in comparison to the reaction rate, and b) the reaction occurs in a medium whose state is ei ther in or sufficiently close to thermodynamic equilibrium. Under the conditions mentioned, the ratio between the rate constants (coefficients) ofthetwo reaction directions equals its equilibrium constant calculated at the temperature of the medium. This regulari ty is intrinsic not only to the single-stage but also to complex multistage reactions.

2. The ratio between the rate constants (coefficients) of mutually reverse reaction directions' in a spatially homogeneous medium whose state is character ized by two different subsystem temperatures (T 1 and T2) , equals the nconstantw K(T1T 2) characterizing the equilibrium of the reaction in such a not completely equilibrium medium.

3. If the equilibrium of the medium is substantially spoiled during the reaction itself, then the state of the medium depends (not only in quantitative but also in qualitative respects) on the direction of the reaction in the general case. Hence, the ratio between the reaction rate constants (coefficients) is already not a thermodynamic character is t ic of the medium.

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

1. O.K. Rice, J. Phys. Chem., 67, No. 7, 1733 (1965}o 2. N.S. Snider, J. Chem. Phys. , 42, No. 2, 548 (1965). 3. A . I . Osipov, Teor . i Eksp. Khim, 2_, No. 5, 649 (1966). 4. E . V . Stupochenko andM. N. Safaryan, Teor. i Eksp. Khim.,2, No. 6 783 (1966). 5. Yu. L. Klimontovich, Zh. l~ksp. Teor. Fiz. , 52, No. 5, 1233 (1967). 6. L . M . Biberman, V. S. Vorob'ev, and I. T. Yakubov, Zh. l~ksp. Teor. Fiz., 56, No. 6, 1992 (1969). 7. V.N. Kondrat'ev, Kinetics of Chemical Gas Reactions [in Russian] Izd. AN SSSR, Moscow (1958). 8. S .R. Byron, J. Chem. Phys. , 44, No. 4, 1378 (1966). 9. D.C. Hardy and B. S. Rabinowitch, J. Chem. Phys., 48, 1282 (1968).

10. J . H . Kiefer, J. Chem. Phys., 57, No. 5, 1938 (1972). 11. Yu. P. Denisov and N. M. Kuznetsov, Zh. Prikl. Mekh. i Tekh. Fiz., No. 2, 32 (1971). 12. E . E . Nikitin, Dokl. Akad. Nauk SSSI~ 119, No. 3,526 (1958). 13. E. Montroll and K. E. Shuler, Advan. Chem. Physo, No. 1,361 (1958). 14. B. Widom, J. Chem. Phys., 34, 2050 (1961). 15. A . I . Osipov and E. V. Stupehenko, Usp. Fiz. Nauk, 79, No. 1, 81 (1963). 16. L. ]~. Gurevich, Principles of Physical Kinetics [in Russian], Gostekhizdat, Moscow (1940). 17. M.A. Leontovich, Statistical Physics [in Russian], Gostekhizdat, Moscow (1944). 18. H.A. Kramers , Physica, 7, 284 (1940). 19. E . E . Nikitin, Theory of Elementary Atomic-Molecular Processes in Gases [in Russian], Khimiya,

Moscow (1970). 20. L . P . Pitaevskii, Zh. ]~ksp. Teor. Fiz. , 42, No. 5, 1326 (1962). 21. A.V. Gurevich and L. P. Pitaevskii, Zh:-]~ksp. Teor. Fiz. , 46, No. 4, 1281 (1964). 22. T. Bak and J. Lesowitz, Disc. Faraday Sot., 33, 189 (1962).

608

23. T. Bak and J. Lesowitz, Phys. Revo, 131, 1138 (1963)o 24. J. Co Keck and G. Carr ier , J. Chem. Phys., 43, 2284 (1965). 25. S. Ao Losev, Doctoral Dissertation, Institute of Mechanics, Moscow State University (1969). 26. Lo Mo Biberman, V. S. Vorob'ev, and I. T. Yakubov, Usp. Fiz. Nauk, 107, No~ 3, 353 (1972)o 27. F. Io Dalidchik and Yuo S. Sayasov, Zh. Ekspo Teor. Fiz~ 49, No. 7, 302 (1965); 52, 1592 (1967)o 28. Yu. P~ Denisov andNoM. Kuznetsov, Zh. Eksp. Teor. Fiz., 61, No. 12, 2298 (1971). 29. So London and J. Co Keck, J. Chem. Phys~ 48, No. 1, 374 (1968). 30. Vo A. Abramov, Optika i Spektr., 18, No. 6,974 (1965). 31. No Mo Kuznetsov, Dokl~ Akad. Nauk SSSR, 164, No. 5, 1097 (1965). 32. N.M. Kuznetsov, Teor. i Eksp. Khim~ 7, No~ 1, 22 (1971). 33. Ao I. Osipov, Teor. i Eksp~ Khim. 2, No. 5,649 (1966)~ 34~ N.A. Generalov and V. Ya. Ovechkin, Teor. i Eksp. Khim. 4, No. 6, 829 (1968)o 35. V. Ya~ Ovechkin, Candidate's Dissertation, Institute of Mechanics, Moscow State University (1970). 36. S.A. Loser and M. So Yalovik, Khim. Vys. Energ~ 4, No~ 3, 202 (1970). 37~ M~ S~ Yalovik, Candidate's Dissertation, Institute of Mechanics, Moscow State University (1972)~ 38. V~ A~ Maksimenko, Candidate's Dissertation, Physics Department, MoscOw State University (1972). 39. V~ Lo Granovskii, Electrical Current in a Gas [in Russian], Nauka, Moscow (1971). 40. Yu~ M. Kagan, Spectroscopy of a Gas-Discharge Plasma [in Russian], Nauka, Leningrad (1970). 41~ Vo N~ Soshnikov and Vo S. Trekhov, Zh. Tekho FiZo, 37, 1414 (1967)~ 42~ A.I . Lukovnikov, Eo S. Trekhov, and E. P. Fetisov, Physics of a Gas-Discharge Plasma [in Russian],

No~ 2, Atomizdat, Moscow (1969). 43~ V.I . Myshenko and Yu~ P. Raizer, Zh~ Eksp~ Teoro Fiz., 61, No. 5, 1882 (1971). 44~ D~ R~ Bates, A. Eo Kingston, and R. W.P. McWhirter, Proc~ Roy. Soc., A267, 297 (1962)~ 45. D~ R~ Bates and A~ Dalgarno, in: Atomic and Molecular Processes , Academic Press (1962)~ 46~ D.R. Bates andS. P. Khare, Proc. Phys. Soc~ 85. 231 (1965). 47~ Co Collins, Phys~ Rev., 177, No~ 1,254 (1969). 48~ B~ F~ Gordiets, Ao I. Osipov, et alo, Usp. Fizo Nauk, 108, No. 4, 655 (1972). 49~ M. No Safaryan, Teor~ i ]~ksp~ Khim. _8, No. 3, 322; 8, No. 4, 445 (1972)~ 50. Yu~ P~ Denisov and N. M. Kuznetsov, Zh. Prikl. Mekho i Tekh. Fiz., No~ 3, 6 (1971)o 51o K~ Wojaczek, Beitr{ige Plasmaphysik, 5, 3 (1965). 52. K. N~ Ul'yanov, Teplofiz. Vys~ Temp~ 4, No. 3, 314 (1966)~ 53~ L~ Mo Biberman, V. S~ Vorob'ev, and I. T. Yakubov, Teplofiz, Vyso Tempo, 6, No. 3, 369 {1968)~

609