K I N E T I C S O F D I S S O C I A T I O N O F P O L Y A T O M I C

U N D E R N O N E Q U I L I B R I U M C O N D I T I O N S

M O L E C U L E S

N . M . K u z n e t s o v UDC 541.10

In o rde r to r e so lve many p rob l em s of chemical k inet ics and gasdynamics one mus t know the r a t e s of chemica l r eac t ions in a nonuniform envi ronment . The r a t e constant of the reac t ion depends on the absolute quas i s t a t ionary s tate of the surroundings and, as is the nonuniform environment i tself , is not in genera l de- sc r ibed by a single definite t e m p e r a t u r e . Such a nonuniform environment , in which two or m o r e subgroups of deg rees of f r eedom a re c h a r a c t e r i z e d by dif ferent t e m p e r a t u r e s , is r e la t ive ly s imple , but p r e s e n t s p rac t ica l i n t e re s t . A gas of heavy pa r t i c l e s and e l ec t rons with two Maxwell dis t r ibut ion t e m p e r a t u r e s may s e rve as an example .

As a r e su l t of col l is ions with p a r t i c l e s of the surroundings and with each other and of the act ion of ex- t e rna l sou rces of energy, the reac t ing molecu les may also have different t e m p e r a t u r e s for the sepa ra t e types of in t r amolecu la r mot ions . The m o s t commonly studied s y s t e m of th is type is a d ia tomic gas d issocia ted by a s t rong shock wave [1-3] or by r e sonance pumping of vibrat ional energy by in f ra red l a s e r i r rad ia t ion [4]. In the p r o c e s s of dissociat ion, the t e m p e r a t u r e s of v ibra t iona l and t r a n s l a t i o n a l - r o t a t i o n a l mot ion of the mo le - cules a r e different , and the d issoc ia t ion r a t e constant depends on each of them. Severa l approximat ions and aspec t s of the kinet ics of d issocia t ion of polyatomic molecu les having a nonuniform dis t r ibut ion of in t r amolecu- l a r energy have been examined [5-7].

Calculated in the p resen t work is the r a t e constant K for un imolecular decomposi t ion in the indicated cases of a nonisothermal g a s - p h a s e envi ronment or of in t ramolecu la r motion. The t e m p e r a t u r e s T l of the subgroup d e g r e e s of f r eedom a r e cons idered to be given. In each concre te case , t he i r de te rmina t ion p r e sen t s an independent p rob lem, the solution of which r e q u i r e s the knowledge of the dependence of K on T l �9 The kinet ics of un imolecu la r r eac t ions in a uni form envi ronment without external sources of energy may be well desc r ibed qual i ta t ively, and in many ca se s quant i ta t ively, by RRKM sta t is t ical theory . Thus, those conclusions of s ta t i s t ica l theory which do not cont rad ic t the new formula t ion of the question a r e expediently used fo r i ts solution. In pa r t i cu l a r , it per ta ins to the exp re s s ion of the density of l eve l s [see Eq. (11)] and to the s tate of fas t i n t r amolecu la r in teract ion at high ene rg i e s with the r e su l t an t energy dependence of the r a t e constants fo r spontaneous decomposi t ion of the act ivated molecule .

U n i m o l e c u l a r D e c o m p o s i t i o n in a T w o - T e m p e r a t u r e E n v i r o n m e n t

L e t tha act ivat ion and deact ivat ion of r eac t ive molecu les p roceed as a r e su l t of col l is ion with two sub- groups of s t r u c t u r e l e s s pa r t i c l e s of the surroundings having the t e m p e r a t u r e s T i and T2, r e spec t ive ly . The usual s ta t i s t ica l theory of un imolecular decomposi t ion is based on two types of quali tat ively di f ferent models of act ivat ion. The f i r s t of these is the model of s t rong col l is ions, in which a f t e r each col l is ion the molecule has a un i form distr ibut ion of energy throughout all i ts d e g r e e s of f r eedom, with a t e m p e r a t u r e equal to that of the envi ronment . The o ther type is the act ivat ion model , wi th the fo rmat ion of an in te rmedia te s ta t i s t ica l complex [8] o r the s tepwise t rans i t ion model with spaces of the o rde r of kT at each col l is ion [9]. The second type of model s e e m s m o r e rea l i s t i c ; neve r the l e s s , in the very widest t e m p e r a t u r e reg ion both types lead to a lmos t the s ame quantitative r e s u l t s and analyt ical exp re s s ions [9]. Only in the l imi t of very high t e m p e r a - t u r e s 3(n-1}kT > D (where D is the act ivat ion energy, and n is the sum of the number of v ibra t ional and half the number of rota t ional deg rees of f r e edom of the molecule) does the model of s t rong col l is ions give too high a value of the r a t e constant . In the t w o - t e m p e r a t u r e environment , as will be seen below, these types of models lead to quite different r e s u l t s .

During s t rong col l is ions in a t w o - t e m p e r a t u r e envi ronment the quas i s ta t ionary population of s ta tes of the molecu le , x i, is de t e rmined by the equation

Inst i tute of Chemical Phys i c s , Academy of Sciences of the USSR, Moscow. T r a n s l a t e d f r o m T e o r e t i c h e s - kaya i Ekspe r imen t a l ' n aya Khimiya, Vol. 14, No. 3, pp. 311-319, May-June , 1978. Original a r t i c le submit ted

August 18, 1977.

244 0040-5760/78/1403-0244 $07.50 �9 1978 Plenum Publishing Corpora t ion

~,x-~ ( tO + n x , (r~) - x, ( , + &) = o, (1)

where v 1 and v~. a r e the gas kinetic n u m b e r s of col l is ions of the molecule with pa r t i c l e s of subgroup 1 and 2, r e spec t ive ly ; ~i(T l ) is the equi l ibr ium population of the i - th quantum sta te at t e m p e r a t u r e T l ; v = v I + v 2; and k i is the r a t e constant fo r spontaneous decomposi t ion of the i - th s ta te . The r a t e constant for un imolecu la r decomposi t ion is e x p r e s s e d in the f o r m [1]

~ . ks ~t~ (T,) + (z2~ (T2) K v "4- let = r (TO + azK (T2), (2) l

where K(T l ) i s the r a t e constant fo r un imoleeu la r decomposi t ion in a s i n g l e - t e m p e r a t u r e env i ronment with the gas kinet ic number v and with act ivat ion by s t rong col l i s ions , a l = v l / v . In the l imi t s of high and low den- s i t i es , f r o m (2) it follows that

K~ = czlK.. (T1) + a2K~. (T2), (3)

Ko --~ ~ziKo (Ti) + cz2Ko ( T2)"

The t r ans i t ion r eg ions f r o m low to high dens i t ies for K(T l) and K(T 2) a r e different . Thus the total t r ans i t ion reg ion in a t w o - t e m p e r a t u r e envi ronment can be very ex tens ive .

We tu rn now to the model of s tepwise t rans i t ions with fixed changes of ~5(~) in in ternal energy. The in ternal energy s will r e p r e s e n t the v ibra t ional and rotat ional energy of the molecule . H e r e we r e s t r i c t the calculat ion of the r a t e constant to the l imi t of low dens i t i es . The quas i s t a t ionary flux through the energy leve l s of an inact ive molecule (g <D) is independent of ~ and is e x p r e s s e d by the fo rmula

J = vN (e) { [~iPt (e) e ~r + a2P2 (s) e -ar x~ - - [~iPt (e) + a2P 2 (8)1 x~+6(~) }, (4)

where xe is the population of a single quantmn state (with an individual s ta t i s t ica l weight) at energy ~; P l (a ) and P2(8) a r e the t h e r m a l motion ave raged probabi l i t i es of the t r ans i t ion z + 5(z) -~ e due to col l is ion of the molecu le with pa r t i c l e s of subgroup 1 or 2; and N(e) is the number of the state in the energy in terval

+ 6(~) . With the normal i za t ion ~ x ~ = l , the unknown r a t e constant equals the f lux (4).

Le t us introduce in place of x e the new va r i ab le

X (@ = x J A (~),

A (8).= [ ' ] [ (z,P~ (8~)e-~(~)/~r'+ =,P~ (e~)e -~(~t)/~r" ] (5) K I L siP, (~) + %P2 (%) J

I

[the product is under taken by s teps with energy 80=0, e l = 8(0), e2= 8(0)+ 6(E l) .. . to the point n e a r e s t to el. In th is r ep resen ta t ion , in place of (4) we have

K = J = A [e + 6 (e)] ~N (e) [a~Pi (e) + a2P2 (e)] [X (e + 6 (e)) - - X (s)]

o r

x (o) - x (D) K =

R

A -1 [e~ + 8 (~i)] • R - - ~ / ' N (e~) [~P~(eO + ~ P ~ (s~)]

where the summat ion is under taken at the ve ry s ame points as in the product (5), but up to e = D - - 8 ( 8 ) .

In the l imi t of the lowes t p r e s s u r e s , the population of s t a t e s of the act ive molecule is vanishingly smal l and, consequently,

X (0) 1 (6) X(D)=O, K o ~ - - - ~ - - Z(T) R '

where Z(T) is the s ta t i s t i ca l sum of all in ternal d e g r e e s of f r e edom of the molecu le . If T 1 =T2, then A(c)= exp( - ~ /kT1) , and (6) t akes the f o r m analogous to the known ladder approximat ion to the d issocia t ion r a t e constant for a d ia tomic [N(~) =1] molecu le [10]. H e r e the r a t e constant in approximat ion (6), with the t r a n s i - t ions 5(s ~-kT at each col l is ion P l (~) = 1, coincides within the r e l a t ive co r rec t ion ( n - 1 ) k T / D with the resu l t (3) obtained based on the model of s t rong col l is ions [9]. However , for T 1 ~ T 2 the d i f ference in the a p p r o x i m a - t ions (3) and (6) m a y be ve ry l a r g e . F o r example , fo r T2>>T1, P1 ( ~ ) - 1 and 5(~)=kT2, Eqs. (3) and (6) give, r e spec t ive ly ,

245

K(0 3) _-- o:2K o (T2),

So sharp a d i f ference in the e f fec ts of the two examined mode l s m a y be used as an exper imenta l ver i f ica t ion of the function of s tepwise and d i r ec t t r ans i t ions in act ivat ion kinet ics . The object of invest igat ion might be, fo r example , a r a t i f i e d gas r eac t i ng in a volume and act ivated by col l is ions with the su r face of a vesse l whose opposi te walls have ve ry different t e m p e r a t u r e s .

U n i m o l e c u l a r D e c o m p o s i t i o n w i t h N o n u n i f o r m E n e r g y

D i s t r i b u t i o n in I n t e r n a l D e g r e e s o f F r e e d o m

Le t us ca lcula te the r a t e constant for un imolecu la r decomposi t ion assuming that fo r the molecu la r in- t e rna l energy e -< ~ *, one subgroup of internal deg rees of f r e edom is populated at t e m p e r a t u r e T 1 and the second subgroup, consis t ing of the r ema in ing internal deg ree s of f r eedom, at t e m p e r a t u r e T 2. The vibra t ional (VT) re laxa t ion of the molecule is provided for by in terac t ion with the environment , of t e m p e r a t u r e T, which in genera l d i f fers f r o m T 1 and T 2. F o r specif ic i ty we take T 2 > T 1. The point e = e * is cons idered given. I t m a y be de te rmined f r o m the condition of equali ty of the r a t e s v 1 of the VT t r ans i t i ons and the r a t e s v 2 of those p r o c e s s e s by which the given va lues of t e m p e r a t u r e s T 1 and T 2 a r e maintained. Usual ly v 1 and v 2 depend s t rongly on e , and change in opposi te d i rec t ions [2]. Thus the approx ima te substi tution is also poss ib le of the t rans i t iona l ene rgy reg ion at one side of which v2>>vl, and at the other v2<<vi, by the dividing line e = ~ *. If the energy ~ * is so l a rge that due to the anharmonic i ty of the i n t r amolecu la r mot ion it is mean ing less to d i f ferent ia te between the t e m p e r a t u r e s T 1 and T 2 for l eve l s c lose to e *, then the quest ion at hand is meaning- ful without fu r the r r e f inemen t s in the case T 1 =T 2. (For T l CT 2 in the lower l eve l s and for a ve ry g r ea t value of e * one m u s t introduce stil l another energy l imit , defining the reg ion of smal l v ibra t ions and quas iergodic motion.)

The populat ions of the quantum s ta tes xi for e i > ~ * depend on the r a t e of VT t rans i t ions , propor t ional to the gas kinet ic number of col l is ions v (T), and on the r a t e constant for spontaneous decomposi t ion k i. These populat ions and the cumulat ive decomposi t ion r a t e constant K(T, T1, T 2) m a y be de te rmined by the s ame method as in the usual "equi l ibr ium" ~f s ta t i s t ica l t heory . The di f ference in the ca lcula t ions l i es only in that mot ion upwards through the energy l eve l s of the molecule mus t now be cons idered beginning not f r o m e = 0, but f r o m

In the "equ i l ib r ium" theory , the r a t e constant for un imolecu la r decomposi t ion is p resen ted in the form, analogous to (1)

K ---- v (T) --.X-~ + (r) ~- ki " (7) l

E x p r e s s i o n (7) follows s t r i c t ly f r o m the model of s t rong col l is ions, but, as was noted in the preceding section, within the r e l a t i ve co r rec t ion ~ ( n - 1 ) k T / D it r e s u l t s also within the scope of m o r e r e a l i s t i c models of the ac t ivat ion of monoatomic molecu les with the t r a n s m i s s i o n of energy of the o r d e r of kT at each coll ision. In the p rob lem examined in th is sect ion, express ion (7) r e m a i n s valid if the magnitude ~i r e p r e s e n t s the popula- t ion not of the whole t he rm odynam i c equi l ibr ium, but of the equi l ibr ium r e l a t i ve to the ave r age value of popu- la t ion x* of one quantum s ta te with ene rgy ~ *. Inasmuch as the act ivat ion of a s tate for which ~ > e *, a c - cording to a g r e e m e n t with the p rob lem, i s de te rmined by the in te rac t ion with an envi ronment having t e m p e r a - t u r e T, the specif ied r e l a t i ve equi l ibr ium impl ies that

x-i ~ x* exp [(e* - - ei)/kT], e i > e* (8)

Inasmuch as at comple te equi l ibr ium

xi -~ x* exp [(e* - - ei)/kT ], x* ~_- exp (-- s*/I~T)/Z (T), (8a)

f r o m (7) and (8) we find

K (T, T~, 7'2) = K (T) x*/~*, (9)

tEqui l ibr ium, in the sense that the t e m p e r a t u r e of i n t r amo lecu l a r mot ion in the p r o c e s s of an i r r e v e r s i b l e reac t ion , where it m a y genera l ly be introduced, is not d i f ferent f r o m the t e m p e r a t u r e of the envi ronment .

246

where K(T) is the r a t e constant for T 2 = T 1 =T . Th i s constant may be de termined e i ther by exper iment or based on the RRKM stat is t ical theory (with co r r ec t i on f r o m con'servation of momentum of the amount of move- ment during spontaneous decomposi t ion) . Thus to de te rmine K(T, Tl, T2) it is sufficient to calculate x * .

By de terminat ion of the average , $r

1 - (10) ~_~ x~ (e,) x~ (~* - - ~) Q~ (80 ~ (~) A (q), (~*)

x~ (g) --~ exp (-- y/kTz)/Z~ (Tz), ~ (e~) - - g~ (@/A (@,

where ~21 (e l ) and Z l (TI ) a r e the densi ty of leve ls and the s ta t is t ical sum of subgroups I , for l = 1, 2; gl (~ l ) is the s ta t is t ical weight of the state of subgroup l with energy e l ; A(e I ) is the distance between neighboring ene rgy leve ls of the subgroup I at the point e l ; and ~ ( e * ) is the density of levels of all internal degrees of f r eedom at the point e *. So that the s ta t is t ical sum Z l (T l ) should be de termined by only one t e m p e r a t u r e T l , we r e s t r i c t the case of sufficiently low t empera tu res : k(nlTi+naT~ ) < ~ *, where n I is the sum of the number of vibrat ional and half the number of rota t ional degrees of f r eedom of the subgroup l . The densi ty of l eve l s 121 for e t<ng~z (where w I is the geomet r ica l average of the vibrat ional quanta of the osc i l l a to rs of subgroup l ) is de te rmined by the cor responding sum over quantum states, and with good accuracy for et>nzh~z, by the express ion

1 ( ez+eo~ )"t-t Z~(T) ~, (e,) = ~ ........ k r /~Y ' (11)

where F(ni) is the gamma function, ~0l is the total energy of null vibrat ions of osc i l l a to rs of the subgroup, and Z l k(~) is the c lass ica l approximat ion to the s ta t is t ical s u m Z l (T).

Substituting express ion (11) for ~22(~*-~ i) in (10) and pe r fo rming the calculation, we obtain with an e r r o r no g r e a t e r than 20% (see Appendix)

x* = ~z (e*) Zt (T +) - - S (T+i e_~./kr,, (e*) Z i (Tt) Z z (T2)

1 1 kT+~(~--I-~q)-l; ~ kT t kT z '

1,13 ~1 - - - - In [ 1 - - 8max/(t~* Jr- Poe2) ], gmax

(-~-/L ~L z (n2- - I )~ :Ot - (rt, ._ 1)(g* ..~ 802 ) ~ ]I/2 01, em~ ~ = max -- [ - ~ + e01 (e* + %2) Jr

L--~- nt + n 2 - - 2

(12)

(13)

(14)

(15)

(16)

e*+%t ) S (T +) ~ Zth (T +) e~~ nt; / r (n 0, , kT +

o o

where F (~; 8) ~ I t~-le-tdt is the par t ia l gamma function, ~ and Z (T +) is the s ta t i s t ica l sum for subgroup 1 at

.Q~ (e*) Zl (T +) - - S~ (T +) Z (T) ~* [ l -k'----I ~ e ~ ~ r T~ ], (17) ~ (e*) Z t (Tt) Z 2 (T~)

the " t e mpe ra tu r g" (13).

F r o m (9) and (12) i t follows that

K (T, Tt, T2) K (T)

o r using (11) for e ( e *) and 122(~ *) and cancell ing out common fac to r s in the rota t ional s ta t is t ical sums (which is possible in the approximation of the r ig id ro tor ) ,

K (r, Tt, Tz) Y (n t + nz) Q2~ (T2) Qi (T +) - - qt (r+) ' • K (T) - - F (n2) Q~ (T) QI (T)

;(+++. ,. ++f1 1)1 +, • \ T t / e* + % e* + % ] exp [--~- T T 2 '

oo o a

? Fo r detai led tables of the in tegra ls ~ t~-le-~tdt =_ f~l-g .I tct-le-~tdt see [ I i I.

247

s ~E0t "~- g02, where r 1 is half the num ber of rota t ional degrees of f r e edom of subgroup 1. The substi tut ion of Z fo r Q and S for q impl ies that ins tead of a v i b r a t i o n a l - r o t a t i o n a l s ta t i s t ica l sum, the cor responding vibra t ional s t a t i s t i - cal s tuns a r e used.

The dependence in (17) of the r a t e constant K(T, TI, T 2) on T 1 is included only in the fac tor [Z~(T +) - S1CT+)]/ZI(Ti). The a s y m m e t r y of the r e s u l t with r e f e r e n c e to t e m p e r a t u r e s T t and T 2 is r e la ted to the initial condition T2 -> TI-

The exp res s ions (12), (17), and (18) a r e significantly s impl i f ied in the case of l a rge o r smal l ft.

1. F o r rice* + e g 1) > 2 ( h i - 1 } in Eqs . (12), (17), and (18), one may d i s r e g a r d Si(T +) and ql(T+). Also, if > ( n t - l ) / e 0 i - - ( n 2 - 1)/(e* + %e), then s = 0 and ~/= 1,1 3 (n 2 - 1 ) / ( ~ * + e 02)-

2. F o r rice * + s we take e - f i e l -~ 1 and calculat ion of (10) leads to the obvious r e su l t

e -8"/kr' K(T,T . T2) Z(T) [e* ( 1 1 )] x * - - - - ~ ; K(T) = ~ e x p ~ T r 2 "

Th i s same r e su l t a lso follows, of course , f r o m (16) as the pa r t i cu l a r case n 1 = 0 [all internal deg rees of f r eedom a r e c h a r a c t e r i z e d by the t e m p e r a t u r e T2, and the values with subsc r ip t 1 in (16) and (17) mus t be dropped].

I t may be shown that fo r T2---TI> T and l a r g e enough e * so that the energy region sma l l e r than e * in- t roduces the bas ic contr ibution to ZCT2) , the ra t io K(T, T1, T2)/KCT) > 1 is found. This ra t io at the given t e m - p e r a t u r e s T 1 and T 2 does not depend on v ; i .e. , it has the s ame value in the l imi t s of l a rge and smal l dens i t ies and in the t rans i t ion reg ion . In the impl ic i t f o r m this ra t io may of course depend on the density i f the t e m - p e r a t u r e s T 1 and T 2 a r e functions of v . Fo r example , in the l im i t v ~ oo for any finite and constant intensity of external influence, T2-*T 1 -*T.

F r o m a compar i son of the der ived r e s u l t s with the exper imenta l data on the r a t e of un imolecu la r de- composi t ion in a local ly noniso thermal envi ronment and on the t e m p e r a t u r e s T1, T2, and T, the p a r a m e t e r ~ * may be de te rmined . This p a r a m e t e r is one of the c h a r a c t e r i s t i c s of i n t e rmo lecu l a r dynamics and, in p a r t i cu - l a r env i ronments , of i n t r amolecu l a r motion.

APPENDIX

F o r T2->T1, the bas ic contribution to the summat ion (10) is int roduced by the energy reg ion ~1 sufficiently smal l so that ~22(s * - el) may be e x p r e s s e d in approximat ion (11) f r o m which, for

si< (8* + ~o,)/2 (n~-- I) (A. I)

with an e r r o r of seve ra l pe rcen t , it follows that

flz (8* - - ei) = f12 (e*) e - ' le ' , ~l = (nz - - l ) / ( e * + Co2)- ( A . 2)

The exponential r ep r e s en t a t i on CA. 2) substant ia l ly s impl i f ies the calculat ion of the summat ion (10). TMs express ion may also be used without the r e s t r i c t i on (A. 1) if the coefficient in the exponential is de te rmined such that in the neighborhood el = e m a x of the max imum value of the function which is the a rgument of the summat ion (10), the equali ty (A. 2) is a l m os t exact . F o r th is it should be

a (n~-- 1) In [1 - - emax/(e* + e02)]. (A. 3) Smax

The value of the coeff icient a [near unity; fo r a = 1 the equali ty (A. 2) is fulfil led exact ly at the point of the maximum] should be se lec ted such as to min imize the e r r o r in the approx imate calculat ion of the summat ion (10) where the e r r o r fo r a = 1 is l a r g e . We calcula te f i r s t of all x* and K(T, T1, T2) /K(T) , consider ing ~ given and leaving unde te rmined for the m om en t the values of e m a x and a (see below).

The summat ion (10), a f t e r substis of (Sa) and (A. 2) into it, t akes the f o r m

x* ~ Q2(e*) exp(-- s*/kT2) ~-~ Ql (%)e-e'/kr+A(et), (A. 4) Q (e*) Z t (TO Z2 (T2)

r

where T + is de te rmined by Eq. (13). Noting that the sum f r o m 0 to ~ * is equal to the d i f ference in the sums f r o m 0 to~o and f r o m ~ * to ~o, using for el > e * exp res s ion (11) for P~l(el), and rep lac ing the sum f r o m s to ~o by an integral , we find

248

e*

fit (el) e-e'/kr+A (ei) ---~ Zi (T +) - - S i (T+). (A. 5) ~t=0

where Sl(T+) is de te rmined by the identi ty (16). The substi tut ion of (A. 5) into (A. 4) l eads to the des i r ed r e s u l t (12).

De te rmina t ion of P a r a m e t e r s s and a . The point s m a y be found approx imate ly , by identifying it with the point of the m a x i m u m value of the function

T = (St ~ - 80t) nz-I (8" "4- $02 - - $0 n t - le - [ lg l , (A. 6)

which, c o r r e c t within a constant f ac to r , r e s u l t s in the a rgument of the summat ion (10) upon subst i tut ion in (10) of the e x p r e s s i o n s (8a) aad (11). Such an approximat ion of a rea x does not lead to a l a r g e e r r o r in the ca lcu- la t ion of (10). Indeed, if the exact value of e m a x is of the o r d e r of ~ * (small d i f fe rence T 2 - T1), then the r e p r e s e n t a t i o n (11) is appl icable to the calculat ion of (10) no l e s s than it i s appl icable fo r the density of l eve l s ~(~ * ) of all s y s t em s , and thus, the function (A. 6) is nea r ly exact . If emax<<e * ( large d i f fe rence T 2 - T 1 ) , then even a l a r g e e r r o r in e m a x [which m a y follow f r o m a low accu racy in the approx imat ion (11) fo r f~l(s in (A. 6) is p e r m i s s i b l e for calculat ion of ~22(e * - el) by Eqs . CA. 2) and (A. 3), inasmuch as in th is case

depends v e r y weakly on e m a x and, m o r e o v e r , in the reg ion of substantial contr ibution to (10), the density of l eve l s f~2(e* - el) is c lose to the value of ~22(s which is independent of e l .

The des i r ed approx imate value e m a x is equal to the sma l l e s t posi t ive roo t of the quadrat ic equation r e - sult ing f r o m the e x t r e m u m condition d r / d e 1 = 0. In the opposi te case , the function (A. 6) is m a x i m u m at the lowes t l imi t of the summat ion (10), and e m a x = 0. The complete reso lu t ion for e m a x obtained in this way is e x p r e s s e d in the f o r m of (15).

The e r r o r in the calculat ion of the summat ion (10) using approx imat ions (A. 2) and (A. 3) to fix the va lues of n 1 and n 2 is m a x i m u m if T 1 =T 2 and ~ = ~02 = 0 [since in this case , ~ m a x / ( ~ * + ~02) and the e r r o r in the approx imat ion (A. 2) a r e m a x i m u m ] . Thus , the p a r a m e t e r a in Eq. (A. 3) mus t be set such that, i n p a r - t icular , in this case the e r r o r in the calculat ion of the summat ion (10) may be r educed to a min imum. The p r o b l e m is substant ia l ly s impl i f ied s ince in the examined case , a f t e r subst i tut ion of the summat ion (10) by an in tegra l , it is eas i ly calculated exactly: The in tegra l obtained leads to the f o r m

I i x"'-x (I - - x)n2-1dx ~ F (ni) F (n2)/F (n t + n 2 - - 1). (A. 7) 6

A compar i son of the in tegra l s (A. 7) and the approx imate exp res s ions

1 .f x~'-le-nXdx' (A. 8) 0

in which ~ is de te rmined by the re la t ionsh ip (A. 3) [where in the p r e s e n t case ~max = ( n l - 1) ~ * / (n t +n 2 - 2)], showed that fo r a = 1.13, the e r r o r in approximat ion (A. 8) fo r all va lues of n 1 and n 2 in the l imi t s f r o m 1 to 10 does not exceed 20%0 Th i s g ives a bas i s to p ropose that fo r T2>T1 and fo r r ea l va lues of e ~ and e02, the e r r o r of the r e s u l t s (12)-(18) also does not exceed 20% and on the a v e r a g e should be significantly s m a l l e r . Sample numer i ca l ca lcula t ions of (10) for a s e r i e s of unequal T l and T2, fo r e01 = e02 = 0, supported this ap- p r a i s a l .

LITERATURE CITED

1. N . M . Kuznetsov, " In t e r r e l a t ion between the p r o c e s s e s of v ibra t ional re laxa t ion and the d issoc ia t ion of d ia tomic mo lecu l e s , " Dokl. Akad. Nauk SSSR, 164, No. 5, 1097-1100 (1965).

2. N . M . Kuznetsov, Kinet ics of Dissoc ia t ion of Molecules in Molecu la r Gase s . Second All-Union Symposium on Combust ion and Explosion [in Russian] , Izd. Akad. Nauk SSSR and Armen ian SSR (1969), pp. 184-188.

3. N . M . Kuznetsov, "Kinet ics of the d issoc ia t ion of mo lecu le s in mo lecu l a r gases , n Teo r . Eksp. Khim., 7, No. 1, 22-23 (1971).

4. Yu. M. Gershenzon and N. M. Kuznetsov, "Quas i s ta t ionary d issocia t ion of d ia tomic molecu les during the l a s e r exci ta t ion of lower v ibra t iona l l e v e l s , , Dokl. Akad. Nauk SSSR, 218, No. 5, 1128-1131 (1974).

5. N . M . Kuznetsov, "Kinet ics of po lya tomic m o l e c u l a r d issoc ia t ion under conditions of nonuniform d i s t r ibu- t ion of the v ibra t ional ene rgy , " Dokl. Akad. Nauk SSSR, 202, No. 6, 1367-1370 (1972).

6. N . M . Kuznetsov, "Theory of monomolecu l a r decomposi t ion of a single component gas and the carbon dioxide d issoc ia t ion r a t e constant at high t e m p e r a t u r e s , " Zh. P r i ld . Mekh. Tekh. F iz . , 3, 46-52 (1972).

249

7. N .M. Kuznetsov, nKinetics of monomolecular decomposition of a one-component gas," Dokl. Akad. Nauk SSR., 208~ No. 1, 145-148 (1973).

8. J . Keck and A. Kaielcar, "Statistical theory of dissociation and recombination for moderately complex molecules, ~ J . Chem. Phys., 49, No. 7, 3211-3223 (1968).

9. N .M. Kuznetsov, "High-temperature features of the decomposition of polyatomic molecules and dis- sociation of carbon dioxide, ~ Teor. Eksp. Khim., 13, No. 2, 185-190 (1977).

10. E . E . Nikitin, The Theory of Elemental Atomic Molecular P rocesses in Gases [in Russian], Khimiya, Moscow (1970).

11. Yu. A. Kruglyak and D. R. Whitman, Tables of integrals for Quantum Chemistry [in Russian], Vychisl. Tsentr Akad. Nauk SSSR, Moscow (1965).

A D E S C R I P T I O N OF THE C O O X I D A T I O N K I N E T I C S OF T E R N A R Y

S Y S T E M S USING AN E Q U A T I O N F O R B I N A R Y M I X T U R E S

R . V. K u c h e r , I . A . O p e i d a , and A. G. M a t v i e n k o

UDC 541.127

An investigation of the cooxidation of various organic substances by oxygen in the liquid phase is of great theoretical and practical importance. Although the theory of such processes has been developed for systems containing any number of components [1], because of the complexity of the final equation obtained, the investi- gations are usually confined to binary systems.

The scheme for the cooxidation of a binary mixture can be written in the form

R~H, R2H-~R,, R2, (i)

Rs -k" Oz "*" R,O2, (2)

R z -{- O~ -*- R202, (3) kpll

RiO2 + RiH ---'~ RiOzH + Rt, (4)

RiOz q- R~H ~P!~ RIO~H -k R~, (5)

R202 + R,H ~ R2OzH + R,, (6)

RzO2 -k RzH ~ R~O~H + R~, (7)

kt. (8) 2RiO 2 ---~

R,O z.-}- R2Oe k_~t'$ inactive products. (9)

kt2~ 2R2 O~- ~ (I 0)

For a particular system the dependence of the cooxidation rate wo~ on the composition is e~pressed by the relationship

(r, [R,H] 2 -~ 2 [Rill] [R2H] + r~ [R2H] ~) w~ I~ (11) r2~2rD Hm~I/2 ' w~ = ( r~ [R,H] z + 2~rirzS,6 ~ [R,H] [R2H] "k- 2t,2 t,x2 j J

w h e r e r l �9 = kt 2/r kt 2; = = A c c o r d i n g to t h e e eri- m e n t a l dependence of the oxidation rate on the composition, it is possible by means of an electronic computer

Institute of Physicoorganic Chemistry and Carbon Chemistry, Academy of Sciences of the Ukrainian SSR, Donetsk. Translated from Teoreticheskaya i l~ksperimental'naya Khimiya, Vol. 14, No. 3, pp. 320-325, May- June, 1978, Original article submitted May 27, 1977.

250 0040-5760/78/1403-0250507.50 �9 1978 Plenum Publishing Corporation