KINETICS FOR THE DISSOCIATION

IN A MOLECULAR GAS

OF MOLECULES

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

The thermal dissociat ion of molecules takes place mainly from the upper, vibrational levels which are within an energy range of the o rder of kT near the dissociat ion threshold. Therefore the dissociation rate , which is proport ional to the population Xn, depends on the rate of decomposi t ion of the molecules : in the dissociat ion p roce s s some quas is ta t ionary value is established for the population, which is determined by the balance in the ra tes for the removal of the highly excited molecules f rom the sys tem because of d issoc i - ation and the mechanism of vibrat ionalexci ta t ion which makes up these losses . Thus, the vibrational re lax- ation and dissociat ion of the molecules are interrelated. During thermal dissociat ion both p rocesses are descr ibed by an inseparable sys tem of kinetic equations.

The study of the dissociat ion kinetics for diatomic molecules which are presented as an impuri ty in a monatomic gas showed [1-3] that because of a decrease in the population Xn during dissociation, the rate constant of the la t ter appears to be much less in a number of cases (by severa l t imes or an order of mag- nitude) than its equilibrium value, which cor responds to the equilibrium population of the vibrational levels. The distribution for x n which exists in the course of the p rocess of the dissociat ion of molecules in a molec- ular gas is also nonequilibrium, but it differs qualitatively f rom the distribution of the vibrational levels for molecules in a monoatomic gas medium. During vibrational relaxation in a one-component molecular gas or in a mixture of gases whose molecules have s imi lar vibrational frequencies, as a f i r s t step, after a pe r - iod of t ime which is m u c h l e s s than the t ime required to establish complete thermodynamic equilibrium, an equilibrium is established with respec t to the exchange of vibrational quanta. Such an equilibrium means that the rate of t r ans fe r of a vibrational quantum f rom one of the molecules (A, B) to another is the same in the d i rec t and r eve r s e direct ions for the p r o c e s s e s :

A n + B i Z A n+-x + B ]~' (1~

(n and J are the numbers of the vibrational levels).

The p r o c e s s e s (1 ~ cannot by themselves lead to a complete thermodynamic equilibrium since they do not change the number of vibrational quanta in the system. The incomplete equilibrium to which the pro- c e s se s (1 ~ lead is an equil ibrium with a given number of vibrational quanta. In the following such a state for the sys tem will be called quasi-equil ibrium.

The quasi -equi l ibr ium in the sys tem of like harmonic osci l la tors is descr ibed by a Boltzmann d is t r i - bution function; the difference between the quasi-equi l ibr ium state and the equilibrium state is expressed in the fact that the t empera tu re for the Boltzmann distribution T k is not equal to the tempera ture T of the p rog res s ive and rotational motion of the molecules . In this case the rat io of the dissociat ion rate constants k(T k) to its equil ibrium value k(T) can be given in the form [4]

k(T) - - z -~-~ j e ~ (1)

Institute of Chemical Physics , ,~cademy of Sciences of the USSR, Moscow. Trans la ted f rom Teore t i - cheskaya i ]~ksperimental 'naya Khimiya, Vol. 7, No. 1, pp. 22-33, J anua ry -Februa ry , 1971. Original ar t ic le submitted December 1, 1969.

�9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available [rom the publisher for $15.00.

17

where z is the vibrational statist ical sum and ed is the dissociat ion energy.*

The dependence of the tempera ture for the Boltzmann distribution, T k, and the relationship of the rate constants for the dissociat ion of a harmonic osci l la tor (1) to ~ d /kT were studied in [4] where, in p a r - t icular , it was shown that for ~ d /kT > 20 to the ra t io (1) is equal to unity with high accuracy, and for e d / k T < 20 it dec reases markedly (by one to two orders of magnitude) with a dec rease in e d/kT.

In this work the dissociat ion is studied for anharmonic osci l la tors (diatomic molecules) in a gas con- s is t ing of molecules or of molecules and atoms. The problem differs considerably f rom the problem for harmonic osci l la tors . F i r s t of all, the quas i -s ta t ionary distr ibution of the anharmonic osci l la tors is not a Boltzmann distribution [6]. The difference in the distribution f rom the Boltzmann is the greater the grea ter the anharmonici ty of the oscillation, and for the upper vibrational levels of the molecule, on the population of which the dissociat ion rate depends, it can be quite significant. Secondly, the rate of the p rocess for the vibrational excitation of the anharmonic osci l la tors (real molecules)

A i § B ~ A ~+' § B (2 0)

for sufficiently la rge numbers of vibrational levels is in no way small in compar ison with the quantum ex- change rate (15. Therefore the quasis ta t ionary distribution, due to the rapid exchange, r e fe r s to level num- be r s l imited f rom above by some value n = n*. Both of the p roper t i es for the dissociat ion of anharmonic osci l la tors that were mentioned affect the value of the ra t io k(Tk)/k(T) [and, consequently, the value of the dissociat ion rate constant k(Ttd also].

Q u a s i - e q u i l i b r i u m D i s t r i b u t i o n f o r the E n e r g y of t h e A n h a r m o n i c O s c i l l a t o r s

T he quasi-equil ibrium distribution of the population for the energy levels of an anharmonic osci l la tor has the form

8it

x. = xoe - ~ - Y ~ , (2 )

where ~ n is the energy of the n-th vibrational level. This distribution, besides the t empera tu re T of the medium (the tempera ture for the p rog res s ive and rotational movement), is charac te r i zed by the pa ramete r T, the value of which for a given osci l la tor energy of E = E ~ nXn and a t empera tu re T is determined in the implicit fo rm by the relationship

~(Olnz \ ~ e - ~ ~ E : k T [ ~ ) , z : _ _ kT. n

Formula (2) was obtained in [6] by the analysis of a sys tem of kinetic equations for the population of all of the vibrational levels for the conditions that the number x n changes only in the p rocesses (15. How- ever, it can be obtained by a much simpler method. We will show below that (2) follows di rec t ly f rom the principle of a detailed equilibrium and also f rom the canonical distribution of the sys tem with respec t to energy and number of par t ic les , and then we will t r ans fo rm the distribution (2) into a form suitable for cal- culating the dissociat ion rate constant. In this case the pa ramete r ,/ is expressed in explicit form in t e r m s of the osci l lator energy and the t empera tu re of the sys tem T.

The p rocesses for the exchange of vibrational quanta (10 ) are accomplished by means of collisions whereby the initial coordinates and momenta which charac te r ize the p rogress ive and rotational movement of the par t ic les are distributed in an equilibrium manner (according to Gibbs) with a tempera ture T. The re - fore, f rom the principle of detailed equilibrium it follows that the equilibrium for the react ion (15 takes

* During dissociation, in addition to the quasi-equil ibr ium diminution in the population xn, which leads to the relationship (1), there is also, finally, a destruction of the quasi-equil ibr ium distribution Xn near the dissociat ion threshold. The la t ter is the one reason for the dec rease in the dissociat ion rate constant in a monatomic gas medium. In a molecular medium, however, such a destruct ion plays a secondary role and leads to a much smal le r change in the rat io k(Tk)/k{T ) as compared with (1) [5]. F rom the resul ts in [5] we see, for example, that in a medium of harmonic osci l la tors the deviation of the population for the upper vibrational levels f rom the quasi-equil ibr ium values dec reases the dissociat ion rate constant by approxi- mate ly a factor of 2.

18

p l a c e f o r t h e cond i t i ons (A%. -- As~ ) x~+tx-----L. : exp

x~xj+, kT ' As l = el+ , - - s t (l : i,'13

o r in a d i f f e r e n t d e s i g n a t i o n

f t - - [ / = A e ~ - - A e i , f ~ k T l n xt X t + i

The r e l a t i o n s h i p s (3) a r e fu l f i l l ed fo r a l l v a l u e s of i and j. F r o m th i s i t f o l l ows e x p l i c i t l y tha t

f~ =- a § Aet, a = const,

Tha t is

(3)

_ a"i-Aei X i + l ~ e kT .

Xt (4)

D i s t r i b u t i o n s (2) and (4) c o i n c i d e f o r a = kTT.

F o r m u l a (2) c a n a l s o b e t r e a t e d as a c a n o n i c a l d i s t r i b u t i o n f o r a s y s t e m of o s c i l l a t o r s with r e s p e e t to the e n e r g y and the n u m b e r of v i b r a t i o n a l quanta , N,

] iN- -8 N

W(N, ezc)=Aoe kr , (5)

where # is the chemical potential. If the vibrational quanta are considered as quasiparticles, then the ex-

citation of the n-th level of the oscillator indicates that the oscillator is populated with n quasiparticles.

The values of the energy and number of quasiparticles in the oscillator are interdependent. If ~N = ~ n

then N = n. Thus, distribution function (5) for a single oscillator has the form

ngt--8 n

W : Ale , r (6)

Going f r o m (6) to the p o p u l a t i o n s of t he l e v e l s we f ind n ~ - - e n

Xn =- xoe kr (7)

F r o m a c o m p a r i s o n of (7) wi th (2) o r (4) we s e e t h a t - T k T ( o r - a ) i s t he c h e m i c a l p o t e n t i a l of the q u a s i p a r - t i c l e ( v ib r a t i ona l quanta) . F o r c o m p l e t e t h e r m o d y n a m i c e q u i l i b r i u m and at a g i v e n t e m p e r a t u r e the n u m b e r of quan ta in the o s c i l l a t o r is not an i ndependen t p a r a m e t e r . In th i s c a s e p -- 0 and (7) i s c o n v e r t e d into the ]3 o l t z m a n n d i s t r i b u t i o n wi th a t e m p e r a t u r e T.

In the c a s e of a h a r m o n i c o s c i l l a t o r Cn = n~w0 and (7) i s the B o l t z m a n n d i s t r i b u t i o n f o r any va lue of T . T h e B o l t z m a n n d i s t r i b u t i o n in t h i s c a s e i s c h a r a c t e r i z e d b y the t e m p e r a t u r e T which s a t i s f i e s the r e l a t i o n - sh ip

&0. "T~ 7" ~ = - - F (S)

and i s r e l a t e d wi th the m e a n e n e r g y of t he o s c i l l a t o r b y m e a n s of the t h e r m o d y n a m i c f o r m u l a

/~o~. I n ( l + /ion0 ) kT~ -- ~ " (9)

Let us find an explicit expression for T at a given value of the mean energy for the anharmonic oscil-

lator E and temperature T for the progressive-rotational movement. For this we used the fact that for

E << ~ d* the vibrational statistical sum for a diatornic molecule differs very little from the static sum for the harmonic oscillator, i.e.,

1 ho 0 z --~-~ E==

-v - kr +~ (10) 1 - - e e - - 1

* The dissociation kinetics is most interesting in the oscillator energy range E << ~ d. For E -~ ~ d several

collisions are sufficient for the dissociation of diatomic molecules.

19

In the approximation (! 0) the paramete r 3/ is determined by the relationship (8) and distr ibution (2) has the form

Xn=:X~ T TK -- -~- ' (11)

where TKiS the t empera ture of the harmonic osci l la tor determined by Eq. (9). The absolute e r r o r in 3/ in this approximation has the order of magnitude of the anharmonicity constant Xe~which enters into the ex- p ress ion for the energy of the vibrational level of the molecule

8 n - - nti(o o - - n2x]~% -}- nSg]~% k . . . . (12)

It can be shown that, with an accuracy up to t e rms of a higher order with respec t to x e,

]k% ( 1 ~.~_)_{_ 2x, (2 E.~T) - 1 ) . - Y = - K - \ - ~ - (13)

The values of the anharmonicity constant x e are usually less than 0.01 (for example, for the molecules 02, N2, CO, $2, C12, Br2, and 12 the values of 2x e are equal to 0.015, 0.013, 0.008, 0.008, 0.014, 0.0066, and 0.0056, respectively). Thus the correction to 3/ for the first approximas (8) is very small. The popula- tion of the upper vibrational levels changes, upon taking into account the correc t ions for y , much more strongly. According to (13) and (2) we have

_Irdk~176 1,T T k'l ) - ~ - e n ( E )] xo -_= x0 exp [ - ~ - + 2x,n 2e- ~ - 1 (14)

Never theless , the cor rec t ion factor

does not differ f rom unity in its order of magnitude even for values of n equal to severa l tens. [In the d is- sociation p roces s E _< ECr).]

A still smal le r cor rec t ion to 3/ and x n [which d isappears completely for E = E(T)] is obtained if in the f i r s t approximation we assume

o0(, 't Y -- T Ta r '

instead of (8), where T a is the t empera ture of the anharmonic osci l la tor which has an energy E for a Bol tz- mann distribution. In this case in the following approximation

T a k i n g (9) into account, formula (14) can be written in the form:

. = x0 (1 + E--~-T~ )" e n - ; ~-~-~-g~ n "e P , - - -~- q- 2xen (2 E-~T) -- 1)] ,

(16)

, ( o ) - 1 0 ~% X o = z = + 1 - E ' 0 . . . .

k

Let us also de termine the ratio of the quasi-equil ibrium population to its equilibrium value x n {T) which will be needed la ter :

Sn 1 _ e---~- n

xn (r) = x0 (T) e kr ~ ~ e ~r.

It follows from (14) and (17) that

~ ) = ~ exp r I 2E

(17)

(18)

20

an accuracy of the fac tor exp r[2xen (E(T)2E 1 /_! ~]which is c lose to unity, express ion (18)formal ly coin- With

cides with the express ion for a harmonic osci l lator . However, it is significant that the value exp (n~c00)/kT and the population Xn, which is proport ional to it at vibration energies close to the dissociat ion threshold

d for the anharmonic osci l la tor (molecules), are much grea te r than for the harmonic osci l lator .

B o u n d a r y f o r t h e Q u a s i - e q u i l i b r i u m D i s t r i b u t i o n

Distribution (14) is valid for conditions of a fast exchange of vibrational quanta with respec t to the p r o c e s s e s of vibrational excitation (2o). This condition is usually fulfilled v e r y well for vibrational levels with small numbers n, which make the main contribution to the vibrational energy of the molecule (excluding the case for collisions with light par t ic les , for example, with hydrogen molecules) . If the condition for the rapid exchange of the vibrational quanta is satisfied for small vibrational amplitudes, then for the harmonic osc i l la tor it is also valid for any amplitudes. However, for real molecules, because of the anharmonici ty of the vibrations, some l imiting value exists for the number of the vibrational level n = n* above which the condition for rapid exchange is not fulfilled and, consequently, the population of the levels does not sat isfy distr ibution (14) in the p rocess of vibrational relaxation.

To determine the boundaries for n* let us compare the probabil i t ies for p rocesses (15 and (2o). The probabil i ty for the adiabatic t ransi t ion (2~ f rom the n + 1 level to the n level, averaged over the thermal movement rates, is exponentially dependent on the value of the quantum energy ~/c0 n = Sn+l - - e n and is ex- p res sed by the Schwar tz-Slavski i -Herzfe ld formula [7]

8 (n 4- 1) V2A 2" :/2e-aX. Pn+l ,n = Z--o Ln '

2 2 2 )2/3 7 n COnref f ~1 V 2 - h~, 2 X. = 2kT 2 ' 2mcooreff

A = 2~;~l(~ . ~2 _ 1 rn 2 @ m~ z o ~ 3. h ' 2 (m I § m2)~ '

(19)

Here m and p 1 a re the reduced m a s s e s of the osc i l la tors and the colliding part icle , respectively, m 1 and m 2 a re the m a s s e s of the atoms of the molecule (oscillator), and ref f -~ 2 �9 10 -9 cm is the effective radius for the action of the exchange forces.

According to (19), the rate for the p r o c e s s e s (2~ increases ra the r rapidly as the number of the v ibra- tional level n increases , i.e., with a dec rea se in the vibrational f requency Wn. The rate of exchange of the vibrational quanta, conversely, dec reases as n increases . This is re lated in the same way with the anhar-

j, j+l monicity of the vibrations. The probabil i ty Qn+l ,n for p rocesses (1~ is re la t ive ly la rge if the conditions

for coll is ion are close to resonance conditions, i.e., the number of the vibrational levels for the colliding molecules are not too different. However, the number of molecules which a re in the vibrational levels, c lose to a given level n, dec reases exponentially with an increase in n according to the distr ibution (14).

The probabil i ty for the transi t ion of molecule A f rom the n + 1 level to the n level in p roces se s (1~, averaged over the vibrational s tates of the par t ic le B, is de termined by the sum

EQi,i+, x - n+',n i ~ Qn+I,n �9 ]

The distribution xj is given by formula (14). The value of Qn+l, n dec reases rapidly with an increase in n.

Because of the rapid and direct ional ly opposite change in the probabil i t ies Pn+l ,n and Qn+l,n the rat io

q~ ~ (N A + N M) P,+z,,I(NMO-,+I., ) (20)

changes in a narrow range of numbers An f rom r >> 1 to r << 1; N A and N M are the number of atoms and molecules in 1 cm ~ respect ively . (For s implic i ty it is assumed that the effect iveness of the atoms and mo- lecules as par t ic les B in (2~ p roces se s is the same.)

In cases where ~ << 1 during vibrational relaxation and dissociation the quasi-equi l ibr ium distribution (14) is real ized. For r >> 1 the Boltzmann distr ibution is maintained in the vibrational relaxation p roces se s

21

with a tempera ture for p rogress ive motion of T. If dissociat ion takes place, then this distribution, as was a l ready noted, is des t royed on the upper vibrational levels.

F o r the levels with n < n*, formulas (14) and (18) are valid. However, for n > n* the rat io Xn/Xn{T) is a lready independent of n and is equal (with an accuracy up to the destruct ion of the Boltzmann distr ibu- tion in the energy band (~d - kT) + s d during dissociation)

x x n. Xn ( T ) - - x . (T) , n > n*. (21)

This expression, along with (18), also determines the rat io of the dissociat ion rate constants k(Tk) / k(T), which are proport ional to the populations of the upper vibrational levels. Let us give it in the explicit form

k (T) z (7') exp k ( T ) z (T~ ) T TK " (21a)

The true value of the dissociat ion rate constant k differs f rom k(TK) by the factor c:

k---cle(TK), l < c ~ 0 - 1 - (22)

The factor c, which depends only slightly on temperature , takes into account the destruction of the Boltzmann distribution on the upper vibrational levels during dissociation. The problem of calculating it is analogous with the case of the dissociat ion of molecules in a monatomic gas and will not be considered here.

In order to solve Eq. (20) the probabil i ty must be known of the t ransfer of a vibrational quantum j, j+l

Qn+l, n" This probabil i ty has a ra ther simple analytical expression within the l imits of large and small

differences in the frequencies of the vibrational quanta which correspond to the adiabatic and nonadiabatic (almost resonance) coll isions.

In the case of a small f requency difference w n - 0Jj, the probabili ty for the t r ans fe r of a vibrational quantum in the exothermal direct ion for the reaction has the form

Qi,i+: 8[x/eTr2eff V4 n+l,n /~2 (n + 1) (] -~- 1) sech ~ (Am/),

. ~ t 1

A : 2reff 3kT ' toni : o ) - - r i.

(23)

The coefficient in front of sech2(Awnj) in (23) expresses the probabil i ty of exchange for total resonance [7]. The difference f rom resonance is taken into account by the factor sech2(A~n]) [8]. Formula (23) is obtained by the approximate averaging of the probabili t ies over the velocities of the particles, which, as can be shown, is valid for the condition Awnj < 1.5. F r o m the following it will be apparent that the solution to Eq. (20) is found in the range of Wn] in which formula (23) is still applicable.

After substituting (19) and (23) in (20), assuming # 1 = 2m, we come to the following equation for de te r - mining r :

zokT~ ~ ( NA "~ "n ' Zo~3"

i (24)

Considering that the region of small values of j makes the main contribution to the sum (24) it is pos- sible to assume in (24) that

exp ( - lio)o!/kT~) xl = z (T,,) , sech ~ (A(%l) = 4 exp (-- 2A%,i),

o~ i ----- o ~ - - 2ix o) o.

22

TABLE I. Values of an*

0,72 0,74 0,7~6 0,79

0.73 0,76 0,78 0,82

0,74 0,78 0,80 0,86

0,75 0,79 0,83 0,91

0,73 0.81 0,85 0,97

T A B L E 2

r N~ o2 $2 c12 Br2 J~ Parameter I

n* [ 22 n*hoo/ea 0,7

i8 0,7

34 0,7

19 0,5

42 0,8

47 0,8

This g ives

M o r e o v e r

1 ~t(] + 1)xisech2(Aton]),~ Z--~K) 4exp(~--2A(ono) 2 ( j 4-1)exp ( - - - i i

to;--~ too (1 4xeAkT ) /i ] "

i~toOkT )'

( j~too kT~ (i + ~) ~xp ~ - - V r /z (7"~) ~ z (r~) + 1 ~ - ~ o 1. ]

As a resu l t , in p l ace of Eq. (24) we ge t

~to; \ -~-~ -k 1 exp (-- 2Atono ) = 1 + ~ xn exp (-- 3~). (25)

Since the values 2A~ono and A 2 a re e x p r e s s e d in t e r m s of • o-- • (C~ in the f o r m

4 / A 2 = 16 6 ( leT ~a2. 2Atono=~g -}x30'2(1-a~); ~ r x 0 \ ~too J ~' (26) 2 2 2 )1/3

to n g tooreff ~tl too 2/~T

we find, ignor ing the va lues (4xeAkTK)/~i , Kw0/kT , and (T -TK) /T , in the p r e - exponen t i a l f ac to r in (25), which a r e sma l l in c o m p a r i s o n with uni ty,

exp [ 3~(o 2n/3 4 2~TX~/2(l__an)l = 4 ( I + N A ) 7/a.~/2 r~ n2z0k~ ~ a ~0 �9 (27)

The t r a n s c e n d e n t a l equation (27) d e t e r m i n e s the dependence of ~n* on X 0. This dependence fo r a g iven value of k has the v e r y s a m e f o r m for any gas . (In the c a s e of hom. onuc lea r m o l e c u l e s X = 0.5.) The c o n c r e t e p a r a m e t e r s of the m o l e c u l e and the t e m p e r a t u r e en te r into Eq. (26) f o r X 0. The solut ion to Eq. (27) f o r d i f f e ren t r a t i o s of the n u m b e r of a toms and m o l e c u l e s NA/N M is given in Tab le 1.

An impor t an t f e a t u r e of the so lu t ion is the fac t that the value of ~ n * , and consequen t ly the l imi t ing n u m b e r for the v ib ra t iona l leve l n* , depend v e r y weakly on ~ and even m o r e weatdsr on the t e m p e r a t u r e , changing by a f ac to r of a p p r o x i m a t e l y 1-3 in the 3 < )/0 < 7 range . We note that a l m o s t all of the range of t e m p e r a t u r e changes in which the in t e rac t ion of the p r o c e s s e s of v ib ra t iona l r e l axa t ion and d i s soc ia t ion a r e of i n t e re s t is included within the l im i t s of the inequal i ty 3 < • 0 < 7.

23

F r o m the data given in Table 1 we see that a d e c r e a s e in the number of molecu les in the d issoc ia t ion p r o c e s s by two to th ree t imes has l i t t le effect on the posi t ion of the boundar ies for the quas i -equ i l ib r ium dis t r ibut ion. The s l ight change in the boundary posi t ions for a significant change in N A / N M also indicates d i rec t ly that the solution to Eq. (27) is not v e r y sens i t ive to changes in the p re -exponent ia l fac tor in Eqs.

(19) and (23) for the probabi l i t i es Pn+l, n and O j ' j+l �9 ~n+l, n"

The numbers of the v ibra t ional levels n*, which cor respond to the values of COn*/a: 0 that were found, a r e detelTained b y the fo rm u l a

n * = 1 (1 _ a)n" ) 2x~ o) o '

which follows f rom the expansion of (12) for the conditions 2Xe >> 3Yen* (this inequali ty is fulfilled for w n , / w 0 > 0.5).

In con t ras t to the ra t io COn,/W o, the numbers n* depend on the concre te shape of the molecule for a given value of X0. The values of n* which co r respond to Wn*/W 0 = 0.73 a re given in Table 2 for some m o - lecules . The ra t io of the values of n* hw 0 which enter into the express ion for the d issocia t ion ra t e constant (21) to the d issocia t ion ene rgy of the molecule ed a re also shown there .*

We see f rom Table 2 that the value of n~liw0 is 20-50% l e s s than the d issocia t ion energy. It is noted that the energy of the boundary level is approx ima te ly 20% below the energy n*l~w0.r

P o p u l a t i o n o f t h e V i b r a t i o n a l L e v e l s i n t h e D i s s o c i a t i o n P r o c e s s

a n d t h e R a t i o o f t h e R a t e C o n s t a n t s

In o rder to de t e rmine the quas i s ta t ionary population of the vibrat ional ene rgy l eve l s and the ra t io of the r a t e constants k (TK) /kT , i t is n e c e s s a r y to find the ene rgy E or the t e m p e r a t u r e T K of the m o l e c u l e ' s v ibrat ions in the quas i s t a t ionary s tage of the dissociat ion. The change in the ene rgy of the v ibra t ions and the concentra t ion of d ia tomic molecu les is desc r ibed by a s y s t e m of equations [4]

_ _ = dNM dE E (T) - - E.(Na~ + NA) ~,- (ed -- E) N~dt dt %

(2s) : _ ( + N A / dNM k (T~) (N A -~ N~) + k'N2A 1 ,

NMd~ \ NM ]

where E (T) is the equi l ibr ium vibra t ional ene rgy of the molecule , which depends on the t e m p e r a t u r e T, t is the t ime, k ' is the recombina t ion ra t e constant, T 0/(NM + NA) i s the v ibra t ional re laxat ion t ime (for s i m - p l ic i ty it is a s sumed that the probabi l i ty of d issocia t ion and the vibrat ional re laxat ion a r e not dependent on what par t ic le the coll ision takes p lace with - an a tom or a molecule) .

The method for the fur ther study of the s y s t e m of equations (28) is comple te ly analogous to that de- sc r ibed in [4]. It cons i s t s of the fact that the p r o c e s s of v ibra t ional re laxa t ion and dissociat ion, f a r f r o m the d issoc ia t ion equi l ibr ium (where the recombina t ion reac t ion need not be considered) in a gas which is suddenly heated to s o m e suff ic ient ly high t e m p e r a t u r e T, b r e a k s up at the outlet s tage into a quas i s t a t ionary r e g i m e during which the ene rgy of the v ibra t ions changes f rom i ts initial value to a value E de te rmined f rom Eq. (28) for the condition for a quas i s ta t ionary s tate

dE dt - - O, (29)

and a quas i s t a t ionary s tage of d issocia t ion during which E does not change. This s tage in a mo lecu la r gas is cha rac t e r i zed by the osc i l l a to r d is t r ibut ions (18) and (21). If in the final equi l ibr ium s ta te the gas is

* The data on the d issocia t ion energ ies and the anharmonic i ty constants x e given in [9] were used in set t ing up the table. t Fo r example , fo r the Morse osc i l la tor for Wn,/W 0 = 0.73 the values of n*~iw0/e d and e n , / e d a r e equal to 0.54 and 0.46, r espec t ive ly .

24

dissociated to any marked extent, then almost all of the molecules dissociate in the quasis ta t ionary stage of the p rocess . It is this stage of the dissociat ion that is studied in the following.*

After introducing the dimensionless var iables

TK 1 X == ed/kT, Y -- T

and simplifications which have only a slight effect on the final resu l t and which consis t of the fact that the nonexponential pa r t s of Eqs. {21a) and (28) are taken to be

z(T) _ T E~kTK, E(7") :kT , z(TK)-- T---('

the following equation is obtained for y(x) f rom sys tems (21a), (28), and (29), and a formula is obtained for the ra t io of the ra te constants k(TK/k(T) :

( l + y - - x ) %k (T) ex p [~xg Y - l + y " l + y '

(30) k(TK) z(T,,) exp .f~xy , ~_~h~on*

In the case of a harmonic oscil lator, fl = 1. In the preceding section it was shown that for diatomic mole- cules the value of fl is usually 0.8-0.5.

The bcundar ies of the region in which there is a marked deviation of k(TK)/k(T) f rom unity can be found f rom the t ranscendental equation (30) without solving it numerical ly. The boundary being sought is determined by the condition

xy ~ 1 . 1 + Y (31)

Moreover, it is not difficult to show that in the vicinity of the boundary the inequalities :

x)) 1 ) )y . (32)

are fulfilled. We find the following express ion for the boundary being considered f rom (30), (31), and (32)

%k (T)x ~ ~--- 1. (33)

Thus, the ra t io k(TK)/k{T) begins to differ significantly f rom unity not under conditions of equality for the t ime of vibrational and chemical relaxation

%k (T) ~--- 1,

but for the much more r igorous condition (33) and x >> 1. The appearance of the fac tor x 2 in cr i te r ion (33) is due to the fact that for each act of dissociat ion approximately x t imes more vibrational energy is lost than the mean vibrational energy E and that the smal l relat ive change in the vibration t empera tu re A T K / T resul t s in a dec rea se in the dissociat ion ra te constant by exp (XATK/T) t imes.

The resul ts of the numerical solution to Eq. (30) for fi = 1-0.6 are given in Fig. 1. Equation (30) was solved with the same data t for 70k(T ) as in [4]:

%k (T) = 10 exp (-- x + 2.9xt/3). (34)

We see f rom Fig. 1 that the ra t io of the rate constants k(TK)/k(T) becomes much less than unity as the gas t empera tu re is increased, beginning with g d / k T = 17-20. This resu l t is retained fo r all of the va r - iations that were studied for the pa r ame te r ft.

* The t rans i t ion f rom the quasi-equi l ibr ium distribution of the osc i l la tors to the equilibrium distribution is accomplished in the final stages of the dissociat ion p rocess when the ra tes of dissociat ion and recombina- t i o n b e c o m e values of the same order . It is noted that Eq. (29) is approximately valid at this stage too if recombinat ion is taken into account in it. t In [4] there is a mis sp r in t : instead of exp(x(y - 1) /y + 1) in Eq. (14) it should read exp ( - ~ / y + l ) . The co r rec ted Eq. (14) coincides with Eq. (30) in this work for fl = 1, x >> I>> y after substituting (34) in it.

25

k(r.) t,o. ~'7-ff IJ~q ~

e,8'

O , O ~ q4, a2

5 ~b

f.0.

0,4.

O,2-

Ed ed t~v io ~-- f 5 ~b is i o ~ - f b

Fig. 1

Y Z,\",~ = o,a

Some sl ight i nc r ea s e in the r a t i o k(TK}/k(r) and dec r ea se in the t e m p e r a t u r e TKfor the quas i -equi l i - b r i u m dis t r ibut ion (14) for a d e c r e a s e in the p a r a m e t e r / 3 is re la ted to the fact tliat the des t ruc t ion of the Bol tzmann dis tr ibut ion for the population of the upper v ibra t ional l eve l s was not taken into account in the calculat ions. In fact , however, any d e c r e a s e in the boundar ies fo r the quas i -equi l ib r ium n* (a d e c r e a s e in /3 ) r e t a r d s the v ibra t ional re laxa t ion in the ene rgy region e > e n*. This in turn r e su l t s in a g r e a t e r de- s t ruc t ion of the Bol tzmann dis t r ibut ion fo r the population of the upper levels during dissocia t ion (the coef f i - cient c in fo rmula (22) depends on n*, in which case dc/dn* > 0). Es t ima te s show that upon taking into account the des t ruc t ion of the Bol tzmann dis t r ibut ion the ra t io ck(TK)/k(T) d e c r e a s e s and the t e m p e r a t u r e T K i n c r e a s e s along with a d e c r e a s e in the p a r a m e t e r 13. The dependence ck(TK)/k(T) and TKOn/3, however, is ve ry slight and the t rue values of these quantit ies differ only s l ight ly f rom the r e su l t s shown in Fig. 1.

The fact that in t r ea t ing the exper imenta l data on the d issocia t ion ra t e constants in the fo rm

k ~ TPe -~ in the region of high t e m p e r a t u r e s 20 kT ~ ~d, l a rge negat ive values a re usual ly obtained for the exponent p, much g r ea t e r than in the case of the d issoc ia t ion of molecu les in a monatomic gas (see, for example, [10]), explains the obse rved t e m p e r a t u r e dependence of the ra t io of the d issocia t ion ra t e constant to its equi l ibr ium value.

1o E, 2. E. 3. A. 4. N. 5. A. 6. C. 7. 8. 9.

10.

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

E. Nikitin, Dokl. Akad. Nauk SSSR, 119, 526 (1958). Montroll and K. Schuller, Adv. Chem. Phys. , 1, 361 (1958). I. Osipov and E. V. Stupoehenko, Usp. Fiz. Nauk, 79, 81 (1963). M. Kuznetsov, Dokl. Akad. Nauk SSSR, 164, 1097 {1965). I. Osipov, Theor . i ]~ksper. Khim., 2_, 649 {1966). T reano r , J. Rich, and R. Rehm, J. Chem. Phys. 48, 1798 (1968).

R. Schwartz and K. Herzfeld, J. Chem. Phys. , 22, 767 (1954). D. Rapp and E. Golden, J. Chem. Phys. , 40, 573 (1964). Thermodynamic P r o p e r t i e s of Individual Substances [in Russian], edited b y V. P. Glushko, L. V. Gurvich, G. A. Khachkuruzov, I. V. Veits, and V. A. Medvedev, Izd. AN SSSR, Moscow (1962). C. A. Losev and O. P. Shatalov, Dokl. Akad. Nauk SSSR, 185, 293 {1969).

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