Role of multi-quantum transitions in kinetics of dissociation of diatomic molecules with an arbitrary rate of exchange of quanta

ROLE OF MULTI-QUANTUM TRANSITIONS IN KINETICS OF DISSOCIATION OF DIATOMIC

MOLECULES WITH AN ARBITRARY RATE OF EXCHANGE OF QUANTA

N. M. Kuznetsov and A. M. Samusenko UDC 541.124/13

Much work has been devoted recently to investigation of the thermal dissociation of diatomic molecules with the presence of vibrational exchange of quanta, this work including both theoretical analyses [1-5] and experimental studies [6-10]. In studying the dissociation of molecules in their own medium, the model most frequently used is =hat of a truncated harmonic oscillator. With such modeling and with fulfillment of the inequality* Q~'~ ,o >> P1~o, which for N2, 02, CO, and other molecules is valid over a wide range of temperature, the populations x n of the vibrational levels are represented in the form of a Boltzmann distribution

x. (T 0 = Xo (Tv) exp (-- ~./Tv)

(approximation of infinitely fast exchange of quanta), where Cn is the energy of the n-th le- vel of the oscillator; T v is the vibrational temperature. This gives a very substantial simp- lification in the calculation of the dissociation rate constant K, both in the case of preferential dissociation from the upper vibrational levels [ll] and with an accounting for mul- tiquantum vibrational and translational transitions into the continuum [5, 12, 13].

For a real molecular oscillator, however, there is a narrow region of values of n where the inequality O~:~_ I >> Pn,n-: changes to the reverse inequality [14]. Moreover, the process of irreversible thermal dissociation in a not too rarefied gas takes place at such high temperatures of translational motion that the difference between the rate constants of the VV and VT processes is not large enough that we can assume that the VV transitions are infinitely fast. Thermal dissociation with an arbitrary rate of quanta exchan@e was studied in [1-4]. The first three of these studies were based on the assumption of preferential dissocia=ion from the upper vibrational levels. It was shown in [2, 4] that with stepwise dissociation, VV exchange always leads to an increase in K; for a real oscillator, however, this increase is insignificant. Along with one-quantum vibrational transitions, multi-quantum transitions into the continuum are significant in the dissociation process [15, 16]. An accounting for such transitions in the dissociation of molecules in a medium with exchange of vibrational quanta was made by Ramakrishna and Babu [3].

No equation was given in [3] for the number of quanta ~. Therefore, in determining x n and along with this the value of K, these investigators had to solve a system consisting of d nonlinear algebraic equations. For the probabilities of multi-quantum transitions Pnd, they used an expression proposed by Kiefer and Hajduk [17, 18]

P~a = Pa-l,a exp [(e. - - D) (~/D q- l/T)], (1)

where D is the dissocia=ion energy; d is the number of levels of the oscillator; Pd-~,d and A are variable parameters that are determined by matching the calculated dissociation rate constants to the experimentally determined values. Equation (i) is a "matching" equation, and it does not give any physically based representation of what sort of elementary processes lead to dissociation in the lower vibrational levels.

In our work, we also investigated the influence of VV exchange on the kinetics of diatomic molecule dissociation with multi-quantum transitions into the continuum. New formulas are proposed for the dependence of the rate constants of the elementary transitions on the vibrational levels, these formulas being obtained by treatment of existing trajectory calcula-

�9 0#1 ~Q1,o and P~,o are the rate constants of vibration-vibration (W) and vibration-translation (VT) transitions, respectively; 0 and 1 are the sequence numbers of the vibrational levels.

Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Institute of Phxsics, Academy of Sciences of the Belorussian SSR, Minsk. Translated from Teoreticheskaya i Eksperimental'naya Khimlya, Vol. 18, No. 3, pp. 259-269, May-June, 1982. Original article submitted April 4, 1981.

0040-5760/82/1803-0223507.50 �9 1983 Plenum Publishing Corporation 223

tions. A system of two equations has been s determining the dissociation rate constant and the average number of vibrational quanta per molecule in the quasi-stationary stage of dissociation. The reduction of the system of d equations for the populations of the levels to two equations greatly simplifies the problem of calculating the dissociation rate constant and analysis of its dependence on the probabilities of the elementary transitions.

DERIVATION OF KINETIC EQUATIONS

In the general case, the expression for the dissociation rage constant has the form

d~

K = ~; ~ Pnaxn. a=O

(2)

where ~ is the gas-klnetic number of collisions as calculated for one particle. In the quasi- stationary process of dissociation, the total number of transitions through a section that are performed in the energy space between the levels n and n - 1 must remain constant, and it will be equal to K. Therefore,

,,-1 (3) K = In + v ~ P~exi, Io ---- K,

i----O

where I_ is the total number of transitions between the n -- 1 and n levels, referred to one molecul~ and to one partner with respect to a collision in unit volume in unit time. Accord- ing t o [2],

In = vP,.,-IRnAI t (X.-i -- X.),

m n ~ I , ,~n ,n - - l lFn ,n_ l ;

n

An ~-- e ~/r l-1 1 + m ~ (I + ~) .

(4a)

d--I : ~ nx . ;

.=0-- ( 4b )

X n ~ Aaxn ,

where w e is the frequency of the vibrational transition 0 § I; w is the relative concentration of molecules in the reacting mixture.

Using Eqs. (2) and (3) and the expression for In, from (4a) we obtain a system of algebraic equations for Xn,

d--I

X . - I - - X~ = an Z b~Xt' i n n

(5)

b~ - - P,d/A~, a~--- A.IP~R~, P. ~ Pn,.-1.

Solution of the system (5) gives X n = XoAn+~/Ax , n = O, ..., d -- 2; A d E i,

An..[_ 1 =

1 "t- an+lbn+l an+lbn+2 . . . aa+lbd.-2 a,~+lbd_l

-- 1 1 -Jr an+~bn+2 . . . an+~ba_2 an+2bd-I

0 - - 1 . . . an+abe-.2 an+3bd-1

0 0 . . . 1 .q- ad--2bd--2 ad--2bd--1

0 0 . . . - - 1 1 -if- aa- lba-1

(6)

with A n satisfying the recurrence relationships

An_1 ==- (1 ~- an_lb ._ 1 -~- an_l/a,~) An - - an- ,An+l/an.

224

From (6) we find the distribu=ion of populations

x~=-X~/A~ = xoA.+dA~At. (7)

Summing (7) with respect to n and using the condition of normalization of Xn~ we express xo in terms of the number of quanta = (both A n and A n depend on s):

From (2), form

/d-i ~-! (8) xo -- & ( Z A.+dA~ ) �9 \n=O /

(6), and (8), we obtain the expression for the dissociation rate constant in final

d--I )--i ~--~ K = ( ~ &+dA~ b.A.+~. (9)

n=O n=O

Entering impllcitly into (9) is the unknown number of vibrational quanta ~. In order to obtain a second relationship linking ~ and K, we use the cond• =ha~ =he process is quasi- stationary. Equating the total rate of supply of vibrational quanta to the oscillator and the rate of their withdrawal on account of dissociation, we find [5]*

d--1 d--1

n = l n=O ( lo )

Using the relationship of the detailed equilibrium, we transform the left-hand side of the equality (i0),

d--I d--1

(Pn---l,nXn--I - - Pmn--lXn) = Z Pn'n--le--en/T (een--ffTXn--I - - e%llXn)" n ~ l n ~ l

If, in accordance with the definition (4b), we replace the relative populations x n by Xn, sum in the right-hand side of the last equality assumes the form

the

d--1

r t~ l

where ~ is the equilibrium value of ~. After replacing the difference Xn_~ - X n in (11) by its expression from (5) and using the identity

X._: __ Xn_le--(en--en_,),'r ~Z + O~ (1 -+- a) m.w A~ aR~ '

which follows from ~he definitions of X n and An, we obtain for (ii), after relatively simple transformations, the following expression:

d--1 d--I d-- I E ~ ~ - - ~ ,%_le-t~n-,~,,_9,'v ~Qn,n-I _z. biX i, (12)

n = l n = l i ~ n

rn calculating the rapidly converging first sum in (12), we can set [2]

%1 o,, (13)

Substituting (12) with a!lowanae for (13) into (i0) and using (6) p as well as the identity d--I d-- ] d--1

*The balance equation does not include two-quantum, three-quantum (etc.) transitions. The main contribution to the sum of the left-hand side of (10) is made by a few of the first terms, for which, at almost all temperatures of practical interest Pn,n+: >> Pn,n+s >>...

225

we finally obtain d--1 .... i d--i d.-1

= = Qo:, ~

n=-O n = l t=n

(14)

Equations (7), (9), and (14) completely determine the populations x n and the dissociation rate cons,an= due to one-quan=um vibration-=ranslation and vibration-vibration transitions~ and also multi-quantum transitions into the continuum. If there is no exchange of quanta, there is no need for joint solu=ions of Eqs. (9) and (14). It is evident from (4b)

^O,1 that with ~i,~, Rn and A n degenerate to unity and exp(en/T) ,respec=ively. In this case, X n and consequently K are independent of the number of quanta ~ and are determined from (6) and (9) with ra n = 0.

For the model of stepwise dissociation, all b n with the exception of bd_ ~ vanish, i,e.,

b~ = 6n,d--lbd--b (15)

where 6n,d_ ~ is the Kronecker del~a. Substituting (15) into (9) and (14), we can obtain by means of elementary transformations an expression for K and ~ that accounts for only one- quantum vibrational transitions. The formulas thus obtained for K and a coincide fully with the results of [2],

PROBABILITIES OF ELEMENTARY TRANSITIONS

In investigating the system (9) and (14), the greater difficulty is related to the short- age or inaccuracy of information on the cross sections of the elementary transitions. This refers equally to the VT and VV processes for the upper vibrational levels and to multi- quantum transitions into the continuum, The SSH theory and its subsequent variants (see for example [19]) with large values of n lead to very much overstated values of Pn,n-~ and under- stated values of Q~ An attempt was made recently [20] at a semiclassical calculation of Pn,n-~ and Q~ thenitrogen molecule that would be more nearly correct khan the

calculation within the framework of =he SSH theory. Here, the trajectory calculation method was used with a classical description of translational and rotational motion and a quantum description of the vibrations. The intermolecular potential included both the short-range repulsive forces and the long-range forces of van der Weals attraction and quadrupole repul- sion. Averaging of the cross sections of the VT and VV processes with respect to translational and rotational motions was performed in the respective intervals of 200-8000~ and 200- 2000~ The times of vibrational relaxation for the N= molecules with 8000~ ~T ~ 300~ found by the trajectory calculation method, are in good agreement with the experimental data. This indicates adequa=e reliability of the procedure proposed in [20] for the calculation of

toO| i Pn,n-t and ~n,n-~. The shortcoming of the method is the use of only an operator A(x) that is linear with respect to the vibrational coordinate x in matrix elements of the type ~n_11A(x) l$n> , which is justified only for small values of n.

The workup of the numerical results of [20] can be represented by the following inter- polation expressions for the probabilities of the VT and VV processes:

Pn,n_l = nP !,oexP [ ~ (nexp (-- l ,44T /~oe)-- l ) ] (8000 K ~ T ~ 300 K), (16)

QO,ln,n_l = nQ~;~ exp [--(D/T) ~ (1 - - con_l/O~e)3/21 (2000 K. ~ T ~ 300 K),

% - 1 -~ (% - - ~,~-1)//~ = % - - ~xe%n + .... ( 1 6 a )

where x e is the anharmonicity cons=ant. Following the recommendation of [21], the number of terms in the right-hand side of (16a) was determined on the basis of data from spectroscopic measurements, and also from the two additional conditions

e ,~ = D, ds r/dn ln=d = O.

In application to the N2 molecule, these conditions make it possible to supplement the known experimental values of the anharmonicity constants xe, Ye, and z e with two more values: ~eqe = 3.867o10-" om -I and wer e = 2.597"i0 -7 cm -~,

For all of the temperatures examined by Billing and Fisher in [20]~ the greatest error of the approximation (16) is less than 50%. We should emphasize, however, that in [20] the calculations were performed only for n < 20. Moreover, the probabilities of W exchange were

226

calculated for comparatively low temperatures (T ~ 2000~ With n > 20 and T > 2000~ (for the %~T transitions), these formulas should be regarded as extrapolations. A comparison of the calculations based on Eq. (16) with the results of the SSH theory shows that with T > 3000~ as n is increased, the probabilities Pn,n-~ as determined by Eq. (16) increases far more slowly, and with n = d, they amount to 0.2-0.5. Such values of Pn,n-1 appear more plausible than the values obtained in the SSH theory, which are one or two orders of magnitude greater. It can also be seen from (16) that the probabilities of VV exchange begin to drop off sharply as significantly greater values of n than would follow from the SSH theory.* This leads to an increasing role of exchange of quanta in the kinetic processes. It is particularly important to take into account the indicated behavior of the probabilities of VV exchange in describing kinetic processes in chemical l~Bsers. Let us note that for the CO molecule diluted in an argon mediumj experimental data are available on Pn,n-: over a very wide range of values of n (i ~ n < 40) [22]. According to these data, the dependence on n has the form

Pn . . . . 1 = nPI,o exp (6n), (17)

where ~ is a certain constant. Such a dependence is in good agreement with (16). Considering the closeness of the molecular parameters (the constant in the Morse potential, the frequency of vibrations, and the molecular mass) of CO and Na, and also considering that the main contribution to Pn,n-~ comes from short-range forces, the dependence (17) can be considered as indirect experimental confirmation of the first of the equations (16).~ The SSH theory in the case of a Morse oscillator also leads to an increase, linear with respect to n, of the exponent in the expression for Pn,n-~. However, the values of the probabilities of one-quantum transitions in this case, as noted above, are i-2 orders of magnitude greater than unity with n = d.

On the whole, the values of Pn,n-* that are determined by Eq. (16) in application to such molecules as N2, O~, and CO are more precise and more suitable for calculating the dissociation rate constant. However, it should be kept in view that in those cases in which the collisions are essentially nonadiabatic in character (for the indicated molecules, this is a region of vibrational energies close to D), in the dissociation kinetics it is necessary to take into account, along with one-quantum transitions, two-quantum and three-quantum transitions, etc. The inclusion of such transitions in the scheme of the reaction process makes the description of dissociation much more complicated. In particular, in this case, the kinetic equations are far more complex than the equations derived in the preceding section of this article. Although we can in principle obtain equations describing the dissociation kinetics with the inclusion of bound--bound transitions in the process scheme, this sort of approach is not feasible or expedient at the present time, owing to the lack of any data on the sections of multi-quantum bound--bound transitions. An effective accounting for such transitions can be brought about by a formal increase in the sections of one-quantum and multi-quantum transitions into the continuum.

The numerical values of [20] for Q~ are well reproduced by the formula

Q~:~ = 6.+*O-~T/D. (18)

With a steric factor of 3, Eq. (18) agrees with Eq. (17.87) in [24], but gives values only one- sixth of those recommended by Rapp in [25]. For the probability P~,o, we used the expression [ n ]

PLo = exp [-- 2,9 ( D / T ) I / 3 ] / [ I - - exp (-- B~/~]. (19)

The relationship (19) was obtained by a workup of experimental data on the relaxation times of N~, 0=, and CO.

Very few data are available on the probabilities of multi-quantum transitions into the continuum. Up to quite recently, multl-quantum transitions were considered as insignificant in the kinetics of dissociation (with the exception of transitions in the band D -- T ~ ~n < D)

eFor N2, the SSH theory leads to the relationship Qn,n-~ ~ exp [--I.4(D/T)~/2(I -- ~n-*/~e)] [4, 19]. +As regards ~n,n-~, here, for CO at temperatures below the characteristic temperature, the main contribution is that from long-range dipole--dipole interaction [23]. For this reason, the rate of ~V exchange at these temperatures is considerably greater than the corresponding rate of exchange of homonuclear molecules (for Q~, by approximately 2-3 orders of magnitude).

no; 1 With increasing T, the difference in values of ~n,n-~ between polar and homonuclear molecules decreases, owing to the increasingly important role of short-range forces.

227

TABLE i.

0 3,76 0,1 3,82 0,5 3,96 I ,0 4,09

Veff/v 1,5

Values of P r o d u c t ~'xe D/T K

25 20

1,95 2,08 2,37 2,61

1,5

15

1,27 1,40 1,65 1,86

1,4

12,5

0,96 I ,05 1,22 1,34-

1,3

0,66 0,75 0,77 0,81

1,1

7,5

0,44 0,47 0,49 0,48

0,9

2,75 2,84 3,06 3,28

1,5

0,29 0,32 0,32 0,29

0,6

[24, 26]. Recently, however, studies have appeared in which it was shown that multi-quantum transitions into the continuum are an important llnk in the scheme of elementary transitions determining the dissociation kinetics [3, 5, 15-18, 27, 28]. This circumstance has stimulated the calculation of the sections of multi-quantum transitions by the quasi-classical trajectory method. These calculations are very laborious, and up to the present time, the only results that have been obtained are for the simplest systems: Ha + He, No, Ar [29-31]. In addition, classical trajectory calculations have been carried out on the recombination of oxygen [32], bromine [33], and iodine [34] in inert gases, through which it is possible to draw a number of qualitative conclusions relative to multi-quantum transitions.

Studies of H2 dissociation i~ inert gases and studies of the recombination of oxygen, bromine, and iodine in Ar and Xe have shown ~hat by far the greatest number of the molecules dissociate from the band • of total energy ~ of vibrations and rotations adjacent to the point e = D. Let us emphasize that the vibrational energy in this case may also be considerably less than D. These results of trajectory calculations indicate that centrifugal rupture of molecules (rotational dissociation) is an important mechanism of direct dissociation; this is also confirmed by direct calculations of the cross sections of inelastic collisions that were performed in [35]. In particular, it was shown in [35] that the inelastic rotational channel for the system Ha + Ar with dissociative collisions is far more effective than the vibrational channel,

On the basis of the foregoing discussion, in the calculation of K and u in the present work, we have given preference to the following expression for Pnd that was used previously [16] in a qualitative study of the role of transitions with a large change of rotational energy in the dissociation kinetics:

P.a = v e f f exp {-- [D - - e~ + 8 (e.)]/T}; 6 (e.) = B (1 - - e~lD) 3/2, (20) v

where 9ef f is the effective number of collisions; 6(r n) is the rotational barrier; the parameter B for nitrogen is equal to 0.4. An averaging of the data of [30] on Pnd(J) (where J is the rotational quantum number) with respect to J leads to the value ~eff/u = O.1. This cor- responds to an effective cross section of approximately 3"10 -~' cm a and is in satisfactory agreement with data on the rotational relaxation times of diatomic molecules and with non- adiabaticity of collisions with respect to rotations. Let us note that these same effective cross sections characterize the activation (vibrational and rotational) of more complex molecules [36].

RESULTS FROM SOLUTIONS OF SYSTEM OF EQUATIONS FOR K AND

Equations (9) and (14) were solved numerically in the temperature interval 5~D/T~30. The sections of the elementary transitions were determined on the basis of Eqs. (16) and (18)- (20). In accord with what was stated above regarding the need for an effective accounting for bound--bound transitions, the quantity ~eff/~ took on values such that the calculated dissociation rate constant coincided with the experimental data on the dissociation of N~ in Ar [37]. Another possible means for matching the calculated K with experimental would be variation of the sections of the one--quantum transitions. However, in view of the specific fea- tures of the problem we are investigating, this possibility canno~ be realized; variation of Pn gives significant changes in the value of mn, the basic characteristic of the interaction of VT and VV processes in the kinetics of dissociation of a diatomio molecule. In order to evaluate the effectiveness of multi-quantum transitions, we carried out calculations of the rate constant: a) on the basis of the stepwise dissociation model in accordance with the appropriate formulas [2] with the transition probabilities (16), (18), and (19); b) on the basis of the same model, but including multi-quantum transitions into the continuum with the

value ~eff/~ = 0.i~

228

TABLE 2. Values of Ratio a / ~

30 25 20 ~5 ~2.s m 7.5 5

0 0,1 0,5 t,0

0,99 0,99 0,99 0,99

0,97 0,97 0,96 0,95

0,89 0,88 0,87 0,86

O, 74 0,73 0,70 0,69

0,51 0,50 0,48 0,47

In Tables 1 and 2 we present the results from calculations of K and a. For convenience, the dissociation constant and the number of quanta are measured in respective units of 9e-D/T and ~. The bottom line in Table 1 contains the values of Veff/~ that were used in the calculations of K and ~.* The same as in the case of stepwise dissociation [4], the influence of VV exchange on the rate of thermal dissociation of a real diatomic model is quite small. The ratio Kw=i/Kw= o is maximal with D/T = 15, and amounts to 1.46. The conclusion as to the insignificant effect of W exchange on the dissociation kinetics of Oa and HCI with various concentrations of inert diluent (Ar) also follows from the calculations of [3].

The large values of the parameter 9eff/9 (Table i), approximately an order of magnitude greater than the true values, point out the important role of bound--bound transitions in the kinetics of dissociation of dlatomic molecules. Among such processes are excitations of bound electron states having identical dissociation limits with a ground term, one-quantum vibrational transitions, and multi-quantum vibrational-rotatlonal transitions. Estimates made in [38] show that with D/T < 20, the influence of the electronic factor can be neglected. The variation of Pn that was carried out in the present work indicates a weak dependence of the dissociation rate constant on the probability of one-quantum transitions: a change in Pn by a factor of i0 is accompanied by a change in K by a factor of only 1.7. Thus, the need for using large values of ~eff/v, in calculating K is apparently related to the role played in the kinetics of thermal dissociation by multl-quantum bound--bound transitions. These transitions accelerate the supply of quanta to the upper vibrational levels and tend to re- duce quite substantially the degree of nonequilibrium of the populations of strongly excited vibrational levels. The role of such transitions is particularly important at low temperatures. This is related to the stronger decrease in the rate of outflow of quanta into the continuum in comparison with the decrease in the rate of supply of quanta to the upper vibrational levels.

An alternative possibility in accounting for bound--bound transitions is a decrease in the parameter k in Eq. (i), or a decrease in B in Eq. (20) with a value of 9eff/9 that is independent of T. Such an approach leads to small values of k (approximately 2-4) [3, 18, 27], corresponding to comparable rates of the transitions into the continuum from the upper and lower vibrational levels. Meanwhile, the trajectory calculations show that the sections of the transitions from the lower vibrational levels into the continuum are several orders of magnitude smaller than those from the upper levels [29-31]. Thus, of the two possibilities for effective accounting for bound--bound transitions, variation of the parameter ~eff/~ is better justified than variations of ~.

It can be seen from Table 2 that at high temperatures, the number of quanta ~ deviates significantly from its equilibrium value. However, in this case as well, the dependence of a on the rate of quanta exchange is relatively minor; at all temperatures D/T ~12.5, we observe that as w increases, there is only a slight decrease in a.

A quantity that is far more sensitive to changes in the rate of W exchange is the population of the upper vibrational levels. Thus, with D/T 412.5, the population Xd-~ increases by a factor of 10' when w changes from 0 to i. With D/T ~25, the corresponding difference is quite small, since in this case, a severe disruption of the vibrational equilibrium takes place only with e n = D~ where the role of VV exchange is small. Let us note also that the degree of nonequilibrium of the upper levels, with an accounting for multi-quantum transitions into the continuum, is many orders of magnitude greater than the degree of nonequilibrium due to stepwise dissociation [4].

In Fig. 1 we show a plot of log Xn/Xo exp(en/T) as a function of n/d with w = 1. At high temperatures (D/T = 7.5), we observe a significant emptying of the lower vibrational levels.

*The collision cross section was assumed to be 3-10 -~s cm 2.

229

-12

Fig. i. Population of vibrational states, with following values of D/T: i) 30; 2) 15; 3) 7.5,

This differs substantially from the well-known fact of the extremely weak disruption of the equilibrium populations of the lower levels in the case of stepwise dissociation,

Upon comparing this particular modal with the stepwlse dissociation model, a number of other differences are revealed. As has been noted m W exchange in the one-quantum dissociation modal leads only to a monotonic increase in the dissociation rate constant with increasing w. In contrast, it can be seen from Table 1 that the monotonic dependence of K on w, when rotational dissociation is taken into account, does not occur at all temperatures. Still another distinctive feature is the considerable deviation of the number of quanta from the equilibrium value at high temperatures; the minimal value of ~/~ in the case of step-

(16). f e o i t t c u ng for multi-quantum transitions into the continuum leads to an increase in K by a factor of 40- 60. The factor 5-10, accounting for the contribution of rotations within the framework of stepwise excitation [26], bring this difference down to an order of magnitude.

Let us compare the values found for K and = with the results from calculations for a harmonic oscillator in the approximation of infinitely fast VV exchange and rotational dissociation [5]:

D/T 30 25 20 15 12.5 i0 7.5 5

Tv/T [51 1 i 1 0.99 0.97 0,90 0.74 0,53 Tv/T 1 1 1 0.99 0.96 0.88 0.72 0,51

The values of Tv/T listed in the first line were obtained by solving the equation determining the vibrational temperature in a system of harmonic oscillators with infinitely fast VV exchange, with an accounting for rotational dissociation [5], The value of 9eff/v in the expression for Pnd was selected so that the values of K calculated in the approximation of infinitely fast VV exchange would coincide with the values of K in Table i. It was found that the value Veff/~ = 0.2 satisfies this condition over the entire range of temperatures that was examined. The values of Tv/T shown in the second line are for a real oscillator; these were determined from the data of Table 2 with w = 1 on the basis of the formula ~o/T v = in [(~+i)/~].

An analogous comparison of values of Tv was carried out for values of K reduced by fac- tors of 2, 3, and 5 in comparison with the experimental data, with D/T = 5, 7.5, and i0 (lower temperatures are not of interest; for such temperatures, 1 -- = < 10"a). A comparison of the results from these calculations shows that the values in lines i and 2 will change from variant to variant, but lines i and 2 are always very nearly identical (the corresponding values of Tv/T do not differ by more than 4%). Such a weak dependence of ~ on the rate of exchange of vibrational quanta makes it possible, in evaluating u, to use the simple dissociation model with infinitely fast exchange of vibrational quanta [5] with an appropriate selection of the

ratio ~eff/v.

The nonequilibrium nature of the distribution of vibrational energy in an irreversible dissociation process at high temperatures has been proven in a number of experimental studies [8-10]. These experiments were not performed on a single-component gas, but rather with dif- ferent dilutions with an inert gas (w was varied from 0.i to 1.0). There, the same as in the present work, the values of T v were only very slightly dependent on the concentration of the molecules in the medium.

230

CONCLUSIONS

The exchange of quanta has only a very slight influence on the kinetics of thermal dissociation of diatomic molecules. The dissociation rate constant generally shows a slight increase upon the "inclusion" of VV exchange. At very high temperatures (D/T~-~7.5), VV exchange can lead to a weak, nonmonotonlc dependence of K on w. This nonmonotonicity is related to the complex character of the influence of VV exchange on the contribution of rotational dissociation to the overall kinetics of the reaction. With increasing temperature, there is an appreciable increase in the disruption of the equilibrium populations of the lower vibrational levels. As a result, at high temperatures, the average number of quanta in the quasi-stationary stage of dissociation is considerably less than the equilibrium number. For a real oscillator, the concept of infinitely fast exchange of quanta is not valid. From a practical point of view, however, the extremely simple method for calculating T v and K, which is valid in the case of infinitely fast exchange of quanta, can be used to approximate the real values of u with a selection of the appropriate value of the parameter ~eff/9. An accounting for multi-quantum transitions into the continuum in the kinetics of dissociation of diatomic molecules, together with an effective accounting for multi-quantum bound--bound transitions, leads to the agreement of the calculations of dissociation rate constant, the degree of breakdown of the equilibrium distribution of vibrational energy, and the induction time (time to come into the quasi-stationary regime) [17, 18] with the experimental data. This gives us grounds for the hypothesis that multi-quantum transitions into the continuum, together with bound-- bound vibrational transitions, are the basic mechanism in the kinetics of thermal dissociation of diatomic molecules.

LITERATURE CITED

I. J. H. Kiefer, "Effect of VV transfer on the rate of diatomic dissociation," J. Chem. Phys., 57, No. 5, 1938-1956 (1972).

2. N.M. Kuznetsov, 'Influence of exchange of vibrational quanta on rate of dissociation of diatomic molecules," Teor. Eksp. Khlm., 10, No. 3, 318-326 (1974).

3. M. Ramakrishna and S. V. Babu, "Dissociation in dilute mixtures of shock heated Oa and HCI in Ar," Chem. Phys., 36~ No. 2, 259-270 (1979).

4. N. M. Kuznetsov and A. M. Samusenko, "Influence of exchange of vibrational quanta on rate of dissociation of diatomlc molecules," Dokl. Akad. Nauk SSSR, 246, No. 6, 1401-1404 (1979).

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Role of multi-quantum transitions in kinetics of dissociation of diatomic molecules with an arbitrary rate of exchange of quanta