160 M. Quack and J. Troe: Vibrational Relaxation of Diatomic Molecules etc Berichte der Bunsen-Ge,rlls( haft

X ) cannot he anything other than (pseudo) first order, with the fitted value of i related to rciigcnt concentration [X,] by i = i, + k [ X , ] . ( 5 )

As such, a plot of i. as ;I I'unction of [X,] must yield a straight linc of slope k and intercept i,. This is shown in Ref. [3] for the Mu + Br, reaction, yielding a himolecular rate constant k = (2.4 & 0.3) ' 10'' 1 rnol-' sec-I. Thc rate constants for the reaction or Mu with CI,, H R r , and HCI have been siiiiilarly determined and are given in Table 1 along with the corresponding H atom values. The reaction of M u with H B r is a factor of ten slower than the Mu + CI, reaction. We anticipated that thc rate constant for the Mu + HCI reaction would be of the same order but found it to be very much slower; indeed, to date we have been able to establish only an upper limit of k < 3 . lo7 1 mol- I sec- I - a factor or 100 slower than the HBr reaction!

Table 1 Rate constants') for Mu (and HJ with X, and H X

M u + Br, 24 i 3 2.2 + 1.5 h, 11 M u + CI, 5.2 + 0 . 4 1.2 * 0 . 2 4 4.3') Mu + H B r 0.75 * 0.15 - 0.3 5 - 2.5 Mu -I- H('I r: 0.003 0.003 6 - < 1

'J All at 2 2 C bJ Scc discuaaion in RcZ [3]. ') Our first mention 01' this ratio in Ref. 131 was too large, due to an

ear ly and apparently inuorrcct value for the H + CI, reaction given in Ref. [Sl.

Comparison with H Atom Data I t is hoped that the eventual comparison between a large

amount of M u and 1-1 atom data will provide a unique test of isotope effects as predicted by current theories of reaction dynamics. O q c a n write the rate expression for a bimolecular reaction in a qualitative manner as

k - ~o(l.)f'(rrl ,L')dl~ * O,,F (6)

the "hard sphere" limit of which i s just an energy independent cross section (u,,) multiplied by the r.m.s. velocity ij. The only point to be made here is that the expression for the rate con- stant will always involve a factor containing 6 and hence will exhibit a (l!p)'" dependence, where p i s the reduced mass of the colliding partners. Since for reactions of Mu or H with X, or HX we havc (p,,/pMu)'!2 - 311, we expect k,,/k,, to

exhibit this same isotope effect, at the same temperature and (assuming) the same activation energy; any marked departures from this ratio should be most interesting.

Of the present data (all at - 22") listed in Table 1, the reac- tions of Mu with X, are straightforward in that only one reaction is likely, namely, Mu + X, + MuX + X. However, for reactions with the HX gases, both the abstraction (Mu + HX -+ MuH + X) and exchange (Mu + HX -+ MuX + H) reactions are possible. For reactions at room temperature (where HX is predominantly in the u = 0 vibrational state) the abstraction reaction probably dominates but the exchange reaction is apparently more likely in the t' = 1 level [7].

Unfortunately the amount of gas-phase Mu atom data available to date is too sparse to allow any detailed compari- sons with the H atom data; moreover, the H atom numbers are generally more poorly determined. Nevertheless, the numbers given in Table 1 are already interesting and indicate a considerable variation in the ratio of k,,/k,, which clearly bears the need for more thought. In the near future at TRIUM F, we plan to measure the M u activation energies for these reactions as well as carry out a study of the Mu + F, reaction, which has been predicted theoretically [8] to be enhanced over !he corresponding H + F, reaction by a factor of six at room temperature.

References [I] V. W. Hughes, Ann. Rev. nucl. Sci. 16, 445 (1966); V. Hughes

et al., Physic. Rev. A 1, 595 (1970); R. D. Stambough et al., Physic. Rev. Letters 33, 568 (1974).

[2] J. H. Brewer et al., Muon Physics, V. Hughes and C. S. Wu, eds., Academic Press, 1976.

(31 D. G. Fleming et al., J. chem. Physics 64, 1281 (1976). [4] R. L. Wilkins, J . chem. Physics 63, 2963 (1Y75); H. G. Wagner,

U. Welzbacher, and R. Zellner, Ber. Bunsenges. physik. Chem. 2 8 (1976), private communication from R. Zellner.

[5] From B. A. Thrush, Prog. Rxn. Kinetics 3, 63 (1965). [6] J. E. Spencer and G. P. Glass, J. physic. Chem. 7Y, 2329 (1975). [7] J. Wolfrum, Invited paper, Konigstein Conference on Energy

[ X I J. N. L. Connor, W. Jakubetz, and J. Manz, preprint, submitted Transfer, Sept. 1976.

to Chem. Physics Letters, 1976.

E 3553

Vibrational Relaxation of Diatomic Molecules in Complex Forming Collisions with Reactive Atoms

M. Quack and J. Troe

Inatittit fur Physikalische Chemie der Universitlt Gottingen, Tammannstr. 6, D-3400 Giittingen, Germany

ErierUieubertragiirtg , ' Keaktionskinetik

Whereas vibrational relaxation of diatomic molecules in For 0 + O2 relaxation, it has been tried to explain this collisions with nonreactive atoms is quite slow [l], during behaviour with the aid of classical trajectory calculations [5] the last years several experiments have shown that collisions and also assuming electronically diabatic collisions [6]. with reactive atoms proceeding via bound intermediates are However, the calculated rates were too low by about an order extremely efficient with this respect [2 -43. of magnitude. On the other hand, the statistical adiabatic

M. Quack and J. Troe: Vibrational Relaxation of Diatomic Molecules etc. 161 Bd. 81, Nr. 2 1977

channel model [7, X] has explained quantitatively the results on the vibrational relaxation of 0, with 0-atoms and has correctly predicted the rate coefficients for the relaxation of NO with Cl and 0 atoms [ X I , as has been verified experimen- tally later [3] . So, we may be fairly confident that the basic physics in such strong coupling situations is correctly described by statistical scattering. The range of “statistical” situations has been estimated by the criterion [8]

A E + p ( E , J ) - ’ (1)

p ( E , J ) is the density of metastable states of the quasibound intermediate (e.g. O,, CINO, NO2 etc.), and A E is the energy spread in the center of mass translational energy of the collision partners. A E is of the order of kT in thermal systems. There are many situations for which condition (1) is fulfilled, some also of practical importance for the relaxation processes in the upper atmosphere and in flames.

In many cases the molecular parameters are known, which are necessary for the detailed calculations with the statistical adiabatic channel model. However, sometimes the com- putational effort required would appear to be rather large and one would like a rather simple statistical estimate. This is the subject of the present communication.

We have presented elsewhere [9] a simple method for obtaining the high pressure limiting rate constant k , , , , , for recombination reactions (e.g. AB + C + ABC). This total rate coefficient for complex formation from thermalized reactants is related to the statistical vibrational relaxation rate constants k,,,,, , by [ X I :

In the definition of k,,,. rotational and electronic states of the collision partners are assumed to be thermalized. f ( u ) is the Boltzmann population of vibrational state u. Assuming that the rate constant for complex formation k,(u) depends only slightly on u (this need not be the case [XI) , one has furthermore:

m

In order to obtain the rate constants k,,,, from k, , , , we have to compute the ratios ( k u , v ~ / k v , ~ , , ) . The accurate statistical expression is the ratio of two triple integrals (or sums) which are quite tedious to evaluate numerically (cf. Equation (12) of Ref. [XI) . As a simple approximation to this, one may suggest:

(4)

W(E, tl) is the total number of open channels at a thermally averaged total energy E, leading to the vibrational state u. In Equation (4) the detailed influence of energy, total angular momentum and of the rational state of the diatomic molecule has been completely neglected. I? may be assumed to be about EL, + 3kT/2 ( E , = vibrational energy in level u).

Let us consider now three limiting cases: (i)The channel maxima occur at large separations q of

the atom and diatomic molecule, with free rotation of the

diatomic molecule. The corresponding number of channels equals the number of rotational states of the diatomic molecule. In the limit that .fl? - E& B (the rotational constant of the diatomic molecule in energy units), this becomes approx- imately:

and therefore:

k,,,. E - E,. ---. kO+”,, E - Eon

(ii) The channel maxima occur at small q and the rotation of the diatomic molecule is strongly hindered. If we describe this motion by a classical harmonic bending oscillator (frequency vh), in order to get the number of channels, we have to convolute the density of states in this bending vibration with the number of states of a one dimensional rotor, with the large rotational constant A of the nearly symmetric top structure assumed for this case. We assume, here, that the average quantum number J of the total angular momentum in the collision is large and that therefore there is no restriction on the symmetric top quantum number K. We obtain then with E - E, 4 A, approximately:

W(E,U) = $ ( h v ) - ’ A - ” 2 ( E - E”)3’2 (7)

and :

(iii) If in case (ii) the bending vibration is strongly quantized and v,, 4 v (the stretching frequency of the diatomic molecule), we have to compute only the number of rotational states corresponding to the quantum number K of the assumed quasi symmetric top. We therefore obtain:

(9)

Considering these three cases as dynamical limiting cases, we would summarize the estimate to be:

with 112 I n I 312. Very slightly hindered rotation with n 1 1 is probably quite near to reality for most cases [9].

In Table 1 we compare estimates based on this assumption with experiments (in parentheses) and with calculations from the statistical adiabatic channel model [in brackets]. The relaxation rate coeflicients are three to four orders of magnitude higher than for collisions with inert species [1,8] and the general agreement between experimental and calculated values in Table 1 is within a factor of 2. It should be stressed that the calculated values are based on our simple calculation of the values of k,,, , (Ref. [S]). These have been obtained by the use of the maximum free energy criterion and an empirical a priori estimate of the only parameter y of the model. The present values are, therefore, by no means “fitted” to the

Berichte dcr Bunsen-Gerellschaf! 162 D. A. Micha: Role of Molecular Momentum Distributions in Impulsive Collisions

~- - -_ Table 1 this in mind, our estimate appears to give reasonable pre-

dictions of the very large rate coeflicients for the vibrational relaxation of diatomic molecules in complex forming col-

Estimated rate coefficients k/lO-i'crn's I

Collision ( T / K ) k,ec ,, (Ref. [9]) k , -o k z - o lisions with reactive atoms. k 2 - o kz.-L

Helnful discussions with Dr. K. Glanzer and financial sumor t H + OH (300)

o+o, (300)

(21W

0 + N O (300)

(2100)

H + N O (300)

F + N O (300)

CI + NO (300)

(21 00)

(2100)

(2100)

Br + NO (300)

22. (218 . ) ' )

0.63 (0.28) [ l l ] [0.25] h,

1 .o [0.57]

[2.01 11.8 18.3

3.4 3.4

7.9 (5.8) [13]

7.4 r6.21

~ 5 . 2 1 6.2

21. (27.)')

0.54 [0.70] 0.57 (0.71)")

[0.33]

1.8

1.3 (2.4)') ~ 1 . 7 1

r1.21 10.3 1 1 . 3

3.0 2.1

6.9 L5.41 4.6 (2.2)')

5.4 [ 3 4

I + NO (300) 4.0 (>1.7) [141 3.5 (2100) 4.6 2.8 (1.8)')

14.3

0.38

0.45 [0.33]

1.3

1 .o

7.3 8.7

2.1 1.6

4.9

3.5

C1.11

~1.31

L1.01

~3 .81

~2 .31 3.8

2.5 2.2

1.9

1.8

1.4 C1.461

1.86 [ 1.71) 1.46

[1.47]

1.86 1.46

1.86 1.46

1.86 [1.75] 1.46

C1.411

1.86

1.86 1.46

[1.83]

") From isotope exchange, Ref. [lo]. b, Values in [brackets] are from the statistical adiabatic channel model

" ) S e e Ref. [4]. also the sum k , , , + k , , , = 3 3 ~ 1 0 ~ " c m 3 s - ' has

d, From Ref. [2], assuming a two state model to be valid, approximately. ') From Ref. [3], assuming a two state model to be valid, for 0 + NO

[7,81.

been reported herein.

near 2700 K and for CI + NO near 1700 K.

experiments. Furthermore, one must not forget that the simplistic statistical considerations are not equivalent t o thc correcf siritistical culiuhutic chunnel model. Particularly, for large differences in (1' and v", the ratio k,,,,,./k,,,- from Equation (10) with n = 1 may be considerably in error. With

.. from the Deutsche Forschungsgemeinschaft are gratefully acknow- ledged.

References [ l ] J. P. Toennies, C'hem. SOC. Reviews 3, 407 (1974); J. E. Dova

and €1. Teitelbaum, Chem. Physics 6 , 43 (1974); Transfer and Storage of Energy, Vol. 2, Vibrational Energy, G. M. Burnelt and A. M. North,eds. Wiley, London 1969.

[2] J. H. Kiefer and K. N. Lutz. XIth Symposium on Combustion, The Combustion Institute, Pittsburgh 1967, p. 6 7 - 7 6 ; J. E. Breen, R. B. Quy. and G . P. Glass, J . chem. Physics 5Y, 556 (1973).

[3] K. Gllnzer and J . Troe. J. chem. Physics 63, 4352 (1975); 65, 4324 (1976).

[4] J . E. Spencer and G . P. Glass, Chern. Physics 15, 35 (1976); J. E. Spencer, J . Endo, and G . P. GI (International) on Combustion, Pittsburgh 1976/77 to be published.

[5] E. Is. Breig, J. chem. Physics 5 / , 4539 (1969). [h] E. E. Nikitin and S. Ya. Ilmanski, Faraday Disc. chem. Soc.

53, 7 (1972) and Lloklady Akad. Nauk SSSK l Y 6 , 145 (1971). [7] M. Quack and J . Troe, Ber. Bunsenges. physik. C'hem. i X , 240

(1974). [8] M . Quack and J . Troe, Ber. Bunsenges. physik. Chem 79, 170

( 19 75). 191 M . Quack and J . Troe, Ber. Bunsenges. physik. Chem., i n press.

[ lo] J . J. Margitan and P. Kaufman, Chem. Physics Letters 34, 485

[ I l l H. Hippler and J. Troe. Ber. Bunsenges. physik. Chcni. 75. 27

1121 J Troe, Ber. Bunsenges. physik. Chem. 73, 906 (1969). 1131 t l . Hippler and J . Troe, Int. J . Chem. Kin. 8, 501 (1976). [14] H. E. van den Bergh and J . Troe, J . chem. Physics 64, 736

E 3554

(1975).

(7971).

(1976).

Role of Molecular Momentum Distributions in Impulsive Collisions

David A. Micha *)

Alom- und Molekulurstruhlen I Energieiihertrugung I Isntopenqfekte I Quantenmechanik I Rwktionskinetik

1. Introduction

Impulsive molecular collisions of atoms or atomic ions with diatomics have frequently been interpreted in terms of simple kinematical models which make use of conservation of energy and momenta, and of simple pair interactions 11 -4). Impulsive collisions may also be treated within one of the quantal approaches to molecular dynamics 151, as a few-body problem.

Our aim here is to briefly show how a multiple-collision expansion of the coupled quantal equations for. three inter- acting atoms [16,7] can provide a foundation for the simple models, while further showing the important role played by the momentum distributions of reactant and product di-

atomics. Detailed applications havc been made to reactions in Ar+ + H,, K + I, and related systems, while energy transfer has been discussed in Art + D,, 0; + He, and Li' + H, 17-91. Other recent studies in the same spirit have dealt with reactions of halogen atoms 1101, and collisional excitation, particularly by Li+ [ll].

A very interesting application of momentum distributions is to the clarification of electron transfer in reactions. Depend- ing on whether transfer occurs or not, one must use the momentum distributions of reactant diatomic ions or of

*)Visiting under the "U.S. Senior Scientist" program of the Alexander v. Humboldt Foundation. Permanent address: Chemistry Department, University of Florida, U.S.A.