I.W.M. Smith- Reactive and Inelastic Collisions involving Molecules in Selected Vibrational States

I Reactive and Inelastic Collisions involving Molecules in Selected Vibrational States

~~

BY I . W. M. SMITH

1htdUCtl ‘on State Selected Kinetics and R d o n Dylramics.-For many years chemical kineticists have sought to observe and understand the processes that bring about macroscopic chemical and physical changes at the level of individual molecular events. Unfortunately, the detailed microscopic information that can be extracted from the results of conventional ‘bulb’ experiments is necessarily limited, since the parameters that characterize the intermolecular collisions, such as relative translational energy, impact parameter, orientation, etc., have, under these conditions, a full spread of values in accofdance with statistical laws. Over about the past 15 years therefore, increasing use has been made of experimental techniques which provide results whose connection with fundamental molecular collision dynamics is less obscured by the many ‘layers’ of averaging [see Section 2 below and Figure 1 in ref. l(a)] that play their part in determining the magnitude of the thermal rate constant for a chemical reaction, k(T), and its dependence on temperature. For example, molecular beam and ‘hot atom’ experiments can yield information about the excitatwn function, i.e. how the crosssection for reaction varies with collision energy, whilst i.r. chemilumine~cence,~~~ chemical l a ~ e r , ~ ~ ~ and molecular beam techniques allow the experimenter to investigate how the energy that is released in an exoergic chemical reaction is distributed among the degrees of freedom of the separating products. The experiments referred to in the second half of the previous sentence reveal

something about the specificity of energy disposal in elementary exoergic reactions. The other side of this coin is the selectivity of energy consumptwn; for example, whether a reaction with a high activation energy is promoted more effectively by providing the reactants with excess translational energy or by providing the same energy to an internal degree of freedom. A measure of these selective energy requirements may be obtained by comparing the results of experiments which yield

1 (a) J. P. Tocnnies, in ‘Physical Chemistry : An Advanced Treatise’, cd. H. Eyring, D. Henderson, and W. Jost, Academic Press, New York and London, 1974, vol. VIA, Chap. 5; (b) Faraday Discuss. Chem. SOC., 1973, SS, on ‘Molecular &am Scattering; (c) R. Grice, Adv. Ctrem. Phys., 1975,3U, 247.

s (a) I. W. M. Smith, Adv. Chem. Phys., 1975,28, 1; (b) J. M. Farrar and Y. T. Let, Ann. Rev. Phys. Chem., 1974, U, 357.

8 (u) J. Dubrin, Ann. Rev. Phys. Chem., 1973,24,97; (b) G. A. Oldershaw, this volume, Chap. 3. 4 T. Carhgton and J. C. Polanyi, in ‘Reaction Kinetics’, ed. J. C. Polanyi, MTP International

Review of Science, Physical Chemistry. Series One, Butterworths, Oxford, 1972, chap. 5. 6 (a) M. J. Berry, in ‘Molecular Energy Transfer’, ad. J. Jortner and R. D. Levine, Wiley, New York,

p. 114; (b) M. J. Berry Ann. Rev. Phys. Chem., 1975,26,259. 1

2 Gas Kinetics and Energy Transfer

an excitation function with those where the rate of reaction is determined for selected internat quantum states of the reactants. In practioe, the Boltzmann laws actually impose some degree of state selection

on a molecular system at thermodynamic equilibrium. This is because the separation of electronic and vibrational states is usually much greater than kT at low temperatures, so that the great majority of intermolecular collisions under these conditions must involve molecules in their lowest vibronic states. Photochemical methods provide the simplest means of disturbing the Boltzmann distribution over states and hence of studying the kinetics of processes involving species in excited states. The photochemical investigation of electronically excited species has, of course, been carried on for many years. However, the process of excitation alters the electronic structure of the atom or molecule that has absorbed light and the results of collisions involving these species cannot be directly related to those of the corresponding ground-state species since the chemical forces controlling the collision dynamics will be quite different. In relatively large molecules, for example cycloheptatriene, the energy supplied initially as electronic excitation can rapidly be transformed into vibrational excitation via a process of internal conversiun. In this way, unimolecular processes can be studied as a function of internal energy supplied viaphotuchemikaZactivatiun.6 Such experiments are considered in Chapter 5.

V i h t i d PhotocbemWy.-In contrast to electronic photochemistry, direct vi6rationalphutochemistry has really only become possible quite recently with the development of powerful i.r. lasers capable of exciting molecules in their relatively weak vibration-rotation bands. The commonest such application has been to the study of vibrational energy transfer.' Molecules are promoted to excited vibrational levels by the absorption of pulsed laser radiation The requirement that frequencies emitted by the laser correspond with lines in the absorption spectrum of the molecule is most easily satisfied when the laser oscillates on lines in the (1,O) fundamental band of the molecule that one wishes to excite, although chance coincidences and tunable laser radiation have also been used. The rate of relaxation of molecules that have been excited in this way is followed by observing how the intensity of the vibrational fluorescence (Zf,) decays with time. In the simplest case, where relaxation occufs predominantly v i a collisions with a single component (Q) of the gas mixture,

where kzo is the rate constant for

BC(w = 1) + Q +BC(v = 0) + Q The method of laser-induced vibrational fluorescence has yielded a great many

results on the transfer of energy from molecules such as the hydrogen and deuterium halides, CO, NO, COz, and other triatomic molecules to chemically stable collision partners. This subject has been reviewed more than once recently and will not be considered here. However, the technique is now being used in several laboratories

(a) D. W. Setser in ref. 4, Chap. 1; (b) P. J. Robinson, in 'Reaction Kinetics', ed. P. G. Ashmore, (Specialist Periodical Reports), The Chemical Society, London, 1975, Vol. 1, Chap. 3; (c) J. Troe and M. Quack, this volume, Chap. 5. ' (a) C. B. Moore, Adv. Chem. Phys., 1973,U, 41; (b) E. Weitz and G. Flynn, Ann. Rev. Phys. Chem., 1974,25, 275; (c) I. W. M. Smith in ref. 5 (a), p. 85.

Reactive and Inelastic Collisions involving Molecules 3

to investigate the result of collisions between vibrationally excited molecules and potentially reactive species, particularly atomic free radicals such as H, N, 0, and halogen atoms. In many such cases chemical reaction, as well as energy transfer, is energetically possible. These alternative channels for removal of the excited molecules, which may be written as

/A + BC(v’ < v ) A + BC(v)

\ A B + C

are not distinguished in a laser-induced vibrational fluorescence experiment, since this only provides a direct measure of the total rate constant for removal of BC(V), i.e. k3 = kSr + ksb. To determine ksb it is necessary to observe one or other product directly and to relate its concentration to the initial concentration of the excited reactant. These three-atom systems are clearly the simplest in which one can study the effect of enhancing the vibrational energy of a molecular reactant and are amenable to the most detailed theoretical interpretation.

So far in this introduction, emphasis has been placed on the part that studies of stateselected processes can play in helping to elucidate the factors that control molecular collisions in cases where there is the possibility of ‘chemical’ interaction between the collision partners. However, the impetus for the recent upsurge of interest in this subject has not been entirely engendered by purely ‘academic’ motives. More mundane, or more important - depending on one’s point-of-view - considerations have also been at work.

The desire to understand, and hence improve, the performance of chemical lasers has served as one such stimulant. If high laser powers are to be extracted from these devices it is necessary to tolerate high concentrations of reactive atoms in the reacting gas that constitutes the laser medium. Unfortunately, these species may deactivate the excited, laser-active molecules at an unusually rapid rate and these processes can then be a crucial factor in limiting the efficiency of the laser. This appears to happen in the laser fuelled by the Hz-CI2 chain reaction. Partly because the Cl + Hz + HCl + H reaction is not particularly fast, the concentration of Cl atoms is likely to be high and these atoms rapidly relax the vibrationally excited HCl formed in the laser pumping reaction, H + Clz + HCl + C1.

hser-induced Chemistry.-There is a second ‘practical‘ reason for interest in vibrational photochemistry which is generating a great deal of excitement at the present time. This concerns the possibility of inducing novel chemical reactions by means of selective vibrational excitation resulting from the absorption of i.r. laser radiation.” The equivalent of the visible-u.v. dye laser is badly needed but no comparably powerful, and relatively cheap laser, providing tunable, narrow bandwidth, radiation in the i.r. yet exists. Nevertheless, a considerable amount

R. G. Macdonald, C. B. Moore, I. W. M. Smith, and F. J. Wodarczyk, J. Chem. Phys., 1975, 62,2934.

9 R. D. H. Brown, G. P. Glass, and I. W. M. Smith, J.C.S. Furuduy 11, 1975,71, 1963. 10 Z. Karny and B. Katz, Chem. Phys. Letters, 1976,38, 382. 11 (a) N. G. Basov, A. N. Oraevsky, and A. V. Pankratov, in ‘Chemical and Biochemical Applica-

tions of Lasers’, ed. C. B. Moore, Academic Press, New York and London, 1974, Chap. 7; (b) J. T. Knudtson and E. M. Eyring, Ann. Rev. Phys. Chem., 1974,2S, 255.


has been achieved using the specific laser sources that are currently available, particularly the COz laser.

However, even where an absorption frequency in one of the potential molecular reagents in a gas mixture does coincide with a laser line, several factors may prevent the laser energy from being used effectively in promoting chemical reaction. The first of these factors is the fairly limited amount of energy that is acquired by a molecule when it absorbs a single photon in a fundamental vibrational band. For example, the P(30) line from the CO, laser at 9.6 pm has a photon energy equivalent to 12.5 kJ mol-l, and the Pl(6) line from an HF laser at 2.71 pm corresponds to 44.2Wmol-l. These energies are comparable with the activation energies of many thermoneutral or exothermic atom-transfer reactions involving a simple free radical and a ‘stable’ molecule, but it now seems clear (see below) that in most cases not all of the vibrational excitation energy can be used to lower the activation energy of the reaction. Consequently, the enhancement of the chemical reaction rate that is brought about by promoting molecules to the first excited level associated with a particular vibrational mode may be fairly small.

Another major problem may be that the high selectivity of the initial excitation is lost rapidly in inelastic collisions. This is certainly true, for example, of any rotational disequili b r im brought about by the excitation process. Consequently, there is rather little direct experimental information about the influence of rotational excitation on chemical reaction rates, although what evidence there is (see below p. 36) suggests that such effects are usually small. In contrast to rotational energy transfer, V-T energy transfer, that is, the transfer of energy between the vibration of a small molecule and the relative translation of it and a second species, is usually extremely slow, unless there are specific intermolecular forces between the colliding species. Thus, the probabilities ‘per collision’ of Ar deactivating HCl (v = 1) l2 and CO (v = 1) at room temperature are 1.7 x and < 1.8 x lo-*, respectively.

The fastest vibrational energy transfer processes are likely to be those involving near-resonant vibrational-vibrational (V-V) energy exchange.’ Thus for a diatomic molecule, processes such as

BC(w = 1) + BC(w = 1) # BC(w = 2) + BC(v = 0) (4)

occur on a timescale characteristic of about 1O-lOOO intermolecular collisions. For polyatomic molecules, similar exchange processes occur, the excitation being retained for some time within the levels associated with one or a limited number of vibrational modes, before ‘leaking away’ into other degrees of freedom.

For some purposes the V-V processes can be extremely useful, since they provide a means of achieving significant excitation to levels high above the ground vibrational state without recourse to direct promotion from v = 0 in extremely weak overtone bands, or without relying on sequential (i.e. v = 0 J - L v = 1 hy\ v = 2, etc.) or multiphoton pumping. A simple example can be provided as an illustration of this. Consider a harmonic oscillator that is pumped sufiiciently strongly on lines in its (1,O) band for 50% of the molecules to be raised ‘instantaneously’ to the

li R V. Steele, jun., and C. B. Moore, J. Chcm. Phys., 1974,60,2794. W. H. Graen and J. K. Hancock, J. Chem. Phys., 1973,59,4326.


first excited level. If then V-T energy transfer can be ignored, once the V-V relaxation has occurred cu. 1.2% of the molecules will be in levels with u < 4. Furthermore, if some reaction removes molecules preferentially from these states, the reaction yield may be very much higher than is suggested by this ‘equilibrium’ figure, since molecules will continue to be excited to these higher levels (but at a continually decreasing rate) as the system attempts to establish a Boltzmann distribution over vibrational states.

The selective activation of a molecular reagent by V-V ‘ladder-climbing’ processes following primary excitation by powerful, singlephoton, optical pumping is one technique for inducing chemical reaction by i.r. laser irradiation. A number of reactions have been promoted in this way and some of these are discussed later. However, this method is, at best, only ‘mode-selective’ rather than ‘state-selective’, since the occurrence of V-V energy exchange prior to chemical reaction destroys the state selectivity of the initial photochemical act.

The rapidity of V-V energy exchange can make it extremely difficult to carry out high& selective experiments of two kinds. The first is measurements of reaction (or relaxation) rates out of specified vibrational levels. It is necessary that the reactive process occurs faster than the redistributim of vibrational quanta via V-V energy exchange if the rate of the former is to be determined. This problem is particularly severe when molecules are excited directly in an overtone absorption band for a study of the kinetics of processes involving species with u > 1. The intrinsic feebleness of the absorption cannot be countered by using high concentrations of the absorbing species, since this will only accelerate relaxation via processes such as

BC(u) + BC(u = 0) + BC(V - 1) + BC(u = 1) ( 5 )

Despite the difficulty caused by rapid V-V energy exchange (and the process

HCl(u = 2) + HCl(u = 0) +- 2HCl(v = 1)

occurs with a rate constant of 2.9 x 10-l2 cm3 molecule-’ s-l, 14--16 corresponding to a probability of 1.4 x Moore and his co-workers 15917 have succeeded in measuring directly the rates of a number of processes involving HCl(u = 2), the excitation being provided by the tuned output from an optical parametric oscillator. Among their experiments is one showing that Br atoms remove HCl(u = 2) 6.4 times more rapidly than HCl (u = 1). In a particularly elegant experiment, Amoldi, Kaufman, and Wolfrum have shown that this is primarily due to the ‘opening up’ of the reactive channel

Br + HCl(v) -+ HBr + C1 (7) once HCl is excited as far as the u = 2 level. These results confirm earlier observations made by Polanyi’s group using a non-laser, ‘i.r. chemiluminescence depletion’ method (see p. 42).19p20 For reaction (7), AEo = +65.6 kJ mol-l, the energy of l4 I. Bur&, Y. Noter, A. M. Ronn, and A. Szoke, Chern. Phys. Letters, 1972, 17, 345.

S. R. Leone and C. B. Moore, Chem. Phys. Letters, 1973, 19, 340. la B. M. Hopkins and H.-L. Chen, J. Chem. Phys., 1972,57,3816. l7 S. R. Leone, R. G. Macdonald, and C. B. Moore, J. Chem. Phys., 1975,63,4735.

D. Amoldi, K. Kaufman, and J. Wolfrum, Phys. Rev. Letters, 1975, 34, 1597. D. J. Douglas, J. C. Polanyi, and J. J. Sloan, J . Chern. Phys., 1973, 59, 6679.

*O D. J. Douglas, J. C. Polyani, and J. J. Sloan, Chern. Phys., 1976, 13, 15.

6 Gas Kinetics and Ehergy Transfer

activation S70 W mol-', and the vibrational excitation energy of HCl(v = 2) corresponds to 67.8 kJ mol-'. The efficient utilization of the vibrational excitation in overcoming the activation barrier appears to be characteristic of endoergic reactions and contrasts sharply with the results expected for exoergic reactions and, to a lesser extent, thermoneutral reactions. The reasons for this difference in behaviour are discussed in Section 3.

Isotope Separation.-Rapid V-V energy exchange also interferes with highly selective experiments which have a different objective and one which could be of immense technological value. These are experiments designed to separate isotopes via selective excitation with lasers. Several schemes have been suggested and the fundamental principles have been discussed by Moore21 and by Letokhov.22 These schemes have four basic requirements in common : (a) isotopically different starting materials that have some discrete spectral absorptions that do not overlap; (b) a laser that is sufficiently tunable and monochromatic to excite only one of these species; (c) a chemical or photochemical method that selectively removes the excited species; and (d) the elimination of processes that destroy the isotopic selectivity during excitation and subsequent reaction.

In the context of this article, we are principally concerned with the case where the laser causes selective vibrational excitation and (c) is a chemical reaction that occurs preferentially with vibrationally excited reactants. In order to achieve a useful isotopic enrichment it will be necessary that the rate of the thermal reaction, averaged over the time the reactants are together, must be appreciably less than that of the laser-enhanced - and therefore isotopically selective -reaction. The latter will depend on the average rate of photochemical excitation as well as on the relative values of the rate constants for reaction of the excited and unexcited molecules and for the competing processes of relaxation. One process that can clearly destroy the isotopic selectivity of a scheme of this kind is V-V energy exchange between species that differ only in their isotopic composition. Except for species that vary in their D and H atom content, vibrational transition energies for different isotopic species are quite similar. Consequently, V-V energy exchange between these species occurs almost as rapidly as between identical molecules.

Although the formidable difficulties associated with isotope separation schemes based on photochemical vibrational excitation plus chemical reaction continue to attract considerable attention, some earlier hopes appear to have been dashed. An experiment performed in 1970 by Mayer et aZ.23 has been much quoted. They reported irradiating mixtures of CH30H aild CD30D in the presence of Br2 with lines from an HF laser that are absorbed only by CH30H. Product analysis indicated the selective depletion of the CHIOH. This observation was interpreted in terms of a selective reaction between vibrationally excited CH30H and Br2, and it appeared to point the way to an economic method for the production of heavy water. However, the results of a careful reexamination of this system have just

C. B. Moore, Accounts Chem. Res., 1973, 6, 323. 22 V. S. Letokhov. Science. 1973. 180. 451. 2s S. W. Mayer, M. A. Kwok, R W. F. Gross, and D. J. Spencer, Appl. Phys. Letters, 1970,

17, 516.


been published. In contrast to the earlier findings, Willis et al?4 have been unable to discover any significant effect brought about by the laser, or filtered block-body, radiation and conclude that the quantum yield for the photochemically induced reaction must be

The news on this front is not, however, all bad. Thus, Amoldi, Kaufman and Wolfrum l8 not only showed that reaction (7) occurs rapidly once HCl is raised to v = 2, but also demonstrated that this system can be made isotopically selective in a particularly simple manner. They excited molecules to w = 2 by sequential optical pumping using a pulsed HCl chemical laser that simultaneously emitted lines in the (1,O) and (2,l) vibrational bands. However, because of the much higher gains on HJ5Cl lines than on Hj7Cl lines, the laser oscillated predominantly on transitions associated with the more abundant isotopic species. Consequently, the overwhelming majority of molecules that were excited were HJSCI. Under the conditions of the experiment, the C1 atoms formed in reaction (7) reacted rapidly with Br2, and the isotopic selectivity of the system was confirmed by the mass spectrometric observation of 3sC179Br and the absence of any mass peak st 118 a.m.u. corresponding to 37C18‘Br. Unfortunately it appears that this enrichment will not usually survive for long. Leone et aZ.,17 using pulses from an optical parametric oscillator to excite H37Cl(v = 2) and the simultaneous frequency doubled (A = 562nm) output from a YAG laser to dissociate Brz in a static system, failed to detect any ClBr in their products, and they suspected that this was due to its removal in a reaction catalysed at the walls of their reaction vessel.

Other schemes for isotope separation or enrichment based on photochemical excitation with i.r. lasers, but not involving chemical reaction, have also been successful. As these fall outside the mainstream of this chapter they will be described only briefly.

One such mechanism for isotopic enrichment that has been successfully demonstrated relies on two-step photodissociation. This method is typified by the selective photodissociation of ammonia to produce molecular nitrogen enriched in 1sN25-27 The experiment depends upon the chance coincidence of a line in the v 2 fundamental band of 15NH3 with the P(16) line in the 10.6 pm branch of a COz laser. Simultaneously with the laser pulse, the sample is irradiated with light from a conventional discharge flash lamp which is passed through a 14NH3 gas filter. This secondary radiation only photodissociates molecules that are vibrationally excited as a result of transitions to the predissociated upper levels of the U.V. ‘hot bands’. In this way a mixture containing equal proportions of lSNH, and 14NH3 has been shown to yield N2 containing 80% ISN and 20% 14N.

Perhaps the most intriguing photochemical isotope separation that has been reported is that which occurs when SFs is subjected to high intensity pulses from a

per infrared photon.

C. Willis, R. A. Back, R. Corkum, R. D. McAlpine, and F. K. McCusky, Chem. Phys. Letters, 1976,38, 336.

*s R. V. Ambartsumian, V. S. Letokhov, G. N. Makarov, and A. A. Puretskii, J.E.T.P. Letters, 1972, 15, 501. ** R. V. Ambartsumian, V. S. Letokhov, G. N. Makarov, and A. A. Puretskii, J.E.T.P. Letters, 1973, 17, 63. *’ R. V. Ambartsumian, V. S. Letokhov, G. N. Makarov, and A. A. Puretskii, in ‘Laser Spectro- scopy’, Plenum Press, 1975, p. 61 1.

8 Gas Kinetics and Energy rl-ansfer

COz This photolyses 34SF6 selectively, via a multiphoton process, leaving the SF6 enriched in WF,. The mechanism for this extraordinary result has not been fully established. Two important factors appear to be that the isotopic selectivity is introduced in the first few absorption steps and that the internal states form a quasicontinuum at moderate to high internal energies. The superficial resemblance of SFs to UF6 has not escaped notice.

In this introduction an attempt has been made to indicate how studies of state selected processes relate to other areas of interest in modern reaction kinetics. In the remainder of the chapter, the main emphasis will be put on the relationship between the observed results, €or systems that have been at least moderately well characterized, and features in the molecular collision dynamics - and ultimately, therefore, in the intermolecular potential - that give rise to different types of detailed behaviour. First, however, it is useful to establish the formal connections between the parameters that define the specificity of energy disposal when reaction proceeds in one direction and those that describe the selectivity of energy consumption for reaction in the opposite direction. This is done in the next section.

2 The hplieations of Microscopic Reversibility Microscopic and Macroscopic II[Inetics.-Any bimolecular collision must lead to one of three general results: elastic scattering, energy transfer, or chemical reaction. Here we are concerned with systems where chemical reaction is at least possible. The simplest such system comprises an atom (A) colliding with a diatomic molecule (BC) and collisions of this type are featured prominently in the remainder of this article.

In conventiond, ‘macroscopic’ chemical kinetics, the rate of a reaction is defined in terms of the change in concentration with time of a single (reactant or product) chemical entity, and observation of this rate yields the rate constant, k(T). At the other, ‘microscopic’, end of the scale, the ultimate, and still distant, goal in reaction dynamics is to study the scattering between species whose velocities and internal states are all accurately defined and to measure diferential reaction cross-sections, a(n’1 n ; wR, 6), for processes that connect fully specified reactant and product states (denoted by n and n’) in collisions of defined relative velocity, vR, where 6 is the scattering angle in the centre-of-mass frame of reference.

Our concern in this article will be with experiments which yield results that in their depth of detail fall between the ‘fully microscopic’ a(n’In; v,, 8) and the ‘totally averaged’ k(T). Introduction of the fully specified differential cross-section serves as a basis from which to consider the principle of microscopic reversibility which, in tum, leads to relationships between detailed rate constants, i.e. rate constants associated with processes connecting species in defined quantum states, for reactions in ‘forward’ and ‘reverse’ directions. These equations are extremely useful since they allow detailed rate constants for reaction from selected reactant

*a R. V. Ambartsumian, A. Gorokhov, V. S. Letokhov, and G. N. Makarov, J.E.T.P. Letters, 1975,21, 375. J. L. Lyman, R. J. Jensen, J. Rink, C. P. Robinson, and S. D. Rockwood, Appl. Phys. Letters, 1975, 27, 87.

‘0 G. Hancock, J. D. Campbell, and K . H. Welge, Optics Comm., 1976,16, 177.


states to be derived from those for the corresponding reverse reaction into specijic product states. Partially detailed rate constants, for reaction into particular product states from thermally equilibrated reagents - or in some cases, from partially selected reagents - have now been obtained for a number of exothermic atom- transfer reactions, particularly via the 'arrested relaxation' version of the i.r. chemiluminwnce t ~ h n i q u e . ~ Application of the reversibility arguments to these data has provided considerable insight into the selective consumption of reactant energy in endothermic reactions.

Microscopic Revmibility and Detailed- BaIa~ce.~~-The principle of microscopic reversibility arises from the invariance of the laws of motion - quantal as well as classical - under time reversal. Because of this the probability of a transition between fully specified states per unit time is independent of the direction in which time is chosen to move, i.e.

P(n', 8 c n, v,) = P(n, 8 c n', v,) (8) and the ratio of the &tailed rate coeflcients, connecting any pair of fully specified reactant and product energy levels, and given by the product of v, and S(v,), the total reaction cross-section, must be equal to the ratio of the 'phase space volumes' associated with the specified products and reactants. Denoting quantities associated with the reactants by unprimed symbols and those for the products by primed symbols, it is possible to derive the expression

pzg S(n'l n; v,) = pt2g' S ( n I n'; v:) (9)

where p and g represent momentum and degeneracy, respectively. For a reaction of the type

A + BC(v,J)# AB(v',J') + C

(g'lg) = (W' + 1 ) W + 1)

(10)

if the only degeneracy is that associated with the rotational levels of BC and AB

(1 1)

(1 2)

then, remembering that '2

E = p 2 / 2 p = and E' = ~ ' ~ / 2 p ' = $p'vR

it is easy to derive the equation

S(v',.J'Iv,J; E ) W' + 1 p'

S'(v,JIv',J'; E') = (m) (F) (r) The corresponding equation in the detailed rate coefficients is

k(v',J'Iv,J; E ) - VRS(U',J'IV,J; 6)

k'(v,Jlv',J'; E') v,'S'(v,JIv',J'; 6') -

Detailed rate constants, which describe the rates of reaction in collisions between species in defined internal states but occurring with a thermal spread of relative

*I J. C. Polyani and J. L. Schreiber, in ref. 1 (a), Chap. 6.


translational velocities (or collision energies), can be obtained by carrying out the integration

where f(u,; T) is the function describing the distribution of relative translational velocities at temperature 7‘. Since for any pair of connected states, v’,J’ and u,J, AEint = E’ - E, solution of (15) and the analogous equation for k’(v,Jlv’,J’; T) yields

In many experiments only partially detailed information is obtained. For example, i.r. chemilumin-nce experiments under conditions of ‘arrested relaxation’ can provide rovibrational state distributions of reaction products. Where, as has usually been the case, the reagents are thermally equilibrated, the corresponding rate constants are defined by

k(v‘,J’( ; T ) = C f(v,J; T)k(v‘,J’Iv,J; T) (1 7)

where f(v,J; T) is the Boltrmann distribution function over v,J levels. A further summation over J’ yields k(v’/ ; T), which are a set of rate constants that define the vibrational specificity of energy disposal in a reaction occurring between equilibrated reagents. If these procedures are applied to equation (16),”2 the result is an equation which relates the rate constants k(v’1; T) to k( Iv’; T) :

”* J

where Qtn, is the total internal partition function of BC, Q:’p‘ is the rotational partition function of AB(v’), and A E , , , ~ is the energy of AB(v’) minus that of BC(v = 0).

Equation (18) is of considerable value since it allows relative values of k’(lw’; T ) for endothermic reactions to be obtained from the values of k(v’1; T) that are determined in i.r. chemiluminescence and chemical laser experiments.2* 4s Examples of its use will be quoted later. A final summation in equation (18) over v’ yields the equation which encapsulates the principle of detailed balancing

where K, is the chemical equilibrium constant expressed in terms of concentrations. Polanyi and his co-workers 33* 34 were the first to use equations based on the

principle of microscopic reversibility to examine the selectivity of energy consumption in elementary reactions. From measurements of the products’ i.r. chemi-

s2 H. Kaplan, R. D. Levine, and J. Manz, Chem. Phys., 1976,12,447. 83 K. G. Anlauf, D. H. Maylotte, J. C. Polyani, and R. B. Bernstein, J. Chem. Phys., 1969, 51,

5716. 34 J. C. Polanyi and D. C. Tardy, .I. Chem. Phys., 1969, 51, 5717.


luminescence, they had obtained relative values of k(v’,J’I ;T) for the exothermic reactions :

CI + HI --f HCl(v’,J’) + I, AH: = -132.6 kJ rno1-l

H + Clz --+ HCI(d,J’) + Cl,

F + H Z --+ HF(v’,J’) + H , AH: = -133,s kJ m01-l

(20)

(21)

(22)

AH: = -188., M mol-I

(From here on, we shall adopt Polanyi’s convention whereby primes are always used to denote the energies or states of products of reactions in the exothermic direction or of reactants of reactions in the endothermic direction.)

Anlauf et aZ.j3 argued that provided the distribution of relative detailed cross- sections, i.e. S(v’,J’Iv,J; &), was only moderately dependent on E, then

k(v‘,J’lv,J; e) k(v‘,J‘lC,.?; E) k(v‘,J’I ; T) k’(vYJ1v‘,J’; d) k’(&,Jlv’,J’; E’) k‘(Q,.ilv‘,J‘; 8‘) (23) - - - -

where 6 and J are the most populated vibrational and rotational levels in the diatomic reagent (BC). The average relative translational energy of those collisions that lead to the exothermic reaction is given by

where NA is the Avogadro constant and EaCt is the activation energy for the reaction in the forward direction. The total energy available to the products of the reaction is, on average,

“total = ( -AH: + Eact + Q RT)INA (25)

the last two terms in the numerator corresponding to the mean translational plus rotational energy of reactants undergoing reaction (5 = 0). Finally, 8’ is given by energy balance as

g‘ = - (&“’,J’ - &$,$I (26)

Combining equation (14) with equations (23)-(26) yields

which allows one to calculate detailed rate coefficients for endothermic reaction out of specified states, if the detailed rate constants k(v‘,J’I ; T ) have been observed.

The results of i.r. chemiluminescence experiments are now frequently displayed on triangular contour plots of the type shown in Figure l(a). In this case, contour lines join values of the product vibrational and rotational energies (quantization is ignored) for which k(v‘,J’I; T ) has the same value. Because the total energy is (approximately) defined, the translational energy is established by energy balance and can be represented on these diagrams by diagonal lines. The other diagrams in this figure display, in a similar fashion, the detailed rate coefficients, k’(6,jlv‘,J’; E‘), calculated from the results of i.r. chemiluminescence experiments on reactions (20)-(22) according to equation (27).j3*j4 Those in l(b) and l(c) are typical for endothermic reactions that take place in direct collisions, i.e. without


0 04

HF(V: J')+H-+F+H;L ENDOTHERMIC

c 2 cV I I I - l - - - - b V

0 . o a

-HCI (u .J ' )+CI -t H+CI, E N D O T H ERMlC

-1'-4 14-0151

Figure 1 Triangular contour plots showing the variation of detailed rate constants. Values of vibratwnal energy (V) and rotational energy (R? are plotted, ignoring quantization along the rectilinear axes and those of translational energy (T') are indicated by the dashed diagonal lines. Units are kcal mol-' (1 kcal mol-' = 4.18 kJ rnol-'). Panel (a) shows the variation of the detailed rate constants for reaction (22) in the exothermic direction, i.e. kf (v', J'l8, .f; a) m kf (v', J'I ; T), as determined by i.r. chemiluminescence experiments. Panels (b) to ( d ) indicate the detailed rate constants for reactions (-20) to (-22), i.e. k, (6, .?Id, J' E?, as obtainedjiom application of equation (27). l%e horizontal line on each diagram indicates the energy of the actual vibrational states and the value of kf or k, beside these lines indicates the results of surnming the detailed rate constants over rotational states (Reproduced by permission from J. Chem. Phys., 1969,51,5716,5717)

the formation of a transient collision complex. These diagrams show that exciting the molecular reageat vibrationally is much more likely to promote the endothermic reaction than putting the equivalent amount of energy into relative translation or into rotation. Figure l(d) for reaction (21) looks quite different from l(b) and l(c).


Here, the optimum distribution of energy is -50 % in the HCl vibration with most of the balance in relative translation. Further vibrational excitation at the expense of relative translational energy reduces the likelihood of reaction. This is a manifestation of the ‘light atom an0maly’,3~ that is the unusual dynamical behaviour that appears when the attacking atom in an exothermic reaction or the product atom in an endothermic reaction, i.e. atom A, is much lighter than the other two, i.e. B and C.

Most of the reactions whose specificity of energy disposal has been studied by the technique of i.r. chemiluminescence are exothermic reactions of the A + BC --f AB + C type that proceed directly and have low activation energies. For these reactions the assumption that k(v‘,J’I$,S; d) is proportional to k(d,J’l;T) seems quite reasonable since the thermal spread of initial energies among the degrees of freedom of A and BC is small relative to the energies released in the reactions themselves. This conclusion has been supported 36,37 by the results of quasiclassical trajectory calculations (see below).

There have been fewer spectroscopic studies of direct reactions of the A + BCD type. Those there have been 38*39 suggest that the degree of vibrational excitation of CD in direct reactions of this type will usually be rather low, typically < 10% of the total available energy appears in this degree of freedom. On the basis of reversibility arguments, this observation indicates that selective excitation of the CD vibration will do little to promote endothermic reactions between AB + CD --f A + BCD. The extrapolated conclusion is that selective excitation of vibrational modes that remain essentially unchanged in the transition state for the reaction is unlikely to enhance the rate of that reaction to any great extent.

An increasing amount of information is becoming available regarding the collision dynamics and disposal of energy in reactions that do not proceed directly but rather via the formation of a collision ~ o r n p l e x . ’ - ~ * ~ ~ When such a complex survives for longer than a rotational period, the eventual scattering of the reaction products should be symmetric about the centre-of-mass scattering angle, 8 = 90”. A fierce argument rages concerning the disposal of energy in such If an energy barrier exists in the exit channel for such a reaction, some energy will be disposed of after this barrier has been surmounted and as the products separate. In this case, the product state distributes will not be ‘statistical’. On the other hand, if no such barrier exists, one might expect that the probability of a particular detailed result will be proportional to the volume of phase space associated with that result, as long as this is consistent with the conservation of total angular momentum throughout the collision. The corollary of this would be that the rate of endothermic reaction’s proceeding via complex formation are much less sensitive to whether the necessary energy is supplied to particular degrees of freedom in the reactants.

See ref. 3 1, p. 42 1, and references therein.

D. S. Perry, J. C. Polanyi, and C. W. Wilson, jun., Chem. Phys., 1974,3, 317. M. A. Nazar, J. C. Polanyi, and W. J. Skrlac, Chem. Phys. Letters, 1974, 29, 473.

a* I. W. M. Smith, Discuss. Furaday SOC., 1967,44, 194. 40 J. M. Farrar and Y. T. Lee, J. Chem. fhys., 1976,65, 1414. 4l See, for example, ref. l(b) and ref. 40.

aa D. S. Perry, J. C. Polanyi, and C. W. Wilson, jun., Chem. Phys. Letters, 1974, 24, 484.


Our purpose in briefly considering energy disposal in reactions proceeding via a collision complex is not to enter into the controversy referred to in the last paragraph, interesting though that is. However, this topic has served as a qualitative introduction to the concept of aprior expectation as the one that might be expected in the limit of statistical behaviour.

The Momtion-Theoretic Approach.-An increasing number of kineticists are now tabulating their experimental results by reference to a new analytical method. This approach, which was first formulated by Bernstein and L e ~ i n e , ~ ~ ~ ~ ~ is based on concepts borrowed from information theory. One of its merits is the possibility of characterizing a wealth of detailed information, for example, a large set of k(v’,J’I; T), in terms of a small number of parameters. Furthermore, the method provides a means of obtaining detailed rate data by extrapolation, where an experiment has given only partial information.

The central feature of the information-theoretic analysis is that it proposes a quantitative measure of the surprisal of a particular outcome, for example, the observation that in reaction (20) -42% of the HCl product is formed in v’ = 3. To do this, it is first necessafy to establish a datum line, i.e. to define what result would cause zero surprise. Then an equation must be formulated which determines the deviation of the actual result from the prior expectation.

In Bernstein and Levine’s treatment it is assumed that at a given total energy, and in the absence of other prior information, all energetically accessible product states are equally probable. In an exothermic reaction like (ZO), the smaller the value of v’ then the more energy there is to distribute between AB rotation and relative translation of AB + C. As there are more product states that can be populated for low v‘ than for high v’, this leads to the prior expectation that AB’s vibrational distribution will fall monotonically. This distribution function is shown, for a sample reaction, as a dashed line in Figure 2. It should be noted that in evaluating these prior expectation functions it is customary to count all energetically accessible states; the need to conserve total angular momentum in a collision is viewed as one of the dynamical constraints that may cause the actual distribution to deviate from the prior expectation.

The surprisal of a particular outcome, for example, I(v’), is defined in terms of the observed probability of the specified result, P(w’), and that expected on prior grounds, Po(v’), according to the equation :

I(w’) = -In [P(v’)/P”(v’)]

Surprisal plots of I(v’) against w’ or 6,’ have now been produced for several reactions. With very few exceptions they are linear or nearly so. That shown in Figure 2 for the reaction

0 + CS -+ CO(v’) + S, AH: = -355 W mol-I (29)

provides a good example;44 extrapolation of the straight line appears to provide a

‘2 R. €3. Bernstein and R. D. Levine, J. Chem. Phys., 1972, 57, 434. ‘s R. D. Levine and R. B. Bernstein, Accounts Chem. Res., 1974,7, 393. “ A. Ben-Shaul, Chem. Phys., 1973, 1, 244.

Reactive and

0.2 0.4 0.6 0.8 1.0

15

10 I t V’

f”,

Figare 2 Information-theoretic analysis of detailed rate data for the reaction 0 + CS ?r: Cqv’ ) + S. Panel (a) compares P(fut), the observed distribution over CO vibrational states from the exothermic reaction,45 * 46 with Pocfv~) the distribution expected on prior groundr. Panel (b) shows the surprisal associated with these C a d ) concentrations as a function of f,#. In panel (c), the partially detailed rate constants for the endothermic reaction from selected CO vibrational levels, calculated from detailed balance, are plotted against E,JlkT for T = 300 K (Adapted by permission from Chem. Phys., 1973,1,244; 1976,12,447)

sound basis for obtaining information about the values of P(d) below d = 6, which have been difficult to determine d i r e ~ t l y . ~ ~ - ~ ’

When the variation of I ( d ) with e,,, or f,, = c,,/~’~,,~~,, is linear

and A,, provides a differential measure of the deviation of the observed distribution

G. Hancock, C. Morley, and I. W. M. Smith, Chem. Phys. Letters, 1971,12, 193. G. Hancock, B. A. Ridley, and I. W. M. Smith, J.C.S. Faraday ZZ, 1972,68,2117. K. D. Foster, J. Chem. Phys., 1972, 57, 2451.

H. T. Powell and J. D. Kelley, J. Chem. Phys., 1974, 60, 2191. 48 S. Tsuchiya, N. Nielsen, and S. H. Bauer, J . Phys. Chem., 1973, 77, 2455.

so N. J. Djeu, J. Chem. Phys., 1974,60,4109. 51 J. W. Hudgens, J. T. Gleaves, and J. D. McDonald, J. Chem. Phys., 1976,64, 2528.


from that expected a priori. Clearly, A,,, is a temperaturelike parameter and exp(Ao) plays the role of a partition function, since

c P(fI?d exP@o) = e W o ) c PCfd Y l V’

= exp(Ao)

The surprisal is a measure of the deviation of a single population from its ‘expected‘ value. The two parameters A,,, and exp(A,), or similar parameters for other distributions, characterize the deviation of the whole distribution from prior expectation when this can be described by equation (30). Finally, the average value of this deviation can be calculated and it is known as the entropy of the distribution. The entropy cteficiency 4 2 s 4 3 * 5 3 of a vibrational distribution is then dehed by

ASv/ = S,Ot - S,,,

= R C pcf,/) in rmut)/~o(f,l~~ (32) Y’

The greater the specificity of energy disposal in an exothermic reaction, then the larger is its entropy deficiency. Values of A,, and AS,,, for some well-studied atom-transfer reactions are listed in Table 1; <f:) is the average fraction of available energy that is released as product vibration, i.e.

Table 1 The vibrational specif;city of some exothermic atorn-transfer reactions

Reaction E,,,/kJ mol-‘ < fU.> A,. AS,,p/J mol-’ K-I ReJ: o + a +CO(v’)+S 364 -0.81 -7.7 -18.5 45, 44

a, 43 C1 + HI --f HCl(u’) + I 140 0.73 -8.0 15.3 C1 + DI -+ DCl(u’) + I 140 0.73 -8.0 F + H2

F + DH + DF(w’) + H 149 0.626 -5.4 F + HD -+ HF(v’) + D 141 144 0.588 0.664 -6.9 -7.0 q + HF(u’) + H

F + D2 + DF(u‘) + D 145 0.665 -5.9 13.0 F + HCl + HF(w’) + CI 144 0.58 -4.8 9.8 F + HBr -+ HF(u’) + Br 208 0.55 -4.0 6.5

a Experimental data from D. H. Maylotte, J. C. Polanyi. and K. B. Woodall, J. Chem. Phys,, 1972,57,1547, but Etot.l is adjusted to take account of recent rate measurements byiK. Bergmann and C. B. Moore, J. Chem. fhys., 1975, 63, 643, that indicate that these reactions have zero activation energy; * Experimental data from B. S. Perry and J. C. Polanyi, Chem. Phys., 1976, 13, 1; f Experimental data from N. Jonathan, C. M. Melliar-Smith, S. Okuda, D. H. Slater, and D. Tilman, Mol. Phys., 1971, 22, 561.

c, 44

Reaction (21) is omitted from Table 1 because the product state vibrational distribution does not yield a linear surprisal plot.52 It appears that this is another manifestation of the ‘light atom anomaly’ referred to earlier. In particular it should

64 A. Ben-Shaul, R. D. Levine, and R. B. Bernstein, J , Chem. Phys., 1972,57, 5427.


be noted that in this reaction (p'/p) > 1 and that no account is taken of angular momentum conservation when Po(u') is calculated.

In the light of the discussion of microscopic reversibility that was givea earlier in this section, it should occasion no surprise that the information-theoretic analysis can be applied to the selective energy requirements for reaction 3-s as well as to the specificity of energy disposal. However, the results that are available for this treatment we largely derived either from data obtained for exothermic reactions or from quasiclassical trajectory calculations, rather than from direct experimental measurements. To consider reaction out of specified reactant states it is necessary to modify

slightly the definition of surprisal that was given in equation (28). Kaplan, Levine, and Manz 54 consider the partially detailed rate constants for reaction from specifid vibrational levels, the rotationsdand translational degrees of freedom having thermal distributions defined by T. Now the (vibrational) surprisal is

I ( d ; T ) = -In [k'( Id; T)/k'"( Id; T)] (34) This equation can be shown to correspond to (28) at a given temperature, and Po(u') is the limit of ko(v'/ ; T)/k"(T) when -Aco/kT > 0.54

Because the prior rates, as well as the actual rates, must conform to equation (1 8), I@'; T) is independent of whether the forward or reverse reaction is being considered, so

I ( d ; T ) = -In [ ~ ' ( I v ' ; T)/k'"(ju'; T)]

= -In [k(u'I ; T)/k"(v'I ; T)]

= -In [P(d)/P"(V')] (35)

(36)

(37)

When the surprisal plot is linear, one can write

so that

where A,,, = A,,(kT/&,t,,). For endothermic reactions, for example the reverse of the reactions listed in Table 1, A,,, and therefore A,,, is usually negative and therefore k'( Id; T ) increases more rapidly than k'"( Iv'; T), i.e. vibrational excitation enhances the reaction selectively.

If equation (18) is used to relate k'O(1v'; T) to k"(v'1; T), equation (37) may be written as

I(v'; T ) = Io(T) + &E,r/kT

k'( Id; T ) = k'O( Id; T ) exp[ -Io - A,,c,,/kTl

k'( Iv'; T) = 6) ' (7) Qint k "(v'] ; T ) exp( -Io) Qr,,,l

= 6)' (2) ko(v'I ; T ) exp( -Io)

sa R. D. Levine and J. M u , J. Chem. Phys., 1975,63,4280. 64 H. Kaplan, R. D. Levine and J. Manz, Chem. Phys., in the press. 66 R. D. Levine and J. Maw, J . Chem. Phys., in the press.


where is the zero-point energy of the primed species less that of the unprimed species (this is the usual notation of statistical thermodynamics but differs in sign from that used by XRvine and co-workers). This equation shows that most, but not all, of the variation of k’( lv’; T ) evolves from the second exponential term on the right-hand side. Where -Ahso is large this factor will exert the dominant effect. However, the slope of a plot of k’(1u’; T ) vs. (e,,/kT) will usually not differ very much from unity since A,, is generally much smaller than 1. An example of the variation of k’( lv’; T ) is given in panel (c) of Figure 2. Because I A,,! is generally much less than one for reactions with Aeo << 0, i.e. highly exothermic reactions, it will always be easiest to determine information about the state-selected kinetics of such reactions by observing the state distributions of the products of the reaction proceeding in the exothermic direction.

Finally, equation (38) can be used to determine the contribution to the overall rate constant at thermal equilibrium arising from reaction out of a particular level. If both sides of the equation are multiplied by f(v’; T ) =exp( -e,,/kT)/Q,,, one obtains

Which v’ level makes the maximum contribution to the overall rate of an endothermic reaction depends on exactly how the decrease in k”(v’1; T ) with v’ cancels the increase in exp( -A,,e,,/kT) when A,, is negative.

The effect of vibrational excitation of the reagents on the rate constants for exothermic reactions can be considered 54 in terms of an equation analogous to (37), i,e.

For exothermic reactions, A, is generally positive, i.e. the reaction rate is enhanced less by vibrational excitation than might be expected a priori. Furthermore, the result of the conflicting variations of kQ(lu; T ) and exp ( -A,e,/kT) may not lead to a simple exponential variation of k( 10; T ) with v. At present, it is too early to say whether equation (40) will accurately describe the variation of k(lv; T), particularly for those v levels whose energy is less than the activation energy. Nevertheless, the above analysis does suggest that it is scarcely surprising that Birely and Lyman 56 were unable to find any pattern to the utilization of reactant vibrational energy to enhance reaction rates despite a careful examination of the available experimental data.

k(lo; T) = k”(lv; T)exp( -lo - A,e,/kT) (40)

3 Theoretical consid~tions Potential Energy Hypemrfh~.--SO far in this chapter the emphasis has been on how one &scribes the results of reactive encounters between species in selected quantum states. Now it is time to examine what the fundamental factors are that control the collision dynamia, and therefore lead to different detailed results. This discussion will be based on the foundation of the Born-Oppenheimer assumption: that is, that the motions of the nuclei during a collision are determined at all (or

66 J. H. Birely and J. L. Lyman, J. Photochem., 1975, 4, 269.


almost all) points by a single electronic potential, which is a function only of the relative nuclear posit ions.

For our representative system of three atoms, the nuclear framework is defined by three internuclear distances and the potential energy can be written as V(rAB, r,,, rcA). For a given potential, the result of an individual collision (or, in quantum mechanics, the probabilities of various allowed results) depends on parameters such as the relative translational energy, impact parameter, etc., that specify the situation at the ‘start’ of that collision. It is the form of the potential hypersurface, e.g. V(rAB, r,,, rca), that distinguishes one molecular system from another and governs the principal features of the results. Thus, the potential for three H atoms permits an atom-transfer reaction to occur in H + Hz collisions at quite moderate energies, whereas this cannot happen when He atoms collide with Hz.

V(rAB, rBC, rcA) can only be represented diagrammatically if the number of independent variables is reduced. Usually, the familiar contoy line plots of energy surfaces are drawn for collinear configurations, i.e. with rcA = rAB + rBc. Those in Figure 3 typify five different types of intermolecular interaction. In all but the first of these, there is a strong ‘chemical’ interaction between A and BC; as well as BC, AB and, in some cases, ABC are chemically bound. In each of these four cases, as A approaches BC, r,, tends to increase. The reverse is true for surface 3(a), where the dominant forces are repulsive, Since A repels B, the atom closer to it, more strongly than C, BC is compressed as A approaches.

X TAB TAB

TAB ~ A B

Figure 3 Potential surfaces representing diferent types of molecular interaction: (a) potential between a diatomic molecule BC and inert atom A as a function of re, and x, the separation of Af iom the centre of muss of BC; @)potential for a thermoneutralatom-transfer reaction with A 3 C so the barrier is symmetrically located; (c)potential for an exothermic reaction A + BC + AB + C with only a low barrier to reaction; ( d ) here ABC, as well as AB and BC, is bound and since A = C there is a symmetrically placed ‘well’ ; (e) again ABC is stable, but now A $ C so the ‘well’ is not symmetrically placed. The black dots indicate the lowest points on the energy surfaces


In actual fact, surface 3(a) is oversimplified since it neglects that, even for He + H2, there is some attraction between He and H, and between He and H, if only that provided by the weak dispersion forces. Indeed, there is now a body of evidence 57 indicating that ‘three-body’ atomic recombination at low temperatures proceeds predominantly via atom-chaperon complexes that are held together by forces much weaker than normal chemical bonds. Therefore, the crucial step in this mechanism of atomic recombination is a highly exothermic bimolecular reaction. Consequently, it is not surprising that recombined diatomic molecules are apparently formed principally in internal states just below the dissociation limit.58-62 Conversely, the reverse dissociation reactions can be regarded as highly endothermic processes whose rates are enhanced preferentially by internal excitation of the molecular reactant rather than by high collision energy. One result of dissociation occurring mainly from internal states close to the dissociation limit is that these levels have steady-state concentrations during dissociation that are appreciably lower than their thermodynamic equilibrium

It may not only be the form of the intermolecular potential that is different when chemical reaction is possible. In addition, A + BC may correlate with more than one electronic state. For collisions between He(lSo) and H2(‘C;), there is just one singlet electronic state. This has lZt symmetry for collinear (Cav) configurations and ‘A’ character in non-linear (C,) geometries, the latter symmetry being more appropriate when considering intermolecular collisions. For H(2S+) + H2(lZfg), the corresponding term symbols are 2E+ and 2A‘; only spin degeneracy is involved, and the interaction can still be described by a single potential hypersurface. This situation changes, however, when either A has L > 0 or BC has A > 0. Now more than one hypersurface correlates with the separated species. For example, the collinear interaction of F(’P+,*) with H2(lZ;) gives rise to zT; and 211 states and in C, geometries, the near degeneracy in the state splits, creating three electronic states (22A‘ + 2A”) in all.

In the great majority of atom-transfer reactions between species in their ground electronic states, even if several non-degenerate states do correlate with the separated reactants, their existence is of limited importance. Chemical reaction usually proceeds via the potential hypersurface that is at all points lowest in energy. The process is then said to be electronically adiabatic. The only effect of there being more than one overall electronic state is the introduction of a statistical f a ~ t o r , ~ ~ , ~ ~ which will be temperature dependent if the spin-orbit terms are split by -kT, to allow for the fact that not all collisions will occur on the lowest potential. In assessing the results of dynamical calculations, one should determine whether any allowance has been made for this effect as no standard practice has been established.

Although reactive collisions in which a system switches between electronic states are unusual, electronically non-adiabutic collisions can provide a quite eficient

67 J. Troe and H. Gg. Wagner, Ann. Rev. Phys. Chem., 1972,23, 31 1. m V. H. Shui, J. P. Appleton, and J. C. Keck, J . Chem. Phys., 1970,53, 2547. m V. H. Shui and J. P. Appleton, 1. Chem. Phys., 1971, 55, 3126.

A. G. Clarke and G. Burns, J. Chem. Phys., 1971,55,4717. A. G. Clarke and G. Burns, J . Chem. Phys., 1972,56,4636. W. H. Wong and G. Burns, J . Chem. Phys., 1973,SS, 4459.

as D. G. Truhlar, J. Chem. Phys., 1972,56, 3189. J. T. Muckerman and M. D. Newton, J. Chem. Phys., 1972,56, 3191.


mechanism for energy transfer. There is now rather strong evidence that when this can occur, i.e. when multiple hypersurfaces correlate with the electronic ground states of A + BC(v), the vibrational relaxation rate can be several orders of magnitude faster than when A and BC are both species with closed electron shells. This topic is considered in greater detail in the last part of this section.

The Dynamics of Electronically Adiabatic Collisions.-There are three parts to a detailed rate theory of processes occurring in electronically adiabatic collisions. First, the potential describing the molecular interaction must be calculated or estimated. Secondly, the equations of motion have to be solved for individual, fully specified, collisions. Finally, the results of calculations on single collisions must be averaged correctly to yield the required result: for example, a reactive cross-section or a detailed rate constant. The procedures for the third stage were outlined in Section 2. In the ‘forward’ direction, i.e. from a(n’ln; v,, 0) to k(T), this averaging presents no problems, but it is the difficulty of reversing this process which makes it impossible to obtain detailed information about the collision dynamics or potential from experimental measurements of thermal rate constants.

Strictly, the dynamics of intermolecular collisions should be treated quantum mechanically 65 but there are formidable difficulties associated with three-dimensional calculations on reactive systems. Only one fully quanta1 study, on H + H2, has been completed.66 One problem is that the trial solution to the Schrodinger equation is expressed as a sum of basis functions, and this should include all the rovibrational states that are coupled during the strongest part of the collision. For molecules with moments of inertia greater than that of H2, many more states have to be included in the basis set and the size of the computation increases rapidly.67 This difficulty is similar to that in calculations of electronic energies in molecules, when for many-electron systems, the basis set of atomic orbitals that is required for accurate calculations becomes too large to handle.

In contrast to quantum scattering calculations, those using the classical equations to describe the collision dynamics have become almost routine. It is usual to select all, or almost all, of the starting parameters for individual collisions by Monte Carlo sampling techniques, so that any necessary averaging is built into the calculation and the results for comparison with experiment are obtained in a relatively small number of trajectories. The initial internal energies are normally chosen to correspond to those of rovibrational levels in the reactant molecule, and the calculations are then referred to as quasicZassicaZ (QCL). The h a 1 energies are not, of course, restricted to quantized levels, so products are assigned essentially to those states which lie closest to the calculated classical energy.68

There is now a good deal of evidence that the QCL trajectory method provides a generally satisfactory description of reactive collision dynamics. Semi-classical calculations 69-71 have been important in defining the situations where purely

66 T. F. George and J. Ross, Ann. Rev. Phys. Chem., 1973,24, 263. 66 G. Wolken, jun., and M. Karplus, J. Chem. Phys., 1974,60, 351. R. G. Gordon in ref. I@), p. 22.

6B J. T. Muckerman, J . Chem. Phys., 1971,54, 1 155. W. H. Miller, J. Chem. Phys., 1970,53, 1949.

70 R. A. Marcus, Chem. Phys. Letters, 1970, 7, 525. 7l W. H. Miller, Adv. Chem. Phys., 1975,30, 77.


classical calculations may be inadequate. The latter provide reasonably accurate results for ‘classically allowed’ processes, that is when the classically calculated probability of a particular outcome is quite large. There may be serious errors, however, in ‘threshold regions’. This will be the case when the cross-section is computed classically for collision energies close to the threshold energy for overall reaction, and similar effects arise, for example, near threshold energies for the formation of products in specified states. These examples can both be considered as situations in which quantum mechanical tunnelling is important.

Undoubtedly, the greatest obstacle to purely theoretical attempts to calculate kinetic data is the lack of accurate ab initio potentials for all but a handful of systems containing more than two atoms. Usually QCL trajectories are calculated using functions for the potential that incorporate a high degree of empiricism. Where the objectives are of a fairly general kind this does not matter. The valuable work of Polanyi’s group, that has done so much to establish connections between various features of the potential and particular aspects of the reaction dynamics,31* 72

head this category. At the other extreme are studies where an attempt is made to construct the potential surface for a particular system by continually adjusting the potential function until the calculated and experimental results coincide. Only if the experimental data are detailed and extensive is this matching procedure likely to succeed.

Of course, many investigations fall somewhere between the two limiting types. Thus, a potential may be chosen on limited experimental information and a Monte Carlo trajectory study carried out to predict the values of quantities that have not been observed experimentally. These results should not be accepted unreservedly, since the collision dynamics are determined by the form of the assumed potential. Therefore, the evidence on which the potential was selected should be carefully scrutinized. Recent calculations which were designed to provide information about the relaxation of HF by H atoms, i.e.

A H + FH(d < u)

b H F ( d < v) + H H f FH(v)

illustrate this point. The potentials on which trajectories have been calculated 7 3 m 74

have barriers to F atom transfer ranging from 6 to 210 kK mol-I; ub initio calculations 75 suggest that the height of this barrier is at least 165 kJ mol-l.

The classical trajectory method and the results that have been obtained from its application have been reviewed in depth recently by Polanyi and Schreiber 31 and by Porter.76 In the next few paragraphs an attempt is made to summarize the results that are especially important from the point of view of interpreting rates of reaction and relaxation in collisions between molecules in excited internal states and potentially reactive, i.e. free radical, species. Trajectory studies that relate to particular systems will be referred to later when experimental data are under discussion.

72 J. C. Polanyi, Accorrnts Chem. Res., 1972, 5, 161. T a R. L. Wilkins, J . Chem. Phys., 197Z57.912; Mol. Phys., 1975,29, 555. T 4 D. L. Thompson, J. Chem. Phys., 1972,57,4170. 76 C . F. Bender, B. J. Garrison, and H. F. Schaeffer, tert. ,J. Chem. Phys., 1975,62, 1188. 76 R. N. Porter, Ann. Rev. Phys. Chem., 1974, 25, 317.


General Results of Classical Trajectory Studies.-The original theoretical efforts of Polanyi and his co-workers 77 were designed to complement their experimental measurements of product state distributions from substantially exothermic reactions such as (20) to (22). In their trajectory calculations, the Toronto group have usually employed an extended form of the London-Eyring-Polanyi-Sat0 (LEPS) potential

+

(42) - - (l + sBC)(l + sCA) ( l + sCA)(l + ’AH)

which allows one to alter the nature of the hypersurface by adjusting the values of SAB, SBc, and ScA. A special concern has been to identify those features of the potential that cause energy to be released preferentially into certain degrees of freedom of the products. Polanyi’s conclusions echo suggestions that were made many years earlier:78 the proportions of energy deposited into (a) the vibration of the molecular product AB and (b) into the relative translation of AB and C can broadly be correlated with the fractions of the total energy that are released as (a’) the reactants A and BC approach one another and (b’) as the products AB and C separate.

A more quantitative picture requires one to define how the reaction path is divided into sections (a’) and (b’). This can be done by reference to diagrams showing the variation of electronic potential energy along the reaction path of minimum energy leading from reactants to products. Examples of these potential energy profiles 72 are given in Figure 4. In both diagrams the regions where

I I Re-Products ADDroach SeDarate

Figure 4 Potential energy profiles along the path of minimum energy shown for reactions proceeding in the exothermic direction. On an attractive surface, as in (a), most of the energy is released as the reactants approach, i.e. before rABlra,AB = rBClrc,BC. In case (b), the swface is more repulsive, the bulk of the energy being released as the products separate (Adapted by permission from Accounts Chem. Res., 1972, 5, 161)

77 P. J. Kuntz, E. M. Nemeth, J. C. Polanyi, S. D. Rosner, and C. E. Young, J . Chem. Phys., 1966,44, 1168.

78 M. G . Evans and M. Polanyi, Trans. Furaduy SOC., 1939,35, 178.


‘reagents approach’ and ‘products separate’ are divided by a vertical line. Polanyi and Schreiber 31 define this boundary by the equation (rAB - rs,AB) = (rBC - re,,,), where ro is the equilibrium internuclear separation of the specified molecule, although a better definition where # ro,Bc may be (rAB/re,AB) = (rBc/r,,Bc).

The potential energy profile in Figure *a) is of the ‘early downhill’ variety: most of the energy is released as the reactants approach. This inclines the system to attractive energy release 77 and high vibrational excitation of AB. In the case represented by 4(b) however, energy is released much later along the reaction path. The resultant repulsive energy release 77 between the separating products leads to comparatively modest internal excitation of AB.

Features of the collision dynamics have just been described in terms of diagrams that show how V varies along a single co-ordinate describing progress along a minimum energy route. Even in collinear collisions in which the system only just surmounts the potential barrier, inertial effects on the ‘downhill run’ ensure that the representative point will not follow this path. Nevertheless, the results of full three-dimensional Monte Car10 studies can frequently be understood - at least, by hindsight - by reference to simplified representations. A further example of this is provided when one considers how the dynamics of reactive collisions are altered by changing the relative masses of the atoms A, B, and C.

In order to interpret these mass effects, it is useful to consider the dynamics of collinear collisions in terms of the motion of a single sliding m a s point on a properly modified potential s~rface.~’ For this analogy to be correct, the equation for the kinetic energy must be diagonalized so that, for example, it takes the form

T = *(22 + Y2) (43)

this representing the motion of a point of unit mass in a new Cartesian frame. The co-ordinates X and Y are related to rm and rBC by

Y = ,u/*rBc cos p,

and so the modified potential surface is constructed by skewing the re, axis away from the Y-axis by an angle 8 which is defined by the equation

and by scaling the r,, and rBc axis by multiplying them by ,u* and ,d* (the square roots of the reduced masses of A + BC and AB + C), respectively.

Figure 5 illustrates the effect of the above procedure on the same rectilinear surface for two very different mass combinations. When mA< m, and m,, the mass point is scarcely diverted from its original path, and little energy is released before it reaches the head of the narrow elongated exit valley. There the trajectory turns sharply and runs down the exit valley. Energy is released mainly as repulsion between the products and after AB has essentially formed; this causes relatively modest vibrational excitation. The surface and dynamics are quite different when nz, and m, >> m,. Now the entry valley is long and narrow, whilst the exit valley is broad and falls steeply near its head. After crossing the barrier, the sliding mass


0.4

(a)

0.3 rBclnm

0.2

0.1 0.2 0.3 r*B I nm

I I I I I 0.1 0.2 0.3

rA6 / nm

01 0.2 03

Figure 5 Diogram illustrating the efect of skewing and scaling the potential surface for collinear collisions. The rectilinear surface (a) was constructed using the LEPS equation (42) with S2 = 0 and spectroscopic parameters for H, C1, C1. The other surfaces show the transformations that occur when (b) m, = 1 a.m.u, mB = mc = 35 a.m.u, and (c) m, = mB = 35 a.m.u, mc = 1 a.m.u. The diagnostic trajectories on these swfaces, run with zero vibrational energy and just sufficient translational energy to surmount the barrier, show much greater mixed energy release in (c) than in (b) and consequently greater vibrational excitation [ f,# = 85% in (c) as against f u * = 58% in (b)]

has time to curve away from its original path and a good deal of energy is released before r,, becomes equal to This constitutes mixed energy release 77 and channels energy into AB’s vibration.

The above two cases indicate how different mass combinations can give quite different reaction dynamics on the same potential. In fact, these examples represent opposite extremes of behaviour, the former corresponding to the light atom anomaly that was referred to earlier. Thorough investigations of mass effects for exothermic reactions has led to the general conclusion that the degree of vibrational excitation in the product of a direct atom-transfer process reflects how much attractive plus mixed energy release occurs. A limit to this correlation appears as the potential is made increasingly attractive. At first this leads to secondary encounters, that is A B having formed, B (or A) strikes atom C one or more times before the products finally separate. Systems where B is much the lightest atom are particularly prone to this. The effect tends to broaden the vibrational energy distribution. As the


potential is made still more attractive, a ‘well’ or ‘basin’ can form on the surface, leading eventually to the formation of genuine collision complexes and a reduction of specificity in energy disposal and of selectivity in energy requirements.

It is now clear that the way in which energy is disposed of or used in reactions depends crucially on the location of the maximum or crest in the energy barrier, although other effects, such as the steepness of the potential energy profile 7 7 * 7 9 9 8 0

and the curvature of the reaction path in the X- Y plane, may also influence the reaction dynamics. The relationships between barrier location and the overall energetics 82 and collision dynamics 83-86 of reactions have now been systematically studied. Using LEPS and BEBO potentials for related series of reactions, Mok and Polanyi 82 found that the maximum in the potential energy profile moved to a progressively earlier position on the reaction path as the exothermicity increased. As far as they were able to test, the crest of the (low) potential barrier for substantially exothermic reactions was always positioned in the entry valley on the surface.

Direct trajectory studies of the selective consumption of reactant energy are of more recent origin than those concerned with specific excitation of products. The calculations of Perry, Polanyi, and Wilson36*37 to check on the accuracy of the detailed rate coefficients obtained by application of equation (27) were mentioned earlier. Polanyi and Wong,*j and later Hodgson and P ~ l a n y i , ~ ~ have examined the energy requirements for reactions on hypersurfaces that differed in the location of the energy barrier. With the crest displaced into the entry valley, relative translational energy was much more effective in promoting reaction than vibrational excitation. However, when the barrier was shifted into the exit valley the situation was reversed ; vibrational energy was more effective than translational energy. These general conclusions were essentially independent of the relative masses of A, B and C. Furthermore, it has been shown that they can be carried over to four-centre reactions of the type AB + CD -+ AC + BD.84

The surfaces used in Polanyi and Wong’s calculations were for thermoneutral reactions, but since the system has no way of anticipating the shape of the surface past the maximum in the potential energy profile, the results reflect the form of the potential up to and including the crest of the barrier. Together with the results of Mok and Polanyi’s investigation 82 this strengthens the conclusion that substantially endothermic, direct, reactions will usually be promoted preferentially by selective excitation of the reactants’ vibration, whereas substantially exothermic reactions, if they have an appreciable activation energy, are more likely to be enhanced by relative translational energy, rather than vibrational excitation.

Reactions that are thermoneutral, or nearly so, form an important category for two reasons. The first is that atom-transfer reactions of this type frequently have activation energies which, although they are appreciable, are less than the energy

’* G. Miller and J. C. Light, J . Chem. Phys., 1971, 54, 1635. G. Miller and J. C. Light, J . Chem. Phys., 1971, 54, 1643. J. W. Duff and D. G. Truhlar, J . Chem. Phys., 1975, 62,2477.

82 M. H. Mok and J. C. Polanyi, J. Cham. Phys., 1%9,51, 1451. 8s J. C. Polanyi and W. H. Wong, J . Chem. Phys., 1969,51, 1439. 84 M. H. Mok and J. C. Polanyi, J . Chem. Phys., 1970,53,4588. 85 B. A. Hodgson and J. C. Polanyi, J . Chem. Phys., 1971,55, 4745. 86 D. S. Perry and J. C. Polanyi, Cunad. J . Chem., 1972, 9, 3916.


acquired by a molecule of reactant excited in a fundamental vibrational transition. Consequently, quite large enhancements of the reaction rate might be observed in relatively simple, direct experiments. A second reason for interest is that efficient vibrational relaxation may occur as a result of either reactive or non-reactive collisions of the types represented in equation (41), and this could be important in a number of chemical lasers.

In Polanyi and Wong’s calculations the potential barriers were artificially displaced into the exit or entry valley on the potential surface. A more reasonable position for the barrier crest in thermoneutral reactions is at a point approximately halfway along the reaction path. Several trajectory studies have now been made of systems where A and C are equivalent so that the barrier is symmetrically placed with its crest at a point where r,, = rBc.

was similar in spirit to those of Polanyi’s group. Calculations were carried out for different mass combinations: (i) m, = rn, = rn, = 1 a.m.u.; (ii) m,, = rn, = 1 a.m.u., m, = 35 a.m.u.; (iii) rn, = rn, = 35 a.m.u., m, = 1 a.m.u. The potentials were ‘equivalent’, in that the barrier to transfer of the B atoms was, in each case, half of the (classical) energy of BC(v = 1). Not surprisingly, the effectiveness of vibrational excitation in promoting reaction was less than that on surfaces with a ‘late’ barrier, but greater than that on surfaces with an ‘early’ barrier.

The results for different mass combinations differed in two interesting ways. First, vibrational energy promoted reaction most effectively in case (iii) and least effectively for case (ii). Secondly, the tendency for multiple crossing of the surface rAe = rBC (in four-dimensional space) was much greater with BC(v = 1) than with BC(v = 0) and was greatest for case (iii), l a s t for (ii). Both of these effects can be understood in terms of the dynamics on the diagnostic skewed surfaces for collinear collisions; in each of the three cases the scaling of the rAB and rBc axes is the same. The skewing angle increases in the order (ii), (i), (iii). For case (iii),

is 80.4” and the initial vibrational motion of BC is almost at right-angles to the line rAB = rBC. Consequently, it is especially effective at carrying the mass point over this line but, at the same time, the system is particularly prone to being reflected back off the further wall of the exit valley.

Smith and Wood’s work also showed that both non-reactive and reactive processes, e.g. channels (a) and (b) in equation (41), could remove vibrationally excited molecules in electronically adiabatic collisions. However, the non-reactive contri bu- tion came almost entirely from trajectories which crossed the surface r,, = rBc an even number of times, so that the motions of the three-atom system became strongly coupled. The product vibrational distributions both from these non- reactive trajectories and from reactive collisions were broad, showing that multiquantum transfers, i.e. (v - v’) > 1, are probable. The majority of trajectories did not, of course, cross r,, = r,, and the transfer of a substantial amount of vibrational energy in these collisions was extremely rare. These general findings have been confirmed in a study modelled on the system Br + HBr 8s with the potential chosen to have a barrier to H atom transfer of N 16 kJ mol-’ and one

Smith and Wood‘s investigation

I. W. M. Smith and P. M. Wood, Mol. Phys., 1973,2!5,441. 88 J. M. White and D. L. Thompson, J. Chern. Phys., 1974, 61, 719.


on F + HF collisions 89 with the corresponding height equal to 22.6 kJ mol-l. The criterion that the ‘effective’ collisions are those in which the trajectory crosses TAB = rec at least once has recently been shown to hold even when the potential has a symmetrically placed ‘well’ rather than a barrier, although then the total cross-sections for reaction and energy transfer are larger.90

ThompSon91*92 has carried out calculations on F + H F collisions, using a potential with a barrier of 105 kJ mo1-I to exchange of the H atom. In spite of this barrier being much higher than the internal energy of an HF molecule in v = 1, vibrational relaxation was found to be quite rapid, certainly it was more efficient than would be expected in collisions involving atoms of the same masses but intermolecular forces typified by the surface in Figure 3(a). This indicates that even if the energy barrier is too high for reaction to occur in potentially reactive collisions, the coupling between the internal and relative translational motions may be sufliciently strong to facilitate energy transfer.

Thompsong3 has also carried out trajectory calculations on H + FH. Once again the potential had a high barrier 75 (1 17 kJ mol-’) to transfer of the ‘central’, F, atom and again vibrational energy transfer was found to be moderately fast. In this case, however, the interpretation is less clear, since it seems possible that collisions in which H atoms strike one another may be the most effective. The surface for linear HHF will be like that shown in Figure 3(c). Strong coupling between the motions should occur if the trajectory ‘turns the corner’ in the collision path of minimum energy. Furthermore, the mass combination in collisions of this type with the light attacking atom impinging on the light end of the diatomic molecule is particularly favourable for energy transfer.

The possibility of efficient vibrational relaxation, as distinct from chemical reaction, in systems where the potential allows for an endothermic reaction has been explored explicitly by Douglas et aL20 and by the Using somewhat different surfaces to represent the Br + HCl interaction they reached slightly different conclusions. The potential of Douglas ef al. gave a better energy distribution among the products of the exothermic reaction but an overall rate constant that was appreciably less than that measured e ~ p e r i m e n t a l l y . ~ ~ ~ ~ ~ Smith’s more attractive surface reproduced the total rate of reaction better, but appears to channel too much energy into the vibration of product HCl. In the endothermic reaction, the calculated increase as w’ was raised from 2 to 4 was similar in both cases to the experimental observations, although again the absolute rate constants derived from Smith’s calculations were larger. Both studies show that HCI molecules in levels v’ > 2, as well as reacting, can be relaxed in collisions where reaction ‘nearly’ occurs. A good diagnostic test for these collisions was whether the trajectory crossed the surface defined by (rAB/re,AB) = (rBc/re,ec). With HCl(d = 2) and translational energies corresponding to room temperature thermal

R. L. Thommarson and G . C . Berend, Internat. J . Chem. Kinetics, 1974,6, 597. I. W. M. Smith, J.C.S. Faraahy ZI, 1975, 71, 1970. D. L. Thompson, J . Chem. Phys., 1972,57,4164.

@* D. L. Thompson, J. Chem. Phys., 1974,60,2200. * D. L. Thompson, J . Chem. Phys., 1972,57,4170. O4 I. W. M. Smith, to be published. *6 F. J. Wodarczyk and C. B. Moore, Chem. Phys. Letters, 1974,26,484.

K, Bergmann and C. B. Moore, J. Chem. Phys., 1975,63,643.


distributions, reaction and relaxation occurred at similar rates. For HCl(v = 3,4), Douglas et al. found that reaction became faster than relaxation whereas Smith found that the two rates remained comparable. From each set of calculations, it was concluded that the removal of HCI(v' = 1) in electronically adiabatic collisions with Br atoms is likely to be very slow.

Energy Transfer in Electronically Non-adiabatic Collisions.-It was pointed out earlier that in collisions involving either atoms with non-zero orbital angular momentum, i.e. with L > 0, or molecules with a non-zero axial component of this momentum, i.e. A > 0, multiple electronic energy states arise. The energy between these states will be a function of the relative positions of the nuclei and, at various points, the splitting between any two electronic states may be equal to the energy of a vibrational transition. The proposal that non-adiabatic transitions at, or close to, these points might provide a mechanism for efficient vibrational relaxation was first put forward by Nikitin97,98 to explain some anomalously rapid processes involving vibrationally excited NO. He has siiice applied this idea to other systems 99 and has recently written an excellent brief review loo relating the electronically adiabatic and non-adiabatic mechanisms for vibrational relaxation to one another.

Before dealing with electronically non-adiabatic processes, it is useful to consider the treatment of vibrational relaxation in encounters between species with closed electronic shells, say A('So) + BC(lCf).lol The collision dynamics will be controlled by a single electronic potential which can be represented by a surface like that shown in Figure 3(a). The dashed line on this surface shows the path of minimum energy and its curvature indicates the degree to which A perturbs the vibrational motion of BC. The probability that energy is transferred between vibration and relative translation (V-T) in diagnostic collinear collisions depends on the extent to which trajectories penetrate the region where the curvature of this path becomes appreciable.

In most of the numerous theoretical treatments of V-T energy transfer in 'non- chemical' collisions,1o2 a simplified intermolecular potential is assumed ; for example, angular anisotropy is usually ignored. Furthermore, the coupling between the relative translational motion of A and BC and the vibration of BC is assumed to be weak. In the theory of Schwartz, Slawsky, and H e r ~ f e l d , ~ ~ ~ . ~ ~ ~ for example, only collinear collisions are treated explicitly and A is assumed to interact only with B, so that the potential is

V ( x , X ) = Cexp[ -a(x - X)] where x and X are' the distances of A and B from the centre-of-mass of BC and a defines the steepness of the repulsion between A and B.

(46)

97 E. E. Nikitin, Optika i Spectroskopiya, 1960,9, 8 . S8 E. E. Nikitin, Optika i Spectroskopiya, 1961, 11, 452. 9* E. E. Nikitin and S. Ya. Umanski, Faraday Discuss. Chem. Soc., 1972,53, 1. loo E. E. Nikitin in ref. l(a), chap. 4. lo1 I. W. M. Smith, Accounts Chem. Res., 1976,9, 161. lo* D. Rapp and T. Kassal, Chem. Rev., 1969,69, 61. loS R. N . Schwartz, 2. I. Slawsky, and K. F. Herzfeld, J. Chem. Phys., 1952, 20, 1591. lo' K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves, Academic

Press, New York, 1959.


Following a first-order perturbation treatment in which terms depending on x and X are separated, the probability of BC being transferred between initial and final states i and f, can be written as

Hi,f ( v i b ) is an off-diagonal matrix element given by

It is then usual to assume that a-I is appreciably greater than the vibrational amplitude so that

a2

Hi’,f ( v l b ) .To (aX)vrf(X)~l(X)dX (49)

although the approximation is a particularly poor one for diatomic hybrides. The second matrix element,

a,

H i , f ( t r ) = 1 - oo c ~XP( -ax)Ff(x)Fi(x)h (50)

can be regarded as the ‘overlap’ of the final and initial translational state wave- functions, Ff(x) and F,(x) - for which pr = h/Af = fikf and pi = h / l , = tik, - on the intermolecular potential. If parallel curves, representing the potential C exp( -ax) for different vibrational levels within the same electranic state, are drawn, as in Figure 6, it can be seen that vibrational transitions require a horizontal ‘tunnelling’ process. HIf (lr) is extremely small, because the classical turning points associated with relative translational motion on the two curves are well displaced from one another and also because pf and p , , and therefore Af and A‘, are very different.

For collision energies within the ‘thermal’ range, evaluation of equation (47) leads to the following general predictions: (i) Pi,f is very small, especially where the duration of the collision is much longer than the vibrational period; (ii) the probabilities of molecules losing two or more quanta are very much smaller than those for Aw = 1 processes; (iii) for harmonic oscillators and, to a good approximation, for real molecules, Pv,v-l = V P ~ , ~ . To compare the theoretical results with experimental data, averaging over collision energy, impact parameter, orientation, etc., has to be carried out. Although this can only be done approximately, the resultant prediction that the thermally averaged probability ‘per collision’ of relaxing a molecule from ZI = 1 to zt = 0 varies with temperature according to the relationship

has been confirmed in many systems. The basis to Nikitin’s theory of vibrational energy transfer in electronically

non-adiabatic collisions is that any degeneracy, or near degeneracy, associated with spin-orbit terms in an isolated atom or molecule is removed as another species interacts with it in a collision. In several respects, his treatment parallels that outlined above for electronically adiabatic collisions. In particular, it is assumed that the two (or more) intermolecular potentials of concern are not orientation

logp a T-+ (51)

Reactive and Inelastic Collfswns involving Molecules 31

x- Figure 6 Potential curves representing the interaction of dyerent vibronic states of a

molecule with a collision partner whose close approach causes the electronic states j and k , which are nearly degenerate when x + 00, to diverge. Vibrational relaxation in electronically adiabatic collisions requires ‘tunnelling’ between parallel curves as indicated by the horizontal arrows. Electronically non-adiabatic collisions can lead to relaxation via transitions at the crossing points, i.e. at xo for (j, 1) + (k, 0), as indicated by the broken arrows. (Based on figures given in papers by Nikitin 97-100)

dependent and that, in each of the states represented by these potentials, the coupling between the vibrational motion of the excited molecule and its translation relative to that of the collision partner is weak.

With these approximations, it is possible to represent the intermolecular electronic states simply by potential curves, V(x), which diverge as x decreases. Furthermore, each vibronic state can be represented by curves which are parallel for the vibrational levels within each electronic state. Examples of these curves are shown in Figure 6, it being assumed that the interaction in each electronic state has an exponential repulsive form, but that the coefficients in the exponents are different for each.

Where the intermolecular potentials are of the kind shown in Figure 6, the expression for the transition probability, again includes a vibrational matrix element of a similar form to that in equation (48). In the electronically adiabatic processes that were considered earlier, transition probabilities are extremely small, because energy transfer requires a tunnelling process to carry the system between the parallel curves representing neighbouring vibronic states within the same electronic manifold. Now, however, curves ‘cross’ and H,,f(tr) is replaced by a term that expresses the probability that, in an effective two-body collision initially controlled by curve (j,l) in Figure 6, the system crosses on to curve (k,O).

Calculating the probability of an electronically non-adiabatic transition in two- particle collisions is the problem first treated by Landau and Zener and described in many If the coupling between the two electronic states is reasonably strong, the resultant transition probability can be much greater than that in elec-


tronically adiabatic collisions. In two respects, the dependence of PI,E on u and Av, Nikitin's theory makes similar predictions to theories of electronically adiabatic vibrational relaxation. In non-adiabatic collisions multiquantum transitions are less likely than Av = 1 processes, first because H;,f(v,b) decreases rapidly as Av increases, secondly because there will be a higher energy threshold to processes with Av > 1, as it will be necessary for the electronic states to be split to a greater extent. Nikitin has suggested that thermal rate constants can be evaluated using either a simple collision treatment or a transition-state theory approach. The temperature dependence of these rate constants is likely to be determined by the forms of the intermolecular potentials and, in particular, whether an appreciable collision energy is required to reach the regions where the vibronic states cross.

4 Experimental Measurements d their Interpretation In this section, results that have been obtained fromdirect kineticstudies of processes involving vibrationally excited molecules are considered in relation to the factors that can determine the dynamics of molecular collisions. The number of investigations that could be mentioned in a review of this kind is already rather large. However, in order to be able to discuss some systems in reasonable depth, selective, rather than comprehensive, coverage of the literature is provided. The selection concentrates on systems where reasonably accurate kinetic data have been measured and on examples that illustrate the range of detailed processes that can occur when molecules in vibrationally excited states collide with potentially reactive species.

Vibrationally Excited Hydrogen.--Systems involving hydrogen occupy a special place in the hearts and minds of theoretically inclined kineticists, since they offer the best opportunity of correlating experimentally determined data with results from truly ab initio calculations. This is, of course, especially true of the HJ system and its isotopic variants.'05 Furthermore, because the vibrational level spacing in Hz is unusually wide, dramatic changes in kinetic behaviour might be anticipated for even quite low levels of excitation.

To excite H2 vibrationally by photochemical means, the Raman effect must be used. Although the stimulated Raman process has been used for this purpose in studies of vibrational energy transfer from Hz (and Dz) to a variety of collision partners,'06 and for an investigation of the effect of vibrational excitation on the molecular isotope exchange rea~tion,'~'

H Z + Dz + 2HD (52)

(which will be referred to later), this method has not been applied to the study of collisions between H2 (v > 0) and potentially reactive atoms. Experiments on these systems have, however, been performed using discharge-flow systems.

When H2, either pure or diluted in He or Ar, is passed through the cavity of a

lo5 D. G. Truhlar and R. E. Wyatt, Ann. Rev. Phys. Chem., 1976,27, in press. Io1 M. M. Audibert, C. Joffrin, and J. Ducuing, J. Chem. Phys., 1974,61, 4357, and references

therein. S. H. Bauer, D. M. Lederman, E. L. Resfer, jun., and E. R. Fisher, Internut. J. Chem. Kinetics, 1973, 5, 93.


microwave discharge, as well as H atom production there is a sinall yield (- 1 %) of vibrationally excited H2. With such a single-discharge system, Heidner and Kasper lo8 have determined the rate constant for deactivation of H2(v = 1) by H atoms, using isothermal calorimetry to measure the concentration of atoms and vacuum U.V. absorption to determine [H2(v = 111. The kinetics were complicated by the surface-catalysed removal (at different rates) of H2(v = 1) and H, but Heidner and Kasper's careful analysis took account of these effects and yielded the rate constant given in Table 2.

Recently, Gershenzon and Rozenshtein log have reported the results of a similar study on D + D2 (v = l), although in their experiments [D] was determined by titration with Not, and [D2(v = 111 was estimated by observing i.r. emission from C02 which was added just upstream of an i.r. detector and was excited by vibrational-vibrational (V-V) energy transfer from Dz. Their results are also given in Table 2.

Interpretation in the HB and D3 systems is not complicated by the possibility of an electronically non-adiabatic mechanism for the removal of the vibrationally excited molecules. Heidner and Kasper suggested that it was likely that relaxation occurs predominantly in reactive collisions, i.e. by

rather than

This proposal was in accord with the results of classical trajectory calculations 110-87

on approximate potential hypersurfaces, although Smith and Wood pointed out that as v increases, there is an increasing tendency for the system having once crossed the surface in phase space defined by rAB = rec to recross, so that a significant proportion of these collisions in which the motions couple strongly, and appreciable energy is transferred, end up by being non-reactive.

Very recently a classical trajectory investigation of H2(C) + H and D2(v) + D collisions has been carried out ll1 on the Yates-Lester hypersurface which is a parameterized best fit to the accurate ab initio surface of Liu '13 for collinear H3. The results of these calculations, listed in Table 2, are not in very good agreement with the experimental results, particularly those for D2( v= 1) + D. It seems unlikely that the rate constants for reaction of molecules in levels with v > 1 will be greatly underestimated by the classical trajectory methods as quantum tunnelling effects will be less than for reactants in v = 0. A possible explanation of the discrepancy is that the classical trajectory method fails to provide an adequate estimate of the vibrational transition probability associated with those unreactive collisions - which for v = 1 amount to a high proportion of the whole sample - in which the motions do not couple strongly and classically only very small amounts of energy (much less than the vibrational transition energy) are transferred from molecular vibration to other degrees of freedom.

HA + HBHC(V = 1) 4 H,H,(v = 0) + H,

€I, + H,Hc(v = 1) HA + H,H,( v = 0)

(53a)

(53b)

108 R. F. Heidner, tert. and J. V. V. Kasper, Chem. Phys. Letters, 1972, 15, 179. Yu. M. Genshenzon and V. B. Rozenshtein, Doklady Phys. Chem., 1975,221,644.

1l0 M. Karplus and I. Wang, unpublished results quoted in ref. 108. 111 I. W. M. Smith, to be published. 11* A. C. Yates and W. A. Lester, jun., Chem. Phys. Letters, 1974, 24, 305.

B. Liu, J. Chem. Phys., 1973, 58, 1925.

Tabl

e 2

Com

paris

on o

f exp

erim

enta

l and

cal

cula

ted

rate

con

stan

ts (l

o'*

k/cm

3 mol

ecul

e-' s

') fo

r re

laxa

tion

of H

~v

)

by H

and

of

D2(

v) b

y D

CalC

.'"

300

600

H2(

8 =

1) +

H +

H2(

v =

0) + H

0.

3( f0

.15)

-

OM

( f0

.013

) 1.

1 ( f0

.3)

H~

(v

= 2

) + H

--+ H

~(w

=

1,0

) + H

H2(

v =

3) +

H +

H~

(v

= 2

,1,0

) + H

D2(

u =

1) +

D +

Dz(

v =

0) +

D 0.

12( f

0.05

) 0.w

f0.1

) 0.

0038

( fO.

OO1)

0.22

( f0.

05)

-

-

-

-

1.7(

f0.4

), 1

.O( f

0.3)

8.q

f1.6

), 5

.5(

f1.3

),4.

7( f

1.2)

12( f

2), 5

.5(

f1.5

)

15( f

3.9,

15(

f3.

5),7

.6( f2.5)

Q

D~

(w

= 2

) + D

--+ D

~(u

=

1,0

) + D

-

-

0.09

3( rt

0.02

), 0.060(

f0.0

2)

1.3(

f0.2

), 0

.70(

f0.2

2)

g D

~(v

=

3) + D

+ D

~(w

=

2,1

,0) +

D -

-

0.56

( f0

.17)

,0.5

6( f0

.17)

, 0.5

2( f0

.16)

4.

2( f1

.5),

5.2(

f1.3

), 2.6( f0.9)

3 a

The

rat

e con

stan

t quo

ted

at 6

00 K

is o

btai

ned

by e

xtra

pola

tion f

rom

thos

e re

port

ed in

ref

. 109

for

300 <

T <

520 K.

s 3


The data in Table 2 emphasize that the rate constants associated with the reactive and inelastic processes removing molecules from excited vibrational levels may increase rapidly with v if the system is potentially reactive. This has an important consequence. If experimental measurements are made under conditions where V-V energy exchange is much faster than the overall relaxation of vibrational energy, only the total loss of vibrational quanta is observed. In a system, such as H + H2 (v), where chemical reaction cannot be distinguished, the rate of loss of quanta is equal to zz (v - v’)kV,JX2(v)] [XI, where k,,,, is the detailed rate constant for

If the values of k,,,, increase sharply with v, the rate constants extracted from the experiments will only correspond to kl,o if the degree of vibrational excitation is very small, i.e. if [x2(v = l)] >[X2(v > l)]. This possibility should always be borne in mind when the results of experiments on potentiallyptive systems are being evaluated.

Discharge-flow experiments have also been carried out to investigate the effect of vibrational excitation of reactant H2 on the reactions*

v v /

X&) + x --f X,(v/) + x (54)

H + HX c Hz(v’) + X ( = Cl or Br) AH: = -3.9 or -69.5 kJ mol-’ (55a,b)

For X = C1, Stedman et al.li4 estimated that the rate constant for H2(v’ = 1) is about two orders of magnitude greater than the thermal rate constant. However, their finding that the ‘enhanced‘ production of HCl was unaffected when the distance between the discharge producing H2t and the C12 inlet was altered is inconsistent with Heidner and Kasper’s observation of relaxation of H2t by H atoms, as these authors have pointed out.io8

Sims et ~ 1 . ~ ~ ~ have carried out trajectory calculations on the endothermic reactions (-55a) and (-55b), and on the analogous reaction with X = I. These also cast some doubt on the interpretation of Stedman’s experiments, since they indicate a rate enhancement for Hz(v’ = 1) five times less than that deduced from the earlier experiments. In addition to their calculations, Sims and his colleagues performed experiments similar to those of Stedman et aZ.li4 but with X = Br and the temperature of the reactor raised to 712 K. They found that the presence of vibrationally excited H, enhanced the rate of HBr production, and attributed this to an increase in rate of reaction (-55b) of ca. 7 x lo3 as H P is raised from v’ = 0 to w’ = 1.

It should perhaps be emphasized that quantitative interpretation of both Stedman’s and Sims’ experiments is not straightforward. Furthermore, in the Br + H2(v’) system, it is again not clear that reaction of molecules with v‘ > 1 can be entirely discounted. For reaction between Br and HJv’ = 1) there remains an activation energy of -29.8 kJ mol-l, so a further large enhancement in k(lu’; T) * Chemical equations are written with the products of the exothermic reaction on the right-hand

side and AH” is given for reaction in this direction. In the text, endothermic reactions are referred to by a negative number; e.g. reaction ( -55a) is that between H, and C1.

11‘ D. H. Stedman, D. Steffenson, and H. Niki, Chern. Phys. Letters, 1970,7, 173. 116 L. B. Sims, L. R. Dosser, and P. S. Wilson, Chern. Phys. Letters, 1975,32, 150. 11( J. H. Birely, J. V. V. Kasper, F. Hai, and L. A. Darnton, Chern. Phys. Letters, 1975, 31,

220.

36 Gas Kinetics and .Energy Transfer

is to be expected as v‘ increases from 1 to 2. Consequently, and in spite of the relatively low level of vibrational excitation, it may be that reaction of Hz(v’ = 2) contributes significantly to the observed enhancement of reaction rate.

Experiments of a similar kind to those already described have been carried out 116

on the reaction :

This nearly thermoneutral reaction is slow at room temperature (kWs6 = 2.7 x an3 molecule-’ s-’); although the activation energy is high (42 kJ mol-I), it is less than the excitation energy of Hz in v’ = 1 (49.8 kJ mol-l). Birely et were unable to detect any evidence for enhanced reaction with H2 containing - 1.0 x lOI4 molecule in v’ = 1 (i.e. -0.2%), and they concluded that k(ld = 1 ; T ) <

The wide spacing of rotational energy levels in HZ and its existence in ortho andpara forms provides an almost unique opportunity to study the effect of reagent rotational state on total reaction rates and on the disposal of energy among reaction products. There have been three studies in which results for the reactions of normal and para Hz with F atoms [see equation (22)] have been compared. Using the reaction between F and DZ as a reference reaction, Klein and Persky lL7 found the thermal rate constant for the reaction of p - Hz to be only 2.5 % greater than that for reaction of normal Hz at 175 K, and no difference could be distinguished at 237 or 298 K. Chemical laser and i.r. chemiluminescence techniques have been used to contrast the state distributions of HF formed in these reactions. Douglas and Polanyi ‘I8 find that reaction of H2(u = 0, J = 1) give rise to less vibrational excitation of the HF product than reaction of molecules in either J = 0 or J = 2. This finding confirms Coombe and Pimentel’s ‘19 earlier observation that reaction of p - Hz increases the specific excitation of the HF vibration.

H + HO t Hz + 0, AH: = -7.9 kJ mol-’ (56)

cm’j molecule-’ s-’ at T = 300 K.

Vibratiody &cited Hydrogen Halides-Because it is relatively easy to construct pulsed chemical lasers which emit lines in the (1,O) bands of €IF, DF, HCl, DC1, HBr, and DBr, collisional processes involving these molecules in their (u = 1) states have already been investigated quite extensively. An unusually elegant experiment has been carried out by Odiorne, Brooks, and Kasper.120 They compared the production of KCI in the slightly endothermic reaction

H + ClK t HCl + K, AH: = -4 kJ mol-I (57)

in a ‘conventional’ crossed-beam experiment, with its formation when the HC1 beam was irradiated with an HCl chemical laser. Under the conditions of their experiment, the effective reaction cross-section was increased from s(v’ = 0) = 0.15 A’ to s(v’ = 1) - 20 A’. Recently, Brooks and his colleagues1z1 have also determined the variation of the reaction cross-section for HCl(u = 0) with collision energy in the range 8.8-50.6 kJ mol-’. The excitation function rises

11‘ F. S. Klein and A. Pensky, J. Chem. Phys., 1974,61, 2472. 118 D. J. Douglas and I. C. Polanyi, Chem. Phys., 1976,16, 1. 11* R. D. Coombe and G. C, Pimeatel, J. Chem. Phys., 1973,59, 1535. ln0 T. J. Odiorne, P. R. Brooks, and J. V. V. Kasper, J. Chem. Phys., 1971,55, 1980.

J. G. Pruett, F. R. Grabiner, and P. R. Brooks, J. Chem. Phys., 1974,60,3335; (b) J. G. Pruett, F. R Grabiner, and P. R. Brooks, J, Chem. Phys., 1975,63, 1173.


from a threshold energy at 8 kJ mol-1 in a form similar to that predicted by the familiar ‘line-of-centres’ collision model. However, the reaction cross-section at a value of the collision energy that corresponds to the excitation energy of HCl(v’ = 1) remains about an order of magnitude less than %(v‘ = 1). These experiments constitute a beautiful, direct, and almost unique study of the relative importance of translational and vibrational excitation of the reactants in promoting chemical reaction.

The rate constants that are listed in Table 3 have all been determined in ‘bulk’, i.e. non-beam, experiments. Most of these results have been obtained using the technique of laser-induced vibrational fluorescence that was alluded to in Section 1. In experiments of this kind with radical atoms present, it is necessary for the fluorescence cell to form part of a discharge-flow system which includes some means of monitoring the concentration of the atoms. In their simplest form the total rate of loss of the excited molecules is found by observing the decay of the fluorescence, but this measurement does not distinguish between non-reactive and reactive removal, i.e. between processes (3a) and (3b).

The data obtained for systems involving H and D atoms will be considered first, since, for all these cases, the mechanism responsible for removal of the vibrationally excited molecules must be electronically adiabatic. Furthermore, the reactions of HF(v = 1) or DF(v = 1) with H or D to yield F atoms and H2, HD, or D2 are too endothermic to play any significant role. Consequently, when H atoms remove HF(v = l), the process may be written as

H + HF(v = 1) + H t HF(v = 0) (58)

although there may still be doubt as to whether the (indistinguishable) F atom- transfer reaction is important.

Unfortunately, there are substantial discrepancies between the rate constants that have been determined for the process represented by equation (58). The two direct studies using laser-induced fluorescence differ more in their interpretation than in the actual results. The analysis is complicated by relatively rapid V-V energy exchange, i.e.

H ~ ( v = 1) + HF(w = 0) # H~(v’ = 0) + HF(v’ = l), AE, = -2.4 kJ mol-l (59)

for which k-,9 = 6.2 x cm3 molecule-’s-l at room temperature. In the absence of atoms, the fluorescence signal decays rapidly to a level associated with equilibrium according to equation (59) and then more slowly, as vibrational energy is dissipated into otter degrees of freedom. With the discharge on the H2 line switched on, Quigley and Wolgalz2 detected no change in either of these decay times and deduced the upper limit of kss < 1.5 x 10-14cm3 molecule-ls-l. However, when the H2 is partly dissociated, not only can the atoms deactivate both HF(u = 1) and H2(w = l), but also the rates of other relaxation processes and the position of the V-V equilibrium are changed. Heidner and Bott lZ3 also saw no change in the faster relaxation time when H atoms were present but, under their conditions, the longer relaxation was decreased. However, the value of kss that

lea G. P. Quigley and G. J. Wolga, Chem. Phys. Letters, 1974,27, 276. laa R. F. Heidner, tert. and J. F. Bott, J. Chem. Phys., 1975, 63, 1810.

w

00

Tabl

e 3

Rat

e co

nsta

nts

(101

3k/c

m3 m

olec

ule-

' s-

l) for

rem

oval

of

vibr

atio

nally

exci

ted

hyho

gen

and

deut

eriu

m h

alid

es b

y re

activ

e at

oms a

t 298

K

Exc

ited

mol

ecul

e

HF(

u =

1)

DF

(v =

1)

HC

I(V

= 2

)

HC

l(v =

1)

DC

l(v =

1)

HBr

(v =

1)

H

5 12

K

O.15

2.3

( f0.7)

1.1

(f0.

3)

-

-

65 ( f2

2)

-

-

76 ( f2

3)

-

18 (

f5)

-

-

D

-

-

0.3

( f0

.27)

< 1.

3 -

-

-

- I

108 (

33)

-

20 ( f

7)

-

-

Br

-

310

(k15

0)

-

-

18 ( f3)

2.8

( fO.

5)

2.7

( f0.7)

4.8 (

f0.9)

2.6

( ItO

.5)

-

9.4

( 5t1

.8)

-

16 ( f7)

2.4

( f0.6)

0

-

31 (

f6)

c

-

41 (

f3

)

8.6

( f1.

4)

37 ( f1

2)

10 (

f2)

35 (

f2

) -

13 (

f4)

-

-

Ref

.

a 122,

127

123

123

176

8,17

6

125,

18

9, 1

368,

134

10,1

35

Q E

124

9,13

6,13

4

124

C d

3 a M

. A. Kwok

and R. L

. Wilk

ins,

J. C

hern

. Phy

s., 1

974,

60,2

189;

b

R. G

. Mac

dona

ld an

d C.

B. M

oore

, to

be p

ublis

hed;

c

R. J

. D

onov

an, D

. H

usai

n,

and

C. D

. Ste

vens

on, T

rans

. Far

aday

SO

C., 1

970,

66,2

148;

d

Z. K

arny

and

B.

Kat

z, C

km

. Phy

s., 1

976,

14,

295.

f


they deduced was based on a fuller analysis of the HF(v = 1) decay at short times, in particular allowing for the decrease in the rate of V-V exchange when the hydrogen is substantially dissociated. At present it appears that their value for kS8, along with their rate constants for the other isotopic variations of this process, should be preferred.

There remains the question of what detailed mechanism causes relaxation to occur at the rates determined by Heidner and Bott. If the ab initio calculations 75

on the HFH potential are correct, then F atom transfer can certainly be ignored and only Thompson’s trajectory calculations 74 are relevant. In spite of using a potential function yielding a high barrier obstructing the transfer of the F atom, Thompson found quite rapid relaxation. Extrapolated to room temperature, his results yield kS8 - 4 x cm3 molecule-I s-l, in fair agreement with Heidner and Bott’s experimental value. Unfortunately, the details of the trajectories that led to energy being transferred were not identified. One possibility is that this happens in collisions where the ‘attacking’ atom approaches the H or D end of the molecule so that the dynamics are controlled by a potential of the type shown in Figure 3(a). Although the vibrational excitation is insufficient to promote reaction, it may enable the system to reach a region of the potential hypersurface where the motions become strongly coupled and energy transfer is facilitated. In this configuration the two light atoms impinge on one another, providing a favourable mass combination for energy transfer, and the higher excitation energy of HF(v = l), compared with DF(v = l), could carry the HF system more nearly to reaction, and hence explain why HF(v = 1) relaxes more rapidly with H and D than DF(v = 1).

Heidner and Bott 124 have also measured the total removal rates of HCl(v = 1) and DCl(v = 1) with H and D atoms. Here, the interpretation of the experimental observations appears more straightforward, but the mechanistic explanation of the results is no more clearcut than in the HF/DF + H/D systems. The excitation energy of HCl(v = l), and probably of DCl(v = 1) as well, may be sufficient to promote both the H atom and C1 atom transfer reactions, i.e.

H + HCl(v = 1) -+ Hz + C1

H + ClH(v = 1) 3 HCl(v = 0) + H The total rate constant, ksO, obtained by Heidner and Bott agrees very well with the earlier value of Arnoldi and W o l f r ~ m . ~ ~ ~ They detected no decrease in the H atom concentration following excitation of the HCl and therefore concluded that reaction (ma) does not occur to any appreciable extent. Unfortunately, the activation energy of reaction (60b) is not established with any certainty.126 The rapid relaxation rates that have been observed and the differences between HCl(v = 1) + H/D, on the one hand, and DCl(v = 1) + D/H, on the other, are consistent with relaxation via this process, with a low, but significant, barrier to the transfer of the C1 atom. However, it is certainly too early to eliminate other possibilities.

l*‘ J. F. Bott and R. F. Heidner, tert., J. Chem. Phys., 1976, 64, 1544. ISs D. Arnoldi and J. W o l f m , Chem. Phys. Letters, 1974, 24, 234.

(a) R. F. Heidner, tert. and J. F. Bott, J. Chem. Phys., 1976,64, 2267; (b) D. L. Thompson, H. H. Suzukawa, jun., and L. M. Raff, ibid.. p. 2269, and references therein.

40 Gas Kinetics and &rgy Transfer

The remainder of the processes for which rate constants are tabulated involve atoms in P states, and therefore an electronically non-adiabatic mechanism for relaxation becomes possible. This appears to be the likeliest route for relaxation of HF(v = 1) by F, Br, and 0 atoms. The HF(v -- 1) + Br system is especially interesting, since the (1,O) vibrational transition energy in HF and the spin-orbit splitting in Br(2P) are nearly resonant. As a result relaxation may occur by vibrational-electronic (V-E) energy transfer, i.e.

HF(v = 1) + Br(2P$ 7+ HF(v = 0) + Br(Zp3), AEo = -3.30 kJ mol-I (61)

Quigley and Wolga 127 observed a double exponential decay in the laser-induced vibrational fluorescence from HF(v = l), which is consistent with this equilibrium being established and then relaxing by slower processes. From an analysis of the fast decay, they derived the value k6’ = 3.1( f1.5) x cm3 molecule-’ s-’.

The reverse process has been studied by Wodarczyk and Sackett.12* They produced Br(2P+) atoms by laser photolysis of Br2 and observed the i.r. fluorescence from either the excited atoms or HF(v = 1). Under their conditions, the back transfer of energy from HF(v = 1) to Br(’P+)) was unimportant and their analysis gave k-61 = 3.4 (~0.6) x lo-” cm3 molecule-’ s-’.

Application of the law of detailed balancing to reaction (61) yields

kb iIk-6 1 = (214) exp( -AEoIRT) (62) where the factor (2/4) arises from the degeneracies of the atomic states. It is therefore clear that Wodarczyk and Sackett’s measurements lead to a value of ksl about twice that determined by Quigley and Wolga. If HF(v = 1) was also relaxed without excitation of the Br atoms this would only widen the discrepancy. Wodarczyk and Sackett argue convincingly, on the basis both of their experimental measurements and comparison with other systems, that Br(2P+) quenching leads to HF vibrational excitation at close to 100% efficiency. If this is so, the reason for the discrepancy may be connected with Quigley and Wolga’s assumption, in the analysis of the rapidly decaying portion of their fluorescence signals, that k s , w k-61. Where rotational equilibrium is established, and this seems likely in their experiments, equation (62), yielding ksl = 1.88 k-61, determines the ratio of the rate constants for reactions in the forward and reverse directions. However, without knowing the experimental conditions in detail, it is impossible to estimate how large an effect Quigley and Wolga’s assumption might have on their result.

Using pulsed chemical lasers to excite HCl and DCI, systems have been investigated where chemical reaction is thermoneutral

C1‘ + HCl” -N Cl’H + C1” (63)

(64) nearly so

or substantially endothermic

Cl + HO t HCI + 0, AH: = -3.8 W mol-1

Cl + HBr t HCl + Br, AH: = -65.6 kJ mol-1 (65)

lt7 G. P. Quigley and G. J. Wolga, J. Chem. Phys., 1975,62,4560. la* F. J. Wodarczyk and P. B. Wkett, Chem. Phys., 1976, 12,65.


The rate constant for relaxation of HCl(w = 1) by C1 atoms at 298 K has been measured in three different laboratories I3-l0 and by two different techniques I3 and now seems firmly established. The process is rapid and, as mentioned earlier, is probably crucial in limiting the performance of chemical lasers based on the H2-C12 chain reaction. The detailed mechanism for this rapid relaxation process is not, however, established. "he nature of the potential for Cl--H--Cl is un- certain, since there is experimental evidence to suggest the presence of a potential 'well' for the symmetrical linear configuration lZ9 which seems incompatible with kinetic observations 130 that suggest an activation energy for reaction (63) of 27.5 kJ mol-I. Trajectory calculations indicate that if the relaxation of HCl(o = 1) and DCl( o= 1) occurs via-reaction (63) then the barrier to C1 atom transfer is 5 10 kJ mol-I.

The experimental rate constants for removal of HCl( v = 1) by 0 atoms are not in good agreement. Since oxygen atoms are probably the easiest of all free radicals to 'handle' in flow systems this is rather surprising, but the discrepancy remains unresolved. The thermal reaction ( -64) has an activation energy of 25 kJ mo1-1,131 whilst that for the equivalent reaction involving DCl is 29 kJ These values compare with excitation energies of 34.5 kJ mol-l for HCl( o = 1) and 24.7 kJ mol-' for DCl(v = 1). Initially, Arnoldi and Wolfrum 125 suggested that HCl(v = 1) was removed mainly by reaction ( -64) induced by the vibrational excitation. This conclusion was based on the observation of a decrease in the NO2 afterglow following laser excitation of HCl(o = 1) in the presence of 0 atoms and small amounts of NO. However, more recent experiments in their laboratory have altered this view.133 Brown et all3* argued for a predominantly non-reactive mechanism and relaxation in electronically non-adiabatic collisions, because DCl(o = 1) was relaxed more rapidly than HCl(o = l), whereas one would expect HCl(v = 1) to react substantially faster than DCl(w = l), as a result of a kinetic isotope effect, which would be normal in direction (Le. k , > k,,) and probably enhanced in magnitude because the difference in energy between HCl(o = 1) and DCl(w = 1) is greater than that between HCl(v = 0) and DCl(o = 0), both being measured relative to the minimum in the potential curve. This view now appears to be supported both by Wolfrum's recent measurements and by experiments carried out by Karny et ~ 1 . l ~ ~ They observed formation of C1 atoms by resonance fluorescence measurements in the vacuum U.V. but estimated that only N 10% of the total removal of HCI(o = 1) occurred via reaction ( -64). The rate constant for total removal of HCl(v = 2) is apparently about five times the value for HCl(v = 1). This increase is probably associated with the enhanced probability for chemical reaction. Some of the results for the Br + HCl(o') system were discussed earlier. The rate

constants for relaxation of HCl(o = 1) that are given in Table 3 refer to the process

lz9 P. N. Noble and G. C. Pimentel, J. Chem. Phys., 1968,49, 3165. lS0 F. S. Klein, A. Persky, and R. E. Weston, jun., J. Chem. Phys., 1964,41, 1799. lS1 R. D. H. Brown and I. W. M. Smith, Internut. J . Chenr. Kinetics, 1975, 7, 301. la2 R. D. H. Brown and I. W. M. Smith, unpublished results.

J. Wolfrum, private communication. la' R. D. H. Brown, G. P. Glass, and I. W. M. Smith, Chem. Phys. Letters, 1975,32, 517. 185 Z. Karny, B. Katz, and A. Szoke, Chem. Phys. Letters, 1975,35, 100.


HCl(v = 1) + Br(2P+) HCl(w = 0) + Br(2P+) (66)

Reaction ( -65) remains too endothermic to contribute appreciably to the observed rate of removal of HCI(v = 1) at room temperature. The V-E transfer of energy from HCl(v = 1) to Br(2P4), i.e.

HCI(v = 0) + Br(2PS) f- HCl(v = 1) + Br(lP+), AEo = -9.56 kJ mol-' (67)

is endothermic, but it can play a significant role in removing HCI(w = l), if the excited atoms are quenched rapidly. This was the situation in Brown et aZ.3 experi- m e n t ~ , ' ~ ~ where relatively large concentrations of 0 2 were present, and their observed rate constant has been reduced to k66 using value for k-67 based on the directly measured value of k67.137 Trajectory calculations 94 indicate that energy transfer is unlikely to occur at the observed rate in electronically adiabatic collisions. Therefore, HCI(v = 1) is probably removed predominantly in collisions in which the system undergoes a transition between the hypersurfaces corresponding to the two doublet electronic states that correlate with Br(ZP3) + HCl('C+).

The rate of the electronically non-adiabatic process which appears to remove HCI(v = 1) should approximately double when HCl is raised to v = 2. In addition, the endothermic reaction (-65) becomes energetically possible and results that confirm that this reaction takes place 17-20 are referred to in Section 1. However, there does appear to be a discrepancy between the various kinetic results that have been obtained for

C1 + HBr(v = 0) Ft HCl(v' = 2) + Br

Moore and his colleaguesi3* have measured the thermal rate constant for the C1 + HBr reaction, ke5, at 295 K as 7.4 x cm3 molecule-' s-'. Combining this with the measurements of Douglas et ~ 1 . ~ ~ 9 ~ ~ on the relative yields of HCl product in different vibrational levels yields :

kse = k(v' = 21. = 0; T) = 1.5 x

(68)

cm3 molecule-'s-' (69)

Application of equation (1 8), using well-known thermodynamic and spectroscopic parameters, yields

k-68 = k(u = 01.' = 2; T) = 4.0 x cm3 rnolec~le-~ s-l (70) but this rate constant is more than twice that obtained by Leone et aZ.I7 from measurements of the total removal of HCl(v = 2) by Br using the laser-induced vibrational fluorescence method.

Studies based on the observation of i.r. chemiluminescence from reactions where the reagents are prepared as uncollimated beams or molecular 'sprays' have provided most of the little available information about how selective excitation of the reactants alters the specificity of energy disposal. In experiments in Polanyi's laboratory 1399140 the spectra emitted by the reaction products are observed as a

R. D. H. Brown, I. W. M. Smith, and S. W. J. Van der Merwe, Chem. Phys., 1976,14, 143. S. R. Leone and F. J. Wodmczyk, J. Chern. Phys., 1974,60, 314.

13* (a) F. J. Wodarczyk and C. B. Moore, Chem. Phys. Letters, 1974,26,484; (b) K. Bergmann and C. B. Moore, J . Chem. Phys., 1975,63, 643.

la@ L. J. Kirsch and J. C. Polanyi, J . Chem. Phys., 1972, 57, 4498. A. M. G. Ding, L. J. Kirsch, D. S. Perry, J. C. Polanyi, and J. L. Schreiber, Furuduy Discuss. Chem. Soc., 1913,55,252.


function of the temperature of each reagent in turn. Only the average collision energy is changed by altering the temperature of the atomic reagent, whilst the influence of internal excitation - and especially reactant vibration - can be studied by changing the source temperature of the molecular reagent and subtracting out the effect of translational excitation.

The exothermic reaction

F + HCl(v) + HF(v’) + C1, AH: = -135.9 kJ mo1-I

has been studied in this way.13gs140 The major findings are that as v increases from 0 to 1, the total rate constant increases by ca. 3.7 times and that the extra vibrational excitation energy of the reactant is transformed efficiently into extra vibrational excitation of the product.

Relaxation via the Formation of Collision Complexes.-For all of the bimolecular, A + BC(v), systems that have been considered up to this point, it has been implicitly assumed that the observed results are determined by the outcome of direct collisions. However, a number of systems have now been studied where ABC is known as a stable triatomic molecule, so that the potential for the lowest electronic state of the system has a deep ‘well’ of the type shown on surfaces (d) and (e) in Figure 3. When the dynamics of molecular collisions are controlled by such a potential, the encounters are no longer direct but collision complexes form. This means that the three atoms usually remain together for a time that is characteristic of at least several vibrational periods, all ‘memory’ of the initial distribution of energy is lost, and the eventual sharing of energy among the degrees of freedom of the products is largely determined by statistical factors and the conservation laws. Because more phase space is associated with states of relative translation than with vibrational levels at the same energy, in the absence of any high barrier in the exit channel, the vibrational state distribution of molecules produced by break-up of a collision complex falls monotonically with increasing vibrational quantum nurnber.l4I Clearly then, if there is no potential barrier preventing the formation of collision complexes, this type of mechanism can provide a highly effective means for vibrational relaxation.

The results in Table 4 are subdivided, according to whether the bound electronic ground state of the triatomic species is directly accessible to the colliding species by a ‘spin-allowed’ route or not. Where both the atomic and diatomic species possess unpaired electrons, as with the systems in part (a) of Table 4, formation of an ABC collision complex is generally allowed. Of course, the ground state may be only one of several electronic states that correlate with the states of the separated species; for example, the ‘A ground state of 0 3 is only one of the 27 states that arise from the interaction of O(3P2,1,0) with 02(3C;). Although each bound state may contribute to relaxation via complex formation, their role will be less than that of the ground state due to smaller state density at energies close to the dissociation limit. For the systems in part (b) of Table 4, the triatomic molecular ground states, i.e. COz (‘X;) and N2Q (lZt), do not correlate directly with the ground states of the collision partners, i.e. with CO (lZt) + 0 (”) and

141 R. A. Marcus, J. Chem. Phys., 1975,62, 1372.


Table 4 Rate constants for removal of vibrationally excited molecules (AB) by atoms (C) where ABC is stable AB+C 1011k/cm3 molecule-’ s- ’ TlK Ref.

(a) Formation of ABC is ‘spin-allowed’ OH(U = 1,2) + H 27,33 295 142 OH(u = 1,2) + NO 7.5,12 295 142 NO(u = 1,2) + 0 3.7,4.0 - 2700 145 NO(W = 1) + C1 2.8 N 1700 145 0 2 (u = 1) + 0 0.8 -2Ooo 150,151 CN(u = 1 4 ) + 0 2.1 298 153 CN(u = 7) + 0 10 298 153

co (v = 1) + 0 0.4-2.7 18004000 155 Nz ( W = 1) + 0 3.2 x 10-44.0 x 300-723 156 N2 (U = I ) + 0 -42 x 10-2 12004000 157,158

(b) Formation of ABC is ‘spin-forbidden’

Nz (‘EL) + 0 (”). This reduces the rate constant associated with the formation of collision complexes and as a result other mechanisms for vibrational relaxation may become important.

The rates of vibrational relaxation of OH radicals by H atoms and by NO have beem determined very recently by Spencer and Excited OH radicals were produced in a fast flow system by means of the rapid reaction

H + NO2 -+ OH(u’) + NO, AH: = -123.5 kJ mol-l (72)

changes in the concentrations of OH (v’ = 0, 1, and 2) were observed under different experimental conditions, using quantitative e.p.r. spectrometry. The rapid deactivation of OH? by H and by NO, and the relative insensitivity of the rate constants for relaxation to v’ is consistent with mechanisms involving the initial formation of HOHT and HONOt collision complexes. Further evidence concerning the role of collision complexes may be sought by comparing the observed rate constants for relaxation with those for either the corresponding radical association reaction or, in some cases, an isotope exchange reaction. Thus the rate constants for relaxation of OH (v’ = 1 and 2) by NO are similar to the second-order rate constant for association of OH with NO in the limit of high pressure.143 Likewise the rate constants for relaxation of OH (w’ = 1 and 2) by H are close to that for the reaction 144

D + OH + O D + H

It therefore seems reasonable to propose a similar mechanism involving complex formation in each of these cases.

have made a quantitative comparison of their experimentally observed rate constants for relaxation of NO (u = 1 and 2) by 0 and Cl atoms with those predicted assuming that complex formation provides the major pathway.146 The basis of their calculations was the ‘statistical adiabatic channel model’

(73)

Glanzer and Troe

us J. E. Spencer and G. P. Glass, Chem. Phys., 1976,15, 35. R. Atkinson, D. A. Hansen, and J. N. Pitts, jun., J. Chem. Phys., 1975,62, 3284. J. 1. Margitan, F. Kaufman, and J. G. Anderson, Chem. Phys. Letters, 1975,34, 485. K. Glanzer and J. Trw, J. Chem. Phys., 1975,63,4352. M. Quack and S. Troe, Ber. Bunsengesellschaft phys. Chem., 1975,79, 170.

Reactive and Ine Iastic Collisions involving Molecules 45

that was originally developed to treat unimolecular dissociation and radical recombination reactions.147 The agreement of the computed rates and the experimental results within a factor of 2 is very satisfactory and confirms the importance of complex formation in the relaxation mechanism in these systems. Their calculations also emphasize the importance of vibrational transitions in which v - v' = Av > 1, where relaxation proceeds via complete formation. Thus, with both 0 and C1 atoms, they estimate that NO(v = 2) is transferred directly to v = 0 about 1.5 times faster than to v = 1.

There are no measurements at low temperatures of the relaxation rates for low-lying levels of O2 by O(3P). The rate of strong coupling collisions can be estimated from two pieces of kinetics data: (i) the rate constant for the isotope exchange reaction,

is given by k a 4 = 6.5 x exp( -550/T)cm3 molecule-1 s-' at 298 < T < 402 K,148 and (ii) the rate constant for association of 0 and O2 in the limit of high

(74) 1 8 0 + 1 6 0 1 6 0 + 1 8 0 1 6 0 + 1 6 0

pressure, 0 + 0 2 ( + M ) +O,(+M) (75)

is k;' = 1.7 x cm3 molecule-' s" at 298 K.149 One would therefore expect relaxation of 02(v) by 0 via formation of O3 collision complexes to proceed with a rate constant of - 1-2 x

Measurements in shock tubes 150*151 yield a relaxation time for 0 2 by 0 atoms that is essentially independent of temperature between 1000 and 3500K and corresponds to a rate constant of -0.8 x cm3 molecule-1 s-' at 2000 K. In addition some results have been obtained for O2 (14 Q v Q 19) in room- temperature flash photolysis The interpretation of these latter experiments was not straightforward, but Webster and Bair lS2 were able to determine a rate constant of -3 x cm3 molecule-i s-' for the sum of all processes removing molecules from the v = 19 level. The interpretation of these experimental results must, at present, be tentative. However, there seems little doubt that the association-dissociation mechanism alone provides for efficient relaxation. At higher temperatures, or for higher vibrational levels of 0 2 , additional reactive mechanisms over surfaces with moderate barriers to atom-exchange may also contribute to the overall rate of relaxation. At present there does not appear to be any compelling evidence to suggest that electronically non-adiabatic collisions are important.99

The reaction between O(P3) atoms and CN(X2Z+, v = 1-7) has been studied by Wolfrum and co-workers using a combination of discharge-flow and flash photolysis techniques. C2N2 could be mixed with atomic oxygen in a flow system since their reaction is slow. Photolysis of C2Nz then produced CN radicals in

cm3 molecule-' s-' at room temperature.

u7 M. Quack and J. Troe, Ber. Bunsengesellschaft phys. Chem., 1974,78, 240. 148 S. Jaffe and F. S. Klein, Trans. Faraday SOC., 1966,62, 3135.

J. Troe and H. Gg. Wagner, Ann. Rev. Phys. Chem., 1972,23, 31 1. 150 J. H. Kiefer and R. W. Lutz, 11th Symposium on Combustion, 1967, p. 67. lS1 J. E. Breen, R. B. Quy, and G. P. Glass, J. Chem. Phys., 1973, 59, 556. 152 H. Webster, tert. and E. J. Bair, J. Chem. Phys., 1972, 56, 6104. 153 H. Schacke, K. J. Schmatjko, and J. Wolfrum, Ber. Bunsengesellschaftphys. Cheni., 1973,77,

248.


states up to v = 7 and the removal of these species was observed by kinetic spectroscopy. Interpretation of their results, which are shown in Table 4, is complicated by the existence of different reactive pathways. Formation of NCO in its 211: ground state is spin-allowed but this route would lead to N(2D) + CO('Z+) as products. Alternatively, reaction may proceed across a quartet hypersurface and yield N(4S) + CO('C+). Schmatjko and Wolfrum lS4 have investigated these two different electronically adiabatic mechanisms by means of classical trajectories. Their conclusion is that the second mechanism is better able to explain the observed facts, although this conclusion is likely to be sensitive to the details of the surfaces chosen to represent the two adiabatic potentials. The increase in reaction rate between CN(v) and 0 when v = 7 has been attributed to the opening up of the reaction channel leading to NO + C.ls3

The rate constant for association of CO with O(3P) atoms in the limit of high pressure,

is given by 149 kq6 = 2.7 x exp( - 1475/T) cm3 molecule-1 s-l, the low value of the pre-exponential factor in this rate expression reflecting the need for the total spin to change if COa('E+) is to be formed. A quite different Arrhenius expression has been obtained

o(3~) + CO(+M) +CO,(+M) (76)

for the isotope exchange reaction,

(77) 1 8 0 + 12C160 + 12C180 + 1 6 0

Between 298 and 398 K, k,, = 1.03 x lo-'* exp( -3500/T) cm3 molecule-1 s-l, which suggests that there is a spin-allowed route for C atom transfer with only a moderate potential barrier on this adiabatic surface. If this is correct it seems likely that the spin-allowed, direct, reactive mechanism will be the most important of these two in collisions involving vibrationally excited CO, particularly at high temperatures.

The only measurements on relaxation of CO by 0 atoms that have been reported come from shock tube experiment^.'^^ At 2000K, the rate constant is 6.8 x cm3 molecule-' s-' and increases with increasing temperature. It seems that these observations are consistent with a reactive, i.e. C atom transfer, mechanism, but once again further measurements over a wider range of temperature and for several excited vibrational states are required, before the relative importance of different possible mechanisms can be established with any certainty.

Measurements on the relaxation of vibrationally excited Nz by atomic oxygen cover a very wide range of temperature^.^^^-^^^ As the results in Table 4 show, the rate constant increases by about two orders of magnitude as the temperature is raised from 300 to 4OOO K. Association of O(jP) atoms with N2 has never been observed. Not only is the pathway to NzO ('X+) spin-forbidden, but also Troe and Wagner 149 suggest that the triplet and singlet hypersurfaces cross one another at anenergy morethan 100 kJ mol-l above theenergy of separated NZ(lCg+) + O(3P).

15' K. J. Schmatjko and J. Wolfrum, Ber. Bunsengesellschaft phys. Chem., 1975, 79, 696. 155 R. E. Center, J . Chem. Phys., 1973, 58, 5230. 156 R. J. McNeal, M. E. Whitson, jun., and G. R. Cook, J . Geophys. Res., 1974,79, 1527. 15' W. D. Breshears and P. F. Bird, J . Chem. Phys., 1968, 48,4768. 158 D. J. Eckstrom, J . Chem. Phys. , 1973, 59, 2787.


This certainly seems to preclude formation of N2O('C+) in collisions between Nz(w) and O(T) - at least, for low values of u and moderate temperatures. An alternative is that energy transfer occurs as a result of electronically adiabatic collisions on the triplet hypersurface across which the reaction

N(4S) + N0(211) Nz('Z:,+) + O(3P), AH," = -314.6 kJ mol-I (78)

takes place. However, reaction between N$X:g+) + q 3 P ) is so endothermic that there appears little likelihood of any significant proportion of collisions reaching regions of the potential surface where vibrational and translational motions become strongly coupled. The most acceptable interpretation seems to be that relaxation is caused by electronically non-adiabatic transitions between the Z3A' + 3A" states that correlate with N2('Zf) + O(3P).

Vibrationally Excited Ozone.-Ozone can be promoted to its (001) vibrational level as a result of chance coincidences between lines in the (001,OOO) fundamental band of O3 and lines from a C02 laser, the strongest overlap being with the P(30) line in the 9.6 ,urn band. This has been the basis for a number of experimental studies of the enhanced reactions of 0,t with N0,159-164 02(1Ag),161 and SO.'65

The reaction between NO and O3 is especially interesting since it can proceed by two distinct routes 166,167

NO + O3 -+ NOz(A2B1) + 02,

NO + Os 3 NOz(%A1) + 02,

AH," = -20.1 kJ mol-'

AH: = -199.2 kJ mol-'

(79a)

(79b)

The Arrhenius expression for the thermal rate constants for these reactions are: k79a = 1.3 x exp( -2100/T)cm3 molecule" s-' and k79,, = 7.1 x exp( - 1 170/T) cm3 molecule- s- l . The chemiluminescence from NOI,(A2Bl) provides an extra means of observing what happens when Os is excited vibrationally by absorption of a C02 laser pulse. In addition to this, the effect on reactions (79a) and (79b) of exciting the NO reactant can be observed, since NO can be excited vibrationally with radiation from either a frequency-doubled COz laser 168 or a

The enhancement of reactions (79a) and (79b) when 0 3 is vibrationally excited was first observed by Gordon and Lin 159 using a repetitively pulsed C02 laser. Since that time they have extended their measurements.16o With the low 0 3 concentrations and high laser powers in their latest experiments, Gordon and Lin believe that equilibration of the initial vibrational excitation is unimportant so lS9 R. J. Gordon and M. C. Lin, Chem. Phys. Letters, 1973,22, 262. le0 R. J. Gordon and M. C. Lin, J . Chem. Phys., 1976,64, 1058.

M . J. Kurylo, W. Braun, A. Kaldor, S. M. Freund, and R. P. Wayne, J. Phofochem., 1974, 3, 71.

Ie2 W. Braun, M. J. Kurylo, A. Kaldor, and R. P. Wayne, J. Chem. Phys., 1974,61,461. 169 M. J. Kurylo, W. Braun, C. N. Xuan, and A. Kaldor, J . Chem. Phys., 1975, 62, 2065; J .

Chem. Phys., 1975,63,1042. le4 S . M. Freund and J. C. Stephenson, Chem. Phys. Letters, 1976,41, 157. 165 A. Kaldor, W. Braun, and M. J. Kurylo, J. Chem. Phys., 1974,61,2496. lSe M. A. A. Clyne, B. A. Thrush, and R. P. Wayne, Trans. Faraday Soc., 1964,60, 359. 16' P. N. Clough and B. A. Thrush, Trans. Faraday SOC., 1967,63,915. lB8 J. C. Stephenson, J. Chem. Phys., 1973, 59, 1523. lB9 P. A. Bonczyk, Chem. Phys. Letters, 1973, 18, 147.

co iaser.169

(48 Gas Kinetics and Energy Transfer

that the observed enhancement of reaction is due entirely to the reaction of molecules in the (001) level. This view is supported by Freund and S t e ~ h e n s 0 n . l ~ ~ Based on their observations of the time dependence of NO2 emission following pulsed excitation of 03, Gordon and Lin have reported values for the rate constants for reaction between NO and 03(001) at 308K of kJ9, = 7.1 x 10-’5cm3 molecule-’ s-I and k;9b = 9.0 x cm3 molecule-’ s-’, these representing increases of 6 and 5.7 over the corresponding thermal rate constants. The main uncertainty in the interpretation of the experimental results appears to be the role played by non-reactive, energy transferring, collisions between 0 3 (001) and NO that could occur in parallel to reactions (79a’) and (79b’).

The reactions between NO and vibrationally excited 0 3 have also been studied by Braun and his colleague^,^^^'^^^ but under somewhat different conditions from those in Gordon and Lin’s experiments. In their experiments, the ozone is excited by the square wave modulated output from a cw C02 laser. Although they also observed enhanced reaction, their conclusions appear to differ in detail from those of Gordon and Lin. They believe that under their conditions, rapid V-V equilibration among the O3 modes precedes reaction. Their results can then either be interpreted on the basis that all these modes are equally active, in which case at 300 K the ratio of the enhanced rate constant for (79a) to the thermal rate constant is 4.2, or by assuming that only the vt and v3 stretching modes effectively promote reaction when the ratio of rate constants becomes 16.4. Kurylo et aZ.163 prefer the former of these two explanations.

Although the rates of reactions (79a) and (79b) are increased by vibrational excitation of the 0 3 reagent, this enhancement is not pronounced. Gordon and Lin I 6 O have compared it with the effect of distributing the same amount of energy, 12.5 kJ mol-’, evenly among all the degrees of freedom of the reactants. This would be equivalent to raising the temperature to 575 K, which would produce a greater acceleration in the rate of reaction (79a) and about the same increase in k7gb as Gordon and Lin determined. This lack of specificity is consistent with a fairly early position of the potential barrier along the reaction co-ordinate for each of these reactions.

Very recently Stephenson and Freund 170 have investigated the effect on reactions (79a) and (79b) of exciting NO to v = 1. This puts almost twice as much energy into the reactants per photon absorbed as excitation of O3 (Ool), but the energy is supplied to a bond that does not break during the reaction. Nevertheless, they found that the rates of both reactions (79a) and (79b) were increased and determined (k79,/k79n) = 4.7 (T;::) and (ky9b/k79b) < 18, where the ‘double primes’ signify reactions of N q v = 1). This, albeit mild, effect of vibrational excitation of the ‘unchanged bond’ may be related to the (small) increase in the NO bond length that must occur during these reactions.

Although the reactions of 0 3 f with NO have received most attention, results have also been reported for the vibrational enhancement of the reactions

0 + 202 c O3 + O2(*AS), AH: = -5.9 kJ mol-I (80)

and SO + Os + SO$(lBl) + 0 2 , AH; = -92.0 kJ mol-I (81)

J. C. Stephenson and S. M. Freund, J. Chem. Phys., 1976,65, 1893.

Reactive and Inelastic Colliswns involving Molecules 49

The progress of reaction (80) when 0 3 was excited vibrationally was followed by resonance fluorescence detection of the q 3 P ) atoms that were formed as product. On the assumption that V-V equilibration was rapid and that the laser enhancement corresponded to reaction of O3 (OlO), Kurylo et aZ.161 deduced (k&eo) - 38 ( f20). If more highly excited states were responsible for the accelerated reaction rate this ratio would increase. With the same assumption regarding the v2 mode as active, Kaldor et ~ 2 Z . l ~ ~ found (k&/kel) = 2.5. In this case, as with the N0-03 reaction, chemiluminescence from an electronically excited product could be observed and was used to follow the kinetics of the enhanced reaction.

Four-centre Reactions.--The fact that the partially detailed rate constants associated with reaction from defined excited vibrational levels of a molecule may increase rapidly with increasing v, even where the excitation energies of all these states are greater than the activation energy for the thermal reaction has already been pointed out. Furthermore, it has been emphasized that if V-V relaxation is rapid compared with the rate of the process actually removing vibrationally excited species, then it may be difficult to identify the detailed process, i.e. specified with regard to v, that i s predominantly responsible for the observed results. It is to be anticipated that such effects will be especially large when the activation energy for reaction actually exceeds the excitation energy of one or more of the excited levels of a molecular reactant, This is the usual situation for four-centre reactions between molecules with closed electronic shells. Tht$e reactions, e.g.

AB + CD +AC + BD (82)

typically have much higher activation energies than the three-centre reactions involving at least one free radical that we have considered so far, although some exceptions to this general rule are k n 0 ~ n . l ' ~ The role that vibrational excitation of AB and CD can play in promoting reactions like (82) has been the subject of lively interest for several years and this topic will be reviewed briefly here.

Due to fourcentre reactions usually having high activation energies most experimental studies of their kinetics have been carried out in shock tubes. As early as 1964 Bauer and Re~ler,"~ in connection with experiments on the isotope exchange reactions between D2 and a series of H-containing molecules, proposed that these reactions occurred via a molecular mechanism which required the D2 reactant to be excited as far as some critical vibrational level. They used a single pulse shock tube (or a matched pair) with product analysis by mass spectrometry. Their interpretation was based on their observations of (a) the activation energies, which were sufficiently low to eliminate a radical mechanism resting on the initial dissociation of Dz, and (b) non-integral orders of reaction with respect to various components of the reaction mixture, which suggested that the bimolecular exchange reaction did not simply occur in collisions of high relative kinetic energy.

Quite a large number of shock tube studies of the kind carried out by Bauer and Resler have now been performed, both in Bauer's laboratory and elsewhere. Among

171 D. R. Herschbach, Faraday D~scuss. Chem. Soc., 1973,55,233. 179 S. H. Bauer and E. L. Resler, JU., Science, 1964,146, 1045.


these are experiments on isotope exchange between H2 + D2172'175, j 2 0 2 + 3602,176

and 12C180 + 13C180.177-178 In each case similar results to those described in the last paragraph have been obtained, confirming that reaction proceeds via a molecular mechanism which probably involves reactants that have reached some critical level of vibrational excitation.

Naturally, the exchange reaction between Hz and D2,

H2 + D2 d 2HD (83)

is especially interesting, since the results offer the best opportunity for detailed theoretical interpretation. This model system was one of those initially investigated by Bauer and Resler,s72 and further shock tube experiments have been carried out. 173 - 175 In addition, Bauer et aZ.179 have performed a series of novel experiments at room temperature in which either H2 or D2 in a high-pressure mixture of these gases was excited to v = 1, using the stimulated Raman effect. The extent of conversion into HD in this mixture after several pulsed laser shots was compared with that in an identical mixture which had been contained in a matched cell but had not been subjected to the stimulating radiation. Significant HD enrichments were found in the irradiated sample. This result was explained in terms of a molecular mechanism, in which the important features were rapid V-V energy exchange feeding molecules into levels above some critical state at which the molecular reaction becomes possible. These reactive processes were assumed to be:

However, it was not possible to rule out reactions involving H2 and D2 molecules both of which were excited, e.g.

Dz (V = 3) + Hz (W = 1) + 2HD (85)

or an atomic mechanism with dissociation induced by V-V exchange processes, such as

H2 (w = 5) + H2(v = 5 ) -+ Hz(w = 0) + H ~ ( v = 15) (86a)

and H2 (V = 15) -+ 2H (86b)

The second of these alternative mechanisms, involving large changes in the vibrational quantum numbers of both molecules, appears highly improbable,' but detailed calculations by Poulsen 179 suggest that processes like (85) are important in determining the overall rate of exchange.

There remains one important point of controversy in regard to the experiments on reaction (83) and their interpretation. The shock tube experiments determine

17* S. H. Bauer and E. Ossa, J . Chem. Phys., 1966,4S, 434. 17* R. D. Kern, jun., and G. G. Nika, J. Phys. Chem., 1971,75, 1615. 175 R. D. Kern, jm., and G. G. Nika, J. Phys. Chem., 1971,75,2541. 174 H. F. Carroll and S. H. Bauer, J. Amer. Chem. SOC., 1969, 91, 7727. lr7 A. Bar-Nun and A. Lifshitz, J . Chem. Phys., 1969,51, 1826.

A. F. Bopp, R. D. Kern, jun., and B. V. O'Grady, J . Phys. Chern., 1975,79, 1483. 17* L. L. Poulsen, J . Chem. Phys., 1970,53, 1987.


an energy of activation of 169 kJ mol-' for 1000 < T 6 3000 H owever, fairly extensive quantum mechanical calculations 180-182 of the H4 potential have failed to reveal a geometry which could serve as the transition state for reaction (83) with an energy below 456 kJ mol-'. It has been suggested 183 that this discrepancy may be resolved if the transition state has a T- or Y-shaped structure.

The reaction between hydrogen and iodine can serve as a final illustration of the effect of vibrational energy on the rate of a reaction which is somewhat more complex than an atom-transfer reaction involving only three atoms. For several decades, this reaction served as the textbook example of a fourantre metathesis, which proceeded in a single elementary step represented by

H2 + I2 2H1, AH: = -8.2 W mol-' (87)

Sullivan's classic experiments lS4 in the 1960's put an end to that, demonstrating conclusively that above 800 K the reaction proceeded via the normal hydrogen- halogen chain reaction, whereas below that temperature the following mechanism was operative

I2 * 21 (8W

21 + Hz Ft 2HI

Jaffe and Anderson Isti explored the energy requirements for reaction ( -88b) by looking for the products of reaction when accelerated beams of HI were directed into a long chamber containing DI at low pressure. The mean collision energy was thereby varied in the range 8 4 4 5 6 kJ mol-' but in no case could HD be detected in the products. This result indicated that for reaction to proceed some vibrational - or possibly rotational - excitation of the reactants is necessary. These reactions have also been the subject of theoretical studies 186-188 using classical trajectories, although the conclusions to be drawn from these calculations are a matter of some controversy.188

Ion-Molecule Reactions.-It is a sad fact of life that the field of gas-phase reaction kinetics has become so large that those subdivisions which deal, on the one hand, with reactions between neutral species and, on the other, with processes where one or both of the reactants are electrically charged have become almost completely divorced from one another. A small band of researchers do retain a foot in both camps, but reviewers whose knowledge extends equally over the two areas are rare. The present Reporter is, unfortunately, no exception to the general rule, but this final section is an attempt at least to draw attention to how much information about the influence of selective reactant excitation on reaction rates has been obtained through studies of ion-molecule systems. A recent review by Dubrin

H. Conroy and G. Malli, J. Chem. Phys., 1969, 50, 5049. ls1 C. W. Wilson, jun. and W. A. Goddard, jun., J. Chem. Phys., 1969,51,716. lS8 M. Rubinstein and I. Shavitt, J. Chem. Phys., 1969, 51, 2014. la* B. M. Gimrac, J. Chem. Phys., 1970,53, 1623. lS4 J. H. Sullivan, J . Chem. Phys., 1967, 46, 73 and references therein. lS6 S. B. Jat€e and J. B. Anderson, J . Chem. Phys., 1968,49,2859. 186 L. M. M, D. L. Thompson, L. B. Sims, and R. N. Porter, J . Chem. Phys., 1972,56, 5998.

J. M. Henry, J. B. Anderson, and R. L. JafTe, Chem. Phys. Letters, 1973, 20, 138. ls8 J. B. Anderson, J. M. Henry, and R. L. JafTe, J. Chem. Phys., 1974,60, 3725.


and Henchman provides an excellent introduction to the subject of ion-molecule reactions, and emphasizes those results which illuminate kinetic problems of general interest.

When it comes to performing experiments designed to probe molecular reaction dynamics, investigations af ion-molecule reactions possess several advantages over comparable studies of neutral-neutral systems. First, charged species can be detected at much lower concentrations than neutral molecules. Secondly, it is quite easy to accelerate charged species so that they all possess a uniform translational energy and this can be varied over wide limits. Paradoxically, it has been the lowest energy regime that has proved hardest to investigate by beam techniques, but this problem has now been largely solved by the use of merged beams. Finally, and of most importance from the viewpoint of this chapter, the distribution over vibrational states of the reactant molecular ions produced by photoionization can be varied by altering the wavelength of the vacuum U.V. photolysing radiation. Despite the low absorption cross-sections associated with direct and indirect photoionization, sufficient ions are produced fur detection.

The features just described have been combined in a series of experiments that have provided excellent detailed information about the energy requirements for a number of ion-molecule reactions. Chupka, who with his ca-workers has k n responsible for several of the most important of these experiments, has reviewed the results in detail.Ig0 Here, the results for only a few systems will be discussed explicitly and even these will be treated briefly. Before doing this, the factors that are generally considered to be important in controlling the dynamics of ion-molecule reactions are summarized so as to provide a context for subsequent discussion of the quoted experimental results.

The interactions between two species one or both of which carry a charge are characterized by much stronger, and longer range, attractive forces than those that exist between two neutral species. In the simple Langevin mode1 for ion-molecule reactions, this attraction is considered to arise through the interaction of the ion with the dipole induced by it in the neutral species, and the magnitude of this induced dipole is estimated from the experimentally determined isotropic polariza- bility. It is further assumed that this component in the intermolecular potential is sufficient to counterbalance any increase in the potential energy arising from ‘rearrangement of the chemical bonding’. Consequently, the rate of collisions in which the species come into ‘close’ contact is determined solely by centrifugal effects. The cross-section associated with these collisions varies as e-*, and exothermic reactions are assumed to take place at every close collision.

The essential validity of the above model for many ion-molecule reactions at thermal energies Igf is demonstrated by the magnitude of the reaction cross-sections and rate constants, and the variation of these quantities with collision energy and temperature, respectively. At low collision energies this model can then serve as a basis for calculating the rate of formation of collision complexes. In order to include the prediction of detailed kinetic data, however, the model must be

130 J. Dubrin and M. J. Henchman, in ref. 4, Chap. 7. loo W. A. Chupka, in ‘Ion-Molecule Reactions’, ed. J. L. Franklin, Plenum, New Yotk, 1972,

Chap. 3. lml M. J. Henchman, in ref. 190, Chap. 5.

Reactive and Inelastic Collisions involving Mdecules 53

extended. The simplest way to do this is to assume that the probability of a collision complex, of defined total energy and angular momentum, decomposing along a particular channel, is equal to the ratio of the phase-space volumes associated with (a) that individual channel, and (b) all channels that are accessible at that energy and angular momentum. This phase-space theory has been applied with some limited success to ion-molecule reactions.192 Unfortunately, its detailed application is most difficult for multiatomic systems for which statistical theories might be expected to work best.

The above, much simplified, treatment suggests the following effects of enhanced vibrational energy in reactive, ion-molecule, collisions at thermal energies. First, the rate of complex formation should be little affected. However, the relative importance of various breakdown channels could be altered. Thus, for an exothermic reaction, as the reactant excitation energy becomes appreciable relative to the exothermicity, the complex may increasingly dissociate back to reactants rather than forward to products, causing the reaction cross-section to decrease as the reactant vibrational excitation is increased. For endoergic reactions, a major effect should be a lowering in the translational energy threshold by an amount corresponding to the internal excitation energy. As reactant translational energy is increased the cross-section for complex formation will fall, but the Langevin model is not appropriate at high collision energies lgl (where many studies of ion-molecule reactions are carried out) as direct reaction mechanisms become increasingly important.

Particularly extensive data exist for the endothermic reaction between H: and He: 190,193

The experimental results, which are summarized in Table 5, show that vibrational excitation of Hl is much more effective than relative translational energy in promoting this reaction. This work is especially important because the system is

Table 5 Relative probability for the reaction H i + He HeH+ + H as a function of total reagent energy (translational and vibrational) and of the vibrational quantum number of the HZ (taken from ref. 190)

Relative probability" D' total energyb = 1.0 eV 2.0 eV 3.0 eV 4.0 eV 0 0.06 0.10 0.13 0.17 1 0.49 0.35 0.31 0.25 2 1.95 0.93 0.55 0.34 3 - 1.70 0.99 0.56 4 - 2.35 1.22 0.68 5 - 2.49 1.70 0.89

a These values have been corrected to take account of the dependence of the collision cross- section on relative translational energy E , assuming S a c-h b 1 eV = 96.5 kJ mol-'.

102 J. C. Light and J. Lin, J. Chem. Phys., 1965,43, 3209. 1'' W. A. Chupka and M. E. Russell, J . Chem. Phys., 1968,49, 5426.


electronically sufficiently simple to allow accurate calculations of the p0tentia1.I~~ These suggest a well depth for H2He+ of only - 10-20 kJ mol-1 relative to the asymptotic limit for Hf + He. This is much less than the collision energies in Chupka's experiments, so the influence of the well is small, reaction proceeds in direct collisions, and the model considered in the previous few paragraphs is inappropriate.

Kuntz and Whitton 19s have calculated three-dimensional classical trajectories for this system on a diatomics-in-molecules hypersurface196 that matches the ab initio surface 193 quite well. These potentials show the late barrier characteristic of endothermic reactions, and the effective use of vibrational excitation in overcoming this barrier is, of course, consistent with the findings of Polanyi and his co-workers 83*8s refmed to in Section 3. In addition, Kuntz and Whitton suggest that the acceleration of reaction (-89) as v' is increased is also caused by an enhancement in attraction between the reagents as the H i bond is extended. This influence declines to high collision energies as the reagents separate before this attraction can pull them together.

The effect of reactant vibrational excitation on reaction cross-sections and rate constants has also been studied for a number of exothermic, ion-molecule reactions. The reaction 197

Hz + Hf(v) -+ H: + H, AH: = -37 kJ mo1-I (90) provides an example. At low collision energies, the reaction cross-section falls gradually and monotonically as w is increased from 0 to 4, the value for H:(u = 4) being 0.79 times that for HZ(V = 0). These results are, at least qualitatively, consistent with the model described earlier: as v is increased, the rate of collisions in which complexes form may not change, but there is an increasing probability that the complex disintegrates back to reactants, thereby detracting from the reaction cross-section. At higher collision energies, Chupka et found evidence that the complex mechanism was replaced by one involving direct collisions, for which vibrational excitation improved the chance of reaction.

Similar results to those just described, with the reaction cross-section independent of, or declining slowly with, v, have been found in experiments at low collision energies on a number of exothermic ion-molecule reactions. These include reactions where a neutral molecular reagent is vibrationally excited, such as that between He+ and Nz(v),198 as well as those like HZ + Hz,197 NH3 + NH3,lg9 NH: + Hz0,199 and CHf + CH4?O0 where it is the ionic reagent that is excited internally. One exception to the general rule is provided by the reaction:lg8

O+ + Nt(v) +NO+ + N, AH," = -107.s kJ mol-* (91)

This important reaction is unusual in a number of ways; for example, its thermal

lS4 P. J. Brown and E. F. Hayes, J. Chem. Phys., 1971,55,922. lm6 P. J . Kuntz and W. N. Whitton, Chem. Phys. Letters, 1975,34, 340.

P. J. Kuntz, Chem. Phys. Letters, 1972, 16, 581. ln7 W. A. Chupka, M. E. Russell, and K. Refaey, J. Chem. Phys., 1968,48, 1518. lm8 A. L. Scbmeltekopf, E. E. Ferguson, and F. C. Fehsenfeld, J. Chem. Phys., 1968,48, 2966. lSm W. A. Chupka and M. E. RuSsel1,J. Chem. Phys., 1968,48, 1527. W. A. Chupka and J. krkowitz, 1. Chcm. Phys., 1971,54,4256.


rate constant at room temperature, kgl = 1.3 x 10-l2 cm3 molecule-l s-l, is roughly three orders of magnitude less than the value normally associated with exothermic ion-molecule reactions, and it has been shown that vibrational excitation of NZ is much more effective than translational excitation of the reagents in promoting this reaction.

Schmeltekpof et Q Z . ' ~ * suggested that the low-temperature rate constant would be small if reaction had to occur via a spin-forbidden, non-adiabatic transition from a 4Z- state (in C,, symmetry) to the ground state potential for NzO+. This view is supported by Kaufman.2O' The increase in rate constant with v is consistent with the presence of a barrier to the reaction in the product channel. In view of the 'normal' rate constants that are associated with reaction from levels with v 5 4, i.e. k(lv; T ) - 4 x cm3 molecule-' s-l, it is tempting to suppose that in these circumstances reaction proceeds adiabatically via the 4Z- potential with this surface having an unusually late barrier. Other possibilities, suggested by Kaufman,201 are that the reaction still proceeds by way of the state, but vibrational excitation of N, enables the system more easily to decompose to products via a second radiationless transition back to the 4X- surface, either directly or via a 411 intermediate. However, this mechanism is less appealing - at least to the Reporter - since it appears more difficult to explain the preferential effect of reactant vibrational excitation relative to translational excitation if reaction is supposed to take place via a strongly bound collision complex which is what N20+(X211J corresponds to.

The coverage provided in this section on ion-molecule reactions has been extremely restricted. However, it is hoped that it has been sufficient to demonstrate that the factors which have to be considered when attempting to explain or predict the effect of reagent vibrational excitation on a reaction rate are largely independent of whether or not one of the reactants is electrically charged.

5 snmmary The study of bimolecular, potentially reactive processes involving molecules in selected quantum states is in its infancy. Besides reviewing this early childhood, this article has had two principal objectives: firstly, to relate this emerging subject to other, better established areas in reaction dynamics, and secondly, to outline a theoretical and analytical context in which any experimental results should be considered. Some disagreements and discrepancies have been referred to, not from critical motives, but in the hope that it may help to stimulate fresh work. An attempt is now made to summarize briefly some of the general points that have been made in the main body of the article.

In the first part of Section 2, it was shown that detailed rate constants for reactions proceeding in opposite directions can be related quantitatively by application of the principle of microscopic reversibility. This provides a powerful method for deriving rate constants for endothermic reactions, detailed as regards reactant states - i.e. k(ln'; 7') - when the product state distribution from the reverse, exothermic reaction has been measured. In addition, these relationships

*O1 J. J. Kaufman, Adv. Chern. Phys., 1975,28, 113.

56 Gas Kinetics and Energy Transfer can be used to check experimental results where partially detailed rate constants have been determined directly for processes occurring in both the forward and reverse direction, as for reaction (68) on p. 42. When only partial information about the specificity of energy disposal or the selectivity of energy requirements in a particular reaction is available, the information-theoretic approach provides a means of predicting detailed rate constants by extrapolating surprisal plots, although this procedure should be used cautiously.

In Section 3, it was pointed out that for many potentially reactive systems more than one potential hypersurface can correlate with the electronic states of the separated collision partners and that this can provide a route for accelerated vibrational relaxation via electronically non-adiabatic collisions. Collisions of vibrationally excited molecules with atoms on a single reactive hypersurface can also lead to relaxation, as well as enhanced reaction. The problem of discriminating experimentally between the reactive and inelastic pathways for removal of excited molecules was also discussed. It is, of course, a quite different problem from that which arises, for example, in i.r. chemiluminescence measurements of product state distributions. There one must eliminate relaxation of the original distribution in collisions following that in which the excited products are formed. In studies of reactant states, however, relaxation can occur parallel with reaction, rather than subsequent to it. The branching ratio for reaction and energy transfer can only be obtained by quantitatively relating the concentration of chemical product to that of the excited reagent.

The method of classical trajectories indicates that vibrational excitation of the molecular reagent in a three-atom system will have the greatest effect in promoting reaction, the further the crest in the energy profile along the reaction path of minimum energy is displaced towards the products. Since, as another general rule, the barrier moves to later positions along this path as reactions become successively less exothermic, thermoneutral, and, finally, more endothermic, vibrational excitation is expected to promote endothermic reactions most selectively. Any fuxn theoretical prediction for particular systems is, however, frequently inhibited by the absence of even limited information about the details of the intermolecular potential.

Finally, we return to the point that unless V-V energy exchange is eliminated, experimental measurements are likely to provide a rate constant which describes the net result of removal of molecules from more than one excited level. If the state-specified rate constants increase rapidly with v then the observed rate constant may depend on the extent of the initial excitation. This problem is most likely to arise with reactive systems, since the rate constants for removal of molecules from successively higher levels can increase rapidly for such systems.

One prediction which can be confidently made is that collisions between stateselected species will be studied increasingly - and with growing sophistication - over the next few years. This increased activity will result from the usual combination of factors: ‘pure’ and ‘practical’ interest, on the one hand, and on the other, increased experimental capability. The latter will result particularly from the increasing availability and improving performance of tunable lasers able to provide large numbers of photons in bandwidths comparable to the Doppler width of spectral transitions in gas-phase species.


The Reporter wishes to acknowledge the help of a number of researchers who have kindly supplied results prior to publication, particularly 0.. P. Glass, G. Hancock, R. F. Heidner, tert., R. D. Levine, C. B. Moore, J. C. Polanyi, J. C. Stephenson, J. Troe, J. Wolfrum, and 0. J. Wolga.

I.W.M. Smith- Reactive and Inelastic Collisions involving Molecules in Selected Vibrational States

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Transcript of I.W.M. Smith- Reactive and Inelastic Collisions involving Molecules in Selected Vibrational States