Statistical Theory of Bimolecular Exchange Reactions*BY JOHN C. LIGHT?Dept. of Chemistry and The James Franck Institute,The University of Chicago, Chicago, Illinois 60637Received 26th June 1967A new concise derivation of the statistical theory of exchange reactions is given. The resultsof the theory are compared wih experiment for several reactions and for several types of information,and the reliability of such statistical calculations is assessed.The area of gas-phase collision phenomena, both energy transfer and chemicalkinetics, has become the subject of intensive experimental and theoretical investigationin recent years. Since the recent review articles 1-3 are both excellent and up to date,we shall concentrate, in this paper, only on the statistical (or phase space) theory ofsuch phenomena.It is, however, the detailed experimental information about theseprocesses, provided by various complementary experimental techniques, which hasprovided the impetus for further theoretical investigations.The experimental information available about some reactions such as the alkalimetal +halogen 4-6 and the hydrogen +halogen exchange reactions is rapidlyapproaching the ultimate of providing cross sections for reaction as a function ofvelocity from a spscified state (vibration-rotation) to a specified product state. Inany case, it is quantitative theoretical understanding on this level which now seemsto be the problem. If reasonable quantitative estimates of these cross sections couldbe made, all the information desired could be obtained by appropriate averages.The theoretical approaches to this problem fall into three rather well-definedcategories : quantum calculations,* * classical dynamical calcuIations, O - l andapproximate statistical calc~lations.~~-~ Although quantum mechanics and theformal apparatus of scattering theory provides the most rigorous approach to theproblem, they have not yet provided a large number of accurate results.There aretwo basic reasons for this: (a) the quantum dynamics of three-body (atomic size)systems are complex, often involving large numbers of states and strong interactions,and (b) the potential energy surfaces (presuming electronically adiabatic reactions)can only be estimated in a gross manner.The most successful approach to date has been the second; following classicaltrajectories on various potential energy surfaces to determine the most essentialfeatures affecting the dynamics of chemical reactions.Since the pioneering work ofWall, Hiller, and Mazur,l and Bunker,l O considerable information and confidencein the method have been produced. Although this approach has been successful,there are two limitations inherent in the method. First, specific quantum effectscannot be obtained. This is, almost certainly, a spurious criticism for most chemicalreactions. The fact that many quantum states are normally involved and availablealmost guarantees that classical mechanics is a very good approximation. Thesecond, more serious limitation is to electronically adiabatic potential energy surfaces.* This research was supported by grants from the National Science Foundation.-t Alfred P.Sloan Research Fellow.1JOHN C. LIGHT 15If more than one surface is present, a transition probability between surfaces must beassigned, normally zero or one. Although this has not been a problem in the reactionsconsidered to date, there is a large class of chemical reactions which are not adiabaticelectronically and which do produce electronically excited products.lgThe statistical approach to chemical kinetics is based on the hope that it may bepossible to bypass the complexities of quantum scattering theory and the uncertaintiesand labour of classical calculations (which consume non-trivial amounts of computertime) and yet to obtain quantitative results in good agreement with experiment.Thestatistical approach substitutes a hypothesis for much calculation, and the resultsmust be subjected to comparison with experiment in order to test the validity of thehypothesis. The comparison must be made in such a way that at worst the hypothesiswill be shown to be completely wrong, and at best it can be shown to be reasonablefor a well-defined class of reactions.There are several reasons for doing this. The most obvious is that it may providea relatively simple method of obtaining reasonable answers, particularly for complexsituations in which the other methods cannot be applied. In addition, the hypothesisis complementary to the other approaches. If, for a particular reaction, or class ofreactions, the statistical approach seems to yield valid results, it implies that thesystem can in no way be considered as weakly coupled.Thus the quantum approachand the classical trajectory approach probably will be even more difficult to carrythrough successfully for these reactions.There have now been enough calculations using the statistical theory and coin-parisons with experiment to provide a somewhat ambiguous answer as to its validity.In the following sections we shall try to provide a comprehensive outline of thefoundations, applications and results of the statistical approach, summarizing, atthis date, its validity and applicability.FOUNDATIONSAll statistical theories of dynamical events, from nuclear reactions to chemicalkinetics, involve two distinct types of approximations.The first is a definition,normally in phase space, of the limits of a strong coupling complex. The second is astatistical prescription, again usually involving phase space, for partitioning theproducts of the collision among the various possible products (or states).Statistical models differ from thermodynamic models of rate constants (such as" absolute " reaction rate theory 20) in several important respects. First, thestatistical theory is set up to describe the probabilities of possible outcomes of singlewell-defined collisions rather than the averages over equilibrium ensembles of theseprobabilities. Thus, in a statistical theory one may ask for the cross section at aparticular initial velocity for the transition from initial state i to final state j .Thisquestion has no meaning in a thermodynamic theory of kinetics. Secondly, in astatistical theory one is forced to make assumptions only on the limits of a strongcoupling complex but not on the lifetime or the configuration involved. By contrast,in a thermodynamic treatment, the complex or activated state must formally betreated as a well-defined " molecule ", having a particular unobservable equilibriumconfiguration with known (but unobservable) vibration-rotation states. Thus theabsolute reaction rate theory is parameterized to a much higher degree than isnecessary in a statistical theory. This makes a meaningful comparison of thepredictions of the statistical theory with experiment easier than that for the absolutereaction rate theory.A statistical theory of dynamical events was first proposed by Fermi 21 to describemultiple particle production in high energy nuclear collisions.This theory has bee16 STATISTICAL THEORYthe basis for much subsequent work in nuclear physics. It is, however, different indetail from the statistical theory of chemical kinetics described below.In 1958 Keck,14 while investigating the kinetics of recombination, gave a clear,correct description of the elements of a statistical theory of chemical reactions in termsof the fluxes in phase space. This theory is equivalent to the corrected version of thephase space (or statistical) theory presented by the author and co-workers in 1965,22-26although the language is different.The statistical theory of kinetics may be constructed in several ways : classically,from the S-matrix, etc.We shall give here a brief derivation of the quantummechanical version, emphasizing only the roles of symmetry and conservation require-ments about which there has been some More detailed derivations canbe found in the literature.23* 28We shall consider the problem of a bimolecular exchange reaction involving onlythree atoms. We denote by a, p, y, etc., the possible chemical species (channels)which can be formed in this reaction (a denotes AB+C, for instance). We denotethe quantum numbers of the internal states of the diatomic molecule in channel a bysa, the total rotational and orbital angular momenta by L, and L respectively, and therelative translational energy in this channel by Ea.To proceed to the essence of the statistical theory we now need to make onedefinition and one assumption. First, a “ strong coupling complex ” is defined by asingle property :(a) the mode of decomposition of a ‘‘ strong coupling complex ” is uncorrelatedwith the mode of formation except through conservation laws and detailed balancing.The conservation laws are conservation of energy, total angular momentum and itsprojection on one axis, and linear momentum.If so desired, one could also includeelectron spin angular momentum as being (approximately) separately conserved.Whether or not such complexes exist, even as an average over unobserved quantitiesis open to question (and experimental verification). The lack of correlation expressedin (a) is, however, a fundamental assumption of a statistical, as opposed to deter-ministic, theory.In the work of Keck and the author (although not in that of Euand Ross 7), one further assumption is made which, although not absolutely necessary,seems reasonable and greatly simplifies (indeed permits) calculations. It can bestated as follows :(b) the probability of formation of a “ strong coupling complex” from everychannel a is either unity or zero, depending upon the values of Ea, L, L a and Sa. ThusThe total cross section for formation of a “ strong coupling complex ” from thecollision specified by a, E,, L,, s,, is thenThe classical analogue of this equation is obtained by equating, for large I,,,,, theclassical and quantum mechanical expressions for orbital angular momentum(hL = p u b ) ,o=(classical)(a,Eu,La,s=) = nb&x(a,Ea,La,Sa), (2)where b is the impact parameterJOHN C.LIGHT 17Assumption (b) is certainly an approximation. Quantum mechanics, showsclearly that except for singular potentials, probabilities are " never " one or zero,i.e., there is always some penetration of a finite potential barrier into a non-classicalregion and a finite probability of reflection of a particle with enough energy to passover a barrier. However, for sufficiently massive particles, such as atoms, approxi-mations such as (b) are numerically very accurate except for L-LmaX, i.e., except fornearly orbiting collisions which constitute a minor fraction of the total number.Thus (b) is, in essence, a " classical " approximation.It implies that the density ofstates of the " complex " is so high that all collisions satisfying (b) can be accommo-dated.*We shall now indicate briefly how statements (a) and (b) are sufficient to determinethe statistical theory of chemical kinetics, specifically for bimolecular exchangereactions. The argument would be trivial in a two-dimensional world in which onlyone orientation of every angular momentum were allowed. In real space of threedimensions the coupling of L and La to form J J, make the algebra more complicatedbut do not change the basis of the derivation. We shall show that statements (a)and (b), together with the principle of detailed balancing, uniquely determine theprobabilities of decomposition of the complex into any final state.On this basis, detailed balancing requires thatwhere p a is the reduced mass of the atom-diatomic molecule in channel a, L a therotational angular momentum, and ( a ] stands for the entire set of quantum numbersEa, Sa, La.Because of the definition (a), we may write the cross sections as productsof a cross section for formation of the complex and the probabilities of decompositionof the complex. Thus, for a given total energy,where J is the total angular momentum of the complex.The cross section for formation of a complex of specified J is, from eqn. (l),nh2where the last factor is the fraction of collisions with L, L a which couple to yield atotal angular momentum J.The summation over L is restricted by the stronger of the two conditions (a) thatCombining eqn. (5), (4) and (3), we havei L-La I <J< I L+La 1, and (b) that LSL,,, (01, Ea, s,, L a ) .Since P(J+(B)) is, by assumption (a), independent of (a), we haveThe last sums are equal, by definition, to the numbers of states available in thespecified channel which can form, or be formed from, a complex of specified energyand angular momentum :* This is not true for electron-atom collisions in which well-defined resonances have been observed.However, no such resonances have been observed in chemical reactions18 STATISTICAL THEORYIt may now be proved by induction, starting with J = 0, that the only reasonablesolution of (7) is the obvious one :The requirement that the sum of the probabilities of decomposition of any complexbe unity provides the normalization and leads to the final result :where the summation in the denominator runs over all states accessible from thespecified complex.Thus Ntot is the total number of states available from a complexof given total energy and angular momentum. Thus the cross section for the overallprocess isIn the derivation the relative density of translational states enters only throughthe energy dependence of the cross sections and the microscopic reversibility relation.This is because we have dealt with probabilities rather than fluxes. However, thetwo results would be equivalent.Let P, be the radial relative momentum (theonly free variable since the angular momenta are specified). The flux into each stateper unit energy range will be the velocity times the probability times the density ofradial translational states per unit energy. Thus we haveThe above theory could be modified by changing one or both of the statements(a) and (b) to allow, for instance, for preferential coupling of certain initial and finalstates. This could most easily be done in the S matrix formulation of the theory.28However, unless such modifications were made with a firm theoretical basis theirvalue would be open to question since it would correspond to ad hoc parameterization.The purpose of such a theory is not to fit every experiment but to give a generalmodel for which a priori calculations can be made and used for comparison and(hopefully) reasonably accurate prediction.Large scale tampering with the modelserves neither purpose.THE “STRONG COUPLING COMPLEX”In the preceding section we used only the formal properties of the ‘‘ strong couplingcomplex ” in the general derivation of the statistical theory. In this section we shalldescribe briefly how the limits of the complex, which are needed before any computa-tion can be done, can be chosen. Since the choice of the limits to be used in state-ment (b) completely determines all the cross sections, this is the only point at whichthe physics of the process under consideration can be inserted. Again one wants tochoose the simplest limits which seem physically reasonable.For reactions without activation energy we have reasoned as follows, using theclassical impact parameter for simplicity.There should be a distance ro(ol) for eacJOHN C . LIGHT 19channel within which the atom and molecule are strongly coupled. Since there is noactivation energy, ro should be approximately independent of the internal and trans-lational energy and the rotational angular momentum. Thus the limit of the strongcoupling complex required in (b) is the maximum impact parameter, bmax(EJ forwhich the atom will approach within r,(a) of the molecule. For low energies withlong-range attractive forces between the atom and molecule and with ro about equalto the range of the repulsive (if any) forces, this leds towhere C, and 6 are determined by the form of the attractive potential (in eachchannel).23 This form seems appropriate for ion-molecule reactions and for someneutral reactions such as K+Br,, C1+Na2, and Hg*(3Po)+H2.At higher energies,however, b,,, given above may be less than ro. This should not be allowed to happenand the correct limits for unit probability of formation of the strong coupling areSeveral calculations using eqn, (13) have been carried out for ion-molecule and neutralreactiorrs. Some of the results will be given in the next section.For reactions with activation energy, the limits for formation ofthe strong couplingcomplex must depend on translational energy and probably internal energy asOnly a few calculations have been made for these reactions and the definitions usedwere :Here the effective activation energy Eeff(ct) was defined as the minimum translationalenergy necessary to have reaction for the molecule in a given internal state.Thespecific dependence on internal energy of the molecule was linear :26where E i is the activation energy from the ground state of the reactants, Evib andErot are the vibrational and rotational energies of the reactant molecule and theA are positive constants between 0 and 1. Although the parameterization in thiscase is higher than for reactions without activation energy, this form, which followsfrom simple collision theory of kinetics, seems as simple as is physically reasonable.Before going on to results, a physical description of what a “ strong couplingcomplex ” might be seems in order. The mathematical properties are clear : once asystem enters such a complex it ‘‘ forgets ” where it came from.In fact this does nothappen. In practice, however, there are certain quantities associated with a collisionwhich one cannot measure such as the z component of the orbital angular momentum,the magnitude of the orbital angular momentum, the exact translational energy, etc.If the coupling between all open channels is strong, the one might expect that crosssections, averaged over the unobserved quantities, to behave statistically. This wouldseem to be particularly likely if the complex lasts a reasonable length of time classically,i.e., for periods larger than the rotational period.Alternatively, if the potentialgradients are large and not smooth, rather random scattering could occur. Thusthe strong coupling approximation is essentially the oppsite of an adiabatic approxi-mation in which the system never changes state. We shall see in the next sectionthat the adequacy of the theory depends, to a large extent, on the type of informationdesired. For highly averaged quantities the statistical theory works well but fo20 STATISTICAL THEORYmore detailed information the deviations from the statistical predictions becomemore apparent.Finally the limits in statement (b) are not the actual limits of the strong couplingregion itself. They are the limits, in terms of the asymptotic states of the system,within which a strong coupling complex will, at some time and in some place, be formedwith unit probability.Thus they define asymptotic trajectories which, if followed,will lead through a strong coupling region. No information is required aboutthe time spent in this region-only that it has the property defined in (a).RESULTSThere has been a number of specific applications of the statistical theory tovarious reactive and inelastic proces~es.~~~ 30 We shall present here only fourexamples in any detail. These examples are chosen to illustrate the types of systemsto which the theory can be applied, and the types of information which can be obtainedand compared with experiment.The four cases are H +Cl,, K+ HBr, He + H;, and Hg* + CO. These illustratethe application to distinct types of reactions : exchange reactions with activationenergy, exchange reactions with negligible activation energy, ion-molecule reactions,10. I I I t I 1 \ I0 I 2 3 4 5 6nFIG.1.-Relative reaction rates kn/ka into vibrational states n for the reaction H+C12 -tHCl(n)+ C1. - - El - - calculated, Xo = Ar = 1.0 ; - - 0 - - d c . , hu = Ar = 0.97. (The experimental values forn<2 are lower than for n = 3 (ref. (34)).and excited atom-molecule reactions. These cases were all studied by the authorand co-workers 22-26 in an attempt to define the limits of applicability of the theoryby comparison with experiment.A. H+C1226The luminescent studies of this reaction by Polanyi 31-33 and co-workers haveprovided unparallelled information about the internal excitation of the products.The reaction has an activation energy (of about 2.0 kcal/mole) and thus the definitionof the complex in eqn.(15) was used in the statistical theory. The calculation wasdone with only the vibrational degree of freedom quantized, the angular momentabeing treated classically. Since the experimental information was for thermal systems,all quantities were averaged over the appropriate Maxwell-Boltzmann distribution.The only significant parameters to enter the calculation are 1, and &, which measurJOHN C. LIGHT 21the fractional contribution of internal energy in reducing the effective activationenergy, and the ro for each channel-the radii of the complex itself. The values ofro apparently have little effect in the statistical calculation except the gross crosssections.However, the values of ;I affect the internal energy distributions consider-ably. The internal calculated energy distributions of the product are plotted in fig.1 and 2.These results illustrate one apparently general result. With complexes definedonly by a single inequality on the impact parameter as a function of the energy variables,the statistical theory never predicts complete population inversions of products. ThereFIG. 2.-Rotational distributions of HCl. -a- calculated for HC1 (n = 2), hu = hr = 1.0,EOr = 0.1 eV; 0 , expt. (ref. (31)-(33)).are always more states available from a lower vibrational state than from a high oneand thus the populations of vibrational states of products fall monotonically withvibrational quantum number.This result has appeared in every statistical calculationto date. Since there is a large class of reactions proceeding with and without activa-tion energy for which the best indications are that vibrational population inversionsare produced, we find that the statistical theory is unreliable with respect to this typeof information.The rotational distribution given by the statistical theory cannot be directlycompared with experimental results since no firm experimental quantities are known. 34It seems, however, to be a reasonable distribution since high rotational excitation,after some relaxation, is still observed.33 The overall rate constants as a functionof temperature were computed.The result, from the statistical theory, isk(T) = 2.1 xThis seems in line with rates of similar reactions, but no good absolute values areavailable from experiment. One interesting point is that the value of 2.2 kcaljmoleappearing in the exponential factor of the rate coefficient is 10 larger than theactivation energy Eg assumed in the definition of the complex. This effect was alsofound in the studies of Karplus, Porter, and Sharrna.l3B. K+HBr 2 5 * 26This reaction, studied experimentally by molecular beam rnethod~,~~’~’ proceedswith an activation energy of less than 0.4 kcaljmole. Considerable information hasexp (-2200/RT) cm3 sec-’ molecule-l22 STATISTICAL THEORYbeen deduced from the experiments about the details of the reaction: probabilityof reaction against impact parameter, rotational, vibrational, and total energydistributions of the products, etc.In this reaction, the value of Ex = 0.15 kcal/mole(best experimental estimate) was used and A, = 0-8 and Lr = 0.4 were chosen to giveFIG.0.2 0.4 0.6 0-8R3.-Probability of reaction against reduced impact parameter for K+ HBr.experimentally deduced points, ref. (35). Ez = 0.08 eV.1 40 r-A0 = 0.8. 0,Et AeVjFIG. 4.Reaction cross section for KfHBr against translational energy, EG (eV). A, = 0.8,Ar = 0-4. -0- experimentally deduced, ref. (35). -@-, calc.the best agreement with the " experimental " probability of reaction against impactparameter. The value of ro = 3.8A for the product channel was taken fromS~plinskas.~~ The aim of the calculation was to see how much of the experimentaldata could be reproduced by the statistical theory with one set of 5 parameters(&, 4, G, e&,, r&))JOHN C .LIGHT 23In fig. 3-7 we plot the comparisons with experiment. It can be seen that for theprobability of reaction against impact parameter, the reaction cross section againsttranslational energy, and the rotational energy distribution, the agreement between--I---l-1-- rFIG. same04.maximum.nFIO. 6.-Vibrational distribution of KBr produced. -A- caIc., hv = 0-8, Ar = 0.4;-0- Cak. hv = 0.9, hr = 0.4; - " Typical " Monte Car10 result, ref. (10)24 STATISTICAL THEORYthe statistical calculation and the experimental estimates are good. We again find,however, that the vibrational distribution of the product molecule is probably notgiven correctly by the statistical theory, and this leads to a shift of the total internalenergy distribution of the product to lower energy.This is sensitive to the assumedvalue of Av and a value of A, = 0.9 improves the agreement considerably, but doesnot reproduce the curve of the probability of reaction against impact parameterquite as well. Since the only quantities varied were A, and A?, we can say that forthis reaction, with E' and the r0 given, the values = 0-8 or 0-9 and 3.r = 0.4reproduce well all the experimental information except, probably, the vibrationalenergy distribution.&ItCeVFIG. 7.-Total internal energy distribution of KBr produced. -D- --CI calc., Xu = 0.9,hr = 0.4; -0-0- calc., hu = 0-8, Ar == 0.04. Curves normalized to maximum = IEOr = 0.08 eV.Arrow indicates experimental maximum (ref. (36)).C. Hg*+CO 26The internal (vibrational) energy distribution for the CO produced in the quenchingreaction of Hg(3Po) has been studied by Karl, Kruse, and P ~ l a n y i . ~ ~ The statisticaltheory, with a value of A, = 0.98 and an activation energy of about zero (0.005 eV)gives moderately good agreement with the experimental results as shown in fig. 8.That this model is not unique in giving reasonable results has been shown by Karlef al., who used a model of impulsive energy release in a Hg*CO complex. Additionalcredibility is lent to the statistical calculation for this type of reaction by the recentwork of K.Yang and co-workers 30 on the Hg*+H, reaction. They used thestatistical theory to predict accurately the quantum yield of H atoms. Thus we mayhope that excited atom-molecule interactions approximate the strong coupling com-plex with some degree of accuracy.D. He+Hz24As one of the simplest ion-molecule reactions, this reaction, He + H:-+ HeHf $. H,has been studied in detail e~perirnentally,~~-~~ although no information concerningenergy distributions of products is available. For this reaction the limiting impactparameter for formation of the complex was taken from the Langevin 44 theory,i.e., it is determined by the requirement that the particles be able to pass over theangular momentum barrier. Since this is determined only by the polarizability oJOHN C.LIGHT 25the neutral species, there are no adjustable parameters. The initial vibrational dis-tribution of the H i was taken from the caIculations of W a c k ~ . ~ ~ It is critical sincethe reaction is endothermic from the ground state by 0.92 eV. Thus at low trans-lational energies, only vibrationally excited H i ions can react.FIG.W I 2 4 6 8 10 12 I4 16 18 2 0V8.-Vibrational energy distribution of CO from the reaction Hg(3Po)+ CO +Hg+ CO*. - calc., hu = 0-98. - - - - calc., hu = 1.0. - - -, expt., ref. (39).6l ' ' \ G + S iEcM(ev)FIG. 9.-Cross section for He+Hz -+HeH++H against Ea. Initial vibrational distribution ref. (45). - - - d c . , - ..-- total " strong coupling " cross section, -0-0 expt.ref. (41)26 STATISTICAL THEORY30-- 2.0-3230\" ir.0Xw* to-0'The cross sections for reactions from each vibrational state of H i were evaluated,and the overall cross section was obtained from the sum, weighted by the fractionalpopulation of each state. The results are compared with two sets of experimentaldata in fig. 9 and 10. In fig. 10, the calculated results were appropriately averaged1 I 1 I I 1-Q QQ --I I I I -0 1 2 3 4 5 6 71.0 2.0 3.0 4.0 5.0ECM(WFIG. 1 1.-Average isotope ratio against average translational energyHeH++D.He+HD++{ HeD++H7 calc., - .. 0 - - expt., ref. (42)JOHN C . LIGHT 27to conform to the experimental velocity distribution in a conventiona'l mass spectro-meter. Of interest is the fact that the calculated cross sections fall off at low energybecause of the purely statistical factor that more inelastic than reactive channelsare open there.Thus the correct result is given without assuming that a minimumtranslational energy (activation energy) is necessary.46 The isotope ratios for productsof the We-!-HD+ reaction were also calculated and compared with experiment.Fig. 11 shows that the qualitative dependence of this ratio on energy is reproducedwell but the absolute value of the ratio is in error by approximately, 20%.In general, the application of the stgtistical theory to low energy ion-mo!eculereactions has been productive. The most important effects seem to be predictedwith reasonable accuracy (such as ratios of possible products 29 and velocity depmdence of cross sections).It is likely that, because of the strong long range attractiveforces involved, the ion-molecule reactions at ZOW energies proceed through a complexwhich is strongly coupled. A good discussion of these processes is given by Futrelland Abrarnsox~.~~SUMMARY AND DISCUSSIONThe results presented in the previous section, while typical, represent only a smallfraction of the cases studied by the statistical method. As in the examples above,the results, in comparison with experiment, vary from being qualitatively andquantitatively in error to being quantitatively correct. It is now possible to setcertain limits on the applicability of the statistical theory. There are two factorswhich must be considered: first, the type of information desired, and second, thetype of reaction involved.In order to summarize the experience to date, we give, in table 1, the evaluationof the accuracy of the statistical theory for the specific types of reactions and infor-mation.The agreement is best for highly averaged quantities in ion-molecule reactionsTABLE 1 .-RELATIVE ACCURACY OF STATISTICAL THEORYatom-molecule atommolecule \ reaction ion-moIecule excited reactions reactionsreactions molecule no activation activation \ (Ex 10 eV) reactions energy energyinformationthermal rate constants good good faircross section against energy good goodisotope and product ratios fair to goodgoodtotal internal energy of products fair fair fairrotational energy distributions good fair( ?)vibrational energy distributions fair poor poorwhich are likely to have the strongest coupling, and worst for detailed informationabout neutral reactions with activation energy, even though one parameter Av wasvaried to some extent.There are three questions which it seems pertinent to attempt to answer (i) Arethere any types of reactive systems for which the basic assumptions of the statisticaltheory, (a) and (b), are approximately valid in detail, i.e., on a state for state basis?(ii) Are there any types of information about specific types of reactions for whichthe statistical theory gives reliable quantitative results ? (iii) When, if ever, should oneuse the statistical theory?The answer to the first question is easy, but ambiguous.No reactive system hasbeen shown to behave statistically in detail, but many have been shown to react in 28 STATISTICAL THEORYnon-statistical manner. The only types of reactions which might be truly statisticalare low energy ion-molecule reactions for which adequate experimental informationto give a definitive answer is not available. Thus we may conclude that it is possiblebut not likely that any real systems react statistically in detail.The answer to the second question is yes. Averaged information about ion-molecule reactions and excited atom-molecule reactions has usually been givenaccurately by the statistical theory.48 Quantities such as rate constants, crosssections against energy, and, perhaps most important, ratios of products can becomputed reliably by the statistical theory.For reactions of neutral, unexcitedspecies this holds true to a lesser degree.The third question is, perhaps, the most important. There appear to be tworeasonable uses of the statistical theory : as a model against which to compare experi-mental results, and as a predictive tool in the absence of much experimental informa-tion. The utility as a model was illustrated for the HefH; reaction. The fall-offin cross section for reaction as the translational energy is decreased is explained bythe statistical theory as the statistical behaviour for an endothermic reaction pro-ceeding from excited vibrational states. Although it cannot be asserted that thisis the correct explanation, its simplicity and accordance with known facts of thresholdbehaviour give it added weight vis-a-vis an alternative explanation in terms of akinetic energy threshold for ion-molecule reaction~.~~ In the opposite sense, thefailure of the statistical theory to account for the vibrational energy distribution inthe H-kCl, reaction means that a more deterministic model is both necessary andreasonable as the accurate approach.Finally, the author believes that the statistical theory can be used as a valuablepredictive tool in some cases.It is fairly simple, flexible, and not highly para-meterized so that it can be applied to systems for which little information is available.For example, the reactions H+HX (X = Cl, Br, I) yield some excited halogenatoms as product.The fraction of excited atoms produced is unknown. Thestatistical theory was applied to yield the following per cent excitation in thermal(300°K) reactions: Cl ("p,) 29 %; Br ("+) 17.5 %; I ("p,) 12 %.The statistical theory appears to be useful for those cases where alternative productsare available and one is interested in determining the ratios of possible products. Theinformation desired is highly averaged and so the results should be fairly reliable.In addition, little information about potential energy surfaces or their crossing pointsis needed so the computation can be carried out quickly and efficiently.We may thus consider the statistical theory as an '' interim " theory-a semi-quantitative model to be used in a comparative or predictive way until we have theability to make accurate a priori calculations of all kinetic quantities of interest.The author acknowledges the real contributions to the foundations, development,and applications of the theory made by Dr.Philip Pechukas and Dr. Jeong-long Lin.The author also gratefully acknowledges the general support of the Institute for theStudy of Metals (James Franck Institute) by the Advanced Research Projects Agency.D. L. Bunker, Theory of Elementary Gas Reaction Rates, (International Encyclopedia ofPhysical Chemistry and Chemical Physics, Topic, 19, Vol. 1) (Pergamon Press, New York,1966).K. J. Laidler and J. C. Polanyi, Prog. Reaction Kinetics 1965, 3, 1.J. Ross, ed., Ado. Chem. Physics, vol. X, (Interscience, New York, 1966).S.Datz and R. E. Minturn, J. Chem. Physics, 1964,41, 1153.K. R. Wilson, G. H. Kwei, J. A. Norris, R. R. Hem, J. H. Birely and D. R. Herschbach,J. Chem. Physics., 1964, 41, 1154.4T. T. Warnock, R. B. Bernstein, and A. E. Grosser, J. Chem. Physics, 1967, 46, 1685JOHN C . LIGHT 29J. C. Polanyi, Chem. in Britain, 1966, 2, 151.R. J. Suplinskas, Thesis, (Brown University, 1965).J. L. Magee, J. Chem. Physics, 1940,8, 687.lo N. C. Blais and D. L. Bunker, J. Chem. Physics, 1962, 37, 2713; ibid, 1963, 39, 315.l 1 J. C. Polanyi and S . D. Rosner, J. Chem. Physics, 1963, 38,1028.l2 M. Karplus and L. M. RaE, J. Chem. Physics, 1964,41, 1267.l 3 M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1964, 40, 2033.l4 J. C. Keck, J.Chem. Physics, 1958,29,410.l5 0. B. Firsov, Zhur. Eksperim. i. Teor. Fiz. 1962, 42, 1307, (Eng. trans. ; Soviet Phys.-JEPTl 6 J. C. 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