A New Collision Theory for Bimolecular Reactions Ian W. M. Smith University of Cambridge, Lensfield Road, Cambridge CB2 IEP England (1, 2) in THIS JOURNAL Over 13 vears aeo, two papers . . outlined way3 in which u n d c r g m d ~ ~ icuursl.. ~tr in i htmical kinetics mirht bc modified to takr account oithc (lewcr ulidcrotmdini 01chelniciil rwctloll. ohtnintd tlir~ugh;lie nppllcition of modern experimenrai technique,. In the d e d e &nce those papers were written, research into reaction dynamics has continued to flourish. Not only have there been further improvements in molecular heam experiments, in studies of the infrared chemiluminescence from excited reaction products, and in the use of lasers to prepare excited reagent states and monitor product states, but also classical scattering calculations have become routine and quanta1 scattering calculations feasible. Despite the intensity of this research activity and the success of Levine and Bernstein's monograph ( 3 ) ,the impact of reaction dynamics on undereraduate courses in chemical kinetics has been rather ;light. One reason for this seems to he the absence of a suitable thtweticul model rout too easy nor ruo dift'iculr~with whi(.h diazuss reactive molecular cullisiuni. Thl. defect.i of " i i l n ~ l ~ collision theory" are so glaring that it tends to he hurried p& in embarrassed haste. Transition state theory is presented as being more satisfactory, although its subtler difficulties and assumptions are frequently glossed over; but it is-at least, at an undergraduate level-simply a "theory for rate constants": no more detailed information is predicted or explained. In a recently published monograph (4), I have proposed a new "modified simple collision theory". Several features of simple collision theory are retained. Thus, the reagents are assumed to interact with one another only when they reach some critical separation (D)and, at this point, one asks whether the energy associated with motion along the line joining the centers of mass of two species exceeds some critical value. If it does, reaction occurs. The modification from simple collision theory is that the critical energy which bas to he exceeded a t the critical seoaration denends on the anele d heween the bond whirh h& ro he h m k h dnd the line (;; ;,lining the centers ill' mas, cjithe reaeents "at impact". A tunction is chosen to describe this variaiion of critical potential energy which is reasonable and which yields equations that can he solved analytically. The result of the modification is that "steric effects" are allowed for approximately and can be estimated if rudimentary information is available ahout the potential energy surface. The main purpose of the present paper is to evaluate this modified version of simple collision theory hy comparing its predictions with those of full-scale quasiclassical (QCL) trajectory calculations for the following atom-transfer reactions H+Hz+Hz+H F
+ Hz
H + F2
0 + Hz
-
HF + H
+F OH + H
HF
Simple Collision Theory
In manv basic texts on nhvsical e . , chemistrv. the s i m ~ l colli.im thiwr) of rracri~mrates i > prc,t.nted :n u ~ abl~rrvi.~ted i furm. It iiareued that the rat(,of re.ictlon iieausi t u thetutd rate of collkons between the reagents mukiplied by the fraction of those collisions in which the energy in two degrees of freedom exceeds some critical value. Actually, since the mathematics involved is quite simple, there seems little reason why students should not he exposed to a fuller treatment which introduces the idea of an energy-dependent reaction cross-section and which illustrates how the thermal rate constant constitutes a highly averaged quantity (2,4). In simple collision theory, the reagents (A and B) are both treated as structureless spheres. The relative motion of such a two-particle system-whatever the potential between them-can, of course, be represented by a single point of mass p = rnarn~l(rna m ~and ) relative velocity v = v~ - VB moving in a central potential (2,4). More conveniently, one can discuss individual collisions in terms of the initial magnitude of the relative velocity, u and the impact parameter b , which is the shortest distance that would separate the particles during the collision in the absence of any interaction or, alternatively, the closest the representative point would get to the origin of the scattering system (see Fig. l(a)). In simple collision theory, A and B are not only "structureless" hut also "hard"; that is, they are assumed not to interact (V(r > d ) = 0) until they reach some separation r = d, corresponding to the hard-sphere collision diameter. The requirement for reaction is that, a t r = d , the energy, EI,, associated with the components of motion directed along the line-of-centers of A and B, exceeds a critical value eo. The component of the relative velocity directed toward 0 at r = d, i.e., vrc = VAJ, - VBJC, can he deduced easily from Fig. l ( a )
+
(1)
AH! = -31.9 kcal/mole
(2)
kcallmole
(3)
t1.9 kcallmole
(4)
AH! = -98 =
These four reactions differ considerahlv in their kinematics and their potrtitial iuriaces.'l'he cornpariwn hri~iys,,utclearly h ~ n hthe vir1w.i and limitiitiuns uf this cxtended rdli&n model for bimolecular reactions.
Figure 1. Parameters in (a) simple collision theory, and (b) modified simple COlliSlOn theory.
Volume 59
Number 1 January 1982
9
Simple collision theory is inadequate largely because the form of the critical surface is too simple. As we shall see, it is not reasonable to define the critical surface at a constant value of r; however, it is unreasonable to suppose that the potential energy on the critical dividing surface is independent of 4, the angle between r and RBC(from hereon, I shall consider a three atom reaction of the type: A BC AB C), and one might also question whether it depends o n R m A second unsatisfactory feature of the critical surface in simple collision theory is that it is positioned (at least, implicitly) at r = d , the "hard-sphere" separation of the reagents, but calculations on potential energy surfaces for reactive systems demonstrate that energy harriers to reaction are invariably at much smaller values of r. For example, the most extensive calculations on the surface for the H Hzreaction (5) giver = 1.39 A a t the barrier, whereas for H Hz,d = 3.5 A,' Both these defects of the simule collision model are removed in the modified version of the theory. The revised model is illustrated in Fieure l(h). As before,
-
+
+
+ +
where r is the collision energy, i.e., e = '/z!.u~~. For any collision energy, as b is increased ci, decreases. As long as ri, > r$ reaction occurs. For each value of r > rO,there is a value of the impact parameter, b, at which rl, = ro. For b < b,, reacthere is no reaction at this collision tion occurs; for b > b, energy. Figure 2 illustrates this unreasonable step-function behavior for P(r,b), the opaclty function, which simple collision theory predicts. Equation (6)with ti, put equal to rOcan be used to find b, and hence the reaction cross-section: S,(r) frequently referred to as the excitation function, rises monotonically from zero at rOto approach 8 d 2asymptotically as r m. In order to derive an exuression for the thermal rate constant, uS,(u) must be averaged over the thermal distribution of relative speeds ( 2 , 4 )i.e.,
energy along the line-of-centers must exceed some critical value if reaction is to occur. Now, however, the critical energy is not constant hut is assumed to depend on 6 according to the equation Clearly, reaction is favored if, at impact, A, B and C are collinear so that 6 = 0. The condition for reaction to occur in collisions of specified energy and impact parameter is that ci, =
r ( l - b2/D2)a c0
+ 2 ~ ' ( 1- cos 9)
(12)
-
The second equality defines the maximum value of d, for which reaction can take place, i.e.
leading to:
To find P,(r,b), all possible orientations of the sphere r = D must be considered: P,(c,b) is simply the fraction of the surface area of the sphere over which eqn. (12) can he satisfied. Here, BC is assumed to be a homonuclear diatomic, so that (the same) reaction is equally likely for d, as for (w - 6). The case where only B (or C ) can react with A is straightforward and is dealt with elsewhere (4). For the "double-ended" case k ( T ) = s d 2 ( 8 k T / n p ) 1 1 Z e x (-c"/kT) p
(10)
If d is set equal t o the hard-sphere collision diameter, n d 2 ( 8 h T / ~ p ) ' /defines ~ a rate constant associated with the total rate of collisions between the hard-spheres A and B, while the exponential factor gives the fraction of those collisions in which e[, > r0. Simple collision theory yields expressions for P,(r,b), S,(E), and h(T). Because of the amumptions in the model one cannot expect these equations to hold accurately for a real collisional reaction. The best publicized shortcoming of the theory is that the reagents are assumed to be structureless spheres so that their orientation when they collide has no effect on the probability of reaction. This defect is often recognized by inserting a steric factor, P , into eqn. (4). However, without any means of estimating P, this procedure is of doubtful value and it misleadingly suggests that the neglect of a steric effectis the only-or, a t least, the major flaw in simple collision theory. On closer inspection (see below), several other factors are seen to influence the probability of reaction in collisions. Modified Simple Collision Theory In anything less than a full dynamical theory of reaction rates, one chooses a "critical surface" in configuration space which separates reagents from products and then estimates the overall rate at which reacting systems cross this surface. 10
Journal of Chemical Education
P,(e,b) = 0 for r
< rn
.= ( ~ (-1 b2/D2)- c0//2~'
(14)
(15)
for
ro < r ( l - b Z / D 2 6 ) (21' + LO) P,(f,b
< b,,)
= 1for 4
1 - b2/D2)> (2cT+cO)
(16)
where the maximum impact parameter for which reaction can occur is given by a similar equation to before b,
= D ( 1 - cOI~)''~
(17)
As the curves in Figure 2 show, eqns. (13)-(15) predict a much more reasonable form for the onacitv . . function than simule collision theory. Expressions for the reaction cross-section and the rate
'
The hard-sphere diameter for an H atom is estimated fromthe 32: potential curve for HZ.
Table 1. Summary of Potential Surface Properties P Reagent3
H
+ HI
(kcallmole) 9.13
fa (kcallmole) 5.3
(A)
R& (A)
D (A)
(ergslradianz)
0.900
0.900
1.35
7.7
RB ;
fb,"&.~&
X
2c' (kcallmole)
Ref.
8. 9
24.9
Figure 5. Excitation function predicted by modified simple collision theory compared with results of lhree-dimensional QCL lraiectory calculalions (10) onH+Fdv=O,J=9).
"shaoes" of the ouacitv and excitation functions are excel: lent6 reproduced;ln this regard, it is worth emphasizing that a maior difference between simple collision theory and the modified version proposed here & their predictions about the form of the excitation function near threshold. According to the modified collision model, at E = E", (dS,/&) = 0 and (a2Sr/&) is positive. In contrast, eqn. (71, besides yielding much larger reaction cross-sections, predicts that the excitation function has its largest value of (as,/&) at E = ~ 0 QCL . trajectories, not only on H + H z , strongly support the threshold behavior suggested by the modified collision model for reactions with significant barriers. Fioure 4. Ooacitv and excitation functions oredicted bv modified s i m ~ l ecollision
Table 1summarizes some properties of potential surfaces for reactions (1)-(4) and lists the values of €0, D, and 2 1 which have been derived from these orooerties. Comparisons be-
7 and now will be considered in turn. The Porter-Kar~lussurface for Hq (7) was used in a classic investigation of tge dynamics of the^ + Hz reaction by Porter, Karplus, and Sharma (8).Besides this QCL trajectory study, there have been numerous quantum scattering calculations on this surface. culminatine in the accurate. threedimensional, results of Schatz anb Kupperman (6). The agreement between the quasiclassical and quantum results is remarkable, there only being appreciable differences at collision energies close to threshold. The compa&on shown in Figure 3 between Karplus, Porter, and Sharma's results and calculations using the modified collision model is encouragingly good. In particular, the 12
Journal of Chemical Education
the values of V* and €0 in able 1 shows that, in the most favorable case. 3.8 kcallmole of the vihrational euerev . . can be releaaea to a s h t I hr ys1t.m IU aurnmuunr t hr barrier. On t h ~ uthrr hand. rrhcn the harrier is "~arl!." as i n t' + H, atid H + F., tht. initiitl ~il~rat:m:rl m18titm rt~n;tin;; ~ l n i ~urthvgunul ~st to the re3ctim cwrdin31e UI) tc, the U(~:IU tit wllith the hirrier is crossed. Consequently, the initial vibrational energy provides little energy for barrier crossing and €"is much closer to V*. I t therefore seems that, at least in regard to this aspect of the collision dynamics, the modified collision model should work best for reactions with "early" barriers. This is the usual situation for exoergic reactions (IZ),such as F H 2 and H
F".
+
+
Figures 4 and 5 compare predictions of the modified collision model with the results of QCL trajectory calculations on these reactions (9,101.Unfortunately, no opacity function was reported by Jonathan et al. ( l o ) ,but otherwise the agreement is almost entirely satisfactory. There is, however, one discrepancy which can be used constructively to illustrate a common failing of any theory of reaction rates which includes less than the full collision dynamics. At higher collision energies, the reaction cross-sections calculated for F He rise
+
Table 2.
Comparison of Rate Constants and Arrhenius Parameters
(i) Modified Collision Model
(ii) QCL Trajectory Calculations
(iii) Standard Transition State Theorv
H t H2 ( T = 500 K) k/cm3 molecule6 s-' A/cm3 molecule-' s-' E,,/kcal mole-' F+H2(T=350K) k/cm3 molecule-' s-' A/cm3 moleculeC s-' E,,,/kcai mole-' 0tHp(T=500K) k/cm3 molecule-' s-'
Figure 6. Actual varlatlon of the potential energy barrier with d (- - - -1 compared with 26'(1 - cos+) (-) on the surfaces for (a) F Hp. and lb) H F2.
+
+
less steeply than the model predicts. A similar effect is evident for H Hz at energies greater than those shown in Figure 3. The reason is that, at high collision energies, some trajectories having passed through the region of the potential barrier fail to "turn the corner" on the potential hypersurface leading to the product's valley (13) The transferred atom rebounds and rejoins its original partner; the critical dividing surface is crossed twice: and the orieinal reagents seoarate. This effect
+
" .
.
the modified collision model makes no allowance for the nhenomenon of "recrossine" and will therefore tend to overestimate the reaction ratewhen it is present. The effect dependsnot only on the potential hypersurface, but also (13) on the relative masses of the atoms involved in the reaction. It is aowarentlv warticularlv marked for F H1 but almost ahsent for H < F (~1 5 ) . The success of the modified simple collision theory, as revealed by the comparisons in Figures ( 3 ) - ( 5 ) , is quite remarkable. However, hearing in mind the crudity of the model, this success should he examined closelv. it is . In warticular. . valuable to see how the energy harriers on surfaces for nonzero values of @ compare with those assumedin the modified collision model calculations. Figure 6 shows this comparison for the F Hz and H Fz potentials. In both systems, the actual barriers increase more steeply with (d) than the function 2 r ' ( l - cos d). If a better revresentation of the votential energy at the critical dividing surface were incorporked into the model. the reaction cross-sections would he lowered leading to poorer agreement with the results of QCL trajectories. Clearly, there must he some balancing effect, and it is not difficult to identify. The model, like simple collision theory, neglects any interaction between the reagents until they reach the critical separation. In practice, one result of such forces will he to deflect the reagents into a more favorahle
+
+
+
Figure 7. Excitation function predicted by modified simple collision theory compared with results of three-dimensional QCL trajectory calculations (11) on 0 Hz(" = 0, J = 0).
+
conformation for reaction. It amears that this diredine effect neatly hut fortuitously &npe&es for the assumed variation of V(d) beine too shallow. It is interestine to note that it is the form' 'df the potential surface which determines the magnitudes of the cross-sections and thermal rate constants for a reaction. These properties are often less well-matched in QCL trajectories than the distribution of product energies. The last example for which model predictions and the results of QCL trajectories ( 1 1 )are compared is the 0 Hz reaction. (R&IRk2) = 1.17 < ( R e , ~ ~ / R e=, 1.31, ~ 2 ) so the energy barrier is in a "late" position on the potential surface. The results displayed in Figure 7 show that the modified collision model predicts reaction cross-sections roughly twice those found in the trajectory calculations over a wide range of collision energies. This is because initial vihrational energy plays a different and larger role in the collision dynamics of reactions with late barriers ( 1 2 ) . As the data in Table I demonatrare, une eit'ect i, chat rhe tra~lslatiwalenergy thrrihdd ior rt.actiun < is now much hwrr than \". In addiri.m, the phaie of the \,ibratiunal m d w l i i much morL. imuortan~.In OCI. trajectory calculations on collinear collisions, this shows itself in a relativelv slow increase in the wrobabilitv of reaction vast threshold: in contrast, when the barrier is early, there is an almost step-function rise in the probability of reaction. The modified collision model takes no explicit account of BC's vibrational motion, other than recornizine that r0 is lowered slightly by its most favorahle c o n k b u t i k However, this maximum contribution can be made onlv when the vibrational
+
in promoting reaction and they are resp&sible for the discrepancy shown in Figure 7. Finally, Table 2 compares thermal rate data for reactions (11, (21, and (4) calculated by means of (i) modified simple Volume 59
Number 1 January 1982
13
system passes the critical surface. One result is that no prec d l i s i ~ mtheury, [ i i , (-)(?I. tr:qt~tc,ry~ Y I ~ c I ~ , I I : ~ ~,111d I I : , 1i1i1 t i ~ n d : ~ rrr;m-:tiuu (l xnretheory. 11) perimr$:~!& t I t i . . c o ~ ~ ~ p ~ r dictions are made about the product attributes; for example, !:MI F 7 H and 0 t H . . i r is lit,c,>;try I U ~ ; ~ k c ~ v ~ m ~ i s l t ~the n t distributions of scatterine aneles or enemies amone the account of the electronic degeneracy and near degeneracy in the atomic free radical. In comparing opacity and excitation allowance is mad;for trajectories which "turn hack". functions, reaction was assumed to proceed on a single surface However. to be useful. and only collisions on this surface were counted. In reality, the . . a sim~lified.uhvsical " model need not reproduce accurately the results of more extensive calculations interaction of FpP) Hz and of 0(3P) H 2 gives rise to three under all conditions. The comwarisons which have been made distinct surfaces, so a factor allowing for electronic effects should he included in all the expressions for the thermal rate constants. However, this does not amear to have been done of the subtler forces that infl&ce the dynamics of reactive in the QCL trajectory calculations o d 6 Hz (91,so this factor collisions. These are not swamped, as in simple collision has been omitted from each of the calculations on this reactheory, by the neglect of steric effects. Furthermore, for atom tion. For 0 Hz, Johnson and Winter (11) included the electronic factor, and all the numbers in Table 2 for this replus diatomic molecule reactions, the estimates of rate conaction incorporate this factor. All the activation energies listed stants appear to he at least as good as those from standard transition state theory and, in addition, the opacity and exin Table 2 have been calculated properly by evaluating citation functions are predicted. In teaching, the modified -din kld(1lRT) a t the specified temperature. collision model should serve as a useful bridge between the Once again the results of the model calculations and those crudities of simple collision theory and the complexities of of QCL tFajectory calculations agree very well. Certainly, the full-scale scattering calculations. In research, two possible uses agreement is better than that between QCL trajectories and standard transition state theory. This is especially encourcan he foreseen. Predicted opacity and excitation functions aging in view of the finding ( 6 )that the transition state rate could serve as the basis for more efficient pseudorandom sampling of impact parameters and collision energies in traexpression is in poor agreement with that derived from acjectory calculations. Finally, the excitation function could act curate quantum scattering calculations, whereas the agreeas a valuable yardstick against which measured reaction ment between the QCL and quantum results is significantly cross-sections could he compared. I t can be confidently prebetter. dicted that this kind of information will become available for Summary and Conclusions more and more reactions over the next few years. Like anv mudel of Dhvsical realitv the one which is me. . Literature Cited wnltd liere hil- its .t.,umptiwn and linlir31izma. S e \ v r ~ of I the+ h,.w I m n ~ i d e ddread^ h t will l x ~ rtr wtit:m. First, Woifgmg, R., J. CHEM EDUC.,46, 359 (19681. G1eene.E. F.,andKuppermann. A.J. CHBM.EIXJC..45, 361 (1968). the model in its simplest form does not reco&e that the 1.evine. R. D.. and Bernltein, R. B., "Molecuia~Reaction Dynamics," Oxford University collision dvnamics mav he governed bv more than one elecPress, New Yurk, 1974.
+
+
+
+
agents before the critical surface is reached and the internal motions of the molecular reanent are unimportant. The first of these effects is relatively ;light and see& to he compensated for if the critical potential is chosen, hy a simple approximate method, to vary less steeply with rp than i t does in practice. The second effect seems to be large only when the surface has a late barrier. The final limitations of the model are both related to its indifference to what happens once a
14
Journal of Chemical Education
Smith, I. W. M.,"Kinetica and Dynamics of Eiementary Gss Reactiord Buttemorths, London. 1980. Siegbuhn. P., and Liu, 8..J. Chem Phyhys , fi8,2457 (1978). Schatr, G. C., and Koppermann. A,, J. Chem Phys., 65,4642 and 4668 (1878). Porter.R. N., and Karp1us.M.. J. Chem. Phys., 40,1105 (1964). Karplus,M.,Pocter,R. N.,and Sharma, R. D., J . W r m . Phys.,43,5259 (1965).
Muekerman,J.T.,J.ChemPhys., 54,1165 (1871).
Jonathan, N., Okuda, S., andTimlin,D.,Mol. Phys.. 24,1153 (1972). Johnson, B.R., and Winter,N.W., J. Chem. Phys.,66,4116 (1977). Polanyi. J. C., Acct Chrm. Res.. 6,161 (19721.
Mahan,B. H.,d. CHEM.EDUC.51.30Rand377 11974). ( a ) Miller, W. H.,Accl. Chrm. Ras , 9 , 306 119761: (b) Pechukas.P..and McLafferty, F. J.. J. Chem. Phys., 58,1622 11973). Smith, I. W. M.. unpubiiihed results.
