EFFECT OF DOUBLE-IMPACT COLLISION ON DISSOCIATION RATECONSTANTS
It will be noted that the limiting linear form of this equation is in excellent agreement with the limiting linear equation obtained by direct measurement. An analogous equation with weight fraction as the independent variable Bz =
1.17409 - 0 , 5 3 4 3 ~ +~8.2824,722.20 (10)
represents the data with nearly the same reliability.
2411
Both equations, however, predict values of the mole fraction a t the minimum which are slightly high, 0.0131 and 0.0134 for eq. 9 and 10, respectively. Equation 9 is, therefore, perhaps the better of the two.
Acknowledgment. The author wishes to express his appreciation to St. John’s University for it,s support of this research in the form of a St,. John’s University summer research stipend.
On the Effect of the Double-Impact Collision on Dissociation Rate Constants
by H.Shin Department of Chemistry, University of Nevada, Reno, Nevada
(Received February 19, 1966)
The effect of the double-impact collision on the dissociation rate constant of the collision B, where RA is a molecule and B is an incident particle, is studied by calcusystem RA lating the probability of the momentum transfer between the collision partners. When the masses of the system satisfy mg/mR = (mA/??zR l)/(mR/mA - l), the exact analysis of the impact process between A of RA and B and, in turn, the explicit formulation of the transition probability are found to be possible. Relatively few energetic particles which just missed the second impact with the molecules predominantly control the magnitude of the rate constant.
+
+
Introduetion In a recent paper’ we have developed an idealized model of the colinear collision between a hydrocarbon R H and a hydrogen atom H to formulate the classical probability of free-radical formation. In this model we have assumed that the mass of R , mR, is much greater than that of H , mH, so that the position of R is practically unchanged during the collision between two hydrogen atoms, that H is attached to R through an inextensible string, and that the interaction is impulsive. Although the model is not realistic, it has revealed some important features of the radical-formation reaction. In this collision system, the bound H atom can suffer at most two impacts by the incident H atom. Therefore, when R H is hit twice by H, the momenta of the two hydrogen atoms are unchanged. In collision theories, the rate expressions are com-
monly discussed without properly considering the effect of the impact m u l t i p l i ~ i t y , ~which -~ invariably brings considerable mathematical difficulties. In the collision between a polyatomic molecule and an inert heatbath molecule (usually a monatomic molecule), the molecule may suffer many impacts before it produces a small fragment(s) if the mass of the fragment is much smaller than that of the heat-bath molecule. On the other hand, if the magnitudes of the masses are reversed, the fragment can most likely be hit only once by the incident particle before it dissociates. (1) H. Shin, J . Phys. Chem., 6 8 , 3410 (1964). (2) T. A. Bak and J. L.Lebowita, Discussions Faraday SOC.,33, 189 (1962); S.E. Nielsen and T. A. Bak, J . Chem. Phys., 41, 665 (1964). (3) E. W. Montroll and K. E. Shuler, Advan. Chem. Phys., 1, 361 (1958). (4) I. Prigogine and T. A. Bak, J . Chem. Phys., 31, 1368 (1959).
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H. SHIN
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Therefore, in the former case, it is necessary to include all possible impacts to formulate the correct rate expression. If all impacts are not taken into account in the formulation, the resulting rate expression may give values which are much lower than the experimental values. In the present paper we consider the dissociation of RA by the collision with B, where R can be either a radical or atom, A is an atom ( = H , D, . . .), and B is an incident atom. When the masses of R, A, and B satisfy certain relations, RA can suffer either one or two impacts by B but not more than two. In such cases we can solve the collision process of the threebody problem analytically. The purpose of this paper is to study the effect of the two-impact collision between A and B on the values of the rate constants for the dissociation of RA. Although the study of the collision process with more than two impacts will give more detailed information about the effect, we feel that the analytical solution of the two-impact process will suffice to draw some conclusions about the importance of the impact multiplicity in collision theories.
t-------PP
t;=O
+
Figure 1. Collision coordinates for RA B. At the moment of the first impact B always moves leftward, while A moves either rightward or leftward.
Method The model is the same as the one considered in ref. 1 except in the present case the mass ratio mR/mA is riot infinite, and mA and mB are not in general identical. The exceptions make the present method different from ref. 1 although the same potential energy surface can be used. To present a more complete account, the main features of the earlier model will be briefly recapitulated. We use the notations and terminologies shown in ref. 1 whenever possible. Consider the collision system of RA B, where A is hit impulsively along its line of oscillation by the incident particle, B. The linear collision coordinate is shown in Figure 1. The atom A oscillates between r1 = 0, the closest approach A can make toward R , and rl = 1. The number of impacts between A and B is dependent on the initial phase of A, Le., the position of A when H reaches rl = 1 from right, on the magnitudes of their momenta, and on the masses of R, A, and B. There are assumed to be three regions in which the potential energies are
+
rl> mA, this expression represents a two-body collision problem with mB = mA, which is equivalent to the origin of the plot of mA/mR vs. mB/mR. The kinematic diagram of the present case, ie., Figure 2, is essentially identical with that of the twobody collision case.' Therefore, it can be readily shown from ref. 1 that the transition probability for the interchange 3i. -+ j per collision between A and B is
+
3 1 a > : > 1
Y
P ( % , j )=
x9
x
(6)
1>:>0 Y
Here, in the 1 > k / j > 0 region, some molecules are hit twice by the incident particles. The fractional P(x,y) (6) D. W. Jepsen and J. 0. Hirschfelder, J . Chem. Phys., 30, 1032 (1969). ( 6 ) R. J. Rubin, ibid.. 40, 1069 (1964).
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value in this region results from the effect of the twoimpact collision.
Rate Constant and Discussion The rate constant for the dissociation of RA can be obtained by integrating the product of the transition kernel z(j)f'(i,y) and the equilibrium fraction of A, which appears in terms of the x-component of the configuration point, over all possible values of j and over values of i > (2MVo)'/'. The collision frequency z ( j ) may be written
formulating K for the one-impact collision case. If we consider only one impact, which corresponds to P(x,j) = 1 regardless of the relative magnitudes of x and j , the rate constant is simply K*
= ze-Vo/kT
(12)
Since the second and third terms in the right-hand side of eq. 11 are significantly smaller than the first
(13)
where c is the concentration of B along the line of motion of each A atom. The total number of collisions per unit time suffered by an A atom is
The rate constant may then be expressed as
When we substitute eq. 6 into eq. 9
With eq. 7 , eq. 10 can be solved in terms of the error function 4 ' ( @ ) where , 6 = (VO/kT)'/'. Although for small 6-values the error function becomes particularly simple, we consider the solutions with large p-values. In such a case the error function may be written
With this approximate expression we integrate eq. 10 to obtain
where - E i ( - z )
=
L-
e-udu, the exponential integral.
An insight into the impact multiplicity is gained by The Journal of Physical Chemistry
from which we see immediately the dependence of K on the transition kernel in the momentum ratio range, 1 > x/j > 0, by comparing with eq. 12. The term ( aVo/kT)'/'is the leading factor obtained by including the collisions in this momentum ratio range. Physically, R should be excited to j to cross the potential barrier of Vo. Therefore, in this model, the conditions for the dissociation of RA are that the incident particle with an energy greater than that of the vibrational energy of A must hit the latter only once and that A is not too close to the origin r1 = 0. If r1 'v 0 (or z 'v 0), the collision process will predominantly involve two impacts; this situation results in the difficulty of the excitation of RA. As Vo/lcT + 03, eq. 11 simplifies to
which shows that only the incident particles which reached the momentum ratio range 1 > x/y > 0 but barely missed two impacts contribute to the limiting value. By comparing eq. 11 and 12, we then obtain
i e . , the inclusion of the contribution of the impacts from the 1 > i / j > 0 region increases K , in the asympwhich is much totic limit, by a factor of (aVo/kT)'/', greater than unity. From the above results it can be said that the most energetic incident particles, which correspond to the motion of the configuration points with small ei values, dominate the over-all magnitude of K . Thus, in the present collision model, the rate may be small a t the most probable energy of the incident particles but can be large a t the high energies in the "tail" of the Boltzmann distribution curve. Acknowledgment. The author wishes to thank M. P. Hanson for making several helpful suggestions.
