0. K. RICE
A
Vol. 67
TABLE I1 Acid
Malonic Succinic Glutaric
Conen. (moles/ 1.) PH
0.5 .1 .3 .1 .3 .I
K,
b/Au
x
108
K1
Kz
2.54 15.0 & 0 . 3 1.397 X low3 2 . 1 X 1.90 6 . 5 f . 2 2.29 6 . 7 f . 2 6 . 6 3 X 10-5 2.8 X 2.52 4 . 3 f .I 2.35 7.0 f .2 4.54 X 4.54 X IO-; 2.00 1 . 2 f .1
t'hc values of the constants are presented in Table 11. To calculate these constants, models of the complex have been assumed. The model assumed determines the value of x to be used in eq. VI11 and IX. The models which may be assumed for each acid are limited by the results presented in Fig. 3 and 4. The diionized species of malonic acid could, conceivably, form a six-membered ring with uranyl ion. The constants presented in Table I1 are calculated from eq. IX with x equal t o one. Attempts to calculate constants from eq. IX with .2: equal to two ( i e . , a 2 :1 complex) yielded negative results. If a 3 : 2 complex is hypot'hesized-x equal to 1.5-a non-linear p cs. U curve is predicted ; hence, calculations for this model were not att,empted. Figure 7 presents experimental reduced rate us. diionized malonic acid measurements for two concentrations of total uranyl. The solid lines were computed from eq. IX using the constants given in Table I1 with x equal to one. The reasonable fit indicates the validity of the 1: 1 complex. The monoionized species of glutaric and succinic acids (see Fig. 3 and 4) could form a charged 1:1 com-
,--------Acid 1:I
:uranyl---------2:l
k
----Acid 1:l
x
103
:uranyl----. 2:l
x
105
15.3 f 0 . 3
1.3 f 0.6 X
lo2
2 . 2 i 0 . 4 X lo5 20.5 f 8 . 0
8.8rt0.6
8 . 7 f 1 . 1 X 10; 50.4 f 30.4
7 . 8 f0 . 3
4 . 0 st 0 . 2
53.1 f 44.5
plex, or an uncharged 2 :1 complex. The constants presented in Table I1 for glutaric and succinic acids were calculated from eq. VI11 for both x equal to one and x equal to two. Hypothesis of a 3 :2 complex predicts a non-linear pus. U curve. Figure 8 presents experimental reduced rate vs. monoionized acid measurements for two concentrations of total uranyl in solutions of succinic and glutaric acids. The solid lines were calculated from eq. VI11 using the constants for x equal to two in Table 11,and the dashed lines were similarly calculated for x equal to one. From these results it is impossible to conclude the precise nature of the complex; however, it seems probable that both photosensitive species are present. The measurements presented in Fig. 7 and 8 represent a pH range from less than 0.5 to near 3.0 a t the upper limit. Within this pH range, the reasonable fit of the calculated reduced rate to that measured shows that the photosensitive species is 1: 1 (A-* :U027 2, in malonic acid. However, a mixture of 1 : 1 and 2:1 (HA-: is indicated in glutaric and succinic acids.
nTON-EQUILIBRIU~XEFFECTS I N THE DISSOCIATIOX OF DIATORXIC NOLECULES BY A THIRD BODY1 BY 0. IC. RICE Department of CheTnistry, University of North Carolina, Chapel Hill,North Carolina Received M a x h 3, 1966 I n the dissociation of diatomic molecules, the population of energy levels near the dissociation limit falls below the equilibrium population, I n the present paper this is treated on the assumption that equal amounts of energy are removed in all collisions, but attention is given to the change of density of energy levels near dissociation. The effect of the mass of the third body and the effect of the temperature on the average amount of energy removed a t each collision are considered. It is found that a reasonably good qualitative discussion of the relative effect of the different rare gases on the dissociation of 1 2 (or association of I atoms) can be given using these ideas. The mesent considerations are comuared with the calculations of Heck, using the so-called variational theory of chemical reactions.
The recombination of atoms in the presence of a third body has been the subject of numerous experimental studies in recent years. It has been found, in general, that the rate constant for the reaction 2x
+M
----f
xz + ilI
(-4)
has a negative temperature coefficient. This can be explained readily if the reaction goes in steps
x + 2M X.hI + M x.31+ x +xz + 31
(B)
where X . M is a loosely bound complex (generally resulting from van der Waals forces) ; however, in cases where the complex X - M is very unstable (as, for ex(1) Work supported by th? National Science Foundation.
ample, where M is a helium atom), the second mechanism may be expected to be unimportant, and in such cases another cause for the negative temperature coefficient must be sought. It has been suggested2that such a negative temperature coefficient could be explained by redissociation of the vibratsionally excited molecules first formed in the recombination, and some det,ails of such a mechanism have been worked out by Polanyi. It also has been noted by Nikitin and Sokolov4 that (2) (a) R. L. Strong, J. C. V. Chien, P. E. Graf, and 5. E. Willard, J . Chem. Phys., 26, 1287 (1957); (b) B. Kidom, ibid., 34, 2050 (1961). (3) J. C. Polanyi, ibid., 31, 1338 (1959). (4) E. E. Nikitin, Dokl. Akad. Nauk S S S R , 116, 584 (1957); 121, 991 (1958) [translations, Soviet P h y s . Dokladzj, 2 , 453 (1957); 3, 701 (1958)l; E. E. Nikitin and N. D. Sokolov, J . Chem. Phys., 31, 1371 (19B9); E. E. Nikitin, Dokl. A k a d . iVauk SSSR. 132, 395 (1960) [translat>ion,Physical Chemistry Section, Conailltants Ri!reaii, 417 (1961)l.
Jan., 1963
DrSSOCIATIOK O F DTATOMIC MOLECULES BY A
the rate of dissociation of a diatomic molecule, thle reverse of reaction A , will be lowered by the lack of equilibrium in the upper vibrational levels when the reaction takes place, and a discussion, uith some cdculatioris based on yuantunt mechankal transition probabilities, has been given by P r i t ~ h a r d . ~These phenomena are essentially equivalent, any lowering in the rate constant for association being in exactly the same proportion as that in the rate constant for dissociation.6 Most recently, Benson and Fueno' have macle a detailed study of the rate of recombination, taking into account the effect of redissociation of excited vibrational states. Their work is based upon the assumption that the transitions take place only between adjacent vibrational levels, and that the transition probabilities depend upon the vibrational le~7elas in the case of a harmonic oscillator. In formulating their result?,they considered a l'species)' Xz* which consisted of a pair of atoms close enough to be bound together, but not actually bound together, ie., having enough energy to dissociate. Then their formula for the rate constant for association, k,, was
k , = XZK*G(T) Here K" was the equilibrium constant for 2x
(1)
xz*
Z was the collisiou frequency of X2* with 14, X, assumed to be unity, was the probability that the collision would result in association, and G(T)was the .fraction of those which associate which are not dissociated again in subsequent collisions. K* was evaluated by some rough but reasonable geometrical considerations, and G( T ) was found to be roughly equal to the reciprocal of the number of vibrational states within a range of kT of the dissociation threshold. As pointed out by Benson and Fueno, this formula is the logical extension of the collisional theory of atom recombination developed by the writer.8 Actually the formula is equivalent to one in the later article,8bexcept for the factor G ( T )and for the method of evaluating K". Since my method of evaluating K* makes it about proportional to the number of vibrational states within a range of kT from the dissociation threshold, the presence of G(T) about cancels this particular factor. However, Benson and Fueno's estimate of the number of vibrational states in the range noted is muoh smaller than mine. New Calculation of the Dissociation Constants.This writer's preference generally has been to deal directly with the dissociation constant kd, since this involves a binary collision. The contant ha then can be obtained, since the equilibrium constant for reaction (A)
K = k,/kd
(2)
can be found. This turns out to have an especial advantage when one wishes to consider the effect of lack of equilibrium in the higher vibrational states, since the problem then can be formulated much more simply, and ( 5 ) H. 0. Pritehard, J Phgs. Chem., 65, 504 (1961) (6) 0. K. Rice, zbzd , 65, 1972 (1961). (7) S W. BenBon 5 n d T Fueno, J. Chem. I'hys., 86, 1597 (1962). The author is indebted t o Dr Denson for a preprint of thls paper ( 8 ) (a) 0. K. Rice, abad , 9, 258 (1941), (b) Monnlsh. Chem , 9 0 , 3 3 0 (1B59).
THIRD BODY
7
a number of relationships which apparently have been overlooked then can be readily seen. We shall, therefore, proceed in this way, and we shall make somew-hal different assumptions than those made previously [but compare Nikitin4 (1960)l. I n particular, we shall not assume that the transitions occur from one vibrational level to the next, but that they take place with a certain energy increment. Of course, the energy increment in X2will not, by any means, be the same in all collisions ' with M, and a change in rotational energy (which would result from a collision crosswise to, rather than along, the line of centers of X2) also can cause dissociation. In the latter case the effect occurs through a change in the effective potential energy, which is shifted with respect to the vibrational energy of the molecule, so again we may consider that the vibrational state is changed relative to the dissociation energy. We may hope that our assumption of equal vibrational energy increments will give a reasonable approximation if we take the increment equal to the average change of energy actually occurring in collisions in which XZis in the range of energy levels close to the dissociation limit. We believe that these are the important levels in determining the rate of dissociation; in general, the dissociating molecule must ascend the ladder of vibrational levels, but not just one rung at a time, since the rungs are not uniformly spaced; instead, the size of the steps is assumed to be uniform. The collision of 31 with X2 when the latter has enough energy to be nearly dissociated, a t least a t the end of the motion when the X's are widely separated (which will be the most probable situation for a collision), is very much like a collision of M with a free atom X. We may suppose that it can be handled classically, and that, roughly speaking, on the average, exchange of a certain amount of energy takes place. For this reason, it appears that the assumption of uniform energy increments is more reasonable than the assumption that the transitions take place only between adjacent vibrational states; actually, as we will see, the truth may lie somewhere between. I n any case, it is to be understood that we are not attempting to present a finished theory, but only what appears to be a convenient basis for discussion of the experimental results. Let us denote the average transfer of energy as q. Then we may formulate the series of reactions or collisions by means of which the molecule ascends the vibrational ladder by equations of the sort
Xz,i
+
ki
~'Xz,i+l
+M
(C)
k-(i+l,
I n this equation Xz,, represents a molecule in a certain range of energy of extent 7, the energy being marked off in zones of width q, with i representing a higher energy 1. We will let i = 0 for the zone of width 7 than i just below the dissociation limit of Xz. If the reaction (C) comes to equilibrium, then the following relations hold
+
x ~ , i + l / X "= ~ , i(gi+Jgi)ea (3) where the italicized symbols X2,,, etc., represent concentrations, where g, is the number of energy levels in the range i, and where a = q/kT. From the law of microscopic reversibility, a t equilibrium kiXz,i =
k-(i+l)xz,i+l
(4)
0. IC R1m
S From (3) and (4) the usual relation follows
Ici/JLn-(i+l) = (gi+l/gi)ea (5) In general, of course, we do not have equilibrium established, and it is exactly the departure from equilibrium which we wish to investigate. The departure from equilibrium occurs because in the case of the reaction which takes place from the uppermost range, i = 0, which is the one resulting in dissociation x2,o
+ &I
lc-0 --t
2x
+ nr
(D)
Vol. 67
in the relative values of the ci in order to compare with the equilibrium case, for which, by eq. 8, all the ci would be equal. Thus we may as well set co = 1, and if we let c-1 = 0 (equivalent to using eq. 6 to start the iteration), we see that we may obtain all the following c, if e-": is known for each i. Only if special assumptions be made can the problem be solved analytically. If e-*: be assumed to be constant, and if we set b = e-ai, then it is readily seen, by substitution in eq. 11 starting with i = 1, that ci= 1 + b + b 2 +
... + b '
= (1 - b'+"/(l - b) (12) the back reaction does not occur a t the beginning when no X atoms are present. Though the system is not a t Unless ai is quite small, this converges fairly rapidly, equilibrium, it will quickly come to a steady ~ t a t e , ~ and for all i greater than that a t which convergence is and, in the light of the above remarks, we may write practically complete, the levels will have essentially their equilibrium quota. The ratio cO/cmwill give us dXz,o/dt = k-lXz,~- (ko k-o)Xz,o = 0 (6) the fraction of the equilibrium quota in the interval i = and, in general, for all other Xz.i(except the last one) 0, and the ratio of the actual rate constant IC, to k,,,,., the constant which would be observed if equilibrium dXz,i/dt = lC-(i+lJZ,i+l Ici-lXz,i-1 prevailed clear up the vibrational ladder. We see a t (ki k-i)Xz,i = O (7) once, since b < 1, that
+
+
+
The advantage of considering the problem from the point of view of the association reaction is that it is now possible to get all the Xz,iin terms of X2.0 by simple iteration, provided the k , are known. We know that for the lower energy ranges (large i) where the deviation from equilibrium is expected to be small, X2,ishould be proportional to giex,with x = (en - e)/ICT (where B is the energy and eo the dissociation energy). We find that if we suppose, and only if we suppose, that Xo is proportional to ex, then all the higher Xi also will be proportional to ex but with different proportionality constants. We expect the quantum weight g, to be an important part of this proportionality constant ; therefore we set
X Z , ~ Cigie' (8) We substitute this in eq. 7, noting that the exponential factor in Xi+l will differ from that in X, by the factor ea, and that in Xi-l will differ from that in X , by e-5.
We also make use of eq. 5. Thus we obtain kici+lgi
+ k-ici-lgi
- (hi + k-i)Cigi
= 0
(9)
which may be rewritten
+ y-ici-1 - ci = 0 + L i ) and Y-.~ =
yici+l
(10)
+
where yi = IC,/(lcl JLn-.i/(k, k-,). Since we wish to use this equation for iteration, starting a t the top of the vibrational ladder (small i), it is convenient to rewrite the equation once more by writing i for i 1, which yields
+
ci = yi-1-'Ci-1 =
(1
- (q-(i-1)/~i-1)ci-2
co/c, = ks/ka,es. = 1 - b = 1
- epai
(13)
The special case of a "truncated" harmonic oscillator, with uniformly spaced energy levels (but a last one, beyond which dissociation occurs) and in which transition takes place only between adjacent levels and always with the same probability ( i e . , all 16, equal and all IC-, equal) has been worked out for the case of the association reaction by Benson and Fueno. They find that the rate of dissociation is decreased by a factor 1 e-qlkT due to redissociation from the higher levels. We get eyactly the same factor for the decrease in the rate of dissociation if we set a, = a = a/kT, which corresponds to the case of the truncated harmonic oscillator with all kt equa1,l'Jas in this case we would have to have y-ii-l,/y,-l = e-a. It is seen that the result embodied in eq. 13 is by no means confined to the special case of the truncated harmonic oscillator, but has a much greater range of application, the only necessary assumption, aside from the basic aDproximation that the energy is transferred in definite increments, being that aI is independent of i. It is a little difficult to see just how restrictive this assumption is, but it seems likelv that it is sufficiently good to give us at least a rough idea of the behavior of actual systems. Furthermore, it seems reasonable to suppose that ai is fairly close to a. For our subsequent purposes we shall set e-al = fe-a, wheref is not far from unity. Within the limitations imposed by the approximations we have made, we are now in a position to find a formal expression for ICd, as
+ e-'"i)~i-i - e-%i-z
(11) where we haveset y-(i-l,/qi-l = e-@!. This, it will be noted, is the ratio of the probability of a jump upward to a jump downward from the same state. It cannot be obtained from microscopic reversibility, but, in general, we may expect ai to be positive so that the probability of a transition to the higher energy level will be less than that to the lower level. We will be interested only (9) See 0. IC. Rice, J . Phys. Chem., 64, 1851 (1960).
I n this expression, the first bracket represents the collision frequency for collisions between Xs and A I , having (10) From this we see that the detailed working out of this special C ~ R C provides another example, showing, a s previouslyP that the relation K = &,/lCd will not he affected by lack of equilibrium in the higher levels.
Jan., 1963
DISSOCIATIONOF DIATOMIC MOLECULES BY A THIRD BODY
an effective collision distance r ; the quantity ,u iei the reduced mass for such a pair. The second bracket is the equilibrium fraction of the molecixles Xzin the zone i = 0 just below the dissociation; the average enlergy of the levels in this region will be close to ED - 7 / 2 = ED akT/2, so the expression in the numerator represents the partition function for the molecules in this zone, while the expression in the denominator is the part ition function for the unexcited molecu.leP; n is the nurnber of electronic levels which have attractive potential curves with asymptotes a t the same energy, and from which transitions ultimately occur to the ground state (actually the various quantities in the equation must be averaged over these electronic states). The third bracket allows for the lack of equilibrium in the higher vibrational states. The fourth bracket is the probability that a collision will result in an increase in energy, with 0. consequent dissociation of those Xz in zone i Up to this point the expression would be correct for non-rotating molecules. When we average over all rotational statesxbl 2 the factor rm2/re2appears, where rm is the average distance between the nuclei a t the maximum in the effective potential energy curve due to the rotational potential, and re is the equilibrium distance. (This now assumes, which may not be quite correct, that f and a do not depend on the rotational state.) For comparison we write down the expression previously derived, la slightly rearranged -1
Aside from the lack of the third arid fourth brackets this differs from the other expression in the numerator of the second bracket. The quanttity 8ev is the energy between adjacent levels a t an energy 1cT below the dissociation energy. A rough relation between go and kT/ 86" readily can be found. The behavior of energy levels near the dissociation limit is such that, roughiy14 where vD is the quantum number at the dissociation limit, e,. is the energy corresponding to quantum number v, and a is a proportionality constant. For particular values of eD - ev let us designate v by a subscript (Le., v vkT at f D - EV = kT and 2) == vRkT at ED - Ev = alzT). We see that du - VkT = (2kT/c~)'/~,and, by differentiation of eq. 16, 6e, = O(UD - VkT) = (2akT)"z. Noting that go = VD - UakT = ( 2 ~ k T / o c )we ~ / ~see ! thah
Estimation of the Parameters and Discussion of the Data on Iodine.-The quantity f,as we have remarked, will not be expected to differ greatly from unity, and we shall assume f = 1. We now turn our attention to a or 9. It seems certain that 9 will be closely related to the energy of the colliding particle, M. To the extent that the collision may be considered to be a collision between M and one of the atoms X, we may make (gome estimate of the possible transfer of energy from the conservation laws of momentum and energy. The most probable situation will be that in which the atoms in Xzare near their positions of maximum displacement, for over a considerable region in this neighborhood the relative velocity will be slow. Let us therefore consider what happens when an atom il/I of mass mM,moving with velocity vM, strikes head on an atom X of mass mx which a t first has zero v e l o ~ i t y . * ~It J ~ is readily shown that in a free collision (no force field), the atom X will acquire a kinetic energy equal to 4mMmx/(mM mx)2times the original kinetic energy of M. A glancing collision certainly will not be as effective as a head-on collision; on the other hand, some collisions in which the relative velocity of the X atoms is large may be expected to be more efficient than those in which their velocity is small. We therefore shall assume tliat16
+
+
a = 4 m ~ r n x / ( m mA2 (18) This is equivalent to the assumption that, if the atoms have equal mass, the average energy transferred is equal1' to kT, whereas it falls off according to (18) when the masses are different. This is a rough assumption, but seems reasonable for a working hypothesis in such a complicated situation. If it be accepted, it permits an evaluation of r from the experimental values of IC,, (or k,, using eq. 2 ) by means of ec;. 14. Careful experiments on the recombination of iodine atoms in the presence of the inert gases a t room temperature have been made by Christie, Harrison, Norriah, and Porter,'8 and we show in Table I the values of u calculated from eq. 14, as well as the values previously obtained from eq. 15. The calculations using eq. 15 have been discussed thoroughly,8b and the values of u appropriate to eq. 14 have been obtained from them by correcting for the effect of the factors involving a. TABLE I CALCULATION OF COLLISION DISTANCE u IN GASESWITH It u
go = 2a'/'kT/6~iv
Thus the relation between eq. 14 and eq. 15 can be found easily if a and 1are known. If we can estimate these quantities, we can use eq. 14 to obtain, from the experimental data, a somewhat more meaningful v a h e of u,which can then be compared with what we know about the range of the intermolecular forces. (11) This is the classical partition function; but if the same approximation is made in calculating K ,the resulting expression for ha will be correct. (12) 0. K. Rice, J . Chem. Phys., 21, 750 (1953). (13) Reference 8b, eq. 6. (14) This is the type of relation used to extrapolate band spectra to convergence. I t would follow precisely from a Morse potential. Actually, of course, the average molecule has some angular momentum, and its potentialenergy curve has an effective rentrifugal barrier. A curve with a centrifugal barrier actually may be bel ter described by a Morse function, which is oomparatively short-ranged, than would be the curve for a non-rotating molecule. In the latter, the asymptotio approach to the dissociation limiQprobably is controlled largely by the long-ranged van der Waals forces.
9
a
(eq. 15)
-
1 e-" e-"/(l 4-e-") 2a1/2ea/2 u (eq. 14)
A.
FOR
VARIOUS
He
Ne
Ar
Kr
Xe
1.1 0.119 ,1120
1.9 0.472 .376 .384 1.737 3.7
3.1 0.729 .518 .325 2.45 4.8
3.9 0.957
4.9 0.999 .632 ,270 3.29 6.6
.470
,735 5.5
,616
,277 3.16 5.4
The unreasonably small value of r obtained for He has been eliminated; the comparatively large value of u obtained for He by use of eq. 15 suggests that the correc(15) See 0. Oldenberg, Phys. Rev., 37, 194 (1931). (16) This equation should not be used if the molecule X Pis lighter than the atom M, since the translational energy of Xz always could he readily converted to internal energy by collision with E heavy body regardless of eq. 18. (17) Some recent classical calculations by J. C. Keck, Discussions Fapaday Soc., to be published, indicate that most pairs formed by association will be within kT of the dissociation energy. The author is indebted to Dr. Keck for a preprint of this work. (18) M. I. Christie, A. J. Harrison, R. G. W. Norrish, and G. Porter, Proc. Roy. SOC.(London), A281,446 (1055).
0. K. RICE
10
tions involving a are somewhat too large. This is especially so, since the values of u for Ar, Kr, and Xe might have to be reduced when one takes into account the contribution to the reaction due to "sticky" collisions, mechanism (€3). It was shown in ref. 8b that contributions to the reaction rate from this "complex" mechanism should, in these cases, be comparable to the contribution from the collision mechanism, (A). The contribution from the complex mechanism might have been slightly overestimated in ref. 8b because no account was takenof the effect of the rotational potential in making a complex X.31 less stable, but the contribution aurely should be of considerable importance for xenon.lg Although the corrections for lack of equilibrium as obtained by using eq. 21 to calculate a may be somewhat exaggerated, there seems to be no doubt that they should be made, and that our calculation gives the right order of magnitude. It is quite interesting that a major part of the effect of a limited transfer of energy due to difference in mass arises from the factor 1 - e-a, which expresses the effect of lack of equilibrium in the higher energy levels. The Temperature Coefficient.-We return now to a consideration of the temperature coefficient of kd, and in this respect eq. 14 appears at first to be something of a disappointment. Using eq. 17, it readily may be seen that if n, a, u, f, and rm are independent of temperature, the expression (14) also will be temperature independent aside from the factor e-eD/kT. Actually, r, may be expected to decrease slowly with temperature. If the more distant part of the potentialenergy curve may be supposed to have the form -c/r6, where c is a constant, then for any particular value of the rotational quantum number j the value of rmcan be found by setting
The average value could t,hen be approximated (Le., t'he corresponding value of j found) by setting j(j l)h2/ 87r2p1,rm2 = IcT, which makes r m proportionalz0 to T-'16, Thus rm2is proportional to T-'/8 and the increase in the rotational potent'ial probably also will cause a slight decrease in go, so that we may say that there is an over-all fact,or of about, T-'I2 if n,a, u, and f are constant. When we calculate IC, from led, using K , t'he factors T-'/%and e - Q / k T both are cancelled, leaving IC, t,emperature independent. However, actually, as noted in the Introduction, k, appears to decrease with temperature, varying at least inversely proportionally to temperature with helium as the inert gas, and being somewhat more dependent on temperature with argon. I n the case of argon this could be in part
+
(19) I t may be t h a t all the values of u should be increased by a factor (5/3)'/2, since n,the number of attractive potential energy curves t o whicli association can occur, was taken as 5. It is possible that n should have been taken as 3, since the potential energy curve for thea& state (of degeneracy 2 ) is very shallow, so t h a t redissociation from i t may occur before stabilization b y transition t o another potential-energy curve with a deeper minimum. I n general, transitions between potential-energy curves should he relatively easy, even if requiring change in multiplicity, in a case like iodine, where the coupling is largely j - j coupling. The argument of D. L. Bunker, J . Chern. Phys., 82, 1001 (1960), t h a t TI should be unity, with only the ground state, contributing, does not seem convincing, since the B r h 3 probably has a considerably deeper mininnwisb than he suggests and it will only be necessary for the rate of non-adiabatic transfer t o be rapid for a range of energies a few times n below the dissociation limit for any given potential-energy curve to make a n effective contribution. (20) See B. Widom, J . Chem. Phys., 31, 1027 (1959).
Vol. 67
accounted for by the decreasing contribution of the complex mechanism. This, however, apparently cannot account for all the effect. Benson and Fueno believe that the temperature coefficient can be accounted for by increasing lack of equilibrium at higher temperatures, this being based on the assumption that 7 is independent of temperature, rather than a. It appears to this writer, however, that the average amount of energy transferred at a collision must increase with the amount of energy available. It has been seen, however, that our assumptions are not completely correct, and it may, indeed, be possible that 17 increases less rapidly with T than was supposed. The classical description of the collision occurring between a rare gas atom and an iodine atom breaks down when the iodine atom moves toward the other iodine atom and comes strongly into its field. It is well known that, because of increasingly poorer overlap of the wave function, large transfers of energy are less probable than small transfers. Thus, the transfer of energy may well fail to keep up with the temperature, causing a to decrease with temperature. This, together with a possible decrease of f and of CT with increasing temperature (the latter because glancing collisions are likely to be less effective at high temperatures where they last a shorter time) may well account for the temperature coefficient. Also, some of the temperature coefficient may, as suggested in reference 8b, arise from an effective decrease in n,since redissociation from an excited electronic level would become more likely the higher the temperature. Unfortunately, it is difficult to make quantitative estimates. It is probably the collisions, which take place when the iodine atoms are close enough together to be in the region where their mutual potential energy changes relatively rapidly and quantum effects are of some importance, which make a greater than given by eq. 21. Therefore, it may be relevant to consider the effect of overlap of wave functions over the range of molecular attraction as a whole to see whether a decrease in a in the relevant range of temperatures, say from 300 to 1500°K., is to be expected. The de Broglie wave length for the vibration of Iz is given by the well known where p, and E , are, formula X = h / p , = h/(2pCI,~,)1/2, respectively, the radial momentum and the radial kinetic energy. We thus find dX/de,
=
-'/PX/E~
whence we estimate that if we change er by 1cT a t T = 3OOOK. and take er as the dissociation energy of 12, we would get a relative change in X of approximately 0.008. However, the average kinetic energy of a pair of iodine atoms in their mutual field in the ground state near the dissociation limit is less than the dissociation energy, so we should perhaps estimate dh/X as about 0.02. Now the number of energy levels in the ground state is about 150, meaning that there are about 75 de Broglie wave lengths near the dissociation, and the change of phase will be about 1.5 wave lengths, which gives relatively good overlapping of the wave function over a range comparable to that in which interatomic forces change appreciably. However, with de, = kT a t T = 15OO0K.,the overlapping would be much less, SO the transition mould be considerably less probable. Similar calculations would be expected to give similar results for the excited states, since both and the
Jan., 1963
FOBMATION OF BEXZENE IN
~ ~ A D I O L Y S I SOF
ACETYLENE
11
of iodine atoms with benzene as a third body indicates number of quantum levels would both be smaller, but that the mutual potential energy of iodine atoms and these factors would not compensate completely. I n benzene is almost the same as that of iodine molecules any case, the suggestion that a should decrease appreciand benzene. ably with temperature appears to be reasonable. Although the mass of the third body does enter one Comparison with Theory of Keck.-It seems desirof his boundary conditions, Keck states that the results able to conclude this paper with a comparison with the are independent of the mass of the third body. There recent work of 1
