H. 0. PRITCHARI)
504
Vol. 65
THE KINETICS OF DISSOCIATIOX OF A DIATOMIC GAS BY H. 0. PRITCHARD Chemistry Department, University of Manchester, Manchester I S , England Received September 21, 1960
The collision theory of chemical reactions is developed to a point where it is capable of describing the mechanics of the dissociation and i*ecombinationof a diatomic gas. The experimental observation of activation energies for dissociation which are substantially less than the heat of dissociation appears to be due t o a failure to maintain an equilibrium population amongst the highest vibrational energy levels. Since the dissociation process is thermodynamically irreversible, there is no simple relation between the dissociation and recombination rate constants and therefore no correspondence betuTeen the heat of dissociation and the activation energies for the two processes.
One of the most unsatisfactory aspects of the under pressure of experimental observation, to theory of chemical kinetics has been the failure to develop a more detailed form of collision theory. A Simple Extension of Collision Theory.-The describe reasonably the simplest of all reactions, the dissociation of a diatomic gas. The rates of dis- normal collision treatment of a diatomic dissociasociation (and their temperature dependences) tion reaction has no fine structure, i.e., it makes no have been measured at high temperatures for three assumptions about the mechanism by which the diatomic molecules, Iz, Br2 and O2 by the shock- ground-state molecule Xz eventually breaks up X. The diatomic molecule has n bound wave technique.' The reactions are biomolecular into X vibrational states,and theirpopulationsareNoe-E*/RT and may be represented as where i runs from 0 to n and all energies are Xp M +X X M measured relative to the ground state. If we make It is convenient to define a standard rate constant for the naive assumption that dissociation may take dissociation, K = Nzoe-Do'RT where N is the total place from any vibrational state, then the relative number of ICz molecules, zo is the number of colli- motion of Xz and the third body I\il must suppiy sions with 2cf per second suffered by a n individual the remainder of the dissociation energy (DO-E,) ; Xz molecule in its zeroth vibrational state, and Do the probability of the relative energy along the line is the dissociation energy of Xz; this is the usual of centres being in excess of this value is e-(Do-fl*)/RT. simple collision theory rate. The experimentally If P i , is an a priori probability which tells us what observed rates of dissociation are 3 . 4 ~for Iz a t fraction of the suitably energetic collisions will lead 1300"1(., 3.8K for Brz a t 1600"K., and 1 5 0 ~ to dissociation, then the rate of dissociation from for O2 a t 3000°K. Furthermore, the temperature any state i will be dependences of these rate constants (i.e., activation P e, .Aloe- E s / R T .z,e -( Do - E I ) / RT = pi A'oz+e-Do/RT energies) art' in all cases substantially less than the respective dissociation energies : for 1 2 the discrep- where zi is calculated using the collision diameter of ancy ranges from 4.2 to 9.5 kcal. according to the X z in its ith vibrational state. However, we cannot nature of M. for Brz from 5.9 to 15.5 kcal., and for write the total rate of dissociation as the sum over O2 it is about 11 kcal. I n addition, the rates of the all i from 0 to n since if any significant dissociation reverse processes have been measured for 1 2 and takes place from the low-lying states, this will Br2: as is well known, the recombinations are have the effect of depopulating the higher states third order i e. and thereby reducing their contribution to the ratesum. It is fairly well established that the approX + X + M+Xz M priate value of P,, is only appreciably for small and the rates vary substantially with the nature of energy jumps (e.g., see later) so that dissociation is &L2 In both cases the recombination proceeds going to be most favourable for the nth state, less more slowly a t higher temperatures3 and we have an so for the (n - 1)th state, and so on. Hence, the apparent negative activation energy. I n a rever- activation of Xz from its ground state to the dissible system, the difference between the recombina- sociated state takes place by a predominantly steption and dissociation activation energies should wise process and the rate constant for the reaction, equal the heat of the reaction, but as the observed provided the Boltzmann distribution is maintained, recornbination temperature dependences are only will be of the order of P,, hio.zne-Do/HT; it will about - 2 kcal., there is still a large discrepancy not be exactly equal to this expression because of between them and the high-temperature dissocia- the effects of some dissociation from lower energy tion results. states. Thus, the maximum rate that we can expect Simple coilision theory accounts for none of to get is with Pnm = 1, Le., k = NOzne-Do/RT, these facts. Nevertheless, the ultimate descrip- and since a diatomic molecule in its topmost vibration of such reactions can only be given in terms of tional state is considerably bigger than the normal collisioii processes, and it is therefore necessary, molecule, this can be greater than the standard rate (1) (a) D. Brilton, N. Davidson, W. Gehman and G. Schott, J. K. It is not easy to calculate the appropriate value Chem. P h y s , 26, 304 (1956); (b) D. Britton and N. Davidson tbzd., of z,, but some idea of its magnitude can be assessed 26, 810 (1956); (c) €1. R. Palmer and D. F. Hornig, zbkd., 28, 98 by considering 5 3 2 : in the ground state, the bond (1957); ld) D.Britton, J . Phyr. Chem.. 64,742 (1960); (e) 9. 4 . Losev, Doklady Translations, Pkys. Chem. S e c f m n , 120,467 (1959). length is 0.74 fi. whereas in the topmost vibrational (2) K. E. Russell and J. Simons, P m c . Roy. Soc. (London), 8217, state (according to the calculation described below), 271 (1953). the bond length is about 3.5 A.; in heavier mole( 3 ) la) D I, Bunker and N. Davidson, J . A m . Chem. Soc., 80,5085 cules having lower force constants, the length of the (195R); (1,) W.G . Givens and .J E. Willard, zhzd.. 81, 4773 (1959).
+
+
+ +
-
+
KINETICS O F DISSOCIATION OF A DIATOMIC GAS
March, 1961
bond in the topmost state will be relatively somewhat greater than this, but in any event, it is unlikely that z, would exceed zo by more than an order of magnitude. Thus we could account for rates up to say ~ O K covering , the experimental data for IZ and Brz, but not the value of 150 for 02. However, sye would still predict a temperature dependence equal to Do and we are not, therefore, much nearer the solution of this problem. For this reason, a more detailed description of the activation process is put forward in this paper, based on an attempt to calculate the probability per collision of a vibrational transition v = i -t j, for all i and j including the continuum. The Elements of the Calculation.-The model chosen for consideration was the reaction Hz+ M + H + H +
M
because the hydrogen molecule has only 15 bound vibrational levels. The molecules of experimental interest have many more, Le., IZ has 125-175, Brz has 100-145 and 0 2 has 50-60, making the necessary calculations impracticably long. The molecule was assumed to be a Morse oscillator, and the necessary constants were chosen to make the spacings of the topmost levels approximately correct (i,e., De = 38284 cm.-', Q = 1.942 X lo8em.-' and lc = 15.25); the energy of the v = 0 -t 1 transition Aol is then overestimated by about 6%. The probability of transition between any two states v = i -+ j was then calculated using the equations given by Jackson and hlott4 for the treatment of the collision of an atom with an oscillator, i.e. 1 pii(w) = 47r2 ( Y i i ) 2sinh ui sinh uj
[cash:: 1li.:l~ -1 uj
where Y;j is the matrix element of the collisional perturbation potential over the Morse wave functions and l3of the two states, and u = 4a2 X mwlah with m = the reduced mass of the colliding pair, w = the relative velocity of approach of the pair, and h = Planck's constant: a is a constant defining the perturbation potential as discussed below. The two subscripts i and j refer to the values of u before and after the transition from v = i -+j . The probability p i j ( w ) is a function of the relative approach velocity w and therefore, to give the required probability p i j , this function has to be integrated over the Maxwell distribution of approach velocities w appertaining to the particular temperature under consideration. For downward transitions, the integral was taken over all energies of approach from 0 to 0 3 , but for the upward transitions, the integral can only run from Ai? to : note that pij refers to a fraction of all colliszons, whereas P,j refers only to s u f i i e n t l y energetic collisions. I n this way, an array of transition probabilities can, in principle, be built up for all transitions between states i and j , whether bound or unbound. In the case of a dissociation process, it is necessary to integrate again over all possible transitions into the continuum, and in the case of a recombination, the second integration must be performed over all values of the approach energies ~3
J. M. Jackson and N. F. Mott, Proc. R o y . SOC.(London), 8137, 701 (1932). (4)
505
of the recombining pair. Knowing this matrix of probabilities, one can then calculate the number of transitions per unit time taking place a t equilibrium, and it is an easy matter to see whether or not equilibrium is likely to be maintained: if it is, the calculation of the over-all rate is straightforward, but if it is not, one can only draw qualitative conclusions because an adequate description of the real situation would then require an exact knowledge of the various probabilities. It should be pointed out that the application of the Jackson and Mott procedure to a light molecule is subject to severe limitations. Some of these have been discussed by Schwartz and Hersfeld,5 and others will be mentioned in this paper. The present treatment therefore contains many drastic approximations, but in all cases these have been made with due consideration to the physics of the situation. These approximations have been made in order to try to make progress. More sophisticated treatments have been given of the u = 0 -t 1 transition, and more formal treatments of dissociation by collisional excitation have been examined; however it is so much easier to understand the mechanics of a complex process like this when numerical magnitudes can be assigned to the basic steps which contribute to the over-all dissociation. The Neglect of Angular Momentum.-In a gas a t a temperature of several thousand degrees, there are many molecules having high angular quantum numbers and it is necessary to consider the effect of rotation on the dissociation process, and what errors are likely t o be introduced by using wave functions corresponding to zero angular momentum. In the first place, if we consider only first-order terms in the angular perturbation, there will be a selection rule limiting changes in angular quantum number to z k l ; in the real case therefore we can expect changes of angular momentum by =!=I unit to predominate. Hence we are justified in neglecting dissociation by the acquiring of a large increment in rotational energy, and can reduce the problem essentially to one of vibrational excitation. We may then consider two separate cases. The first of these is when the molecules have so much rotational energy that a positive energy barrier is present between the bound and the dissociated states. Consideration of these states is ruled out of the present discussion on the grounds that the net energy required for these molecules to dissociate is larger than Do; tunnelling through the barrier will be infinitesimal,6 particularly for Brz and IS,and so these processes will tend to increase the activation energy whereas we are trying to account for the experimental reduction of E below Do. We may therefore confine ourselves to a consideration of the second case, z.e., to molecules having only moderate amounts of rotational energy. Here, the vibrational wave functions are qualitatively similar tn those for zero angular momentum, and as we have ruled out large changes in rotational energy, the over-all ( 5 ) R. N. Schwarta and K. F. IIerzfeld, J . Chern. P h y e . , 22, 707 (1954). (6) D. E. Stogryn and J. 0. Hirschfelder, zbtd., S i , 1531 (1959); 33, 942 (1960).
1%.0. PRITCHARD
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probabilities for any individual state to add up to unity, and we must accept that they are correct relative to each other, and obtain their absolute magnitudes by suitable renormalization. The matrix elements
-v=14
-v=13
L
v
=
B
-V=8
-=,, M - v = 6
, , , , : 2 Ly
= 5
J#Iil-
y =
J & -\
v=3
J:: d 8Z d
Vol. 65
“:1 6
n ..
v
=
4
2
v = l
: . 2
2, = 0 Fig. 1.-Morse wave functions (p2[,2) for the bound states of H2 (all functions shown to the same scale). This diagram was constructed using the Graphical Output of the Manchester University Mercury Computer. The functions extend over 5 4. from p = -0.6 A. to p = +4.4 A . ; the point p = 0 is marked by a dot below the zeroth order wave function.
description of the dissociation process will not be grossly wrong- if we use the simple Morse wave functions. Details of the Calculation.-The Jackson and Mott treatment assumes that the interaction potential between the colliding particles is exponential and purely repulsive, ie., C$ = Ce-ar. The constant a was chosen from the best fit of the experimental data for H2/He collisions7 to an exponential over the range of +/k from 0 to 2000’K. A reasonable value of a is about 5 x lo8 em.-’. A further assumption of the Jackson and Mott treatment is that one member of the vibrating pair has an infinite mass, but of necessity we are limited to a consideration of Hz. The principal effect of violating this assumption will be to alter a in some unknown way and therefore we can only expect to get, a t best, a semi-quantitative description of the situation: we cannot expect the sum of the transition (7) E. A. Mason and W. E. Rice, J . Chem. Phus., 22, 522 (1954).
where p is (T - re) and pz is the average value of p for the ith state, were calculated by a 250-strip Simpson’s Rule quadrature. The wave functions are shown as a plot of p2Ea2in Fig. 1; note that for the state u = 14, the left-hand maximum is almost non-existent and the wave function virtually has one large peak near 3.3 A. internuclear separation. The only difficulty arose in the cases of Y14,14 and Y13,14which diverged (as they must since a > a ) , but it was found that logarithmic plots of Y,,, Ym-1,,, etc. against m gave straight lines, and suitable values for the divergent cases were chosen by extrapolation. Morse wave functions are not known for the continuum, and although various approximations were tried they were unsatisfactory and the following very crude assumption was made: since only small energy jumps are favored, the important part of the continuum will be very close to the discrete states, and so the wave functions will not be too unlike that for the topmost bound state; therefore the matrix elements involving the continuum were taken to be the same as those for the state u = 14. The probabilities ioi, (w)were integrated over the three-dimensional Maxwell distribution of approach velocities using a 10-point Laguerre quadratures ; this integration yielded directly the over-all P , ~for discrete transitions, but for transitions into the continuum, the total probability was found by integration of the resulting pt, over all possible continuum energies E . The individual rates for various processes were then found by applying these probabilities to the collision numbers appropriate to the individual initial states, the collision diameters ui being estimated from the pz. The reason for using the three-dimensional Max.vvel1 distribution was to obtain the maximum possible value for the rate of dissociation, having the value of 1 5 0 ~ for O2 in mind; however, in no case was an increase in rate of excitation of more than about 50% found over that which would be obtained using the normal two-dimensional distribution. Since we have to renormalize the relative transition probabilities, it is necessary to know the probability that no transition mill take place upon collision. This was taken to be pzz(w)= limp,,(w) u>+ urn
=
1
9(YaJ2uz2
-
and integrated over the hlaxwell distribution as before. The transition probabilities were then rep,, = 1 with the added condinormalized so that
x j
tion at high temperatures that none of the f p,,de should exceed e-(DO - E ) / R t T ; this latter condition is due to an overestimate of the probability of (8) H. E. Salaer and R. Zucker, Bull. Am. Math. Soe.. 55, 1004 (1949).
hlarch, 19G1
KINETICS OF DISSOCIATION OF A DIATOMIC Gas
dissociation relative to other processes occasioned by our arbitrary choice of Yte. h short discussion of the failure of detailed halancing in this calculation is relevant a t the present juncture. If this calculation were performed rigorously, one would start with a three-dimensional interaction between the approaching particles, and then what one means by a collision is fully defined in terms of the nature and magnitude of the interaction potential. In the present case we are trying to use a one-dimensional interaction potential to describe a three-dimensional problem. One can, of course, average over all directions of approach, as has been done in previous treatments, but this only applies a constant factor to all probabilities, and is irrelevant since we have to renormalixe. And one is still left with the problem of defining what one means by a collision, and therefore a collision diameter. Thus, if x e consider the transitions v = i + .j and v = j --t i, we should find the number of such transitions taking place a t equilibrium to be equal. If n-e assumed equal collision diameters, equal values of Yi,and YJ2and a two-dimensional Maxwell distribution, this would be so; however, we have already found it necessary on experimental and theoretical grounds to increase the collision diameter n-ith increasing excitation, therefore favoring the jump v = j --t i;however, this loading is to some extent off-set by the non-equality of Y,j and YJ2, and the use of the three-dimensional Maxwell distribution. The states v = i and v = j also have slightly different renormalization constants, with the result that detailed balancing only holds to an order of magnitude. The calculations were carried out for temperatures of 300, 500, 1000, 2000 and 3000OIC A sample of the results is given in Table I as a matrix of numbers of transitions per cc. per second, based on the model of a mixture of 3.5 X 1016molecules of HB and 3.5 X atoms of He per cc. (i.e., 1 mm. Hz and 1000 mm. He a t 273OK.). It is assumed that there is no recombination taking place, and the rate constants obtained are therefore analogous to the initial rates derived from shock-wave measurements. However, bearing in mind the various approximations that have been necessary in this calculation, we cannot expect the results to do more than serve as a useful guide to our thinking on the problem. Transitions between Bound Levels.-By inspection of Table I and similar tables calculated for other temperatures, one can immediately write down two rules: (i) the smaller the energy gap, the more likely i s there to be a transition between two states; (ii) the higher the temperature, the more likely i s a n y given transition. Examining the first rule in detail, we find two factors are involved. Consider the v = 0 --t 1 transition in Table I: we note that out of about 1.3 x lo2' collisions per sec., about 5 x 1020transitions take place, i.e., about 1 in 3 X lo6; however, at collisions have this temperature only 1.5 X sufficient energy to cause the transition to go, so that in fact the efficiency for suficiently energetic collisions is about 1 in 3 x lo3. If we next consider the transition v = 10 -F 11, of 1.5,X 10' collisions
507
2
7 +++++ Nrn.3u)N
a
00
h
$
a
'E
*a
U
v)
505
H. 0. PRITCHARD
per sec., about 2 X lo6 are sufficiently energetic and 3.2 X 106 transitions occur, i.e., we have an efficiency of 1 in 50 of all collisions, and 1 in 6 of sufficiently energetic collisions. I n other words, the number of transitions taking place from u = i +j is P1,N2eze--A%~/RT and both Pij and e - A * l / R T increase as Ai3 decreases. As evidence of the second rule, we observe that a t 30OO0I
