134
A. W. Yau and H. 0. Pritchard
The Journal of Physical Chemistry, Val. 83,No. 1, 1979
Toward a Unified Master-Equation Theory of Thermal Decomposition Reactions. Analytic Solution for Diatomic Dissociation Andrew W. Yau and Huw 0. Pritchard” Centre for Research in Experimental Space Science, York University, Downsview, Ontario, Canada M3J 1P3 (Received June 28, 1978) Publication costs assisted by the National Research Council of Canada
The master equation is formulated and solved analytically for a ladder-climbing model of the dissociation of a diatomic molecule diluted in a heat bath. The dissociation rate is explicitly related to the normal modes of relaxation of the internal degrees of freedom of the molecule giving, in closed form, an expression for the pseudo-second-order dissociation rate in terms of the energy-levelstructure and collisional transition probabilities. This form of the rate constant shows clearly the criteria for the occurrence of “bottleneck” and “network” effects. The dissociation of Hzin Ar is studied in detail between 2000 and 6000 K. The principal qualitative conclusions in respect of the rate of dissociation are that (a) nonequilibrium effects are significant even at 2000 K; (b) rotation enhances the reaction rate, although it is only at very high temperatures (above 6000 K) that rotational energy contributes equally with vibrational energy in causing dissociation; (c) the presence of tunneling gives only a very modest enhancement in the rate. The relative contributions to the Arrhenius temperature coefficient of the rate of the following theoretical constructs are delineated: (i) temperature dependence of the internal relaxation rate; (ii) vibrational disequilibrium; (iii) rotational averaging; (iv) rotation-vibration interaction; (v) rotational disequilibrium; (vi) tunneling. The predicted rate of dissociation including all of these effects agrees well with experiment over the entire temperature range studied. Effect (i) has a tendency to increase the Arrhenius temperature coefficient of the rate whereas all the others decrease the temperature coefficient, the order of importance being (ii) Z (iii) > (iv) > (v) >> (vi) in the range 2000-6000 K. A parallel set of calculations on the dissociation of D2 in Ar also gives results in good agreement with experiment. The principal determinants of the kinetic isotope effect are the differences between H2 and Dzin the internal relaxation rates and in the energy-level densities. The effect of energy-level density in itself is rather complicated in the sense that, given the same internal relaxation rates for both molecules, the heavier isotope would dissociate faster at low temperatures, but more slowly at high temperatures. It is shown that in diatomic recombination a t room temperature and below the bottleneck occurs in the region of the dissociation limit, making it possible to solve very simply for the reaction-rate eigenvalue of the relaxation matrix by truncating the matrix and by assuming that all levels lying more than 3000-4000 cm-’ below the dissociation limit remain in equilibrium at all times. Carrying through this calculation with an assumed set of transition probabilities confirms that both the total rate and the ortho-para ratios in recombination are virtually unaffected by the presence of extensive tunneling into certain states. This state of affairs is most easily rationalized as the removal of the top of the rotational barrier when the tunneling rate is very high, and it is clear from the results of this calculation that recombination will always lead to an ortho-para ratio of approximately 3 to 1. These conclusions, although they are in agreement with the experiments of Bonhoeffer and Harteck, are in direct conflict with the results of orbiting resonance theory, and it is shown by performing the orbiting resonance calculations with the same set of transition rates that there is a fundamental fallacy in the orbiting resonance formulation which makes its predictions in respect of both total rates and ortho-para ratios simply artefacts of the model. New experiments and new theoretical work, required for a final and complete reconciliation between theory and experiment for the low-temperature recombination of hydrogen atoms in the presence of an inert gas as third body, are proposed.
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Introduction The internal relaxation of a molecule, or to be more precise, the bulk isothermal relaxation of the rotational and vibrational energy of an assembly of molecules in an inert heat bath, can be described very simply and compactly by a set of normal modes of relaxation. Using these normal modes, ic is an easy matter to calculate the evolution of the population distribution vector and of the total energy for the relaxing molecules, whence it follows that the measured relaxation time must depend upon the initial population distribution and thereby on the method of measurement ~ s e d . l - ~ If n(t) is the population distribution vector (i.e., no,nl, n2,...) at time t for the relaxing molecules, the evolution of the population distribution is determined by
d -n(t) =An(t) dt
-
where A is a matrix in which the off-diagonal elements a,, are the transition rate constants for processes j i, and the diagonal elements aji are chosen in such a way as to 0022-3654/79/2083-0134$01 .OO/O
make the asymptotic behavior n(t) ii where 5 (i.e,, Eo, AI,E z , ...) is the equilibrium d i ~ t r i b u t i o n .The ~ solution of eq 1 proceeds by symmetrizing the matrix A using the detailed-balancing relationships, viz. E E e,, = 8,,AL
B = E-1/2AE1/2
(2)
This matrix can be diagonalized by a matrix transformation
B = SAST to give a series of eigenvalues 0 = >> L
Xi.-3
(3)
... L A0
(4)
where N is the order of the matrix (Le., the number of levels in the molecule) and the X are the time constants associated with each normal mode of relaxation. The zero eigenvalue and its corresponding eigenvector
SL,J+l=
A,l/2
(5)
ensure the attainment of the Boltzmann equilibrium
0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83,
Diatomic Dissociation and Recombination
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distribution as t m; this we call the conservative (or nondissipative) case. In the dissilpative case, i.e., when molecules are lost from the assembly by reaction, eq 1 is modified to
No. 1,
1979
135
precisely than blefore, using numerical experiment.l')
The Rate Constant We summarize here the results of our two earlier
paper^^^^ and reciast them in a form suitable for treating where q ( t ) is the (now decaying) population vector and D is a diagonal matrix whose elements D,, = d , are (minus) the rate constants for annihilation of molecules from state i of the system. The eigenvalues given by eq 4 are now perturbed to a new set y,where (7) 0 > y&-1>> yN-2 2 YN-3 ... LYo and5-7the raite constant for the reaction kobsd
=
(8)
-YN-1
Thus, the rjolution of the problem of determining the rate constant for a thermal dissipative reaction reduces to the calculation of the perturbation of the normal modes of internal relaxation (determined by eq 1) due to the presence of the dissipative terms d, in eq 6. The nature of the dissipaitive term d, can of course be very different from case to case. For example, in a thermal unimolecular reaction it could represent the rate constant for the first-order microscopic process CH,NC(i) CH3CNG)
-
whereas in the thermal ionization of hydrogen atoms, it would represent the rate constant for one of the second-order processes H(i) -t e-
-
H+
+ e- + e
In the particular case of diatomic dissociation, d, must represent th(2 sum of the collisional and (if any) the tunneling processes, viz. M H,(u,J) M +H +H
+
-
Hz*(v,J)
--+
H
the energy-transfer component of the diatomic dissociation-recombination problem. The reactant molecule Xz, possessing a sei of rotation-vibration states (u,J;l, is considered to be diluted in a heat bath of structureless inert-gas moleculles M, and transitions among the various internal states are induced by collisions with M. Molecules in the topmost vibrational level of any particular rotational state may dissociate, either by collision-induced diseociation or, in the case of a quasi-bound level, by rotational predissociation in competition with collisional dissociation and deactivation, The model is dissipative, Le., the recombination reaction is ignored, a simplification justified throughout a relatively long period of time following the initial transients (Le., t > I Y N - ~ - ~ ~but ) , before the backreaction becomes importantagThe temperature of the heat bath remains unchanged during the reaction, and the heat-bath molecules maintain their translational equilibrium; also, the system is sufficiently diluted that the mutual interactions between the X2 molecules can be neglected. T o begin with, rotational equilibrium is assumed among the reactant molecules. Hence, the master equation is formulated in terms of the set of levels forming a "vibrational lad der" of fixed J,I1 and the total rate will be averaged over all J . We shall relax this rotational restriction later when we discuss the implications of these assumptions and their validity. For a vibrational ladder with N vibrational levels, we have
V,V'
+H
where Hz*((u,.r) is a quasi-bound state, and H2(u,j)is any state, bound or quasi-bound. Consequently, the whole range of explerimental behaviors observed in thermal decomposition reactions, all the way through from the collisional ioiiizationi of atoms to the conventional unimolecular reaction, depends solely upon the nature of the d , terms in eq 6 and upon the way in which they perturb the normal mlodes of internal relaxation of the molecule. We have shown that in certain cases, eq 6 can be solved analytically, igiven a knowledge of the solutions of eq 1. The first such case is when the elements of A are statistical and obey the "sum rule", and in this way we have recently been able to re-formulate the theory of unimolecular reactions in a wily which is conceptually simpler than RRKM theory, and which gives slightly better agreement with e ~ p e r i m e n t . ~A second case is when A contains only nearest-neighbor transitions (the step-ladder model) nnd there is only one decay term d,, and using this, we have been able to treat very effectively the ionization of hydrogen atoms in dense plasman8 It is by now well established that, especially a t lower temperatures, the dissociation of a diatomic molecule is reasonably well represented as a ladder-climbing p r o c e s ~ , and ~ J ~ in this paper we apply our perturbed normal-mode technique to this problem. Not only does this approach give excellent results for tht: calculated rate constants where sufficient ancillary data are available for the method, but the existence of an analytic expression for the rate constant enables us to examine the causal relationships much more
M
...,N -1
= 0, 1, 2 ,
+ X.z(N- 1) &(N - 1)
QAT)
X
+X +M
-x + x kN-l
where quu,(T) is the rate constant for the transition of X2 from state u t o sitate v'at temperature T likewise, qa.(T) is the rate constant for collision-induced dissociation, and kN-l is the rate constant for predissociation. In these telrms, the elements of the matrix A and D are
a,] =
[iWM - 4Jq,,
- &C(1.- S,k)qihJ k
(9)
and d, = 0 for i
cUl > -500 cm-') states, their total contribution to the recombination rate exceeds the total contribution from the (four to) six tunneling states singled out in the orbiting resonance theory. As we argued in the preceding section (for lowtemperature dissociation) the bottleneck occurs at the ( N - 1)th level when tunneling is absent, but at the ( N - 2)th
,I 0
2
4
6
IO -
0 -_ 8 M 12
14
16
ROTAT!ONAL OUNTOM
I8
20 22 24 26 28 30 32
NUMBtR J
Energy-level diagram for paraH, in the ground electronic state. The vibrational quantum number vis given for each vibrational level. The energy-level diagram would be appropriate for calculations in the temperature range of 50-100 K. Figure 7.
level when tunneling is present. Thus, activation-deactivation ladders terminating in short-lived quasi-bound states are at no significant advantage with respect to other ladders since, as we have pointed out the only net effect of the existence of a short-lived quasi-hound state is to cut off the top of the rotational harrier a t the energy level corresponding to the state in question. The fundamental fallacy in the existing formulation of the orbiting resonance theory lies in its failure to take explicit account of the mutual interactions between the reactive states. The simultaneous presence of reactions from different states leads to nonzero fluxes between these states unless all states are in equilibrium. Thus, it is not permissible to assume that some reactive states are in equilibrium (and treat them, for example, using eq 45) and to ignore the other states altogether. Rather, an explicit nonequilihrium treatment is essential?? The main task remaining, therefore, is the identification of these mutual interactions, which can only he achieved through the calculation of the transition probabilities in the interesting region. Fortunately, we can now pinpoint fairly closely from the results presented in this section the
148
The Journal of Pbysical Cbemistry, Vol. 83, No. 1, 1979
particular transition probabilities which need to be calculated in order to achieve a final reconciliation between theory and experiment for the recombination of hydrogen atoms in the presence of an inert gas as third body. We may deduce from the results in Table IV that the only collisional probabilities which must be calculated are between states lying in the range -4000 < cud < 5000 cm-l for 300 K and in the range -2500 < cuJ < 500 cm-' for 77 K. The most important transitions can be identified from the relevant energy-level diagram. We present only the diagram for para-H, in Figure 7 . First, there are the simple rotational transitions, lAJl = 2, Au = 0. Then there are two very important ladders having A J = lt2, Lu = ~1 commencing a t J = 20, u = 6 and J = 16, u = 9, respect i ~ e l y .Next, ~ ~ we have to consider the importance of near-resonant transitions for which IAq > 2, Au # 0. This can be shown by repeating the full nonequilibrium calculation using, instead, the set of transition probabilities eq 16 of ref 11 (in which the [-51 in the exponent of eq 44 is replaced by [-2.51AJ1]thereby penalizing transitions of large IAJI severely). whence the recombination rates are reduced by about 10%. Several such cases can be identified in Figure 7 , for example, J = 16, u = 8 to J = 22, u = 5 or J = 4, u = 13 to J = 8, u = 12. Finally, two kinds of collisional dissociation process must be considered first, the traditional dissociation process in which dissociation takes place vertically, Le., a t constant J , and second a rotational dissociation process in which the quasi-bound state is dissociated at constant energy by removal of rotational energy.34 Looking, for example, a t the state J = 16, u = 10, we might expect the latter process to be much more important for this particular state. Note however that in a proper treatment of collisional dissociation, these two specific processes are no more than two special cases in a whole continuum of possible processes. Unfortunately, because of the proximity between the bottleneck and the dissociation process for each ladder, these relatively few probabilities will need to be known, in general, quite accurately (we are already within a factor of 2.5 by sheer guesswork, eq 44 or its modified form just mentioned) but, as we have explained previously,1° the precision of the remainder of the -(m2/2) required probabilities is not very critical.
Acknowledgment. This work was supported by the National Research Council of Canada. References and Notes H. 0. Pritchard and N. I.Labib, Can. J . Chem., 54, 329 (1976). H. 0. Pritchard, Can. J . Chem., 54, 2372 (1976). A. W. Yau and H. 0. Pritchard, Can. J . Chem., 55, 737 (1977). E. W. Montroii and K. E. Shuler, Adv. Chem. Phys., I , 361 (1958). D. L. S. McElwain and H. 0. Ptitchard, Symp. (Inti.) Combust., [Proc.], 73tb, 37 (1971). H. 0. Pritchard, Can. J . Chem., 55, 284 (1977). A. W. Yau and H. 0. Pritchard. Can. J . Chem., 56, 1389 (1978). A. W. Yau and H. 0. Pritchard, h o c . R . Soc. London, Ser. A , 362, 113 (1978). D. L. S. McElwain and H. 0. Pritchard, J. Am. Chem. Soc., 91, 7693 (1969). H. 0. Pritchard, React. Kinet., I , 243 (1975). T. Ashton, D. L. S. McElwain, and H. 0. Pritchard, Can. J . Chem., 51, 237 (1973). E. Kamaratos and ti. 0.Pritchard, Can. J . Chem., 49, 2617 (1971). D. L. S. McElwain and H. 0. Pritchard, Can, J . Chem., 49, 3915 (197 1). N. I.Labib, D. L. S. McElwain, and H. 0. Pritchard, Can. J . Chem., 50, 897 (1972). K. J. Laidler, "Theories of Chemical Reaction Rates", McGraw-Hill, New York, 1969. L. Landau and E. Teller, Phys. 2. Sowjetunion, IO, 34 (19361. This calculation assumes Do = 36117.5 cm'l, w, = 4162.1 cm- , whence there are nine vibrational levels, with the topmost level v = N- 1 = 8 at cN-l = 33297 cm-l and the absorbing barrier is placed at cN = Nu, = 37459 cm-l; the qv,v+lobey the Landau-Teller rule.
A. W. Yau and H. 0. Pritchard For example, for the H,-He system, quantal resultsig give 91,2/90,~ 4 for J = 0 at 500 K, instead of 2 as predictedby the Landau-Teller rule. H. Rabitz and G. Zarur, J . Chem. Pbys., 62, 1425 (1975). R. J. LeRoy, Theoretical Chemistry Institute, University of Wisconsin Reoort WIS-TCI-387. Jan 1971. R. N. Schwartz, Z. I.Slawsky, and K. F Herzfeid, J . Chem. Phys., 20, 1591 (1952). J. Keck and G. Carrier, J . Chem. Phvs., 43, 2284 (1965). In a recent model calculation, Sniderz4speculated that the energy-level spacings near the dissociation limit would play an important role in determining the overall reaction rate. The present analysis shows, however, that the peak of the bottleneck shifts toward the lower vibrational levels, meaning that the high-lying vibrational levels become less important at high temperatures, as we have argued before. He also concludes that nonequilibrium effects can be neglected below 3000 K, which is inconsistentwith our analysis; nonequilibrium effects are important even at 2000 K with the SSH model of transition 0.15), as can be confirmed by inspection of curves probabilities(4 I and I1 of Figure 2. N. S. Snider, Can. J . Chem., 55, 3464 (1977). E. Kamaratos and H. 0. Pritchard, Can. J . Chem., 51, 1923 (1973). T. Ashton, Ph.D. Thesis, York University, Toronto, 1972. J. E. Dove, D. G. Jones, and H. Teitelbaum, J . Phys. Cheni., 81, 2564 (1977). H. Rabitz and S. H. Lam, J . Chem. Phys., 63, 3532 (1975). J. E. Dove and D. G. Jones, Chem. Phys. Lett., 17, 134 (1972). H. 0. Pritchard, J . Phys. Chem., 65, 504 (1961). H. S. Johnston and J. Birks, Acc. Chem. Res,, 5, 327 (1972). B. J. McCoy and R. G. Carbonell, J . Chem. Phys., 66, 4564 (1977). H. 0. Pritchard, Can. J . Chem., 51, 3152 (1973). H. 0. Pritchard, Acc. Chem. Res., 9, 99 (1976). N. C. Blais and D. G. Truhlar, J . Chem. Phys., 65, 5335 (1976). C. S. Lin and D. Secrest, J . Chem. Phys., 67, 1291 (1977). J. E. Dove and S. Raynor, Chem. Phys., 28, 113 (1978). R. E. Roberts, Ph.D. Thesis, University of Wisconsin, Madison, Wis, 1968. A. I.Osipov and E. V. Stupochenko, Sov. Phys. Usp.,6, 47 (1963). J. E. Dove and H. Teitelbaum, Chem. Phys., 6, 431 (1974). M. M. Audibert, C. Joffrin, and J. Ducuing, J , Chem. Phys., 61, 4357 (1974). M. Warshay, J . Chem. Phys., 54, 4060 (1971). K. N. C. Bray, J . Phys., B1, 705 (1968). I n eq 30, g = n2co,l(28p)1'2 = 0.4589coel(/t/ T)"' where l is the interaction distance in A, p is the reduced mass of the colliding pair in amu, and w, is in cm-l. For H,/Ar, l = 0.297 A45 and o, and x, are 4395.2 cm-l and 0.02829, respectively." @(v) = [3 exp(-2yV/3)]/[3 - exp(-2yo/3)], where yv = g[ 1 - 2(v+ I)x,], and may be taken as unity for all temperatures of interest. The only variation we have made to this model is to use the actual energy levelsz0of H, rather than those given c, = vco,(I - x , - vx,) as the latter formula does not represent very well the energy levels near the dissociation limit. J. G. Barker, J . Chem. Phys., 41, 1600 (1964). For very high J ( J = 36-39 in H2) the ladder-ciimbing model is no longer appropriate because there is only one bound ievei in each state, and the (1 term in eq 14 diverges; fortunately, at the temperatures of interest, these states make an infinitesimal contribution to the overall rate, and can be ignored. There is also a small variation of x, with J , which also tends to increase the rate of higher J , but for simplicity we have ignored this, and used the constant value given in the preceding footnote. A. L. Myerson and W. S.Watt, J . Chem. Phys., 49, 425 (1968). J. P. Rink, J . Chem. Phys., 36, 262 (1962). E. A. Sutton, J . Chem. Phys., 36, 2923 (1962). W. D. Breshears and P. F. Bird, Symp. ( I n t l . ) Combust., [Proc.], Mtb, 211 (1973). R. W. Patch, J . Chem. Phys., 36, 1919 (1962). A typographical error was introduced into the penultimate line of p 244, ref 10, during the final editorial corrections: Ivshould, of course, read A€. J. E. Dove and D. G. Jones, J . Chem. Phys., 55, 1531 (1971). M. H. Alexander, Chem. Phys. Lett., 40, 101 (1976). For D,. I = 0.297 A, w e = 3118.1 cm-', and x , = 0.0199. J. H. Kiefer and R. W. Lutz, J . Chem. Phys., 44, 658 (1966). J. B. Moreno, Phys. Fluids, 9, 431 (1966). J. P. Rink, J . Chem. Phys., 36, 1398 (1962). D. Appel and J. P. Appleton, Symp. (Intl.) Combust.,[Proc.], 15th, 701 (1975). On two previous occasion^^,^^ the Toiman criterion has been used to estimate the activation energy for these dissociation reactions. However, in calculating the average energy of those molecules reacting, the component of the activation energy contributed by M in the reactive collision was incorrectly neglected. Equation 38 does not neglect this contribution, but, nevertheless, the relationship between D,, and e,, remains to be evaluated because of additional terms in the nonequilibrium expression for the Tolman activation energy.6z D. G. Truhlar and N. C. Blais, J . Am. Chem. SOC.,99, 8108 (1977). L. S. Kassel, "Kinetics of Homogeneous Gas Reactions",The Chemical Catalog Co., New York, 1932.
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Diatomic Dissociation and Recombination
S. H. Bauer and C. S.Tsang, Phys. Nuids, 6, 1783 (1963). A. J. Stacf?, Chem. Phys. Leff., 55, 77 (1978). W. L. Hogarth and D. L. S. McElwain, Proc. R . SOC.London, Ser A , 345, 251, 265 (1975). At low temperature, k,,, obtained by combining eq 16, 17, and 42 reduces to the steady-state solution of Keck and Carrier." Using eq '16 and 19-23, together with the observed vibrational relaxation rates4' and the tunneling rates of Whitlock et aLe8values at 300 K, N at 77 K! of q5 N ( k d , J k e ) N P.A. Whitlock, J. f. Muckerman, and R. E.Roberts, J . Chem. Phys.. 60, 3658 (1974). The recent study of C:spdissociation near room temperature7' should not be conddered as a "low-temperature" dissociation-recombination reaction in this context: DolkT N 16 in these reactions, and the activation energy for dissociation should be less than Do, just as it is for H, at 3500 K, and for all the same reasons, cf. Figure 3. H.-J. Glas ,and H. G. Weber, Z. Phys. A , 284, 253 (1978). D. W. Trainor, D. 0. Ham, and F Kaufman, J Chem. Phys,, 58, 4599 (1973). L. P Walkauskas and F. Kaufman, J Chem. Phys., 64,3885 (1976). D. N. Mitchell and D. J. LeRov. J . Chem. Phvs.. 67. 1042 (19771 R. E. Roberts, R. B. Bernstein, and C. F. Curiiss, J . Chem.'Phys., 50, 5163 ('1969). R. E. Roberts, J . Chem. Phys., 54, 1422 (1971). M. M. Menzinger, Chem. Phys. Lett., 10, 507 (1971). V. H. Shui and J. P. Appleton, J . Chem. Phys., 55, 3126 (1971). The thermodynamic ratio (Le., pp /,f,f where the pf's are the partition functions for odd and even rotation-vibration states, respectively) is sometimes referred to as the "statistical" ratio in the literature. 'To avoid any further confusion, we propose to make the explicit distinction beiween the thermodynamic ratio as defined here and the ratio of the statistical recombination rates which arises from the nuclear spin degeneracy. A. Farkas, "Orthohydrogen, Parahydrogen and Heavy Hydrogen", Cambridge University Press, Cambridge, 1935. K. F. Bonhoeffer and P. Harteck, Naturwissenschaften, 17, 182 (1929). K. F Bonhoeffer and P. Harteck, Sitzungsber. Preuss. Akad. Wiss., 103 (1929). The deactivation cross section a,( T ) is defined as &( T)/ v( T ) where CfvJ( T ) is the sum of all downward transition rate constants from ( v , J ) to all other (v',J') below, and v( T ) is the thermal velocity. ~ they were (83) This was pointed out by Shui and A p p l e t ~ n , 'although not able to demonstrate the magnitude of the nonequilibrium effect.
Discussion J. TROE(Institut fur Physikalische Chemie der Universitat
The Journal of Physical Chemistry, Vol. 83,
No. 1, 1979
149
Gottingen). My question and comment concern the applicaition of this theory to unimolecular reactions of polyatomic molecules, (a) The specific rate constants h ( E )have been forced earlier, e.g., in Slater's book on unimolecular reactions, to a form as used here by postulating k , == A exp(-E,/hT) (where Eo is the threslhold energy at 0 K). Whu:re does the present approach go beyond what Slater has given earlier? It has been shown experimentally that k , does not have this form. The apparent activation energies Ea, have been shown to (differ from Eo for simple bond fission reactions where Eo = A H o 0 is known independently. Therefore, it seems to me that the proposed technique is unrealistic. For complex bond fission reactions, unfortunately, we have no handle on the barrier height Eo;however, it appears highly improbable to me that E,, = Eo is fulfilled here in contrast to simple bond fissions. (b) Chemical and optical activation experiments which cover larger energy ranges than thermal activation experiments show that k ( E ) values as derived from forcing k , to 4 exp(-Eo/RT) with E,, = Eoare not adequate. Therefore, if a falloff curve from this me1 hod agrees with experiments at a given temperature, it will differ from it for experiments at very different temperatures. The RRKM technique will do much better there, since it uses a physically more reasonable h(E).
H. 0. PRITCHARD. Professor Troe's comment is really addressed to our earlier paper on unimolecular reactions' rather than to this one, which deals with diatomic dissociation. Summarizing earlier discussions of this p r ~ b l e r nwe , ~ would ~ ~ like to re-emphasize the following point.' 'The Arrhenius expression is not an exact representation of the high-pressure rate constant. Nevertheless, it is an extremely good approximation4 and, because of the linearity properties of the Laplace transform, does leadl to a good approximation to the exact inverse transform, k ( E ) . Notice that both A, and E , , in the Arrhenius expression are empirical quantities. N o assumption is made that E , , is equal to Eo. The merits of the RRKM vs. the Slater-Forst procedures, for constructing falloff curves have also been discussed pre~iously.~ (1)A. W. Yau and H. 0. Pritchard, Can. J . Chem., 56, 1389 (1978). (2) P. J. Robinson, React. Kinet., 1, 93 (1975). (3) Comment by Professor W. Forst and author's reply in J. Troe, Symp. (Zntl.) Combust., [Proc.]Isth, 667 (1974). (4)Appendix 1 3f ref 1.
